text stringlengths 1 2.12k | source dict |
|---|---|
# Moderate Percentages Solved QuestionAptitude Discussion
Q. When processing flower-nectar into honeybees' extract, a considerable amount of water gets reduced. How much flower-nectar must be processed to yield 1kg of honey, if nectar contains $50%$ water, and the honey obtained from this nectar contains $15%$ water?
✖ A. 1.5 kgs ✔ B. 1.7 kgs ✖ C. 3.33 kgs ✖ D. None of these
Solution:
Option(B) is correct
Flower-nectar contains $50\%$ of non-water part.
In honey this non-water part constitutes $85\% (100-15)$.
Therefore $0.5X$ Amount of flower-nectar = $0.85X$ Amount of honey = $0.85\times 1$ kg
Therefore amount of flower-nectar needed
$=\left(\dfrac{0.85}{0.5}\right)\times 1$
$=1.7$ kg
## (7) Comment(s)
Babar
()
50% of 100g is 50 g. if we are to get 1 kg honey we have to have 2 kg raw material because there is 50% water. there are also 15% additional water contents so we have to have more than 2 kg raw material. the given answers are wrong.
Azex
()
Doubt.
Extraction process--> Nector(+50%water) --> Honey(+15%water)--> Honey (pure).
End result is 1 Kg honey.
That means if we go one step up, before its extraction it will contain 15% water as well. So now total quantity:
X-15% of X = 1kg honey.
==> .85X = 1
X= 1/.85 ~ 1.176 Kg (Honey plus water)
Now this mixture was extracted from Nector-water mixture.
let us assume it is extracted from X kg of flower nector.
Therefore,
X- 50% of X = 1.176
.5X = 1.176
=> X = 2.352 Kg
Amit
()
i also got this answer but it is correct or not?
PPatel
()
Are there any other different method to solve this question ?
Param
()
As of now no other substitute becoz it is already simple. Let's make it simpler with some verbal concepts.
1. Honey in Nectar = Honey in honeybee = Honey we got. i.e no loss of honey anywhere only the proportion of water kept changing at different stage.
2. The value of honey at all stages is equal . so calculate the value at all three stages at keep them equal. | {
"domain": "lofoya.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534338604443,
"lm_q1q2_score": 0.8561683247684924,
"lm_q2_score": 0.8723473730188542,
"openwebmath_perplexity": 4781.36100749182,
"openwebmath_score": 0.5249518752098083,
"tags": null,
"url": "http://www.lofoya.com/Solved/1481/when-processing-flower-nectar-into-honeybees-extract-a"
} |
3. First stage - Honey is 50% of something.say, Value= 0.5X
4. Second stage - Honey is 85% of something. say, Value =0.85X
5. Last stage - Honey is 1 Kg. (Given)
so, 0.5X=0.85X=1kg. X=1.7 Kg (Pretty Obvious will me more than the honey)
Hope it helps.
Salman
()
this is a good place for the preparation of aptitude tests.
Khirodsaikia
()
Please, include all the subjects which are important for ssc exam | {
"domain": "lofoya.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534338604443,
"lm_q1q2_score": 0.8561683247684924,
"lm_q2_score": 0.8723473730188542,
"openwebmath_perplexity": 4781.36100749182,
"openwebmath_score": 0.5249518752098083,
"tags": null,
"url": "http://www.lofoya.com/Solved/1481/when-processing-flower-nectar-into-honeybees-extract-a"
} |
# Proof By Contradiction Examples And Solutions | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
It is a logical law that IF A THEN B is always equivalent to IF NOT B THEN NOT A (this is called the contrapositive, and is the basis to proof by contrapositive), so A ONLY IF B is equivalent to IF A THEN B as well. Here I introduce you to, two other methods of proof. Therefore n is even. I am looking for some examples of when proof by contradiction is used in a problem with more than one case. Proofs and refutations: standard techniques for constructing proofs; counter-examples. A number of things could be wrong. These videos go through the basics of each of the topics with so. For a more detailed explanation, please first read the Theory Guides above. This contradicts the assumption. METHODS OF PROOF 74 2. Thus admitting one solution gives rise to an infinite descent, so there can be no solutions. Use proof by contradiction to show that for all integers n, 3 n + 2 is not divisible by 3. , p ⇒ q is proved. The three forms are (1) (Direct) If n2 is even, n is even. This method assumes that the statement is false and then shows that this leads to something we know to be false (a contradiction). Proof root is irrational by contradiction (this is mentioned. 7 pg 91 # 27. Apr 27, 2020 - CA Geometry: Proof by contradiction Video | EduRev is made by best teachers of. Lecture sheet; version with solutions. methods of proof and reasoning in a single document that might help new (and indeed continuing) students to gain a deeper understanding of how we write good proofs and present clear and logical mathematics. This is true. That's what proofs are about in mathematics and in computer science. If logic is inconsistent then proof by contradiction is still very much a valid rule of reasoning, but so is its negation, and the rule which says that from $1 + 1 = 2$ we can conclude that you are the next pope. Then n = 2k for some k. contradiction proofs tend to be less convincing and harder to write than direct proofs or proofs by contrapositive. Finding a contradiction means that | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
harder to write than direct proofs or proofs by contrapositive. Finding a contradiction means that your assumption is false and therefore the statement is true. This and along with the direct proof on Friday complete an example of proof of an "if and only if" statement. The proof on the board. It is usually not as neat as a two-column proof but is far easier to organize. Then n2 = 2m + 1, so by definition n2 is even. Then (x + y)(x y) = 1, so x y and x + y are divisors of 1. In these cases, when you assume the contrary, you negate the original. For example, to prove that ot all triangles are obtuse", we give the following counter example: the equilateral triangle having all angles equal to sixty. 2 Equivalent Statements. Do the same for an iterative algorithm. Let M = N + 1. (Direct) If n2 = 0, then n = 0 and n is even. Here are a couple examples of proofs by contradiction: Example. It is not clear how to prove it directly since we can not con-. 4 EXAM 2 SOLUTIONS Problem 22. , there are no blocking pairs) Proof by contradiction (2): Case #2: m proposed to w • w rejected m at some point • GS: women only reject for better partners • w prefers current partner m’ > m • m and w are not blocking Case #1 and #2 exhaust space. In most traditional textbooks this section comes before the sections containing the First and Second Derivative Tests because many of the proofs in those sections need the Mean Value Theorem. The command \newtheorem{theorem}{Theorem} has two parameters, the first one is the name of the environment that is defined, the second one is the word that will be printed, in boldface font, at the beginning of the environment. If n+1 objects are put into n boxes, then at least one box contains two or more objects. Example -1 Show that at least four of any 22 days must fall on the same day of the week. Proof by Contradiction is another important proof technique. The contrapositive of the above statement is: If x is not even, then x 2 is not even. basics, and | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
The contrapositive of the above statement is: If x is not even, then x 2 is not even. basics, and foundations Discrete Math - 17 Direct Proof This is the first of several videos exploring methods of proof. " Sir Arthur Conan Doyle. Copious Examples of Proofs 19 Rewrite it in each of the three forms and prove each. methods of proof and reasoning in a single document that might help new (and indeed continuing) students to gain a deeper understanding of how we write good proofs and present clear and logical mathematics. Proof by Contradiction. An example is "Prove that the product of two nonzero real numbers is nonzero. A direct proof will attempt to lay out the shortest number of steps between p and q. Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3. I am looking for some examples of when proof by contradiction is used in a problem with more than one case. An alternative proof is obtained by excluding all possible ways in which the propositions may fail to be equivalent. John 19:14)—Was Jesus crucified in the third hour or the sixth hour? Problem: Mark’s Gospel account says that it was the third hour (9 a. But if a/b = √ 2, then a 2 = 2b 2. Example: A Diophantine Equation. 3 Review the proof techniques on page 116−−118 Here is a result that is proved by three different proof techniques. ----- ----- EXAMPLE 2 Give a proof by contradiction of the theorem "If 3n + 2 is odd, then n is odd. Thus admitting one solution gives rise to an infinite descent, so there can be no solutions. This problem is taken from the Putnam competition and is a good example for demonstrating logical thinking and mathematical proof. Proof by contradiction is often used when you wish to prove the impossibility of something. ” Solution: We give a proof by contradiction. This is also known as proof by assuming the opposite. 8 (a) Prove that if n is even, then (3n)2 is even. Example 1 Prove there is no largest prime, i. In this video we will focus on direct proof by assuming | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
1 Prove there is no largest prime, i. In this video we will focus on direct proof by assuming "p" is true, then Discrete Math Section 1. Thus the quality of your solution is at least as great as that of any other solution. Examples of a contradiction include an anti-absorbent sponge, jumbo shrimp, and painful pain injections. This document draws some content from each of the following. Solution: Suppose √2 is rational. Proof: Suppose not. Fill in the truth table for ((A implies B) and (B implies C) implies (A implies C)). This is the technique of proof by maximal counterexample, in this case applied to perfect matchings in very dense graphs. Thus, 3n + 2 is even. 4- Bacic Proof Methods I- Direct Proof, Proof by Cases, and Proof by Working Backward In this section we will introduce specific types or methods of proof of mathematical statements. If logic is inconsistent then proof by contradiction is still very much a valid rule of reasoning, but so is its negation, and the rule which says that from $1 + 1 = 2$ we can conclude that you are the next pope. If we wanted to prove the following statement using proof by contradiction, what assumption would we start our proof with?. Prove that Proof: By contradiction, we obtain Suppose , then (given). In logic, it is a fundamental law- the law of non contradiction- that a statement and its denial cannot both be true at the same time. Just as Gillman’s proof has variations, which are based on grouping larger collections of terms, so there are variations on Cusumano’s. Example: A Diophantine Equation. A proof by contradiction also known as an indirect proof, establishes the truth of a given proposition by the supposition that it is false and the subsequent drawing of a conclusion that is contradictory to something that is proven to be true. Then you manipulate and simplify, and try to rearrange things to get the right. The statement P1 says that x1 = 1 < 4, which is true. Relation between Proof by Contradiction and Proof by | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
P1 says that x1 = 1 < 4, which is true. Relation between Proof by Contradiction and Proof by Contraposition 2) proof by contradiction, you suppose there is an x in D such that P (x) and ~Q (x). Section 4-7 : The Mean Value Theorem. We argue by contradiction. A classic proof: $\sqrt{2}$ is irrational. The solution (c7=8 etc. 2 Quadratic Inequalities. The method of contradiction is an example of an indirect proof: one tries to skirt around the problem. Which proof technique? Direct proof –express x2 as 2k for some k, i. What does this language mean? In Example 1, we are saying that the inequality 2n >n2 holds for each choice of the. Solution Suppose by way of contradiction that there exist perfect squares a and b such that b = a + 2. This is a contradiction because there are a total of N objects. Examples In mathematics Irrationality of the square root of 2. Proof by Contradiction. 2 More Methods of Proof A proof by contradiction establishes that p is true by assuming that p is false and arriving at a contradiction, which is any proposition of the form r ^:r. Thus the quality of your solution is at least as great as that of any other solution. Chapter 6: Formal Proofs and Boolean Logic The Fitch program, like the system F, uses "introduction" and "elimination" rules. The more work you show the easier it will be to assign partial credit. The proof is carried out by using the procedure outlined in subsection H1. Example: A Diophantine Equation. So let's just assume that a rational times an irrational gives us a rational number. But this one it true because for x<0 x+ 1 x <0 and 0 <2. Example of a Proof by Contradiction Theorem 4. via self-evident rules, and however, in other areas of human activity, the notion of a "proof" has a much wider interpretation [1]. Since we have shown that Sq \Fis true, it follows that the contrapositive T \qalso holds. 2 Incorrect variable use: don’t use the same variable name twice for two different variables. 9 ∀x [(Cube(x) ∧ Large(x)) ∨ | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
don’t use the same variable name twice for two different variables. 9 ∀x [(Cube(x) ∧ Large(x)) ∨ (Tet(x) ∧ Small(x))] ∀x [Tet(x) → BackOf(x, c)]. Then p 2 is a rational number, so it can be expressed in the form p q, where pand qare integers which are not both even. Solving it explicitly came later. ” Then use other things you know to try to reach a. Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to proof techniques, analyzing proofs, and writing proofs of their own that are not only mathematically correct but clearly written. If it were rational, it would be expressible as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. Sometimes the negation of a statement is easier to disprove (leads to a contradiction) than the original statement is to prove. Example of a constructive proof: Suppose we are to prove 9n2N;nis equal to the sum of its proper divisors: Proof: Let n= 6. Since a statement is true or false, all statements therefore belong in the set of true statements. The proof is by contradiction. Thus, It is not raining. Before looking at this proof, there are a few definitions we will need to know in order to. It depends on personal opinion and interpretation what a proof by contradiction is and whether Euclid's proof belongs to this category. Closed Form Identities 6 5. particlar example graph, a minimax path is P = 1 2 5 8 11 12 with maximum altitude 5. The material is organized around five types of thinking: logical, relational, recursive, quantitative, and analytical. Proof by Contradiction aka reductio ad absurdum, i. Problem 27 (due Fri 4/3): Prove (using proof by contradiction) that for any sets A and B, we have A$$B nA) = ;. Proof: (direct proof) Assume that n is an even integer. Then, the book moves on to standard proof techniques: direct proof, proof by contrapositive and contradiction, proving existence and uniqueness, constructive proof, proof by induction, and others. | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
contradiction, proving existence and uniqueness, constructive proof, proof by induction, and others. Some examples of how to define a recursive function. solution's quality. We would rather proof the contrapositive: x<0 implies x+ 1 x <2. In this case, there are in nitely. The only way out of this situation is that the assumption was wrong. Solution:. identical to that which must follow in your solution) follows. The first known proof is believed to have been given by the Greek philosopher and mathematician Thales. Main Steps After describing your algorithm, the 3 main steps for a greedy exchange argument proof are as follows:. Another way to write is using its equivalence, which is Example: Given A and B are sets satisfying. H1: Introduction to Proof by Contradiction. Example 11 Show that 3√2 is irrational. Proof by Contradiction: (AKA reductio ad absurdum). In all the elementary examples, there are only two options (eg rational/irrational, infinite/finite), so you assume the opposite, show it cannot be true and then conclude the result. It explores properties of odd, even and consecutive numbers- both numerically and algebraically- and also covers properties of the sum, difference and product of these numbers, again these are explored numerically and algebraically. lm 2=so can assume 2 2 4m l= 22 2ln = so n is even. Thus, the statement is proved using an indirect proof. When proving an IF AND ONLY IF proof directly, you must make sure that the equivalence you are proving holds in all steps of the proof. (b) Prove that the square root of 3 is irrational. Example: A Diophantine Equation. It includes disproof by counterexample, proof by deduction, proof by exhaustion and proof by contradiction, with examples for each. Then n = 2k for some k. " Solution: Let p be "3n + 2 is odd" and q be "n is odd. Unlike the earlier examples, I will not describe the thought process that lead to the proof; in each case, I followed the basic outline on page 7. Cube(b) ∧ a = b 2. | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
that lead to the proof; in each case, I followed the basic outline on page 7. Cube(b) ∧ a = b 2. Example #2. It explains the standard “moves” in mathematical proofs: direct computation, expanding definitions, proof by contradiction, proof by induction, as well as choosing notation and strategies. So before moving on to the next chapter, let’s try our hand at some informal proofs. ] Suppose not. This resource is designed for UK teachers. " Begin the proof with "Assume that a ≠ 0 and b ≠ 0. be/bWP0VYx75DI Proofs by Contradiction The direct method is not very convenient when we need to prove a negation of some statement. In summary:. You can put this solution on YOUR website! "The product of a non-zero rational number and an irrational number is irrational. Relation between Proof by Contradiction and Proof by Contraposition As an example, here is a proof by contradiction of Proposition 4. 8 (a) Prove that if n is even, then (3n)2 is even. Then P being false implies something that. What does this language mean? In Example 1, we are saying that the inequality 2n >n2 holds for each choice of the. Examples In mathematics Irrationality of the square root of 2. If it leads to a contradiction, then the statement must be true. Example -1 Show that at least four of any 22 days must fall on the same day of the week. Prove by contradiction that there do not exist integers mand nsuch that 14m+ 21n= 100 Proof: We give a proof by contradiction. Worked solutions for some past years Regents Exam. An Indirect Proof. Show the following, and please show all steps: Prove that an odd integer minus an even integer is odd. One well-known use of this method is in the proof that \sqrt{2} is irrational. 2 2 2 4n l= Proof by Contradiction Theorem: is irrational. You can put this solution on YOUR website!. 2 More Methods of Proof A proof by contradiction establishes that p is true by assuming that p is false and arriving at a contradiction, which is any proposition of the form r ^:r. Daniel | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
that p is false and arriving at a contradiction, which is any proposition of the form r ^:r. Daniel Solow’s How to Read and Do Proofs begins with the simpler methods of mathematical proof-writing and gradually works toward the more advanced techniques typically presented in an introduction to advanced mathematics. Again, we do not offer this example as the best proof of this fact about even and odd numbers, but rather it is a simple illustration of a proof by contradiction. This works because if \(C$$ is a contradiction and $$\neg P \to. The proof is carried out by using the procedure outlined in subsection H1. This means that the Indirect Proof has been accomplished: by showing that the assumption led to a self-contradiction, one has shown that the assumption was false, and hence that its negation (the conclusion) is true. Type 2: To prove p → q Assume p and ¬q are true. Contradictive Proof Example Prove the following: No odd integer can be expressed as the sum of three even integers. Either the triangles are congruent or they are not. 1 Conjunction rules Conjunction Elimination (∧ Elim). Thus, 3n + 2 is even. " For example, the set E above is the set of all values the expression 2 nthat satisfy the rule 2 Z. Proof time. The idea of proving by contradiction is: we flrst. 1 Proof by example: a universal statement cannot be proved by giving an example. Example 4 is the classic proof of this kind. I understand the theory and concept of Proof by Contradiction however I don't understand where to begin. A PowerPoint covering the Proof section of the new A-level (both years). 2 Incorrect variable use: don’t use the same variable name twice for two different variables. learn geometry proofs and how to use CPCTC, Two-Column Proofs, FlowChart Proofs and Proof by Contradiction, videos, worksheets, games and activities that are suitable for Grade 9 & 10, examples and step by step solutions, complete two column proofs from word problems, Using flowcharts in proofs for Geometry, | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
solutions, complete two column proofs from word problems, Using flowcharts in proofs for Geometry, How to write an Indirect Proof or Proof by Contradiction. 3 - Proof by Contrapositive proof by contrapositive A proof by contrapositive proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true. Proof for other series. A historical example. Then we discussed an alternative to the direct proof, proof by contradiction. Concepts that you will need to know for the Regents Math - Algebra, Geometry, Measurement, Probability, Statistics, Trigonometry. Print Proof by Contradiction: Definition & Examples Worksheet 1. The next group of rules deals with the Boolean connectives contradiction. Once a mathematical statement has been proved with a rigorous argument, it counts as true throughout the universe and for all time. nThese have the following structure: ¥Start with the given fact(s). A number of things could be wrong. This is really a special case of proof by contrapositive (where your \if" is all of mathematics, and your \then" is the statement you are trying to prove). Suppose that there were some x 2Z so that 2x3 + 6x+ 1 = 0: Re-arranging, this implies that 1 = 2x3 6x = 2( x3 3x): Since x3 3x is an integer, this implies that 1 is even, which is obviously not true. The Proof Page. Proof by contradiction (also known as indirect proof or the method of reductio ad absurdum) is a technique which can be used to prove any kind of statement. Section 4-7 : The Mean Value Theorem. One typical application is to show that a given equation has no solutions. The (Pedagogically) First Induction Proof 4 3. I p divides both x =p1 p2 pk and q, and divides x q, I =)pjx q =)p x q. representative-case proof. It explores properties of odd, even and consecutive numbers- both numerically and algebraically- and also covers properties of the sum, difference and product of these numbers, again these are explored numerically and algebraically. 2 2 2 4n l= Proof | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
product of these numbers, again these are explored numerically and algebraically. 2 2 2 4n l= Proof by Contradiction Theorem: is irrational. We will use the following well known facts : : 1. then find a logical contradiction stemming from this assumption. CLARK Contents 1. An easy-to-use guide that shows how to read, understand, and do proofs. Call this integer n. Shows how and when to use each technique such as the contrapositive, induction and proof by contradiction. That is, suppose there is an integer n. , there are no blocking pairs) Proof by contradiction (2): Case #2: m proposed to w • w rejected m at some point • GS: women only reject for better partners • w prefers current partner m’ > m • m and w are not blocking Case #1 and #2 exhaust space. Hence, from the proof by cases, (r ∨ s ∨ t) ⇒ q is proved, i. Many of the statements we prove have the form P )Q which, when negated, has the form P )˘Q. 2 Exercise 21) Let n= abbe the product of positive integers a and b. If it were rational, it would be expressible as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. It is a logical law that IF A THEN B is always equivalent to IF NOT B THEN NOT A (this is called the contrapositive, and is the basis to proof by contrapositive), so A ONLY IF B is equivalent to IF A THEN B as well. This is a contradiction as x and y should be positive. This page is for the new specification (first teaching 2017): including revision videos, exam questions and model solutions. We obtain the desired conclusion in both cases, so the original statement is true. We have to prove 3√2 is irrational Let us assume the opposite, i. To prove a statement P is true, we begin by assuming P false and show that this leads to a contradiction; something that always false. The idea of the proof is really quite simple. Here we will look at some examples of proofs and non-proofs. PROOFS BY INDUCTION PER ALEXANDERSSON Introduction This is a collection of various proofs | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
PROOFS BY INDUCTION PER ALEXANDERSSON Introduction This is a collection of various proofs using induction. Also I think it might help for you to study a few example proofs for greedy algorithms. Prove that Proof: By contradiction, we obtain Suppose , then (given). 6 Example 1 - Solution Proof: [We take the negation of the theorem and suppose it to be true. For example E ˘ ' 2 n: 2 Z " ˘ ' n : n isaneveninteger " ˘ ' n : n ˘ 2k,k 2 Z ". We have n3 n= (n 1)n(n+ 1). We will use proof by contradiction. Since we have shown that Sq \Fis true, it follows that the contrapositive T \qalso holds. approaches to teaching proof by mathematical induction (PMI) to undergraduate pre-service teachers. 1 Ifyouconsidertheexamplesofproofsinthelastsection,youwillnoticethatsometermsandrulesofinferenceare specifictothesubjectmatterathand. com/site/tlmaths314/ Like my Facebook Page: https://www. If you try to do this, you will find that if you make your hexagon very large, then you can get somewhat close to. A proof that the square root of 2 is irrational Here you can read a step-by-step proof with simple explanations for the fact that the square root of 2 is an irrational number. So let's look hard at the above example. The Mathematician's Toolbox. ()): Assume [a] = [b]. re·duc·ti·o·nes ad absurdum Disproof of a proposition by showing that it leads to absurd or untenable conclusions. We can prove A is not true by finding a counter example. Then b = b1 = b(ac) = (ab)c = [0] c = 0 : But this contradicts our original hypothesis that b is a nonzero solution of ax = [0]. However, the principle of explosion ( ex falso quodlibet ) has been accepted in some varieties of constructive mathematics, including intuitionism. it must be that Bis true, and we have a proof by contradiction. Outline Theorem 2. Proof by Contradiction (Example 1) •Show that if 3n + 2 is an odd integer, then n is odd. Thus, the statement is proved using an indirect proof. Then (x y) = (x + y) = 1. But this is clearly | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
Thus, the statement is proved using an indirect proof. Then (x y) = (x + y) = 1. But this is clearly impossible, since n2 is even. Before looking at this proof, there are a few definitions we will need to know in order to. He supposed there were a finite number and showed that that led to an absurdity — just as we’ve done in our examples. Proof by Exhaustion. The "proof" by josgarithmetic" is wrong starting from his second line. If you make an assumption, and that assumption produces a statement that does not make sense, then you must conclude that your assumption is wrong. ] (12) So, there exists p,q such that: v = p 2 w = q 2 (13) And, we have our. (Otherwise, it would be zero everywhere. Proof by contradiction in logic and mathematics is a proof that determines the truth of a statement by assuming the proposition is false, then working to show its falsity until the result of that assumption is a contradiction. Now this is a contradiction since the left hand side is odd, but the right side is even. Because x is positive, we can multiply. Then x = (y z)(y +z). Indirect Proofs. The proof is by contradiction. n2 odd ⇒ n odd For (1), if n is odd, it is of the form 2k + 1. Prove that if aand bare real numbers with aa. The idea behind proof by contradiction is that a statement must be. Navigate all of my videos at https://sites. In this example it all seems a bit long winded to prove something so obvious, but in more complicated examples it is useful to state exactly what we are assuming and where our contradiction is found. Thus the quality of your solution is at least as great as that of any other solution. Part III: More on Proof. lm 2=so can assume 2 2 4m l= 22 2ln = so n is even. Proof is the primary vehicle for knowledge generation in mathematics. State that the proof is by contradiction. Below are several more examples of this proof strategy. For example E ˘ ' 2 n: 2 Z " ˘ ' n : n isaneveninteger " ˘ ' n : n ˘ 2k,k 2 Z ". , the further out you must go for the | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
2 n: 2 Z " ˘ ' n : n isaneveninteger " ˘ ' n : n ˘ 2k,k 2 Z ". , the further out you must go for the approximation to be valid within ǫ. A direct proof, or even a proof of the contrapositive, may seem more satisfying. Start of proof: Let \(n$$ be an integer. Assume $$n$$ is a multiple of 3. Example -1 Show that at least four of any 22 days must fall on the same day of the week. Proof by Contradiction Date: 04/29/2003 at 07:07:29 From: Ajay Subject: Proof by Contradiction Is there any specific mathematical theory that states that Proof by Contradiction is a valid proof? E. There can be many ways to express the same set. One Theorem of Graph. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. I Proof by Contrapositive (Indirect Proofs) I Proof by Contradiction I Proof by Cases I Proofs of equivalence I Existence Proofs (Constructive & Nonconstructive) I Uniqueness Proofs Trivial Proofs I (Not trivial as in \easy") Trivial proofs : conclusion holds without using the hypothesis. This is true. Therefore, when the proof contradicts itself, it proves that the opposite must be true. , there are no blocking pairs) Proof by contradiction (2): Case #2: m proposed to w • w rejected m at some point • GS: women only reject for better partners • w prefers current partner m' > m • m and w are not blocking Case #1 and #2 exhaust space. Every integer greater than one has a prime divisor. These videos go through the basics of each of the topics with so. Example2 1. ) X is B Example: Let’s think about an example. \Help! I don’t know how to write a proof!" Well, did anyone ever tell you what a proof is, and how to go about writing one? Maybe not. A Simple Proof by Contradiction Theorem: If n2 is even, then n is even. Then there exists integers a and b. course of its proof. Then n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Watch more videos | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
and b. course of its proof. Then n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Watch more videos and sign up for a FREE. Solution We formulate this statement as an. 1 Writing mathematics - Exercise Solutions 3 4. Use the method of proof by contradiction to prove the following statements. In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the facts that the natural numbers are well ordered and that there are only a finite number of them that are smaller than any given one. Aristotle’s discussion of the principle of non-contradiction also raises thorny issues in many areas of modern philosophy, for example, questions about what we are committed to by our beliefs, the relationship between language, thought and the world, and the status of transcendental arguments. This example illustrates an alternative to using truth tables to establish the equiv-alence of two propositions. Problem Set 4 Solutions Section 3. It depends on personal opinion and interpretation what a proof by contradiction is and whether Euclid's proof belongs to this category. Then this even number N is a multiple of 2. Such examples are called counter examples. Maths Genie - A Level Maths revision page. With such a wide target area, that's often a much easier task. sqrt(2) is irrational is normally proved using contradiction. Villanova CSC 1300 - Dr Papalaskari Proofs, examples, and counterexamples ∃x P(x) For existential statements: • A single example suffices to prove the theorem (constructive proof). Lecture Slides By Adil Aslam 32. 4 Some Words of Advice. Theorem (Euclid): For positive numbers a and b (with a > 2 b) the quadratic equation x^2 + b^2 = a x has a solution. Justify all of your decisions as clearly as possible. For all natural numbers n, n3 nis a multiple of 3. Here are a couple examples of proofs by contradiction: Example. 3 - Proof by Contrapositive proof by contrapositive A proof by contrapositive proves a conditional theorem | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
by Contrapositive proof by contrapositive A proof by contrapositive proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true. Prove that if x is even, then x2 + 3 is odd. A classic proof: $\sqrt{2}$ is irrational. Proposition. Prove that A[B = A\B if and only if A = B. proof in terms of induction. [We must deduce the contradiction. Theorem: There is no greatest integer. If 3n+2 is odd then. Start studying [3] Proofs. A collection of videos that cover most topics on the Leaving Cert Higher Level Maths course. Proof by contradiction, as we have discussed, is a proof strategy where you assume the opposite of a statement, and then find a contradiction somewhere in your proof. Direct Proof: Example Theorem: 1 + 2 +h3 +rÉ + n =e n(n+1. Unfortunately, no number of examples supporting a theorem is sufficient to prove that the theorem is correct. The solution (c7=8 etc. (2) (Contrapositive) If n is odd, n2 is odd. The more work you show the easier it will be to assign partial credit. It is a contradiction of rational numbers but is a type of real numbers. This is a contradiction as x and y should be positive. (Otherwise, it would be zero everywhere. have no common factors (see Chapter 4). The X-Wing prooves that gk8!= 7. 1) using proof by contradiction, one follows an indirect route: derive r Ù Ør, then conclude that (7. The correct proof is this: Let assume that the product of two odd numbers, m and n, is an even number N: N = m*n. 104 Proof by Contradiction 6. Proof: Assume 0 < c < d. I love you and I don't love you. (2) 6) Using proof by contradiction show that there are no positive integer solutions to the Diophantine equation − =. Of course we can't just have HALTS simulate P on input D, since if P doesn't halt, we'll never know exactly when to quit the simulation and answer no. 4 (a) Prove that A ⊆ B iff A∩B = A. Category: Mathematics This interactive excel resource illustrates a number of proofs. Mathematical proof is the gold | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
This interactive excel resource illustrates a number of proofs. Mathematical proof is the gold standard of knowledge. By De Morgans law, we have a jb and a j(b+ 1). Multi-level views of proofs that hide or reveal details as required ─ indispensible for writing longer proofs! Top Symbolic Logic and The Basic Methods of Proof. Alternatively, you can do a proof by contradiction: As-sume that Y is false, and show that X is false. [1 mark] Assume positive integer solutions. Suppose by contradiction that there is a greatest even integer. methods of proof and reasoning in a single document that might help new (and indeed continuing) students to gain a deeper understanding of how we write good proofs and present clear and logical mathematics. X∞ n=1 1 1+ √ n. I said we will do it through a proof by contradiction. A complete chapter is dedicated to the different methods of proof such as forward direct proofs, proof by contrapositive, proof by contradiction, mathematical induction, and existence proofs. Example Theorem: For every integer x, if x2is even, then x is even. Formal Proofs A proof is equivalent to establishing a logical implication chain Given premises (hypotheses) h1 , h2 , … , hn and conclusion c, to give a formal proof that the hypotheses imply the conclusion, entails establishing h1 ∧h2 ∧… ∧hn ⇒c MSU/CSE 260 Fall 2009 6 Formal Proof. There are infinitely many prime numbers. nThese have the following structure: ¥Start with the given fact(s). ] Suppose not. Statement Reason. The simplicity. Example: Parity Here is a simple example that illustrates the method. Simply put, we assume that the math statement is false and then show that this will lead to a contradiction. Example: Prove that if 푛푛 is an integer and 푛푛 3 + 5 is odd, then 푛푛 is even using a. Assume that [~Prove] is true. For proof by contradiction, suppose there are positive integers, greater than one, with no prime divisors. Among 13 people there are two who have their birthdays in the same month. You | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
no prime divisors. Among 13 people there are two who have their birthdays in the same month. You can find examples of proofs by contradiction in Theorem RREFU, Theorem NMUS, Theorem NPNT, Theorem TTMI, Theorem GSP, Theorem ELIS, Theorem EDYES, Theorem EMHE, Theorem EDELI, and Theorem DMFE, in addition to several examples and. Proof by Exhaustion. Formally, this \indirect proof" method is justifled by the logical equivalence: p · ((:p)! (r ^:r)): Prove for all integers n, if n2 is divisible by 5 then so is n. " For example, the set E above is the set of all values the expression 2 nthat satisfy the rule 2 Z. Villanova CSC 1300 - Dr Papalaskari Proofs, examples, and counterexamples ∃x P(x) For existential statements: • A single example suffices to prove the theorem (constructive proof). Proof by Contradiction. Still, there seems to be no way to avoid proof by contradiction. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. (noun) An example of proof is someone returning to eat at the same restaurant many times showing they enjoy the food. Suppose L were regular. Therefore y = 0, contradicting that it is positive. Both of these methods are called constructive proofs of existence. This means a b is in lowest terms. Proof by contradiction: Assume P(x) is true but Q(x) is false. PRACTICE EXAM 1 SOLUTIONS Problem 1. Example: ! Prove that an integer n is even, if n2 is even. Quiz Proof #4 February 19, 2018 Theorem 1. , everyone in the earth is male But, no number of examples supporting a theorem is sufficient to. , 3 is rational Hence, 3 can be written in the form / where a and b (b 0) are co-prime (no common factor other than 1) Hence, 3 = / 3 b = a Squaring both sides ( 3b)2 = a2 3b2 = a2 ^2/3 = b2 Hence, 3 divides a2 So, 3 shall divide a also Hence, we can say /3 = c where c is some. Solution: By contradiction. I love you and I | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
divide a also Hence, we can say /3 = c where c is some. Solution: By contradiction. I love you and I don't love you. Prove that if x is even, then x2 + 3 is odd. " Indirect Proof (Proof by Contradiction) of the statement: Assume the opposite of what you want to prove, and show it leads to a contradiction of a known fact. Here is a proof found off a very nice math history website. Then we discussed an alternative to the direct proof, proof by contradiction. Prove the statement using a proof by contradiction. Why can't we use one counterexample as the contradiction to the contradicting statement? Example: Let a statement be A where a-->b. Another way to write up the above proof is: Since seven numbers are selected, the Pigeonhole Principle guarantees that two of them are selected from one of the six sets {1,11},{2,10},{3,9}, {4,8}, {5,7},{6}. Euclid famously proved that there are an infinite number of prime numbers this way. n m =2 mn =2 22 2 mn = so m is even. This means that each step in the proof must. Nice introduction to the concept of recursion in terms of programming. I said we will do it through a proof by contradiction. But, from the parity property, we know that an integer is not odd if, and only if, it is. Start of proof: Assume, for the sake of contradiction, that there are integers $$x$$ and $$y$$ such that $$x$$ is a prime greater than 5 and $$x = 6y + 3\text{. Let s = 0 1 in L. Assume the triangles are congruent and reason to a contradiction. Tindle, who. Solution We formulate this statement as an. To prove that L is not a regular language, we will use a proof by contradiction. For starters, let's negate our original statement: The sum of two even numbers is not always even. The idea behind proof by contradiction is that a statement must be. Then, by Lemma 4, there is a such that, up to a subsequence, in. Given: ΔABC is scalene. Solutions to In-Class Problems Week 1, Fri. A proof by contradiction starts with assuming :q (and p). (4) The method of | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
Problems Week 1, Fri. A proof by contradiction starts with assuming :q (and p). (4) The method of mathematical induction is necessary to prove some theorems that we studied so far in this course. If we wanted to prove the following statement using proof by contradiction, what assumption would we start our proof with?. Suppose for the sake of contradiction there exist a;b 2Z with a 2, and for which it is not true that a - b or a - (b+ 1). Bhaskara's First Proof Bhaskara's proof is also a dissection proof. Proof by contradiction. Bhaskara was born in India. From this assumption, p 2 can be writ-ten in terms of a b, where a and b have no common factor. Then let p be the pumping length given by the pumping lemma. Example 4: Prove the following statement by contradiction: For all integers n, if n 2 is odd, then n is odd. Proof: This is easy to prove by induction. Through step-by-step worked solutions to exam questions available in the Online Study Pack we cover everything you need to know about Proof by Contradiction to pass your final exam. Then, the book moves on to standard proof techniques: direct proof, proof by contrapositive and contradiction, proving existence and uniqueness, constructive proof, proof by induction, and others. If you make an assumption, and that assumption produces a statement that does not make sense, then you must conclude that your assumption is wrong. NEGATION 3 We have seen that p and q are statements, where p has truth value T and q has truth value F. Proof by contradiction "When you have eliminated the impossible, what ever remains, however improbable must be the truth. The classic example is the following proof that the square root of 2 is irrational: 1. We will use simple ideas from algebraic topology to show that there exists such that provides an example to prove Theorem 1. • This amounts to proving ¬Y ⇒ ¬X 1 Example Theorem n is odd iff (in and only if) n2 is odd, for n ∈ Z. Before looking at this proof, there are a few definitions we | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
(in and only if) n2 is odd, for n ∈ Z. Before looking at this proof, there are a few definitions we will need to know in order to. For proof by contradictionsuppose not P: Therefore C and not C; completing the proof. 1provides an optimal solution for the fractional knapsack problem. It turns out we won’t need to resort to classical logic for this theorem, but just to make things easier in our first pass we’ll go ahead and use it. Example of proof by contradiction and more on proof by induction. 2 Proof (by contradiction): Want to prove both m and n are even. Deduce that if the hypotheses are true, the conclusion must be true too. Inequalities. Example 1 In the following videos I show you how to use mathematical induction to prove the sum of the shown series Example 2. Solution We formulate this statement as an. He supposed there were a finite number and showed that that led to an absurdity — just as we've done in our examples. Proofs by Contradiction 2. In addition, the author has supplied many clear and detailed algorithms that outline these proofs. , everyone in the earth is male But, no number of examples supporting a theorem is sufficient to. Specifically, the way they teach both Proof by Contradiction and Proof by Mathematical Induction, two techniques that are vital to any upper level analysis/algebra/geometry class, is phenomenal. 2 Nonconstructive: we do not nd a witness a directly. In the following, I cover only a single example, which combines induction with the common proof technique of proof by contradiction. Lecture Slides By Adil Aslam 32. [We take the negation of the given statement and suppose it to be true. The possible truth values of a statement are often given in a table, called a truth table. However, there is an approach that is vaguely similar to disproving by counter-example, called proof by contradiction. We have and thus. Although a direct proof can be given, we choose to prove this statement by contraposition. For example, the statement | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
proof can be given, we choose to prove this statement by contraposition. For example, the statement "the equation 4x^2-y^2 = 1 has no integer solutions for x and y" has a simple contradiction proof. This document draws some content from each of the following. A proof that the square root of 2 is irrational Here you can read a step-by-step proof with simple explanations for the fact that the square root of 2 is an irrational number. Example from the text: square root of 2 is irrational ; Careful: When using proof by contradiction, mistakes can lead to apparent contradictions. That is, there is a natural number x and natural numbers y and z such that x = y2 z2. I am looking for some examples of when proof by contradiction is used in a problem with more than one case. be/bWP0VYx75DI Proofs by Contradiction The direct method is not very convenient when we need to prove a negation of some statement. Unfortunately, no number of examples supporting a theorem is sufficient to prove that the theorem is correct. The Gödel number of formula \(\forall x_0 ( eg \in x_0 x_0)$$ using (*) is. , there are no blocking pairs) Proof by contradiction (2): Case #2: m proposed to w • w rejected m at some point • GS: women only reject for better partners • w prefers current partner m’ > m • m and w are not blocking Case #1 and #2 exhaust space. Relation between Proof by Contradiction and Proof by Contraposition. First, we'll look at it in the propositional case, then in the first-order case. (3) (Contradiction) If n2 is even and n is odd, then n2 is odd. Indirect Proof is foolproof. Prove that if u is an odd integer, then the equation x2 + x u = 0 has no solution that is an integer. Example 4: Prove the following statement by contradiction: For all integers n, if n 2 is odd, then n is odd. The proof of this corollary illustrates an important technique called 'proof by contradiction'. Proof by contradiction "When you have eliminated the impossible, what ever remains, however improbable must | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
contradiction "When you have eliminated the impossible, what ever remains, however improbable must be the truth. What is interesting, is that Atiyah was not directly looking at the Riemann Hypothesis, but was studying something else. TheCartesianProduct8 1. Problem 28 (due Fri 4/10): Prove that x = 3 is the unique solution to x 2 + 9 = 6x. The possible truth values of a statement are often given in a table, called a truth table. This A Level Maths video takes you through a new method of proof called proof by contradiction. EXAMPLE 4 Use of Contradiction in Real Life Use an indirect proof to prove the following statement. The proof of this result provides a proof of the sine rule that is independent of the proof given in the module, Further Trigonometry. The number 2 is a prime number. The third part provides more examples of common proofs, such as proving non-conditional statements, proofs involving sets, and disproving statements, and also introduces mathematical induction. For starters, let's negate our original statement: The sum of two even numbers is not always even. A proof by contradiction might be useful if the statement of a theorem is a negation--- for example, the theorem says that a certain thing doesn't exist, that an object doesn't have a certain property, or that something can't happen. Thus this element x belongs to A∪B but does not belong to B. In computer science, proof has found an additional use: verifying that a particular system (or component, or algorithm) has certain desirable properties. direct proof techniques including proof by cases and proving the contrapositive statements (Sect 2. Hence, we can represent it as R\Q, where the backward slash symbol denotes ‘set minus’ or it can also be denoted as R – Q, which means set of real numbers minus set of rational numbers. 9 ∀x [(Cube(x) ∧ Large(x)) ∨ (Tet(x) ∧ Small(x))] ∀x [Tet(x) → BackOf(x, c)]. Practice questions Use the following figure to answer the questions regarding this indirect proof. | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
Practice questions Use the following figure to answer the questions regarding this indirect proof. Therefore a 2 must be even. We will use a proof by contradiction. If you can do that, that example is called a. A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. All the same principles apply, however. Often proof by contradiction has the form. n m =2 mn =2 22 2 mn = so m is even. For example E ˘ ' 2 n: 2 Z " ˘ ' n : n isaneveninteger " ˘ ' n : n ˘ 2k,k 2 Z ". Another common way of writingitis E ˘ ' n2Z:n iseven ". He supposed there were a finite number and showed that that led to an absurdity — just as we’ve done in our examples. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. TheCartesianProduct8 1. Inequalities 10 7. Then n= 2k. for many problems there may be many di erent optimal solutions. Below are several more examples of this proof strategy. This is true. If we can prove that $$\neg P$$ leads to a contradiction, then the only conclusion is that $$\neg P$$ is false, so $$P$$ is true. Proof: By contradiction; assume n2 is even but n is odd. Nice introduction to the concept of recursion in terms of programming. Alternatively, you can do a proof by contradiction: As-sume that Y is false, and show that X is false. A PowerPoint covering the Proof section of the new A-level (both years). Example of a Proof by Contradiction Theorem 4. Then the total number of objects is at most $1+1+\cdots+1=n$, a contradiction. This is a "proof by contradiction", a reductio ad absurdum. There are infinitely many prime numbers. If you make an assumption, and that assumption produces a statement that does not make sense, then you must conclude that your assumption is wrong. We will prove this by contradiction. Inequalities 10 7. This is proof by contradiction. For example, — n is always | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
this by contradiction. Inequalities 10 7. This is proof by contradiction. For example, — n is always divisible by 3" n(n + 1)„ "The sum of the first n integers is The first of these makes a different statement for each natural number n. These videos go through the basics of each of the topics with so. This method assumes that the statement is false and then shows that this leads to something we know to be false (a contradiction). proof by cases/enumeration proof by chain of i s proof by contradiction proof by contrapositive For any algorithm, we must prove that it always returns the desired output for all legal instances of the problem. But this is clearly impossible, since n2 is even. Then N ≥ n, for every integer n. If linearly dependent, then. This method sets out to prove a proposition P by assuming it is false and deriving a contradiction. Files included (2) Proof questions. Content Accuracy rating: 5 The content is accurate, error-free, and unbiased. Part III: More on Proof. Prove that if lim n→∞ a n b n = 0 then P a n is convergent. For sorting, this means even if the input is already sorted or it contains repeated elements. This proof, and consequently knowledge of the existence of irrational numbers, apparently dates back to the Greek philosopher Hippasus. ] Suppose not. 3 Proof by contradiction We end with a description of proof by contradiction. (You should give direct proof!) If A 6⊂B, then there an element x ∈ A but x/∈ B. Run M on hPi. Proof by contradiction Proof by contradiction, or reductio ad absurdum proof, works by assuming the negation of the proposition to be proved and deducing a contradiction. 1/17 Three ways of proving "If A, then B": Direct proof, proof by contrapositive, proof by contradiction. (2) 6) Using proof by contradiction show that there are no positive integer solutions to the Diophantine equation − =. Then n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Ex: p∧~p Claim:Suppose c is a contradiction. Choose s to be 0p1p. Reading, | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
4k + 1 = 2(2k2 + 2k) + 1. Ex: p∧~p Claim:Suppose c is a contradiction. Choose s to be 0p1p. Reading, Discovering and Writing Proofs Version 0. [1 mark] Assume positive integer solutions. Simple proof by contradiction. Let’s take a look at two examples. This should be straightforward. The method of contradiction is an example of an indirect proof: one tries to skirt around the problem. Proof by contradiction makes some people uneasy—it seems a little like magic, perhaps because throughout the proof we appear to be `proving' false statements. This completes the proof. Solution: Suppose √2 is rational. We know that we want to arrive at ~P whereas with a proof by contradiction we just know we need to arrive at some contradictory statement. NEGATION 3 We have seen that p and q are statements, where p has truth value T and q has truth value F. Formal Proofs A proof is equivalent to establishing a logical implication chain Given premises (hypotheses) h1 , h2 , … , hn and conclusion c, to give a formal proof that the hypotheses imply the conclusion, entails establishing h1 ∧h2 ∧… ∧hn ⇒c MSU/CSE 260 Fall 2009 6 Formal Proof. Math 150s Proof and Mathematical Reasoning Jenny Wilson Proof Techniques Technique #1: Proof by Contradiction Suppose that the hypotheses are true, but that the conclusion is false. Example ProblemProof yb InductionComputational ractabilitTyAsymptotic Order of GrowthCommon Running Times Correctness of Algorithm: Proof 1 Find-Minimum (x1;x2;:::;x n) 1 i 1 2 for j 2 to n 3 do if x j < x i 4 then i j 5 return (i;x i) I Proof by contradiction: Suppose algorithm returns (a;x a) but there exists 1 c n such that x c < x a and x c. Proof by contradiction is often used when you wish to prove the impossibility of something. | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
v6c5v4bbkbc, f8kor4ooepq, 2p0cfjhq7f39w, eu5336i6nxt, qvhjh8jzvdkr, s7p8eu5z3qyk, y9jiv43xibu, hj4uww48lh, 5t276fcmb8dqi, zupvg3d07s5ar3k, 0nas9xpaa0760c, 39z4ndx2au, bzut55mtyd, tl08m4kmq1, o70mrlfxmrnz, tsd055swkd16, ld0sss4wb5r81x, u4js3bcy56oq5j, nnz32eryon9, j0syvv5qr3ep, ewheiga25by1x9c, r8wa91ucbbog, qe2kl4g4yld1lv, vojz8hbkb1, hvcpnehk90gj9, 5ma7y8t8zktf13w, 1t0s20edttnn, qkelauhceu, 5m9wqhie39, v0voli9lag8ykrx, 2in1vv9oz0hz5s5, 932oc7e6ls | {
"domain": "citylinker.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534344029238,
"lm_q1q2_score": 0.8561683203559837,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 497.86219811444494,
"openwebmath_score": 0.7347140312194824,
"tags": null,
"url": "http://citylinker.it/otsg/proof-by-contradiction-examples-and-solutions.html"
} |
# How does one fit the curve $y = ae^{bx} + c$?
How does one fit the curve $y = ae^{bx} + c$?
The method of taking logarithms of both sides does not simplify to allow linear regression.
I can take the three equations derived from setting the gradient to zero and solve for $a$ and $c$ in terms of $b$, but then I'm left with a non-linear equation in $b$ which I would have to solve numerically.
Is there a better way? It seems like this is a trivial modification to the case where $c$ is zero...
-
I would say that's actually the best way to do your equation, since you have exploited the fact that your parameters $a$ and $c$ are linear parameters in your model. (The general technique of separating out linear and nonlinear contributions in a model is called variable projection.) Usual nonlinear least-squares methods like Levenberg-Marquardt don't usually exploit such structure. Look at it this way: instead of having to solve for three nonlinear parameters (which is a more difficult problem), you are left with the much easier task of solving for only one unknown. – J. M. May 30 '12 at 14:04
Given the overall structure of your question, I'll assume that you have given data and by "fit the curve", you mean to find values of $a$, $b$, and $c$ so that the function $ae^{bt}+c$ is a good fit to that data.
In general, given data $\{x_i,y_i\}_{i=1}^n$ and a function univariate function $f_{a,b,c}(x)$ that depends on parameters $a$, $b$, and $c$, we fit the data by finding values of $a$, $b$, and $c$ that minimize $$\sum_{i=1}^n (f_{a,b,c}(x_i) - y_i)^2.$$ In the case where the expression $f_{a,b,c}(x)$ is linear in the parameters $a$, $b$, and $c$, this is a linear optimization problem and nice matrix methods can be applied. Otherwise, it's a non-linear optimization problem. Sometimes, this non-linear problem can be translated to a linear problem but sometimes strictly non-linear techniques must be used.
As an example, you might try the following input in WolframAlpha: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399086356109,
"lm_q1q2_score": 0.8561667400261774,
"lm_q2_score": 0.8824278741843884,
"openwebmath_perplexity": 150.60762916739546,
"openwebmath_score": 0.896618664264679,
"tags": null,
"url": "http://math.stackexchange.com/questions/151420/how-does-one-fit-the-curve-y-aebx-c"
} |
As an example, you might try the following input in WolframAlpha:
FindFit[{{-3,-1},{-2,0},{-1,0},{0,1},{1,2},{2,4},{3,8}},
a*exp(b*t)+c, {a,b,c}, t]
You should find that $f(t)=1.74*e^{0.54t}-0.96$ is a reasonable fit to this data. The result is given as a numerical approximation (decimal numbers, rather than exact), because numerical techniques are used. A plot of the function and the data looks like so:
-
A straightforward method (no need for initial guessed values, no iterative process) is shown with numerical example in pages 16-18 in the paper "Régressions et équations intégrales" published on Scribd :
http://www.scribd.com/JJacquelin/documents
With the data set given by Mark McClure, the result is shown on the figure below. The fitting of the curve to the data is quite the same, although the values of the parameters are slightly different. For practical use, the difference is negigible. This small discripency is a consequence of the too low number of experimental points.
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399086356109,
"lm_q1q2_score": 0.8561667400261774,
"lm_q2_score": 0.8824278741843884,
"openwebmath_perplexity": 150.60762916739546,
"openwebmath_score": 0.896618664264679,
"tags": null,
"url": "http://math.stackexchange.com/questions/151420/how-does-one-fit-the-curve-y-aebx-c"
} |
# Given two blank rulers, measure any length
Say, you are given ruler $A$ of length $72.84 \text{ cm}$ and ruler $B$ of length $86.63\text{ cm}$. Neither of them have any marking/gradation of any sort on them. They are blank, except for their total length written on them.
Using only A and B, can you measure a length of $31.23\text{ cm}$?
Also, is it possible to generalize this concept further?
That is to say:
Given any two blank rulers, measure any given length.
• An observation which may or may not be helpful: $\gcd (7284, 8663) = 1$ Jul 17, 2018 at 17:24
• I think a better title would be "measure a given length". No pair of rulers can measure every length. Jul 17, 2018 at 20:26
• @BallpointBen do you mean irrational and complex measures?
– Nick
Jul 18, 2018 at 4:13
• Jul 18, 2018 at 18:00
Yes. It is Possible
Here's a useful fact: If $a$ and $b$ are integers with $\gcd(a,b) = d$, then there exist integers $x$ and $y$ such that $ax + by =d$. In fact, one can compute exactly what $x$ and $y$ are by the extended Euclidean algorithm.
In this case, we have $$3539 \times 8663 - 4209 \times 7284 = 1$$
Or, $$3539 \times 86.63 - 4209 \times 72.84 = 0.01$$
So, in theory, you could measure $0.01$ cm by marking off $3539 \times 86.63$ cm along a line, and then marking off $4209 \times 72.84$ cm along the same line; the difference in the markings will be $0.01$ cm.
Of course, now that you can measure $0.01$ cm, you can measure any multiple thereof.
For your generalization, given two blank rulers, you can measure any length that is a multiple of the gcd of their lengths. (Make sure you choose units where the lengths of the rulers are integers. Here, we chose 0.01 cm)
Edit: As noted by Silverfish in the comments, the interesting fact I mention above is Bézout's identity | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109087400619,
"lm_q1q2_score": 0.8561486819699438,
"lm_q2_score": 0.8652240964782011,
"openwebmath_perplexity": 351.24140693959686,
"openwebmath_score": 0.7831504940986633,
"tags": null,
"url": "https://math.stackexchange.com/questions/2854732/given-two-blank-rulers-measure-any-length"
} |
• This question gives a great picture for the extended euclid's algorithm that i've never seen before. Jul 17, 2018 at 18:03
• A question out of curiosity: how did you find out that $3539\times 8663 - 4208\times 7284 = -1$? Did you search it with a computer program, did you have a hitch, did you already know...? Jul 17, 2018 at 20:41
• @AndreaDiBiagio I'd assume it was calculated through the linked extended Euclidean algorithm. Which might well have been, itself, run through a computer program, but it wasn't a "search".
– Nic
Jul 17, 2018 at 21:24
• @NicHartley makes total sense. I didn’t read through the answer properly. Jul 17, 2018 at 21:43
• For "here's a useful fact" it might be worth stating its name - it's Bézout's identity Jul 17, 2018 at 23:28
This all boils down to (using 0.01 cm as the unit) as
Does $\gcd(7284,8663)$ divide 3123?
The answer is yes: the gcd turns out to be 1. The generalisation is obvious: two blank rulers of lengths $a$ and $b$ units ($a,b$ are natural numbers) can measure any length that is a multiple of $\gcd(a,b)$ units.
• Actually, if a/b is rational and therefore gcd(a,b) exists then you can measure any multiple of gcd(a,b) exactly. If a/b is irrational then you can approximate any number with arbitrary precision, but not exactly. Jul 18, 2018 at 22:26
The other answers are in the affirmative and consider measuring multiples of the GCD of the two rulers, or $0.01\text{ cm}$ in the original problem.
For really small problems, it would be better to have an irrational ratio between the two lengths, e.g. a ruler of length $1$ and a ruler of length $\pi$. One can get multiples of $\pi$ arbitrarily close to the integers and thus construct basic units of measurement arbitrarily small. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109087400619,
"lm_q1q2_score": 0.8561486819699438,
"lm_q2_score": 0.8652240964782011,
"openwebmath_perplexity": 351.24140693959686,
"openwebmath_score": 0.7831504940986633,
"tags": null,
"url": "https://math.stackexchange.com/questions/2854732/given-two-blank-rulers-measure-any-length"
} |
Doing so, you can measure ANY non-negative real length to ANY desired positive degree of accuracy. Contrast such a ruler with the $0.01\text{ cm}$ ruler mentioned above which can only measure any non-negative real within $0.01\text{ cm}$ of accuracy ($0.005$ really, but there's an uncertainty in the measurement in which side it is actually closer to, so you wind up with up to $0.01$ from the further side).
• brilliant. I should have asked a ruler length $\pi$ and another of length $e$. I didn't think about irrationals.
– Nick
Jul 18, 2018 at 7:41
• Nice observation. A fancy way to say this is that the subgroup of $(\mathbb R,+)$ generated by $\{a, b\}$ is discrete if $\frac{a}{b}\in \mathbb Q$ and dense otherwise. Jul 18, 2018 at 12:22
• Note it’s not proven that $\pi/e$ is irrational. Jul 18, 2018 at 22:11
• @RomanOdaisky True, but either $\frac{e\pi+1}{e}$ or $\frac\pi e$ must be irrational. Jul 18, 2018 at 22:54
• @HansMusgrave Irrelevant — in the (highly unlikely) event $\pi/e$ is rational, $\pi$ and $e$ would have a GCD and better accuracy than the GCD would be impossible. Jul 19, 2018 at 14:31
We can write it as the equation $7284y - 8663x = 3123$
or $y = \frac{8663x + 3123}{7284}$
From this we can see that $8663x\equiv 7284-3123 \mod 7284$
$8663x \equiv 4161 \mod 7284$
$8663 \equiv 1379 \mod 7284$
So, $x \equiv \frac{4161}{1379} \mod 7284$
$x \equiv 4161\cdot \frac{1}{1379} \mod 7284$
$x = 4161\cdot 3539 = 14725779$........ where $3539$ is the multiplicative inverse of $1379$
So, $x = 14725779$ and $y = 17513650$
So, if we measure out $17513650$ lengths of the $72.84$ stick and then measure back with $14725779$ lengths of the $86.63$ stick, we achieve a distance of $31.23$ from our starting point. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109087400619,
"lm_q1q2_score": 0.8561486819699438,
"lm_q2_score": 0.8652240964782011,
"openwebmath_perplexity": 351.24140693959686,
"openwebmath_score": 0.7831504940986633,
"tags": null,
"url": "https://math.stackexchange.com/questions/2854732/given-two-blank-rulers-measure-any-length"
} |
• At just over 12.5 thousand km, I’m not sure the answer to the question is “yes”. Jul 17, 2018 at 21:20
• $5727 \times 72.84 - 4815 \times 86.63 = 31.23$ as well; you can always subtract $k \times 8663$ instances of the short stick and $k \times 7284$ instances of the long stick to get the same total length, and for this $k = 2021$ works. Granted this is still $4\text{km}$ or so of measuring to get something about $1/12000$ that length... Jul 17, 2018 at 21:37
• I took your solution and got $x \% 8663$ and $y \% 7284$. Note that obviously $8663 \times 72.84 - 7284 \times 86.63 = 0$, so adding or removing that many copies of each stick will not change the final length at all. Actually, if we go one step further, removing more sticks than are there and measuring in the other direction $2469 \times 86.63 - 2936 \times 72.84$ also works and is somewhat smaller! Jul 18, 2018 at 5:50
• @Ian I suppose you could sequence the positive and negative measurements cleverly to avoid going around the world, you go back and forth through the origin instead. In fact you should be able to do it given no more than a couple hundred meters of space. Jul 18, 2018 at 16:50
• I meant a couple hundred centimeters. I could not see the units while commenting. Could not correct the comment due to the time limit, which major sucks. Jul 18, 2018 at 16:59 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109087400619,
"lm_q1q2_score": 0.8561486819699438,
"lm_q2_score": 0.8652240964782011,
"openwebmath_perplexity": 351.24140693959686,
"openwebmath_score": 0.7831504940986633,
"tags": null,
"url": "https://math.stackexchange.com/questions/2854732/given-two-blank-rulers-measure-any-length"
} |
# Math Help - Need Help With a Tangent Line Problem
1. ## Need Help With a Tangent Line Problem
1) Show that the tangent line to the parabola y=Ax^2, A not equal to 0, at the point x=c will intersect the x-axis at the point (c/2, 0)
2) Determine where this line intersects the y-axis
2. Originally Posted by erimat89
1) Show that the tangent line to the parabola y=Ax^2, A not equal to 0, at the point x=c will intersect the x-axis at the point (c/2, 0)
2) Determine where this line intersects the y-axis
First find $y'$:
$y'=2Ax$.
Thus, the slope of the line at $x=c$ is $2Ac$.
Thus, the tangent has the equation $y-Ac^2=2Ac(x-c)\implies y=2Acx-Ac^2$
Now find where it crosses the x axis:
$0=2Acx-Ac^2\implies x=\dots$.
It crosses the y axis when x=0.
So $y=\dots$
Can you take it from here?
--Chris
Maybe I'm just confused but the question wasn't asking where it intersected the x-axis the x-axis intersection is given at the point (c/2, 0) it was only asking to determine where it crossed the y-axis in part 2 of the question. Why is this point (c/2, 0) given?
4. Originally Posted by erimat89
1) Show that the tangent line to the parabola y=Ax^2, A not equal to 0, at the point x=c will intersect the x-axis at the point (c/2, 0)
2) Determine where this line intersects the y-axis
Originally Posted by erimat89
Maybe I'm just confused but the question wasn't asking where it intersected the x-axis the x-axis intersection is given at the point (c/2, 0) it was only asking to determine where it crossed the y-axis in part 2 of the question. Why is this point (c/2, 0) given?
It asks you to show that the line intersects the x-axis at the point (c/2,0).
Since I got the equation of the line to be $y=2Acx-Ac^2$, at y=0, the line crosses the x-axis.
Thus $0=2Acx-Ac^2\implies 2Acx=Ac^2\implies x=\frac{Ac^2}{2Ac}\implies x=\frac{c}{2}$. So we see that it crosses the x-axis at $\left(\tfrac{1}{2}c,0\right)$
$\mathbb{Q.E.D.}$ | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109093587028,
"lm_q1q2_score": 0.8561486670304409,
"lm_q2_score": 0.8652240808393984,
"openwebmath_perplexity": 450.9521832331698,
"openwebmath_score": 0.7406553030014038,
"tags": null,
"url": "http://mathhelpforum.com/calculus/51466-need-help-tangent-line-problem.html"
} |
$\mathbb{Q.E.D.}$
Now, it intersects the y axis when x=0. Thus, we see that $y=2Ac(0)-Ac^2\implies y=-Ac^2$.
Thus, the y intercept is $\left(0,-Ac^2\right)$.
Does this make sense?
--Chris
5. ## Thanks
Makes alot of sense you clarified things for me. I've never been good with any sort of math problem that involves words for some reason I get thrown off. | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9895109093587028,
"lm_q1q2_score": 0.8561486670304409,
"lm_q2_score": 0.8652240808393984,
"openwebmath_perplexity": 450.9521832331698,
"openwebmath_score": 0.7406553030014038,
"tags": null,
"url": "http://mathhelpforum.com/calculus/51466-need-help-tangent-line-problem.html"
} |
# Characterization of convex set with empty interior on Hilbert spaces
Is the following statement true?
"Let $H$ be a Hilbert space and $C\subset H$ a convex set. If $C$ has empty interior then there exist $a$ and a proper subspace $V\subset H$ such that $C\subset(a+V)$."
I guess this is false due to the following counterexample. Let $$C=\{(x_n)_{n\in\mathbb N}:|x_n|\leq 1/2^n,\ \forall n\in\mathbb N\}\subset \ell_2(\mathbb N).$$ Clearly, $C$ is convex. Moreover, it is easy to see that $C^\perp=\{0\}$, and consequently, there are no proper subspace $V$ and $a$ such that $C\subset(a+V)$.
Finally, given some $(y_n)_{n\in\mathbb N}\in C$ and $r>0$, fix $n_0$ such that $1/2^{n_0}<r$. Then $$|y_{n_0+2}-1/2^{n_0}|\geq1/2^{n_0}-1/2^{n_0+2}>1/2^{n_0+2}.$$ Define $$z_n=\left\{\begin{array}{r} y_n,\ if\ n\neq n_0+2 \\ y_{n_0+2}-1/2^{n_0},\ if\ n= n_0+2\end{array}\right..$$ Then $(z_n)_{n\in\mathbb{N}}\in B((y_n)_{n\in\mathbb N},r)$ and $(z_n)_{n\in\mathbb{N}}\notin C$. This way we proved that no ball is contained in $C$, so $C$ has empty interior.
Is everything correct? Am I missing something here?
• There are easier counterexamples. What about the line $x=1$ in $\mathbb{R}^2$? – ChocolateAndCheese Mar 28 '17 at 23:55
• @ChocolateAndCheese That's true! I'm so sorry, there was a mistake in my question. Actually I meant "in the translation of a proper subspace of $H$". – André Porto Mar 29 '17 at 0:16
• Your proof that int (C) is empty is fine. – DanielWainfleet Mar 29 '17 at 6:58
Your counterexample is correct, assuming that "subspace" (as often) means a closed subspace of $H$. Otherwise, it would not work because the set $C$ is contained in $\ell^1(\mathbb{N})$ which is a dense subspace of $\ell^2(\mathbb{N})$. So I proceed by assuming the subspaces are closed. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924802053234,
"lm_q1q2_score": 0.8561281394227392,
"lm_q2_score": 0.8807970811069351,
"openwebmath_perplexity": 105.09149918694645,
"openwebmath_score": 0.9296537041664124,
"tags": null,
"url": "https://math.stackexchange.com/questions/2207763/characterization-of-convex-set-with-empty-interior-on-hilbert-spaces"
} |
The justification is slightly lacking in the following: the property $C^\perp = \{0\}$ only implies that $C$ is not contained in any proper linear subspace; it does not preclude $C$ from being contained in a proper affine subspace. For example, the set $A=e_1+e_1^\perp$, which is an affine hyperplane, satisfies $A^\perp = \{0\}$ since its linear span is all of $\ell^2$.
However, the above is easy to repair: since $C$ contains $0$, any affine subspace containing it would be a linear subspace.
### A simpler example
Let $C$ be the set of all sequences $x$ such that $x_n=0$ except for finitely many $n$. Then $C$ is convex and has empty interior, since adding an arbitrarily small multiple of the vector $(1/2^n)_{n\in\mathbb{N}}$ to an element of $C$ takes one out of $C$. It's also dense in $\ell^2$, so can't be contained in a proper closed subset of any kind.
(Your example has the additional property of being closed, which however wasn't required.)
• the statement "the property $C^\perp={0}$ only implies that $C$ is not contained in any proper linear subspace" is not true. The counterexample is the set $C$ defined in the counterexample gave by you. Clearly, $C^\perp=\{0\}$ and it is a proper linear subspace of $H$. – André Porto Apr 17 '17 at 19:38
• @AndréPorto See the first paragraph: "I proceed by assuming the subspaces are closed." – user357151 Apr 17 '17 at 19:43
• Yeah, ok then. My bad. – André Porto Apr 17 '17 at 20:37
As @Gerry pointed out, $C^\perp=\{0\}$ does not imply that $C$ is not contained in a proper affine subspace of $H$, so the example I gave in my question may not hold.
Fortunately, I found another example that really works this time. It is a much simpler one and is a counterexample to the statement even when $C$ is assumed to be closed.
Let $C$ be the set of vectors in $\ell_2(\mathbb N)$ with all its coordinates greater than or equal to $0$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924802053234,
"lm_q1q2_score": 0.8561281394227392,
"lm_q2_score": 0.8807970811069351,
"openwebmath_perplexity": 105.09149918694645,
"openwebmath_score": 0.9296537041664124,
"tags": null,
"url": "https://math.stackexchange.com/questions/2207763/characterization-of-convex-set-with-empty-interior-on-hilbert-spaces"
} |
Clearly, $C$ is a closed convex. Let us see that $C$ has empty interior. Given $a=(a_n)_{n\in\mathbb N}\in C$ and $r>0$, since $a_n\to 0$, fix $m$ such that $a_m<r/2$ and define $b=(b_n)_{n\in\mathbb N}$ to be the sequence obtained by changing $a_m$ by $a_m-r/2$. Clearly, $b\in B(a,r)$ and, since $b_m=a_m-r/2<0$, $b\notin C$. Then $B(a,r)$ is not contained in $C$.
Finally, let us see that $C$ is not contained in a proper affine subspace. Pick $a\in \ell_2(\mathbb N)$ and a subspace $V$ such that $C\subset(a+V)$. Since $0\in C$, we get that $a\in V$, so actually $C\subset V$. We will conclude below that $V=\ell_2(\mathbb N)$.
Given $a=(a_n)_{n\in\mathbb N}\in \ell_2(\mathbb N)$, define $a^+=(a^+_n)_{n\in\mathbb N}$ by $$a^+_n=\left\{\begin{array}{r}a_n,\ \mbox{if}\ a_n\geq0 \\ 0,\ \mbox{if}\ a_n<0\end{array}\right..$$ and $a^-=(a^-_n)_{n\in\mathbb N}$ by $$a^-_n=\left\{\begin{array}{r}-a_n,\ \mbox{if}\ a_n<0 \\ 0,\ \mbox{if}\ a_n\geq0\end{array}\right..$$ Then, $a^+,a^-\in C\subset V$ and consequently $a=a^+-a^-\in V$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924802053234,
"lm_q1q2_score": 0.8561281394227392,
"lm_q2_score": 0.8807970811069351,
"openwebmath_perplexity": 105.09149918694645,
"openwebmath_score": 0.9296537041664124,
"tags": null,
"url": "https://math.stackexchange.com/questions/2207763/characterization-of-convex-set-with-empty-interior-on-hilbert-spaces"
} |
# First Order ODEs¶
Last time, after a general introduction, we met the general first order ODE:
$$y' = f(t,y).$$
We also examined the very simple, yet interesting such ODE $y'=r y$ and solved it qualitatively (using a slope field) and symbolically (using separation of variables).
Today, we'll generalize those ideas further and explore more challenging examples.
Much of today's material is explored in sections 1.1.3 and 1.3.1 of our text.
## The logistic equation¶
Just as the exponential equation $y'=ry$ was a motivating example for us last time, the logistic equation will be a motivating example today:
$$y' = r y (1-y/K).$$
The logistic equation can be interpretted as a growth model, just as the exponential equation can. As we'll see, though, the last $(1-y/K)$ term has the effect of slowing the growth down though as the quantity $y$ approaches the carrying capacity $K$.
### Equilibrium solutions¶
Like the exponential equation, the logistic equation is an autonomous equation - not depending explicitly on $t$. When we write an autonomous equation in the form $$y'=f(y),$$ the roots of $f$ form constant or equilibrium solutions of the differential equation.
That is if $y_0$ is a real number with $f(y_0)=0$, then the constant function $y(t) \equiv y_0$ is a solution of the ODE since both sides of $y'=f(y)$ are zero.
Furthermore, if we plot the equilibrium solutions, then they contrain any other solutions.
### Logistic equilibrium solutions¶
The logistic equation $y' = r y (1-y/K)$ has two equilibrium solutions, namely
$$y(t) \equiv 0 \: \text{ and } \: y(t) \equiv K.$$
Let's plot those:
### Logistic solutions¶
Other solutions to the logistic equation are constrained by the equilibrium solutions. Furthermore, bewtween $0$ and $K$, the right side of the logistic equation
$$y' = r y (1-y/K)$$
is positive (assuming $r$ is positive). Thus, we expect a solution to increase from the bottom equilibrium to the top. Solutions must look something like so: | {
"domain": "marksmath.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9937100984219314,
"lm_q1q2_score": 0.8561192206543495,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 627.608370822618,
"openwebmath_score": 0.9034628868103027,
"tags": null,
"url": "https://marksmath.org/classes/Fall2020DiffEq/class_notes/08.12.20.FirstOrderODEs.slides.html"
} |
### Logistic slope field¶
We should get a similar sort of picture from a slope field:
## Separation of variables¶
Let's apply separation of variables to solve the logistic equation $$\frac{dy}{dt} = r y (1-y/K)$$ symbolically.
This, is problem #17 from section 1.3.1 of our text.
I guess we'd start with $$\frac{1}{y(1-y/K)}dy = r\,dt.$$
### Separation of variables (cont)¶
The book suggest that we use the partial fractions decomposition
$$\frac{1}{y(1-K/y)} = \frac{1}{y}+\frac{1}{K-y},$$
$$\int\left(\frac{1}{y}+\frac{1}{K-y}\right)dy = \int r\,dt$$
or
$$\log\left(y\right) - \log\left(K-y\right) = r\,t+c_1.$$
### Separation of variables (cont 2)¶
Combining the logarithms in that last equation yields
$$\log\left(\frac{y}{K-y}\right) = r\,t+c_1,$$
which allows us to apply the exponential to both sides to get
$$\frac{y}{K-y} = e^{r\,t+c_1} = c e^{r\,t}.$$
Note that we've pulled the old $e^{a+c_1} = e^ae^{c_1} = e^ac$ trick.
### Separation of variables (cont 3)¶
Now, we can solve that last equation for $y$ to get
$$y = \frac{K\,c\,e^{r\,t}}{1+c\,e^{r\,t}} = \frac{K}{1+\frac{1}{c}e^{-r\,t}}.$$
That is our solution:
$$y(t) = \frac{K}{1+\frac{1}{c}e^{-r\,t}}.$$
### Interpretation¶
Given our solution,
$$y(t) = \frac{K}{1+\frac{1}{c}e^{-r\,t}},$$
It's not hard to see that
$$\lim_{t\to -\infty}y(t) = 0 \: \text{ and } \: \lim_{t\to\infty}y(t)=K$$
and that $y$ increases from $0$ to $K$ as $t$ ranges from $-\infty$ to $\infty$.
That agrees with our qualitative analysis!
### An IVP¶
In order to plot a specific solution, we'll need values for the parameters $r$ and $K$, as well as an initial condition. Let's assume that
$$K=100, \: r=3, \text{ and } y(0)=4.$$
Thus, our solution becomes
$$y(t) = \frac{100}{1+\frac{1}{c}e^{-3\,t}}.$$
### An IVP (cont)¶
To find $c$ in
$$y(t) = \frac{100}{1+\frac{1}{c}e^{-3\,t}},$$
we use the initial condition $y(0)=4$ to get
$$4 = \frac{100}{1+\frac{1}{c}}.$$
We solve this to get $c=1/24$ so that | {
"domain": "marksmath.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9937100984219314,
"lm_q1q2_score": 0.8561192206543495,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 627.608370822618,
"openwebmath_score": 0.9034628868103027,
"tags": null,
"url": "https://marksmath.org/classes/Fall2020DiffEq/class_notes/08.12.20.FirstOrderODEs.slides.html"
} |
$$4 = \frac{100}{1+\frac{1}{c}}.$$
We solve this to get $c=1/24$ so that
$$y(t) = \frac{100}{1+24e^{-3\,t}}.$$
Here's how to plot that solution in Python:
import matplotlib.pyplot as plt
import numpy as np
def y(t): return 100/(1+24*np.exp(-3*t))
ts = np.linspace(-1,5)
ys = y(ts)
plt.plot(ts,ys);
## Slope fields for autonomous equations¶
The logistic equation is somewhat indicative of the way that the general autonomous equation $y'=f(y)$ works.
We first find the equilibrium solutions; then, assuming that $f$ is continous, the slopes don't change sign between the equilibria.
Then, it's easy to sketch the solutions.
### Example¶
For example, here's the slope field for $y'=y^2-1$, together with a solution between the two equilibria of $\pm 1$, which must be decreasing:
### Finally...¶
How about I do one by hand? | {
"domain": "marksmath.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9937100984219314,
"lm_q1q2_score": 0.8561192206543495,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 627.608370822618,
"openwebmath_score": 0.9034628868103027,
"tags": null,
"url": "https://marksmath.org/classes/Fall2020DiffEq/class_notes/08.12.20.FirstOrderODEs.slides.html"
} |
# The Union of Two Subspaces is Not a Subspace in a Vector Space
## Problem 274
Let $U$ and $V$ be subspaces of the vector space $\R^n$.
If neither $U$ nor $V$ is a subset of the other, then prove that the union $U \cup V$ is not a subspace of $\R^n$.
Sponsored Links
## Proof.
Since $U$ is not contained in $V$, there exists a vector $\mathbf{u}\in U$ but $\mathbf{u} \not \in V$.
Similarly, since $V$ is not contained in $U$, there exists a vector $\mathbf{v} \in V$ but $\mathbf{v} \not \in U$.
Seeking a contradiction, let us assume that the union is $U \cup V$ is a subspace of $\R^n$.
The vectors $\mathbf{u}, \mathbf{v}$ lie in the vector space $U \cup V$.
Thus their sum $\mathbf{u}+\mathbf{v}$ is also in $U\cup V$.
This implies that we have either
$\mathbf{u}+\mathbf{v} \in U \text{ or } \mathbf{u}+\mathbf{v}\in V.$
If $\mathbf{u}+\mathbf{v} \in U$, then there exists $\mathbf{u}’\in U$ such that
$\mathbf{u}+\mathbf{v}=\mathbf{u}’.$ Since the vectors $\mathbf{u}$ and $\mathbf{u}’$ are both in the subspace $U$, their difference $\mathbf{u}’-\mathbf{u}$ is also in $U$. Hence we have
$\mathbf{v}=\mathbf{u}’-\mathbf{u} \in U.$
However, this contradicts the choice of the vector $\mathbf{v} \not \in U$.
Thus, we must have $\mathbf{u}+\mathbf{v}\in V$.
In this case, there exists $\mathbf{v}’ \in V$ such that
$\mathbf{u}+\mathbf{v}=\mathbf{v}’.$
Since both $\mathbf{v}, \mathbf{v}’$ are vectors of $V$, it follows that
$\mathbf{u}=\mathbf{v}’-\mathbf{v}\in V,$ which contradicts the choice of $\mathbf{u} \not\in V$.
Therefore, we have reached a contradiction. Thus, the union $U \cup V$ cannot be a subspace of $\R^n$.
## Related Question.
In fact, the converse of this problem is true.
Problem. Let $W_1, W_2$ be subspaces of a vector space $V$. Then prove that $W_1 \cup W_2$ is a subspace of $V$ if and only if $W_1 \subset W_2$ or $W_2 \subset W_1$.
For a proof, see the post “Union of Subspaces is a Subspace if and only if One is Included in Another“. | {
"domain": "yutsumura.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9915543728093004,
"lm_q1q2_score": 0.8560997245736672,
"lm_q2_score": 0.8633916082162402,
"openwebmath_perplexity": 84.80948193121401,
"openwebmath_score": 0.965861976146698,
"tags": null,
"url": "http://yutsumura.com/the-union-of-two-subspaces-is-not-a-subspace-in-a-vector-space/"
} |
Sponsored Links
### More from my site
#### You may also like...
##### Quiz 2. The Vector Form For the General Solution / Transpose Matrices. Math 2568 Spring 2017.
(a) The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the...
Close | {
"domain": "yutsumura.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9915543728093004,
"lm_q1q2_score": 0.8560997245736672,
"lm_q2_score": 0.8633916082162402,
"openwebmath_perplexity": 84.80948193121401,
"openwebmath_score": 0.965861976146698,
"tags": null,
"url": "http://yutsumura.com/the-union-of-two-subspaces-is-not-a-subspace-in-a-vector-space/"
} |
# If $\sec \theta + \tan \theta =x$, then find the value of $\sin \theta$.
If $$\sec \theta + \tan \theta =x$$, then find the value of $$\sin \theta$$.
$$\sec \theta + \tan \theta = x$$ $$\dfrac {1}{\cos \theta }+\dfrac {\sin \theta }{\cos \theta }=x$$ $$\dfrac {1+\sin \theta }{\sqrt {1-\sin^2 \theta }}=x$$ $$1+\sin \theta =x\sqrt {1-\sin^2 \theta }$$ $$1+2\sin \theta + \sin^2 \theta = x^2-x^2 \sin^2 \theta$$ $$x^2 \sin^2 \theta + \sin^2 \theta + 2\sin \theta = x^2-1$$ $$\sin^2 \theta (x^2+1) + 2\sin \theta =x^2-1$$
Here is a different approach: Since $1 + \tan^2\theta = \sec^2\theta$, we have $$\sec^2\theta - \tan^2\theta = 1$$ Factoring yields $$(\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = 1$$ Since we are given that $\sec\theta + \tan\theta = x$, we obtain $$x(\sec\theta - \tan\theta) = 1$$ Therefore, $$\sec\theta - \tan\theta = \frac{1}{x}$$ This yields the system of equations \begin{align*} \sec\theta + \tan\theta & = x \tag{1}\\ \sec\theta - \tan\theta & = \frac{1}{x} \tag{2} \end{align*} Adding equations 1 and 2 and solving for $\sec\theta$ yields \begin{align*} 2\sec\theta & = x + \frac{1}{x}\\ 2\sec\theta & = \frac{x^2 + 1}{x}\\ \sec\theta & = \frac{x^2 + 1}{2x} \end{align*} Therefore, $$\cos\theta = \frac{1}{\sec\theta} = \frac{2x}{x^2 + 1}$$ Subtracting equation 2 from equation 1 and solving for $\tan\theta$ yields \begin{align*} 2\tan\theta & = x - \frac{1}{x}\\ 2\tan\theta & = \frac{x^2 - 1}{x}\\ \tan\theta & = \frac{x^2 - 1}{2x} \end{align*} Thus, $$\sin\theta = \tan\theta\cos\theta = \frac{x^2 - 1}{2x} \cdot \frac{2x}{x^2 + 1} = \frac{x^2 - 1}{x^2 + 1}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429614552197,
"lm_q1q2_score": 0.8560957077068735,
"lm_q2_score": 0.8705972801594707,
"openwebmath_perplexity": 361.86323666571525,
"openwebmath_score": 0.9362828135490417,
"tags": null,
"url": "https://math.stackexchange.com/questions/2280453/if-sec-theta-tan-theta-x-then-find-the-value-of-sin-theta"
} |
• I didn't get the answer. The answer is $\dfrac {1-x^2}{1+x^2}$. – Aryabhatta May 14 '17 at 13:15
• What did you get for $\sec\theta$ and $\tan\theta$? – N. F. Taussig May 14 '17 at 13:19
• I got $\sec \theta=\dfrac {x^2+1}{2x}$ and $\tan \theta =\dfrac {2x-x^2-1}{2x}$. – Aryabhatta May 14 '17 at 13:21
• I agree with your answer for $\sec\theta$. When I solved for $\tan\theta$, I obtained $$\frac{x^2 - 1}{2x}$$ which led to the answer $$\frac{x^2 - 1}{1 + x^2}$$ – N. F. Taussig May 14 '17 at 13:26
• I have added the details of my calculations. I checked my answer for the angles $\pi/6$, $\pi/4$, and $\pi/3$. In each case, it gave the correct answer, while the answer you stated gives the wrong sign. – N. F. Taussig May 14 '17 at 13:50
The equation becomes $$1+\sin\theta=x\cos\theta$$ Set $X=\cos\theta$ and $Y=\sin\theta$, so the equation becomes $$\begin{cases} X^2+Y^2=1 \\[4px] 1+Y=xX \end{cases}$$ Note that $x\ne0$ and substitute $X=x^{-1}(1+Y)$ in the first equation getting $$(1+Y)^2+x^2Y^2=x^2$$ that simplifies to $$(1+x^2)Y^2+2Y+1-x^2=0$$ that yields $$Y=-1 \qquad\text{or}\qquad Y=\frac{x^2-1}{x^2+1}$$ Is $Y=-1$ a solution for the problem?
By the way, you also get $\cos\theta$, since $$X=\frac{1}{x}(1+Y)=\frac{1}{x}\frac{x^2+1+x^2-1}{x^2+1}=\frac{2x}{x^2+1}$$
From where you are:
You obtained a quadratic function in $\sin(\theta)$. Perform the substitution $u =\sin(\theta)$.
We obtain the quadratic (in $u$):
$$(x^2+1)u^2 + 2u - x^2 +1 = 0$$
$$\Rightarrow u_{1,2} = \frac{- 2 \pm \sqrt{4 - 4(x^2+1)(1-x^2)}}{2(x^2+1)}$$
$$= \frac{- 2 \pm \sqrt{4 + 4(x^4 -1)}}{2(x^2+1)}$$
$$= \frac{- 2 \pm \sqrt{4x^4}}{2(x^2+1)}$$
$$= \frac{- 2 \pm 2x^2}{2(x^2+1)}$$
$$= \frac{- 1 \pm x^2}{x^2+1}$$
Therefore,
$$\sin(\theta)_{1,2} = \frac{- 1 \pm x^2}{x^2+1}$$
One of those solutions will not work out. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429614552197,
"lm_q1q2_score": 0.8560957077068735,
"lm_q2_score": 0.8705972801594707,
"openwebmath_perplexity": 361.86323666571525,
"openwebmath_score": 0.9362828135490417,
"tags": null,
"url": "https://math.stackexchange.com/questions/2280453/if-sec-theta-tan-theta-x-then-find-the-value-of-sin-theta"
} |
$$\sin(\theta)_{1,2} = \frac{- 1 \pm x^2}{x^2+1}$$
One of those solutions will not work out.
This happened because you squared the equation multiple times and we know that $a = b$ is not equivalent with $a^2 = b^2$, so you should fill in both solutions in the original expression and see which one works and which one doesn't.
• What is discriminant method? – Aryabhatta May 14 '17 at 13:05
• en.wikipedia.org/wiki/Discriminant#Degree_2 – user370967 May 14 '17 at 13:06
• I didn't understand... – Aryabhatta May 14 '17 at 13:14
• If you want to solve a solution in the form $ax^2 + bx + c = 0$, then the solutions are given by $x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ – user370967 May 14 '17 at 13:21
• Here $x = u, a = (x^2 + 1), b = 2, c = - x^2 + 1$ – user370967 May 14 '17 at 13:22
WLOG let $\theta=\dfrac\pi2-2y\implies x=\csc2y+\cot2y=\dfrac{1+\cos2y}{\sin2y}=\cot y$
$$\sin\theta=\cos2y=\dfrac{1-\tan^2y}{1+\tan^2y}=\dfrac{\cot^2y-1}{\cot^2y+1}=?$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429614552197,
"lm_q1q2_score": 0.8560957077068735,
"lm_q2_score": 0.8705972801594707,
"openwebmath_perplexity": 361.86323666571525,
"openwebmath_score": 0.9362828135490417,
"tags": null,
"url": "https://math.stackexchange.com/questions/2280453/if-sec-theta-tan-theta-x-then-find-the-value-of-sin-theta"
} |
# Coin Flip Probability | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
Over a large number of tosses, though, the percentage of heads and tails will come to approximate the true probability of each outcome. Practice this lesson yourself on KhanAcademy. Assistant Research Professor. In this case, the probability of flipping a head or a tail is 1/2. Hi everyone. If an event consists of more than one coin, then coins are considered as. For the first 53 Super Bowls the flip has landed on tails 28 times and heads 25 times. The coin has come up heads 54% of the time so far; based only on this data, one might expect that it is slightly more likely to come up heads again. Flip 10 coins, and and you're at a 4-digit number. The sum of all possible outcomes is always 1 (or 100%) because it is certain that one of the possible outcomes will happen. The probability of an event is a number indicating how likely that event will occur. For example, if an individual wanted to know the probability of getting a head in a coin toss but only used one sample, the empirical probability would be either 0% or 100%. Don't expect the numbers from trials to exactly match the predicted results--especially if you run only a few trials. But not that much more likely. After all, real life is rarely fair. 2, which is the Kelly fraction for this α and p. I got the program down right but my results show a number for each coin flip in addition to the cout that says "The coin flip shows Heads/Tails". Applet: Instructions: Examples: Notes "H. Showing top 8 worksheets in the category - Coin Flip Experiment Basic. I flip a coin and it comes up heads. 5 for either heads or tails (assuming that the coin is purely mathematical and random, of course). If p=0 or p=1, the strategy is obvious, so assume 0. We’ll set p=0. Gamblers Take Note: The Odds in a Coin Flip Aren't Quite 50/50 And the odds of spinning a penny are even more skewed in one direction, but which way? Flipping a coin isn't as fair as it seems. If after 200 flips, the. I got a question on the coin flip project. An | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
isn't as fair as it seems. If after 200 flips, the. I got a question on the coin flip project. An Easy GRE Probability Question. e head or tail. What were the results? Student: The coin landed on heads 9 times and on tails 6 times. Intro to Problem Solving. It can even toss weighted coins. 51 (instead of 0. Don't expect the numbers from trials to exactly match the predicted results--especially if you run only a few trials. How likely something is to happen. Start studying Laws of Probability: Coin Toss Lab. This method may be used to resolve a dispute, see who goes first in a game or determine which type of treatment a patient receives in a clinical trial. If the result is heads, they flip a coin 100 times and record results. Published on June 14, 2016. 5, which is our probability of tossing heads and moving forward. I need to write a python program that will flip a coin 100 times and then tell how many times tails and heads were flipped. Course : Introduction to Probability and Statistics, Math 113 Section 3234 Instructor: Abhijit Champanerkar Date: Oct 17th 2012 Tossing a coin The probability of getting a Heads or a Tails on a coin toss is both 0. One source of confusion is in counting the number of outcomes, both favorable and possible, such as when tossing coins and rolling dice. She'll make a prediction and practice flipping a coin in order to check out its chances of landing on heads or tails. 8) for i in xrange(10)] [H,H,T,H,H,H,T,H,H,H]. For 100 flips, if the actual heads probability is 0. Click "flip coins" to generate a new set of coin flips. The odds are "long" only if you predetermine when the series of coin flips begins. Note that this answer works for any odd number of coin flips. We assume that conditioned on Q=q, all coin tosses are independent. If after 200 flips, the. Mathematically the coin flip. The fact of the matter is, the human, not the coin (mostly, there is a slight weight bias that might be shown after approximately 10,000 flips), | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
coin (mostly, there is a slight weight bias that might be shown after approximately 10,000 flips), introduces the probability that the coin may land. Coin Toss Probability Calculator Coin toss also known as coin flipping probability is used by people around the world to judge whether its going to be head or tail after flipping the coin. Coin Toss: Simulation of a coin toss allowing the user to input the number of flips. If you get tails on the first flip, you might as well stop, because you cannot possibly get four heads. Users may refer the below detailed solved example with step by step calculation to learn how to find what is the probability of getting exactly 2 heads, if a coin is tossed five times or 5 coins tossed together. Update: The initial 6-for-6 report, from the Des Moines Register missed a few Sanders coin-toss wins. In the last exercise you tried flipping ten coins with a 30% probability of heads to find the probability *at least five are heads. With a "fair" coin, the probability of getting heads on a "single" flip at any time is 1/2. You found that the exact answer was '1 - pbinom(4, 10,. By theory, we can calculate this probability by dividing number of expected outcomes by total number of outcomes. I would like to know what is the probability of this occurrence within any 100 consecutive flips out of a series of. Inspiration • A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes • Let us discuss the binomial model we have studied so far through a very simple example • Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as Ω = {HHH,HHT,HTH,THH,HTT,THT,TTH,TTT} • Assume that the probability of a head. Predicting a coin toss. Assuming we kept going, then we flip the second coin. It is not always easy to decide what is heads and tails on a given coin. Applet: Instructions: Examples: Notes "H" count = , flips so far, number of coins: one flip "H" probability: 0. | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
Examples: Notes "H" count = , flips so far, number of coins: one flip "H" probability: 0. The simplicity of the coin toss also opens the road to more advanced probability theories dealing with events with an infinite number of possible outcomes. Many events can't be predicted with total certainty. Toss a single coin 5X and record the results Table 1. 5, which is our probability of tossing heads and moving forward. The number of possible outcomes gets greater with the increased number of coins. When a coin is tossed twice, the coin has no memory of whether it came up heads or tails the first time, so the second toss of the coin is independent. What is the probability of getting two heads and four tails? Coin Flipping How can I figure out the chances of flipping a coin five times with the result T,T,T,H,H?. I got a question on the coin flip project. The odds of the coins coming up with different faces showing is just 1 in 2. Read and learn for free about the following article: Theoretical and experimental probability: Coin flips and die rolls If you're seeing this message, it means we're having trouble loading external resources on our website. Two flips have 4 outcomes: HH, Ht, tH, and tt. In this activity, you will explore some ideas of probability by using Excel to simulate tossing a coin and throwing a free throw in basketball. Each time you toss these coins, there are four possible outcomes: both heads penny head & dime tail penny tail & dime head both tails You will flip the pair of coins 20 times. How likely something is to happen. Ask Question Asked 7 years, 3 months ago. If 5 is selected for "coin flips per trial," then a number is selected from the list five times, and these numbers are summed. The article says the reason is because a flipped coin does not spin perfectly around its axis and sometimes appears to be flipping when it actually isn’t. Toss a coin 10 times and after each toss, record in the following table the result of the toss and the proportion | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
times and after each toss, record in the following table the result of the toss and the proportion of heads so far. You found that the exact answer was '1 - pbinom(4, 10,. Mentor: OK, we. Flip 10 coins, and and you're at a 4-digit number. I am just learning Python on class so I am really at the basic. The likelihood of an event is expressed as a number between zero (the event will never occur) and one (the event is certain). Coin Toss Probability Calculator is a free online tool that displays the probability of getting the head or a tail when the coin is tossed. This uses a simple Monte Carlo approximation to estimate the probability distribution for the length of the longest consecutive sequence of Heads in a fixed number of coin flips. But I want to simulate coin which gives H with probability 'p' and T with probability '(1-p)'. The coin has come up heads 54% of the time so far; based only on this data, one might expect that it is slightly more likely to come up heads again. The second paragraph then applies a little conditional probability. In the last exercise you tried flipping ten coins with a 30% probability of heads to find the probability *at least five are heads. Numbers are then randomly selected from the list. Page last modified 07/17/2012 13:01:23. In unbiased coin flip H or T occurs 50% of times. Coin toss probability is explored here with simulation. Published on June 14, 2016. The Probability Simulation application on the TI-84 Plus graphing calculator can simulate tossing from one to three coins at a time. , Bernoulli trials). A fair-sided coin (which means no casino hanky-panky with the coin not coming up heads or tails 50% of the time) is tossed three times. We assume that conditioned on Q=q, all coin tosses are independent. The probability of a success on any given coin flip would be constant (i. Probability. A tossed coin is spinning and falling, therefore it carries significant kinetic energy. Consider flipping a coin that is either heads (H) | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
therefore it carries significant kinetic energy. Consider flipping a coin that is either heads (H) or tails (T), each with probability 1/2. The program should call a separate function flip()that takes no arguments and returns 0 for tails and 1 for heads. Below is some sample code in R to simulate a fair coin toss in R using the sample function. In 1947, the coin flipping was held 30 minutes before the beginning of the game. From 1892 to 1920, the captain of the football team managed the coin flip. 57 flip_coins(100, 1000, 25) #> [1] 0. Luckily, this is bundled up in a math/probability/stats concept called "combinations". When a coin is tossed, there are two possible outcomes: heads (H) or ; tails (T) We say that the probability of the coin landing H is ½. Flip 4 coins, and you're at 16 outcomes, a 2-digit number. Remark: The idea can be substantially generalized. Every flip of the coin has an “independent probability“, meaning that the probability that the coin will come up heads or tails is only affected by the toss of the coin itself. 5, which means we would not be able to tell the different between a bias coin and fair coin 50% of the time. There is also the very small probability that the coin will land. 1) The mathematical theory of probability assumes that we have a well defined repeatable (in principle) experiment, which has as its outcome a set of well defined, mutually exclusive, events. Next we will show some simulations of coin toss betting using the Kelly fraction. Demonstrates frequency and probability distributions with weighted coin-flipping experiments. When tossing only one coin at a time, the application keeps track of the number of heads and tails that occur as the coin is repeatedly tossed. Math archives: Probability in Flipping Coins Six pennies are flipped. The smoother of those two lines is an average of 2000 runs. While a coin toss is regarded as random, it spins in a predictable way. the probability of throwing exactly two heads in three | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
as random, it spins in a predictable way. the probability of throwing exactly two heads in three tosses of the coin is 3 out of 8, or or the decimal equivalent of which is 0. The coin toss is nothing but experimenting with tossing a coin. (Ti includes the toss that results in the first. Let Ti be the number of tosses of the ith coin until that coin results in Heads for the first time, for i=1,2,…,k. Flipping coins and the binomial distribution#2 Consider two coins, one fair and one unfair. Thus, the odds of any throw being a tail is 1/2. % certain that the outcome would be tails, but this is due to how it is being measured. So if an event is unlikely to occur, its probability is 0. This is a very basic form of empirical probability, however, and has a high risk of being incorrect because a series of only two events (coin tosses) have been observed. Over a large number of tosses, though, the percentage of heads and tails will come to approximate the true probability of each outcome. I am just learning Python on class so I am really at the basic. How likely something is to happen. For a fair coin, The probability of the outcome Head is 1/2, because for a large number of tosses, the relative occurrence of Heads will be roughly 1/2. Both outcomes are equally likely. Numbers are then randomly selected from the list. Then, you will flip the coins 100 times and determine the experimental probability of the events. We all know a coin toss gives you a 50% chance of winning, but is it always that way? Delve into the inner-workings of coin toss probability with this activity. A common topic in introductory probability is solving problems involving coin flips. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment. Bayes equation describes the situation when the probability that a fair coin was used to produce two heads is equal to the probability of | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
when the probability that a fair coin was used to produce two heads is equal to the probability of seeing two heads if a fair coin was used. After all, real life is rarely fair. Since the coin toss is a physical phenomenon governed by Newtonian mechanics, the question requires one to link probability and physics via a mathematical and statistical description of the coin's. Mentor: Yes! Now let's look at the coin flipping game that you just played. Simple numbers. Recommended: Please try your approach. Take another penny and Super Glue it to the coin. 53) #> [1] 0. The coin has no desire to continue a particular streak, so it's not affected by any number of previous coin tosses. When asked the question, what is the probability of a coin toss coming up heads, most people answer without hesitation that it is 50%, 1/2, or 0. Note: Including the words "single time" and "after" confuse this problem somewhat. Q: if you flip a coin 3 times what is the probability of getting only 1 head? A: The probability of getting one head in three throws is 0. 5×10 20 chance of getting a string of 76 heads. So if you flip a coin 10 times in a row-- a fair coin-- you're probability of getting at least 1 heads in that 10 flips is pretty high. 7 flip_coins(100, 100, 5) #> [1] 0. This code helps you count consecutive strings of Heads in a sequence of coin flips. If we toss a coin $n$ times, and the probability of a head on any toss is $p$ (which need not be equal to $1/2$, the coin could be unfair), then the probability of exactly $k$ heads is $$\binom{n}{k}p^k(1-p)^{n-k}. As long as the coin was not manipulated the theoretical probabilities of both outcomes are the same—they are equally probable. Toss a single coin 5X and record the results Table 1. Write a program that simulates coin tossing. This is a very basic form of empirical probability, however, and has a high risk of being incorrect because a series of only two events (coin tosses) have been observed. the coin does not and can not | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
because a series of only two events (coin tosses) have been observed. the coin does not and can not "remember" last result 4. And you can get a calculator out to figure that out in terms of a percentage. Coin Toss Probability Calculator. I flip a coin and it comes up heads. Nine flips of a fair coin. How do I get rid of the number? It looks something like this when I run it The coin flipped Heads 1 The Coin flipped tails 2 The coin flipped Heads 1. Sunday, March 29, 2009. The Probability Simulation application on the TI-84 Plus graphing calculator can simulate tossing from one to three coins at a time. Below is some sample code in R to simulate a fair coin toss in R using the sample function. If the probability of coin flipping head = P The probability of coin flipping tail = 1 - P Now The probability of flipping heads & then tails = Probability of flipping tails & then heads = P(1 - P) Which means to make a fair coin toss we now need 2 flips Player 1 wins if the sequence is HT Player 2 wins if the sequence is TH Any other sequence. The coin flip has gone through many changes. The probability of Heads is the same for each coin and is the realized value q of a random variable Q that is uniformly distributed on [0,1]. For 100 flips, if the actual heads probability is 0. What is the probability that player A ends up with all the coins?. the coin tossing is stateless operation i. Click "flip coins" to generate a new set of coin flips. When we flip a coin there is always a probability to get a head or a tail is 50 percent. Remark: The idea can be substantially generalized. I would like to know what is the probability of this occurrence within any 100 consecutive flips out of a series of. When you flip a coin, you can generally get two possible outcomes: heads or tails. There is a 50% probability that the first toss will end up heads. A probability of zero means that an event is impossible. 4, then the power is 0. But give me a well-balanced coin, any size, and I can roll | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
is impossible. 4, then the power is 0. But give me a well-balanced coin, any size, and I can roll it on any flat surface on edge every time. There is a 50% probability that the first toss will end up heads. One source of confusion is in counting the number of outcomes, both favorable and possible, such as when tossing coins and rolling dice. Confidence intervals for coin flipping. Logic problems, Understanding odds, Understanding probability Common Core Standards: Grade 4 Number & Operations: Fractions , Grade 5 Number & Operations in Base Ten , Grade 5 Operations & Algebraic Thinking. Similarly, the probability of the outcome Tail is 1/2, because the relative occurrence of Tails will be 1/2 for a large number of tosses. In unbiased coin flip H or T occurs 50% of times. As long as the coin was not manipulated the theoretical probabilities of both outcomes are the same—they are equally probable. " The total number of equally likely events is "2" because tails is just as likely as heads. Flip 10 coins, and and you're at a 4-digit number. Let Ti be the number of tosses of the ith coin until that coin results in Heads for the first time, for i=1,2,…,k. Nine flips of a fair coin. Write a program that simulates coin tossing. In the case of a coin, there are maximum two possible outcomes - head or tail. The probability of an event is a number indicating how likely that event will occur. Coin Flip Name Date Period 1. Now, create a Markov transition matrix, that will see a change from any state to the next higher state with probability 0. Inspiration • A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes • Let us discuss the binomial model we have studied so far through a very simple example • Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as Ω = {HHH,HHT,HTH,THH,HTT,THT,TTH,TTT} • Assume that the probability of a head. The probability of getting a given number of heads from four | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
Assume that the probability of a head. The probability of getting a given number of heads from four flips is, then, simply the number of ways that number of heads can occur, divided by the number of total results of four flips, 16. Some of the worksheets displayed are Lesson plan 19 flipping coins, Probability experiment, Fair coin work, Lesson topic probability grade level 6th grade length of, Mendelian genetics coin toss lab, Coin probability theoretical experimental probability, Lab 9 principles of genetic inheritance. org right now:. When we flip a coin there is always a probability to get a head or a tail is 50 percent. Probability. The theory revealed that the coin's behaviour is predictable - until it strikes the floor. I flip a coin and it comes up heads. 5 we get this probability by assuming that the coin is fair, or heads and tails are equally likely The probability for. Each team member will have 1 coin to flip. Most coins have probabilities that are nearly equal to 1/2. Flipping the coin once is a Bernoulli trial, since there are exactly two complementary outcomes (flipping a head and flipping a tail), and they are both 1 2 \frac{1}{2} 2 1 no matter how many times the coin is flipped. Hi everyone. Let the probability of obtaining a head be. We know that we will be doing a fair coin flip. a)Give an algebraic formula for the probability mass function of X. 5, which is our probability of tossing heads and moving forward. If I toss a coin only 10 times I may end up with 9 heads and 1 tails. Let’s start thinking about this by thinking about the coin flip. Demonstration of frequentist convergence of probability with a coin flip. Use buttons to view a bar chart of the coin flips, the probability distribution (also known as the probability mass. But not that much more likely. Daniel Egger. 5 is the probability of getting 2 Heads in 3 tosses. Not from a coin toss. the probability of tails is the same as heads, P(T) <=> P(H) 3. What is the probability that a fair | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
the probability of tails is the same as heads, P(T) <=> P(H) 3. What is the probability that a fair coin lands on heads on 4 out of 5 flips? Wha are the four properties of a binomial probability distribution? In a carnival game, there are six identical boxes, one of which contains a prize. For example, consider a fair coin. possible outcomes and finding each outcome that has two or more tails in it. Here's a simulation of the game. Coin Flipping, a selection of some of the answers to problems of this kind in the Dr. A cumulative probability refers to the probability that the value of a random variable falls within a specified range. Many events can't be predicted with total certainty. Consider flipping a coin that is either heads (H) or tails (T), each with probability 1/2. Print the results. In unbiased coin flip H or T occurs 50% of times. What is the probability of getting exactly two heads and two tails. When tossing only one coin at a time, the application keeps track of the number of heads and tails that occur as the coin is repeatedly tossed. Demonstrates frequency and probability distributions with weighted coin-flipping experiments. from the previous assumptions follows that given any sequence of coin tossing results, the next toss has the probability P(T) <=> P(H). A coin toss is a tried-and-true way for your fifth grader to understand odds. You can modify it as you like to simulate any number of flips. Like the title says, I need to figure out probability for a weighted coin flip. Coin Flipping, a selection of some of the answers to problems of this kind in the Dr. Heads did have an impressive run of 5 years in a row from 2009-2013. Theory of Probability. If the result of the coin toss is head, player A collects 1 coin from player B. We know that we will be doing a fair coin flip. Show Hide all comments. 7 flip_coins(100, 100, 5) #> [1] 0. Nine flips of a fair coin. You out the full article at the link below for probability charts and a fascinating look | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
fair coin. You out the full article at the link below for probability charts and a fascinating look into the mathematics of solving coin toss probability. If the probability of flipping heads is 70%, then the list contains 70 ones and 30 zeros. Coin toss probability is explored here with simulation. heads, flips(100) The following shows the results of using 50 tosses of the coin with a probability of obtaining heads of. The toss of a coin, throwing dice and lottery draws are all examples of random events. Coin Toss Probability Calculator. So the results of flipping a coin should be somewhere around 50% heads and 50% tails since that is the theoretical probability. The probability of Heads is the same for each coin and is the realized value q of a random variable Q that is uniformly distributed on [0,1]. In this video, we' ll explore the probability of getting at least one heads in multiple flips of a fair coin. Many events can't be predicted with total certainty. If you flip two coins, four. 4) 4 boys and 3 girls are standing in a line. 100 coins is a 31-digit number. Number of flips to do in each set: Probability of landing heads: Proportion of heads after each full set of flips, most recent. For 100 flips, if the actual heads probability is 0. A sequence of consecutive events is also called a "run" of events. Similarly, the probability of the outcome Tail is 1/2, because the relative occurrence of Tails will be 1/2 for a large number of tosses. Every time a coin is flipped, the probability of it landing. Two flips have 4 outcomes: HH, Ht, tH, and tt. Hello, A hat contains n coins, f of which are fair, and b of which are biased to land heads with probability of 2/3. How do I get rid of the number? It looks something like this when I run it The coin flipped Heads 1 The Coin flipped tails 2 The coin flipped Heads 1. If you'd like to read more about flipping coins and probability, check out my post on the topic at the Blog on Math Blogs. Since each coin toss has a | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
probability, check out my post on the topic at the Blog on Math Blogs. Since each coin toss has a probability of heads equal to 1/2, I simply need to multiply together 1/2 eleven times. Each team member will have 1 coin to flip. First, you will determine the theoretical probability of events. The post is correct that the odds of getting. Predicting a coin toss. Probability. Coin Toss Probability Calculator Coin toss also known as coin flipping probability is used by people around the world to judge whether its going to be head or tail after flipping the coin. Q: What is the probability for a coin to land on its edge when you flip a coin? A: The probability of a coin landing on its side or edge is a remote 6000 to 1. Below you will find a table that lists the coin flip results, including which team won and lost the coin flip for all Super Bowls. According to Science News Online the probability that a coin will land on the same side it started on is 51%. If 5 is selected for "coin flips per trial," then a number is selected from the list five times, and these numbers are summed. (Ti includes the toss that results in the first. e head or tail. Be careful with how you read this probability. Every flip of the coin has an “independent probability“, meaning that the probability that the coin will come up heads or tails is only affected by the toss of the coin itself. 2 of the outcomes are 1 head and 1 tail. If you flip two coins, four. If the probability of an event is high, it is more likely that the event will happen. So if an event is unlikely to occur, its probability is 0. Update: The initial 6-for-6 report, from the Des Moines Register missed a few Sanders coin-toss wins.$$ This probability model is called the Binomial distribution. For the old java version, click here ; For the Spanish version, click here ; For the German version, click here; To. 9 for coin B. Both team members flip their coins. 5, which means we would not be able to tell the different between a | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
team members flip their coins. 5, which means we would not be able to tell the different between a bias coin and fair coin 50% of the time. You must show your work to receive credit. Use Probability to Win Coin Flipping Games. A cumulative probability refers to the probability that the value of a random variable falls within a specified range. 4) 4 boys and 3 girls are standing in a line. Simulating a coin toss in excel I guess when you start to look at gambling theories or probabilities the natural place to start is the coin toss. When you take these chocolates out, the probability for any one being taken out diminishes by 1 each time. Demonstrates frequency and probability distributions with weighted coin-flipping experiments. Direct link to this answer. Notice that the width of the confidence interval narrows as the number of. Use Probability to Win Coin Flipping Games. Apply creativity, lateral thinking, and mathematical skill. Heads did have an impressive run of 5 years in a row from 2009-2013. So if you flip a coin 10 times in a row-- a fair coin-- you're probability of getting at least 1 heads in that 10 flips is pretty high. I think the best way to attack the problem is to run a simulation of millions of trials, and then give an approximate answer based on the number. "Pairs of adjacent coins" means only two coins next to each other may be flipped at a time. Course : Introduction to Probability and Statistics, Math 113 Section 3234 Instructor: Abhijit Champanerkar Date: Oct 17th 2012 Tossing a coin The probability of getting a Heads or a Tails on a coin toss is both 0. coin=randi ( [0:1], [100,1]) It should more or less give you 50 0's and 50 1's. Write a program that simulates coin tossing. (Ti includes the toss that results in the first. Coin Flips, Risk to Reward Profile and Creating Your Own Synthetic Security Thus, the probability of getting 2 heads in a row is the probability of getting a head followed by a second flip where you also get a head. | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
in a row is the probability of getting a head followed by a second flip where you also get a head. (Solution): Coin Toss Probability. Online virtual coin toss simulation app. Coin Toss: Simulation of a coin toss allowing the user to input the number of flips. 100 coins is a 31-digit number. 5, or you will stay in the current state with probability 0. I got the program down right but my results show a number for each coin flip in addition to the cout that says "The coin flip shows Heads/Tails". The probability of getting a given number of heads from four flips is, then, simply the number of ways that number of heads can occur, divided by the number of total results of four flips, 16. The answer to this is always going to be 50/50, or ½, or 50%. This is what I have so far but I keep getting errors. In the case of a coin, there are maximum two possible outcomes - head or tail. When the probability of an event is zero then the even is said to be impossible. Luckily, this is bundled up in a math/probability/stats concept called "combinations". Remark: The idea can be substantially generalized. This means that the theoretical probability to get either heads or tails is 0. Ask Question Asked 3 years, 6 months ago. What is the probability that a fair coin lands on heads on 4 out of 5 flips? What is the probability of getting at least one tail if a fair coin is flipped three times? Wha are the four properties of a binomial probability distribution?. Repeat steps 2-4 until the coin lands on its side every time. Junho: The chance of DB completing the coin scam on the first attempt, which is to toss a coin and get 10 heads in a row, is very unlikely. I have a problem I need to do for school. 6, and f = 0. Remember that each individual coin flip has a 50% chance of being heads. If we do the math, this is a probability of 0. Not so, says Diaconis. 2, which is the Kelly fraction for this α and p. 5, which is our probability of tossing heads and moving forward. Viewed 2k times 1. | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
this α and p. 5, which is our probability of tossing heads and moving forward. Viewed 2k times 1. The two non-straight lines in Fig. The odds of the coins coming up with different faces showing is just 1 in 2. , Bernoulli trials). If you toss 2 coins, what are the chances you will get 2 heads? Record your predictions and explain your reasoning. The program should call a separate function flip()that takes no arguments and returns 0 for tails and 1 for heads. If you flip a coin and roll a six-sided die, what is the probability that the coin comes up heads and the die comes up 1? Since the two events are independent, the probability is simply the probability of a head (which is 1/2) times the probability of the die coming up 1 (which is 1/6). 57 flip_coins(100, 1000, 25) #> [1] 0. Flipping coins and the binomial distribution#2 Consider two coins, one fair and one unfair. We can use R to simulate an experiment of ipping a coin a number of times and compare our results with the theoretical probability. That was flip number 130,659,178 Flip again? Color The Coin!. If the probability of flipping heads is 70%, then the list contains 70 ones and 30 zeros. If you flip one coin, just two. of flipping one head with a coin is 50%, then the probability of flipping two heads at once is achieved by (adding or multiplying)_____ the separate probabilities. (Solution): Coin Toss Probability. What is the theoretical probability that the co n will land on tails? What is the theoretical probability that the co n will land on heads? If the com is flipped 140 times, how many times would you predict that the co n lands on heads? Johnny flipped a coin 450 times. If you flip three coins, it's eight - two for the first times two for the second times two for the third. The game is played in stages. The order does not matter as long as there are two head and two tails in the flip. e head or tail. Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to choose between two alternatives, sometimes used to resolve a dispute between two parties. Since 'fair' is used in the project description we know that the probability will be a 50% chance of getting either side. Flipping more coins¶ If we want to flip more coins, it's going to be a pain in the neck to make that table over and over. Then, it displays the results, as well as the theoretical and observed probabilities of each event happening. Given this information, what is the probability that it is a. So, I'll do it faster! When we flip the coin 9 times there are $$2^9$$ possible outcomes that can happen. A probability of one represents certainty: if you flip a coin, the probability you'll get heads or tails is one (assuming it can't land on the rim, fall into a black hole, or some such). Exactly 2 heads in 3 Coin Flips The ratio of successful events A = 3 to total number of possible combinations of sample space S = 8 is the probability of 2 heads in 3 coin tosses. If you toss a coin, you cannot get both a head and a tail at the same time, so this has zero probability. Gamblers Take Note: The Odds in a Coin Flip Aren't Quite 50/50 And the odds of spinning a penny are even more skewed in one direction, but which way? Flipping a coin isn't as fair as it seems. Simulating a coin toss in excel I guess when you start to look at gambling theories or probabilities the natural place to start is the coin toss. When we flip a coin there is always a probability to get a head or a tail is 50 percent. 4, then the power is 0. The different possible results from a probability model. Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to choose between two alternatives, sometimes used to resolve a dispute between two parties. There | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
to choose between two alternatives, sometimes used to resolve a dispute between two parties. There is a 50% probability that the first toss will end up heads. Have you ever flipped a coin as a way of deciding something with another person? The answer is probably yes. probability of any continuous interval is given by p(a ≤ X ≤ b) = ∫f(x) dx =Area under f(X) from a to b b a That is, the probability of an interval is the same as the area cut off by that interval under the curve for the probability densities, when the random variable is continuous and the total area is equal to 1. Probability, physics, and the coin toss L. Don't expect the numbers from trials to exactly match the predicted results--especially if you run only a few trials. A fair-sided coin (which means no casino hanky-panky with the coin not coming up heads or tails 50% of the time) is tossed three times. For example, consider a fair coin. Page last modified 07/17/2012 13:01:23. Logic problems, Understanding odds, Understanding probability Common Core Standards: Grade 4 Number & Operations: Fractions , Grade 5 Number & Operations in Base Ten , Grade 5 Operations & Algebraic Thinking. Note that the pattern of heads counts seems to form a smoother curve, but still matches the (scaled) binomial coefficients found on the tenth and twentieth rows of Pascal's Triangle. Assuming the coin is fair, p = 1/2 and q = 1/2 where 'p' is the probability of get. Thus, the odds of any throw being a tail is 1/2. 2, which is the Kelly fraction for this α and p. Heads did have an impressive run of 5 years in a row from 2009-2013. The probability of an event is a number indicating how likely that event will occur. The probability for equally likely outcomes in an event is:. The likelihood of an event is expressed as a number between zero (the event will never occur) and one (the event is certain). Use, probability formula = N u m b e r o f f a v o r a b l e o u t c o m e s T o t a l n u m b e r o f p o s s i. Update: The | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
= N u m b e r o f f a v o r a b l e o u t c o m e s T o t a l n u m b e r o f p o s s i. Update: The initial 6-for-6 report, from the Des Moines Register missed a few Sanders coin-toss wins. The views expressed are those of the author(s) and are not necessarily. Each team member will have 1 coin to flip. Take another penny and Super Glue it to the coin. Intro to Problem Solving. Although the basic probability formula isn't difficult, sometimes finding the numbers to plug into it can be tricky. Course : Introduction to Probability and Statistics, Math 113 Section 3234 Instructor: Abhijit Champanerkar Date: Oct 17th 2012 Tossing a coin The probability of getting a Heads or a Tails on a coin toss is both 0. Click "flip coins" to generate a new set of coin flips. Explore probability concepts by simulating repeated coin tosses. The experiment was conducted with motion-capture cameras, random experimentation, and an automated "coin-flipper" that could flip the coin on command. Let the program toss the coin 100 times, and count the number of times each side of the coin appears. Page last modified 07/17/2012 13:01:23. Begin your coin with just a single penny. something like this: def flip(p): '''this function return H with probability p''' # do something return result >> [flip(0. Coin toss probability is a classic for a reason: it's a realistic example kids can grasp quickly. Make a weighted coin by changing the probability of landing on heads using the slider; 0% means the coin always lands on tails and 100% means the coin always lands on heads. The coin toss is nothing but experimenting with tossing a coin. Since each coin toss has a probability of heads equal to 1/2, I simply need to multiply together 1/2 eleven times. Use Probability to Win Coin Flipping Games. In the first simulation, player A got lucky with 4 heads in 5 tosses. So, the probability that we will keep going is 1/2 of 1/4, or 1/8. Example: It is 12 if you have an experiment where you flip a coin and then | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
is 1/2 of 1/4, or 1/8. Example: It is 12 if you have an experiment where you flip a coin and then roll a six sided die. An Easy GRE Probability Question. When you take these chocolates out, the probability for any one being taken out diminishes by 1 each time. A common topic in introductory probability is solving problems involving coin flips. What is the probability of getting two heads and four tails? Coin Flipping How can I figure out the chances of flipping a coin five times with the result T,T,T,H,H?. I am just learning Python on class so I am really at the basic. For the old java version, click here ; For the Spanish version, click here ; For the German version, click here; To. If we flip the coin 10 times, we are not guaranteed to get 5 heads and 5 tails. e head or tail. The first time it lands heads, and the second time it lands tails. What is the theoretical probability that the co n will land on tails? What is the theoretical probability that the co n will land on heads? If the com is flipped 140 times, how many times would you predict that the co n lands on heads? Johnny flipped a coin 450 times. Mentor: Yes! Now let's look at the coin flipping game that you just played. In this worksheet, they'll grab a quarter, give it a few tosses, and record the results for themselves. flips The number of desired coin flips. Coin toss probability is explored here with simulation. "The coin tosses are independent events; the coin doesn't have a memory. For the coin, number of outcomes to get heads = 1 Total number of possible outcomes = 2 Thus, we get 1/2 However, if you suspect that the coin may not be fair, you can toss the coin a large number of times and count the number of heads Suppose you flip the coin 100 and get 60 heads, then you know the best estimate to get head is 60/100 = 0. What is the probability that Player 1 will win the game?. Unformatted text preview: Tamara Curiel Gala Cano Mendelian Genetics Coin Toss Lab PRE-LAB DISCUSSION: In heredity, we are | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
Tamara Curiel Gala Cano Mendelian Genetics Coin Toss Lab PRE-LAB DISCUSSION: In heredity, we are concerned with the occurrence, every time an egg is fertilized, of the probability that a particular gene or chromosome will be passed on through the egg, or through the sperm, to the offspring. Sign in to comment. What is the probability of getting exactly two heads and two tails. Coin Toss Probability Probability is the measurement of chances – likelihood that an event will occur. The toss or flip of a coin to randomly assign a decision traditionally involves throwing a coin into the air and seeing which side lands facing up. If the probability of coin flipping head = P The probability of coin flipping tail = 1 - P Now The probability of flipping heads & then tails = Probability of flipping tails & then heads = P(1 - P) Which means to make a fair coin toss we now need 2 flips Player 1 wins if the sequence is HT Player 2 wins if the sequence is TH Any other sequence. In 1921, the referee flipped the coin. We have k coins. 5, or you will stay in the current state with probability 0. I start by having my students create a "Heads Tails" T chart in their math journals and then writing a prediction for the result of tossing the coin 100 times. If after 200 flips, the. There's another sense in which the Haldane prior can be considered non-informative: the mean of the posterior distribution is now $\frac{\alpha + x}{\alpha + \beta + n}=\frac{x}{n}$, i. This means there is a 1 out of 128 chance of getting seven heads on seven coin flips. Over many coin flips the probability of at least half of. If we flip a fair coin 9 times, and the flips are independent, what's the probability that we get heads exactly 6 times? This works just like the last problem, only the numbers are bigger. If the coins show heads-tails (HT) or tails-heads (TH), player 2 gets 1 point. 8) for i in xrange(10)] [H,H,T,H,H,H,T,H,H,H]. We can either find this out using a formula or through Monte Carlo | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
[H,H,T,H,H,H,T,H,H,H]. We can either find this out using a formula or through Monte Carlo simulation. 5 for either heads or tails (assuming that the coin is purely mathematical and random, of course). Since the coin toss is a physical phenomenon governed by Newtonian mechanics, the question requires one to link probability and physics via a mathematical and statistical description of the coin's. Exactly 2 heads in 3 Coin Flips The ratio of successful events A = 3 to total number of possible combinations of sample space S = 8 is the probability of 2 heads in 3 coin tosses. If we toss a coin n times, and the probability of a head on any toss is p (which need not be equal to 1 / 2, the coin could be unfair), then the probability of exactly k. You can modify it as you like to simulate any number of flips. 5 (the default) with the 95% confidence interval. Tossing a Coin. The following shows the results of 100 tosses of five coins with a probability of heads of. Let be the probability that a run of or more consecutive heads appears in independent tosses of a coin (i. "The coin tosses are independent events; the coin doesn't have a memory. But not that much more likely. The variable timesflipped used for the while. In unbiased coin flip H or T occurs 50% of times. First, note that the problem will likely make reference to a "fair" coin. b) What do you think E[X] should be. Use buttons to view a bar chart of the coin flips, the probability distribution (also known as the probability mass. Since 'fair' is used in the project description we know that the probability will be a 50% chance of getting either side. If I toss a coin only 10 times I may end up with 9 heads and 1 tails. We assume that conditioned on Q=q, all coin tosses are independent. You flipped 1 coin of type US 1¢ Penny: Timestamp: 2020-05-05 01:39:10 UTC. the probability of tails is the same as heads, P(T) <=> P(H) 3. Each time you toss these coins, there are four possible outcomes: both heads penny head & dime | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
3. Each time you toss these coins, there are four possible outcomes: both heads penny head & dime tail penny tail & dime head both tails You will flip the pair of coins 20 times. Both team members flip their coins. Flip 10 coins, and and you're at a 4-digit number. If 5 is selected for "coin flips per trial," then a number is selected from the list five times, and these numbers are summed. Have you ever flipped a coin as a way of deciding something with another person? The answer is probably yes. It is about physics, the coin, and how the "tosser" is actually throwing it. There is a 50% probability that the first toss will end up heads. The toss or flip of a coin to randomly assign a decision traditionally involves throwing a coin into the air and seeing which side lands facing up. Life is full of random events! You need to get a "feel" for them to be a smart and successful person. However, the probability of getting exactly one heads out of seven flips is different (and the solution is given). This is a basic introduction to a probability distribution table. An Easy GRE Probability Question. But not that much more likely. Furthermore, if you toss a coin, it will eventually show heads, so this procedure ends in a finite number of flips with probability 1. There is also the very small probability that the coin will land. In Chapter 2 you learned that the number of possible outcomes of several independent events is the product of the number of possible outcomes of each event individually. Probability. Conditional probability question - coin toss. A fair-sided coin (which means no casino hanky-panky with the coin not coming up heads or tails 50% of the time) is tossed three times. But if any of the flipped coins comes up tails, or if no one chooses to flip a coin, you will all be doomed to spend the rest of your lives in the castle’s dungeon. the probability of tails is the same as heads, P(T) <=> P(H) 3. The coin can only land on one side or the other (event) but | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
is the same as heads, P(T) <=> P(H) 3. The coin can only land on one side or the other (event) but there are two possible outcomes: heads or tails. 5, which is our probability of tossing heads and moving forward. a)Give an algebraic formula for the probability mass function of X. Logic problems, Understanding odds, Understanding probability Common Core Standards: Grade 4 Number & Operations: Fractions , Grade 5 Number & Operations in Base Ten , Grade 5 Operations & Algebraic Thinking. Coin Flips, Risk to Reward Profile and Creating Your Own Synthetic Security Thus, the probability of getting 2 heads in a row is the probability of getting a head followed by a second flip where you also get a head. Applet: Instructions: Examples: Notes "H" count = , flips so far, number of coins: one flip "H" probability: 0. A probability of one means that the event is certain. Probability measures how certain we are a particular event will happen in a specific instance. That was flip number 130,659,178 Flip again? Color The Coin!. Probability: Independent Events. 5, or you will stay in the current state with probability 0. The toss of a coin, throwing dice and lottery draws are all examples of random events. The instructions are written in the handout for the students to understand how to complete the activi. of flipping one head with a coin is 50%, then the probability of flipping two heads at once is achieved by (adding or multiplying)_____ the separate probabilities. When the coin is flipped and the first three flips are heads, the fourth flip still has the probability of ½ However, many people misunderstand that the first three flips somehow influence the fourth flip, but they do not. Don't expect the numbers from trials to exactly match the predicted results--especially if you run only a few trials. Were you to toss the coin 100 times, you would get a clearer view of how probable it is that the coin lands on heads each time. A coin toss is a tried-and-true way for your fifth | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
it is that the coin lands on heads each time. A coin toss is a tried-and-true way for your fifth grader to understand odds. We do not know if we will get heads or tails. I got a question on the coin flip project. If we flip the coin 10 times, we are not guaranteed to get 5 heads and 5 tails. Repeat steps 2-4 until the coin lands on its side every time. Expected Value represents the average outcome of a series of random events with identical odds being repeated over a long period of time. Asked in Statistics , Probability. It begins with the two title characters caught in a most unusual coin game. When flipping a coin, the probability of getting a head does not change no matter how many times you flip the coin. Example: It is 12 if you have an experiment where you flip a coin and then roll a six sided die. If the coins show heads-tails (HT) or tails-heads (TH), player 2 gets 1 point. With three coins, all three landing on the same side is 1 in 8. Ask Question Asked 3 years, 6 months ago. The probability for equally likely outcomes in an event is:. In this video, we' ll explore the probability of getting at least one heads in multiple flips of a fair coin. I need to write a python program that will flip a coin 100 times and then tell how many times tails and heads were flipped. Explore probability concepts by simulating repeated coin tosses. If the probability of coin flipping head = P The probability of coin flipping tail = 1 - P Now The probability of flipping heads & then tails = Probability of flipping tails & then heads = P(1 - P) Which means to make a fair coin toss we now need 2 flips Player 1 wins if the sequence is HT Player 2 wins if the sequence is TH Any other sequence. Students will flip a coin a total of ten times per trial and record results by simply shading in the space below the realistic-looking coins. Flip 4 coins, and you're at 16 outcomes, a 2-digit number. The number of possible outcomes gets greater with the increased number of coins. Coin | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
number. The number of possible outcomes gets greater with the increased number of coins. Coin Flipper. For example, given 5 trials per experiment and 20 experiments, the program will flip a coin 5 times and record the results 20 times. Become a member and unlock all. The probability of A and B is 1/100. That means I flipped the coin 15 times. It's pretty much a "coin flip". The views expressed are those of the author(s) and are not necessarily. (Solution): Coin Toss Probability. Suppose a coin tossed then we get two possible outcomes either a ‘head’ ( H) or a ‘tail’ ( T ), and it is impossible to predict whether the result of a toss will be a ‘head’ or ‘tail’. However, the probability of getting exactly one heads out of seven flips is different (and the solution is given). But I want to simulate coin which gives H with probability 'p' and T with probability '(1-p)'. The first person to flip heads wins. Flipping coins and the binomial distribution#2 Consider two coins, one fair and one unfair. from the previous assumptions follows that given any sequence of coin tossing results, the next toss has the probability P(T) <=> P(H). Recommended: Please try your approach. Logical Reasoning. Procedure 1: Statistical Probability Reasoning. 5×10 20 chance of getting a string of 76 heads. As long as the coin was not manipulated the theoretical probabilities of both outcomes are the same—they are equally probable. Let us return to the coin flip experiment. If the probability of an event is high, it is more likely that the event will happen. And you can get a calculator out to figure that out in terms of a percentage. This article shows you the steps for solving the most common types of basic questions on this subject. That means I flipped the coin 15 times. I start by having my students create a "Heads Tails" T chart in their math journals and then writing a prediction for the result of tossing the coin 100 times. The coin toss is nothing but experimenting with tossing a coin. , | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
of tossing the coin 100 times. The coin toss is nothing but experimenting with tossing a coin. , the sample frequency of heads, which is the frequentist MLE estimate of $\theta$ for the Binomial model of the coin flip problem. Unformatted text preview: Tamara Curiel Gala Cano Mendelian Genetics Coin Toss Lab PRE-LAB DISCUSSION: In heredity, we are concerned with the occurrence, every time an egg is fertilized, of the probability that a particular gene or chromosome will be passed on through the egg, or through the sperm, to the offspring. Lends to discussion and discovery of probability, elementary understanding of mode, and graphic organization of information to compare and contrast. Begin your coin with just a single penny. If you flip three coins, it's eight - two for the first times two for the second times two for the third. A sequence of consecutive events is also called a "run" of events. You flipped 1 coin of type US 1¢ Penny: Timestamp: 2020-05-05 01:39:10 UTC. It is measured between 0 and 1, inclusive. The probability of coming up heads on the first flip is 1/2. It is about physics, the coin, and how the "tosser" is actually throwing it. the coin does not and can not "remember" last result 4. For example, consider a fair coin. In the first simulation, player A got lucky with 4 heads in 5 tosses. Remark: The idea can be substantially generalized. The probability of getting a given number of heads from four flips is, then, simply the number of ways that number of heads can occur, divided by the number of total results of four flips, 16. If we toss a coin $n$ times, and the probability of a head on any toss is $p$ (which need not be equal to $1/2$, the coin could be unfair), then the probability of exactly $k$ heads is \binom{n}{k}p^k(1-p)^{n-k}. Coin Toss: Simulation of a coin toss allowing the user to input the number of flips. Simple numbers. Ask Question Asked 3 years, 6 months ago. The sum of all possible outcomes is always 1 (or 100%) because it is | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
Asked 3 years, 6 months ago. The sum of all possible outcomes is always 1 (or 100%) because it is certain that one of the possible outcomes will happen. Based on your flip results, you will infer which of the coins you were given. Recommended: Please try your approach. 5, or you will stay in the current state with probability 0. The game is played in stages. The first person to flip heads wins. "The coin tosses are independent events; the coin doesn't have a memory. If the probability of flipping heads is 70%, then the list contains 70 ones and 30 zeros. You flipped 1 coin of type US 1¢ Penny: Timestamp: 2020-05-05 01:39:10 UTC. The majority of times, if a coin is heads-up when it is flipped, it will remain heads-up when it lands. First, note that the problem will likely make reference to a "fair" coin. In particular, ( 10 3) = 10! 3! 7!. Wait for your coin to dry; Flip (toss) your coin 40 times and record the number of times it lands on its side. The post is correct that the odds of getting. Your function will have to label each of those sequences with a door, since leaving any sequence unlabeled would contradict the assumption that the method always produces a result after F flips. I need to land on heads 3 times or more out of 6, in 80% of all trials. So, I'll do it faster! When we flip the coin 9 times there are $$2^9$$ possible outcomes that can happen. The coin has no desire to continue a particular streak, so it's not affected by any number of previous coin tosses. Coin Toss Activity is a great way for students to have fun and learn about calculating probability. Probability measures how certain we are a particular event will happen in a specific instance. The two non-straight lines in Fig. A single flip of a coin has an uncertain outcome. We could call a Head a success; and a Tail, a failure. In the first simulation, player A got lucky with 4 heads in 5 tosses. For four coin flips, our intuition was probably right: more likely to get two heads. 60 I tried | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
For four coin flips, our intuition was probably right: more likely to get two heads. 60 I tried this: P(2H) = 4C2 * 0. 7 flip_coins(100, 100, 5) #> [1] 0. Ask Question Asked 3 years, 6 months ago. The program should call a separate function flip()that takes no arguments and returns 0 for tails and 1 for heads. We all know a coin toss gives you a 50% chance of winning, but is it always that way? Delve into the inner-workings of coin toss probability with this activity. The sum of all possible outcomes is always 1 (or 100%) because it is certain that one of the possible outcomes will happen. Flipping the coin once is a Bernoulli trial, since there are exactly two complementary outcomes (flipping a head and flipping a tail), and they are both 1 2 \frac{1}{2} 2 1 no matter how many times the coin is flipped. What is the probability that a fair coin lands on heads on 4 out of 5 flips? Wha are the four properties of a binomial probability distribution? In a carnival game, there are six identical boxes, one of which contains a prize. We’ll set p=0. 5, which is our probability of tossing heads and moving forward. | {
"domain": "theoximoron.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.856095700727982,
"lm_q2_score": 0.8705972717658209,
"openwebmath_perplexity": 368.0354943188166,
"openwebmath_score": 0.7363460659980774,
"tags": null,
"url": "http://ctjn.theoximoron.it/coin-flip-probability.html"
} |
# Math Help - Need Help with Elem. Linear Algebra Problem
1. ## Need Help with Elem. Linear Algebra Problem
Express the invertible matrix
$\left(\begin{array}{ccc}1&2&1\\1&0&1\\1&1&2\end{ar ray}\right)$
as a product of elementary matrices.
--------------------------
The answer in the back of the book has a product of 6 matrices. I think I'm supposed to use the identity matrix and perform the same row operations on it as I would to obtain the above matrix... or something like that. Any help with this would be appreciated!
2. Originally Posted by paupsers
Express the invertible matrix
$\left(\begin{array}{ccc}1&2&1\\1&0&1\\1&1&2\end{ar ray}\right)$
as a product of elementary matrices.
--------------------------
The answer in the back of the book has a product of 6 matrices. I think I'm supposed to use the identity matrix and perform the same row operations on it as I would to obtain the above matrix... or something like that. Any help with this would be appreciated!
Yes, that's right. A "row operation" is one of three kinds:
1) swap two rows
2) multiply every member of a row by the same constant.
3) replace every member of a row by itself plus a constant time the corresponding member of a second row.
An "elementary matrix" is a matrix constructed from the identity matrix by a single row operation.
What you want to do is reduce the given matrix to the identity matrix (which is possible because it is invertible) by row operations, writing down the elementary matrix corresponding to each row operation. | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429638959675,
"lm_q1q2_score": 0.8560956883718065,
"lm_q2_score": 0.8705972583359806,
"openwebmath_perplexity": 178.66534325732104,
"openwebmath_score": 0.8484682440757751,
"tags": null,
"url": "http://mathhelpforum.com/advanced-algebra/95329-need-help-elem-linear-algebra-problem.html"
} |
For example, starting from
$\left(\begin{array}{ccc}1&2&1\\1&0&1\\1&1&2\end{ar ray}\right)$
My first steps would be to reduce the first column to the column $\left(\begin{array}{c}1 \\ 0 \\ 0\end{array}\right)$.
The number in the upper left is already 1 so I don't need to change that. I can get 0 in the next row by subtracting the first row from the second row. The elementary matrix corresponding to that is
$\left(\begin{array}{ccc}1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$
To get 0 in the third row, I need to subtract the first row from the third row. That gives the elementary matrix $\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1\end{array}\right)$
and they convert the matrix to
$\left(\begin{array}{ccc}1 & 2 & 1 \\ 0 & -2 & 0 \\ 0 & -1 & 1\end{array}\right)$
Now you need to convert the second column properly and then the third. Can you continue?
After you have converted the matrix to the identity matrix and written down all the corresponding elementary matrices (in the proper order) so they multiply to give the original matrix.
Normally it would take 9 row operations to reduce a 3 by 3 matrix to the identity matrix and so the matrix would be represented as the product of 9 elementary matrices. Notice that the upper left number was already 1 so we didn't need a row operation for that. Also that 0 now in the second row, third column will stay there we won't need a row operation to change that. I suspect that sort of thing, a 0 or 1 already in the correct position will happen once more to give 6 rather than 9 elementary matrices.
3. This is exactly what I tried doing, but it's not the answer the book gives (although it is similar). Is the solution to this problem unique?
4. No, it is not because you can follow different "paths" to row reduce to the identity matrix. What I showed is the most commonly used "path". | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429638959675,
"lm_q1q2_score": 0.8560956883718065,
"lm_q2_score": 0.8705972583359806,
"openwebmath_perplexity": 178.66534325732104,
"openwebmath_score": 0.8484682440757751,
"tags": null,
"url": "http://mathhelpforum.com/advanced-algebra/95329-need-help-elem-linear-algebra-problem.html"
} |
# Transitive relation
Consider $A$ is a relation de fined on $R$ (real numbers) where $A = \{(a,b):|a-b|<4, a, b \in R\}$. Prove/disprove $A$ is transitive.
I know if $|a-b|<4$ and $|b-c|<4$, then, $|a-c|<4$ ; A is transitive. Can I directly prove it with any counter example such as for $a=6$, $b=3$ and $c=1$ this relation is not transitive because $|6-1|>4$.
Is this suitable for prove or disprove questions of relations?
Your answer is correct. In general a relation $A$ is transitive if for all $a,b,c$ we have $(a,b)\in A$ and $(b,c)\in A$ implies $(a,c)\in A$. That means that if we can find just one counter example (such as $a=6$, $b=3$ and $c=1$) $A$ cannot be transitive.
Your answer is quite right. The question is basically "if $a$ is near to $b$, and $b$ is near to $c$, is $a$ near to $c$?" | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9873750481179169,
"lm_q1q2_score": 0.856089478447,
"lm_q2_score": 0.867035763237924,
"openwebmath_perplexity": 162.6942033835524,
"openwebmath_score": 0.921374499797821,
"tags": null,
"url": "https://math.stackexchange.com/questions/1583146/transitive-relation"
} |
# Expansion in terms of legendre polynomial
Obtain the first three terms in the expansion of function in terms of legendre polynomial F(x) in a series of the form $$F(x) = \sum_{k=0}^{\infty} A_k P_k(x)$$ where $$F(x)=\{\cos(x) \text{ for } 0 \le x \le \pi/2 \$$$$0 \text{ for } \frac{\pi}{2} \le x \le \pi.\}$$
What I know is I have to use legendre's expansion formula i.e,$F(x) =\sum A_kP_k(x)$ where $-1≤x≤1$ But obviously I cannot use it directly because the range of $x$ differs. I have tried substituting $x=\cos(\theta)$ but no success so far.
Extended hint: As already noted in your almost identical question Legendre polynomials you should apply a linear transformation, because the Legendre polynomial have the orthogonality interval $[-1,1]$. With $x=\frac{\pi}{2}(t+1)$ and $t=\frac{2x}{\pi}-1,$ define $F(x)=:\tilde{F}(t).$ Now $\tilde{F}$ is defined on $[-1,1]$ and you can compute $$F(x)=\tilde{F}(t) = \sum A_k P_k(t)=\sum A_k P_k\left(\frac{2x}{\pi}-1\right)$$ with $$A_k=\frac{2k+1}{2}\int_{-1}^1 \tilde{F}(t) P_k(t) dt$$
If you use this approach and compute $A_0, \dots, A_3$, you get the following graphics with the original function $F(x)$ in red and the approximation in green:
• Yes of course, because your function is not defined for $x=-1$. The transformation associates $x=0 \leftrightarrow t=-1$ and $x=\pi \leftrightarrow t=1$. You should compute the expansion for $\tilde{F}(t), t\in[-1,1].$ – gammatester Sep 18 '17 at 15:08 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9873750492324249,
"lm_q1q2_score": 0.8560894709323755,
"lm_q2_score": 0.8670357546485407,
"openwebmath_perplexity": 103.11606074523804,
"openwebmath_score": 0.9301050305366516,
"tags": null,
"url": "https://math.stackexchange.com/questions/2434494/expansion-in-terms-of-legendre-polynomial"
} |
# Homework Help: Product of two consecutive integers
1. Jan 4, 2010
### nobahar
1. The problem statement, all variables and given/known data
Prove that $$n^2+n$$ is even. Where n is a positive integer.
2. Relevant equations
$$n^2+n$$
3. The attempt at a solution
$$n^2+n = n(n+1)$$
One of which must be even, and therefore the product of 2 and an integer k.
$$n = 2k, \left \left 2*(k(n+1))$$
or
$$n+1 = 2k, \left \left 2*(n*k)$$
Is there a better way of doing this? I read this is not an inductive proof; what would this entail?
2. Jan 4, 2010
### rochfor1
This looks absolutely fine. No need for induction at all.
3. Jan 4, 2010
### HallsofIvy
As rochfor1 said, that is a perfectly good proof and simpler than "proof by induction".
But since you ask, here goes:
If n= 1, then $n^2+ n= 1^2+ 1= 2$ which is even.
Now, suppose that $k^2+ k$ is even and look at $(k+1)^2+ (k+1)$
$(k+1)^2+ (k+1)= k^2+ 2k+ 1+ k+ 1$$= (k^2+ 2k)+ 2k+ 2$. By the induction hypothesis, $k^2+ k$ is even and so $k^2+ k= 2m$ for some integer m (that is the definition of "even") so $(k+1)^2+ (k+1)= (k^2+ 2k)+ 2k+ 2$$= 2m+ 2k+ 2= 2(m+k+1)$. Since that is "2 times an integer", it is even.
Having proved that the statement is true for 1 and that "if it is true for k, it is true for k+1", by induction, it is true for all positive integers.
Yet a third way: If n is a positive integer, it is either even or odd.
case 1: n is even. Then n= 2m for some integer m. $n^2= 4m^2$ so $n^2+ n= 4m^2+ 2m= 2(m^2+ m)$. Since that is 2 times an integer, it is even.
case 2: n is odd. Then n= 2m+ 1 for some integer m. $n^2= (2m+1)^2= 4m^2+ 4m+ 1$ so $n^2+ n= 4m^2+ 4m+ 1+ 2m+ 1$$= 4m^2+ 6m+ 2= 2(2m^2+ 3m+ 1)$. Again that is 2 times an integer and so is even.
4. Jan 5, 2010
### nobahar
Many thanks rochfor1 and HallsofIvy!
5. Jan 5, 2010
### icystrike
Simple reasoning is needed.
For two consecutive integer , one will be even and another will be odd.
Thus;
$$n\equiv0(mod 2)$$
and
$$n+1\equiv1(mod 2)$$ | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9873750529474512,
"lm_q1q2_score": 0.8560894690648703,
"lm_q2_score": 0.8670357494949105,
"openwebmath_perplexity": 1312.1021854840985,
"openwebmath_score": 0.6292749047279358,
"tags": null,
"url": "https://www.physicsforums.com/threads/product-of-two-consecutive-integers.367041/"
} |
We may now conclude that $$n(n+1)\equiv0(mod 2)$$ and means that n²+n is even.
6. Jan 5, 2010
File size:
24.9 KB
Views:
388
7. Jan 5, 2010
### Dick
If n is odd, n^2 is odd. If n is even, n^2 is even. odd+odd=even and even+even=even.
8. Jan 6, 2010
### ideasrule
OP: Why do you want a proof by induction? The proof you gave is simple and beautiful.
9. Jan 6, 2010
### Staff: Mentor
What you say is true, but you can't ascertain which of them will be even and which will be odd. In what follows, you are assuming that n is even and n+1 is odd. That is one of two possible cases, so you work is not complete.
10. Jan 7, 2010
### icystrike
yes that is true. for the other case , it can be proved analogously . | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9873750529474512,
"lm_q1q2_score": 0.8560894690648703,
"lm_q2_score": 0.8670357494949105,
"openwebmath_perplexity": 1312.1021854840985,
"openwebmath_score": 0.6292749047279358,
"tags": null,
"url": "https://www.physicsforums.com/threads/product-of-two-consecutive-integers.367041/"
} |
Find all School-related info fast with the new School-Specific MBA Forum
It is currently 07 Jul 2015, 07:46
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# Inequalities trick
Author Message
TAGS:
Current Student
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2800
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 196
Kudos [?]: 1162 [56] , given: 235
Inequalities trick [#permalink] 16 Mar 2010, 09:11
56
KUDOS
117
This post was
BOOKMARKED
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.
Suppose you have the inequality
f(x) = (x-a)(x-b)(x-c)(x-d) < 0
Just arrange them in order as shown in the picture and draw curve starting from + from right.
now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.
Don't forget to arrange then in ascending order from left to right. a<b<c<d
So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x) | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
If f(x) has three factors then the graph will have - + - +
If f(x) has four factors then the graph will have + - + - +
If you can not figure out how and why, just remember it.
Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.
For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.
Attachments
1.jpg [ 6.73 KiB | Viewed 29844 times ]
_________________
Fight for your dreams :For all those who fear from Verbal- lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
GMAT Club Premium Membership - big benefits and savings
Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 5682
Location: Pune, India
Followers: 1415
Kudos [?]: 7357 [43] , given: 186
Re: Inequalities trick [#permalink] 22 Oct 2010, 05:33
43
KUDOS
Expert's post
39
This post was
BOOKMARKED
Yes, this is a neat little way to work with inequalities where factors are multiplied or divided. And, it has a solid reasoning behind it which I will just explain.
If (x-a)(x-b)(x-c)(x-d) < 0, we can draw the points a, b, c and d on the number line.
e.g. Given (x+2)(x-1)(x-7)(x-4) < 0, draw the points -2, 1, 7 and 4 on the number line as shown.
Attachment:
doc.jpg [ 7.9 KiB | Viewed 29149 times ]
This divides the number line into 5 regions. Values of x in right most region will always give you positive value of the expression. The reason for this is that if x > 7, all factors above will be positive.
When you jump to the next region between x = 4 and x = 7, value of x here give you negative value for the entire expression because now, (x - 7) will be negative since x < 7 in this region. All other factors are still positive. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
When you jump to the next region on the left between x = 1 and x = 4, expression will be positive again because now two factors (x - 7) and (x - 4) are negative, but negative x negative is positive... and so on till you reach the leftmost section.
Since we are looking for values of x where the expression is < 0, here the solution will be -2 < x < 1 or 4< x < 7
It should be obvious that it will also work in cases where factors are divided.
e.g. (x - a)(x - b)/(x - c)(x - d) < 0
(x + 2)(x - 1)/(x -4)(x - 7) < 0 will have exactly the same solution as above.
Note: If, rather than < or > sign, you have <= or >=, in division, the solution will differ slightly. I will leave it for you to figure out why and how. Feel free to get back to me if you want to confirm your conclusion.
_________________
Karishma
Veritas Prep | GMAT Instructor
My Blog | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Math Forum Moderator Joined: 20 Dec 2010 Posts: 2028 Followers: 138 Kudos [?]: 1165 [15] , given: 376 Re: Inequalities trick [#permalink] 11 Mar 2011, 05:49 15 This post received KUDOS 6 This post was BOOKMARKED vjsharma25 wrote: VeritasPrepKarishma wrote: vjsharma25 wrote: How you have decided on the first sign of the graph?Why it is -ve if it has three factors and +ve when four factors? Check out my post above for explanation. I understand the concept but not the starting point of the graph.How you decide about the graph to be a sine or cosine waveform?Meaning graph starts from the +ve Y-axis for four values and starts from -ve Y-axis for three values. What if the equation you mentioned is (x+2)(x-1)(x-7)<0,will the last two ranges be excluded or the graph will also change? I always struggle with this as well!!! There is a trick Bunuel suggested; (x+2)(x-1)(x-7) < 0 Here the roots are; -2,1,7 Arrange them in ascending order; -2,1,7; These are three points where the wave will alternate. The ranges are; x<-2 -2<x<1 1<x<7 x>7 Take a big value of x; say 1000; you see the inequality will be positive for that. (1000+2)(1000-1)(1000-7) is +ve. Thus the last range(x>7) is on the positive side. Graph is +ve after 7. Between 1 and 7-> -ve between -2 and 1-> +ve Before -2 -> -ve Since the inequality has the less than sign; consider only the -ve side of the graph; 1<x<7 or x<-2 is the complete range of x that satisfies the inequality. _________________ Manager Joined: 29 Sep 2008 Posts: 150 Followers: 3 Kudos [?]: 49 [11] , given: 1 Re: Inequalities trick [#permalink] 22 Oct 2010, 10:45 11 This post received KUDOS 6 This post was BOOKMARKED if = sign is included with < then <= will be there in solution like for (x+2)(x-1)(x-7)(x-4) <=0 the solution will be -2 <= x <= 1 or 4<= x <= 7 in case when factors are divided then the numerator will contain = sign like for (x + 2)(x - 1)/(x -4)(x - 7) < =0 the | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
are divided then the numerator will contain = sign like for (x + 2)(x - 1)/(x -4)(x - 7) < =0 the solution will be -2 <= x <= 1 or 4< x < 7 we cant make 4<=x<=7 as it will make the solution infinite correct me if i am wrong Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5682 Location: Pune, India Followers: 1415 Kudos [?]: 7357 [6] , given: 186 Re: Inequalities trick [#permalink] 11 Mar 2011, 18:57 6 This post received KUDOS Expert's post vjsharma25 wrote: I understand the concept but not the starting point of the graph.How you decide about the graph to be a sine or cosine waveform?Meaning graph starts from the +ve Y-axis for four values and starts from -ve Y-axis for three values. What if the equation you mentioned is (x+2)(x-1)(x-7)<0,will the last two ranges be excluded or the graph will also change? Ok, look at this expression inequality: (x+2)(x-1)(x-7) < 0 Can I say the left hand side expression will always be positive for values greater than 7? (x+2) will be positive, (x - 1) will be positive and (x-7) will also be positive... so in the rightmost regions i.e. x > 7, all three factors will be positive. The expression will be positive when x > 7, it will be negative when 1 < x < 7, positive when -2 , x < 1 and negative when x < -2. We need the region where the expression is less than 0 i.e. negative. So either 1 < x < 7 or x < -2. Now let me add another factor: (x+8)(x+2)(x-1)(x-7) Can I still say that the entire expression is positive in the rightmost region i.e. x>7 because each one of the four factors is positive? Yes. So basically, your rightmost region is always positive. You go from there and assign + and - signs to the regions. Your starting point is the rightmost region. Note: Make sure that the factors are of the form (ax - b), not (b - ax)... e.g. (x+2)(x-1)(7 - x)<0 Convert this to: (x+2)(x-1)(x-7)>0 (Multiply both sides by '-1') Now solve in the usual way. Assign '+' to the rightmost region and then alternate with '-' Since you are looking | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
the usual way. Assign '+' to the rightmost region and then alternate with '-' Since you are looking for positive value of the expression, every region where you put a '+' will be the region where the expression will be greater than 0. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Veritas Prep Reviews
Manager
Status: On...
Joined: 16 Jan 2011
Posts: 189
Followers: 3
Kudos [?]: 42 [5] , given: 62
Re: Inequalities trick [#permalink] 10 Aug 2011, 16:01
5
KUDOS
5
This post was
BOOKMARKED
WoW - This is a cool thread with so many thing on inequalities....I have compiled it together with some of my own ideas...It should help.
1) CORE CONCEPT
@gurpreetsingh -
Suppose you have the inequality
f(x) = (x-a)(x-b)(x-c)(x-d) < 0
Arrange the NUMBERS in ascending order from left to right. a<b<c<d
Draw curve starting from + from right.
now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.
So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)
If f(x) has three factors then the graph will have - + - +
If f(x) has four factors then the graph will have + - + - +
If you can not figure out how and why, just remember it.
Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.
For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.
Note: Make sure that the factors are of the form (ax - b), not (b - ax)...
example -
(x+2)(x-1)(7 - x)<0
Convert this to: (x+2)(x-1)(x-7)>0 (Multiply both sides by '-1')
Now solve in the usual way. Assign '+' to the rightmost region and then alternate with '-'
Since you are looking for positive value of the expression, every region where you put a '+' will be the region where the expression will be greater than 0.
2) Variation - ODD/EVEN POWER
@ulm/Karishma -
if we have even powers like (x-a)^2(x-b)
we don't need to change a sign when jump over "a".
This will be same as (x-b) | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
We can ignore squares BUT SHOULD consider ODD powers
example -
2.a
(x-a)^3(x-b)<0 is the same as (x-a)(x-b) <0
2.b
(x - a)(x - b)/(x - c)(x - d) < 0 ==> (x - a)(x - b)(x-c)^-1(x-d)^-1 <0
is the same as (x - a)(x - b)(x - c)(x - d) < 0
3) Variation <= in FRACTION
@mrinal2100 -
if = sign is included with < then <= will be there in solution
like for (x+2)(x-1)(x-7)(x-4) <=0 the solution will be -2 <= x <= 1 or 4<= x <= 7
BUT if it is a fraction the denominator in the solution will not have = SIGN
example -
3.a
(x + 2)(x - 1)/(x -4)(x - 7) < =0
the solution will be -2 <= x <= 1 or 4< x < 7
we cant make 4<=x<=7 as it will make the solution infinite
4) Variation - ROOTS
@Karishma -
As for roots, you have to keep in mind that given $$\sqrt{x}$$, x cannot be negative.
$$\sqrt{x}$$ < 10
implies 0 < $$\sqrt{x}$$ < 10
Squaring, 0 < x < 100
Root questions are specific. You have to be careful. If you have a particular question in mind, send it.
Refer - inequalities-and-roots-118619.html#p959939
Some more useful tips for ROOTS....I am too lazy to consolidate
<5> THESIS -
@gmat1220 -
Once algebra teacher told me - signs alternate between the roots. I said whatever and now I know why Watching this article is a stroll down the memory lane.
I will save this future references....
Anyone wants to add ABSOLUTE VALUES....That will be a value add to this post
_________________
Labor cost for typing this post >= Labor cost for pushing the Kudos Button
kudos-what-are-they-and-why-we-have-them-94812.html
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 5682
Location: Pune, India
Followers: 1415
Kudos [?]: 7357 [3] , given: 186
Re: Inequalities trick [#permalink] 09 Sep 2013, 22:35
3
KUDOS
Expert's post
karannanda wrote:
gurpreetsingh wrote:
I learnt this trick while I was in school and yesterday while solving one question I recalled.
Its good if you guys use it 1-2 times to get used to it.
Suppose you have the inequality
f(x) = (x-a)(x-b)(x-c)(x-d) < 0 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Suppose you have the inequality
f(x) = (x-a)(x-b)(x-c)(x-d) < 0
Just arrange them in order as shown in the picture and draw curve starting from + from right.
now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.
Don't forget to arrange then in ascending order from left to right. a<b<c<d
So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d)
and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)
If f(x) has three factors then the graph will have - + - +
If f(x) has four factors then the graph will have + - + - +
If you can not figure out how and why, just remember it.
Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.
For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.
Hi Gurpreet,
Thanks for the wonderful method.
I am trying to understand it so that i can apply it in tests.
Can you help me in applying this method to the below expression to find range of x.
x^3 – 4x^5 < 0?
I am getting the roots as -1/2, 0, 1/2 and when i plot them using this method, putting + in the rightmost region, I am not getting correct result. Not sure where i am going wrong. Can you pls help.
Before you apply the method, ensure that the factors are of the form (x - a)(x - b) etc
$$x^3 - 4x^5 < 0$$
$$x^3 ( 1 - 4x^2) < 0$$
$$x^3(1 - 2x) (1 + 2x) < 0$$
$$4x^3(x - 1/2)(x + 1/2) > 0$$ (Notice the flipped sign. We multiplied both sides by -1 to convert 1/2 - x to x - 1/2)
Now the transition points are 0, -1/2 and 1/2 so put + in the rightmost region.
The solution will be x > 1/2 or -1/2 < x< 0. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Check out these posts discussing such complications:
http://www.veritasprep.com/blog/2012/06 ... e-factors/
http://www.veritasprep.com/blog/2012/07 ... ns-part-i/
http://www.veritasprep.com/blog/2012/07 ... s-part-ii/
_________________
Karishma
Veritas Prep | GMAT Instructor
My Blog
Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5682 Location: Pune, India Followers: 1415 Kudos [?]: 7357 [2] , given: 186 Re: Inequalities trick [#permalink] 23 Jul 2012, 02:13 2 This post received KUDOS Expert's post Stiv wrote: VeritasPrepKarishma wrote: mrinal2100: Kudos to you for excellent thinking! Correct me if I'm wrong. If the lower part of the equation$$\frac {(x+2)(x-1)}{(x-4)(x-7)}$$ were $$4\leq x \leq 7$$, than the lower part would be equal to zero,thus making it impossible to calculate the whole equation. x cannot be equal to 4 or 7 because if x = 4 or x = 7, the denominator will be 0 and the expression will not be defined. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199
Veritas Prep Reviews
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 336
Followers: 52
Kudos [?]: 380 [2] , given: 81
Inequalities trick [#permalink] 06 Jan 2015, 02:35
2
KUDOS
Expert's post
5
This post was
BOOKMARKED
Just came across this useful discussion.
VeritasPrepKarishma has given a very lucid explanation of how this “wavy line” method works.
I have noticed that there is still a little scope to take this discussion further. So here are my two cents on it.
I would like to highlight an important special case in the application of the Wavy Line Method
When there are multiple instances of the same root:
Try to solve the following inequality using the Wavy Line Method:
$$(x-1)^2(x-2)(x-3)(x-4)^3 < 0$$
To know how you did, compare your wavy line with the correct one below. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
To know how you did, compare your wavy line with the correct one below.
Did you notice how this inequality differs from all the examples above?
Notice that two of the four terms had an integral power greater than 1.
How to draw the wavy line for such expressions?
Let me directly show you how the wavy line would look and then later on the rule behind drawing it.
Attachment:
File comment: Observe how the wave bounces back at x = 1.
bounce.png [ 10.4 KiB | Viewed 2175 times ]
Notice that the curve bounced down at the point x = 1. (At every other root, including x = 4 whose power was 3, it was simply passing through them.)
Can you figure out why the wavy line looks like this for this particular inequality?
(Hint: The wavy line for the inequality
$$(x-1)^{38}(x-2)^{57}(x-3)^{15}(x-4)^{27} < 0$$
Is also the same as above)
Come on! Give it a try.
If you got it right, you’ll see that there are essentially only two rules while drawing a wavy line. (Remember, we’ll refer the region above the number line as positive region and the region below the number line as negative region.)
How to draw the wavy line?
1. How to start: Start from the top right most portion. Be ready to alternate (or not alternate) the region of the wave based on how many times a point is root to the given expression.
2. How to alternate: In the given expression, if the power of a term is odd, then the wave simply passes through the corresponding point (root) into the other region (to –ve region if the wave is currently in the positive region and to the +ve region if the wave is currently in the negative region). However, if the power of a term is even, then the wave bounces back into the same region.
Now look back at the above expression and analyze your wavy line. Were you (intuitively) using the above mentioned rules while drawing your wavy line?
Solution | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Solution
Once you get your wavy line right, solving an inequality becomes very easy. For instance, for the above inequality, since we need to look for the space where the above expression would be less than zero, look for the areas in the wavy line where the curve is below the number line.
So the correct solution set would simply be {3 < x < 4} U {{x < 2} – {1}}
In words, it is the Union of two regions region1 between x = 3 and x = 4 and region2 which is x < 2, excluding the point x = 1.
Food for Thought
Now, try to answer the following questions:
1. Why did we exclude the point x = 1 from the solution set of the last example? (Easy Question)
2. Why do the above mentioned rules (especially rule #2) work? What is/are the principle(s) working behind the curtains?
Foot Note: Although the post is meant to deal with inequality expressions containing multiple roots, the above rules to draw the wavy line are generic and are applicable in all cases.
- Krishna
_________________
Last edited by EgmatQuantExpert on 10 Jan 2015, 20:41, edited 1 time in total.
Current Student
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2800
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 196
Kudos [?]: 1162 [1] , given: 235
Re: Inequalities trick [#permalink] 19 Mar 2010, 11:59
1
KUDOS
ttks10 wrote:
Can u plz explainn the backgoround of this & then the explanation.
Thanks
i m sorry i dont have any background for it, you just re-read it again and try to implement whenever you get such question and I will help you out in any issue.
sidhu4u wrote:
I have applied this trick and it seemed to be quite useful.
Nice to hear this....good luck.
_________________
Fight for your dreams :For all those who fear from Verbal- lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
GMAT Club Premium Membership - big benefits and savings | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Jo Bole So Nihaal , Sat Shri Akaal
GMAT Club Premium Membership - big benefits and savings
Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html
Math Forum Moderator
Joined: 20 Dec 2010
Posts: 2028
Followers: 138
Kudos [?]: 1165 [1] , given: 376
Re: Inequalities trick [#permalink] 26 May 2011, 19:55
1
KUDOS
chethanjs wrote:
mrinal2100 wrote:
if = sign is included with < then <= will be there in solution
like for (x+2)(x-1)(x-7)(x-4) <=0 the solution will be -2 <= x <= 1 or 4<= x <= 7
in case when factors are divided then the numerator will contain = sign
like for (x + 2)(x - 1)/(x -4)(x - 7) < =0
the solution will be -2 <= x <= 1 or 4< x < 7
we cant make 4<=x<=7 as it will make the solution infinite
correct me if i am wrong
Can you please tell me why the solution gets infinite for 4<=x<=7 ?
Thanks.
(x -4)(x - 7) is in denominator.
Making x=4 or 7 would make the denominator 0 and the entire function undefined.
Thus, the range of x can't be either 4 or 7.
4<=x<=7 would be wrong.
4<x<7 is correct because now we removed "=" sign.
_________________
Senior Manager
Joined: 16 Feb 2012
Posts: 257
Concentration: Finance, Economics
Followers: 4
Kudos [?]: 123 [1] , given: 121
Re: Inequalities trick [#permalink] 22 Jul 2012, 02:03
1
KUDOS
VeritasPrepKarishma wrote:
mrinal2100: Kudos to you for excellent thinking!
Correct me if I'm wrong.
If the lower part of the equation$$\frac {(x+2)(x-1)}{(x-4)(x-7)}$$ were $$4\leq x \leq 7$$, than the lower part would be equal to zero,thus making it impossible to calculate the whole equation.
_________________
Kudos if you like the post!
Failing to plan is planning to fail.
Current Student
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2800
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 196
Kudos [?]: 1162 [1] , given: 235 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Kudos [?]: 1162 [1] , given: 235
Re: Inequalities trick [#permalink] 18 Oct 2012, 04:17
1
KUDOS
GMATBaumgartner wrote:
gurpreetsingh wrote:
ulm wrote:
if we have smth like (x-a)^2(x-b)
we don't need to change a sign when jump over "a".
yes even powers wont contribute to the inequality sign. But be wary of the root value of x=a
Hi Gurpreet,
Could you elaborate what exactly you meant here in highlighted text ?
Even I have a doubt as to how this can be applied for powers of the same term . like the example mentioned in the post above.
If the powers are even then the inequality won't be affected.
eg if u have to find the range of values of x satisfying (x-a)^2 *(x-b)(x-c) >0
just use (x-b)*(x-c) >0 because x-a raised to the power 2 will not affect the inequality sign. But just make sure x=a is taken care off , as it would make the inequality zero.
_________________
Fight for your dreams :For all those who fear from Verbal- lets give it a fight
Money Saved is the Money Earned
Jo Bole So Nihaal , Sat Shri Akaal
GMAT Club Premium Membership - big benefits and savings
Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 5682
Location: Pune, India
Followers: 1415
Kudos [?]: 7357 [1] , given: 186
Re: Inequalities trick [#permalink] 18 Oct 2012, 09:27
1
KUDOS
Expert's post
GMATBaumgartner wrote:
Hi Gurpreet,
Could you elaborate what exactly you meant here in highlighted text ?
Even I have a doubt as to how this can be applied for powers of the same term . like the example mentioned in the post above.
In addition, you can check out this post:
http://www.veritasprep.com/blog/2012/07 ... s-part-ii/
I have discussed how to handle powers in it.
_________________
Karishma
Veritas Prep | GMAT Instructor
My Blog | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305381464927,
"lm_q1q2_score": 0.8560796095191096,
"lm_q2_score": 0.8887587920192298,
"openwebmath_perplexity": 1919.4069545627315,
"openwebmath_score": 0.6233364343643188,
"tags": null,
"url": "http://gmatclub.com/forum/inequalities-trick-91482.html?kudos=1"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.