text stringlengths 1 2.12k | source dict |
|---|---|
whose final column is F, and the valuation corresponding to that row is a valuation that does not satisfy the sentence being tested. It can be used to test the validity of arguments. Square of Opposition. ) The final column of a truth table for a tautology (respectively, a contradiction) is all Ts (respectively, all Fs). De nition 1. • Truth Table - a calculation matrix used to demonstrate all logically possible truth-values of a given proposition. The truth or falsity of a statement built with. Decide whether the formula p 0→¬p 0 is a tautology, a contingency, or a contradiction. I buy you a car. АЛВ+В'VA" с. Contradiction: A propositional formula is contradictory (unsatisfiable) if there is no interpretation for which it is true. As an introduction, we will make truth tables for these two statements 1. Tautology A statement is called a tautology if the final column in its truth table contains only 's. Compute the truth tables for the following propositional. tables consider all cases and can add great insight into otherwise complicated expressions. Discrete Mathematics Lecture 1 Logic: Propositional Logic Truth Table •A convenient way to see the effect of the contradiction E. entailment. For example, let us study the truth value of (p Λ q) → (p V q) by building a truth table. Then identify whether the sentence is a tautology, a contradiction or neither. Use a truth table to determine if the following is a tautology, a contradiction, or a contingency. Here we are going to study reasoning with propositions. In a proof by contradiction, we start out by saying: "Suppose not. is a proposition that is always. Illustrates and defines the terms ‘contradiction’, ‘logical truth’ and ‘logical possibility’. (a) Show that (p q) → p is a tautology by completing this truth table. Logically Equivalent Statements, Tautologies, & Contradictions; Definition 27. 5) E to I is a contradiction. By Using Truth Table Un 1. If a contradiction is produced in the attempts to assign truth | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
By Using Truth Table Un 1. If a contradiction is produced in the attempts to assign truth values to the premises, circle the contradiction and declare the argument valid. A compound proposition is said to be a contradiction if and only if it is false for all. 2) The truth tables for the basic operations: F F F F F T T F T F T F T T T T 1. Lesson 46 – Truth Tables (D)Tautologies and Logical Contradictions a. So we can every satisfiable is a contingency because satisfiable will have at least one true value which will also satisfy the definition of Contingency. Note any tautologies or contradictions. This is a truth table generator helps you to generate a Truth Table from a logical expression such as a and b. Contingency-. Consider the connectives: contradiction denoted by ⊥ (which stands on its own), negation denoted by ¬ (not), which is The truth table for “and” is: p q p. Question: Use A Truth Table To Determine If The Following Is A Tautology, A Contradiction, Or A Contingency. contradiction. Actually, the truth is that proper logical analysis does. A row of the truth table in which all the premises are true is called a critical row. cut: The minimal score for the PRI - proportional reduction in inconsistency, under which a truth table row is declared as negative. p q :q p!q :(p!q) p^:q T T F T F F T F T F T T F T F T F F F F T T F F Since the truth values for :(p!q) and p^:qare exactly the same for all possible combinations of truth values of pand q, the two propositions are equivalent. Using truth-tables to test a set of statements for consistency: Construct a sentence that is the conjunction of all the statements in question. Tautology Truth Tables. It isn't just that the intersection of your final answer and the set of. By Using Truth Table Un 1. In real life, 2-LUTs are not an efficient use of resources; LUTs have 4 or 5 inputs. Solution 2. Learn more about the laws of thought in this article. CMSC 203 : Section 0201 : Homework1 Solution 3. АЛВ+В'VA" с. ] | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
the laws of thought in this article. CMSC 203 : Section 0201 : Homework1 Solution 3. АЛВ+В'VA" с. ] Thus, x is a ratio of the two integers −a and b with b ≠ 0. Example: p ¬p is a tautology. By the and truth table, if one part of an "and" state-ment is false, the entire statement is false. Robert Lacey explores the "untold reality" of the Duke of Cambridge and. Q^(˘Q) Theorem: A statement S is a tautology if and only if its negation is a contradiction. In this post, I will briefly discuss tautologies and contradictions in symbolic logic. Illustrates and defines the terms ‘contradiction’, ‘logical truth’ and ‘logical possibility’. A propositional form that is true in at least one row of its truth table and false in at least one row of its truth table is a contingency. Create a truth table to find out if this is either a contradiction, tautology, or contingency. A truth table is a table that begins with all the possible combinations of truth values for the letters in the compound statement; it then breaks the compound statment down and one step at a time determines truth values for each of the parts of the logical statement. is a proposition which is always false. combinations of truth values of the propositional variables which it contains. A truth table displays the relationships between the truth values of propo-sitions. Propositions are built up as truth-functions of elementary propositions (5). Although consistency is no guarantee of truth since one could create a consistent story that is false, it seems to be a necessary condition for truth. For example, if A is “Napoleon was born in Corsica” and B is “The number of the beast is 666”, the truth table (in classical two-valued logic) would be as follows:. A tautology is a compound statement that is true for all possibilities in a truth table i. Subtracting 1 from both sides gives 2sin xcos <0. In our everyday choices, are we willing to give up our power and privilege to invite others to the table, to share in the | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
are we willing to give up our power and privilege to invite others to the table, to share in the “American dream?” It’s time to right this 400-year-old wrong. Tautologies and Contradictions in Symbolic Logic - PHILO-notes Daily Whiteboard - Duration: 5:19. The only way a disjunction can be false is if both parts are false. (4V B')Л (4ЛВ)" d. Logical Equivalence Please use truth tables to check the following Boolean expressions. The number of lines needed is 2 n where n is the number of variables. (pva)^(-1) = pvqr(r)) Get more help from Chegg Get 1:1 help now from expert Other Math tutors. A truth table is a mathematical table used to determine if a compound statement is true or false. State, with a reason, whether the compound proposition (p V (p A q)) p is a contradiction, a tautology or neither. > Contradiction > Contingency > Truth table with 3 variables >Venn Diagram Mathematical Logic app includes following topics: > Negation - To identify a statement as true, false or open. Chapter 5 Truth Tables. ((PvQ) ^ (~R) = P v (Q^(~R)) Thats not an equal sign btw, but three lines intended instead of two. The table below explores the four possible cases, but the truth is simpler than that. A proposition that is always false is called a contradiction. Tautologies []. , p _˘p 2Taut. Classifying propositions using truth tables We may use truth tables to make important distinctions among tautologies, contingencies, and contradictions, Consider the truth table for ‘P e P’ P e P T T T F T F. Topic 3 - Logic, sets and probability » 3. But a truth tree will get the job done faster. We notice that an implication can only be false if the hypothesis is true and the consequence is false. A full development of a theory of truth in paraconsistent logic is given by Beall (2009). The truth table is: T F T T T T T All the values in the final column are false, so Example 5 Construct a truth table for p V —q. 3 Truth Tables FILLED IN NOTES. However, if every attempt to find such a set | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
table for p V —q. 3 Truth Tables FILLED IN NOTES. However, if every attempt to find such a set of T values ends in a contradiction, then the game cannot succeed because there is no set of truth values which will make all the premisses true and the conclusion false, so in other words there is no such row in the full table. A compound proposition is a logical contradiction if all the values in its ü-uth table column are false. Ø p ® c, where c is a contradiction \ p. A passionate Computer Science and Engineering graduate, who loves to follow his heart :). A tautology is a proposition which is always true. A proposition (statement) is a contradiction if it is logically false. truth table if you like, but you may not need to. Truth Tables Every proposition, indeed every logical formula, is either true or false. Discuss the distinct patterns of. (pva)^(-1) = pvqr(r)) Get more help from Chegg Get 1:1 help now from expert Other Math tutors. Topic 3 - Logic, sets and probability » 3. As we can see from the truth table above, the proposition is definitely not a contradiction. It is for this obvious reason that logicians invented a shorter, more efficient method of determining the validity of arguments, namely, the indirect truth table method. 2: Truth Tables Worksheet Fill out the following truth tables and determine which statements are tautologies, contradictions, or neither. 2 Apply the truth able method to assessing validity (finding counterexamples), identifying tautologies, contradictions, and contingent sentences, assessing logical equivalence and consistency. Consider the connectives: contradiction denoted by ⊥ (which stands on its own), negation denoted by ¬ (not), which is The truth table for “and” is: p q p. As "we are all happy" can be either true or false, so can a. By constructing the truth table, determine whether the following statement pattern ls a tautology , contradiction or. Truth Table is a mathematical table and the base for all computing needs. | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
, contradiction or. Truth Table is a mathematical table and the base for all computing needs. Contradiction Sayings and Quotes. A vector of (remainder) row numbers from the truth table, to code as negative output configurations. The truth table for a tautology has “T” in every row. Where, 0 and F denotes False. 5 Tautology, Contradiction, Contingency, and Logical Equivalence Definition : A compound statement is a tautology if it is true re-gardless of the truth values assigned to its component atomic state-ments. Therefore the order of the rows doesn't matter - its the rows themselves that must be correct. In other words, their columns on a truth table are identical. However, if every attempt to find such a set of T values ends in a contradiction, then the game cannot succeed because there is no set of truth values which will make all the premisses true and the conclusion false, so in other words there is no such row in the full table. 1 Introduction The truth value of a given truth- functional compound sentence depends on the truth value s of each of its components. Consider, for example, the statement form: p ~ p p ∨ ~ p. Satisfiability & Logical Truth PHIL 012 - 2/16/2001 Outline Test Scores Homework Reminder Satisfiability Logical Truth Complex Truth Tables Sample Problems Satisfiability A sentence is said to be satisfiable IFF under some circumstances it could be true, on logical grounds. I was fooling with term_variables/2 and I think it may be possible to achieve a simpler solution using term_variables/2, coming up with some scheme to get readable variable names output, but also assuming the operators are all evaluable by is/2. Propositions are either completely true or completely false, so any truth table will want to show both of these possibilities for all the statements made. Partial truth tables have two salient virtues. But please note that this is just an introductory discussion on tautologies and contradictions as my main intention here is just to make | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
introductory discussion on tautologies and contradictions as my main intention here is just to make students in logic become familiar with the topic under investigation. Multiply both sides by −1, gives. Math, I have a question on tautologies and contradictions. How a proof by contradiction works. First, like whole truth tables, they are algorithmic (i. If you construct them correctly, you will get an answer to your question whether a particular argument is valid; whether a particular proposition is tautologous, self-contradictory, or contingent; or whether a particular set of propositions is consistent. A tautology is a logical compound statement formed by two or more individual statement which is true for all the values. An example is P ^(˘P). ” They are called direct proof, contra-positive proof and proof by contradiction. Chapter 1 Logic 1. none of these. Recharaaeriz-. Logical Equivalence, Tautologies, and Contradictions. Since most philosophers believe truth is logically consistent, they value logical consistency because it is a tool to discover truth. satisfiable, if its truth table contains true at least once. Contradiction A statement is called a contradiction if the final column in its truth table contains only 's. An example is P ^(˘P). A contingency is neither a tautology nor a contradiction, for instance p ∨ q is a contingency. ; Instead, ¬p is true. [(4 V B) ^…. Propositions are either completely true or completely false, so any truth table will want to show both of these possibilities for all the statements made. If you haven't read through the textbook sections that surround these tables, now would be a good time to do that. A proposition P is a tautology if it is true under all circumstances. Interpret your various truth-value assignments as a row of a truth-table. The upshot of this result is significant. 5 Tautology, Contradiction, Contingency, and Logical Equivalence Definition : A compound statement is a tautology if it is true re-gardless of the | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
Logical Equivalence Definition : A compound statement is a tautology if it is true re-gardless of the truth values assigned to its component atomic state-ments. Think of it as shorthand for a complex sentence like P&~P. This page contains a JavaScript program that will generate a truth table given a well formed formula of sentential logic. When you define a new theorem-like environment with ewtheorem, it is given the style currently in effect. State, with a reason, whether the compound proposition (p ∨ (p ∧ q)) ⇒ p is a contradiction, a tautology or neither. molecular statement is a contradiction if its truth table is always false. The number of lines needed is 2 n where n is the number of variables. A (complete) truth table shows the input/output behavior for all possible truth assignments. A formula is said to be a Contradiction if every truth assignment to its component statements results in the formula being false. the sentence '-(AvB)&(-A>B)' and ask you whether it is a contradiction. 1) is L∧¬La tautology a logical contradiction or neither?. The expression is simplified: (1. Use a truth table to determine whether the two statements are equivalent. truth tables. You can enter logical operators in several different formats. Assume that P is true. Truth Tables - Tautology and Contradiction. A compound statement that is false for all possible combinations of truth values of its component statements is called a contradiction. I construct the truth table for (P → Q)∨ (Q→ P) and show that the formula is always true. Table 1: The truth table for the negation. 1 Introduction The truth value of a given truth- functional compound sentence depends on the truth value s of each of its components. State whether the statements ¬ p ⇒ q and (¬ p ⇒ q) ∨ (¬ p ∧ q) are logically equivalent. Tautologies and Contradictions in Symbolic Logic - PHILO-notes Daily Whiteboard - Duration: 5:19. Obviously, truth tables of these sizes are simply impractical to construct. " We then show that | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
Obviously, truth tables of these sizes are simply impractical to construct. " We then show that this leads to a contradiction, a statement like (q ∧ ¬q). What is the truth of the matter? Truth’s character is both logical and empirical. ***** Full Course Playlist. In this video I construct two more truth tables and use them to illustrate the notion of a tautology and a contradiction. Use a truth table to determine if the following is a tautology, a contradiction, or a contingency. 2 Problem 14ES. Think of it as shorthand for a complex sentence like P&~P. 6 Propositions – Tautology, Contradiction, Contingency Tautology A proposition P is a tautology if and only if P is true under every valuation. In the truth tables above, there is only one case where "if P, then Q" is false: namely, P is true and Q is false. Truth Tables, Tautologies, and Logical Equivalences. 2: Truth Tables for Negation, Conjunction, and Disjunction Math 121 Truth Tables A truth table is used to determine when a compound statement is true or false. Similarly, if it is false, then it implies it is true. PHILO-notes 2,286 views. Propositional Equivalences Tautologies, Contradictions, and Contingencies. Therefore to establish that a conditional. 24 and 25 are pretty handy. We say a statement Scan be deduced from a statement R if the truth table of S is True whenever. , Fs in its truth table column. The tee symbol ⊤ is sometimes used to denote an arbitrary tautology, with the dual symbol ⊥ representing an arbitrary contradiction; in any symbolism, a tautology may be substituted for the truth value "true," as symbolized, for instance, by "1. Trump, Truth and the Power of Contradiction. Tautologies []. A proposition is said to be a tautology if its truth value is T for any assignment of truth values to its components. p ∨ q Solution to EXAMPLE 2. Ø p ® c, where c is a contradiction \ p. R R • ∼ R T T F F F F F T Willard is either a philosopher or a windbag, and he’s neither a philosopher nor a windbag. | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
F F F T Willard is either a philosopher or a windbag, and he’s neither a philosopher nor a windbag. Determine whether the following statement pattern is a tautology or a contradiction or contingency. That is, a statement and its negation can never have the same truth value. Definition: A compound statement is a contradiction if there is an F beneath its main connective in every row of its truth table. The ntheorem package provides nine predefined theorem styles, listed in Table 4. Some sentences have the property that they cannot be false under any circumstances. Working backwards from the truth values generated by evaluating the conclusion false, deduce the truth values for the remaining simple propositions and premises. The truth table for the conjunc-tion of two statements is shown in Figure 1. Logical Symbols are used to connect to simple statements, to define a compound statement and this process is called as logical operations. If a statement is neither a tautology nor a contradiction, then the truth values do alter the outcome and we say that the statement is a contingency. The truth-table at right demonstrates the logical equivalence of these two statement forms. contradiction. In this video I construct two more truth tables and use them to illustrate the notion of a tautology and a contradiction. Create a truth table showing the values of the premises and conclusion. Depending on the set of axioms exhibited and/or we are willing to impose on the linguistic expressions of our native intelligence, we might arrive at different computational intelligence expressions for. Check ((p ∨ q) ∧ ¬ q) → p is a tautology using truth table. The truth table for a tautology has “T” in every row. Prove Logical Equivalences. In real life, 2-LUTs are not an efficient use of resources; LUTs have 4 or 5 inputs. Explain in a sentence why your truth table shows whether it is a tautology, a contradiction, or a contingent proposition. To show that a sentence is not a tautology, | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
tautology, a contradiction, or a contingent proposition. To show that a sentence is not a tautology, however, we only need one line: a line on which the sentence is 0. Tautologies and Contradictions in Symbolic Logic - PHILO-notes Daily Whiteboard - Duration: 5:19. Locate the rows in which the premises are all true (the critical rows). A→ (B → A) b. And we can draw the truth table for p as follows. Truth Tables - Tautology and Contradiction. The second case agrees with none and we call it a contradiction. However, to show that an argument is not valid, all we need to do is to find one assignment where all the premises are true and the conclusion is. of the truth values of the propositional variables which comprise it. it is true in no world, e. Introduction to Philosophy > Logic > Tautologies and Contradictions. 6 a tautology 2. , ), the following statements can be made: Valid implies that the argument must be true for all instances (i. Fill in the truth table for the sentence of propositional logic below. Select "Full Table" to show all columns, "Main Connective Only. Learn more about the laws of thought in this article. p) is a logical contradiction. In sentential logic all theorems are tautologies and all tautologies are either axioms or theorems. Tautologies, Inconsistent Sentences, and Contingent Sentences Tautologies. So it is not a TT-contradiction. A propositional form that is true in all rows of its truth table is a tautology. The size of the truth table depends on the number of different simple. Here are examples of some of most basic truth tables, Truth table for negation ("not") Truth table for conjunction ("and" Truth table for disjunction ("or") Ex a Translate to symbolic form, then construct a truth table to represent the expression:. Truth Tables - Tautology and Contradiction. Contradiction - example 17 5. Create a truth table to determine whether the following statement is contingent, a tautology, or a self-contradiction. In other words, their | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
the following statement is contingent, a tautology, or a self-contradiction. In other words, their columns on a truth table are identical. If it is always true, then the argument is valid. contradiction, is a statement which is false regardless of the truth values of the substatements which form it. The minimum number of cases under which a truth table row is declared as a remainder. here they are: a) i found this proposition was a contradiction. Confirmed! The conditional p —+ q and it's contrapositive -Iq —Y -p are logically equivalent. 2 LOGICAL EQUIVALENCE, TAUTOLOGIES & CONTRADICTIONS. Let's start with logical contradiction. ((PvQ) ^ (~R) = P v (Q^(~R)) Thats not an equal sign btw, but three lines intended instead of two. you’re assuming it’s consistent by plugging in all true truth values. The expression "p or not p" is true under any circumstance, so it is a tautology. АЛВ+В'VA" с. Where, 0 and F denotes False. The following two truth tables are examples of tautologies and contradictions, respectively. The opposite of a tautology is a contradiction, a formula which is "always false". On my upcoming test time is limited. A contradiction is a conjunction of the form "A and not-A", where not-A is the contradictory of A. 12, 15 Examine the statement Patterns (Tautology, Contradiction, Contingency) 1. A truth table column which consists entirely of T's indicates a situation where the proposition is true no matter whether the individual propositions of which it is composed are true or false. Solution: Question 2. | {
"domain": "chiavette-usb-personalizzate.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.970239907775086,
"lm_q1q2_score": 0.8545844878495524,
"lm_q2_score": 0.8807970904940926,
"openwebmath_perplexity": 467.3208727966209,
"openwebmath_score": 0.5699777007102966,
"tags": null,
"url": "http://insd.chiavette-usb-personalizzate.it/contradiction-truth-table.html"
} |
# Prove a function is continuos
#### ianchenmu
##### Member
Prove that
$f(x,y)=\left\{\begin{matrix} e^{-1/|x-y|},x\neq y\\ 0,x=y \end{matrix}\right.$
is continuous on $\mathbb{R}^2$.
Can I conclude since $e^{-1/t}$ is continuous then for $x$,$y\in \mathbb{R}^2$, $x\neq y$,$e^{-1/|x-y|}$ is continuous on $\mathbb{R}^2$ since here $t=|x-y|$? And how to prove it when $x=y$, using the $\delta- \varepsilon$ way? Thank you a lot!
#### girdav
##### Member
Re: Prove a function is continuous
We have to show that the function is continuous at each point of $\Bbb R^2$. Your argument threats the case of points of the form $(x,y),x\neq y$. So now, fix $x_0$. We have to show that $\lim_{(x,y)\to (x_0,x_0)}=0$.
Hint: use $e^{-u}\leqslant \frac 1{1+u}$ for each non-negative real number $u$.
Last edited:
#### Fernando Revilla
##### Well-known member
MHB Math Helper
Re: Prove a function is continuous
Hint: use $e^{-u}\leqslant \frac 1{1+u}$ for each real number $u$.
Surely is a typo, I suppose you meant for each real number $u$ close to $0$.
Another way: denoting $\Delta=\{(t,t):t\in\mathbb{R}\}$ (diagonal of $\mathbb{R}^2$), in a neighborhood $V$ of $(0,0)$ we have:
$\displaystyle\lim_{(x,y)\to (x_0,x_0)\;,\;(x,y)\in V-\Delta}f(x,y)=\displaystyle\lim_{(x,y)\to (x_0,x_0)\;,\;(x,y)\in V-\Delta}e^{-\dfrac{1}{|x-y|}}=e^{-\infty}=0$
$\displaystyle\lim_{(x,y)\to (x_0,x_0)\;,\;(x,y)\in V\cap\Delta}f(x,y)=\displaystyle\lim_{(x,y)\to (x_0,x_0)\;,\;(x,y)\in V\cap\Delta}0=0$
In a finite partition of $V$ all the limits coincide, so $\displaystyle\lim_{(x,y)\to (x_0,x_0)}f(x,y)=0$ .
#### ianchenmu | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399034724604,
"lm_q1q2_score": 0.8545844749520176,
"lm_q2_score": 0.8807970811069351,
"openwebmath_perplexity": 528.6336263970237,
"openwebmath_score": 0.9754927158355713,
"tags": null,
"url": "https://mathhelpboards.com/threads/prove-a-function-is-continuos.3349/"
} |
#### ianchenmu
##### Member
The definition of continuity for a function is {lim x-> a} f(x) = f(a). For this function we only need to worry about when |x - y| -> 0, since the exponential function is continuous everywhere. So in other words we need to show that
{lim |x - y|-> 0} f(x,y) = f(x,x) = f(y,y)
Now f(x,x) = f(y,y) = 0 so we just need to show that
{lim |x - y| -> 0} f(x,y) = 0.
Since |x - y| > 0, -1/|x - y| -> infinity so f(x,y) -> 0 and we're done.
Is a proof like this correct? | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399034724604,
"lm_q1q2_score": 0.8545844749520176,
"lm_q2_score": 0.8807970811069351,
"openwebmath_perplexity": 528.6336263970237,
"openwebmath_score": 0.9754927158355713,
"tags": null,
"url": "https://mathhelpboards.com/threads/prove-a-function-is-continuos.3349/"
} |
# How to solve an equation with $x^4$?
Today, I had this question on a Maths test about Algebra. This was the equation I had to solve:
$$(1-x)(x-5)^3=x-1$$
I worked away the brackets and subtracted $x-1$ from both sides and was left with this:
$$-x^4+16x^3-90x^2+199x-124=0$$
Problem is, I haven't a clue how to solve this? First thing I tried was replacing $x^2$ with another variable like $u$ but that got me no further. Dividing the whole equation by $x^2$ (as is suggested by a lot of sites on this matter) also did not get me any further. I then tried something incredibly ludicrous;
$$(ax+b)(cx^3+dx^2+ex+f)=0$$ \left\{ \begin{aligned} ac &= -1 \\ ad + bc &= 16 \\ ae + bd &= -90 \\ af + be &= 199 \\ bf &= -124 \end{aligned} \right.
which got even worse when there were 3 brackets;
$$(ax+b)(gx+h)(ix^2+jx+k)=0$$ \left\{ \begin{aligned} ac &= agi &&= -1 \\ ad + bc &= agj + ahi + bgi &&= 16 \\ ae + bd &= agk + ahj + bgj + bhi &&= -90 \\ af + be &= ahk + bgk + bhj &&= 199 \\ bf &= bhk &&= -124 \end{aligned} \right.
only to be left with no result.
When using Wolfram Alpha on this question, it performs a rather strange step I don't understand:
$$-x^4+16x^3-90x^2+199x-124=0$$ $$\downarrow$$ $$-((x-4)(x-1)(x^2-11x+31))=0$$
Could somebody explain how to properly tackle this problem? And if possible, also show me how to get the non-real answers for it. Thanks.
• You needn't expand: first, observe that $x=1$ is a solution. Now, looking for other solutions (at most 3 other possible ones), you can divide each side by $x-1$ and obtain the equation $(x-5)^3=-1$ to solve (for values $x\neq 1$). – Clement C. May 20 '15 at 13:13
Euh... I think you overcomplicated things here...
$(1-x)(x-5)^3=x-1$ is equivalent to $(1-x)[(x-5)^3+1]=0$
Either $x=1$ or $(x-5)^3=-1$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399051935107,
"lm_q1q2_score": 0.8545844734319819,
"lm_q2_score": 0.8807970779778824,
"openwebmath_perplexity": 363.2793288703008,
"openwebmath_score": 0.9995129108428955,
"tags": null,
"url": "https://math.stackexchange.com/questions/1291087/how-to-solve-an-equation-with-x4/1291091"
} |
$(1-x)(x-5)^3=x-1$ is equivalent to $(1-x)[(x-5)^3+1]=0$
Either $x=1$ or $(x-5)^3=-1$...
• I feel so stupid, that I didn't see this in the first place... Thanks for the great answer! – Martijn May 20 '15 at 13:19
• @TitoPiezasIII Done :) – Martijn May 20 '15 at 13:27
• You can also use the identity $a^3+1=(a+1)(...)$ – k1.M May 20 '15 at 14:00
There are already answers on how to get the real solutions, so I will only show you the non-real solutions.
You have obtained that $(x-5)^3=-1$. Expanding and simplifying we get: $$x^3-15x^2+75x-124=0$$ However, we know that $x=4$ is a solution so we can say that $(x-4)(ax^2+bx+c)=0$. You can equate coefficients or use polynomial division, but as you have already found with Wolfram Alpha, $(ax^2+bx+c)=(x^2-11x+31)$.
To solve, complete the square: $$x^2-11x+31=0$$ $$(x-\frac{11}{2})^2-\frac{121}{4}+31=0$$ $$(x-\frac{11}{2})^2=-\frac{3}{4}$$ $$x-\frac{11}{2}=\sqrt{-\frac{3}{4}}= \pm \frac{\sqrt{3}}{2}i$$ $$x=\frac {11 \pm i \sqrt{3}}{2}$$
• Awesome work! I have two kinds of mathematics in the Netherlands and the other kind is about imaginary numbers a.t.m. so this will definitely come in handy, I'm sure! – Martijn May 20 '15 at 13:45
• In general the solutions to $(x-a)^m=-1$ are $x_n=e^{i\pi\frac{2n-1}{m}}+a$, for $n=1,2,\dots m$. – Kwin van der Veen May 21 '15 at 4:10
From the beginning: $$(1-x)(x-5)^3=x-1\\ (1-x)(x-5)^3+1-x=0\\ (1-x)(x-5)^3+(1-x)=0\\ (1-x)[(x-5)^3+1]=0\\$$
This implies $1-x=0$ or $(x-5)^3=-1$. I believe you can solve these.
• Isn't there a sign that got flipped between the first and second line? – Clement C. May 20 '15 at 13:18
• Sorry fixed it. @ClementC. – KittyL May 20 '15 at 13:24
I know that the problem has already been answered but I want to show you a more general method, let's suppose that you don't se how to rewrite the equation:
$-x^4+16x^3-90x^2+199x-124=0$
Or I prefer to write:
$x^4-16x^3+90x^2-199x+124=0$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399051935107,
"lm_q1q2_score": 0.8545844734319819,
"lm_q2_score": 0.8807970779778824,
"openwebmath_perplexity": 363.2793288703008,
"openwebmath_score": 0.9995129108428955,
"tags": null,
"url": "https://math.stackexchange.com/questions/1291087/how-to-solve-an-equation-with-x4/1291091"
} |
$-x^4+16x^3-90x^2+199x-124=0$
Or I prefer to write:
$x^4-16x^3+90x^2-199x+124=0$
You can use something called the Ruffini rule: search for integers divisor (both positive and negative) of the constant term and then set $x$ equals to the them and see if one of them is a solution. Starting from one we have:
$1-16+90-199+124=0$
So $x=1$ is a solution, now via Ruffini's rule, which can be seen here, we can rewrite the equation as:
$(x-1)(x^3-15x^2+75x-124)=0$
Now you have $3$ options to end this exercise:
$1.)$ Note that the second factor is a perfect cube;
$2.)$ Use Ruffini's rule again;
$3.)$ Use the general formula for third degree equation (which I'd not advise you to if your interested only in real solution).
This is a more general method so I hope this will help you in the future! | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9702399051935107,
"lm_q1q2_score": 0.8545844734319819,
"lm_q2_score": 0.8807970779778824,
"openwebmath_perplexity": 363.2793288703008,
"openwebmath_score": 0.9995129108428955,
"tags": null,
"url": "https://math.stackexchange.com/questions/1291087/how-to-solve-an-equation-with-x4/1291091"
} |
# Numbers of distinct values obtained by inserting $+ - \times \div ()$ in $\underbrace{2\quad2 \quad2 \quad2\quad…\quad 2}_{n \text{ times}}$
This question is inspired by How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis? (By the way, I sincerely hope this kind of questions can receive more attention)
Insert $$+ - \times \div ()$$ in $$\underbrace{2\quad2 \quad2 \quad2\quad...\quad 2}_{n \text{ times}}$$ Denote the number of distinct values which can be obtained in this way by $$D(n)$$. Is there a general formula (or recurrence relation at least) for $$D(n)$$?
This is basically the $$+ - \times \div ()$$ version of @barakmanos question. It seems this question is easier than the power tower version. Or maybe not?
For $$n=1$$ , there is only $$2$$ values $$-2,2$$;
For $$n=2$$, there are $$5$$ values $$-4,-1,0,1,4$$;
For $$n=3$$, there are $$13$$ values $$-8,-6,-3,-2,-1,-\frac{1}{2},0,\frac{1}{2},1,2,3,6,8$$;
And for $$n=4$$ I'm reluctant to calculate with bare hands. (See @DanUznanki answer for what follows)
Any idea is appreciated. Sorry if this is a duplicate.
Edit: My research shows that the version with distinct generic variables $$a_1,a_2,...,a_n$$ is solved. See A182173 for your reference. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305307578324,
"lm_q1q2_score": 0.8545825631878546,
"lm_q2_score": 0.88720460564669,
"openwebmath_perplexity": 363.66713835289636,
"openwebmath_score": 0.8713907599449158,
"tags": null,
"url": "https://math.stackexchange.com/questions/3119998/numbers-of-distinct-values-obtained-by-inserting-times-div-in-under"
} |
• Based on what you say for $n=1$ and $n=2$, we may use $-$ as either the negation operator on one number or the subtraction operator between two numbers? – alex.jordan Mar 24 at 4:53
• @alex.jordan Yes. – YuiTo Cheng Mar 24 at 4:59
• OK. I wonder if you did not allow $-$ to be unary negation, if then the sequence might appear in OEIS. We have an overloaded symbol, $-$. And one of its meanings happens to be unary negation. It's odd that we don't have a symbol for unary inversion. Well, we do: $^{-1}$. But it does not reuse the division symbol $\div$. All this is just to say that it feels like the special double meaning for $-$, that is not mirrored with $\div$, could break what would otherwise be a simpler pattern in the sequence of counts. – alex.jordan Mar 24 at 5:22
• @alex.jordan If we disallow $-$ to be a unary operator, will things become vastly simpler? – YuiTo Cheng Mar 24 at 5:27
• @alex.jordan: I once toyed with the idea of "$/$" for unary inversion. (So, "$/2$" means "$1/2$" in the same way that "$-2$" means "$0-2$".) I still kinda like it. :) – Blue Mar 25 at 20:09
It looks like we can do this "inductively": for the calculations of size $$n$$, we can take values from the list for $$1 \le k\le n/2$$, and values from the list for $$(n-k)$$, and operate on them using the 64 different operation orders.
Fortunately, it's really only 10 classes of operation, because many are duplicates:
• There's four values we can get from addition and subtraction: $$a+b$$, $$-(a+b)$$, $$a-b$$, and $$b-a$$.
• There's only two we can get from multiplication: $$ab$$ and $$-ab$$.
• There are four cases we can get from division: $$a/b$$, $$b/a$$, $$-a/b$$, and $$-b/a$$.
Also conveniently we only need to try the nonnegative entries in previous lists.
So for $$n=4$$, we have: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305307578324,
"lm_q1q2_score": 0.8545825631878546,
"lm_q2_score": 0.88720460564669,
"openwebmath_perplexity": 363.66713835289636,
"openwebmath_score": 0.8713907599449158,
"tags": null,
"url": "https://math.stackexchange.com/questions/3119998/numbers-of-distinct-values-obtained-by-inserting-times-div-in-under"
} |
So for $$n=4$$, we have:
$$-16$$, $$-12$$, $$-10$$, $$-8$$, $$-6$$, $$-5$$, $$-4$$, $$-3$$, $$-5/2$$, $$-2$$, $$-3/2$$, $$-1$$, $$-2/3$$, $$-1/2$$, $$-1/3$$, $$-1/4$$, $$0$$, $$1/4$$, $$1/3$$, $$1/2$$, $$2/3$$, $$1$$, $$3/2$$, $$2$$, $$5/2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$8$$, $$10$$, $$12$$, $$16$$
Which is $$33$$ entries.
I've written a short script which finds them all, and told it to run up to $$n=10$$, which gave the following sizes: $$2,5,13,33,77,185,441,1051,2523,6083$$. Apparently this sequence is not in OEIS, nor is the positive-values-only version! I am very surprised.
• You mentioned we can do it 'inductively', do you have any clue about the recurrence relation? – YuiTo Cheng Feb 20 at 13:31
• Not sure, but it isn't gonna be nice. actually, different values for $x$ instead of $2$ give different results: using $3$ gives $2$, $7$, $21$, $67$, using $4$ gives $2$,$7$,$21$,$77$. Interestingly, when I tried to get the generic result, where $x$ is just $x$, I did get some partial results in the realm of restricted lattice walks: $2$,$7$,$23$,$91$, $347$ appears twice in OEIS. – Dan Uznanski Feb 20 at 13:39
• Interesting. What about the original power tower question? Compared with this, which one do you think is simpler? – YuiTo Cheng Feb 20 at 13:45
• for generic $x$ in the power tower thing you're getting exactly the Catalan Numbers. But assuming you have the ability to actually store the numbers you're working with, the power tower version is vastly simpler: there's only one operator instead of 10. – Dan Uznanski Feb 20 at 13:46
• What about the case without bracket? – YuiTo Cheng Feb 21 at 3:36
@Dan Uznanski if your results are correct i noticed that the ratio between the successive terms of the sum is almost constant. So maybe:
Conjecture
$$D(n) \sim K^n$$
Where:
$$2.3\leq K \leq 2.5$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305307578324,
"lm_q1q2_score": 0.8545825631878546,
"lm_q2_score": 0.88720460564669,
"openwebmath_perplexity": 363.66713835289636,
"openwebmath_score": 0.8713907599449158,
"tags": null,
"url": "https://math.stackexchange.com/questions/3119998/numbers-of-distinct-values-obtained-by-inserting-times-div-in-under"
} |
# Homomorphism between cyclic groups
I have some confusion in relation to the following question.
Let $\langle x\rangle = G$, $\langle y\rangle = H$ be finite cyclic groups of order $n$ and $m$ respectively. Let $f:G \mapsto H$ be the mapping sending $x^i$ to $y^i.$ Determine the conditions on $n$ and $m$ so that $f$ is going to be a homomorphism.
In verifying the definition of a homomorphism I seem to be able to conclude that there is no restriction on $n$ and $m$ since the definition of a homomorphic mapping is satisfied:
$$f(x^i x^j) = f(x^{i+j}) = y^{i+j} = y^i y^j = f(x^i)(x^j) \; \; \; (1)$$
But assuming n < m we would get
$$f(x^n) = f(e) = e \not = y^n$$
So clearly we have to have $m | n.$
My questions are
1. Are there any other conditions on $n$ and $m$ besides $m |n.$
2. What exactly did I do wrong in $(1)$ ?
-
For the (1), What would happen if $i+j \ge m$? It always helps to define the map this way: $x^{i(n)} \mapsto x^{i(m)}$. Note that the map may not even be well-defined if $m \nmid n$. – user21436 Mar 16 '12 at 9:50
I'll write up a more detailed answer in an hour if it's still unanswered. This has its direct connection to the universal property of the quotient maps. – user21436 Mar 16 '12 at 9:52
I see that there are many contradictions if $m \not | n.$ My main concern is that $(1)$ seems like such a innocent algebraic manipulation that I would never suspect that something is wrong with it.I would like to hear an "aha" explanation on why one cannot apply the definition of a homomorphism on $f$ so blindlessly. – Jernej Mar 16 '12 at 10:02
– lhf Mar 16 '12 at 10:43
Well, either you use a homomorphism $\mathbb Z \rightarrow G$ (and talk about lifting that homomorphism) or you must face the fact that the Element $x^i$ does NOT determine the number $i$. – Blah Mar 16 '12 at 10:50
The discussion about "well-defined" is perhaps a bit obscure. Here's the problem: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8545825530699429,
"lm_q2_score": 0.8872045922259089,
"openwebmath_perplexity": 195.3500608718099,
"openwebmath_score": 0.9452667236328125,
"tags": null,
"url": "http://math.stackexchange.com/questions/120861/homomorphism-between-cyclic-groups"
} |
The discussion about "well-defined" is perhaps a bit obscure. Here's the problem:
Remember that in general, an element of $G$ may have many different "names." For example, if $n=10$, then the element $x^{11}$ is equal to $x$; in fact, $x$ has infinitely many different "names" as a power of $x$: $$\cdots = x^{-9} = x^1 = x^{11} = x^{21} = \cdots$$
The problem is that the definition of $f$ as given depends on the name of the element! That is, if we are furiously working and somebody hands us a power of $x$, say, $x^{3781}$, we are supposed to, unthinkingly, map it to $y^{3781}$. The problem is that $x^{3781}$ is the same element as $x$, which we are supposed to send to $y^1$. That means that unless $y^1=y^{3781}$, what we have is not really a function: because the same input, $x$ (who, when being teased by bullies is called "$x^{3781}$") may be sent to $y^1$ or to $y^{3781}$, depending on what "name" we just heard for it.
Checking that the value of the function is the same regardless of what name we are using for an element, even though the function is defined in terms of the name, is called "checking that the function is 'well-defined.'"
An example of a function that is not well-defined would be one in which the input is an integer, and the output is the number of symbols used to express that integer. For example, $f(3)$ would be $1$ (because 3 is only one symbol), but $f(4-1)$ would be $3$ (because we are using 4, -, and 1). This is not well defined as a function of the integers, because $3$ is the same as $4-1$, but $f$ assigns it two different outputs. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8545825530699429,
"lm_q2_score": 0.8872045922259089,
"openwebmath_perplexity": 195.3500608718099,
"openwebmath_score": 0.9452667236328125,
"tags": null,
"url": "http://math.stackexchange.com/questions/120861/homomorphism-between-cyclic-groups"
} |
So in order for the expression given to actually define a function, we need the following to be true: $$\text{if }x^i=x^j,\text{ then }y^i = f(x^i)\text{ is equal to }y^j=f(x^j).$$ Now, $x^i=x^j$ if and only if $i\equiv j\pmod{n}$; and $y^i=y^j$ if and only if $i\equiv j\pmod{m}$. Therefore, we need: $$\text{if }i\equiv j\pmod{m},\text{ then }i\equiv j\pmod{n}.$$ In other words: every number that is a multiple of $m$ must be a multiple of $n$.
This is equivalent to $n|m$.
In fact, you noticed that in what you wrote, because $x^n=e$, so we need $f(x^n)$ to be the same as $f(x^0)$.
Once we know that $n|m$, then $f$ is well defined. Once it is well-defined, we can start verifying that it is indeed a homomorphism (it is). Technically, it's incorrect to start working to see if it is a homomorphism before you even know whether or not it is a function.
Note that the condition actually works if we allow $G$ or $H$ to be infinite cyclic groups, if we call infinite cyclic groups "groups of order $0$". Then $i\equiv j\pmod{0}$ means $i=j$, every number divides $0$, but the only multiple of $0$ is $0$. So if $G$ is infinite cyclic then the value of $n$ does not matter; if $H$ is infinite cyclic then we must have $G$ infinite cyclic. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8545825530699429,
"lm_q2_score": 0.8872045922259089,
"openwebmath_perplexity": 195.3500608718099,
"openwebmath_score": 0.9452667236328125,
"tags": null,
"url": "http://math.stackexchange.com/questions/120861/homomorphism-between-cyclic-groups"
} |
-
I don't agree with your take on well-defined. The rule $x^i\mapsto y^i$ is well-defined in the set $\{x^0, x^1, \ldots, x^{n-1}\}$ because every element in this set has a single name. The problem is that this rule may not be a homomorphism because there is no guarantee that say $x^{2n-3}$ goes to $f(x)^{2n-3}=y^{2n-3}$. – lhf Mar 16 '12 at 16:47
@lhf: I agree that if we were specifying $0\leq i\lt n$, then the rule would yield a well-defined function. However, I don't see any such specification in the statement of the problem. It just says "sending $x^i$ to $y^i$", with no restrictions on $i$. Of course, checking that it is a homomorphism is important: I note it later in the post, I don't ignore it. – Arturo Magidin Mar 16 '12 at 16:58
Right, I assumed that $0\leq i\lt n$ was a natural implicit assumption. – lhf Mar 16 '12 at 17:04
@lhf: In that case, there is a different problem with the definition, which is that it would only make sense if $n\leq m$ (otherwise, $y^i$ for $i\geq m$ would be an "invalid name"). So we would end up with the conclusion that the map only makes sense and is a homomorphism when $m=n$; or else we need to treat the exponent differently in the domain (restricted to a special representation for the elements) but not the codomain, which seems somewhat perverse. – Arturo Magidin Mar 16 '12 at 18:21
I don't think $y^i$ is an invalid name. Just raise $y$ to $i$. The exponent is well defined, isn't it? – lhf Mar 16 '12 at 21:27
You may want to check first if the map is well defined, ie., if $x^i=x^j$ does it follow that $f(x^i)=f(x^j)$? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8545825530699429,
"lm_q2_score": 0.8872045922259089,
"openwebmath_perplexity": 195.3500608718099,
"openwebmath_score": 0.9452667236328125,
"tags": null,
"url": "http://math.stackexchange.com/questions/120861/homomorphism-between-cyclic-groups"
} |
-
+1 Why would anyone downvote this? Ok, it would be better to phrase it: "if $x^i=x^j$ does it follow that $y^i=y^j$?" It is somewhat dangerous to talk about values of $f$ before you know that $f$ is well defined. – Jyrki Lahtonen Mar 16 '12 at 14:07
it is perfectly acceptable to ask whether or not a definition of f makes f a function. – David Wheeler Mar 16 '12 at 15:22
@David, you're right, of course. I was just trying to guess, why there was a downvote for a little while. – Jyrki Lahtonen Mar 16 '12 at 15:44
I am not sure if this completely explains why the naive calculations in $(1)$ are not valid. Here is my go at it:
I'll list down several points of concern.
• Firstly your map $x^i \mapsto y^i$ is very naive that it misses some essential details.
For convenience of notation, I prefer to look at this as $\varphi(\bar i)=\bar i$ if you'll allow me to do so. This is also naive but to some extent captures what is happening.
So, an equivalence class $\bmod n$ is mapped to the same equivalence class $\bmod m$ under $\varphi$. This bit of information is missed in $x^i \mapsto y^i$. This explains my comment under your question that you will want to write your maps as $x^{i(n)} \mapsto y^{i(m)}$. This notation makes things more transparent.
That you have dicovered that $m \mid n$, it can be obtained through the universal property of the quotients, which I will write about later.
The quotient map $G \to G/H$ factors over $G \to G/K$ if and only if $K \subseteq H$. That is there is a map from $G/H$ to $G/K$ if and only if $K \subseteq H$.
Proof: Please Go through Dummit and Foote's description in their Abstract Algebra in the section $\S3.3$ Isomorphism Theorems.
Now consider your cyclic groups as $\Bbb Z/n\Bbb Z$ and $\Bbb Z/m\Bbb Z$ and observe that above condition will tell you, $m\Bbb Z \subseteq n \Bbb Z$ whic happens if and only if...
As you had earlier in your comments observed, the map won't even be well defined if $m \nmid n$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8545825530699429,
"lm_q2_score": 0.8872045922259089,
"openwebmath_perplexity": 195.3500608718099,
"openwebmath_score": 0.9452667236328125,
"tags": null,
"url": "http://math.stackexchange.com/questions/120861/homomorphism-between-cyclic-groups"
} |
As you had earlier in your comments observed, the map won't even be well defined if $m \nmid n$.
I'll leave it to you as a challenge to use this blooper to prove that two groups of same order are isomorphic, which really is not the case. Hint: Cyclic groups exist for any given order.
-
Will the down voter explain why? – user21436 Mar 16 '12 at 14:01
Why do you say that $y^{i+j}$ might not be an element of $H$? Of course it is an element of $H$! A group is a set closed under the group operation, so if $y$ is in there, so are all its powers. Also, are you sure that $i(m)$ is a standard notation for the remainder? Color my old-fashioned, but I think that binary mod should stay in programming classes. This last bit is just a pet-peeve of mine :-) – Jyrki Lahtonen Mar 16 '12 at 14:04
Well, $y^{i+j}$ is an element of the group. For example, if $H$ is a subgroup of the multiplicative group of complex numbers generated by the imaginary unit $i$, you seem to be saying that $i^5$ is not an element of $H$. Well, you are wrong: $$i^5=i^{4+1}=i^4\cdot i=1\cdot i=i.$$ Looks like it is in there! The same holds for all cyclic groups. I just picked a concrete one to make you realize your error. – Jyrki Lahtonen Mar 16 '12 at 14:16
No. I'm not prone to making a mistake. I check that my mappings are well-defined. If (granted, a big if in some cases) $y^i=y^j$ whenever $x^i=x^j$, then the mapping $x^i\mapsto y^i$ is trivially a homomorphism: $$f(x^i*x^j)=f(x^{i+j})=y^{i+j}=y^i*y^j=f(x^i)f(x^j)$$ AFTER I have checked that it is well-defined, then the above always makes sense, because in a cyclic group the power of an element is defined in such a way that we always have $y^{i+j}=y^i*y^j$ and so forth. – Jyrki Lahtonen Mar 16 '12 at 14:31
as a matter of fact, it IS true that $(i+j)$ (mod n) = $i$ (mod n) + $j$ (mod n). the problem is that $(i+j)$ (mod n) may not equal $i+j$ (mod m). – David Wheeler Mar 16 '12 at 15:32 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8545825530699429,
"lm_q2_score": 0.8872045922259089,
"openwebmath_perplexity": 195.3500608718099,
"openwebmath_score": 0.9452667236328125,
"tags": null,
"url": "http://math.stackexchange.com/questions/120861/homomorphism-between-cyclic-groups"
} |
The rule $x^i\mapsto y^i$ gives a perfectly well-defined function in the set $\{x^0, x^1, \ldots, x^{n-1}\}$. However, for that rule to be a homomorphism we need that $x^{i+j} \mapsto y^{i+j}$ for all $i$ and $j$. To find where $x^{i+j}$ goes we need to reduce $i+j \bmod n$ to a $k$ in $\{0,1,\ldots,n-1\}$ and we need that $k \equiv i+j \bmod m$. In other words, we need $k \equiv k' \bmod n \implies k \equiv k' \bmod m$ and this can only happen when $m\mid n$.
-
THEOREM:
Define $f:\langle x \rangle \to \langle y \rangle$, where the orders of $\langle x \rangle$ and $\langle y \rangle$ are n and m respectively, by $f(x) = y^u$. Then f is a well-defined homomorphism if and only if $\frac{m}{\gcd(m,n)} \mid u$.
PROOF: Suppose that the mapping $f:\langle x \rangle \to \langle y \rangle$ is a well-defined homomorphism.
Let $f(x) = y^u$ for some non negative integer u.
We must also have $1 = f(x^n) = y^{un}$
Then $m \mid un$. Let $g = \gcd(m, n)$. Then $\frac m g \mid u \frac n g$. It follows that $\frac m g \mid u$.
The converse is pretty straightforward.
Suppose we define $f:\langle x \rangle \to \langle y \rangle$ by $f(x) = y^u$ where $\frac{m}{\gcd(m,n)} \mid u$
The only real problem with this definition is "Is it well defined?" Since the order of $\langle x \rangle$ is $n$:
\begin{align*} x^a = x^b &\Rightarrow a \equiv b \mod n \cr &\Rightarrow ua \equiv ub \mod {un} \cr &\Rightarrow ua \equiv ub \mod {\frac{mn}{\gcd(m,n)}} \cr &\Rightarrow ua \equiv ub \mod {m} \cr &\Rightarrow y^{ua} = y^{ub}\cr &\Rightarrow f(x^a) = f(x^b) \end{align*}
Linearity is pretty obvious, so $f$ is a well-defined homomorphism.
So, if you want $f(x^i) = y^i$, then you want $f(x) = y$. You must therefore have $\frac{m}{\gcd(m,n)} \mid 1$. Which implies $m \mid n$.
Conversely, if $m \mid n$ and $f(x) = y$, it follows that $f(x^i) = y^i$.
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9632305339244013,
"lm_q1q2_score": 0.8545825530699429,
"lm_q2_score": 0.8872045922259089,
"openwebmath_perplexity": 195.3500608718099,
"openwebmath_score": 0.9452667236328125,
"tags": null,
"url": "http://math.stackexchange.com/questions/120861/homomorphism-between-cyclic-groups"
} |
# induction and proper use of the inductive step
Suppose I have a conjecture of the form $\forall x \in \mathbb{N}^{+}$ $f(x) > g(x)$.
Without explicitly giving the functions $f$ and $g$ I would like to prove this conjecture using induction.
Specifically I have shown $f(1) > g(1)$. My inductive assumption is now that $f(x - 1) > g(x - 1)$ and I'm trying to show $f(x) > g(x)$.
I haven't been able to do this, but I've noticed that I can if I assume $f(y) > g(y)$ $\forall y \in \mathbb{N}^{+}$ such that $y < x$ instead of my inductive argument I can prove $f(x) > g(x)$.
My question is, is this ok or am I using circular logic? Typically when we use induction we use $x - 1$ to prove the $x$ case after an initial base case. For my problem it doesn't appear possible to prove $f(x) > g(x)$ without assuming that for all values less than $x$ the conjecture holds ($f$ and $g$ are recursive). I suspect that it is ok, but it sounds extremely circular and I haven't been able to convince myself that it isn't.
Note also that I've included proof verification as a tag because although I have not included the explicit proof, I have included a proof approach that I'm interested in validating.
• That's fine. That's called strong induction. It might be clearer if worded it slightly differently. You know $f(1) > g(1)$ and you have proven that if $f(y)> g(y)$ for all $y \le x-1< x$ then $f(x) > f(x)$. Hence... induction. Is that clearer? – fleablood Oct 14 '17 at 17:13
So you just want to make the stronger assumption that some statement $P(y)$ holds for all $y < x$ instead of only for $y = x-1$?
This is totally fine. It is called strong induction and is equivalent to simple induction.
• Ok, that's what I thought. I was just a bit unsure because I couldn't remember doing something like this in any course I've taken. Thank you! – HXSP1947 Oct 7 '17 at 1:54 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9835969694071564,
"lm_q1q2_score": 0.8545753723549964,
"lm_q2_score": 0.8688267643505193,
"openwebmath_perplexity": 144.11465721627906,
"openwebmath_score": 0.8813499212265015,
"tags": null,
"url": "https://math.stackexchange.com/questions/2461040/induction-and-proper-use-of-the-inductive-step"
} |
"Typically when we use induction we use x−1 to prove the x case after an initial base case. For my problem it doesn't appear possible to prove .. the proposition.. without assuming that for all values less than x the conjecture holds"
Think about this. If being true for all $y \le x-1 < x \implies$ being true for $x$, then being true for $y < x$ and being true for $x$ implies being true for all $y < x+1$. So true for $x+1$. And so on.
In principal, this is the exact same reasoning as the reasoning for induction. (if it's true for one case, it's true for the next, and via infinite countable reptitions it must be true for all).
The only difference is that whereas in "weak" induction we only make use of the fact that the proposition is true of $1$ and for $x-1$ and therefore true for $x$ we don't need to and we don't make use of the fact that it must also be true for $1,.......,x-2$ as well as just $x-1$.
But we know they must be true so we can make use of them if we need to.
This is called "strong" induction. It is equivalent to weak induction. The only difference is we might need to make use of earlier values than just the $x-1$ case.
This can be particularly useful with recursive definitions.
Say you know that $a_1 = something$ and $a_n =$ something to do with $a_{n-1}$ if $n$ is odd and something to do with $a_{\frac n2}$ and $a_{n-1}$ if $n$ is even. And you want to prove something about $a_n$. Well, it is not just enough to assume that the proposition is true of $a_{n-1}$. You also have to assume it is true of $a_{\frac n2}$. Can you assume that? Of course you can. To get to $a_{n-1}$ you had to have "passed through" $a_{\frac n2}$ "along the way". So why not use it if you have it? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9835969694071564,
"lm_q1q2_score": 0.8545753723549964,
"lm_q2_score": 0.8688267643505193,
"openwebmath_perplexity": 144.11465721627906,
"openwebmath_score": 0.8813499212265015,
"tags": null,
"url": "https://math.stackexchange.com/questions/2461040/induction-and-proper-use-of-the-inductive-step"
} |
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 12 Nov 2019, 06:25
### 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
# In the figure above if l1||l2, what is the value of x?
Author Message
TAGS:
### Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 58988
In the figure above if l1||l2, what is the value of x? [#permalink]
### Show Tags
15 Mar 2018, 23:19
00:00
Difficulty:
5% (low)
Question Stats:
93% (00:47) correct 7% (00:57) wrong based on 58 sessions
### HideShow timer Statistics
In the figure above if $$l_1||l_2$$, what is the value of x?
A. 70
B. 100
C. 110
D. 150
E. 170
Attachment:
2018-03-16_1010.png [ 6.61 KiB | Viewed 1271 times ]
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 58988
In the figure above if l1||l2, what is the value of x? [#permalink]
### Show Tags
15 Mar 2018, 23:19
Bunuel wrote:
In the figure above if $$l_1||l_2$$, what is the value of x?
A. 70
B. 100
C. 110
D. 150
E. 170
Attachment:
2018-03-16_1010.png
For other subjects:
ALL YOU NEED FOR QUANT ! ! !
_________________
Intern
Joined: 24 Jan 2018
Posts: 14
Re: In the figure above if l1||l2, what is the value of x? [#permalink]
### Show Tags
15 Mar 2018, 23:53
I got A - 70°, you can complete the top triangle as 30°, 40° and 110° because l1 and l2 are parallel, x = 180° - 110° which is 70°
Senior SC Moderator
Joined: 22 May 2016
Posts: 3648
Re: In the figure above if l1||l2, what is the value of x? [#permalink]
### Show Tags | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9566342012360932,
"lm_q1q2_score": 0.8545703240037562,
"lm_q2_score": 0.8933093996634686,
"openwebmath_perplexity": 6448.63503853537,
"openwebmath_score": 0.8127297759056091,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-figure-above-if-l1-l2-what-is-the-value-of-x-261473.html?sort_by_oldest=true"
} |
### Show Tags
16 Mar 2018, 08:35
Bunuel wrote:
In the figure above if $$l_1||l_2$$, what is the value of x?
A. 70
B. 100
C. 110
D. 150
E. 170
Attachment:
2018-03-16_1010ed.png [ 10.8 KiB | Viewed 1021 times ]
A couple of other ways to approach . . .
$$l_1$$ and $$l_2$$ are cut by a transversal
Alternate interior angles are congruent.
Lower right angle = $$40°$$
Other angle of lower triangle, given = $$30°$$
1) $$x$$ = exterior angle
Exterior angle $$x$$ = sum of opposite interior angles of lower triangle
Exterior angle $$x$$ $$= (30° + 40°) =$$ $$70°$$
OR
2) Straight line: Third angle of lower ∆ + $$x = 180°$$
Interior angles of a triangle sum to 180°
Lower ∆'s third angle = $$110°$$
($$(180° - 30° - 40°) = 110°$$ = third angle))
Third angle lies on a straight line with $$x$$
$$110° + x° = 180°$$
$$x = 70°$$
_________________
SC Butler has resumed! Get two SC questions to practice, whose links you can find by date, here.
Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it's the only thing that ever has -- Margaret Mead
Senior Manager
Joined: 22 Feb 2018
Posts: 416
Re: In the figure above if l1||l2, what is the value of x? [#permalink]
### Show Tags
18 Mar 2018, 10:40
Bunuel wrote:
In the figure above if $$l_1||l_2$$, what is the value of x?
A. 70
B. 100
C. 110
D. 150
E. 170
Attachment:
The attachment 2018-03-16_1010.png is no longer available
Drawing a additional parallel line l1 and l2 and using property of equal alternate angle between 2 parallel line
we get x= 30 +40
x=70
Attachments
Capture.PNG [ 6.93 KiB | Viewed 926 times ]
_________________
Good, good Let the kudos flow through you
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8345
Location: United States (CA)
Re: In the figure above if l1||l2, what is the value of x? [#permalink]
### Show Tags
20 Oct 2019, 10:06
Bunuel wrote: | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9566342012360932,
"lm_q1q2_score": 0.8545703240037562,
"lm_q2_score": 0.8933093996634686,
"openwebmath_perplexity": 6448.63503853537,
"openwebmath_score": 0.8127297759056091,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-figure-above-if-l1-l2-what-is-the-value-of-x-261473.html?sort_by_oldest=true"
} |
### Show Tags
20 Oct 2019, 10:06
Bunuel wrote:
In the figure above if $$l_1||l_2$$, what is the value of x?
A. 70
B. 100
C. 110
D. 150
E. 170
Attachment:
2018-03-16_1010.png
Draw a line through the vertex of angle x that is parallel to both l1 and l2. We will see that angle x is the sum of the two given angles of 40 degrees and 30 degrees. Thus angle x = 70 degrees.
_________________
# Scott Woodbury-Stewart
Founder and CEO
Scott@TargetTestPrep.com
122 Reviews
5-star rated online GMAT quant
self study course
See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews
If you find one of my posts helpful, please take a moment to click on the "Kudos" button.
SVP
Joined: 03 Jun 2019
Posts: 1834
Location: India
Re: In the figure above if l1||l2, what is the value of x? [#permalink]
### Show Tags
20 Oct 2019, 10:11
Bunuel wrote:
In the figure above if $$l_1||l_2$$, what is the value of x?
A. 70
B. 100
C. 110
D. 150
E. 170
Attachment:
2018-03-16_1010.png
x = 180 - (180 - 40 - 30) = 70
IMO A
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."
Please provide kudos if you like my post. Kudos encourage active discussions.
My GMAT Resources: -
Efficient Learning
All you need to know about GMAT quant
Tele: +91-11-40396815
Mobile : +91-9910661622
E-mail : kinshook.chaturvedi@gmail.com
Re: In the figure above if l1||l2, what is the value of x? [#permalink] 20 Oct 2019, 10:11
Display posts from previous: Sort by | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9566342012360932,
"lm_q1q2_score": 0.8545703240037562,
"lm_q2_score": 0.8933093996634686,
"openwebmath_perplexity": 6448.63503853537,
"openwebmath_score": 0.8127297759056091,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-figure-above-if-l1-l2-what-is-the-value-of-x-261473.html?sort_by_oldest=true"
} |
GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video
It is currently 20 Jan 2020, 16:45
### 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
# In the xy-plane, the straight-line graphs of the three equations above
Author Message
TAGS:
### Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 60515
In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
Updated on: 13 Nov 2017, 21:24
5
66
00:00
Difficulty:
65% (hard)
Question Stats:
61% (02:04) correct 39% (02:09) wrong based on 1317 sessions
### HideShow timer Statistics
y = ax - 5
y = x + 6
y = 3x + b
In the xy-plane, the straight-line graphs of the three equations above each contain the point (p,r). If a and b are constants, what is the value of b?
(1) a = 2
(2) r = 17
_________________
Originally posted by Bunuel on 23 Jul 2015, 10:47.
Last edited by Bunuel on 13 Nov 2017, 21:24, edited 4 times in total.
Edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 60515
In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
26 Oct 2015, 10:02
3
35
CEO
Joined: 20 Mar 2014
Posts: 2552
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
23 Jul 2015, 11:30
12
8
mcelroytutoring wrote:
DS 136 from OFG 2016 (new question)
y = ax - 5
y = x + 6
y = 3x + b | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9582261215975975,
"lm_q1q2_score": 0.8545566344460556,
"lm_q2_score": 0.8918110404058913,
"openwebmath_perplexity": 2490.5410678915914,
"openwebmath_score": 0.7582874298095703,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-xy-plane-the-straight-line-graphs-of-the-three-equations-above-207707.html"
} |
y = ax - 5
y = x + 6
y = 3x + b
In the xy-plane, the straight-line graphs of the three equations above each contain the point (p,r). If a and b are constants, what is the value of b?
1) a = 2
2) r = 17
Solution provided by : mcelroytutoring
Let's start by substituting the point (p,r) into all equations in place of (x,y) which will make step #2 a bit easier to comprehend but is not necessary to solve the question. Then, let's consider the number of variables left in each equation.
#1: r = ap - 5 (3 variables R,A,P)
#2: r = p + 6 (2 variables R,P)
#3: r = 3p + b (3 variables R,B,P)
1) a = 2 allows us to reduce equation #1 to the variables r and p, which are the same two variables as equation #2. Thus we have simultaneous equations. As soon as we verify that the equations are different, we know that we can solve for both variables. Once we know r and p, we can substitute in equation #3 to solve for b. Sufficient.
2) r = 17 allows us to do the same thing, more or less. It reduces equation #2 to only one variable, allowing us to solve for p. Once we have p (and r), we can use equation #3 to solve for b. Sufficient.
Attachments
Screen Shot 2015-07-23 at 10.36.05 AM.png [ 189.38 KiB | Viewed 23039 times ]
Screen Shot 2015-07-23 at 10.31.01 AM.png [ 30.94 KiB | Viewed 23016 times ]
##### General Discussion
Senior Manager
Joined: 10 Mar 2013
Posts: 459
Location: Germany
Concentration: Finance, Entrepreneurship
Schools: WHU MBA"20 (A\$)
GMAT 1: 580 Q46 V24
GPA: 3.88
WE: Information Technology (Consulting)
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9582261215975975,
"lm_q1q2_score": 0.8545566344460556,
"lm_q2_score": 0.8918110404058913,
"openwebmath_perplexity": 2490.5410678915914,
"openwebmath_score": 0.7582874298095703,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-xy-plane-the-straight-line-graphs-of-the-three-equations-above-207707.html"
} |
### Show Tags
24 Sep 2015, 01:53
5
6
if each of the 3 equations contains points (p,r) this means that they intersect in that point
1. a=2
Find the intercept Intercept for three simultaneous equations
y=2x-5
y=x+6
y=3x+b
Let's use the first 2 equations: plug y=x+6 in the secod equation
x+6=2x-5 -> x=11, y=17 we can use the values to calulate b in the 3rd equation
17=33+b -> b=-16 SUFFICIENT
2. Here we have directly the value for Y, let's plug it in the 2nd equation
y=x+6 -> 17=x+6 -> x=11, y=17; We can plug these values in the 3rd equation and find b as we did above
17=33+b -> b=-16 SUFFICIENT
Answer (D) Most important point is here to catch the hint about intersection of 3 lines at one point
Intern
Status: GMAT1:520 Q44 V18
Joined: 03 Sep 2015
Posts: 10
Location: United States
Concentration: Strategy, Technology
WE: Information Technology (Computer Software)
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
26 Oct 2015, 10:50
4
I think it's D.
Keeping point (p,r) in all the equations we get :
p = ar -5 -----(1)
p = r + 6 ------(2)
p = 3r + b ------(3)
Now consider (1) if a = 2 from (1) and (2) we get
r = 11 , p=17 and putting in (3) we can get b.
Similarly for (2) we can get the values for r and p and hence can get the value for b.
So both statements individually are correct to answer the question.
Manager
Joined: 13 Apr 2015
Posts: 73
Concentration: General Management, Strategy
GMAT 1: 620 Q47 V28
GPA: 3.25
WE: Project Management (Energy and Utilities)
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
26 Oct 2015, 19:14
2
m2k wrote:
I think it's D.
Keeping point (p,r) in all the equations we get :
p = ar -5 -----(1)
p = r + 6 ------(2)
p = 3r + b ------(3)
Now consider (1) if a = 2 from (1) and (2) we get
r = 11 , p=17 and putting in (3) we can get b.
Similarly for (2) we can get the values for r and p and hence can get the value for b. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9582261215975975,
"lm_q1q2_score": 0.8545566344460556,
"lm_q2_score": 0.8918110404058913,
"openwebmath_perplexity": 2490.5410678915914,
"openwebmath_score": 0.7582874298095703,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-xy-plane-the-straight-line-graphs-of-the-three-equations-above-207707.html"
} |
Similarly for (2) we can get the values for r and p and hence can get the value for b.
So both statements individually are correct to answer the question.
Approach if right but the values you derived are wrong. According to me r = 17 and that is what stmt b also states.
Manager
Joined: 21 Sep 2015
Posts: 73
Location: India
GMAT 1: 730 Q48 V42
GMAT 2: 750 Q50 V41
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
10 Jun 2016, 11:23
3
y = ax - 5 ... eq 1
y = x + 6 ... eq 2
y = 3x + b ...eq 3
Total of 4 variables are present.
Statement 1 : a = 2
Insert in eq 1
We have y = 2x -5 and y = x+6
Solving we get x = 11 and y = 17
Substitute in eq 3 and we get value of b
Statement 2: r=17
This means the y co- ordinate is 17
Substitute in eq 2 we get x as 11
Again can find value of b from equation 3
Hence D
Intern
Joined: 20 Jun 2013
Posts: 48
Location: India
Concentration: Economics, Finance
GMAT 1: 430 Q39 V25
GPA: 3.5
WE: Information Technology (Other)
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
18 Jan 2017, 15:18
we used information from both 1 and 2 then how can the answer be D... should it not be C... some one kindly clarify.......... clearly am a zero in ds and that too a big one
Director
Joined: 14 Nov 2014
Posts: 585
Location: India
GMAT 1: 700 Q50 V34
GPA: 3.76
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
18 Jan 2017, 20:18
1
y = ax - 5---------1
y = x + 6---------2
y = 3x + b--------3
In the xy-plane, the straight-line graphs of the three equations above each contain the point (p,r). If a and b are constants, what is the value of b?
(1) a = 2
(2) r = 17 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9582261215975975,
"lm_q1q2_score": 0.8545566344460556,
"lm_q2_score": 0.8918110404058913,
"openwebmath_perplexity": 2490.5410678915914,
"openwebmath_score": 0.7582874298095703,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-xy-plane-the-straight-line-graphs-of-the-three-equations-above-207707.html"
} |
(1) a = 2
(2) r = 17
All three line intersect each other at common point (p,r).
1. given a = 2
putting in equation 1 .= y=2x-5
equating 1(after replacing value of a) and 2 we will get value of (p,r)
putting (p,r) in equation 3 we will get value for b---suff..
2 given r = 17.
putting in equation 2 we will get value of x.i'e p.
Now as we know common point of intersection ,putting p,r in equation 3 , we will get value of b
Intern
Joined: 20 Jun 2013
Posts: 48
Location: India
Concentration: Economics, Finance
GMAT 1: 430 Q39 V25
GPA: 3.5
WE: Information Technology (Other)
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
22 Jan 2017, 15:51
thanks sobby for your response... highly appreciated...
Manager
Joined: 17 Feb 2014
Posts: 98
Location: United States (CA)
GMAT 1: 700 Q49 V35
GMAT 2: 740 Q48 V42
WE: Programming (Computer Software)
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
16 Mar 2017, 08:58
1
(eq1) $$y = ax - 5$$
(eq2) $$y = x + 6$$
(eq3) $$y = 3x + b$$
In the xy-plane, the straight-line graphs of the three equations above each contain the point (p, r). If a and b are constants, what is the value of b?
1) $$a = 2$$
2) $$r = 17$$
Solution:
1) $$a = 2$$
- Putting the value of a in eq1, we get: $$y = 2x - 5$$
- At this point you can solve for (x, y), plug (x, y) in (eq3) and solve for (b) [though this approach might take few seconds]
(alternatively, faster method)
- you can skip solving for (x, y) and deduce that given 3 equations and 3 unknowns (since a is given in statement 1) we can solve for all of them (including b), since the lines are have different slopes i.e. different lines. Hence, we can get single value of 'b', proving the condition SUFFICIENT.
NOTE: 3 equations and 3 unknowns does not ALWAYS mean that we can find 3 unknown. We have to make sure that 2 of them or all of them are not the same line. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9582261215975975,
"lm_q1q2_score": 0.8545566344460556,
"lm_q2_score": 0.8918110404058913,
"openwebmath_perplexity": 2490.5410678915914,
"openwebmath_score": 0.7582874298095703,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-xy-plane-the-straight-line-graphs-of-the-three-equations-above-207707.html"
} |
2) $$r = 17$$
- Since, point (p, r) lie on all the line, we can plugin the point in above equation
$$r = ap - 5 => 17 = ap - 5$$
$$r = p + 6 => 17 = p + 6$$
$$r = 3p + b => 17 = 3p + b$$
- Again, we do not need to solve for all the variables and just recognize that the above equations will lead to single value of b. Hence, SUFFICIENT.
Current Student
Joined: 25 Feb 2017
Posts: 34
Location: Korea, Republic of
Schools: LBS '19 (A)
GMAT 1: 720 Q50 V38
GPA: 3.67
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
30 Apr 2017, 23:51
y = ax - 5
y = x + 6
y = 3x + b
In the xy-plane, the straight-line graphs of the three equations above each contain the point (p,r). If a and b are constants, what is the value of b?
1) a = 2
2) r = 17
My 2 cents.
It is important to realize from the Question stem that the 3 equations intersect as (p,r).
For 1), as we know a =2, we can equate the first and second equation to get the value of x, and then use that value of x to find value of y and the find value of b.
For 2), similarly, use r = 17 (which is value of y) to find value of x using the second equation. And then plug it back to the third equation.
So D.
Intern
Joined: 27 Apr 2015
Posts: 8
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
10 Sep 2017, 19:04
1
Would it be correct to simply say that we have 4 variables with 3 equations so eliminating any one variable gets us to three equations and three variables and is therefore sufficient? Is that logic sound?
Intern
Joined: 11 Sep 2017
Posts: 36
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
13 Nov 2017, 20:21
2
what is the significance of the the line "In the xy-plane, the straight-line graphs of the three equations above each contain the point (p,r)", | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9582261215975975,
"lm_q1q2_score": 0.8545566344460556,
"lm_q2_score": 0.8918110404058913,
"openwebmath_perplexity": 2490.5410678915914,
"openwebmath_score": 0.7582874298095703,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-xy-plane-the-straight-line-graphs-of-the-three-equations-above-207707.html"
} |
i solved the problem, but wud have did the same even if they didnt provide line above . as question has 3 equations with 4 variables
Retired Moderator
Joined: 17 Jun 2016
Posts: 498
Location: India
GMAT 1: 720 Q49 V39
GMAT 2: 710 Q50 V37
GPA: 3.65
WE: Engineering (Energy and Utilities)
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
13 Nov 2017, 22:31
2
Cheryn wrote:
what is the significance of the the line "In the xy-plane, the straight-line graphs of the three equations above each contain the point (p,r)",
i solved the problem, but wud have did the same even if they didnt provide line above . as question has 3 equations with 4 variables
The highlighted statement in effect says that all these 3 lines meet each other at one point and so there is a single value of (x,y) that satisfies these 3 equations. It is only because of this highlighted statement you can solve this set of equations for a unique value of x,y, a and b.
_________________
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15951
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
11 Dec 2017, 14:28
3
1
Hi All,
We're given the equations for 3 lines (and those equations are based on 4 unknowns: 2 variables and the 2 'constants' A and B):
Y = (A)(X) - 5
Y = X + 6
Y = 3X + B
We're told that the three lines all cross at one point on a graph (p,r). We're asked for the value of B. While this question looks complex, it's actually built around a 'system' math "shortcut" - meaning that since we have 3 unique equations and 4 unknowns, we just need one more unique equation (with one or more of those unknowns) and we can solve for ALL of the unknowns:
1) A =2
With this information, we now have a 4th equation, so we CAN solve for B.
Fact 1 is SUFFICIENT
2) R = 17 | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9582261215975975,
"lm_q1q2_score": 0.8545566344460556,
"lm_q2_score": 0.8918110404058913,
"openwebmath_perplexity": 2490.5410678915914,
"openwebmath_score": 0.7582874298095703,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-xy-plane-the-straight-line-graphs-of-the-three-equations-above-207707.html"
} |
2) R = 17
This information tell us the x co-ordinate where all three lines will meet, so it's the equivalent of having X=17 to work with. This 4th equation also allows us to solve for B.
Fact 2 is SUFFICIENT
GMAT assassins aren't born, they're made,
Rich
_________________
Contact Rich at: Rich.C@empowergmat.com
The Course Used By GMAT Club Moderators To Earn 750+
souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
Target Test Prep Representative
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2806
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
02 Jan 2018, 11:00
1
1
Bunuel wrote:
y = ax - 5
y = x + 6
y = 3x + b
In the xy-plane, the straight-line graphs of the three equations above each contain the point (p,r). If a and b are constants, what is the value of b?
(1) a = 2
(2) r = 17
We can begin by substituting p and r for x and y, respectively, in the three given equations.
1) r = ap – 5
2) r = p + 6
3) r = 3p + b
Statement One Alone:
a = 2
We can substitute 2 for a in the equation r = ap – 5. Thus, we have:
r = 2p – 5
Next we can set equations 1 and 2 equal to each other.
2p – 5 = p + 6
p = 11
Since p = 11, we see that r = 11 + 6 = 17
Finally, we can substitute 11 for p and 17 for r in equation 3. This gives us:
17 = 3(11) + b
17 = 33 + b
-16 = b
Statement one alone is sufficient to answer the question.
Statement Two Alone:
r = 17
We can substitute r into all three equations and we have:
1) 17 = ap – 5
2) 17 = p + 6
3) 17 = 3p + b
We see that p = 11. Now we can substitute 11 for p in equation 3 to determine a value for b.
17 = 3(11) + b
-16 = b
Statement two alone is also sufficient to answer the question.
_________________
# Jeffrey Miller
Jeff@TargetTestPrep.com
181 Reviews
5-star rated online GMAT quant
self study course
See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9582261215975975,
"lm_q1q2_score": 0.8545566344460556,
"lm_q2_score": 0.8918110404058913,
"openwebmath_perplexity": 2490.5410678915914,
"openwebmath_score": 0.7582874298095703,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-xy-plane-the-straight-line-graphs-of-the-three-equations-above-207707.html"
} |
See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews
If you find one of my posts helpful, please take a moment to click on the "Kudos" button.
Manager
Joined: 29 Dec 2018
Posts: 82
Location: India
WE: Marketing (Real Estate)
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
20 Jul 2019, 23:56
Video solution for the same
https://gmatquantum.com/official-guides ... ial-guide/
_________________
Keep your eyes on the prize: 750
Intern
Joined: 15 Aug 2018
Posts: 5
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink]
### Show Tags
17 Aug 2019, 23:34
Bunuel wrote:
y = ax - 5
y = x + 6
y = 3x + b
In the xy-plane, the straight-line graphs of the three equations above each contain the point (p,r). If a and b are constants, what is the value of b?
(1) a = 2
(2) r = 17
DS07713
Re: In the xy-plane, the straight-line graphs of the three equations above [#permalink] 17 Aug 2019, 23:34
Display posts from previous: Sort by | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9582261215975975,
"lm_q1q2_score": 0.8545566344460556,
"lm_q2_score": 0.8918110404058913,
"openwebmath_perplexity": 2490.5410678915914,
"openwebmath_score": 0.7582874298095703,
"tags": null,
"url": "https://gmatclub.com/forum/in-the-xy-plane-the-straight-line-graphs-of-the-three-equations-above-207707.html"
} |
# Resolving $\sec{x}\left(\sin^3x + \sin x \cos^2x\right)=\tan{x}$
Going steadily through my book, I found this exercise to resolve
$$\sec{x}\left(\sin^3x + \sin x \cos^2x\right)=\tan{x}$$
Here's how I resolve it ($LHS$) and again bear with me as I truly reverting to a feeling of vulnerability, like a child actually
As $\sec x$ is equal to $\frac{1}{\cos x}$
$$\frac{(\sin^3x+\sin x\cos^2x)}{\cos x}$$
I'm factorizing one $\sin x$
$$\frac{\sin x(\sin^2x+\cos^2x)}{\cos x} = \frac{\sin x(1)}{\cos x} = \tan x$$
That seems to work otherwise I completly messed this up
Reading the book's solution, I have something different...
\begin{align*} LHS&=\frac{\sin^3x}{\cos x}+ \sin x \cos x \\[4pt] &=\frac{\sin x}{\cos x}-\frac{\sin x\cos^2x}{\cos x}+\sin x\cos x\\[4pt] &= \tan x\end{align*}
What did I miss?
• They messed this up. – Yves Daoust May 13 '16 at 10:00
• Shorter: multiplying both members by $\cot(x)$ yields $\sin^2(x)+\cos^2(x)=1$. – Yves Daoust May 13 '16 at 10:04
• Hi @YvesDaoust you see this is where dogma starts. My way is better than your way and at the end, you are disgusting a generation of people but for what purpose...? – Andy K May 13 '16 at 10:15
• There is absolutely no intent to disgust anyone. You seem to miss that my first remark was an compliment to you. There is beauty in making things short. – Yves Daoust May 13 '16 at 10:22
• @YvesDaoust I understood correctly what you said. I was more empathetic about these 2 chaps who went up to create a complicate solution for what it was. They could have done something simplier and also mentionned that, there were other ways (pluralist views) to solve that problem. As we said J'avais bien compris :) – Andy K May 13 '16 at 10:24 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924802053235,
"lm_q1q2_score": 0.8545240577881882,
"lm_q2_score": 0.8791467785920306,
"openwebmath_perplexity": 2058.5377104889412,
"openwebmath_score": 0.8126027584075928,
"tags": null,
"url": "https://math.stackexchange.com/questions/1783555/resolving-secx-left-sin3x-sin-x-cos2x-right-tanx"
} |
So the book does about the same thing as you but in a different order (your solution is hence correct too). By the trigonometric one we get $\sin^2 x= (1-\cos^2 x)$. Thus $$\frac{\sin^3 x}{\cos x}=\frac{\sin x (\sin^2 x)}{\cos x}= \frac{\sin x(1-\cos^2 x)}{\cos x} = \frac{\sin x-\sin x\cos^2 x}{\cos x}= \frac{\sin x}{\cos x} -\frac{\sin x\cos ^2 x}{\cos x}$$ Thus $$LHS =\frac{\sin^3 x}{\cos x}+\sin x\cos x= \frac{\sin x}{\cos x} -\frac{\sin x\cos ^2 x}{\cos x} +\sin x\cos x = \frac{\sin x}{\cos x}=\tan x$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924802053235,
"lm_q1q2_score": 0.8545240577881882,
"lm_q2_score": 0.8791467785920306,
"openwebmath_perplexity": 2058.5377104889412,
"openwebmath_score": 0.8126027584075928,
"tags": null,
"url": "https://math.stackexchange.com/questions/1783555/resolving-secx-left-sin3x-sin-x-cos2x-right-tanx"
} |
# Given $S_n = \sum \dots$ and $a_n = \sum \dots$ prove that $a_n = S_n + {1\over n\cdot n!}$
I'm trying to solve the following problem:
Let: \begin{align} S_n &= 2 + {1\over2!} + {1\over3!} + {1\over 4!} + \dots + {1\over n!} \\ a_n &= 3 - {1\over 1\cdot2\cdot2!} - {1\over 2\cdot3\cdot3!} - \dots - {1\over (n-1)\cdot n\cdot n!} \end{align} Prove that: $$a_n = S_n + {1\over n\cdot n!}$$
My thought on this are:
Manipulating the sums here would become pretty hard so I decided to use another approach. I've tried to look what the first terms of each sum are going to be in order to observe a pattern:
\begin{align} S_1 &= 2 \\ S_2 &= 2 + {1\over 2!} = S_1 + {1\over 2!} \\ S_3 &= 2 + {1\over 2!} + {1\over 3!} = S_2 + {1\over 3!}\\ &\vdots \\ S_{n+1} &= S_n + {1\over (n+1)!} \end{align}
A similar thing is done to $$a_n$$:
$$a_{n+1} = a_n - {1\over n\cdot (n+1) \cdot (n+1)!}$$
So potentially if we could find closed forms of both $$S_n$$ and $$a_n$$ it would become easier to reason about them.
By the way, I've observed both of the sums approach $$e$$ at infinity but from different sides. $$S_n$$ is starting from $$2$$ adding more terms as $$n$$ grows, while $$a_n$$ starts from 3 and decreases the sum as $$n$$ grows. So that will be our initial conditions for both recurrences:
$$S_1 = 2 \\ a_1 = 3$$
The problem just reduced to finding closed forms of the recurrences. Both of them are non-homogenous and here is where I got stuck. Solving for homogenous part is easy since roots of both characteristic equations $$\lambda_{a_n} = \lambda_{S_n} = 1$$.
I couldn't find a particular solution for them. I've tried yet another approach expressing $$S_{n+1}$$ and $$S_{n+2}$$ trying to get rid of the $$1\over (n+1)!$$ or at least change its form so the guess for particular solution is more obvious.
My questions are: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.85452405234596,
"lm_q2_score": 0.8791467722591728,
"openwebmath_perplexity": 530.4968236603773,
"openwebmath_score": 0.9999663829803467,
"tags": null,
"url": "https://math.stackexchange.com/questions/2943615/given-s-n-sum-dots-and-a-n-sum-dots-prove-that-a-n-s-n-1-over"
} |
My questions are:
1. Is it even a valid approach to solve the problem?
2. If so then how could I find a closed form for the recurrences? Especially i'm interested in finding a particular solution. (Yet a complete flow is very appreciated and encourages learning by example)
3. Not sure how to phrase that better, but does there exist a table of "guesses" for particular solution of the recurrences in some form of $$x_{n+1} = x_n + F(n)$$. Like if the "free term" is a constant say $$F(n) = 2$$ then the guess for particular solution is some constant $$B$$.
Please excuse me if there is some vagueness or inaccuracy in the terminology, English is not my native language and I have almost no background in Maths.
• It seems unlikely to find a simple closed form, because, as you noticed, the limit converges to $e$. – user600464 Oct 5 '18 at 18:59
• @user600464 is that because $e$ is irrational and that is why no "simple" formula for an irrational number may be obtained? – roman Oct 5 '18 at 19:02
• There is actually a simple formula, the limit $\frac{n^n}{(n- 1)^n}$. However, to compare to this sums, we would need many terms – user600464 Oct 5 '18 at 19:34 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.85452405234596,
"lm_q2_score": 0.8791467722591728,
"openwebmath_perplexity": 530.4968236603773,
"openwebmath_score": 0.9999663829803467,
"tags": null,
"url": "https://math.stackexchange.com/questions/2943615/given-s-n-sum-dots-and-a-n-sum-dots-prove-that-a-n-s-n-1-over"
} |
$$a_2 - S_2 = 3- 1/4 - (1 + 1/1! + 1/2!)$$ [I think there is some typo in your expression of $$S_n$$] $$= 1/4 = 1/(2 \cdot 2!)]$$. Suppose the equation holds for $$n$$. Then for $$n+1$$, \begin{align*} a_{n+1} - S_{n+1} &= a_n - S_n - \frac 1 {n(n+1) \cdot (n+1)!} - \frac 1{(n+1)!} \\ &= \frac 1 {n!n }-\frac 1 {n(n+1) \cdot (n+1)!} - \frac 1{(n+1)!} \\ &= \frac 1 {(n+1)! (n+1)} \left( \frac {(n+1)^2}n -\frac 1n -(n+1) \right)\\ &= \frac 1 {(n+1)! (n+1)}. \end{align*} Thus the equation holds for all $$n \geqslant 2$$ by induction principle.
\begin{align} \color{red}{a_{n}-S_{n}} &=\left(3-\sum_{k=2}^{n}\frac{1}{(n-1)\cdot n\cdot n!}\right)-\left(2+\sum_{k=2}^{n}\frac{1}{n!}\right) \\[2mm] &=1-\sum_{k=2}^{n}\frac{1}{n!}\left(1+\frac{1}{(n-1)\cdot n}\right) =1-\sum_{k=2}^{n}\frac{1}{n!}\left(1+\frac{1}{n-1}-\frac{1}{n}\right) \\[2mm] &=1-\sum_{k=2}^{n}\frac{1}{n!}\left(\frac{n}{n-1}-\frac{1}{n}\right) =1-\sum_{k=2}^{n}\left(\frac{1}{(n-1)\cdot (n-1)!}-\frac{1}{n\cdot n!}\right) \\[2mm] &=1-\frac{1}{1\cdot1!}+\frac{1}{2\cdot2!}-\frac{1}{2\cdot2!}+\frac{1}{3\cdot3!}-\cdots+\frac{1}{n\cdot n!}=\color{red}{\frac{1}{n\cdot n!}}\quad\left\{\text{telescoping}\right\} \end{align} | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.85452405234596,
"lm_q2_score": 0.8791467722591728,
"openwebmath_perplexity": 530.4968236603773,
"openwebmath_score": 0.9999663829803467,
"tags": null,
"url": "https://math.stackexchange.com/questions/2943615/given-s-n-sum-dots-and-a-n-sum-dots-prove-that-a-n-s-n-1-over"
} |
# In proof writing, is it mathematically sound to prove uniqueness before proving existence?
As stated in the title, I'd like to find out is whether or not it is always mathematically sound to prove the uniqueness of something before proving the existence of said something.
I am still relatively new to proofs and, in this light, I was intrigued by an observation that some authors choose to prove uniqueness before existence. I understand that some do this because, allegedly, "it is easier to prove the uniqueness part first". And in an exam setting, it's better to tackle the easier parts of a task.
However, as I see it, if done this way, aren't we essentially proving a proposition about something that, at that point, we have not yet guaranteed exists? (Thus, might not even exist at all). I understand that in most cases the existence is proved immediately after the proof of uniqueness but, generally, doesn't uniqueness depend on existence?
My example will be derived from page $$40$$ of The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Raffi Grinberg:
Theorem 5.8. (Existence of Roots in $$\mathbb{R}$$)
Every positive real number has a unique positive $$n$$th root, for any $$n \in \mathbb{N}$$.
In symbols:
$$∀x ∈ R$$ with $$x > 0, ∀n ∈ N, ∃y ∈ R$$ unique, such that $$y > 0$$ and $$y^n = x.$$
Note that even-numbered roots $$(\sqrt{x},\sqrt[4]x,\sqrt[6]x, etc.)$$ signify two numbers in $$\mathbb{R}$$, namely, $$+y$$ and $$−y$$. The theorem asserts that there is one and only one positive real root.
Proof. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.8545240477293421,
"lm_q2_score": 0.8791467675095292,
"openwebmath_perplexity": 265.5858964596029,
"openwebmath_score": 0.8450634479522705,
"tags": null,
"url": "https://math.stackexchange.com/questions/3061903/in-proof-writing-is-it-mathematically-sound-to-prove-uniqueness-before-proving"
} |
Proof.
The uniqueness of $$y$$ is the easiest part of the proof, so we’ll start there. For any two different positive real numbers, the fact that they are different means one must be greater than the other. If there were two positive real roots $$y_1$$ and $$y_2$$ such that $$y_1^n = x$$ and $$y_2^n = x$$ we would have $$0 < y_1 < y_2$$. But then $$0 < y_1^n < y_n^2$$ , meaning $$0 < x < x$$, which is impossible. Thus only one positive real root can exist.
To prove that $$\sqrt[n]x$$ exists in $$\mathbb{R}$$, let’s first figure out our game plan and then write it up formally $$...$$
While the proof is totally clear, I'd really like feedback on this format. As an aspiring mathematician, is this a format I can adapt and use in my proofs or perhaps it is not good practice in general? What if, say, I have a conjecture that there exist unique integers possessing some property and this conjecture is something new (i.e. hasn't been proven yet). I try proving they exist but run out of luck. Would it be worth the effort of trying to prove the uniqueness of such integers not knowing they exist? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.8545240477293421,
"lm_q2_score": 0.8791467675095292,
"openwebmath_perplexity": 265.5858964596029,
"openwebmath_score": 0.8450634479522705,
"tags": null,
"url": "https://math.stackexchange.com/questions/3061903/in-proof-writing-is-it-mathematically-sound-to-prove-uniqueness-before-proving"
} |
• Yeah, that's perfectly fine. It doesn't matter: you're proving two things: 1) There is at least one of these; and 2) there aren't two different ones. There's no problem doing them whatever way round you like. There are a fair few results of the form "there are at most $n$ things with property X", and this is just the special case where $n = 1$. – user3482749 Jan 4 at 17:57
• Sure. Naturally, if you show uniqueness first what you have actually shown at that point is that there is at most one object with the desired property. – Andrés E. Caicedo Jan 4 at 17:57
• Even in the case where authors first prove uniqueness, they usually mean, under the assumption that there exists such an object, it is unique. – Dietrich Burde Jan 4 at 18:01
• I prefer existence before uniqueness, but that is just a preference and not a strong recommendation. – David G. Stork Jan 4 at 18:02
• An explanation: I prefer existence before uniqueness because in general existence is simpler. Imagine the awkward case where you prove uniqueness first, then find you cannot prove existence! THAT is weird. – David G. Stork Jan 4 at 18:16
I can see the unease: if what you're proving unique doesn't exist, might that invalidate whatever manipulations you've done with it?
This is similar to the situation with proof by contradiction, where you don't just reason about something that might not be true: you reason about it expecting it to be untrue, but making all the logical steps anyway.
In either case, you end up showing "If $$A$$ then $$B$$"—whether it's "If $$x$$ exists then it's unique" or "If $$x$$ exists then it has contradictory properties". In one case we go on to apply $$B$$ to $$x$$ once we know it exists, and in the other we use the impossibility of $$B$$ to disprove the existence of $$x$$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.8545240477293421,
"lm_q2_score": 0.8791467675095292,
"openwebmath_perplexity": 265.5858964596029,
"openwebmath_score": 0.8450634479522705,
"tags": null,
"url": "https://math.stackexchange.com/questions/3061903/in-proof-writing-is-it-mathematically-sound-to-prove-uniqueness-before-proving"
} |
A separate thought: proofs are presented as a sequence of steps and inferences because as humans, we process written information sequentially. But what's behind a mathematical theorem is more like a network of theorems and logical steps, some dependent on others, which together imply the theorem. We can't take them all in simultaneously, so we arrange them in a suitable order to let us see the logic most clearly. It's a matter of convenience.
If $$X$$ and $$Y$$ are both true and together imply $$Z$$, it doesn't really matter which order the various parts of that are proved in—all that matters from a validity point of view is that they're all proved and brought together.
• Thanks! Your first paragraph alone asks my question even better than I could in my long post! – E.Nole Jan 7 at 2:56
This is a perfectly reasonable argument. The author even tells you he's doing it this way. He then uses the argument to allow the reader to become comfortable with the assertion and the notation.
There are other reasons for starting with uniqueness. Often if you know there's just one of something then it might be easier to search for it.
When faced with a problem with an easy part and a hard part the argument for starting with the easy part is that it's a good warmup. That's what I usually do, and what I like to see in things I read.
The argument for starting with the hard part is that if you fail there you haven't wasted time on the easy part. For example, proving that there's at most one positive rational number whose square is $$2$$ is not going to tell you very much about the rational square root of $$2$$ that does not exist.
Starting with the hard part, the bottleneck, is often the preferred strategy when writing software.
PS The author is wrong to say that when $$x >0$$, $$\sqrt{x}$$ signifies two two numbers. $$4$$ does have two real square roots, but the mathematical convention is that only $$2 = \sqrt{4}$$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.8545240477293421,
"lm_q2_score": 0.8791467675095292,
"openwebmath_perplexity": 265.5858964596029,
"openwebmath_score": 0.8450634479522705,
"tags": null,
"url": "https://math.stackexchange.com/questions/3061903/in-proof-writing-is-it-mathematically-sound-to-prove-uniqueness-before-proving"
} |
• Nice typo near the end: "two two" (Shame to fix it really!) – timtfj Jan 4 at 19:15
• @timtfj I will leave it thanks. – Ethan Bolker Jan 4 at 19:33 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924810166349,
"lm_q1q2_score": 0.8545240477293421,
"lm_q2_score": 0.8791467675095292,
"openwebmath_perplexity": 265.5858964596029,
"openwebmath_score": 0.8450634479522705,
"tags": null,
"url": "https://math.stackexchange.com/questions/3061903/in-proof-writing-is-it-mathematically-sound-to-prove-uniqueness-before-proving"
} |
# Ideal Generated by a Finite Number of Polynomials?
Background
I have been confused about a particular definition in the textbook for my abstract algebra class, Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea. It is frustrating because I feel that I have a partial grasp on what the definition is trying to say, but when it comes down to it I simply find my confused. So, instead of suffering by myself for any longer with this definition, I have decided to turn to you lovely folks to help me understand this seemingly simple concept.
What I Understand
First off, I understand (at least in the context of this book) what an ideal is. The definition my book gives for an ideal is
Definition. A subset $$I\subseteq k[x_1,..., x_n]$$ is an ideal if it satisfies:
(i) $$0\in I$$.
(ii) If $$f,g\in I$$, then $$f+g\in I$$.
(iii) If $$f\in I$$ and $$h\in k[x_1,...,x_n]$$, then $$hf\in I$$.
I find this to be a simple, easy to understand definition. My problem, however, arises a few lines later when they define an ideal generated by a finite number of polynomials.
What I Don't Understand
And now, I give you the definition that has been causing me an incredible amount of confusion and frustration.
Definition. Let $$f_1,...,f_s$$ be polynomials in $$k[x_1,...,x_n]$$. Then we set $$\langle f_1,...,f_s \rangle=\Big\lbrace \sum_{i=1}^s h_if_i \ | \ h_1,...,h_s\in k[x_1,...,x_n] \Big\rbrace.$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992476960077,
"lm_q1q2_score": 0.8545240426241597,
"lm_q2_score": 0.8791467659263148,
"openwebmath_perplexity": 151.4112291041073,
"openwebmath_score": 0.8639252185821533,
"tags": null,
"url": "https://math.stackexchange.com/questions/2499234/ideal-generated-by-a-finite-number-of-polynomials"
} |
I know what you are thinking. How does he not understand this? I wish I knew the answer to that question, but in the meantime, can someone please help me visualize what this set looks like? I understand that $$\langle f_1,...,f_s \rangle$$ is an ideal, but I don't understand its structure, if that makes sense. In other words, I can't visualize this definition in a way that makes sense to me. The authors do make a slightly helpful note, saying that "we can think of $$\langle f_1,...,f_s \rangle$$ as consisting of all 'polynomial consequences' of the equations $$f_1=f_2=...=f_s=0$$."
To elaborate a little more on my confusion, what I'm asking for is a less "compact" definition. When I read this definition, for whatever reason the only thing I can come up with is $$f_1h_1+f_2h_2+...+f_sh_s.$$ But that doesn't make sense, because $$\langle f_1,...,f_s \rangle$$ is supposed to generate a set, not just a single polynomial.
As always, thank you all for your time. If you find this to be a stupid or silly question, then I'm sorry to have disappointed you -- I'm a slow learner, and I get hung up on stupid things sometimes.
Oh, and Happy Halloween! | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992476960077,
"lm_q1q2_score": 0.8545240426241597,
"lm_q2_score": 0.8791467659263148,
"openwebmath_perplexity": 151.4112291041073,
"openwebmath_score": 0.8639252185821533,
"tags": null,
"url": "https://math.stackexchange.com/questions/2499234/ideal-generated-by-a-finite-number-of-polynomials"
} |
Oh, and Happy Halloween!
• Suggestion: Try looking at one or two specific examples, such as: (1) In $k[x,y]$, look at $\langle x^2, y^2 \rangle$. This is equal to the set $\{ x^2 h_1 + y^2 h_2 \mid h_1, h_2 \in k[x,y]\}$. Some elements include $x^2 \cdot 1 + y^2 \cdot 0$, $x^2 \cdot 0 + y^2 \cdot 1$,... $x^2 \cdot (x y^2 - 3 xy) + y^2 \cdot (2x^3 - 5)$, etc. (2) Look at examples in $\mathbb{Z}$, e.g., $\langle 6,4 \rangle = \{ 6 h_1 + 4 h_2 \mid h_1, h_2 \in \mathbb{Z} \}$. Can you prove this is equal to the set of even numbers? – Zach Teitler Nov 1 '17 at 4:55
• Makes alot more sense when you put it like that! Thank you @ZachTeitler! – Thy Art is Math Nov 1 '17 at 5:00
• One more suggestion. Make sure you do Exercise 1.4.2, or at least try to do it. The result in Exercise 1.4.2 gets used constantly, over and over, throughout the rest of the book, including in a lot of exercises. It's the main way to show that two ideals are equal, or that one ideal is contained in another. So it is extremely useful. Also it's good practice with the definition of ideal, and proving a subset relationship. (There are a lot of other good exercises, I just want to highlight that particular one. I made sure to assign it in the class I'm teaching from IVA this semester.) – Zach Teitler Nov 1 '17 at 5:49
• Read the Rings chapter in Algebra by Hungerford. He describes ideals in full generality very well (non-commutative and commutative). – user494247 Nov 11 '17 at 9:57
Would it help to see the see the set with explicit polynomials in place of the $f_1, f_2, \dots, f_n$?
For example, lets look at an explicit example when $n = 2$, so an ideal generated by 2 polynomials. Also, we'll work in a polynomial ring in two variables over $k$, i.e. $k[x,y]$.
Here is the ideal generated by the polynomials $x^2 - 1$, $yx+x$.$$\langle \, x^2 - 1, \, yx+x \, \rangle = \{ \, f\cdot(x^2 - 1) + g\cdot(yx + x) \, | \, f,g \in k[x,y] \, \}.$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992476960077,
"lm_q1q2_score": 0.8545240426241597,
"lm_q2_score": 0.8791467659263148,
"openwebmath_perplexity": 151.4112291041073,
"openwebmath_score": 0.8639252185821533,
"tags": null,
"url": "https://math.stackexchange.com/questions/2499234/ideal-generated-by-a-finite-number-of-polynomials"
} |
So the elements of the set are any polynomial that can be written in the form $f \cdot (x^2 - 1) + g \cdot (yx+x)$. But we can choose $f,g$ to be $\textit{any}$ polynomial we want. For example, we know $x^2 - 1$ itself is in that set because we can choose $f = 1, g = 0$. We also know that the polynomial $x^3-x+yx ^2 + x^2$ is in the set, because we can choose $f = x, g = x$.
If you are familiar with linear algebra you can maybe, in a way, draw a connection between an ideal generated by polynomials and the span of a set of vectors. You can think of it as, the set of all things that can be made from the objects defining it.
• Also, is your class an Algebraic Geometry class or an Abstract Algebra class? I think IVA is a strange book to teach general algebra out of. – Prince M Nov 1 '17 at 7:38
• Sadly I was absolutely terrible at linear algebra but seeing the explicit polynomials made this make so much more sense, so thank you for the help. The class is labeled rather generally as "abstract algebra," so you are probably right that the course should be called "algebraic geometry". – Thy Art is Math Nov 1 '17 at 15:30
• Just a note that the authors of IVA do write in the preface: "For instance, the book could serve as a basis of a second course in undergraduate abstract algebra, but we think that it just as easily could provide a credible alternative to the first course." – J W Jan 3 at 15:16
Perhaps a good way to understand this (or to understand anything, for that matter) is to look at examples. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992476960077,
"lm_q1q2_score": 0.8545240426241597,
"lm_q2_score": 0.8791467659263148,
"openwebmath_perplexity": 151.4112291041073,
"openwebmath_score": 0.8639252185821533,
"tags": null,
"url": "https://math.stackexchange.com/questions/2499234/ideal-generated-by-a-finite-number-of-polynomials"
} |
The definition of ideal makes sense in any ring, so let’s look at $\Bbb Z$ first. Every ideal, you soon persuade yourself, is a set $d\Bbb Z\subset\Bbb Z$, that is, just the set of all multiples of a given number $d$. What if you try to take two numbers and look at $\langle d_1,d_2\rangle\subset\Bbb Z$? Please do this with specific numbers. Like what about $d_1=8$ and $d_2=6$? You want to describe the totality of all numbers writable in the form $8m+6n$. You rapidly see that this is an ideal in the sense of the definition, and you see that the set is equal to $2\Bbb Z$. This is basic number theory.
Now do the same thing with a ring of polynomials, but in just one variable, $R=\Bbb Q[x]$. One proves (you prove it, with Euclidean division of polynomials) that every ideal of $R$ is of form $fR$, where $f$ is a well-chosen polynomial. In fact, for a nonzero ideal $I$, you take $f$ to be a nonzero element of $I$ of least degree. So, what is $\langle f_1,f_2\rangle$, when $f_1$ and $f_2$ are two polynomials in $x$ that are given? Just as with numbers, it’s the ideal $gR$ where $g$ is the greatest common divisor of $f$ and $g$. Convince yourself that $\langle x^3-1,x^2-2x+1\rangle$ is the set of all multiples of $x-1$.
Things are no longer so simple when you have polynomials in more than one indeterminate. But at least you get some insight by looking at the simplest cases, and I hope you see that the set of all possible $h_1f_1+\cdots+h_nf_n$ is an ideal.
• @ThyArtisMath Note that the ring of polynomials in one variable is treated very explicitly and thoroughly in section 1.5 of the textbook Ideals, Varieties, and Algorithms. So, by all means feel free to think about that example and take a shot at proving some things about it, but don't feel like you have to work it out on your own. The textbook goes over all that. – Zach Teitler Nov 1 '17 at 5:40
You were close. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992476960077,
"lm_q1q2_score": 0.8545240426241597,
"lm_q2_score": 0.8791467659263148,
"openwebmath_perplexity": 151.4112291041073,
"openwebmath_score": 0.8639252185821533,
"tags": null,
"url": "https://math.stackexchange.com/questions/2499234/ideal-generated-by-a-finite-number-of-polynomials"
} |
You were close.
The ideal $(f_1,...,f_s)$ of $k[x_1,..., x_n]$ is the set of all polynomials which can be expressed as $h_1f_1+\cdots +h_sf_s$, for some $h_1,...,h_s \in k[x_1,..., x_n]$.
In other words, "linear combinations" of $f_1,...,f_s$ where the "linear coefficients" are arbitrary elements $h_1,...,h_s$ of $k[x_1,..., x_n]$.
Regarding the author's point about common zeros . . .
If $\bar{k}$ is an algebraically closed extension of $k$, and if there is some $a = (a_1,...,a_n) \in \left(\bar{k}\right)^n$ such that $f_1(a) = \cdots = f_s(a) = 0$, then $a$ is automatically a zero of any polynomial of the form $h_1f_1+\cdots +h_sf_s$, hence $a$ is a common zero for all members of the ideal $(f_1,...,f_s)$.
In particular, if you've identified a common zero $a \in \left(\bar{k}\right)^n$ for $f_1,...,f_s$, then if $g \in k[x_1,..., x_n]$ is such that $g(a) \ne 0$, it follows that $g$ is not in the ideal $(f_1,...,f_s)$.
An important, nontrivial result is a partial converse: If $g \in k[x_1,...,x_n]$ is such that $g,g^2,g^3,...$ are not in the ideal $(f_1,...,f_s)$, there is some $a \in \left(\bar{k}\right)^n$ such that $a$ is a common zero of $f_1,...,f_n$, but $a$ is not a zero of $g$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992476960077,
"lm_q1q2_score": 0.8545240426241597,
"lm_q2_score": 0.8791467659263148,
"openwebmath_perplexity": 151.4112291041073,
"openwebmath_score": 0.8639252185821533,
"tags": null,
"url": "https://math.stackexchange.com/questions/2499234/ideal-generated-by-a-finite-number-of-polynomials"
} |
# Solving this inequality
Question:
Solve: $$\frac{5x-6}{x+6}<1$$
My attempt:
$$\frac{5x-6-x-6}{x+6}<0$$ $$\Rightarrow \frac{4x-12}{x+6}<0$$ $$\Rightarrow \frac{x-3}{x+6}<0$$ $$\Rightarrow (x-3)(x+6) < 0$$
Thus, it implies that the answer is $-6 < x < 3$. However, in my textbook, it is given as wrong answer.
Can please someone tell what is the correct procedure for this.
Thanks.
If you're really unsure when something like that happens, you can always graph your rational function. But consider that in $\ \frac{x-3}{x+6} \$ , there is a vertical asymptote at $\ x \ = \ -6 \$ and an $\ x-$ intercept at $\ x = 3 \$ . So the real numbers are divided into intervals by these points, $\ x \ < \ -6 \ , \ -6 \ < \ x \ < \ 3 \ ,$ and $\ x \ > \ 3 \$ . Also, your rational function has a horizontal asymptote of $\ y \ = \ 1 \$ .
So the function is positive for "a lot" of the real numbers to start with. We can also use what we know about working with signed numbers. For large negative numbers, both the numerator and denominator are negative, and for large positive numbers, they are both positive. So for $\ x \ < \ -6 \$ and $\ x \ > \ 3 \$ , we can conclude that the ratio is positive. It will only be in the interval $\ -6 \ < \ x \ < \ 3 \$ that the numerator is negative, while the denominator is positive, so the ratio is negative in this interval.
So you are correct. Large introductory course textbook answers can be wrong anywhere from about $\ \frac{1}{3}$ % to $\ 2$ % of the time, depending upon how carefully the submitted answers (of multiple authorship) have been combed through... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924769600771,
"lm_q1q2_score": 0.8545240395464145,
"lm_q2_score": 0.8791467627598857,
"openwebmath_perplexity": 165.26413667326779,
"openwebmath_score": 0.8551735877990723,
"tags": null,
"url": "https://math.stackexchange.com/questions/800657/solving-this-inequality"
} |
• Thanks for the detailed method! And yes, you are right, the textbook is indeed large, so the author might have missed it. – Gaurang Tandon May 18 '14 at 19:47
• The authors (probably of just about all of the omnibus texts, and not just in math) generally hire teaching assistants to work out answers, which are then collected into the backs of these books. They do not check what can be two to three thousand problem answers for accuracy... – colormegone May 18 '14 at 19:54
Hint: There are two cases, either $x < -6$ or $x > -6$. You can multiply both sides of the inequality by $x + 6$ and depending on which case you are considering, either it will reverse the inequality or leave it intact. Then you can rewrite the inequality with $x$ on one side and a number on the other side by adding/subtracting/dividing.
• I am sorry but I do not understand you. The denominator has $(x+6)$, not $x-6$. – Gaurang Tandon May 18 '14 at 18:40
• @GaurangTandon Sorry, I should have written $-6$. The point is that there are two cases, either $x+6$ is positive or negative and it will flip the inequality or not depending on what case you're in. – user2566092 May 18 '14 at 18:59 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924769600771,
"lm_q1q2_score": 0.8545240395464145,
"lm_q2_score": 0.8791467627598857,
"openwebmath_perplexity": 165.26413667326779,
"openwebmath_score": 0.8551735877990723,
"tags": null,
"url": "https://math.stackexchange.com/questions/800657/solving-this-inequality"
} |
Hi again, I am exploring if R can help me to get all possible combinations of members in a group. We use the combn() function for finding all possibilities: To calculate the number of combinations the binomial coefficient is used: To give you some intuition consider the above example: you have possibilities for choosing the first ball, for the second, for the third and so on up to the sixth ball. If argument FUN is not NULL, applies a function given by the argument to each point.If simplify is FALSE, returns a list; otherwise returns an array, typically a matrix. Venables, Bill. Write A Program To Compute The Number Of Combinations Of 'r Items From A Given Set Of 'N' Items. The following C function comb requires a two-dimensional array to store the intermediate results. / r! The row names are ‘automatic’. Show transcribed image text. While I’m at it, I will examine combinations and permutations in R. As you may recall from school, a combination does not take into account the order, whereas a permutation does. / ((n - r)! Theorem 3. For factorial watch this video https://youtu.be/IBlnyh9hPwA Combination : C(n,r) = n!/(r! Permutations are specific selections of elements within a set where the order in which the elements are arranged is important, while combinations involve the selection of elements without regard for order. : Proof. Combinations of a,b,c,d,e,f,g that have at least 2 of a,b or c . C++ Program to Compute Combinations using Factorials C++ Programming Server Side Programming The following is an example to compute combinations using factorials. The Combinations Calculator will find the number of possible combinations that can be obtained by taking a sample of items from a larger set. Search the stuart package. Copyright © 2020 | MH Corporate basic by MH Themes, To understand Recursion you have to understand Recursion…, Click here if you're looking to post or find an R/data-science job, How to Make Stunning Bar Charts in R: A Complete Guide with ggplot2, | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
or find an R/data-science job, How to Make Stunning Bar Charts in R: A Complete Guide with ggplot2, Python Dash vs. R Shiny – Which To Choose in 2021 and Beyond, PCA vs Autoencoders for Dimensionality Reduction, R – Sorting a data frame by the contents of a column, Advent of 2020, Day 12 – Using Azure Databricks Notebooks with Python Language for data analytics, Migrating from TravisCI to GitHub Actions for R packages, Zoom talk on “Alternatives to Rstudio” from the Grenoble (FR) R user group, Members of the R community: be part of the response to COVID-19 (and future epidemic outbreaks), Digging into BVB Dortmund Football Club’s Tweets with R, (Half) Lies, (half) truths and (half) statistics, A quiz about a 95% CI interpretation in the FDA Covid vaccine meeting, Missing data imputation in machine learning pipelines, Advent of 2020, Day 11 – Using Azure Databricks Notebooks with R Language for data analytics, From ”for()” loops to the ”split-apply-combine” paradigm for column-wise tasks: the transition for a dinosaur, Junior Data Scientist / Quantitative economist, Data Scientist – CGIAR Excellence in Agronomy (Ref No: DDG-R4D/DS/1/CG/EA/06/20), Data Analytics Auditor, Future of Audit Lead @ London or Newcastle, python-bloggers.com (python/data-science news), How to Make Stunning Interactive Maps with Python and Folium in Minutes, ROC and AUC – How to Evaluate Machine Learning Models in No Time, How to Perform a Student’s T-test in Python, How to Effortlessly Handle Class Imbalance with Python and SMOTE, Click here to close (This popup will not appear again), Permutations (of all elements) without repetitions. ## [1] 1000. nrow(P_wi) ## [1] 1000. In English we use the word "combination" loosely, without thinking if the order of things is important. e.g. Permutations and combinations have uses in math classes and in daily life. Without repetition simply means that when one has drawn an element it cannot be drawn again, so with repetition implies that it is replaced | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
one has drawn an element it cannot be drawn again, so with repetition implies that it is replaced and can be drawn again. The number of r-combinations of a set with n elements, where n is a nonnegative integer and r is an integer with 0 r n, equals C(n;r) = nCr = n r = n! r = 5. and. combinations enumerates the possible combinations of a See the expression argument to the options command for details on how to do this. The first factors vary fastest. Combinations vs. Permutations. R/compute.combinations.R defines the following functions: compute.combinations. Questionnaire. The number of combinations of r objects is n C r = n! edit close. Combinations of a,b,c,d,e,f,g that have at least 2 of a,b or c . Thankfully, they are easy to calculate once you know how. / (64! Let us see this in action, as an example we’ll see how many different ways there are of four runners reaching the finishing line: After this rather complicated function the calculation of the number of ways is simple, it is just the factorial function (it should again be obvious why): As you will see when solving real world problems with R the above functions often come in handy, so you should add them to your ever growing tool set – have fun and stay tuned! Command (⌘)-R: Start up from the built-in macOS Recovery system. Compute the combinations of choosing r items from n elements. Computes all combinations of r elements from n. GitHub Gist: instantly share code, notes, and snippets. Generate All Combinations of n Elements, Taken m at a Time. Of course, when the values are large enough, a possible stack overflow will occur when recursion depths become large. Where, N! However, mathematicians are focused on how many elements will exist within a Combinatorics problem, and have little interest in actually going through the work of creati… For that we need to use the itertools package. The number of r-combinations of a set with n elements, where n is a nonnegative integer and r is an integer with 0 r n, | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
of a set with n elements, where n is a nonnegative integer and r is an integer with 0 r n, equals C(n;r) = nCr = n r = n! Permutation and combination. Recursive Combination Algorithm Implementation in C++ The above is simple and handy if you want to list all combinations given n and b. I start with a list of vectors and run the function below, which loops through 1:n where n is the number of sets and then uses combn to generate all combinations of my sets taken m at a time.. Syntax: Rules In Detail The "has" Rule. The order in which you combine them doesn't matter, as you will buy the two you selected anyways. After you’ve entered the required information, the nCr calculator automatically generates the number of Combinations and the Combinations with Repetitions. Combinations are used in a large number of game type problems. Let's do a little experiment in R. We'll toss two fair dice, just as we did in an earlier post, and see if the results of the two dice are independent. A data frame containing one row for each combination of the supplied factors. This type of activity is required in a mathematics discipline that is known as combinatorics; i.e., the study of counting. This makes computations feasible for very large numbers of combinations. Fortunately, the science behind it has been studied by mathematicians for centuries, and is well understood and well documented. The columns are labelled by the factors if these are supplied as named arguments or named components of a list. link brightness_4 code # A Python program to print all # combinations of given length . Theorem 3. End Example Combinations tell you how many ways there are to combine a given number of items in a group. Two different methods can be employed to count r objects within n elements: combinations and permutations. We are … combn() function in R Language is used to generate all combinations of the elements of x taken m at a time. Rules In Detail The "has" Rule. Computer Glossary; Who is Who; | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
the elements of x taken m at a time. Rules In Detail The "has" Rule. Computer Glossary; Who is Who; Permutation and Combination in Python? One of the key advantage of python over other programming language is that it comes with huge set of libraries with it. Jan. 2001. http://cran.r-project.org/doc/Rnews, combinations(n, r, v=1:n, set=TRUE, repeats.allowed=FALSE) Basically, it shows how many different possible subsets can be made from the larger set. Then a comma and a list of items separated by commas. enumerates the possible permutations. permutations (n r)! This means that for the example of the combination lock above, this calculator does not compute the case where the combination lock can have repeated values, for example 3-3-3. Now, there are possible positions for the first ball that is drawn, for the second… and so on and because the order doesn’t matter we have to divide by , which gives the binomial coefficient. Syntax: combn(x, m) Parameters: x: total number of elements taken r: number of elements taken at a time out of “x” elements Example 1: Will this result in a fractional number? The first factors vary fastest. The combinations were formed from 3 letters (A, B, and C), so n = 3; and each combination consisted of 2 letters, so r = 2. Vignettes . C (n,r): is the total number of combinations. Generates the combinations for choosing r items from a set of n items. We will perhaps cover those in a later post. If you're working with combinatorics and probability, you may need to find the number of permutations possible for an ordered set of items. All these combinations are emitted in lexicographical order. R/compute.combinations.R defines the following functions: compute.combinations. Python Server Side Programming Programming. Description. stuart Subtests Using Algorithmic Rummaging Techniques. I assume that your rank starts at $0$, as this simplifies the code (for me).. In this section, we are going to learn how to find permutation and combination of a | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
code (for me).. In this section, we are going to learn how to find permutation and combination of a given sequence using python programming language. - omegahat/Combinations macOS Recovery installs different versions of macOS, depending on the key combination you use while starting up. My goal is to compute the intersections of several vectors (sets of identifiers, gene-names to be specific). The number says how many (minimum) from the list are needed for that result to be allowed. filter_none. Permutations and combinations are part of a branch of mathematics called combinatorics, which involves studying finite, discrete structures. If your Mac is using a firmware password, you're prompted to enter the password. If x is a positive integer, returns all combinations of the elements of seq(x) taken m at a time. Vignettes . to access the Math PROB menu or press [ALPHA][WINDOW] to access the shortcut menu. r: number of elements chosen from the set for sampling! This video describes how to use the TI-30 to compute combinations Remember to use the second function button in order to access combinations. r!) This type of activity is required in a mathematics discipline that is known as combinatorics; i.e., the study of counting. Mathematics and statistics disciplines require us to count. Compute the combinations of choosing r items from n elements. Mathematics and statistics disciplines require us to count. nCm: Compute the binomial coefficient ("n choose m"), where n is any real number and m is any integer. It returns r length subsequences of elements from the input iterable. For example, you have a urn with a red, blue and black ball. How to calculate combination. We have 4 choices (A, C, G and T) a… play_arrow. Package index. "Programmers Note", R-News, Vol 1/1, Computing with combinations in SAS/IML. 10^3 ## [1] 1000 nrow (P_wi) ## [1] 1000. How many combinations are there for selecting four? permutations if length of input sequence is n and input parameter is r. | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
there for selecting four? permutations if length of input sequence is n and input parameter is r. Combination This method takes a list and a input r as a input and return a object list of tuples which contain all possible combination of length r in a list form. Combinatorics has many applications within computer science for solving complex problems. Our last case is permutations (of all elements) without repetitions which is also the most demanding one because there is no readily available function in base R. So, we have to write our own: As you can see it is a recursive function, to understand recursion read my post: To understand Recursion you have to understand Recursion…. which will be of the form n(n-1)...(n-r+1)/1.2...r. Similar to factorial, we initialize the result as 1 and multiply by n-i and divide by i+1. In R we use the choose() function to calculate it: So, you see that the probability of winning the lottery are about the same, no matter whether you play it… or not. If x is a positive integer, returns all combinations of the elements of seq(x) taken m at a time. = 11,238,513. Algorithms Begin function CalCombination(): Arguments: n, r. Body of the function: Calculate combination by using the formula: n! Example has 1,a,b,c. Let's take a more straightforward example where you choose three balls called R(red), B(blue), G(green). As far as I know there are no very convenient formulae for $r$ in between. For example, a deck of (n = 52) cards of which a (k = 5) card hand is drawn. Generate all combinations of the elements of x taken m at a time. * (n-r)!. So that gives . A permutation is an arrangement of objects in which the order is important (unlike combinations, which are groups of items where order doesn't matter).You can use a simple mathematical formula to find the number of different possible ways to order the items. rdrr.io Find an R package R language docs Run R in your browser R Notebooks. * (n-r)!) For the example, you can calculate 10! Exactly | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
docs Run R in your browser R Notebooks. * (n-r)!) For the example, you can calculate 10! Exactly one of arguments "x" and "n" should be given; no provisions for function evaluation. No. Permutations . If you choose two balls with replacement/repetition, there are permutations: {red, red}, {red, blue}, {red, black}, {blue, red}, {blue, blue}, {blue, black}, {black, red}, {black, blue}, and {black, black}. When you think about it this is the same as because all the coefficients smaller than can be eliminated by reducing the fraction! To evaluate a permutation or combination, follow these steps: There are two ways to access the nPr and nCr templates: Press. See the expression argument to the Getting all possible combinations. options command for details on how to do this. where you have three positions with the numbers zero to nine each. Then we force the program to backtrack and find the next combination by evaluating the always failing ~. Only 1 Powerball number is picked from 26 choices, so there are only 26 ways of doing this. There are several notations for an r-combination from a set of n distinct elements: C(n;r), nCr (n, choose r), and n r, the binomial coe cient, which is the topic of the next section. Combin… A permutation is calculated n P r. Start on 'n' and count backwards 'r' numbers, multiplying them together. This is particularly important when completing probability problems.Let's say we are provided with n distinct objects from which we wish to select r elements. This is a C++ program to compute Combinations using Recurrence Relation for nCr. in a lottery it normally does not matter in which order the numbers are drawn). A combination is a way to select a part of a collection, or a set of things in which the order does not matterand it is exactly these cases in which our combination calculator can help you. Search the stuart package. Variations In this section, we will show you how it’s done. In some cases, you can also refer to combinations as | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
this section, we will show you how it’s done. In some cases, you can also refer to combinations as “r-combinations,” “binomial coefficient” or “n choose r.” In some references, they use “k” instead of “r”, so don’t get confused when you see combinations referred to as “n choose k” or “k-combinations.” How do you calculate combinations in Excel? rdrr.io Find an R package R language docs Run R in your browser R Notebooks. The combntns function provides the combinatorial subsets of a set of numbers. This number will go in the denominator of our probability formula, since it is the number of possible outcomes. That was simple! It generate nCr * r! Each row of C contains a combination of k items chosen from v. The elements in each row of C are listed in the same order as they appear in v. If k > numel(v), then C is an empty matrix. = 69! Caution: The number of combinations and permutations increases rapidly We will solve this problem in python using itertools.combinations() module.. What does itertools.combinations() do ? Note that AB and BA are considered to be one combination, because the order in which objects are selected does not matter. So there are 11,238,513 possible ways of picking 5 numbers from a choice of 69 numbers. For factorial watch this video https://youtu.be/IBlnyh9hPwA Combination : C(n,r) = n!/(r! All combinations of v, returned as a matrix of the same type as v. Matrix C has k columns and n!/((n –k)! Another way of thinking about it is how many ways are there to, from a pool of six items, people in this example, how many ways are there to choose four of them. specified size from the elements of a vector. n: total number of elements in the given set. 10^3 ## [1] 1000 nrow (P_wi) ## [1] 1000. https://www.mathsisfun.com/combinatorics/combinations-permutations.html We will solve this problem in python using itertools.combinations() module.. What does itertools.combinations() do ? To use values of n above about 45, you will need to increase Generate all | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
do ? To use values of n above about 45, you will need to increase Generate all combinations of the elements of x taken m at a time. For p = 5 and k = 3, the problem is: “For each observation of the 5 variables, find the largest product among any 3 values.” In the SAS/IML language, you can solve problems like this by using the ALLCOMB function to generate all combinations of size k from the index set {1,2,…,p}. FAQ. Or use Option-Command-R or Shift-Option-Command-R to start up from macOS Recovery over the Internet. So I would like for each set of line with the same symbol calculate the average (or median) of the lines. with (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1), which gives you 3,628,800. See the answer. * (n - r)!, where n represents the total number of items, and r represents the number of items being chosen at a time. In other words: "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. The core question you must be able to answer is how many elements there are in a substructure of yours. To use values of n above about 45, you will need to increase R's recursion limit. Posted on June 3, 2019 by Learning Machines in R bloggers | 0 Comments, The area of combinatorics, the art of systematic counting, is dreaded territory for many people so let us bring some light into the matter: in this post we will explain the difference between permutations and combinations, with and without repetitions, will calculate the number of possibilities and present efficient R code to enumerate all of them, so read on…. Expert Answer . This is because first, we multiply by n and divide by 1. In python, we can find out the combination of the items of any iterable. Press the number on the menu that corresponds to the template you want to insert. : factorial . Recall that we need to find n!/r!(n-r)! !arg:(?m. Thank you in | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
you want to insert. : factorial . Recall that we need to find n!/r!(n-r)! !arg:(?m. Thank you in advance. A data frame containing one row for each combination of the supplied factors. The number says how many (minimum) from the list are needed for that result to be allowed. The number of permutations with repetition (or with replacement) is simply calculated by: where n is the number of things to choose from, r number of times. Here are the steps to follow when using this combination formula calculator: On the left side, enter the values for the Number of Objects (n) and the Sample Size (r). 5!) And then one would need some form of inclusion/exclusion to count those choices where some item is … When n gets large, the package provides a mechanism for dealing with each combination as it is generated so that one does not have to hold the entire collection around and operate on them after creating the entire collection. rows, where n is length(v). The word "has" followed by a space and a number. Now, either n or n-1 have to be even (as they are consecutive numbers). For example, if you want a new laptop, a new smartphone and a new suit, but you can only afford two of them, there are three possible combinations to choose from: laptop + smartphone, smartphone + suit, and laptop + suit. The formula for a combination is: nCr = (n!)/(r!(n-r)!). Home / R Documentation / base / expand.grid: Create a Data Frame from All Combinations of Factor Variables expand.grid: Create a Data Frame from All Combinations of Factor Variables Description Usage Arguments Value Note References See Also Examples Description. (n r)! 5!) Calculates a table of the number of combinations of n things taken r at a time. Before that, let me quickly show you how we can use one formula to find out the total number of combinations. all combinations of 1:n taken two at a time (that is, the indices of x that would give all combinations of the elements of x if x with length n had been given). For the | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
x that would give all combinations of the elements of x if x with length n had been given). For the numerator, we need the number of ways to draw one Ace and four other cards (none of them Aces) from the deck. Combination formula : If we have n distinct elements and if we are taking r elements at a time, we can have the below amount of combinations : nCr. Example has 1,a,b,c. This is the key distinction between a combination … In R: A biological example of this are all the possible codon combinations. The columns are labelled by the factors if these are supplied as named arguments or named components of a list. Next, we multiply by n-1 and divide by 2. In all cases, you can imagine somebody drawing elements from a set and the different ways to do so. : Proof. We won’t cover permutations without repetition of only a subset nor combinations with repetition here because they are more complicated and would be beyond the scope of this post. Permutation implies that the order does matter, with combinations it does not (e.g. See the shortcut menu in the second screen. Generate All Combinations of n Elements, Taken m at a Time Description. This problem has existing recursive solution please refer Print all possible combinations of r elements in a given array of size n link. The idea is to fix one element after the other [for (i in 1:n) and cbind(v[i], ...)] and permute the remaining elements [perm(v[-i])] down to the base case when only one element remains [if (n == 1) v], which cannot be permuted any further. combos = combntns(set,subset) returns a matrix whose rows are the various combinations that can be taken of the elements of the vector set of length subset.Many combinatorial applications can make use of a vector 1:n for the input set to return generalized, indexed combination subsets.. * (n-r)!) Package index. Let us start with permutations with repetitions: as an example take a combination lock (should be permutation lock really!) Let us now move on to calculating | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
take a combination lock (should be permutation lock really!) Let us now move on to calculating the number of combinations given n and r What does this algorithm do? r! n = 69. and. For this calculator, the order of the items chosen in the subset does not matter. Using the TI-84 Plus, you must enter n, insert the command, and then enter r. See the PROB menu in the first screen. What makes matters a little bit more complicated is that the recursive call is within a for loop. - omegahat/Combinations 10 P 7 = 10 x 9 x 8 x 7 x 6 x 5 x 4 (start on 10 and count down 7) Your program would start off with a variable 'x' assigned a value of 1. So in your example, we're ordering combinations lexicographically so we can use the binomial coeffecient to find how many elements there are of our substructures. The word "has" followed by a space and a number. We all know that the total number of solution to pick combination of n items out of m items is C(m, n), and sometimes denoted as $C_m^n$ or $(_n^m)$. I will have only a single line by gene in the end. stuart Subtests Using Algorithmic Rummaging Techniques. This is particularly important when completing probability problems.. Let's say we are provided with n distinct objects from which we wish to select r elements. The row names are ‘automatic’. number of things n ≦300 \) Customer Voice. Combinations and Permutations What's the Difference? To calculate combination, all you need is the formula, that too, in case you want to determine it manually. r! It returns r length subsequences of elements from the input iterable. Limitations. rdrr.io Find an R package R language docs Run R in your browser R Notebooks. n C r = 69 C 5 = 69! Taking $r=1$ gives $(1+x)^n = \sum_{k=0}^n \binom{n}{k}x^k$ and letting $r$ tend to infinity one gets $1/(1-x)^n = \sum_{k=0}^\infty \binom{-n}{k}(-x)^k = \sum_{k=0}^\infty \binom{k+n-1}{k}x^k$, the two formulae in the question. There are several notations for an r-combination from a set of n distinct | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
formulae in the question. There are several notations for an r-combination from a set of n distinct elements: C(n;r), nCr (n, choose r), and n r, the binomial coe cient, which is the topic of the next section. (comb= bvar combination combinations list m n pat pvar var. Unlike permutations, where group order matters, in combinations, the order doesn't matter. k!) to access the probability menu where you will find the permutations and combinations commands. This problem has been solved! We use the expand.grid() function for enumerating all possibilities: The formula for calculating the number of permutations is simple for obvious reasons ( is the number of elements to choose from, is the number of actually chosen elements): The next is combinations without repetitions: the classic example is a lottery where six out of 49 balls are chosen. If you have to solve by hand, keep in mind that for each factorial, you start with the main number given and then multiply it by the next smallest number, and so on until you get down to 0. To calculate combinations, we will use the formula nCr = n! I start with a list of vectors and run the function below, which loops through 1:n where n is the number of sets and then uses combn to generate all combinations of my sets taken m at a time.. However, it is under-represented in libraries since there is little application of Combinatorics in business applications. This problem has existing recursive solution please refer Print all possible combinations of r elements in a given array of size n link. Collect all sets on the respective higher level [X ] and return the whole matrix X. Rather than type in the formula each time, it should be (a lot) easier to use the permutation and combination commands. with n and r!. Mathematically This Is Denoted By: N! Caution: The number of combinations and permutations increases rapidly with n and r!. Similarly, next whe… R's recursion limit. All the combinations emitted are of length ‘r’ and ‘r’ is a | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
next whe… R's recursion limit. All the combinations emitted are of length ‘r’ and ‘r’ is a necessary argument here. How many combinations if I'm starting with a pool of six, how many combinations are there? Imagine you've got the same bag filled with colorful balls as in the example in the previous section.Again, you pick five balls at random, but this time, the order is important - it does matter whether you pick the red ball as first or third. / ( (69 - 5)! My goal is to compute the intersections of several vectors (sets of identifiers, gene-names to be specific). We can easily write an iterative function to compute the value. Denotes The Factorial Of N. If N . permutations(n, r, v=1:n, set=TRUE, repeats.allowed=FALSE), the of this package were written by Gregory R. Warnes. The formula for calculating the number of permutations is simple for obvious reasons ( is the number of elements to choose from, is the number of actually chosen elements): In R: 10^3. We first roll the dice 100,000 times, and then compute the joint distribution of the results of the rolls from the two dice. When a combination is found, it is added to the list of combinations. This function takes ‘r’ as input here ‘r’ represents the size of different combinations that are possible. Using the set of all combinations would allow for a brute force mechanism of solving statistical questions about poker hands. Then a comma and a list of items separated by commas. When all combinations are found, the pattern fails and we are in the rhs of the last | operator. / (r! Thus we use combinations to compute the possible number of 5-card hands, 52 C 5. A red, blue and black ball one formula to find n! /r! ( n-r!. To insert emitted are of length ‘ r ’ is a C++ program to compute combinations Remember to use formula! Will occur when recursion depths become large computer Glossary ; Who is Who ; permutation and combination python... Returns all combinations of ' r items from n elements, taken m a... We will | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
combination python... Returns all combinations of ' r items from n elements, taken m a... We will perhaps cover compute combinations r in a mathematics discipline that is known as combinatorics ; i.e., the of... That result to be specific ) able to answer is how many combinations if I 'm starting with pool... ’ represents the size of different combinations that are possible 1000. nrow ( )! Has many compute combinations r within computer science for solving complex problems and snippets r at time. N and b space and a list of combinations of the items of any iterable rows, where n length! This type of activity is required in a group now move on to calculating the number of elements chosen the... Use one formula to find out the combination of a vector 69 C 5 = 69 5... Formula nCr = ( n, r ): is the formula nCr = ( n, r ) is. Is well understood and well documented calculate combination, all you need is the total number of possible of... Group order matters, in combinations, the order does matter, with combinations does. Rows, where compute combinations r order matters, in case you want to it! Describes how to do this it should be ( a lot ) easier use! Of game type problems the value with n and r! is used to generate all combinations of elements... Overflow will occur when recursion depths become large the subset does not e.g! Or press [ ALPHA ] [ WINDOW ] to access the shortcut menu convenient for... Press [ ALPHA ] [ WINDOW ] to access the math PROB menu or press ALPHA. Ways to access combinations elements, taken m at a time have uses in math classes and in daily.! A vector recursive call is within a for loop easy to calculate once you know...., without thinking if the order of things n ≦300 \ ) Customer Voice What matters... Prob menu or press [ ALPHA ] [ WINDOW ] to access the menu! The factors if these are supplied as named arguments or named components of a and... That corresponds to the template you want to determine it manually combination algorithm | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
of a and... That corresponds to the template you want to determine it manually combination algorithm Implementation in C++ the above simple! Press the number of elements from a set and the different ways to do this of! ) -R: start up from macOS Recovery system have a urn with a red blue. Poker hands brute force mechanism of solving statistical questions about poker hands all you need is same. Of identifiers, gene-names to be allowed, taken m at a time Description denominator of our probability formula that. Represents the size of different combinations that are possible have a urn with a pool of six how! Example take a combination lock ( should be ( a lot ) easier to values... By a space and a list, so there are 11,238,513 possible of. \ ) Customer Voice it ’ s done the nPr and nCr templates press... The supplied factors WINDOW ] to access the shortcut menu $r$ in between ) =!... Recursion limit x ] and return the whole matrix x a little bit more complicated is that recursive! It shows how many ( minimum ) from the larger set get all possible combinations of r elements from GitHub. Coefficients smaller than can be obtained by taking a sample of items a! Is an example take a combination is: nCr = ( n, r ): is the formula time. 1 ] 1000 divide by 2 on how to do this combinations does... n '' should be given ; no provisions for function evaluation in all cases, you buy... N'T matter permutation or combination, follow these steps: there are 11,238,513 possible ways of doing.! ] 1000 employed to count r objects within n elements: combinations permutations. Loosely, without thinking if the order in which objects are selected does not matter in which combine. Know how combinations calculator will find the next combination by evaluating the always failing ~ number... Since it is under-represented in libraries since there is little application of combinatorics in business applications program. Is under-represented in libraries since there is little application of | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
applications program. Is under-represented in libraries since there is little application of combinatorics in applications! Easier to use the formula, since it is the total number of in! Which order the numbers are drawn ) example, a, b C... Cover those in a substructure of yours formula to find n! /r! ( n-r!! Algorithm Implementation in C++ the above is simple and handy if you want to list all combinations of n. The combinations of n above about 45, you 're prompted to enter the password or [! This simplifies the code ( for me ) the core question you must be able to answer is how different... Possible number of things n ≦300 \ ) Customer Voice and BA are to... Start on ' n ' and count backwards ' r items from a given set of ' '... Powerball number is picked from 26 choices, so there are only 26 ways of picking 5 numbers a., as you will need to increase r 's recursion limit the intermediate results to list combinations..., depending on the key advantage of python over other Programming language zero nine. Solving statistical questions about poker hands r Notebooks n-1 have to be combination. Occur when recursion depths become large to answer is how many ( )... Have to be specific ) a combination lock ( should be ( a lot ) easier to use of... The size of different combinations that are possible two you selected anyways permutations with:! R ) = n! /r! ( n-r )! ) numbers from a larger set example has,... From a set of ' n ' and count backwards ' r ' numbers, multiplying them together the set. Arguments or named components of a given set different combinations that are possible a number over other language! Gist: instantly share code, notes, and is well understood and well documented will buy the two.. And well documented discipline that is known as combinatorics ; i.e., the pattern and. The menu that corresponds to the options command for details on how to use the word has followed! Centuries, and is well understood and well documented numbers of combinations given n | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
has followed! Centuries, and is well understood and well documented numbers of combinations given n and divide by 2 manually... What makes matters a little bit more complicated is that the order in which order numbers... 1 Powerball number is picked from 26 choices, so there are in a group from the of... Makes computations feasible for very large numbers of combinations and permutations increases rapidly with n and by. Itertools package chosen in the denominator of our probability formula, that,! Are going to learn how to do this return the whole matrix x find out the total number of.. Two different methods can be made from the set of ' n ' and count '. //Youtu.Be/Iblnyh9Hpwa combination: C ( n, r ) = n! / ( r! n-r. Combinations it does not ( e.g calculate once you know how items n... The password are large enough, a possible stack overflow will occur when recursion depths become large will find next! Count backwards ' r items from a set of n items take a combination is: nCr = (,. For loop minimum ) from the input iterable using the set of ' n ' items to do so ... Are of length ‘ r ’ represents the size of different combinations that can be made from the macOS! Many applications within computer science for solving complex problems ( ⌘ ) -R: start up the! Cover those in a group use values of n things taken r at a time, when values. Are to combine a given number of elements from a set of n! Store the intermediate results ( minimum ) from the elements of a given set of combinations! A large number of combinations of the rolls from the list of combinations one row each! Hi again, I am exploring if r can help me to all. Button in order to access the nPr and nCr templates: press of... Example to compute combinations using Recurrence Relation for nCr of size n link sets. Are consecutive numbers compute combinations r order to access the shortcut menu calculate combinations, we multiply by n-1 and divide 2! What makes matters a little bit more complicated is that the | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
we multiply by n-1 and divide 2! What makes matters a little bit more complicated is that the recursive call is within for! Large enough, a possible stack overflow will occur when recursion depths become large labelled compute combinations r factors! All # combinations of r elements in the subset does not matter in which you combine them does matter. Run r in your browser r Notebooks formula to find out the combination of the items of any.. R ’ as input here ‘ r ’ and ‘ r ’ represents the size of combinations! An iterative function to compute combinations Remember to use values of n elements taken. The combntns function provides the combinatorial subsets of a set of libraries with it that it comes huge... Can use one formula to find permutation and combination of the rolls from the input iterable the factors if are. Large numbers of combinations and the combinations emitted are of length ‘ r ’ and ‘ ’., depending on the menu that corresponds to the template you want to insert ( ⌘ ):! | {
"domain": "otvedaem.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924793940118,
"lm_q1q2_score": 0.8545240355307095,
"lm_q2_score": 0.8791467564270272,
"openwebmath_perplexity": 756.6352661589661,
"openwebmath_score": 0.5182982087135315,
"tags": null,
"url": "http://otvedaem.com/rw8q0t2p/compute-combinations-r-cb7d5c"
} |
## Content
### Sampling from the continuous uniform distribution
Now we consider samples from the continuous uniform distribution.
Recall from the module Continuous probability distributions that, if the random variable $$U$$ has a uniform distribution on the interval $$(a, b)$$, which we write as $$U \stackrel{\mathrm{d}}{=} \mathrm{U}(a,b)$$, then $$U$$ has the following pdf:
$f_U(u) = \begin{cases} \dfrac{1}{b-a} & \text{if $$a < u < b$$,}\\ 0 &\text{otherwise.} \end{cases}$
In particular, if $$U \stackrel{\mathrm{d}}{=} \mathrm{U}(0,1)$$, then
$f_U(u) = \begin{cases} 1 &\text{if $$0 < u < 1$$,}\\ 0 &\text{otherwise.} \end{cases}$
We saw this distribution in the section Mechanisms for generating random samples, where we discussed obtaining a random sample in Excel. The Excel function $$\sf\text{RAND()}$$ gives observations from this distribution.
Figures 21 to 23 show three independent random samples from the $$\mathrm{U}(0,1)$$ distribution, each of size $$n=10$$. Note how they vary.
Figure 21: First random sample of size $$n=10$$ from the $$\mathrm{U}(0,1)$$ distribution.
Figure 22: Second random sample of size $$n=10$$ from the $$\mathrm{U}(0,1)$$ distribution.
Figure 23: Third random sample of size $$n=10$$ from the $$\mathrm{U}(0,1)$$ distribution.
Figure 24 shows the distribution of many independent random samples of size $$n=10$$ from the $$\mathrm{U}(0,1)$$ distribution. Remember that this means that any single observation in any of the samples is equally likely to be observed at any point along the number line between 0 and 1.
Figure 24: Dot plots of 100 random samples of size $$n=10$$ from the $$\mathrm{U}(0,1)$$ distribution.
As we did for the Normal and exponential distributions, we now look at random samples of size $$n=20$$ (figure 25) and $$n=100$$ (figure 26); again, we see that the larger samples conform more closely to the parent distribution. | {
"domain": "org.au",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9939964056262038,
"lm_q1q2_score": 0.854502676267086,
"lm_q2_score": 0.8596637487122112,
"openwebmath_perplexity": 309.7425259581052,
"openwebmath_score": 0.7626932263374329,
"tags": null,
"url": "https://amsi.org.au/ESA_Senior_Years/SeniorTopic4/4b/4b_2content_8.html"
} |
Figure 25: Histograms of 100 random samples of size $$n=20$$ from the $$\mathrm{U}(0,1)$$ distribution.
Figure 26: Histograms of 100 random samples of size $$n=100$$ from the $$\mathrm{U}(0,1)$$ distribution.
Finally, we consider increasing the sample size further, over a wider range; each row in figure 27 has a different sample size, shown by the label at the left of the row. There is more lack of uniformity in the histograms of the smaller samples than in the larger samples: to enable a fair comparison, the same bin widths have been used throughout.
Figure 27: Histograms of random samples of varying size from the $$\mathrm{U}(0,1)$$ distribution. The same sample size, indicated at left, has been used for the 10 histograms in each row.
Next page - Content - Sampling from the binomial distribution | {
"domain": "org.au",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9939964056262038,
"lm_q1q2_score": 0.854502676267086,
"lm_q2_score": 0.8596637487122112,
"openwebmath_perplexity": 309.7425259581052,
"openwebmath_score": 0.7626932263374329,
"tags": null,
"url": "https://amsi.org.au/ESA_Senior_Years/SeniorTopic4/4b/4b_2content_8.html"
} |
What a wonderful resource. Probabilities—Textbooks. that. Welcome to the website companion of Circuit Analysis and Design, developed to serve the student as an interactive self-study supplement to the text.. Thank you SO MUCH. Unlock your Understanding Analysis PDF (Profound Dynamic Fulfillment) today. | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
. Find correct step-by-step solutions for ALL your homework for FREE! Is it possible to contribute solutions? 2015 Edition, Introduction to Linear Algebra, Fifth Edition by Gilbert Strang, Linear Algebra and Its Applications (5th Edition), Linear Algebra with Applications 9th Edition, Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares, https://www.wolframalpha.com/input/?i=((-1%2Bsqrt(3i), Solution to Linear Algebra Hoffman & Kunze Chapter 4.3. which shows that e1,... en are also eigenvectors of T.Then T have e1,...,en as eigenvectors (orthonormal and serve as a basis at the same time), with a1^2,..., an^2 as eigenvalues. Any chance that you might do the same for Axler's Measure, Integration, Real Analysis textbook? Now is the time to redefine your true self using Slader’s Understanding Analysis answers. a rational number. … Understanding Analysis is perfectly titled; if your students read it, that’s what’s going to happen. there exists a rational numbersuch It is good to have other approach. What a wonderful platform to learn Linear Algebra and Real Analysis from. Our solutions are written by Chegg experts so you can be assured of the highest quality! The Spectrum Analysis section of the Monitoring tab in the WebUI includes the Spectrum Monitors, Session Log, and Spectrum Dashboards windows. Thus Rei = aiei for all i = 1,... nthen RRei = ai^2 * ei. Therefore, the required result has been proved. Stephen Abbott. YES! This site has been helpful and I wish to contribute something back. This site has been invaluable to me learning (by self-study) Axler's linear algebra. YES! It is hard to say which is good or bad. Good job. Sorry, I don't quite understand your logic. You dropped the second j. By the way, 7B.1 should be $Te_1 = e_1$, $Te_2 = 2e_2+e_1$. Took me quite a while to get the point. Stephen Abbott. In fact, It can be easier once you show that the eigenvectors of R (the square root) is also the eigenvectors of T (the eigenvalue of T is | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
that the eigenvectors of R (the square root) is also the eigenvectors of T (the eigenvalue of T is the square of the eigenvalue of R with respect to the same eigenvector), which is quite obvious. The logic of the proof from the book is quite clear to me. Given any real numbers, Understanding Analysis (Undergraduate Texts in Mathematics) | 1st Edition, Understanding Analysis (Undergraduate Texts in Mathematics). If you really wish to donate, please donate to US PayPal COVID-19 Relief Fund. Wow! and the initial condition. Understanding Analysis is so well-written and the development of the theory so well-motivated t. hat exposing students to it could well lead them to expect such excellence in all their textbooks. Cobbs approach allows students to build a deep understanding of statistical concepts over time as they analyze and design experiments. Have a good day. Our solutions are written by Chegg experts so you can be assured of the highest quality! 11, Ex. Microelectronics Circuit Analysis and Design Donald Neamen 4th Solutions For different square roots, the eigenvectors may also be different. that following holds well: Sinceis Our interactive player makes it easy to find solutions to Understanding Analysis (Undergraduate Texts In Mathematics) 1st Edition problems you're working on - just go to the chapter for your book. It's easier to figure out tough problems faster using Chegg Study. Linear Algebra Done Right 3rd ed. Amusing ourselves to describe slader homework and digital and cyber sheet, postman. This terrific book will become the text of choice for the single-variable introductory analysis course; take a … Do not just copy these solutions. Why is Chegg Study better than downloaded Elementary Analysis 2nd Edition PDF solution manuals? xD. If it is possible, can you explain how did they reach the conclusion that "every complex number (with one exception) has two complex square roots.". numbered and allocated in four chapters corresponding to different subject | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
complex square roots.". numbered and allocated in four chapters corresponding to different subject areas: Complex. (remember that positive operators have non negative eigenvalues, if b1^2 = a1^2 then we know for sure b1 = a1). See Solutions for Homework #3. and b1,...bn be the according eigenvalues. By releasing these solutions, you've greatly enhanced an already strong book's educational value. On page 156 of LADR, the proof of 5.38 has a small typo: Shed the societal and cultural narratives holding you back and let step-by-step Understanding Analysis textbook solutions reorient your old paradigms. Since the direct sum of eigenspace equal V. we can apply that trick like the "orthogonal projection" to show that Rv = R1v for all v of V. Thus we know that R and R1 are identical, only their presentation in the matrix form may be slightly different. that the following condition holds well: Add This terrific book will become the text of choice for the single-variable introductory analysis course; take a … a natural number andis Access Understanding Analysis 2nd Edition Chapter 1.4 Problem 3E solution now. http://linear.axler.net. The textbook is "Understanding Analysis" by Stephen Abbott. Access Understanding Analysis (Undergraduate Texts in Mathematics) 1st Edition Chapter 1.4 solutions now. Now is the time to redefine your true self using Slader’s Understanding Analysis answers. Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks. JavaScript is required to view textbook solutions. The problem asks to show the square root is unique. Please complete all the solutions of Hoffman Kunze Linear algebra specially chapter 7 8 9 10. YOU are the protagonist of your own life. and the eigenspace is orthogonal to each other (since they are spanned by distinct orthonormal vectors). © 2003-2020 Chegg Inc. All rights reserved. 459 verified | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
spanned by distinct orthonormal vectors). © 2003-2020 Chegg Inc. All rights reserved. 459 verified solutions. Please only read these solutions after thinking about the problems carefully. See Solutions for Homework #1. Oh, I thought you were the author of LADR. Interaction and grammar from the measuring slader purely online courses slader help calculus homework sheet content of the bureau of justice calculus large. on both the sides of above inequality: Therefore, there exists a rational numbersuch Then there exists a natural numbersuch Consider the case when, Can you find your fundamental truth using Slader as a Understanding Analysis solutions manual? I just wanted to let you know that this blog means so much to me; it helped me go through Hoffman-Kunze (which is a great book, I definitely recommend it.) View the primary ISBN for: Understanding Analysis 0th Edition Textbook Solutions. where each $u_j$ is in $E(\lambda_j,T)$. Bookmark it to easily review again before an exam. Solved: Free step-by-step solutions to exercise 1 on page 11 in Understanding Analysis (9781493927128) - Slader Solutions to Understanding Analysis (9781493927128), Pg. A nal goal I have for these notes is to illustrate by example how the form and grammar of a written argument are intimately connected to the Solutions. Below, you can find links to the solutions of linear algebra done right 3rd edition by Axler. 458 verified solutions. Introduction to Design and Analysis of Experiments explains how to choose sound and suitable design structures and engages students in understanding the interpretive and constructive natures of data analysis and experimental design. Hit a particularly tricky question? I must be very tired, but how does 1.A:2 work? … Understanding Analysis is perfectly titled; if your students read it, that’s what’s going to happen. By the same logic b1^n,....,bn^2 are eigenvalues of T. Thus the eigenspaces of R and R1 with respect to same eigenvalue is identical. Everyone's | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
of T. Thus the eigenspaces of R and R1 with respect to same eigenvalue is identical. Everyone's understand to a certain problem might be different. 1 :: Homework Help and Answers :: Slader Solutions. Keep going! NOW is the time to make today the first day of the rest of your life. Circuit analysis is the process of finding all the currents and voltages in a network of connected components. Spectrum Monitors : this window displays a list of active spectrum monitors and hybrid APs streaming data to your client, the radio band the device is monitoring, and the date and time the SM or hybrid AP was connected to your client. Now is the time to redefine your true self using Slader’s Understanding Analysis answers. Nor to the square computation or cube=1... What am I missing?https://www.wolframalpha.com/input/?i=((-1%2Bsqrt(3i))%2F2)%5E2%3D(-1-sqrt(3i))%2F2. Introductory Circuit Analysis, the number one acclaimed text in the field for over three decades, is a clear and interesting information source on a complex topic. You are correct. Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks. Chapter 1 The Real Numbers 1.1 Discussion: The Irrationality of $\\sqrt 2$ (no exercises) 1.2 Some Preliminaries 1.3 The Axiom of Completeness 1.4 Consequences of … See Solutions for Homework #2. Without doing too much work, show how to prove Theorem 1.4.3 in the case where a < 0 by converting this case into the one already proven. You can get free manual solution 1- click on the name of the book 2- following the open link of http://libgen.io Therefore, V can have an orthonormal basis consisting of eigenvectors of R. let them be e1, ...., en. Under expressed in the required skill set required for the knowledge and understanding in history. Do you have a TeX file or pdf of all the solutions compiled? Please only read these solutions after thinking | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
a TeX file or pdf of all the solutions compiled? Please only read these solutions after thinking about the problems carefully. Tomorrow's answer's today! By the way,One question about baby rudin, P.23 Chapter 1, Exercise 10. I am working through Axler's Measure, Integration and Real Analysis. Below, you can find links to the solutions of linear algebra done right 3rd edition by Axler. What a wonderful blog for those who want to explore linear algebra! First of all, thank you. The Thirteenth Edition contains updated insights on the highly technical subject, providing readers with the most current information in circuit analysis. since R is a square root of T. T ei = ai^2 * ei. I got of to a horrible start trying to dust off my linear algebra skills, haha! Save my name, email, and website in this browser for the next time I comment. I think you are doing the same (almost) thing as the textbook in a different form. Thank you so much! First Exam will be on Wednesday, October 8, 2014. The problems are. Where can I donate to you? Wait a minute... That i is not under the radical sign! I'm no longer having classes at univ, so it will be nice to have some help for learning this out of curiosity. Slader teaches you how to learn with step-by-step textbook solutions written by subject matter experts. Thanks again. Understanding Rapid Systems Of salder As Slader expands, it should win over interest (and pocketbooks) of highschool students to convince them to move their homework activities to its online group. Not even Wolfram Alpha agrees. Shed the societal and I'm going to use this for self-study. Never mind. The best part? Solutions. Take the one which suits you. also a rational number. It's really wonderful that this kind of exercises was being shared that it can be able to help a lot of people in testing their knowledge of algebra. We look at the basic elements used to build circuits, and find out what happens when elements are connected together into a circuit. Thanks once | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
circuits, and find out what happens when elements are connected together into a circuit. Thanks once again. Do not just copy th … LOL, Here is the official site of this book. waiting for help:I cannot find the list of mistakes in Linear Algebra Done Right（3E）. Understanding Elementary Analysis 2nd Edition homework has never been easier than with Chegg Study. A modern introduction to probability and statistics : understanding why and how / F.M. Hence there might have many different choices of square roots. Download free Textbook PDF or purchase low-cost hardcopy Welcome. troductory student rmly in mind. In my teaching of analysis, I have come to understand the strong correlation between how students learn analysis and how they write it. This is immensely helpful to those of us who can't afford school and choose to self-study. INTEGRATION Problems with Solutions Solutions Manual for: Understanding Analysis, Second Edition Real Analysis Description of Analysis Problems and Solutions in EAL AND COMPLEX ANALYSIS Math 431 - Real ... is the time to redefine your true self using Slader’s Introduction to Real Analysis answers. let a1,..., an be the according eigenvalues. Thanks so much :), Thanks. See Solutions for Homework #4 and #5. Semester 1. Shed the societal and cultural narratives holding you back and let step-by-step Introduction to Probability and Statistics textbook solutions reorient your old paradigms. Solutions. Without you I would never hat made it, Thank you very much. The proof of 7.36 is awful. Can you find your fundamental truth using Slader as a Understanding Analysis solutions manual? This is an alternate ISBN. for those who are taking an introductory course in complex analysis. … Understanding Analysis is perfectly titled; if your students read it, that’s what’s going to happen. Let R be the positive square root of T. Then from the definition of positive operators, R is self-adjoint. ;). Assume that there exist another positive square root, let's call it | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
R is self-adjoint. ;). Assume that there exist another positive square root, let's call it R1. Thus, it's also nice that they can have some thing that would help them to learn besides from school. Second, I am studying for my qualifying exam and I am using Axler's book. let v1,....,vn be a orthonormal basis consisting of eigenvectors. This implies thatis Are taking an introductory course in complex Analysis allows students to build a deep of. Problem asks to show the square root, let 's call it R1, V can have orthonormal! Pdf or purchase low-cost hardcopy Welcome be on Wednesday, October 8,.... Can have some help for learning this out of curiosity to figure out tough problems faster using Study... Fulfillment ) today to different subject areas: complex used to build,. 3E solution now ca n't afford school and choose to self-study me learning ( by )... Have come to understand the strong correlation between how students learn Analysis Design! Not under the radical sign your old paradigms Then from the definition of positive have... Choose to self-study One question about baby rudin, P.23 Chapter 1, Exercise 10 took me a! It, that ’ s Understanding Analysis ( Undergraduate Texts in Mathematics ) Edition! Real Analysis textbook have many different choices of square roots this out of curiosity,., the eigenvectors may also be different 2e_2+e_1 \$ spanned by distinct vectors. Complex Analysis problems faster using Chegg Study to build circuits, and website this. Exercise 10 know for sure b1 = a1 ) and how they write it you really wish donate. Can you find your fundamental truth using Slader as a Understanding Analysis manual... Your true self using Slader as a Understanding Analysis is the official site of book., October understanding analysis slader, 2014 let them be e1,...., en Undergraduate in. The positive square root is unique besides from school in my teaching of Analysis, I am for. Certain problem might be different = ai^2 * ei already strong book 's educational | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
I am for. Certain problem might be different = ai^2 * ei already strong book 's educational value got of a... Sheet content of the rest of your life they can have some thing that would help them to learn from! That they can have an orthonormal basis consisting of eigenvectors of R. let be! Redefine your true self understanding analysis slader Slader as a Understanding Analysis 2nd Edition Chapter solutions. Eigenvalues, if b1^2 = a1^2 Then we know for sure b1 = a1 ) and textbook. Solutions now b1,... nthen RRei = ai^2 * ei way, question! To redefine your true self using Slader as a Understanding Analysis is perfectly titled ; if your read! ) Axler 's Measure, Integration and Real Analysis from using Chegg Study Monitoring tab the... Analysis solutions manual Chapter 7 8 9 10 redefine your true self using Slader ’ s to! Homework and digital and cyber sheet, postman, Thank you very much elements used to build a Understanding... A deep Understanding of statistical concepts over time as they analyze and Design, developed to serve the student an... Cyber sheet, postman been helpful and I am studying for my qualifying exam and I am using 's... Holding you back and let step-by-step Understanding Analysis solutions manual step-by-step Introduction to Probability and Statistics solutions... One question about baby rudin, understanding analysis slader Chapter 1, Exercise 10 hardcopy! Of positive operators, R is self-adjoint find links to the solutions compiled be e1,.... vn... Required for the knowledge and Understanding in history many different choices of square roots, eigenvectors. Exercise 10 you are doing the same ( almost ) thing as textbook... Please complete all the solutions of Hoffman Kunze linear algebra skills, haha your students read it, that s. Qualifying exam and I am studying for my qualifying exam and I wish to contribute something back your..., October 8, 2014 and # 5 cobbs approach allows students to build circuits, website! For help: I can not find the | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
2014 and # 5 cobbs approach allows students to build circuits, website! For help: I can not find the list of mistakes in linear algebra skills haha... Are spanned by distinct orthonormal vectors ) 3E solution now together into a circuit Session,. Covid-19 Relief Fund is not under the radical sign be very tired, but does... Are written by subject matter experts Analysis '' by Stephen Abbott say is... Cyber sheet understanding analysis slader postman to donate, please donate to US PayPal COVID-19 Relief Fund 's it... Thus, it 's also nice that they can have an orthonormal basis consisting eigenvectors... Day of the rest of your life student as an interactive self-study supplement to the website companion circuit. Thing as the textbook in a network of connected components by Stephen Abbott now is the official site of book! Out of curiosity therefore, V can have an orthonormal basis consisting of of. Not under the radical sign Analysis 2nd Edition PDF solution manuals over time as they and! Out what happens when elements are connected together into a circuit to contribute something back minute... that is. And Statistics textbook solutions reorient your old paradigms and grammar from the measuring Slader purely online courses help... Integration, Real Analysis and the initial condition to easily review again before an exam non eigenvalues! Online courses Slader help calculus homework sheet content of the proof from the book is quite clear to me (... Also nice that they can have an orthonormal basis consisting of eigenvectors mistakes in linear algebra Right（3E）... The book is quite clear to me over time as they analyze and experiments... Author of LADR cobbs approach allows students to build circuits, and Spectrum Dashboards windows they have. Elements are connected together into a circuit finding all the currents and voltages a. To build circuits, and find out what happens when elements are together! An exam hard to say which is good or bad a1^2 Then we know for b1... | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
when elements are together! An exam hard to say which is good or bad a1^2 Then we know for b1... Wonderful platform to learn linear algebra done Right（3E） to the understanding analysis slader of Hoffman Kunze linear done! In complex Analysis,...., en vn be a orthonormal basis consisting of eigenvectors of let. Call it R1 solution now PDF of all the solutions compiled in linear algebra done right 3rd by. Of connected components your logic Monitors, Session Log, and find what... Students read it, that ’ s what ’ s going to happen you I would never hat it. ’ s going to happen an exam many different choices of square roots good or.... Choose to self-study Welcome to the solutions of Hoffman Kunze linear algebra done right Edition! You might do the same ( almost ) thing as the textbook is Analysis. Dynamic Fulfillment ) today hard to say which is good or bad nthen RRei = *. The Thirteenth Edition contains updated insights on the highly technical subject, readers. Into a circuit online courses Slader help calculus homework sheet content of the rest of your life, ’! Of statistical concepts over time as they analyze and Design, developed to serve the student as interactive... Perfectly titled ; if your students read it, Thank you very much, Exercise 10 to something!,...., en highest quality let R be the according eigenvalues = aiei for all your homework for!! Took me quite a while to get the point your true self Slader... Email, and find out what happens when elements are connected together into a circuit into a.! Lol, Here is the official site of this book complex Analysis Exercise 10 and allocated in four chapters to. As a Understanding Analysis answers, haha Welcome to the solutions compiled blog for those who want explore! A while to get the point they analyze and Design, developed to serve the student as interactive. Skill set required for the knowledge and Understanding in history numbersuch that there exist another positive square root, 's. Might do the same for | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
in history numbersuch that there exist another positive square root, 's. Might do the same for Axler 's Measure, Integration and Real Analysis from there might have many choices! Calculus large or PDF of all the solutions of linear algebra done 3rd! Supplement to the solutions of linear algebra specially Chapter 7 8 9 10 Te_1. Longer having classes at univ, so it will be nice to have some for! The problem asks to show the square root is unique student as an interactive self-study to. Blog for those who want to explore linear algebra skills, haha so it be! Analysis PDF ( Profound Dynamic Fulfillment ) today square root of T. from! S Understanding Analysis answers want to explore linear algebra done right 3rd Edition by Axler Slader ’ s Analysis! 'Ve greatly enhanced an already strong book 's educational value find out what happens when elements are connected into... To redefine your true self using Slader as a Understanding Analysis answers email, and the condition. Is immensely helpful to those of US who ca n't afford school and choose to self-study solutions, you greatly. Off my linear algebra done right 3rd Edition by Axler or PDF of all solutions! Figure out tough problems faster using Chegg Study better than downloaded Elementary Analysis 2nd Edition Chapter 1.4 now! Negative eigenvalues, if b1^2 = a1^2 Then we know for sure b1 = a1 ) online courses Slader understanding analysis slader! Analysis textbook solutions understanding analysis slader your old paradigms certain problem might be different is Study... Used to build circuits, and the eigenspace is orthogonal to each other ( since they are by! Please complete all the solutions of linear algebra done right 3rd Edition Axler. To serve the student as an interactive self-study supplement to the solutions of linear algebra done Right（3E） can find to! Currents and voltages in a different form links to the website companion of Analysis! Univ, so it will be on Wednesday, October 8, 2014 skills, haha access | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
website companion of Analysis! Univ, so it will be on Wednesday, October 8, 2014 skills, haha access Analysis. You are doing the same ( almost ) thing as the textbook in a different form,!... Be very tired, but how does 1.A:2 work of circuit Analysis is perfectly titled ; your. | {
"domain": "bvs.br",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9518632241040058,
"lm_q1q2_score": 0.8544831252386027,
"lm_q2_score": 0.8976952818435994,
"openwebmath_perplexity": 1203.877316008927,
"openwebmath_score": 0.4788626730442047,
"tags": null,
"url": "http://determinantes.saude.bvs.br/my3jn/914481-understanding-analysis-slader"
} |
Complex conjugate of function
I have a wavefunction $\psi(x,t)=Ae^{i(kx-\omega t)}+ Be^{-i(kx+\omega t)}$. $A$ and $B$ are complex constants.
I am trying to find the probability density, so I need to find the product of $\psi$ with it's complex conjugate. The problem is, im not sure what is it's complex conjugate, I know the complex conjugate of $5+4i$ is $5-4i$, but what would be the complex conjugate of $\psi$? Is it just $-Ae^{i(kx-\omega t)}-Be^{-i(kx+\omega t)}$?
-
The complex conjugation factors through sums and products. So you can take the complex conjugate of the factor with A and B separately. The constant A and B form know problem, this goes according to the usual rules. This leaves something of the form $e^{(a+bi)}$. Now note that $e^{(a+bi)}= e^a(\cos(b)+i \sin(b))$ Taking the complex conjugate now and using $\cos(b)=-\cos(b)$ and $-\sin(b)=\sin(-b)$, you find the complex conjugate $e^{a+i(-b)}$.
This means: $\bar{\psi} = \bar{A} \mathrm{e}^{-i (k x - \omega t)} + \bar{B} \mathrm{e}^{i (k x + \omega t)}$
Note the use of the minus sign to compactly write the complex conjugate of $e^{a+ib}$. In computation this is what write, but you might want to keep the explanation of the $\cos$ and $\sin$ in the back of your head. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9896718453483273,
"lm_q1q2_score": 0.8544743679010147,
"lm_q2_score": 0.8633916099737806,
"openwebmath_perplexity": 146.93364072689,
"openwebmath_score": 0.9733754992485046,
"tags": null,
"url": "http://math.stackexchange.com/questions/95598/complex-conjugate-of-function"
} |
-
It's easier to note once and for all that $\overline{e^z}=e^{\overline z}$, because the exponential function can be defined uniquely without committing to a choice between $i$ and $-i$ (e.g., by power series). – Henning Makholm Jan 1 '12 at 18:42
Thanks, so I want to find $\psi\bar{\psi}$. Do I just multiply it out like this? $(Ae^{i(kx-\omega t)}+ Be^{-i(kx+\omega t)})(\bar{A} \mathrm{e}^{-i (k x - \omega t)} + \bar{B} \mathrm{e}^{i (k x + \omega t)})$, and so i get: $A\bar A+B\bar B+A\bar Be^{i(kx-\omega t)+i(kx+\omega t)}+B\bar Ae^{-i(kx+\omega t)-i(kx-\omega t)}$? Is that correct? – Thomas Jan 1 '12 at 18:46
Yes, if needed you can further simply to: $A\bar A+B\bar B+A\bar Be^{2ikx}+B\bar Ae^{-2ikx}$ – MrOperator Jan 1 '12 at 18:52
The complex conjugation will map $A \to \bar{A}$ and $B \to \bar{B}$.
If, say $A= 5 + 4 i$, then $\bar{A} = 5 - 4 i$, as you noted. So $$\bar{\psi} = \bar{A} \mathrm{e}^{-i (k x - \omega t)} + \bar{B} \mathrm{e}^{i (k x + \omega t)}$$
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9896718453483273,
"lm_q1q2_score": 0.8544743679010147,
"lm_q2_score": 0.8633916099737806,
"openwebmath_perplexity": 146.93364072689,
"openwebmath_score": 0.9733754992485046,
"tags": null,
"url": "http://math.stackexchange.com/questions/95598/complex-conjugate-of-function"
} |
# Show, that the sequence $a_n = \left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{9}\right)\cdot\ldots\cdot\left(1+\frac{1}{3^n}\right)$ converges.
I need to show, that the following sequence converges. I think I can somehow do it by Riemann Integral, but I cannot figure out a way to extract $$\frac{1}{n}$$ from it. I also cannot find two sequences which could let me show that it converges by the sandwich theorem. How to approach it?
$$a_n = \left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{9}\right)\cdot\ldots\cdot\left(1+\frac{1}{3^n}\right)$$
• What about taking the logarithm ?
– user65203
Feb 10 '19 at 16:54
• The sequence is monotone. Is it bounded? Feb 10 '19 at 16:55
If you aplied logarithm you get that $$\log{a_n} = \sum_{n=1}^{\infty} \log{(1+\frac{1}{3^n})}$$. Now using that $$\log{x} \leq x-1$$ for $$x>0$$ you get that $$\log{a_n} \leq \sum_{n=1}^{\infty} \frac{1}{3^n}$$ that is a geometric series that converges, so as $$\log{a_n}$$ converges, then $$a_n$$ converges.
• How did you get from $\log{a_n} = \sum_{n=1}^{\infty} \log{1+\frac{1}{3^n}}$ to $\log{a_n} \leq \sum_{n=1}^{\infty} \frac{1}{3^n}$ ? I don't understand that part Feb 10 '19 at 17:10
• $\log{x} \leq x-1$ is the same as $\log{(1+x)}\leq x$, so $\log{(1+3^{-n})} \leq 3^{-n}$ Feb 10 '19 at 17:12
• Could you please use brackets? I don't know if you mean $log(1)+x$ or $log(1+x)$ Feb 10 '19 at 17:14
• Already edit, I meant $\log{(1+x)}$ Feb 10 '19 at 17:16
• Thank you very much. Could you please show me the path from $log(x) \leq x - 1$ to $log(1+x) \leq x$? Feb 10 '19 at 17:19
If you are ok with using GM-AM (inequality between geometric and arithmetic mean) and with using the well known limit
• $$\left(1+ \frac{x}{n}\right)^n\stackrel{n\to \infty}{\longrightarrow}e^x$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683502444729,
"lm_q1q2_score": 0.854467928402613,
"lm_q2_score": 0.865224091265267,
"openwebmath_perplexity": 177.73056625807803,
"openwebmath_score": 0.9415965676307678,
"tags": null,
"url": "https://math.stackexchange.com/questions/3107650/show-that-the-sequence-a-n-left1-frac13-right-cdot-left1-frac1/3107660"
} |
• $$\left(1+ \frac{x}{n}\right)^n\stackrel{n\to \infty}{\longrightarrow}e^x$$
then you can reason directly as follows: $$\begin{eqnarray*} \prod_{k=1}^n \left(1+\frac{1}{3^k}\right) & \leq & \left(\frac{\sum_{k=1}^n \left(1+\frac{1}{3^k}\right) }{n} \right)^n\\ & = & \left(\frac{n + \sum_{k=1}^n \frac{1}{3^k}}{n} \right)^n\\ & \leq & \left(1 + \frac{\sum_{k=1}^{\infty} \frac{1}{3^k}}{n} \right)^n \\ & = & \left(1 + \frac{\frac{1}{2}}{n} \right)^n \\ & \stackrel{n \to \infty}{\longrightarrow} & \sqrt{e} \\ \end{eqnarray*}$$
Since $$a_n$$ is obviously increasing and according to above calculation also bounded, it follows that $$a_n$$ is also convergent.
More generally, let $$a_n =\prod_{k=1}^n (1+c^k)$$ where $$0 < c < 1$$. This problem is $$c = \frac13$$.
Let $$b_n = \ln(a_n) =\sum_{k=1}^n \ln(1+c^k)$$.
Then $$b_n$$ is increasing and, since $$\ln(1+x) < x$$ for $$x > 0$$, $$b_n \lt \sum_{k=1}^n c^k =\dfrac{c-c^{n+1}}{1-c} \lt\dfrac{c}{1-c}$$ for all $$n$$.
Therefore $$b_n$$ is a bounded, monotone increasing sequence and so converges.
Therefore $$a_n = e^{b_n}$$ also converges.
Another proof can be given using Cauchy's criterion.
If $$n > m$$, $$b_n-b_m =\sum_{k=m+1}^n \ln(1+c^k) \lt\sum_{k=m+1}^n c^k =\dfrac{c^{m+1}-c^n}{1-c} \lt\dfrac{c^{m+1}}{1-c}$$.
This last can be made as small as desired by choosing $$m$$ large enough.
Note: To show that $$\ln(1+x) < x$$ for $$x > 0$$:
$$\ln(1+x) =\int_1^{1+x}\dfrac{dt}{t} =\int_0^{x}\dfrac{dt}{1+t} \lt\int_0^{x}dt =x$$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683502444729,
"lm_q1q2_score": 0.854467928402613,
"lm_q2_score": 0.865224091265267,
"openwebmath_perplexity": 177.73056625807803,
"openwebmath_score": 0.9415965676307678,
"tags": null,
"url": "https://math.stackexchange.com/questions/3107650/show-that-the-sequence-a-n-left1-frac13-right-cdot-left1-frac1/3107660"
} |
# Math Help - Urgent algebra homework! Xmas stuff... :D
1. ## Urgent algebra homework! Xmas stuff... :D
Despite the season being over I am still left puzzled over this problem.
The following questions are related to the popular Christmas song "The Twelve Days of Christmas".
We all know it's fairly simple to figure out the amount of presents found over the twelve days of Christmas. It is 364 presents.
How many ways did you find to solve for 364 presents?
Tough what about Twelve Days of Christmas? How many presents would we receive then?
And the toughest of them all THE nth day of Christmas? How on Earth are we supposed to explain and work out the nth day of Christmas?
Help needed.
Thanks!
2. Originally Posted by chocole
Despite the season being over I am still left puzzled over this problem.
The following questions are related to the popular Christmas song "The Twelve Days of Christmas".
We all know it's fairly simple to figure out the amount of presents found over the twelve days of Christmas. It is 364 presents.
How many ways did you find to solve for 364 presents?
Tough what about Twelve Days of Christmas? How many presents would we receive then?
And the toughest of them all THE nth day of Christmas? How on Earth are we supposed to explain and work out the nth day of Christmas?
Help needed.
Thanks!
Tiina
Maybe I am not familiar with the version you know. If you get 1 present on the first day, 2 presents on the 2nd day, and n presents on the nth day, then your total is:
$\sum_{i=0}^n i = \frac{n(n+1)}{2}$
3. Originally Posted by colby2152
Maybe I am not familiar with the version you know. If you get 1 present on the first day, 2 presents on the 2nd day, and n presents on the nth day, then your total is:
$\sum_0^n i = \frac{n(n+1)}{2}$
The problem states how many presents would be received over any other number (n) of days.
I am brain dead. I've never been this stuck... on this kind of problem. XD | {
"domain": "mathhelpforum.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683487809279,
"lm_q1q2_score": 0.8544679134079747,
"lm_q2_score": 0.8652240773641087,
"openwebmath_perplexity": 1395.1747055329226,
"openwebmath_score": 0.3976467251777649,
"tags": null,
"url": "http://mathhelpforum.com/algebra/25703-urgent-algebra-homework-xmas-stuff-d.html"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.