id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-17500 | Seventeen Horses/General Problem 2 | A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $m$ heirs.
They are to be distributed in the ratio $\dfrac 1 {a_1} : \dfrac 1 {a_2} : \cdots : \dfrac 1 {a_m}$.
Let $t = \dfrac q r = \ds \sum_{k \mathop = 1}^m \dfrac 1 {a_k}$ expressed in canonical form.
Let $t \ne 1$.
Then it is po... | {{ProofWanted|More clarity of exposition is required. Might not even be correct}} | A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $m$ heirs.
They are to be distributed in the [[Definition:Ratio|ratio]] $\dfrac 1 {a_1} : \dfrac 1 {a_2} : \cdots : \dfrac 1 {a_m}$.
Let $t = \dfrac q r = \ds \sum_{k \mathop = 1}^m \dfrac 1 {a_k}$ expressed in [[Definition:Canonica... | {{ProofWanted|More clarity of exposition is required. Might not even be correct}} | Seventeen Horses/General Problem 2 | https://proofwiki.org/wiki/Seventeen_Horses/General_Problem_2 | https://proofwiki.org/wiki/Seventeen_Horses/General_Problem_2 | [
"Seventeen Horses"
] | [
"Definition:Ratio",
"Definition:Rational Number/Canonical Form",
"Definition:Positive/Integer",
"Definition:Negative/Integer"
] | [] |
proofwiki-17501 | Sum from 1 to n of 1 over r(r+1)(r+2)/Corollary | :$\ds \sum_{r \mathop = 1}^\infty \frac 1 {r \paren {r + 1} \paren {r + 2} } = \frac 1 4$ | {{begin-eqn}}
{{eqn | l = \sum_{r \mathop = 1}^\infty \frac 1 {r \paren {r + 1} \paren {r + 2} }
| r = \lim_{n \mathop \to \infty} \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} \paren {r + 2} }
}}
{{eqn | r = \lim_{n \mathop \to \infty} \frac {n \paren {n + 3} } {4 \paren {n + 1} \paren {n + 2} }
| c = S... | :$\ds \sum_{r \mathop = 1}^\infty \frac 1 {r \paren {r + 1} \paren {r + 2} } = \frac 1 4$ | {{begin-eqn}}
{{eqn | l = \sum_{r \mathop = 1}^\infty \frac 1 {r \paren {r + 1} \paren {r + 2} }
| r = \lim_{n \mathop \to \infty} \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} \paren {r + 2} }
}}
{{eqn | r = \lim_{n \mathop \to \infty} \frac {n \paren {n + 3} } {4 \paren {n + 1} \paren {n + 2} }
| c = [... | Sum from 1 to n of 1 over r(r+1)(r+2)/Corollary | https://proofwiki.org/wiki/Sum_from_1_to_n_of_1_over_r(r+1)(r+2)/Corollary | https://proofwiki.org/wiki/Sum_from_1_to_n_of_1_over_r(r+1)(r+2)/Corollary | [
"Sum from 1 to n of 1 over r(r+1)(r+2)",
"Limits of Series"
] | [] | [
"Sum from 1 to n of 1 over r(r+1)(r+2)",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Basic Null Sequence",
"Category:Sum from 1 to n of 1 over r(r+1)(r+2)",
"Category:Limits of Series"
] |
proofwiki-17502 | Heronian Triangle whose Altitude and Sides are Consecutive Integers | There exists exactly one Heronian triangle one of whose altitudes and its sides are all consecutive integers.
This is the Heronian triangle whose sides are $\tuple {13, 14, 15}$ and which has an altitude $12$. | We note that a Heronian triangle whose sides are all consecutive integers is also known as a Fleenor-Heronian triangle.
From Sequence of Fleenor-Heronian Triangles, we have that the smallest such triangles are as follows:
:$\tuple {1, 2, 3}$, which has an altitude of $0$
This is the degenerate case where the Heronian t... | There exists [[Definition:Unique|exactly one]] [[Definition:Heronian Triangle|Heronian triangle]] one of whose [[Definition:Altitude of Triangle|altitudes]] and its [[Definition:Side of Polygon|sides]] are all consecutive [[Definition:Integer|integers]].
This is the [[Definition:Heronian Triangle|Heronian triangle]] w... | We note that a [[Definition:Heronian Triangle|Heronian triangle]] whose [[Definition:Side of Polygon|sides]] are all consecutive [[Definition:Integer|integers]] is also known as a [[Definition:Fleenor-Heronian Triangle|Fleenor-Heronian triangle]].
From [[Definition:Fleenor-Heronian Triangle/Sequence|Sequence of Fleeno... | Heronian Triangle whose Altitude and Sides are Consecutive Integers | https://proofwiki.org/wiki/Heronian_Triangle_whose_Altitude_and_Sides_are_Consecutive_Integers | https://proofwiki.org/wiki/Heronian_Triangle_whose_Altitude_and_Sides_are_Consecutive_Integers | [
"Heronian Triangles"
] | [
"Definition:Unique",
"Definition:Heronian Triangle",
"Definition:Altitude of Triangle",
"Definition:Polygon/Side",
"Definition:Integer",
"Definition:Heronian Triangle",
"Definition:Polygon/Side",
"Definition:Altitude of Triangle"
] | [
"Definition:Heronian Triangle",
"Definition:Polygon/Side",
"Definition:Integer",
"Definition:Fleenor-Heronian Triangle",
"Definition:Fleenor-Heronian Triangle/Sequence",
"Definition:Fleenor-Heronian Triangle",
"Definition:Altitude of Triangle",
"Definition:Degenerate Case",
"Definition:Heronian Tria... |
proofwiki-17503 | Integer Heronian Triangle can be Scaled so Area equals Perimeter | Let $T_1$ be an integer Heronian triangle whose sides are $a$, $b$ and $c$.
Then there exists a rational number $k$ such that the Heronian triangle $T_2$ whose sides are $k a$, $k b$ and $k c$ such that the perimeter of $T$ is equal to the area of $T$. | For a given triangle $T$:
:let $\map \AA T$ denote the area of $T$
:let $\map P T$ denote the perimeter of $T$.
We are given that $T_1$ is an integer Heronian triangle whose sides are $a$, $b$ and $c$.
Let $\map P {T_1} = k \map \AA {T_1}$.
Let $T_2$ have sides $k a$, $k b$ and $k c$.
Then we have that:
{{begin-eqn}}
{... | Let $T_1$ be an [[Definition:Integer Heronian Triangle|integer Heronian triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$.
Then there exists a [[Definition:Rational Number|rational number]] $k$ such that the [[Definition:Heronian Triangle|Heronian triangle]] $T_2$ whose [[Definition:Side of Po... | For a given [[Definition:Triangle (Geometry)|triangle]] $T$:
:let $\map \AA T$ denote the [[Definition:Area|area]] of $T$
:let $\map P T$ denote the [[Definition:Perimeter|perimeter]] of $T$.
We are given that $T_1$ is an [[Definition:Integer Heronian Triangle|integer Heronian triangle]] whose [[Definition:Side of Pol... | Integer Heronian Triangle can be Scaled so Area equals Perimeter | https://proofwiki.org/wiki/Integer_Heronian_Triangle_can_be_Scaled_so_Area_equals_Perimeter | https://proofwiki.org/wiki/Integer_Heronian_Triangle_can_be_Scaled_so_Area_equals_Perimeter | [
"Heronian Triangles"
] | [
"Definition:Integer Heronian Triangle",
"Definition:Polygon/Side",
"Definition:Rational Number",
"Definition:Heronian Triangle",
"Definition:Polygon/Side",
"Definition:Perimeter",
"Definition:Area"
] | [
"Definition:Triangle (Geometry)",
"Definition:Area",
"Definition:Perimeter",
"Definition:Integer Heronian Triangle",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Category:Heronian Triangles"
] |
proofwiki-17504 | 3 Proper Integer Heronian Triangles whose Area and Perimeter are Equal | There are exactly $3$ proper integer Heronian triangles whose area and perimeter are equal.
These are the triangles whose sides are:
:$\tuple {6, 25, 29}$
:$\tuple {7, 15, 20}$
:$\tuple {9, 10, 17}$ | First, using Pythagoras's Theorem, we establish that these integer Heronian triangles are indeed proper:
{{begin-eqn}}
{{eqn | l = 6^2 + 25^2
| r = 661
| c =
}}
{{eqn | o = \ne
| r = 29^2
| c = so not right-angled
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 7^2 + 15^2
| r = 274
| c =
... | There are exactly $3$ [[Definition:Proper Heronian Triangle|proper]] [[Definition:Integer Heronian Triangle|integer Heronian triangles]] whose [[Definition:Area|area]] and [[Definition:Perimeter|perimeter]] are equal.
These are the [[Definition:Triangle (Geometry)|triangles]] whose [[Definition:Side of Polygon|sides]... | First, using [[Pythagoras's Theorem]], we establish that these [[Definition:Integer Heronian Triangle|integer Heronian triangles]] are indeed [[Definition:Proper Heronian Triangle|proper]]:
{{begin-eqn}}
{{eqn | l = 6^2 + 25^2
| r = 661
| c =
}}
{{eqn | o = \ne
| r = 29^2
| c = so not [[Defini... | 3 Proper Integer Heronian Triangles whose Area and Perimeter are Equal | https://proofwiki.org/wiki/3_Proper_Integer_Heronian_Triangles_whose_Area_and_Perimeter_are_Equal | https://proofwiki.org/wiki/3_Proper_Integer_Heronian_Triangles_whose_Area_and_Perimeter_are_Equal | [
"Heronian Triangles"
] | [
"Definition:Proper Heronian Triangle",
"Definition:Integer Heronian Triangle",
"Definition:Area",
"Definition:Perimeter",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side"
] | [
"Pythagoras's Theorem",
"Definition:Integer Heronian Triangle",
"Definition:Proper Heronian Triangle",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Area",
"Definition:Perimeter",
"Heron's Form... |
proofwiki-17505 | Heronian Triangle is Similar to Integer Heronian Triangle | Let $\triangle {ABC}$ be a Heronian triangle.
Then there exists an integer Heronian triangle $\triangle {A'B'C'}$ such that $\triangle {ABC}$ and $\triangle {A'B'C'}$ are similar. | Let $\triangle {ABC}$ have sides whose lengths are $a$, $b$ and $c$.
By definition of Heronian triangle, each of $a$, $b$ and $c$ are rational.
By definition of rational number, we can express:
:$a = \dfrac {p_a} {q_a}$, $b = \dfrac {p_b} {q_b}$ and $c = \dfrac {p_c} {q_c}$
where each of $p_a, q_a, p_b, q_b, p_c, q_c$ ... | Let $\triangle {ABC}$ be a [[Definition:Heronian Triangle|Heronian triangle]].
Then there exists an [[Definition:Integer Heronian Triangle|integer Heronian triangle]] $\triangle {A'B'C'}$ such that $\triangle {ABC}$ and $\triangle {A'B'C'}$ are [[Definition:Similar Triangles|similar]]. | Let $\triangle {ABC}$ have [[Definition:Side of Polygon|sides]] whose [[Definition:Length of Line|lengths]] are $a$, $b$ and $c$.
By definition of [[Definition:Heronian Triangle|Heronian triangle]], each of $a$, $b$ and $c$ are [[Definition:Rational Number|rational]].
By definition of [[Definition:Rational Number|rat... | Heronian Triangle is Similar to Integer Heronian Triangle | https://proofwiki.org/wiki/Heronian_Triangle_is_Similar_to_Integer_Heronian_Triangle | https://proofwiki.org/wiki/Heronian_Triangle_is_Similar_to_Integer_Heronian_Triangle | [
"Heronian Triangles",
"Similar Triangles"
] | [
"Definition:Heronian Triangle",
"Definition:Integer Heronian Triangle",
"Definition:Similar Triangles"
] | [
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Heronian Triangle",
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Integer",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Similar Tria... |
proofwiki-17506 | Triple with Sum and Product Equal | For $a, b, c \in \Z$, $a \le b \le c$, the solutions to the equation:
:$a + b + c = a b c$
are:
:$\tuple {1, 2, 3}$
:$\tuple {-3, -2, -1}$
and the trivial solution set:
:$\set {\tuple {-z, 0, z}: z \in \N}$ | Suppose one of $a, b, c$ is zero.
Then $a b c = 0 = a + b + c$.
The remaining two numbers sum to $0$, giving the solution set:
:$\set {\tuple {-z, 0, z}: z \in \N}$
{{qed|lemma}}
Suppose $a < 0$ and $0 < b \le c$.
Then $a b c \le a < a + b + c$.
Hence equality never happens.
Similarly, for $a \le b < 0$ and $c > 0$:
:$... | For $a, b, c \in \Z$, $a \le b \le c$, the solutions to the equation:
:$a + b + c = a b c$
are:
:$\tuple {1, 2, 3}$
:$\tuple {-3, -2, -1}$
and the trivial solution set:
:$\set {\tuple {-z, 0, z}: z \in \N}$ | Suppose one of $a, b, c$ is [[Definition:Zero (Number)|zero]].
Then $a b c = 0 = a + b + c$.
The remaining two numbers [[Definition:Integer Addition|sum]] to $0$, giving the solution set:
:$\set {\tuple {-z, 0, z}: z \in \N}$
{{qed|lemma}}
Suppose $a < 0$ and $0 < b \le c$.
Then $a b c \le a < a + b + c$.
Hence e... | Triple with Sum and Product Equal | https://proofwiki.org/wiki/Triple_with_Sum_and_Product_Equal | https://proofwiki.org/wiki/Triple_with_Sum_and_Product_Equal | [
"6"
] | [] | [
"Definition:Zero (Number)",
"Definition:Addition/Integers",
"Definition:Contradiction",
"Definition:Strictly Positive/Integer",
"Category:6"
] |
proofwiki-17507 | Triple with Product Quadruple the Sum | Let $a, b, c \in \N$ such that $a \le b \le c$.
Then the solutions to:
:$a b c = 4 \paren {a + b + c}$
are:
:$\tuple {0, 0, 0}, \tuple {1, 5, 24}, \tuple {1, 6, 14}, \tuple {1, 8, 9}, \tuple {2, 3, 10}, \tuple {2, 4, 6}$ | Suppose $a \ge 4$.
Then:
{{begin-eqn}}
{{eqn | l = a b c
| o = \ge
| r = 16 c
| c = as $4 \le a \le b$
}}
{{eqn | o = \ge
| r = 4 \paren {a + b + c + c}
| c = as $a \le b \le c$
}}
{{eqn | o = >
| r = 4 \paren {a + b + c}
| c = as $c > 0$
}}
{{end-eqn}}
hence $0 \le a \le 3$.
F... | Let $a, b, c \in \N$ such that $a \le b \le c$.
Then the solutions to:
:$a b c = 4 \paren {a + b + c}$
are:
:$\tuple {0, 0, 0}, \tuple {1, 5, 24}, \tuple {1, 6, 14}, \tuple {1, 8, 9}, \tuple {2, 3, 10}, \tuple {2, 4, 6}$ | Suppose $a \ge 4$.
Then:
{{begin-eqn}}
{{eqn | l = a b c
| o = \ge
| r = 16 c
| c = as $4 \le a \le b$
}}
{{eqn | o = \ge
| r = 4 \paren {a + b + c + c}
| c = as $a \le b \le c$
}}
{{eqn | o = >
| r = 4 \paren {a + b + c}
| c = as $c > 0$
}}
{{end-eqn}}
hence $0 \le a \le 3$.... | Triple with Product Quadruple the Sum | https://proofwiki.org/wiki/Triple_with_Product_Quadruple_the_Sum | https://proofwiki.org/wiki/Triple_with_Product_Quadruple_the_Sum | [] | [] | [
"Definition:Strictly Positive/Integer"
] |
proofwiki-17508 | Area of Integer Heronian Triangle is Multiple of 6 | Let $\triangle {ABC}$ be an integer Heronian triangle.
Then the area of $\triangle {ABC}$ is a multiple of $6$. | Heron's Formula gives us that:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where:
:$\AA$ denotes the area of the triangle
:$a$, $b$ and $c$ denote the lengths of the sides of the triangle
:$s = \dfrac {a + b + c} 2$ denotes the semiperimeter of the triangle.
We set out to eliminate $s$ and simplify... | Let $\triangle {ABC}$ be an [[Definition:Integer Heronian Triangle|integer Heronian triangle]].
Then the [[Definition:Area|area]] of $\triangle {ABC}$ is a [[Definition:Integer Multiple|multiple]] of $6$. | [[Heron's Formula]] gives us that:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where:
:$\AA$ denotes the [[Definition:Area|area]] of the [[Definition:Triangle (Geometry)|triangle]]
:$a$, $b$ and $c$ denote the [[Definition:Length of Line|lengths]] of the [[Definition:Side of Polygon|sides]] of the... | Area of Integer Heronian Triangle is Multiple of 6 | https://proofwiki.org/wiki/Area_of_Integer_Heronian_Triangle_is_Multiple_of_6 | https://proofwiki.org/wiki/Area_of_Integer_Heronian_Triangle_is_Multiple_of_6 | [
"Heronian Triangles"
] | [
"Definition:Integer Heronian Triangle",
"Definition:Area",
"Definition:Integral Multiple/Real Numbers"
] | [
"Heron's Formula",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Semiperimeter",
"Definition:Triangle (Geometry)",
"Definition:Square/Function",
"Solutions of Pythagorean Equation",
... |
proofwiki-17509 | Proper Integer Heronian Triangle whose Area is 24 | There exists exactly one proper integer Heronian triangle whose area equals $24$.
That is, the obtuse triangle whose sides are of length $4$, $13$ and $15$. | First we show that the $\tuple {4, 13, 15}$ triangle is actually Heronian.
Heron's Formula gives us that:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where:
:$\AA$ denotes the area of the triangle
:$a$, $b$ and $c$ denote the lengths of the sides of the triangle
:$s = \dfrac {a + b + c} 2$ denotes ... | There exists [[Definition:Unique|exactly one]] [[Definition:Proper Heronian Triangle|proper]] [[Definition:Integer Heronian Triangle|integer Heronian triangle]] whose [[Definition:Area|area]] equals $24$.
That is, the [[Definition:Obtuse Triangle|obtuse triangle]] whose [[Definition:Side of Polygon|sides]] are of [[D... | First we show that the $\tuple {4, 13, 15}$ [[Definition:Triangle (Geometry)|triangle]] is actually [[Definition:Heronian Triangle|Heronian]].
[[Heron's Formula]] gives us that:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where:
:$\AA$ denotes the [[Definition:Area|area]] of the [[Definition:Tria... | Proper Integer Heronian Triangle whose Area is 24 | https://proofwiki.org/wiki/Proper_Integer_Heronian_Triangle_whose_Area_is_24 | https://proofwiki.org/wiki/Proper_Integer_Heronian_Triangle_whose_Area_is_24 | [
"Heronian Triangles"
] | [
"Definition:Unique",
"Definition:Proper Heronian Triangle",
"Definition:Integer Heronian Triangle",
"Definition:Area",
"Definition:Triangle (Geometry)/Obtuse",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length"
] | [
"Definition:Triangle (Geometry)",
"Definition:Heronian Triangle",
"Heron's Formula",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Semiperimeter",
"Definition:Triangle (Geometry)",
... |
proofwiki-17510 | Semiperimeter of Integer Heronian Triangle is Composite | The semiperimeter of an integer Heronian triangle is always a composite number. | Let $a, b, c$ be the side lengths of an integer Heronian triangle.
By Heron's Formula, its area is given by:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } \in \N$
where the semiperimeter $s$ is given by:
:$s = \dfrac {a + b + c} 2$
First we prove that $s$ is indeed an integer.
{{AimForCont}} not.
Sinc... | The [[Definition:Semiperimeter|semiperimeter]] of an [[Definition:Integer Heronian Triangle|integer Heronian triangle]] is always a [[Definition:Composite Number|composite number]]. | Let $a, b, c$ be the [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] of an [[Definition:Integer Heronian Triangle|integer Heronian triangle]].
By [[Heron's Formula]], its [[Definition:Area|area]] is given by:
:$\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} } \in \N$
where the [... | Semiperimeter of Integer Heronian Triangle is Composite | https://proofwiki.org/wiki/Semiperimeter_of_Integer_Heronian_Triangle_is_Composite | https://proofwiki.org/wiki/Semiperimeter_of_Integer_Heronian_Triangle_is_Composite | [
"Heronian Triangles"
] | [
"Definition:Semiperimeter",
"Definition:Integer Heronian Triangle",
"Definition:Composite Number"
] | [
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Integer Heronian Triangle",
"Heron's Formula",
"Definition:Area",
"Definition:Semiperimeter",
"Definition:Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
... |
proofwiki-17511 | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Necessary Condition | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\mathscr B$ be the set of bases of the matroid $M$.
Then $\mathscr B$ satisfies formulation $1$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}} | Let $B_1, B_2 \in \mathscr B$.
Let $x \in B_1 \setminus B_2$.
We have:
{{begin-eqn}}
{{eqn | l = \size {B_1 \setminus \set x}
| r = \size {B_1} - \size {\set x}
| c = Cardinality of Set Difference with Subset
}}
{{eqn | r = \size {B_2} - \size {\set x}
| c = All Bases of Matroid have same Cardinality
... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\mathscr B$ be the set of [[Definition:Base of Matroid|bases]] of the [[Definition:Matroid|matroid]] $M$.
Then $\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 1|formulation $1$ of base axiom]]:
{{:Axiom:Base Axiom (Matroid)/F... | Let $B_1, B_2 \in \mathscr B$.
Let $x \in B_1 \setminus B_2$.
We have:
{{begin-eqn}}
{{eqn | l = \size {B_1 \setminus \set x}
| r = \size {B_1} - \size {\set x}
| c = [[Cardinality of Set Difference with Subset]]
}}
{{eqn | r = \size {B_2} - \size {\set x}
| c = [[All Bases of Matroid have same Ca... | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Necessary Condition | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom/Necessary_Condition | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom/Necessary_Condition | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom"
] | [
"Definition:Matroid",
"Definition:Base of Matroid",
"Definition:Matroid",
"Axiom:Base Axiom (Matroid)/Formulation 1"
] | [
"Cardinality of Set Difference with Subset",
"All Bases of Matroid have same Cardinality",
"Cardinality of Singleton",
"Axiom:Matroid Axioms",
"Set Difference with Set Difference is Union of Set Difference with Intersection",
"Intersection With Singleton is Disjoint if Not Element",
"Union with Empty Se... |
proofwiki-17512 | Independent Subset is Base if Cardinality Equals Rank of Matroid | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\rho: \powerset S \to \Z$ be the rank function of $M$.
Let $B \in \mathscr I$ such that:
:$\size B = \map \rho S$
Then:
:$B$ is a base of $M$. | Let $Z \in \mathscr I$ such that:
:$B \subseteq Z$
From Cardinality of Subset of Finite Set:
:$\size B \le \size Z$
By definition of the rank function:
:$\size Z \le \map \rho S$
Then:
:$\size Z = \size B$
From the contrapositive statement of Cardinality of Proper Subset of Finite Set:
:$B = Z$
It follows that $B$ is a... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\rho: \powerset S \to \Z$ be the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
Let $B \in \mathscr I$ such that:
:$\size B = \map \rho S$
Then:
:$B$ is a [[Definition:Base of Matroid|base]] of $M$. | Let $Z \in \mathscr I$ such that:
:$B \subseteq Z$
From [[Cardinality of Subset of Finite Set]]:
:$\size B \le \size Z$
By definition of the [[Definition:Rank Function (Matroid)|rank function]]:
:$\size Z \le \map \rho S$
Then:
:$\size Z = \size B$
From the [[Definition:Contrapositive Statement|contrapositive state... | Independent Subset is Base if Cardinality Equals Rank of Matroid | https://proofwiki.org/wiki/Independent_Subset_is_Base_if_Cardinality_Equals_Rank_of_Matroid | https://proofwiki.org/wiki/Independent_Subset_is_Base_if_Cardinality_Equals_Rank_of_Matroid | [
"Matroid Independent Subsets",
"Matroid Bases",
"Matroid Rank Functions"
] | [
"Definition:Matroid",
"Definition:Rank Function (Matroid)",
"Definition:Base of Matroid"
] | [
"Cardinality of Subset of Finite Set",
"Definition:Rank Function (Matroid)",
"Definition:Contrapositive Statement",
"Cardinality of Proper Subset of Finite Set",
"Definition:Maximal",
"Definition:Matroid/Independent Set",
"Definition:Base of Matroid",
"Category:Matroid Independent Subsets",
"Categor... |
proofwiki-17513 | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Sufficient Condition | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$ satisfying formulation $1$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}}
Then $\mathscr B$ is the set of bases of a matroid on $S$. | Let $\mathscr I = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq B}$
It is to be shown that:
:* $\quad \mathscr I$ satisfies the matroid axioms
and
:* $\quad \mathscr B$ is the set of bases of the matroid $M = \struct{S, \mathscr I}$
From Independence System Induced from Set of Subsets:
:$\mathscr I$ is an... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$ satisfying [[Axiom:Base Axiom (Matroid)/Formulation 1|formulation $1$ of base axiom]]:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}}
Then $\ma... | Let $\mathscr I = \set{X \subseteq S : \exists B \in \mathscr B : X \subseteq B}$
It is to be shown that:
:* $\quad \mathscr I$ satisfies the [[Axiom:Matroid Axioms|matroid axioms]]
and
:* $\quad \mathscr B$ is the [[Definition:Set|set]] of [[Definition:Base of Matroid|bases]] of the [[Definition:Matroid|matroid]] $M ... | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Sufficient Condition | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom/Sufficient_Condition | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom/Sufficient_Condition | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Definition:Base of Matroid",
"Definition:Matroid"
] | [
"Axiom:Matroid Axioms",
"Definition:Set",
"Definition:Base of Matroid",
"Definition:Matroid",
"Independence System Induced from Set of Subsets",
"Definition:Independence System",
"Max Operation Equals an Operand",
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Lemma 1",
"Definition... |
proofwiki-17514 | Morley's Trisector Theorem | Let $\triangle ABC$ be a triangle.
Let the internal angles of $\triangle ABC$ be trisected.
Let the points where these angle trisectors first intersect be $D$, $E$ and $F$.
:500px
Then $\triangle EDF$ is equilateral. | By comparing the '''given triangle''' $\triangle A'B'C'$ with the '''constructed triangle''' $\triangle ABC $, we shall prove that $\triangle X'Y'Z' \sim \triangle XYZ$ where $\triangle XYZ$ is an equilateral triangle.
'''The Given Triangle''' $\triangle A'B'C'$
:File:Morleys-Theorem-Fig1xxxx.png
'''The Constructed Tr... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let the [[Definition:Internal Angle|internal angles]] of $\triangle ABC$ be [[Definition:Trisection|trisected]].
Let the [[Definition:Point|points]] where these [[Definition:Angle Trisector|angle trisectors]] first [[Definition:Intersection (Geomet... | By comparing the '''given triangle''' $\triangle A'B'C'$ with the '''constructed triangle''' $\triangle ABC $, we shall prove that $\triangle X'Y'Z' \sim \triangle XYZ$ where $\triangle XYZ$ is an [[Definition:Equilateral Triangle|equilateral triangle]].
'''The Given Triangle''' $\triangle A'B'C'$
:[[File:Morleys-Th... | Morley's Trisector Theorem/Proof 2 | https://proofwiki.org/wiki/Morley's_Trisector_Theorem | https://proofwiki.org/wiki/Morley's_Trisector_Theorem/Proof_2 | [
"Morley's Trisector Theorem",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Internal Angle",
"Definition:Trisection",
"Definition:Point",
"Definition:Angle Trisector",
"Definition:Intersection (Geometry)",
"File:Morleys-Theorem.png",
"Definition:Triangle (Geometry)/Equilateral"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"File:Morleys-Theorem-Fig1xxxx.png",
"File:Morleys-Theorem-Fig2xx.png",
"Definition:Triangle (Geometry)/Equilateral",
"Law of Sines",
"Law of Sines",
"Triangles with One Equal Angle and Two Sides Proportional are Similar",
"File:Morleys-Theorem-Auxiliary-T... |
proofwiki-17515 | Mean Number of Elements Fixed by Self-Map | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $S$ be a finite set of cardinality $n$.
Let $S^S$ be the set of all mappings from $S$ to itself.
Let $\map \mu n$ denote the arithmetic mean of the number of fixed points of all the mappings in $S^S$.
Then:
:$\map \mu n = 1$ | Let $f \in S^S$ be an arbitrary mapping from $S$ to itself.
Let $s \in S$ be an arbitrary element of $S$.
$s$ has an equal probability of being mapped to any element of $S$.
Hence the probability that $\map f s = s$ is equal to $\dfrac 1 n$.
There are $n$ elements of $S$.
By the above argument, each one has a probabili... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $S$ be a [[Definition:Finite Set|finite set]] of [[Definition:Cardinality|cardinality]] $n$.
Let $S^S$ be the [[Definition:Set of All Mappings|set of all mappings]] from $S$ to itself.
Let $\map \mu n$ denote the [[Defin... | Let $f \in S^S$ be an arbitrary [[Definition:Mapping|mapping]] from $S$ to itself.
Let $s \in S$ be an arbitrary [[Definition:Element|element]] of $S$.
$s$ has an equal [[Definition:Probability|probability]] of being mapped to any [[Definition:Element|element]] of $S$.
Hence the [[Definition:Probability|probability]... | Mean Number of Elements Fixed by Self-Map | https://proofwiki.org/wiki/Mean_Number_of_Elements_Fixed_by_Self-Map | https://proofwiki.org/wiki/Mean_Number_of_Elements_Fixed_by_Self-Map | [
"Combinatorics",
"Probability Theory",
"Mean Number of Elements Fixed by Self-Map"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Set of All Mappings",
"Definition:Arithmetic Mean",
"Definition:Fixed Point",
"Definition:Mapping"
] | [
"Definition:Mapping",
"Definition:Element",
"Definition:Probability",
"Definition:Element",
"Definition:Probability",
"Definition:Element",
"Definition:Probability",
"Definition:Fixed Point",
"Definition:Expectation",
"Definition:Fixed Point"
] |
proofwiki-17516 | Condition for 3 over n producing 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient | Consider proper fractions of the form $\dfrac 3 n$ expressed in canonical form.
Let Fibonacci's Greedy Algorithm be used to generate a sequence $S$ of Egyptian fractions for $\dfrac 3 n$.
Then $S$ consists of $3$ terms, where $2$ would be sufficient {{iff}} the following conditions hold:
:$n \equiv 1 \pmod 6$
:$\exists... | By Upper Limit of Number of Unit Fractions to express Proper Fraction from Greedy Algorithm, $S$ consists of no more than $3$ terms.
Suppose $n$ has our desired property.
Since $\dfrac 3 n$ is proper, $n \ge 4$.
Since $\dfrac 3 n$ is in canonical form, $3 \nmid n$.
We also have that $S$ consists of at least $2$ terms.
... | Consider [[Definition:Proper Fraction|proper fractions]] of the form $\dfrac 3 n$ expressed in [[Definition:Canonical Form of Rational Number|canonical form]].
Let [[Fibonacci's Greedy Algorithm]] be used to generate a [[Definition:Sequence|sequence]] $S$ of [[Definition:Egyptian Fraction|Egyptian fractions]] for $\df... | By [[Upper Limit of Number of Unit Fractions to express Proper Fraction from Greedy Algorithm]], $S$ consists of no more than $3$ [[Definition:Term of Sequence|terms]].
Suppose $n$ has our desired property.
Since $\dfrac 3 n$ is [[Definition:Proper Fraction|proper]], $n \ge 4$.
Since $\dfrac 3 n$ is in [[Definition:... | Condition for 3 over n producing 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient | https://proofwiki.org/wiki/Condition_for_3_over_n_producing_3_Egyptian_Fractions_using_Greedy_Algorithm_when_2_Sufficient | https://proofwiki.org/wiki/Condition_for_3_over_n_producing_3_Egyptian_Fractions_using_Greedy_Algorithm_when_2_Sufficient | [
"Fibonacci's Greedy Algorithm",
"Egyptian Fractions"
] | [
"Definition:Fraction/Proper",
"Definition:Rational Number/Canonical Form",
"Fibonacci's Greedy Algorithm",
"Definition:Sequence",
"Definition:Egyptian Fraction",
"Definition:Term of Sequence",
"Definition:Term of Sequence"
] | [
"Upper Limit of Number of Unit Fractions to express Proper Fraction from Greedy Algorithm",
"Definition:Term of Sequence",
"Definition:Fraction/Proper",
"Definition:Rational Number/Canonical Form",
"Definition:Term of Sequence",
"Fibonacci's Greedy Algorithm",
"Definition:Term of Sequence",
"Definitio... |
proofwiki-17517 | Divisor of Product | Let $a, b, c \in \Z$ be integers.
Let the symbol $\divides$ denote the divisibility relation.
Let $a \divides b c$.
Then there exist integers $r, s$ such that:
:$a = r s$, where $r \divides b$ and $s \divides c$. | Let $r = \gcd \set {a, b}$.
By Integers Divided by GCD are Coprime:
:$\exists s, t \in \Z: a = r s \land b = r t \land \gcd \set {s, t} = 1$
So we have written $a = r s$ where $r$ divides $b$.
We now show that $s$ divides $c$.
Since $a$ divides $b c$ there exists $k$ such that $b c = k a$.
Substituting for $a$ and $b$:... | Let $a, b, c \in \Z$ be [[Definition:Integer|integers]].
Let the [[Definition:Symbol|symbol]] $\divides$ denote the [[Definition:Divisor of Integer|divisibility relation]].
Let $a \divides b c$.
Then there exist [[Definition:Integer|integers]] $r, s$ such that:
:$a = r s$, where $r \divides b$ and $s \divides c$. | Let $r = \gcd \set {a, b}$.
By [[Integers Divided by GCD are Coprime]]:
:$\exists s, t \in \Z: a = r s \land b = r t \land \gcd \set {s, t} = 1$
So we have written $a = r s$ where $r$ [[Definition:Divisor of Integer|divides]] $b$.
We now show that $s$ [[Definition:Divisor of Integer|divides]] $c$.
Since $a$ [[Defi... | Divisor of Product | https://proofwiki.org/wiki/Divisor_of_Product | https://proofwiki.org/wiki/Divisor_of_Product | [
"Divisors"
] | [
"Definition:Integer",
"Definition:Symbol",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer"
] | [
"Integers Divided by GCD are Coprime",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Euclid's Lemma",
"Category:Divisors"
] |
proofwiki-17518 | Equivalence of Definitions of Matroid | Let $M = \struct {S, \mathscr I}$ be an independence system.
{{TFAE|def = Matroid}} | === Definition 1 implies Definition 2 ===
{{:Equivalence of Definitions of Matroid/Definition 1 implies Definition 2}}{{qed|lemma}} | Let $M = \struct {S, \mathscr I}$ be an [[Definition:Independence System|independence system]].
{{TFAE|def = Matroid}} | === [[Equivalence of Definitions of Matroid/Definition 1 implies Definition 2|Definition 1 implies Definition 2]] ===
{{:Equivalence of Definitions of Matroid/Definition 1 implies Definition 2}}{{qed|lemma}} | Equivalence of Definitions of Matroid | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid | [
"Matroid Theory",
"Equivalence of Definitions of Matroid"
] | [
"Definition:Independence System"
] | [
"Equivalence of Definitions of Matroid/Definition 1 implies Definition 2"
] |
proofwiki-17519 | P-adic Norm is Well Defined | P-adic norm $\norm {\, \cdot \,}_p$ is well defined. | {{AimForCont}} $\norm {\, \cdot \,}_p$ is not well defined.
Then, given $r \in \Q$, for two equivalent representations of $r$, $\norm r_p$ will yield two different results.
Let $k_1, k_2, m_1, m_2 \in \Z, n_1, n_2 \in \Z_{\ne 0} : p \nmid m_1, m_2, n_1, n_2$.
Let $\ds r = p^{k_1} \frac {m_1} {n_1} = p^{k_2} \frac {m_2... | [[Definition:P-adic Norm|P-adic norm]] $\norm {\, \cdot \,}_p$ is [[Definition:Well-Defined|well defined]]. | {{AimForCont}} $\norm {\, \cdot \,}_p$ is not [[Definition:Well-Defined|well defined]].
Then, given $r \in \Q$, for two [[Equivalent Representations of Rational Numbers|equivalent representations]] of $r$, $\norm r_p$ will yield two different results.
Let $k_1, k_2, m_1, m_2 \in \Z, n_1, n_2 \in \Z_{\ne 0} : p \nmid... | P-adic Norm is Well Defined | https://proofwiki.org/wiki/P-adic_Norm_is_Well_Defined | https://proofwiki.org/wiki/P-adic_Norm_is_Well_Defined | [
"P-adic Number Theory"
] | [
"Definition:P-adic Norm",
"Definition:Well-Defined"
] | [
"Definition:Well-Defined",
"Equivalent Representations of Rational Numbers",
"Definition:Prime Number",
"Definition:Multiplication/Product",
"Definition:Divisor (Algebra)/Integer",
"Definition:Contradiction",
"Definition:Assumption",
"Definition:Contradiction",
"Definition:Integer",
"Integers form... |
proofwiki-17520 | Equivalence of Definitions of Matroid/Definition 1 implies Definition 2 | Let $M = \struct {S, \mathscr I}$ be an independence system.
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 3
| q = \forall U, V \in \mathscr I
| mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}}
Then $M$ satisfies:
{{begin-axiom}}
{{... | Since:
:$\forall U, V \in \mathscr I : \size U = \size V + 1 \implies \size V < \size U$
If follows that if $M$ satisfies condition $(\text I 3)$ then $M$ satisfies condition $(\text I 4)$. | Let $M = \struct {S, \mathscr I}$ be an [[Definition:Independence System|independence system]].
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 3
| q = \forall U, V \in \mathscr I
| mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}}
... | Since:
:$\forall U, V \in \mathscr I : \size U = \size V + 1 \implies \size V < \size U$
If follows that if $M$ satisfies condition $(\text I 3)$ then $M$ satisfies condition $(\text I 4)$. | Equivalence of Definitions of Matroid/Definition 1 implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_1_implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_1_implies_Definition_2 | [
"Equivalence of Definitions of Matroid"
] | [
"Definition:Independence System"
] | [] |
proofwiki-17521 | Equivalence of Definitions of Matroid/Definition 2 implies Definition 3 | Let $M = \struct {S, \mathscr I}$ be an independence system.
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 4
| q = \forall U, V \in \mathscr I
| mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}}
Then $M$ satisfies:
{{begin-axiom}... | From Independent Set can be Augmented by Larger Independent Set it follows that if $M$ satisfies condition $(\text I 4)$ then $M$ satisfies condition $(\text I 5)$. | Let $M = \struct {S, \mathscr I}$ be an [[Definition:Independence System|independence system]].
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 4
| q = \forall U, V \in \mathscr I
| mr= \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}... | From [[Independent Set can be Augmented by Larger Independent Set]] it follows that if $M$ satisfies condition $(\text I 4)$ then $M$ satisfies condition $(\text I 5)$. | Equivalence of Definitions of Matroid/Definition 2 implies Definition 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_2_implies_Definition_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_2_implies_Definition_3 | [
"Equivalence of Definitions of Matroid"
] | [
"Definition:Independence System"
] | [
"Independent Set can be Augmented by Larger Independent Set"
] |
proofwiki-17522 | Equivalence of Definitions of Matroid/Definition 3 implies Definition 1 | Let $M = \struct {S, \mathscr I}$ be an independence system.
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 5
| q = \forall U, V \in \mathscr I
| mr= \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren {V \cup Z \in \mathscr I} \land \paren {\size {V \cup Z} = \size U}
}}
... | Let $M$ satisfy condition $(\text I 5)$.
Let $U, V \in \mathscr I$ such that $\size V < \size U$.
By condition $(\text I 5)$:
:$\exists Z : \exists Z \subseteq U \setminus V : \paren {V \cup Z \in \mathscr I} \land \paren {\size {V \cup Z} = \size U}$
Then:
:$V \cup Z \ne V$
From Union with Empty Set:
:$Z \ne \O$
Then:... | Let $M = \struct {S, \mathscr I}$ be an [[Definition:Independence System|independence system]].
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 5
| q = \forall U, V \in \mathscr I
| mr= \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren {V \cup Z \in \mathscr I} \land \pa... | Let $M$ satisfy condition $(\text I 5)$.
Let $U, V \in \mathscr I$ such that $\size V < \size U$.
By condition $(\text I 5)$:
:$\exists Z : \exists Z \subseteq U \setminus V : \paren {V \cup Z \in \mathscr I} \land \paren {\size {V \cup Z} = \size U}$
Then:
:$V \cup Z \ne V$
From [[Union with Empty Set]]:
:$Z \ne... | Equivalence of Definitions of Matroid/Definition 3 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_3_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_3_implies_Definition_1 | [
"Equivalence of Definitions of Matroid"
] | [
"Definition:Independence System"
] | [
"Union with Empty Set",
"Singleton of Element is Subset",
"Set Union Preserves Subsets",
"Axiom:Independence System Axioms",
"Definition:Subset"
] |
proofwiki-17523 | Equivalence of Definitions of Matroid/Definition 1 implies Definition 4 | Let $M = \struct {S, \mathscr I}$ be an independence system.
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 3
| q = \forall U, V \in \mathscr I
| mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}}
Then $M$ satisfies:
{{begin-axiom}}
{{... | Let $M$ satisfy condition $(\text I 3)$.
Let $A \subseteq S$.
Let $Y_1, Y_2$ be maximal independent subsets of $A$.
{{WLOG}}, let:
:$\size {Y_2} \le \size {Y_1}$
{{AimForCont}}:
:$\size {Y_2} < \size {Y_1}$
By condition $(\text I 3)$:
:$\exists y \in Y_1 \setminus Y_2 : Y_2 \cup \set y \in \mathscr I$
From Union of Sub... | Let $M = \struct {S, \mathscr I}$ be an [[Definition:Independence System|independence system]].
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 3
| q = \forall U, V \in \mathscr I
| mr= \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I
}}
{{end-axiom}}
... | Let $M$ satisfy condition $(\text I 3)$.
Let $A \subseteq S$.
Let $Y_1, Y_2$ be [[Definition:Maximal|maximal]] [[Definition:Independent Subset (Matroid)|independent subsets]] of $A$.
{{WLOG}}, let:
:$\size {Y_2} \le \size {Y_1}$
{{AimForCont}}:
:$\size {Y_2} < \size {Y_1}$
By condition $(\text I 3)$:
:$\exists y ... | Equivalence of Definitions of Matroid/Definition 1 implies Definition 4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_1_implies_Definition_4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_1_implies_Definition_4 | [
"Equivalence of Definitions of Matroid"
] | [
"Definition:Independence System"
] | [
"Definition:Maximal",
"Definition:Matroid/Independent Set",
"Union of Subsets is Subset",
"Definition:Contradiction",
"Definition:Maximal"
] |
proofwiki-17524 | Equivalence of Definitions of Matroid/Definition 4 implies Definition 1 | Let $M = \struct {S, \mathscr I}$ be an independence system.
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 6
| q = \forall A \subseteq S
| mr= \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality}
}}
{{end-axiom}}
Then $M$ satisfies:
{{... | Let $M$ satisfy condition $(\text I 6)$.
Let $U, V \in \mathscr I$ such that $\size V < \size U$.
Let $W$ be a maximal independent subset of $U \cup V$ containing $U$.
Then:
:$\size U \le \size W$
By condition $(\text I 6)$:
:$V$ is not a maximal independent subset of $U \cup V$
Then:
:$\exists x \in \paren {U \cup V} ... | Let $M = \struct {S, \mathscr I}$ be an [[Definition:Independence System|independence system]].
Let $M$ also satisfy:
{{begin-axiom}}
{{axiom | n = \text I 6
| q = \forall A \subseteq S
| mr= \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality}
}}
... | Let $M$ satisfy condition $(\text I 6)$.
Let $U, V \in \mathscr I$ such that $\size V < \size U$.
Let $W$ be a [[Definition:Maximal|maximal]] [[Definition:Independent Subset (Matroid)|independent subset]] of $U \cup V$ containing $U$.
Then:
:$\size U \le \size W$
By condition $(\text I 6)$:
:$V$ is not a [[Definiti... | Equivalence of Definitions of Matroid/Definition 4 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_4_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid/Definition_4_implies_Definition_1 | [
"Equivalence of Definitions of Matroid"
] | [
"Definition:Independence System"
] | [
"Definition:Maximal",
"Definition:Matroid/Independent Set",
"Definition:Maximal",
"Definition:Matroid/Independent Set",
"Set Difference with Union is Set Difference"
] |
proofwiki-17525 | Cardinality of Set Difference | Let $S$ and $T$ be sets such that $T$ is finite.
Then:
:$\card {S \setminus T} = \card S - \card {S \cap T}$
where $\card S$ denotes the cardinality of $S$. | From Intersection is Subset:
:$S \cap T \subseteq S$
:$S \cap T \subseteq T$
From Subset of Finite Set is Finite:
:$S \cap T$ is finite.
We have:
{{begin-eqn}}
{{eqn | l = \card {S \setminus T}
| r = \card {S \setminus \paren {S \cap T} }
| c = Set Difference with Intersection is Difference
}}
{{eqn | r = \... | Let $S$ and $T$ be [[Definition:Set|sets]] such that $T$ is [[Definition:Finite Set|finite]].
Then:
:$\card {S \setminus T} = \card S - \card {S \cap T}$
where $\card S$ denotes the [[Definition:Cardinality|cardinality]] of $S$. | From [[Intersection is Subset]]:
:$S \cap T \subseteq S$
:$S \cap T \subseteq T$
From [[Subset of Finite Set is Finite]]:
:$S \cap T$ is [[Definition:Finite Set|finite]].
We have:
{{begin-eqn}}
{{eqn | l = \card {S \setminus T}
| r = \card {S \setminus \paren {S \cap T} }
| c = [[Set Difference with Inte... | Cardinality of Set Difference | https://proofwiki.org/wiki/Cardinality_of_Set_Difference | https://proofwiki.org/wiki/Cardinality_of_Set_Difference | [
"Set Difference",
"Cardinality"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Cardinality"
] | [
"Intersection is Subset",
"Subset of Finite Set is Finite",
"Definition:Finite Set",
"Set Difference with Intersection is Difference",
"Cardinality of Set Difference with Subset",
"Category:Set Difference",
"Category:Cardinality"
] |
proofwiki-17526 | Set Difference and Intersection are Disjoint | Let $S$ and $T$ be sets.
Then:
:$S \setminus T$ and $S \cap T$ are disjoint
where $S \setminus T$ denotes set difference and $S \cap T$ denotes set intersection. | From Set Difference Intersection with Second Set is Empty Set:
:$\paren {S \setminus T} \cap T = \O$
and hence immediately from Intersection with Empty Set:
:$\paren {S \setminus T} \cap \paren {S \cap T} = \O$
So $S \setminus T$ and $S \cap T$ are disjoint.
{{qed}}
Category:Set Difference
Category:Set Intersection
5bi... | Let $S$ and $T$ be [[Definition:Set|sets]].
Then:
:$S \setminus T$ and $S \cap T$ are [[Definition:Disjoint Sets|disjoint]]
where $S \setminus T$ denotes [[Definition:Set Difference|set difference]] and $S \cap T$ denotes [[Definition:Set Intersection|set intersection]]. | From [[Set Difference Intersection with Second Set is Empty Set]]:
:$\paren {S \setminus T} \cap T = \O$
and hence immediately from [[Intersection with Empty Set]]:
:$\paren {S \setminus T} \cap \paren {S \cap T} = \O$
So $S \setminus T$ and $S \cap T$ are [[Definition:Disjoint Sets|disjoint]].
{{qed}}
[[Category:Se... | Set Difference and Intersection are Disjoint | https://proofwiki.org/wiki/Set_Difference_and_Intersection_are_Disjoint | https://proofwiki.org/wiki/Set_Difference_and_Intersection_are_Disjoint | [
"Set Difference",
"Set Intersection"
] | [
"Definition:Set",
"Definition:Disjoint Sets",
"Definition:Set Difference",
"Definition:Set Intersection"
] | [
"Set Difference Intersection with Second Set is Empty Set",
"Intersection with Empty Set",
"Definition:Disjoint Sets",
"Category:Set Difference",
"Category:Set Intersection"
] |
proofwiki-17527 | Straight Line has Zero Curvature | A straight line has zero curvature. | From Equation of Straight Line in Plane: Slope-Intercept Form, a straight line has the equation:
:$y = m x + c$
Differentiating twice {{WRT|Differentiation}} $x$:
{{begin-eqn}}
{{eqn | l = \dfrac {\d y} {\d x}
| r = m
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d^2 y} {\d x... | A [[Definition:Straight Line|straight line]] has zero [[Definition:Curvature|curvature]]. | From [[Equation of Straight Line in Plane/Slope-Intercept Form|Equation of Straight Line in Plane: Slope-Intercept Form]], a [[Definition:Straight Line|straight line]] has the [[Definition:Equation|equation]]:
:$y = m x + c$
[[Definition:Differentiation|Differentiating]] twice {{WRT|Differentiation}} $x$:
{{begin-eq... | Straight Line has Zero Curvature | https://proofwiki.org/wiki/Straight_Line_has_Zero_Curvature | https://proofwiki.org/wiki/Straight_Line_has_Zero_Curvature | [
"Straight Lines",
"Curvature"
] | [
"Definition:Line/Straight Line",
"Definition:Curvature"
] | [
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Line/Straight Line",
"Definition:Equation",
"Definition:Differentiation",
"Power Rule for Derivatives",
"Definition:Curvature/Cartesian Form",
"Definition:Line/Curve",
"Definition:Curvature"
] |
proofwiki-17528 | Partial Differential Equation of Spheres in 3-Space | The set of spheres in real Cartesian $3$-dimensional space can be described by the system of partial differential equations:
:$\dfrac {1 + z_x^2} {z_{xx} } = \dfrac {z_x z_x} {z_{xy} } = \dfrac {1 + z_y^2} {z_{yy} }$
and if the radii of these spheres are expected to be real:
:$z_{xx} z_{yy} > z_{xy}$ | From Equation of Sphere, we have that the equation defining a general sphere $S$ is:
:$\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$
where $a$, $b$ and $c$ are arbitrary constants.
We use the technique of Elimination of Constants by Partial Differentiation.
Taking the partial first derivatives {{WRT|Dif... | The [[Definition:Set|set]] of [[Definition:Sphere (Geometry)|spheres]] in [[Definition:Real Cartesian Space|real Cartesian $3$-dimensional space]] can be described by the [[Definition:System of Differential Equations|system]] of [[Definition:Partial Differential Equation|partial differential equations]]:
:$\dfrac {1 +... | From [[Equation of Sphere/Rectangular Coordinates|Equation of Sphere]], we have that the [[Definition:Equation|equation]] defining a general [[Definition:Sphere (Geometry)|sphere]] $S$ is:
:$\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$
where $a$, $b$ and $c$ are [[Definition:Arbitrary Constant|arbitrar... | Partial Differential Equation of Spheres in 3-Space | https://proofwiki.org/wiki/Partial_Differential_Equation_of_Spheres_in_3-Space | https://proofwiki.org/wiki/Partial_Differential_Equation_of_Spheres_in_3-Space | [
"Partial Differential Equations",
"Solid Analytic Geometry"
] | [
"Definition:Set",
"Definition:Sphere/Geometry",
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Differential Equation/System",
"Definition:Differential Equation/Partial",
"Definition:Sphere/Geometry/Radius",
"Definition:Sphere/Geometry",
"Definition:Real Number"
] | [
"Equation of Sphere/Rectangular Coordinates",
"Definition:Equation",
"Definition:Sphere/Geometry",
"Definition:Arbitrary Constant",
"Elimination of Constants by Partial Differentiation",
"Definition:Partial Derivative",
"Definition:Equation",
"Definition:Constant",
"Definition:Partial Derivative",
... |
proofwiki-17529 | Fermat's Right Triangle Theorem | $x^4 + y^4 = z^2$ has no solutions in the (strictly) positive integers. | This proof using Method of Infinite Descent was created by {{AuthorRef|Pierre de Fermat}}.
Suppose there is such a solution.
Then there is one with $\gcd \set {x, y, z} = 1$.
By Parity of Smaller Elements of Primitive Pythagorean Triple we can assume that $x^2$ is even and $y^2$ is odd.
By Primitive Solutions of Pythag... | $x^4 + y^4 = z^2$ has no solutions in the [[Definition:Strictly Positive Integer|(strictly) positive integers]]. | This proof using [[Method of Infinite Descent]] was created by {{AuthorRef|Pierre de Fermat}}.
Suppose there is such a solution.
Then there is one with $\gcd \set {x, y, z} = 1$.
By [[Parity of Smaller Elements of Primitive Pythagorean Triple]] we can assume that $x^2$ is [[Definition:Even Integer|even]] and $y^2$ ... | Fermat's Right Triangle Theorem | https://proofwiki.org/wiki/Fermat's_Right_Triangle_Theorem | https://proofwiki.org/wiki/Fermat's_Right_Triangle_Theorem | [
"Number Theory"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Method of Infinite Descent",
"Parity of Smaller Elements of Primitive Pythagorean Triple",
"Definition:Even Integer",
"Definition:Odd Integer",
"Solutions of Pythagorean Equation/Primitive",
"Definition:Coprime/Integers",
"Definition:Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Pos... |
proofwiki-17530 | Taylor's Theorem/One Variable with Two Functions | Let $f$ and $g$ be real functions satisfying following conditions:
:$(1): \quad f$ is $n + 1$ times differentiable on the open interval $\openint a x$
:$(2): \quad f$ is of differentiability class $C^n$ on the closed interval $\closedint a x$
:$(3): \quad g$ is $k + 1$ times differentiable on the open interval $\openin... | We define $F$ and $G$ as follows:
{{begin-eqn}}
{{eqn | l = \map F t
| r = \map f t + \map {f'} t \paren {x - t} + \dfrac {\map {f^{\prime\prime} } t} {2!} \paren {x - t}^2 + \dotsb + \dfrac {\map {f^{\paren n} } t} {n!} \paren {x - t}^n
}}
{{eqn | l = \map G t
| r = \map g t + \map {g'} t \paren {x - t} + ... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] satisfying following conditions:
:$(1): \quad f$ is $n + 1$ times [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a x$
:$(2): \quad f$ is of [[Definition:Differentiability Class|diff... | We define $F$ and $G$ as follows:
{{begin-eqn}}
{{eqn | l = \map F t
| r = \map f t + \map {f'} t \paren {x - t} + \dfrac {\map {f^{\prime\prime} } t} {2!} \paren {x - t}^2 + \dotsb + \dfrac {\map {f^{\paren n} } t} {n!} \paren {x - t}^n
}}
{{eqn | l = \map G t
| r = \map g t + \map {g'} t \paren {x - t} +... | Taylor's Theorem/One Variable with Two Functions | https://proofwiki.org/wiki/Taylor's_Theorem/One_Variable_with_Two_Functions | https://proofwiki.org/wiki/Taylor's_Theorem/One_Variable_with_Two_Functions | [
"Taylor's Theorem"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Differentiability Class",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Dif... | [
"Definition:Continuous Real Function/Interval",
"Definition:Differentiable Mapping/Real Function/Interval",
"Cauchy Mean Value Theorem",
"Definition:Real Number"
] |
proofwiki-17531 | Set of Even Integers is Countably Infinite | Let $\Bbb E$ be the set of even integers.
Then $\Bbb E$ is countably infinite. | Let $f: \Bbb E \to \Z$ be the mapping defined as:
:$\forall x \in \Bbb E: \map f x = \dfrac x 2$
$f$ is well-defined as $x$ is even and so $\dfrac x 2 \in \Z$.
Let $x, y \in \Bbb E$ such that $\map f x = \map f y$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
| c =
}}
{{eqn | ll= \leadsto
... | Let $\Bbb E$ be the [[Definition:Set|set]] of [[Definition:Even Integer|even integers]].
Then $\Bbb E$ is [[Definition:Countably Infinite Set|countably infinite]]. | Let $f: \Bbb E \to \Z$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in \Bbb E: \map f x = \dfrac x 2$
$f$ is [[Definition:Well-Defined Mapping|well-defined]] as $x$ is [[Definition:Even Integer|even]] and so $\dfrac x 2 \in \Z$.
Let $x, y \in \Bbb E$ such that $\map f x = \map f y$.
Then:
{{begin-... | Set of Even Integers is Countably Infinite | https://proofwiki.org/wiki/Set_of_Even_Integers_is_Countably_Infinite | https://proofwiki.org/wiki/Set_of_Even_Integers_is_Countably_Infinite | [
"Countable Sets",
"Odd Integers"
] | [
"Definition:Set",
"Definition:Even Integer",
"Definition:Countably Infinite/Set"
] | [
"Definition:Mapping",
"Definition:Well-Defined/Mapping",
"Definition:Even Integer",
"Definition:Injection",
"Definition:Inverse of Mapping",
"Definition:Well-Defined/Mapping",
"Definition:Even Integer",
"Definition:Mapping",
"Definition:Injection",
"Cantor-Bernstein-Schröder Theorem",
"Definitio... |
proofwiki-17532 | Basis Expansion of Rational Number | Let $b$ be a number base.
Let $x$ be a real number.
Then $x$ is a rational number {{iff}} the basis expansion of $x$ in base $b$ terminates or recurs. | {{ProofWanted}}
Category:Basis Expansions
Category:Rational Numbers
cjine5yordcl9st3b5sphsip4414gyj | Let $b$ be a [[Definition:Number Base|number base]].
Let $x$ be a [[Definition:Real Number|real number]].
Then $x$ is a [[Definition:Rational Number|rational number]] {{iff}} the [[Definition:Basis Expansion|basis expansion]] of $x$ in [[Definition:Number Base|base]] $b$ [[Definition:Termination of Basis Expansion|te... | {{ProofWanted}}
[[Category:Basis Expansions]]
[[Category:Rational Numbers]]
cjine5yordcl9st3b5sphsip4414gyj | Basis Expansion of Rational Number | https://proofwiki.org/wiki/Basis_Expansion_of_Rational_Number | https://proofwiki.org/wiki/Basis_Expansion_of_Rational_Number | [
"Basis Expansions",
"Rational Numbers"
] | [
"Definition:Number Base",
"Definition:Real Number",
"Definition:Rational Number",
"Definition:Basis Expansion",
"Definition:Number Base",
"Definition:Basis Expansion/Termination",
"Definition:Basis Expansion/Recurrence"
] | [
"Category:Basis Expansions",
"Category:Rational Numbers"
] |
proofwiki-17533 | Integration by Parts/Definite Integral | :$\ds \int_a^b \map f t \map G t \rd t = \bigintlimits {\map F t \map G t} a b - \int_a^b \map F t \map g t \rd t$ | By Product Rule for Derivatives:
:$\map D {F G} = f G + F g$
Thus $F G$ is a primitive of $f G + F g$ on $\closedint a b$.
Hence, by the Fundamental Theorem of Calculus:
:$\ds \int_a^b \paren {\map f t \map G t + \map F t \map g t} \rd t = \bigintlimits {\map F t \map G t} a b$
The result follows.
{{qed}} | :$\ds \int_a^b \map f t \map G t \rd t = \bigintlimits {\map F t \map G t} a b - \int_a^b \map F t \map g t \rd t$ | By [[Product Rule for Derivatives]]:
:$\map D {F G} = f G + F g$
Thus $F G$ is a [[Definition:Primitive (Calculus)|primitive]] of $f G + F g$ on $\closedint a b$.
Hence, by the [[Fundamental Theorem of Calculus]]:
:$\ds \int_a^b \paren {\map f t \map G t + \map F t \map g t} \rd t = \bigintlimits {\map F t \map G t} ... | Integration by Parts/Definite Integral | https://proofwiki.org/wiki/Integration_by_Parts/Definite_Integral | https://proofwiki.org/wiki/Integration_by_Parts/Definite_Integral | [
"Integration by Parts"
] | [] | [
"Product Rule for Derivatives",
"Definition:Primitive (Calculus)",
"Fundamental Theorem of Calculus"
] |
proofwiki-17534 | Integration by Parts/Primitive | :$\ds \int \map f t \map G t \rd t = \map F t \map G t - \int \map F t \map g t \rd t$
on $\closedint a b$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d t} } {\map F t \map G t}
| r = \map f t \map G t + \map F t \map g t
| c = Product Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \int \paren {\map f t \map G t + \map F t \map g t} \rd t
| r = \map F t \map G t
| c = Fundamental Theorem of... | :$\ds \int \map f t \map G t \rd t = \map F t \map G t - \int \map F t \map g t \rd t$
on $\closedint a b$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d t} } {\map F t \map G t}
| r = \map f t \map G t + \map F t \map g t
| c = [[Product Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \int \paren {\map f t \map G t + \map F t \map g t} \rd t
| r = \map F t \map G t
| c = [[Fundamental Theo... | Integration by Parts/Primitive/Proof 1 | https://proofwiki.org/wiki/Integration_by_Parts/Primitive | https://proofwiki.org/wiki/Integration_by_Parts/Primitive/Proof_1 | [
"Integration by Parts"
] | [] | [
"Product Rule for Derivatives",
"Fundamental Theorem of Calculus",
"Definition:Primitive (Calculus)/Integration",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-17535 | Integration by Parts/Primitive | :$\ds \int \map f t \map G t \rd t = \map F t \map G t - \int \map F t \map g t \rd t$
on $\closedint a b$. | Let $\map u x$ and $\map v x$ be integrable functions defined on $\closedint a b$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {u v}
| r = u \dfrac {\d v} {\d x} + v \dfrac {\d u} {\d x}
| c = Product Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = v \dfrac {\d u} {\d x}
| r = \m... | :$\ds \int \map f t \map G t \rd t = \map F t \map G t - \int \map F t \map g t \rd t$
on $\closedint a b$. | Let $\map u x$ and $\map v x$ be [[Definition:Integrable Function|integrable functions]] defined on $\closedint a b$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {u v}
| r = u \dfrac {\d v} {\d x} + v \dfrac {\d u} {\d x}
| c = [[Product Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| ... | Integration by Parts/Primitive/Proof 2 | https://proofwiki.org/wiki/Integration_by_Parts/Primitive | https://proofwiki.org/wiki/Integration_by_Parts/Primitive/Proof_2 | [
"Integration by Parts"
] | [] | [
"Definition:Integrable Function",
"Product Rule for Derivatives",
"Definition:Primitive (Calculus)/Integration",
"Linear Combination of Integrals/Indefinite",
"Fundamental Theorem of Calculus"
] |
proofwiki-17536 | Integration by Substitution/Definite Integral | If $\map \phi a \le \map \phi b$, then the definite integral of $f$ from $a$ to $b$ can be evaluated by:
:$\ds \int_{\map \phi a}^{\map \phi b} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$
where $t = \map \phi u$.
If $\map \phi a > \map \phi b$, then the definite integral of $f$ fr... | Let $F$ be an antiderivative of $f$.
We have:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d u} } {\map F t}
| r = \map {\frac \d {\d u} } {\map F {\map \phi u} }
| c = Definition of $\map \phi u$
}}
{{eqn | r = \dfrac \d {\d t} \map F {\map \phi u} \dfrac \d {\d u} \map \phi u
| c = Chain Rule for Deri... | If $\map \phi a \le \map \phi b$, then the [[Definition:Definite Integral|definite integral]] of $f$ from $a$ to $b$ can be evaluated by:
:$\ds \int_{\map \phi a}^{\map \phi b} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$
where $t = \map \phi u$.
If $\map \phi a > \map \phi b$, ... | Let $F$ be an [[Definition:Primitive (Calculus)|antiderivative]] of $f$.
We have:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d u} } {\map F t}
| r = \map {\frac \d {\d u} } {\map F {\map \phi u} }
| c = Definition of $\map \phi u$
}}
{{eqn | r = \dfrac \d {\d t} \map F {\map \phi u} \dfrac \d {\d u} \map ... | Integration by Substitution/Definite Integral | https://proofwiki.org/wiki/Integration_by_Substitution/Definite_Integral | https://proofwiki.org/wiki/Integration_by_Substitution/Definite_Integral | [
"Integration by Substitution",
"Definite Integrals"
] | [
"Definition:Definite Integral",
"Definition:Definite Integral"
] | [
"Definition:Primitive (Calculus)",
"Derivative of Composite Function",
"Definition:Primitive (Calculus)",
"Fundamental Theorem of Calculus/Second Part"
] |
proofwiki-17537 | Product Formula for Norms on Non-zero Rationals/Lemma | Let $z \in \Z_{\ne 0}$.
Then the following infinite product converges:
:$\size z \times \ds \prod_{p \mathop \in \Bbb P}^{} \norm z_p = 1$ | === Case 1 : $z \in \Z_{>0}$ ===
Let $z \in \Z_{>0}$.
From Fundamental Theorem of Arithmetic, we can factor $z$ as a product of one or more primes:
:$z = p_1^{b_1} p_2^{b_2} \dots p_k^{b_k}$
Then for every prime number $q$:
:$\norm z_q = \begin{cases}
p_i^{-b_i} & : \exists i \in \closedint 1 k :q = p_i \\
1 & : \foral... | Let $z \in \Z_{\ne 0}$.
Then the following [[Definition:Infinite Product|infinite product]] [[Definition:Convergent Real Sequence|converges]]:
:$\size z \times \ds \prod_{p \mathop \in \Bbb P}^{} \norm z_p = 1$ | === Case 1 : $z \in \Z_{>0}$ ===
Let $z \in \Z_{>0}$.
From [[Fundamental Theorem of Arithmetic]], we can factor $z$ as a [[Definition:Integer Multiplication|product]] of one or more [[Definition:Prime Number|primes]]:
:$z = p_1^{b_1} p_2^{b_2} \dots p_k^{b_k}$
Then for every [[Definition:Prime Number|prime number $... | Product Formula for Norms on Non-zero Rationals/Lemma | https://proofwiki.org/wiki/Product_Formula_for_Norms_on_Non-zero_Rationals/Lemma | https://proofwiki.org/wiki/Product_Formula_for_Norms_on_Non-zero_Rationals/Lemma | [
"P-adic Number Theory"
] | [
"Definition:Continued Product/Infinite",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Fundamental Theorem of Arithmetic",
"Definition:Multiplication/Integers",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Absolute Value",
"Eventually Constant Sequence Converges to Constant"
] |
proofwiki-17538 | P-adic Open Ball is Instance of Open Ball of a Norm | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $B \subseteq \Q_p$.
Then:
:$B$ is an open ball in $p$-adic numbers with radius $\epsilon$ and centre $a$
{{iff}}:
:$B$ is an open ball of th... | From P-adic Numbers form Non-Archimedean Valued Field:
:the $p$-adic numbers $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a non-Archimedean valued field.
The definition of an open ball in $p$-adic numbers is identical to the definition of an open ball of a normed division ring with respect to the norm $\norm {\,\cdot\,}_... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $B \subseteq \Q_p$... | From [[P-adic Numbers form Non-Archimedean Valued Field]]:
:the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean valued field]].
The definition of an [[Definition:Open Ball in P-adic Numbers|open ... | P-adic Open Ball is Instance of Open Ball of a Norm | https://proofwiki.org/wiki/P-adic_Open_Ball_is_Instance_of_Open_Ball_of_a_Norm | https://proofwiki.org/wiki/P-adic_Open_Ball_is_Instance_of_Open_Ball_of_a_Norm | [
"Open Balls",
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Strictly Positive/Real Number",
"Definition:Open Ball/P-adic Numbers",
"Definition:Open Ball/P-adic Numbers/Radius",
"Definition:Open Ball/P-adic Numbers/Center",
"Definition:Open Ball/Normed Division Ring",
"Definitio... | [
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:Valued Field of P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Open Ball/P-adic Numbers",
"Definition:Open Ball/Normed Division Ring",
"Definition:Norm/Division Ring",
"Category:Open Balls",
"... |
proofwiki-17539 | P-adic Closed Ball is Instance of Closed Ball of a Norm | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $B \subseteq \Q_p$.
Then:
:$B$ is a closed ball in $p$-adic numbers with radius $\epsilon$ and centre $a$
{{iff}}:
:$B$ is a closed ball of ... | From P-adic Numbers form Non-Archimedean Valued Field:
:the $p$-adic numbers $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a non-Archimedean valued field.
The definition of a closed ball in $p$-adic numbers is identical to the definition of a closed ball of a normed division ring with respect to the norm $\norm {\,\cdot\,... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $B \subseteq \Q_p$... | From [[P-adic Numbers form Non-Archimedean Valued Field]]:
:the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean valued field]].
The definition of a [[Definition:Closed Ball in P-adic Numbers|clos... | P-adic Closed Ball is Instance of Closed Ball of a Norm | https://proofwiki.org/wiki/P-adic_Closed_Ball_is_Instance_of_Closed_Ball_of_a_Norm | https://proofwiki.org/wiki/P-adic_Closed_Ball_is_Instance_of_Closed_Ball_of_a_Norm | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Strictly Positive/Real Number",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Closed Ball/P-adic Numbers/Radius",
"Definition:Closed Ball/P-adic Numbers/Center",
"Definition:Closed Ball/Normed Division Ring",
"D... | [
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:Valued Field of P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Closed Ball/Normed Division Ring",
"Definition:Norm/Division Ring",
"Category:P-adic Numbe... |
proofwiki-17540 | P-adic Sphere is Instance of Sphere of a Norm | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $S \subseteq \Q_p$.
Then:
:$S$ is a sphere in $p$-adic numbers with radius $\epsilon$ and centre $a$
{{iff}}:
:$S$ is a sphere of the normed... | From P-adic Numbers form Non-Archimedean Valued Field:
:the $p$-adic numbers $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a non-Archimedean valued field.
The definition of a sphere in $p$-adic numbers is identical to the definition of a sphere in a normed division ring with respect to the norm $\norm {\,\cdot\,}_p$.
{{qe... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $S \subseteq \Q_p$... | From [[P-adic Numbers form Non-Archimedean Valued Field]]:
:the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean valued field]].
The definition of a [[Definition:Sphere in P-adic Numbers|sphere in... | P-adic Sphere is Instance of Sphere of a Norm | https://proofwiki.org/wiki/P-adic_Sphere_is_Instance_of_Sphere_of_a_Norm | https://proofwiki.org/wiki/P-adic_Sphere_is_Instance_of_Sphere_of_a_Norm | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Strictly Positive/Real Number",
"Definition:Sphere/P-adic Numbers",
"Definition:Sphere/P-adic Numbers/Radius",
"Definition:Sphere/P-adic Numbers/Center",
"Definition:Sphere/Normed Division Ring",
"Definition:Normed Div... | [
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Definition:Valued Field of P-adic Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Sphere/P-adic Numbers",
"Definition:Sphere/Normed Division Ring",
"Definition:Norm/Division Ring",
"Category:P-adic Number Theory"
... |
proofwiki-17541 | Sphere is Set Difference of Closed Ball with Open Ball/P-adic Numbers | Let $p$ be a prime number.
Let $\struct{\Q_p,\norm{\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Let $\map {{B_\epsilon}^-} a$ denote the $\epsilon$-closed ball of $a$ in $\Q_p$.
Let $\map {B_\epsilon} a$ denote the $\epsilon$-open ball of $a$ i... | The result follows directly from:
:P-adic Closed Ball is Instance of Closed Ball of a Norm
:P-adic Open Ball is Instance of Open Ball of a Norm
:P-adic Sphere is Instance of Sphere of a Norm
:Sphere is Set Difference of Closed and Open Ball in Normed Division Ring
{{qed}}
Category:P-adic Number Theory
arvlbze0xra6eabck... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct{\Q_p,\norm{\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a \in \Q_p$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\map {{B_\epsilon}^... | The result follows directly from:
:[[P-adic Closed Ball is Instance of Closed Ball of a Norm]]
:[[P-adic Open Ball is Instance of Open Ball of a Norm]]
:[[P-adic Sphere is Instance of Sphere of a Norm]]
:[[Sphere is Set Difference of Closed and Open Ball in Normed Division Ring]]
{{qed}}
[[Category:P-adic Number Theor... | Sphere is Set Difference of Closed Ball with Open Ball/P-adic Numbers | https://proofwiki.org/wiki/Sphere_is_Set_Difference_of_Closed_Ball_with_Open_Ball/P-adic_Numbers | https://proofwiki.org/wiki/Sphere_is_Set_Difference_of_Closed_Ball_with_Open_Ball/P-adic_Numbers | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Strictly Positive/Real Number",
"Definition:Closed Ball/P-adic Numbers",
"Definition:Open Ball/P-adic Numbers",
"Definition:Sphere in P-adic Numbers"
] | [
"P-adic Closed Ball is Instance of Closed Ball of a Norm",
"P-adic Open Ball is Instance of Open Ball of a Norm",
"P-adic Sphere is Instance of Sphere of a Norm",
"Sphere is Set Difference of Closed Ball with Open Ball/Normed Division Ring",
"Category:P-adic Number Theory"
] |
proofwiki-17542 | Set is Closed in Metric Space iff Closed in Induced Topological Space | Let $M = \struct {A, d}$ be a metric space.
Let $\tau$ be the topology induced by the metric $d$.
Let $F$ be a subset of $M$.
Then:
:$F$ is closed in $M$ {{iff}} $F$ is closed in $\struct {A, \tau}$ | By definition of a closed set in $M$:
:$F$ is closed set in $M$ {{iff}} $A \setminus F$ is open in $M$
By definition of the topology $\tau$ induced by the metric $d$:
:$A \setminus F$ is open in $M$ {{iff}} $A \setminus F$ is open in $\struct {A, \tau}$
By definition of a closed set in $\struct{A, \tau}$:
:$A \setminus... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\tau$ be the [[Definition:Topology Induced by Metric|topology induced]] by the [[Definition:Metric|metric]] $d$.
Let $F$ be a [[Definition:Subset|subset]] of $M$.
Then:
:$F$ is [[Definition:Closed Set (Metric Space)|closed in $M$]] {{iff}}... | By definition of a [[Definition:Closed Set (Metric Space)|closed set in $M$]]:
:$F$ is [[Definition:Closed Set (Metric Space)|closed set in $M$]] {{iff}} $A \setminus F$ is [[Definition:Open Set (Metric Space)|open in $M$]]
By definition of the [[Definition:Topology Induced by Metric|topology $\tau$ induced]] by the [... | Set is Closed in Metric Space iff Closed in Induced Topological Space | https://proofwiki.org/wiki/Set_is_Closed_in_Metric_Space_iff_Closed_in_Induced_Topological_Space | https://proofwiki.org/wiki/Set_is_Closed_in_Metric_Space_iff_Closed_in_Induced_Topological_Space | [
"Closed Sets",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Metric Space/Metric",
"Definition:Subset",
"Definition:Closed Set/Metric Space",
"Definition:Closed Set/Topology"
] | [
"Definition:Closed Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Metric Space/Metric",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition... |
proofwiki-17543 | Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form | :$\ds \int \csch x \rd x = -\map {\coth^{-1} } {\cosh x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cosh x
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d u} {\d x}
| r = \sinh x
| c = Derivative of Hyperbolic Cosine
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = \int \dfrac {\d x} {\sinh x}
| c = {{Defof|Hyperb... | :$\ds \int \csch x \rd x = -\map {\coth^{-1} } {\cosh x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cosh x
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d u} {\d x}
| r = \sinh x
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = \int \dfrac {\d x} {\sinh x}
| c = {{Defof... | Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Inverse_Hyperbolic_Cotangent_of_Hyperbolic_Cosine_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Inverse_Hyperbolic_Cotangent_of_Hyperbolic_Cosine_Form/Proof_1 | [
"Primitive of Hyperbolic Cosecant Function",
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Integration by Substitution",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form"
] |
proofwiki-17544 | Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form | :$\ds \int \csch x \rd x = -\map {\coth^{-1} } {\cosh x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cosh x
}}
{{eqn | ll= \leadsto
| l = u'
| r = \sinh x
| c = Derivative of Hyperbolic Cosine
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = \int \frac 1 {\sinh x} \rd x
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = ... | :$\ds \int \csch x \rd x = -\map {\coth^{-1} } {\cosh x} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \cosh x
}}
{{eqn | ll= \leadsto
| l = u'
| r = \sinh x
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \csch x \rd x
| r = \int \frac 1 {\sinh x} \rd x
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn... | Primitive of Hyperbolic Cosecant Function/Inverse Hyperbolic Cotangent of Hyperbolic Cosine Form/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Inverse_Hyperbolic_Cotangent_of_Hyperbolic_Cosine_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Cosecant_Function/Inverse_Hyperbolic_Cotangent_of_Hyperbolic_Cosine_Form/Proof_2 | [
"Primitive of Hyperbolic Cosecant Function",
"Primitives involving Hyperbolic Cotangent Function"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form"
] |
proofwiki-17545 | Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form | :$\ds \int \frac 1 {\sqrt {a^2 - x^2} } \rd x = -\arccos \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac 1 {\sqrt {a^2 - x^2} } \rd x
| r = \int \frac {\rd x} {\sqrt {a^2 \paren {1 - \frac {x^2} {a^2} } } }
| c = factor $a^2$ out of the radicand
}}
{{eqn | r = \int \frac {\rd x} {\sqrt{a^2} \sqrt {1 - \paren {\frac x a}^2} }
| c =
}}
{{eqn | r = \frac 1 a \int \frac... | :$\ds \int \frac 1 {\sqrt {a^2 - x^2} } \rd x = -\arccos \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac 1 {\sqrt {a^2 - x^2} } \rd x
| r = \int \frac {\rd x} {\sqrt {a^2 \paren {1 - \frac {x^2} {a^2} } } }
| c = factor $a^2$ out of the [[Definition:Radicand|radicand]]
}}
{{eqn | r = \int \frac {\rd x} {\sqrt{a^2} \sqrt {1 - \paren {\frac x a}^2} }
| c =
}}
{{eqn | ... | Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_squared_minus_x_squared/Arccosine_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_squared_minus_x_squared/Arccosine_Form | [
"Arccosine Function",
"Primitive of Reciprocal of Root of a squared minus x squared"
] | [] | [
"Definition:Radicand",
"Integration by Substitution",
"Real Cosine Function is Bounded",
"Shape of Cosine Function",
"Definition:By Hypothesis",
"Powers of Group Elements",
"Negative of Absolute Value",
"Definition:Differentiation",
"Derivative of Cosine Function",
"Derivative of Composite Functio... |
proofwiki-17546 | Negative of Logarithm of x plus Root x squared minus a squared | Let $x \in \R: \size x > 1$.
Let $x > 1$.
Then:
:$-\map \ln {x + \sqrt {x^2 - a^2} } = \map \ln {x - \sqrt {x^2 - a^2} } - \map \ln {a^2}$ | First we note that if $x > 1$ then $x + \sqrt {x^2 - a^2} > 0$.
Hence $\map \ln {x + \sqrt {x^2 - a^2} }$ is defined.
Then we have:
{{begin-eqn}}
{{eqn | l = -\map \ln {x + \sqrt {x^2 - a^2} }
| r = \map \ln {\dfrac 1 {x + \sqrt {x^2 - a^2} } }
| c = Logarithm of Reciprocal
}}
{{eqn | r = \map \ln {\dfrac {... | Let $x \in \R: \size x > 1$.
Let $x > 1$.
Then:
:$-\map \ln {x + \sqrt {x^2 - a^2} } = \map \ln {x - \sqrt {x^2 - a^2} } - \map \ln {a^2}$ | First we note that if $x > 1$ then $x + \sqrt {x^2 - a^2} > 0$.
Hence $\map \ln {x + \sqrt {x^2 - a^2} }$ is defined.
Then we have:
{{begin-eqn}}
{{eqn | l = -\map \ln {x + \sqrt {x^2 - a^2} }
| r = \map \ln {\dfrac 1 {x + \sqrt {x^2 - a^2} } }
| c = [[Logarithm of Reciprocal]]
}}
{{eqn | r = \map \ln {\... | Negative of Logarithm of x plus Root x squared minus a squared | https://proofwiki.org/wiki/Negative_of_Logarithm_of_x_plus_Root_x_squared_minus_a_squared | https://proofwiki.org/wiki/Negative_of_Logarithm_of_x_plus_Root_x_squared_minus_a_squared | [
"Logarithms"
] | [] | [
"Logarithm of Reciprocal",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Difference of Logarithms"
] |
proofwiki-17547 | Supremum Norm on Vector Space of Real Matrices is Norm | Supremum Norm forms a norm on the vector space of real matrices. | Let $M \in \R^{m \times n}: m, n \in \N_{>0}$ be a real matrix.
Denote the $\paren {i, j}$-th entry of $M$ by $a_{i j}$.
Note that the set of matrix elements of $M$ is a finite set of real numbers.
We have that:
:Real Numbers form Totally Ordered Field
:Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Ele... | [[Definition:Supremum Norm|Supremum Norm]] forms a [[Definition:Norm on Vector Space|norm]] on the [[Definition:Vector Space|vector space]] of [[Definition:Real Matrix|real matrices]]. | Let $M \in \R^{m \times n}: m, n \in \N_{>0}$ be a [[Definition:Real Matrix|real matrix]].
Denote the $\paren {i, j}$-th entry of $M$ by $a_{i j}$.
Note that the [[Definition:Set|set]] of [[Definition:Element of Matrix|matrix elements]] of $M$ is a [[Definition:Finite Set|finite set]] of [[Definition:Real Number|real... | Supremum Norm on Vector Space of Real Matrices is Norm | https://proofwiki.org/wiki/Supremum_Norm_on_Vector_Space_of_Real_Matrices_is_Norm | https://proofwiki.org/wiki/Supremum_Norm_on_Vector_Space_of_Real_Matrices_is_Norm | [
"Examples of Norms"
] | [
"Definition:Supremum Norm",
"Definition:Norm/Vector Space",
"Definition:Vector Space",
"Definition:Real Matrix"
] | [
"Definition:Real Matrix",
"Definition:Set",
"Definition:Matrix/Element",
"Definition:Finite Set",
"Definition:Real Number",
"Real Numbers form Totally Ordered Field",
"Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements",
"Definition:Greatest Element",
"Definition:Matrix/Element"... |
proofwiki-17548 | Arccotangent Logarithmic Formulation | For any real number $x$:
:$\arccot x = -\dfrac i 2 \map \ln {\dfrac {i x - 1} {i x + 1} }$
where $\arccot x$ is the arccotangent and $i^2 = -1$. | Assume $y \in \R$, $ -\dfrac \pi 2 \le y \le \dfrac \pi 2 $.
{{begin-eqn}}
{{eqn | l = y
| r = \arccot x
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \cot y
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = i \dfrac {e^{i y} + e^{-i y} } {e^{i y} - e^{-i y} }
| c = Euler's Cotangent Ident... | For any [[Definition:Real Number|real number]] $x$:
:$\arccot x = -\dfrac i 2 \map \ln {\dfrac {i x - 1} {i x + 1} }$
where $\arccot x$ is the [[Definition:Arccotangent|arccotangent]] and $i^2 = -1$. | Assume $y \in \R$, $ -\dfrac \pi 2 \le y \le \dfrac \pi 2 $.
{{begin-eqn}}
{{eqn | l = y
| r = \arccot x
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = \cot y
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = i \dfrac {e^{i y} + e^{-i y} } {e^{i y} - e^{-i y} }
| c = [[Euler's Cotangent Id... | Arccotangent Logarithmic Formulation | https://proofwiki.org/wiki/Arccotangent_Logarithmic_Formulation | https://proofwiki.org/wiki/Arccotangent_Logarithmic_Formulation | [
"Arccotangent Function"
] | [
"Definition:Real Number",
"Definition:Inverse Cotangent/Real/Arccotangent"
] | [
"Euler's Cotangent Identity",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-17549 | Primitive of Reciprocal of x squared plus a squared/Arccotangent Form | :$\ds \int \frac {\d x} {x^2 + a^2} = -\frac 1 a \arccot \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 + a^2}
| r = \frac 1 a \int \frac {\d t} {t^2 + 1}
| c = Substitution of $x \to a t$}}
{{eqn | r = \frac 1 a \int \frac {\d t} {\paren {1 + i t} \paren {1 - i t} }
| c = factoring
}}
{{eqn | r = \frac 1 {2 a} \paren {\int \frac {\d t} {1 + i t} + \int \... | :$\ds \int \frac {\d x} {x^2 + a^2} = -\frac 1 a \arccot \frac x a + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 + a^2}
| r = \frac 1 a \int \frac {\d t} {t^2 + 1}
| c = [[Integration by Substitution|Substitution of $x \to a t$]]}}
{{eqn | r = \frac 1 a \int \frac {\d t} {\paren {1 + i t} \paren {1 - i t} }
| c = factoring
}}
{{eqn | r = \frac 1 {2 a} \paren {\in... | Primitive of Reciprocal of x squared plus a squared/Arccotangent Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_plus_a_squared/Arccotangent_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_plus_a_squared/Arccotangent_Form | [
"Primitive of Reciprocal of x squared plus a squared"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal",
"Sum of Logarithms",
"Logarithm of Reciprocal",
"Arccotangent Logarithmic Formulation"
] |
proofwiki-17550 | Primitive of Reciprocal of x by Root of x squared plus a squared/Reciprocal Logarithm Form | :$\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = \frac 1 a \map \ln {\frac x {a + \sqrt {x^2 + a^2} } } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 + a^2} }
| r = -\frac 1 a \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C
| c = Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: Logarithm form
}}
{{eqn | r = \frac 1 a \map \ln {\frac x {a + \sqrt {a^2 + x^2} } } + C
| c = Logarithm of Recipr... | :$\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = \frac 1 a \map \ln {\frac x {a + \sqrt {x^2 + a^2} } } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 + a^2} }
| r = -\frac 1 a \map \ln {\frac {a + \sqrt {x^2 + a^2} } x} + C
| c = [[Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form|Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$: Logarithm form]]
}}
{{eqn | r = \frac ... | Primitive of Reciprocal of x by Root of x squared plus a squared/Reciprocal Logarithm Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared/Reciprocal_Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared/Reciprocal_Logarithm_Form | [
"Primitive of Reciprocal of x by Root of x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x by Root of x squared plus a squared/Logarithm Form",
"Logarithm of Reciprocal"
] |
proofwiki-17551 | Derivative of Hyperbolic Sine | :$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sinh x}
| r = \map {\frac \d {\d x} } {\dfrac {e^x - e ^{-x} } 2}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \paren {\map {\frac \d {\d x} } {e^x} - \map {\frac \d {\d x} } {e^{-x} } }
| c = Linear Combination of Derivatives
}}
{{eqn | ... | :$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sinh x}
| r = \map {\frac \d {\d x} } {\dfrac {e^x - e ^{-x} } 2}
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \paren {\map {\frac \d {\d x} } {e^x} - \map {\frac \d {\d x} } {e^{-x} } }
| c = [[Linear Combination of Derivatives]]
}}
{{eq... | Derivative of Hyperbolic Sine/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine/Proof_1 | [
"Derivative of Hyperbolic Sine Function"
] | [] | [
"Linear Combination of Derivatives",
"Derivative of Exponential Function",
"Derivative of Composite Function"
] |
proofwiki-17552 | Derivative of Hyperbolic Sine | :$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sinh x}
| r = \lim_{h \mathop \to 0} \frac {\map \sinh {x + h} - \sinh x} h
| c = {{Defof|Derivative}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {2 \map \cosh {\frac {x + h + x} 2} \map \sinh {\frac {x + h - x} 2} } h
| c = Hyperbolic Sine minus Hy... | :$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sinh x}
| r = \lim_{h \mathop \to 0} \frac {\map \sinh {x + h} - \sinh x} h
| c = {{Defof|Derivative}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {2 \map \cosh {\frac {x + h + x} 2} \map \sinh {\frac {x + h - x} 2} } h
| c = [[Hyperbolic Sine minus ... | Derivative of Hyperbolic Sine/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine/Proof_2 | [
"Derivative of Hyperbolic Sine Function"
] | [] | [
"Prosthaphaeresis Formulas/Hyperbolic Sine minus Hyperbolic Sine",
"Derivative of Exponential at Zero"
] |
proofwiki-17553 | Derivative of Hyperbolic Sine | :$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sinh x}
| r = -i \map {\frac \d {\d x} } {\sin i x}
| c = Hyperbolic Sine in terms of Sine
}}
{{eqn | r = \cos i x
| c = Derivative of Sine Function
}}
{{eqn | r = \cosh x
| c = Hyperbolic Cosine in terms of Cosine
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sinh x}
| r = -i \map {\frac \d {\d x} } {\sin i x}
| c = [[Hyperbolic Sine in terms of Sine]]
}}
{{eqn | r = \cos i x
| c = [[Derivative of Sine Function]]
}}
{{eqn | r = \cosh x
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Sine/Proof 3 | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine/Proof_3 | [
"Derivative of Hyperbolic Sine Function"
] | [] | [
"Hyperbolic Sine in terms of Sine",
"Derivative of Sine Function",
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-17554 | Open Ball is Open Set/Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball of $x$ in $M$.
Then $\map {B_\epsilon} x$ is an open set of $M$. | Let $y \in \map {B_\epsilon} x$.
From Open Ball of Point Inside Open Ball, there exists $\delta \in \R_{>0}$ such that:
:$\map {B_\delta} y \subseteq \map {B_\epsilon} x$
The result follows from the definition of open set.
{{qed}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} x$ be an [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] of $x$ in $M$.
Then $\map {B_\epsilon} x$ is an [[Definition:Open Set (Metric Space)|open set]] of $M$. | Let $y \in \map {B_\epsilon} x$.
From [[Open Ball of Point Inside Open Ball]], there exists $\delta \in \R_{>0}$ such that:
:$\map {B_\delta} y \subseteq \map {B_\epsilon} x$
The result follows from the definition of [[Definition:Open Set (Metric Space)|open set]].
{{qed}} | Open Ball is Open Set/Metric Space | https://proofwiki.org/wiki/Open_Ball_is_Open_Set/Metric_Space | https://proofwiki.org/wiki/Open_Ball_is_Open_Set/Metric_Space | [
"Open Ball is Open Set",
"Open Sets (Metric Spaces)"
] | [
"Definition:Metric Space",
"Definition:Open Ball",
"Definition:Open Set/Metric Space"
] | [
"Open Ball of Point Inside Open Ball",
"Definition:Open Set/Metric Space"
] |
proofwiki-17555 | Derivative of Hyperbolic Cosine | :$\map {\dfrac \d {\d x} } {\cosh x} = \sinh x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\cosh x}
| r = \map {\dfrac \d {\d x} } {\dfrac {e^x + e ^{-x} } 2}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \dfrac 1 2 \map {\dfrac \d {\d x} } {e^x + e^{-x} }
| c = Derivative of Constant Multiple
}}
{{eqn | r = \dfrac 1 2 \paren {e^x + \pa... | :$\map {\dfrac \d {\d x} } {\cosh x} = \sinh x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\cosh x}
| r = \map {\dfrac \d {\d x} } {\dfrac {e^x + e ^{-x} } 2}
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \dfrac 1 2 \map {\dfrac \d {\d x} } {e^x + e^{-x} }
| c = [[Derivative of Constant Multiple]]
}}
{{eqn | r = \dfrac 1 2 \paren {e^x +... | Derivative of Hyperbolic Cosine | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosine | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosine | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Cosine Function"
] | [] | [
"Derivative of Constant Multiple",
"Derivative of Exponential Function",
"Derivative of Composite Function",
"Linear Combination of Derivatives"
] |
proofwiki-17556 | Derivative of Hyperbolic Tangent | :$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x = \dfrac 1 {\cosh^2 x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tanh x}
| r = \map {\dfrac \d {\d x} } {\dfrac {\sinh x} {\cosh x} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \dfrac {\paren {\dfrac \d {\d x} \sinh x} \cosh x - \sinh x \paren {\dfrac \d {\d x} \cosh x} } {\cosh^2 x}
| c = Quot... | :$\map {\dfrac \d {\d x} } {\tanh x} = \sech^2 x = \dfrac 1 {\cosh^2 x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tanh x}
| r = \map {\dfrac \d {\d x} } {\dfrac {\sinh x} {\cosh x} }
| c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \dfrac {\paren {\dfrac \d {\d x} \sinh x} \cosh x - \sinh x \paren {\dfrac \d {\d x} \cosh x} } {\cosh^2 x}
| c = [[Qu... | Derivative of Hyperbolic Tangent | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Tangent | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Tangent | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Tangent Function"
] | [] | [
"Quotient Rule for Derivatives",
"Derivative of Hyperbolic Sine",
"Derivative of Hyperbolic Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-17557 | Derivative of Hyperbolic Cotangent | :$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x = \dfrac {-1} {\sinh^2 x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\coth x}
| r = \map {\dfrac \d {\d x} } {\frac {\cosh x} {\sinh x} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\sinh x \dfrac \d {\d x} \cosh x - \cosh x \dfrac \d {\d x} \sinh x} {\sinh^2 x}
| c = Quotient Rule for Derivat... | :$\map {\dfrac \d {\d x} } {\coth x} = -\csch^2 x = \dfrac {-1} {\sinh^2 x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\coth x}
| r = \map {\dfrac \d {\d x} } {\frac {\cosh x} {\sinh x} }
| c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {\sinh x \dfrac \d {\d x} \cosh x - \cosh x \dfrac \d {\d x} \sinh x} {\sinh^2 x}
| c = [[Quotient Rule for Deriv... | Derivative of Hyperbolic Cotangent | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cotangent | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cotangent | [
"Derivative of Hyperbolic Cotangent",
"Derivatives of Hyperbolic Functions",
"Hyperbolic Cotangent Function"
] | [] | [
"Quotient Rule for Derivatives",
"Derivative of Hyperbolic Cosine",
"Derivative of Hyperbolic Sine",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-17558 | Derivative of Hyperbolic Secant | :$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech x}
| r = \map {\frac \d {\d x} } {\frac 1 {\cosh x} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \map {\frac \d {\d x} } {\paren {\cosh x}^{-1} }
| c = Exponent Laws
}}
{{eqn | r = -\paren {\cosh x}^{-2} \sinh x
| c = Derivative of H... | :$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech x}
| r = \map {\frac \d {\d x} } {\frac 1 {\cosh x} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \map {\frac \d {\d x} } {\paren {\cosh x}^{-1} }
| c = [[Exponent Combination Laws/Negative Power|Exponent Laws]]
}}
{{eqn | r = -\paren {\cos... | Derivative of Hyperbolic Secant/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Secant | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Secant/Proof_1 | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Secant Function",
"Derivative of Hyperbolic Secant Function"
] | [] | [
"Exponent Combination Laws/Negative Power",
"Derivative of Hyperbolic Cosine",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Exponent Combination Laws"
] |
proofwiki-17559 | Derivative of Hyperbolic Secant | :$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech x}
| r = 2 \map {\frac \d {\d x} } {\frac {e^x} {e^{2 x} + 1} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 2 {\paren {e^{2 x} + 1}^2} \paren {\map {\frac \d {\d x} } {e^x} \paren {e^{2 x} + 1} - e^x \map {\frac \d {\d x} } {e^{2 x} + 1} }
| c = Quo... | :$\map {\dfrac \d {\d x} } {\sech x} = -\sech x \tanh x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech x}
| r = 2 \map {\frac \d {\d x} } {\frac {e^x} {e^{2 x} + 1} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac 2 {\paren {e^{2 x} + 1}^2} \paren {\map {\frac \d {\d x} } {e^x} \paren {e^{2 x} + 1} - e^x \map {\frac \d {\d x} } {e^{2 x} + 1} }
| c = [[Q... | Derivative of Hyperbolic Secant/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Secant | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Secant/Proof_2 | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Secant Function",
"Derivative of Hyperbolic Secant Function"
] | [] | [
"Quotient Rule for Derivatives",
"Derivative of Exponential Function"
] |
proofwiki-17560 | Derivative of Hyperbolic Cosecant | :$\map {\dfrac \d {\d x} } {\csch x} = -\csch x \coth x$ | It is noted that at $x = 0$, $\csch x$ is undefined.
Hence the restriction of the domain.
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\csch x}
| r = \map {\dfrac \d {\d x} } {\frac 1 {\sinh x} }
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \map {\dfrac \d {\d x} } {\paren {\sinh z}^{-1} }
... | :$\map {\dfrac \d {\d x} } {\csch x} = -\csch x \coth x$ | It is noted that at $x = 0$, $\csch x$ is undefined.
Hence the restriction of the domain.
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\csch x}
| r = \map {\dfrac \d {\d x} } {\frac 1 {\sinh x} }
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = \map {\dfrac \d {\d x} } {\paren {\sinh z}^{-1} }
... | Derivative of Hyperbolic Cosecant | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosecant | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosecant | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Cosecant Function"
] | [] | [
"Exponent Combination Laws/Negative Power",
"Derivative of Hyperbolic Cosine",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Exponent Combination Laws"
] |
proofwiki-17561 | Derivative of Inverse Hyperbolic Sine Function | :$\map {\dfrac \d {\d x} } {\arsinh u} = \dfrac 1 {\sqrt {1 + u^2} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arsinh u}
| r = \map {\frac \d {\d u} } {\arsinh u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {\sqrt {1 + u^2} } \frac {\d u} {\d x}
| c = Derivative of Inverse Hyperbolic Sine
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\arsinh u} = \dfrac 1 {\sqrt {1 + u^2} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arsinh u}
| r = \map {\frac \d {\d u} } {\arsinh u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac 1 {\sqrt {1 + u^2} } \frac {\d u} {\d x}
| c = [[Derivative of Inverse Hyperbolic Sine]]
}}
{{end-eqn}}
{{qed}} | Derivative of Inverse Hyperbolic Sine Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Sine_Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Sine_Function | [
"Derivative of Inverse Hyperbolic Sine"
] | [] | [
"Derivative of Composite Function",
"Derivative of Inverse Hyperbolic Sine"
] |
proofwiki-17562 | Derivative of Real Area Hyperbolic Cosine of Function | :$\map {\dfrac \d {\d x} } {\arcosh u} = \dfrac 1 {\sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$
where $u > 1$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcosh u}
| r = \map {\frac \d {\d u} } {\arcosh u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {\sqrt {u^2 - 1} } \frac {\d u} {\d x}
| c = Derivative of Real Area Hyperbolic Cosine
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\arcosh u} = \dfrac 1 {\sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$
where $u > 1$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcosh u}
| r = \map {\frac \d {\d u} } {\arcosh u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac 1 {\sqrt {u^2 - 1} } \frac {\d u} {\d x}
| c = [[Derivative of Real Area Hyperbolic Cosine]]
}}
{{end-eqn}}
{{qed}} | Derivative of Real Area Hyperbolic Cosine of Function | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine_of_Function | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine_of_Function | [
"Derivative of Real Area Hyperbolic Cosine"
] | [] | [
"Derivative of Composite Function",
"Derivative of Real Area Hyperbolic Cosine"
] |
proofwiki-17563 | Derivative of Inverse Hyperbolic Tangent Function | :$\map {\dfrac \d {\d x} } {\artanh u} = \dfrac 1 {1 - u^2} \dfrac {\d u} {\d x}$
where $\size u < 1$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\artanh u}
| r = \map {\frac \d {\d u} } {\artanh u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {1 - u^2} \frac {\d u} {\d x}
| c = Derivative of Inverse Hyperbolic Tangent
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\artanh u} = \dfrac 1 {1 - u^2} \dfrac {\d u} {\d x}$
where $\size u < 1$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\artanh u}
| r = \map {\frac \d {\d u} } {\artanh u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac 1 {1 - u^2} \frac {\d u} {\d x}
| c = [[Derivative of Inverse Hyperbolic Tangent]]
}}
{{end-eqn}}
{{qed}} | Derivative of Inverse Hyperbolic Tangent Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Tangent_Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Tangent_Function | [
"Derivative of Inverse Hyperbolic Tangent"
] | [] | [
"Derivative of Composite Function",
"Derivative of Inverse Hyperbolic Tangent"
] |
proofwiki-17564 | Derivative of Inverse Hyperbolic Cotangent Function | :$\map {\dfrac \d {\d x} } {\coth^{-1} u} = \dfrac {-1} {u^2 - 1} \dfrac {\d u} {\d x}$
where $\size u > 1$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\coth^{-1} u}
| r = \map {\frac \d {\d u} } {\coth^{-1} u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {-1} {u^2 - 1} \frac {\d u} {\d x}
| c = Derivative of Inverse Hyperbolic Cotangent
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\coth^{-1} u} = \dfrac {-1} {u^2 - 1} \dfrac {\d u} {\d x}$
where $\size u > 1$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\coth^{-1} u}
| r = \map {\frac \d {\d u} } {\coth^{-1} u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac {-1} {u^2 - 1} \frac {\d u} {\d x}
| c = [[Derivative of Inverse Hyperbolic Cotangent]]
}}
{{end-eqn}}
{{qed}} | Derivative of Inverse Hyperbolic Cotangent Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Cotangent_Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Cotangent_Function | [
"Derivative of Inverse Hyperbolic Cotangent"
] | [] | [
"Derivative of Composite Function",
"Derivative of Inverse Hyperbolic Cotangent"
] |
proofwiki-17565 | Derivative of Inverse Hyperbolic Secant Function | :$\map {\dfrac \d {\d x} } {\sech^{-1} u} = \dfrac {-1} {u \sqrt {1 - u^2} } \dfrac {\d u} {\d x}$
where $0 < u < 1$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech^{-1} u}
| r = \map {\frac \d {\d u} } {\sech^{-1} u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {-1} {u \sqrt {1 - u^2} } \frac {\d u} {\d x}
| c = Derivative of Inverse Hyperbolic Secant
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\sech^{-1} u} = \dfrac {-1} {u \sqrt {1 - u^2} } \dfrac {\d u} {\d x}$
where $0 < u < 1$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech^{-1} u}
| r = \map {\frac \d {\d u} } {\sech^{-1} u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac {-1} {u \sqrt {1 - u^2} } \frac {\d u} {\d x}
| c = [[Derivative of Inverse Hyperbolic Secant]]
}}
{{end-eqn}}
{{... | Derivative of Inverse Hyperbolic Secant Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Secant_Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Secant_Function | [
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Secant"
] | [] | [
"Derivative of Composite Function",
"Derivative of Inverse Hyperbolic Secant"
] |
proofwiki-17566 | Derivative of Inverse Hyperbolic Cosecant Function | :$\map {\dfrac \d {\d x} } {\arcsch u} = \dfrac {-1} {\size u \sqrt {1 + u^2} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcsch u}
| r = \map {\frac \d {\d u} } {\arcsch u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {-1} {\size u \sqrt {1 + u^2} } \frac {\d u} {\d x}
| c = Derivative of Inverse Hyperbolic Cosecant
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\arcsch u} = \dfrac {-1} {\size u \sqrt {1 + u^2} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcsch u}
| r = \map {\frac \d {\d u} } {\arcsch u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac {-1} {\size u \sqrt {1 + u^2} } \frac {\d u} {\d x}
| c = [[Derivative of Inverse Hyperbolic Cosecant]]
}}
{{end-eqn}}
... | Derivative of Inverse Hyperbolic Cosecant Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Cosecant_Function | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Cosecant_Function | [
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Cosecant"
] | [] | [
"Derivative of Composite Function",
"Derivative of Inverse Hyperbolic Cosecant"
] |
proofwiki-17567 | Intersection of Closed Sets is Closed/Topology | Let $T = \struct {S, \tau}$ be a topological space.
Then the intersection of an arbitrary number of closed sets of $T$ (either finitely or infinitely many) is itself closed. | Let $I$ be an indexing set (either finite or infinite).
Let $\ds \bigcap_{i \mathop \in I} V_i$ be the intersection of a indexed family of closed sets of $T$ indexed by $I$.
Then from De Morgan's laws: Difference with Intersection:
:$\ds S \setminus \bigcap_{i \mathop \in I} V_i = \bigcup_{i \mathop \in I} \paren {S \s... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then the [[Definition:Set Intersection|intersection]] of an arbitrary number of [[Definition:Closed Set (Topology)|closed sets]] of $T$ (either [[Definition:Finite|finitely]] or [[Definition:Infinite|infinitely]] many) is itself [[De... | Let $I$ be an [[Definition:Indexing Set|indexing set]] (either [[Definition:Finite|finite]] or [[Definition:Infinite|infinite]]).
Let $\ds \bigcap_{i \mathop \in I} V_i$ be the [[Definition:Set Intersection|intersection]] of a [[Definition:Indexed Family of Subsets|indexed family]] of [[Definition:Closed Set (Topology... | Intersection of Closed Sets is Closed/Topology | https://proofwiki.org/wiki/Intersection_of_Closed_Sets_is_Closed/Topology | https://proofwiki.org/wiki/Intersection_of_Closed_Sets_is_Closed/Topology | [
"Intersection of Closed Sets is Closed"
] | [
"Definition:Topological Space",
"Definition:Set Intersection",
"Definition:Closed Set/Topology",
"Definition:Finite",
"Definition:Infinite",
"Definition:Closed Set/Topology"
] | [
"Definition:Indexing Set",
"Definition:Finite",
"Definition:Infinite",
"Definition:Set Intersection",
"Definition:Indexing Set/Family of Subsets",
"Definition:Closed Set/Topology",
"De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Intersection",
"Definition:Closed Set/Topolo... |
proofwiki-17568 | Finite Union of Closed Sets is Closed/Topology | Let $T = \struct {S, \tau}$ be a topological space.
Then the union of finitely many closed sets of $T$ is itself closed. | Let $\ds \bigcup_{i \mathop = 1}^n V_i$ be the union of a finite number of closed sets of $T$.
Then from De Morgan's laws:
:$\ds S \setminus \bigcup_{i \mathop = 1}^n V_i = \bigcap_{i \mathop = 1}^n \paren {S \setminus V_i}$
By definition of closed set, each of the $S \setminus V_i$ is by definition open in $T$.
We hav... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then the [[Definition:Set Union|union]] of [[Definition:Finite|finitely many]] [[Definition:Closed Set (Topology)|closed sets]] of $T$ is itself [[Definition:Closed Set (Topology)|closed]]. | Let $\ds \bigcup_{i \mathop = 1}^n V_i$ be the [[Definition:Set Union|union]] of a [[Definition:Finite|finite]] number of [[Definition:Closed Set (Topology)|closed sets]] of $T$.
Then from [[De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union|De Morgan's laws]]:
:$\ds S \setminus \bigcup_{... | Finite Union of Closed Sets is Closed/Topology | https://proofwiki.org/wiki/Finite_Union_of_Closed_Sets_is_Closed/Topology | https://proofwiki.org/wiki/Finite_Union_of_Closed_Sets_is_Closed/Topology | [
"Finite Union of Closed Sets is Closed"
] | [
"Definition:Topological Space",
"Definition:Set Union",
"Definition:Finite",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology"
] | [
"Definition:Set Union",
"Definition:Finite",
"Definition:Closed Set/Topology",
"De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Set Intersection",
"Definition:Finite",
"Definition:Open Set... |
proofwiki-17569 | Closed Ball is Closed/Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon^-} x$ be the closed $\epsilon$-ball of $x$ in $M$.
Then $\map {B_\epsilon^-} x$ is a closed set of $M$. | We show that the complement $A \setminus B_\epsilon^- \left({x}\right)$ is open in $M$.
Let $a \in A \setminus \map {B_\epsilon^-} x$.
Then by definition of closed ball:
:$\map d {x, a} > \epsilon$
Put:
:$\delta := \map d {x, a} - \epsilon > 0$
Then:
:$\map d {x, a} - \delta = \epsilon$
Let $b \in \map {B_\delta} a$.
T... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon^-} x$ be the [[Definition:Closed Ball|closed $\epsilon$-ball]] of $x$ in $M$.
Then $\map {B_\epsilon^-} x$ is a [[Definition:Closed Set (Metric Space)|closed set]] of $M$. | We show that the [[Definition:Set Complement|complement]] $A \setminus B_\epsilon^- \left({x}\right)$ is [[Definition:Open Set (Metric Space)|open]] in $M$.
Let $a \in A \setminus \map {B_\epsilon^-} x$.
Then by definition of [[Definition:Closed Ball|closed ball]]:
:$\map d {x, a} > \epsilon$
Put:
:$\delta := \map d... | Closed Ball is Closed/Metric Space | https://proofwiki.org/wiki/Closed_Ball_is_Closed/Metric_Space | https://proofwiki.org/wiki/Closed_Ball_is_Closed/Metric_Space | [
"Closed Balls",
"Closed Ball is Closed",
"Metric Spaces",
"Closed Sets (Metric Spaces)",
"Closed Sets (Metric Spaces)",
"Closed Ball is Closed"
] | [
"Definition:Metric Space",
"Definition:Closed Ball",
"Definition:Closed Set/Metric Space"
] | [
"Definition:Set Complement",
"Definition:Open Set/Metric Space",
"Definition:Closed Ball",
"Reverse Triangle Inequality",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Category:Closed Sets (Metric Spaces)",
"Category:Closed Ball is C... |
proofwiki-17570 | Derivative of Sine of Function | :$\map {\dfrac \d {\d x} } {\sin u} = \cos u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sin u}
| r = \map {\frac \d {\d u} } {\sin u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \cos u \frac {\d u} {\d x}
| c = Derivative of Sine Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\sin u} = \cos u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sin u}
| r = \map {\frac \d {\d u} } {\sin u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \cos u \frac {\d u} {\d x}
| c = [[Derivative of Sine Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Sine of Function | https://proofwiki.org/wiki/Derivative_of_Sine_of_Function | https://proofwiki.org/wiki/Derivative_of_Sine_of_Function | [
"Derivative of Sine Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Sine Function"
] |
proofwiki-17571 | Derivative of Cosine of Function | :$\map {\dfrac \d {\d x} } {\cos u} = -\sin u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cos u}
| r = \map {\frac \d {\d u} } {\cos u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = -\sin u \frac {\d u} {\d x}
| c = Derivative of Cosine Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\cos u} = -\sin u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cos u}
| r = \map {\frac \d {\d u} } {\cos u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\sin u \frac {\d u} {\d x}
| c = [[Derivative of Cosine Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Cosine of Function | https://proofwiki.org/wiki/Derivative_of_Cosine_of_Function | https://proofwiki.org/wiki/Derivative_of_Cosine_of_Function | [
"Derivative of Cosine Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Cosine Function"
] |
proofwiki-17572 | Derivative of Tangent of Function | :$\map {\dfrac \d {\d x} } {\tan u} = \sec^2 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\tan u}
| r = \map {\frac \d {\d u} } {\tan u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \sec^2 u \frac {\d u} {\d x}
| c = Derivative of Tangent Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\tan u} = \sec^2 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\tan u}
| r = \map {\frac \d {\d u} } {\tan u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \sec^2 u \frac {\d u} {\d x}
| c = [[Derivative of Tangent Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Tangent of Function | https://proofwiki.org/wiki/Derivative_of_Tangent_of_Function | https://proofwiki.org/wiki/Derivative_of_Tangent_of_Function | [
"Derivative of Tangent Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Tangent Function"
] |
proofwiki-17573 | Derivative of Cotangent of Function | :$\map {\dfrac \d {\d x} } {\cot u} = -\csc^2 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cot u}
| r = \map {\frac \d {\d u} } {\cot u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = -\csc^2 u \frac {\d u} {\d x}
| c = Derivative of Cotangent Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\cot u} = -\csc^2 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cot u}
| r = \map {\frac \d {\d u} } {\cot u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\csc^2 u \frac {\d u} {\d x}
| c = [[Derivative of Cotangent Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Cotangent of Function | https://proofwiki.org/wiki/Derivative_of_Cotangent_of_Function | https://proofwiki.org/wiki/Derivative_of_Cotangent_of_Function | [
"Derivative of Cotangent Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Cotangent Function"
] |
proofwiki-17574 | Derivative of Secant of Function | :$\map {\dfrac \d {\d x} } {\sec u} = \sec u \tan u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sec u}
| r = \map {\frac \d {\d u} } {\sec u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \sec u \tan u \frac {\d u} {\d x}
| c = Derivative of Secant Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\sec u} = \sec u \tan u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sec u}
| r = \map {\frac \d {\d u} } {\sec u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \sec u \tan u \frac {\d u} {\d x}
| c = [[Derivative of Secant Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Secant of Function | https://proofwiki.org/wiki/Derivative_of_Secant_of_Function | https://proofwiki.org/wiki/Derivative_of_Secant_of_Function | [
"Derivatives of Trigonometric Functions",
"Secant Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Secant Function"
] |
proofwiki-17575 | Derivative of Cosecant of Function | :$\map {\dfrac \d {\d x} } {\csc u} = \csc u \cot u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\csc u}
| r = \map {\frac \d {\d u} } {\csc u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = -\csc u \cot u \frac {\d u} {\d x}
| c = Derivative of Cosecant Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\csc u} = \csc u \cot u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\csc u}
| r = \map {\frac \d {\d u} } {\csc u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\csc u \cot u \frac {\d u} {\d x}
| c = [[Derivative of Cosecant Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Cosecant of Function | https://proofwiki.org/wiki/Derivative_of_Cosecant_of_Function | https://proofwiki.org/wiki/Derivative_of_Cosecant_of_Function | [
"Derivatives of Trigonometric Functions",
"Cosecant Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Cosecant Function"
] |
proofwiki-17576 | Derivative of General Logarithm of Function | :$\map {\dfrac \d {\d x} } {\log_a u} = \dfrac {\log_a e} u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\log_a u}
| r = \map {\frac \d {\d u} } {\log_a u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {\log_a e} u \frac {\d u} {\d x}
| c = Derivative of General Logarithm Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\log_a u} = \dfrac {\log_a e} u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\log_a u}
| r = \map {\frac \d {\d u} } {\log_a u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac {\log_a e} u \frac {\d u} {\d x}
| c = [[Derivative of General Logarithm Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of General Logarithm of Function | https://proofwiki.org/wiki/Derivative_of_General_Logarithm_of_Function | https://proofwiki.org/wiki/Derivative_of_General_Logarithm_of_Function | [
"Derivative of General Logarithm Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of General Logarithm Function"
] |
proofwiki-17577 | Derivative of Natural Logarithm of Function | :$\map {\dfrac \d {\d x} } {\ln u} = \dfrac 1 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\ln u}
| r = \map {\frac \d {\d u} } {\ln u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 u \frac {\d u} {\d x}
| c = Derivative of Natural Logarithm Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\ln u} = \dfrac 1 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\ln u}
| r = \map {\frac \d {\d u} } {\ln u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac 1 u \frac {\d u} {\d x}
| c = [[Derivative of Natural Logarithm Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Natural Logarithm of Function | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_of_Function | https://proofwiki.org/wiki/Derivative_of_Natural_Logarithm_of_Function | [
"Derivative of Natural Logarithm Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Natural Logarithm Function"
] |
proofwiki-17578 | Derivative of Constant to Power of Function | :$\map {\dfrac \d {\d x} } {a^u} = a^u \ln a \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {a^u}
| r = \map {\frac \d {\d u} } {a^u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = a^u \ln a \frac {\d u} {\d x}
| c = Derivative of Power of Constant
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {a^u} = a^u \ln a \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {a^u}
| r = \map {\frac \d {\d u} } {a^u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = a^u \ln a \frac {\d u} {\d x}
| c = [[Derivative of Power of Constant]]
}}
{{end-eqn}}
{{qed}} | Derivative of Constant to Power of Function | https://proofwiki.org/wiki/Derivative_of_Constant_to_Power_of_Function | https://proofwiki.org/wiki/Derivative_of_Constant_to_Power_of_Function | [
"Derivatives involving Exponential Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of General Exponential Function"
] |
proofwiki-17579 | Derivative of Exponential of Function | :$\map {\dfrac \d {\d x} } {e^u} = e^u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {e^u}
| r = \map {\frac \d {\d u} } {e^u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = e^u \frac {\d u} {\d x}
| c = Derivative of Exponential Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {e^u} = e^u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {e^u}
| r = \map {\frac \d {\d u} } {e^u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = e^u \frac {\d u} {\d x}
| c = [[Derivative of Exponential Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Exponential of Function | https://proofwiki.org/wiki/Derivative_of_Exponential_of_Function | https://proofwiki.org/wiki/Derivative_of_Exponential_of_Function | [
"Derivatives involving Exponential Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Exponential Function"
] |
proofwiki-17580 | Derivative of Arcsine of Function | :$\map {\dfrac \d {\d x} } {\arcsin u} = \dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcsin u}
| r = \map {\frac \d {\d u} } {\arcsin u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {\sqrt {1 - u^2} } \frac {\d u} {\d x}
| c = Derivative of Arcsine Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\arcsin u} = \dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcsin u}
| r = \map {\frac \d {\d u} } {\arcsin u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac 1 {\sqrt {1 - u^2} } \frac {\d u} {\d x}
| c = [[Derivative of Arcsine Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Arcsine of Function | https://proofwiki.org/wiki/Derivative_of_Arcsine_of_Function | https://proofwiki.org/wiki/Derivative_of_Arcsine_of_Function | [
"Derivatives of Inverse Trigonometric Functions",
"Arcsine Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Arcsine Function"
] |
proofwiki-17581 | Derivative of Arccosine of Function | :$\map {\dfrac \d {\d x} } {\arccos u} = -\dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$ | {{:Graph of Arccosine Function}}
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arccos u}
| r = \map {\frac \d {\d u} } {\arccos u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = -\dfrac 1 {\sqrt {1 - u^2} } \frac {\d u} {\d x}
| c = Derivative of Arccosine Function
}}
{{end-... | :$\map {\dfrac \d {\d x} } {\arccos u} = -\dfrac 1 {\sqrt {1 - u^2} } \dfrac {\d u} {\d x}$ | {{:Graph of Arccosine Function}}
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arccos u}
| r = \map {\frac \d {\d u} } {\arccos u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\dfrac 1 {\sqrt {1 - u^2} } \frac {\d u} {\d x}
| c = [[Derivative of Arccosine Function]]
... | Derivative of Arccosine of Function | https://proofwiki.org/wiki/Derivative_of_Arccosine_of_Function | https://proofwiki.org/wiki/Derivative_of_Arccosine_of_Function | [
"Derivatives of Inverse Trigonometric Functions",
"Arccosine Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Arccosine Function"
] |
proofwiki-17582 | Derivative of Arctangent of Function | :$\map {\dfrac \d {\d x} } {\arctan u} = \dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arctan u}
| r = \map {\frac \d {\d u} } {\arctan u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {1 + u^2} \frac {\d u} {\d x}
| c = Derivative of Arctangent Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\arctan u} = \dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arctan u}
| r = \map {\frac \d {\d u} } {\arctan u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac 1 {1 + u^2} \frac {\d u} {\d x}
| c = [[Derivative of Arctangent Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Arctangent of Function | https://proofwiki.org/wiki/Derivative_of_Arctangent_of_Function | https://proofwiki.org/wiki/Derivative_of_Arctangent_of_Function | [
"Derivatives of Inverse Trigonometric Functions",
"Arctangent Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Arctangent Function"
] |
proofwiki-17583 | Derivative of Arccotangent of Function | :$\map {\dfrac \d {\d x} } {\arccot u} = -\dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arccot u}
| r = \map {\frac \d {\d u} } {\arccot u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = -\dfrac 1 {1 + u^2} \frac {\d u} {\d x}
| c = Derivative of Arccotangent Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\arccot u} = -\dfrac 1 {1 + u^2} \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arccot u}
| r = \map {\frac \d {\d u} } {\arccot u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\dfrac 1 {1 + u^2} \frac {\d u} {\d x}
| c = [[Derivative of Arccotangent Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Arccotangent of Function | https://proofwiki.org/wiki/Derivative_of_Arccotangent_of_Function | https://proofwiki.org/wiki/Derivative_of_Arccotangent_of_Function | [
"Derivatives of Inverse Trigonometric Functions",
"Arccotangent Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Arccotangent Function"
] |
proofwiki-17584 | Derivative of Arcsecant of Function | :$\map {\dfrac \d {\d x} } {\arcsec u} = \dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcsec u}
| r = \map {\frac \d {\d u} } {\arcsec u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \dfrac 1 {\size u \sqrt {u^2 - 1} } \frac {\d u} {\d x}
| c = Derivative of Arcsecant Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\arcsec u} = \dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arcsec u}
| r = \map {\frac \d {\d u} } {\arcsec u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac 1 {\size u \sqrt {u^2 - 1} } \frac {\d u} {\d x}
| c = [[Derivative of Arcsecant Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Arcsecant of Function | https://proofwiki.org/wiki/Derivative_of_Arcsecant_of_Function | https://proofwiki.org/wiki/Derivative_of_Arcsecant_of_Function | [
"Derivatives of Inverse Trigonometric Functions",
"Arcsecant Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Arcsecant Function"
] |
proofwiki-17585 | Derivative of Arccosecant of Function | :$\map {\dfrac \d {\d x} } {\arccsc u} = -\dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arccsc u}
| r = \map {\frac \d {\d u} } {\arccsc u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = -\dfrac 1 {\size u \sqrt {u^2 - 1} } \frac {\d u} {\d x}
| c = Derivative of Arccosecant Function
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\arccsc u} = -\dfrac 1 {\size u \sqrt {u^2 - 1} } \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\arccsc u}
| r = \map {\frac \d {\d u} } {\arccsc u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\dfrac 1 {\size u \sqrt {u^2 - 1} } \frac {\d u} {\d x}
| c = [[Derivative of Arccosecant Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Arccosecant of Function | https://proofwiki.org/wiki/Derivative_of_Arccosecant_of_Function | https://proofwiki.org/wiki/Derivative_of_Arccosecant_of_Function | [
"Derivatives of Inverse Trigonometric Functions",
"Arccosecant Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Arccosecant Function"
] |
proofwiki-17586 | Derivative of Even Function is Odd | Let $f$ be a differentiable real function such that $f$ is even.
Then its derivative $f'$ is an odd function. | {{begin-eqn}}
{{eqn | l = \map f x
| r = \map f {-x}
| c = {{Defof|Even Function}}
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \map f x
| r = \frac \d {\d x} \map f {-x}
| c = differentiating both sides {{WRT|Differentiation}} $x$
}}
{{eqn | ll= \leadsto
| l = \map {f'} x
| r... | Let $f$ be a [[Definition:Differentiable Real Function|differentiable real function]] such that $f$ is [[Definition:Even Function|even]].
Then its [[Definition:Derivative|derivative]] $f'$ is an [[Definition:Odd Function|odd function]]. | {{begin-eqn}}
{{eqn | l = \map f x
| r = \map f {-x}
| c = {{Defof|Even Function}}
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \map f x
| r = \frac \d {\d x} \map f {-x}
| c = [[Definition:Differentiation|differentiating]] both sides {{WRT|Differentiation}} $x$
}}
{{eqn | ll= \leadsto
... | Derivative of Even Function is Odd | https://proofwiki.org/wiki/Derivative_of_Even_Function_is_Odd | https://proofwiki.org/wiki/Derivative_of_Even_Function_is_Odd | [
"Even Functions",
"Differential Calculus"
] | [
"Definition:Differentiable Mapping/Real Function",
"Definition:Even Function",
"Definition:Derivative",
"Definition:Odd Function"
] | [
"Definition:Differentiation",
"Derivative of Composite Function",
"Definition:Odd Function",
"Category:Even Functions",
"Category:Differential Calculus"
] |
proofwiki-17587 | Derivative of Odd Function is Even | Let $f$ be a differentiable real function such that $f$ is odd.
Then its derivative $f'$ is an even function. | {{begin-eqn}}
{{eqn | l = \map f x
| r = -\map f {-x}
| c = {{Defof|Odd Function}}
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \map f x
| r = -\frac \d {\d x} \map f {-x}
| c = differentiating both sides {{WRT|Differentiation}} $x$
}}
{{eqn | ll= \leadsto
| l = \map {f'} x
| ... | Let $f$ be a [[Definition:Differentiable Real Function|differentiable real function]] such that $f$ is [[Definition:Odd Function|odd]].
Then its [[Definition:Derivative|derivative]] $f'$ is an [[Definition:Even Function|even function]]. | {{begin-eqn}}
{{eqn | l = \map f x
| r = -\map f {-x}
| c = {{Defof|Odd Function}}
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \map f x
| r = -\frac \d {\d x} \map f {-x}
| c = [[Definition:Differentiation|differentiating]] both sides {{WRT|Differentiation}} $x$
}}
{{eqn | ll= \leadsto
... | Derivative of Odd Function is Even | https://proofwiki.org/wiki/Derivative_of_Odd_Function_is_Even | https://proofwiki.org/wiki/Derivative_of_Odd_Function_is_Even | [
"Odd Functions",
"Differential Calculus"
] | [
"Definition:Differentiable Mapping/Real Function",
"Definition:Odd Function",
"Definition:Derivative",
"Definition:Even Function"
] | [
"Definition:Differentiation",
"Derivative of Composite Function",
"Definition:Even Function",
"Category:Odd Functions",
"Category:Differential Calculus"
] |
proofwiki-17588 | Form of Prime Sierpiński Number of the First Kind | Suppose $S_n = n^n + 1$ is a prime Sierpiński number of the first kind.
Then:
:$n = 2^{2^k}$
for some integer $k$. | {{AimForCont}} $n$ has an odd divisor $d$.
By Sum of Two Odd Powers:
:$\paren {n^{n/d} + 1} \divides \paren {\paren {n^{n/d}}^d + 1^d} = S_n$
thus $S_n$ is composite, which is a contradiction.
Hence $n$ has no odd divisors.
That is, $n$ is a power of $2$.
Write $n = 2^m$.
{{AimForCont}} that $m$ has an odd divisor $f$.... | Suppose $S_n = n^n + 1$ is a [[Definition:Prime Number|prime]] [[Definition:Sierpiński Number of the First Kind|Sierpiński number of the first kind]].
Then:
:$n = 2^{2^k}$
for some [[Definition:Integer|integer]] $k$. | {{AimForCont}} $n$ has an [[Definition:Odd Integer|odd]] [[Definition:Divisor of Integer|divisor]] $d$.
By [[Sum of Two Odd Powers]]:
:$\paren {n^{n/d} + 1} \divides \paren {\paren {n^{n/d}}^d + 1^d} = S_n$
thus $S_n$ is [[Definition:Composite Number|composite]], which is a [[Definition:Contradiction|contradiction]].... | Form of Prime Sierpiński Number of the First Kind | https://proofwiki.org/wiki/Form_of_Prime_Sierpiński_Number_of_the_First_Kind | https://proofwiki.org/wiki/Form_of_Prime_Sierpiński_Number_of_the_First_Kind | [
"Sierpiński Numbers of the First Kind"
] | [
"Definition:Prime Number",
"Definition:Sierpiński Number of the First Kind",
"Definition:Integer"
] | [
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer",
"Sum of Two Odd Powers",
"Definition:Composite Number",
"Definition:Contradiction",
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Odd Integer",
"Definition:Diviso... |
proofwiki-17589 | Motion of Body Falling through Air | The motion of a body $B$ falling through air can be described using the following differential equation:
:$m \dfrac {\d^2 y} {\d t^2} = m g - k \dfrac {d y} {d t}$
where:
:$m$ denotes mass of $B$
:$y$ denotes the height of $B$ from an arbitrary reference
:$t$ denotes time elapsed from an arbitrary reference
:$g$ denote... | From Newton's Second Law of Motion, the force on $B$ equals its mass multiplied by its acceleration.
Thus the force $F$ on $B$ is given by:
:$F = m \dfrac {\d^2 y} {\d t^2}$
where it is assumed that the acceleration is in a downward direction.
The force on $B$ due to gravity is $m g$.
The force on $B$ due to the air it... | The motion of a [[Definition:Body|body]] $B$ falling through air can be described using the following [[Definition:Differential Equation|differential equation]]:
:$m \dfrac {\d^2 y} {\d t^2} = m g - k \dfrac {d y} {d t}$
where:
:$m$ denotes [[Definition:Mass|mass]] of $B$
:$y$ denotes the [[Definition:Height (Linear ... | From [[Newton's Second Law of Motion]], the [[Definition:Force|force]] on $B$ equals its [[Definition:Mass|mass]] multiplied by its [[Definition:Acceleration|acceleration]].
Thus the [[Definition:Force|force]] $F$ on $B$ is given by:
:$F = m \dfrac {\d^2 y} {\d t^2}$
where it is assumed that the [[Definition:Accelerat... | Motion of Body Falling through Air | https://proofwiki.org/wiki/Motion_of_Body_Falling_through_Air | https://proofwiki.org/wiki/Motion_of_Body_Falling_through_Air | [
"Gravity",
"Examples of Differential Equations"
] | [
"Definition:Body",
"Definition:Differential Equation",
"Definition:Mass",
"Definition:Linear Measure/Height",
"Definition:Time",
"Definition:Local Gravitational Constant",
"Definition:Force",
"Definition:Proportion",
"Definition:Speed"
] | [
"Newton's Laws of Motion/Second Law",
"Definition:Force",
"Definition:Mass",
"Definition:Acceleration",
"Definition:Force",
"Definition:Acceleration",
"Definition:Force",
"Definition:Gravity",
"Definition:Force",
"Definition:Speed",
"Definition:Differential Equation"
] |
proofwiki-17590 | Rational Numbers are Everywhere Dense in Set of Real Numbers/Normed Vector Space | Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers.
Let $\Q$ be the set of rational numbers.
Then $\Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$ | {{WIP|Under brief review}}
We have that Between two Real Numbers exists Rational Number:
:$\forall a, b \in \R : a < b : \exists r \in \Q : a < r < b$
Let $a := x$ with $x \in \R$.
Let $\epsilon \in \R_{\mathop > 0} : r - a < \epsilon$.
Let $b := x + \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = x - \epsilon
| o = <... | Let $\struct {\R, \size {\, \cdot \,}}$ be the [[Real Numbers with Absolute Value form Normed Vector Space|normed vector space of real numbers]].
Let $\Q$ be the [[Definition:Rational Number|set of rational numbers]].
Then $\Q$ are [[Definition:Everywhere Dense in Normed Vector Space|everywhere dense]] in $\struct {... | {{WIP|Under brief review}}
We have that [[Between two Real Numbers exists Rational Number]]:
:$\forall a, b \in \R : a < b : \exists r \in \Q : a < r < b$
Let $a := x$ with $x \in \R$.
Let $\epsilon \in \R_{\mathop > 0} : r - a < \epsilon$.
Let $b := x + \epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = x - \epsilon
... | Rational Numbers are Everywhere Dense in Set of Real Numbers/Normed Vector Space | https://proofwiki.org/wiki/Rational_Numbers_are_Everywhere_Dense_in_Set_of_Real_Numbers/Normed_Vector_Space | https://proofwiki.org/wiki/Rational_Numbers_are_Everywhere_Dense_in_Set_of_Real_Numbers/Normed_Vector_Space | [
"Real Analysis",
"Rational Number Space",
"Denseness"
] | [
"Real Numbers with Absolute Value form Normed Vector Space",
"Definition:Rational Number",
"Definition:Everywhere Dense/Normed Vector Space"
] | [
"Between two Real Numbers exists Rational Number",
"Definition:Everywhere Dense/Normed Vector Space"
] |
proofwiki-17591 | Number of Parameters of Autoregressive Model | Let $S$ be a stochastic process based on an equispaced time series.
Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t -... | By definition of the parameters of $M$:
{{:Definition:Parameter of Autoregressive Model}}
Thus:
:there are $p$ parameters of the form $\phi_j$
:$1$ parameter $\mu$
:$1$ parameter $\sigma_a^2$.
That is: $p + 1 + 1 = p + 2$ parameters.
{{qed}} | Let $S$ be a [[Definition:Stochastic Process|stochastic process]] based on an [[Definition:Equispaced Time Series|equispaced time series]].
Let the values of $S$ at [[Definition:Timestamp of Time Series Observation|timestamps]] $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z... | By definition of the [[Definition:Parameter of Autoregressive Model|parameters]] of $M$:
{{:Definition:Parameter of Autoregressive Model}}
Thus:
:there are $p$ [[Definition:Parameter of Autoregressive Model|parameters]] of the form $\phi_j$
:$1$ [[Definition:Parameter of Autoregressive Model|parameter]] $\mu$
:$1$ [[... | Number of Parameters of Autoregressive Model | https://proofwiki.org/wiki/Number_of_Parameters_of_Autoregressive_Model | https://proofwiki.org/wiki/Number_of_Parameters_of_Autoregressive_Model | [
"Autoregressive Models"
] | [
"Definition:Stochastic Process",
"Definition:Time Series/Equispaced",
"Definition:Time Series/Timestamp",
"Definition:Deviation from Mean",
"Definition:Constant Mean Level",
"Definition:Sequence",
"Definition:Independent Shocks",
"Definition:Time Series/Timestamp",
"Definition:Autoregressive Model",... | [
"Definition:Autoregressive Model/Parameter",
"Definition:Autoregressive Model/Parameter",
"Definition:Autoregressive Model/Parameter",
"Definition:Autoregressive Model/Parameter",
"Definition:Autoregressive Model/Parameter"
] |
proofwiki-17592 | Autoregressive Model is Special Case of Linear Filter Model | Let $S$ be a stochastic process based on an equispaced time series.
Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t -... | We can eliminate $\tilde z_{t - 1}$ from the {{RHS}} of $(1)$ by substituting:
:$\tilde z_{t - 1} = \phi_1 \tilde z_{t - 2} + \phi_2 \tilde z_{t - 3} + \dotsb + \phi_p \tilde z_{t - p - 1} + a_{t - 1}$
Similarly we can substitute for $\tilde z_{t - 2}$, and so on.
Eventually we get an infinite series in $a_{t - j}$.
He... | Let $S$ be a [[Definition:Stochastic Process|stochastic process]] based on an [[Definition:Equispaced Time Series|equispaced time series]].
Let the values of $S$ at [[Definition:Timestamp of Time Series Observation|timestamps]] $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z... | We can eliminate $\tilde z_{t - 1}$ from the {{RHS}} of $(1)$ by substituting:
:$\tilde z_{t - 1} = \phi_1 \tilde z_{t - 2} + \phi_2 \tilde z_{t - 3} + \dotsb + \phi_p \tilde z_{t - p - 1} + a_{t - 1}$
Similarly we can substitute for $\tilde z_{t - 2}$, and so on.
Eventually we get an [[Definition:Infinite Series|in... | Autoregressive Model is Special Case of Linear Filter Model | https://proofwiki.org/wiki/Autoregressive_Model_is_Special_Case_of_Linear_Filter_Model | https://proofwiki.org/wiki/Autoregressive_Model_is_Special_Case_of_Linear_Filter_Model | [
"Autoregressive Models"
] | [
"Definition:Stochastic Process",
"Definition:Time Series/Equispaced",
"Definition:Time Series/Timestamp",
"Definition:Deviation from Mean",
"Definition:Constant Mean Level",
"Definition:Sequence",
"Definition:Independent Shocks",
"Definition:Time Series/Timestamp",
"Definition:Autoregressive Model",... | [
"Definition:Series",
"Definition:Linear Filter"
] |
proofwiki-17593 | Irrationals are Everywhere Dense in Reals/Topology | Let $T = \struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.
Let $\R \setminus \Q$ be the set of irrational numbers.
Then $\R \setminus \Q$ is everywhere dense in $T$. | Let $x \in \R$.
Let $U \subseteq \R$ be an open set of $T$ such that $x \in U$.
From Basis for Euclidean Topology on Real Number Line, there exists an open interval $V_0 = \openint {x - \epsilon} {x + \epsilon} \subseteq U$ for some $\epsilon > 0$ such that $x \in V_0$.
From Between two Real Numbers exists Rational Num... | Let $T = \struct {\R, \tau}$ denote the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\R \setminus \Q$ be the [[Definition:Irrational Number|set of irrational numbers]].
Then $\R \setminus \Q$ is [[Definition:Everywhere Dense|everywhere dense]] in ... | Let $x \in \R$.
Let $U \subseteq \R$ be an [[Definition:Open Set (Topology)|open set]] of $T$ such that $x \in U$.
From [[Basis for Euclidean Topology on Real Number Line]], there exists an [[Definition:Open Real Interval|open interval]] $V_0 = \openint {x - \epsilon} {x + \epsilon} \subseteq U$ for some $\epsilon > ... | Irrationals are Everywhere Dense in Reals/Topology | https://proofwiki.org/wiki/Irrationals_are_Everywhere_Dense_in_Reals/Topology | https://proofwiki.org/wiki/Irrationals_are_Everywhere_Dense_in_Reals/Topology | [
"Real Analysis",
"Real Number Line with Euclidean Topology",
"Irrational Number Space",
"Denseness"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Irrational Number",
"Definition:Everywhere Dense"
] | [
"Definition:Open Set/Topology",
"Basis for Euclidean Topology on Real Number Line",
"Definition:Real Interval/Open",
"Between two Real Numbers exists Rational Number",
"Definition:Real Interval/Open",
"Definition:Real Interval/Open",
"Subset Relation is Transitive",
"Between two Rational Numbers exist... |
proofwiki-17594 | Irrationals are Everywhere Dense in Reals/Normed Vector Space | Let $\struct {\R, \size {\, \cdot \,} }$ be the normed vector space of real numbers.
Let $\R \setminus \Q$ be the set of irrational numbers.
Then $\R \setminus \Q$ is everywhere dense in $\struct {\R, \size {\, \cdot \,} }$ | Let $x \in \R$.
Let $\epsilon \in \R_{\mathop > 0}$
Either $x \in \Q$ or $x \in \R \setminus \Q$.
Suppose $x \in \R \setminus \Q$.
Let $y := x$.
Then:
:$\size {x - y} < \epsilon$
Suppose $x \in \Q$.
Let $n \in \N : n > \dfrac {\sqrt 2} \epsilon$
Let $y := x + \dfrac {\sqrt 2} n$
Then $y \in \R \setminus \Q$.
Furthermor... | Let $\struct {\R, \size {\, \cdot \,} }$ be the [[Real Numbers with Absolute Value form Normed Vector Space|normed vector space of real numbers]].
Let $\R \setminus \Q$ be the [[Definition:Irrational Number|set of irrational numbers]].
Then $\R \setminus \Q$ is [[Definition:Everywhere Dense in Normed Vector Space|ev... | Let $x \in \R$.
Let $\epsilon \in \R_{\mathop > 0}$
Either $x \in \Q$ or $x \in \R \setminus \Q$.
Suppose $x \in \R \setminus \Q$.
Let $y := x$.
Then:
:$\size {x - y} < \epsilon$
Suppose $x \in \Q$.
Let $n \in \N : n > \dfrac {\sqrt 2} \epsilon$
Let $y := x + \dfrac {\sqrt 2} n$
Then $y \in \R \setminus \Q$.
... | Irrationals are Everywhere Dense in Reals/Normed Vector Space | https://proofwiki.org/wiki/Irrationals_are_Everywhere_Dense_in_Reals/Normed_Vector_Space | https://proofwiki.org/wiki/Irrationals_are_Everywhere_Dense_in_Reals/Normed_Vector_Space | [
"Real Analysis",
"Normed Vector Spaces",
"Irrational Number Space",
"Denseness"
] | [
"Real Numbers with Absolute Value form Normed Vector Space",
"Definition:Irrational Number",
"Definition:Everywhere Dense/Normed Vector Space"
] | [
"Definition:Everywhere Dense/Normed Vector Space"
] |
proofwiki-17595 | Real Number Subtracted from Itself leaves Zero | Let $x \in \R$ be a real number.
Then:
:$x - x = 0$
where $x - x$ denotes the operation of real subtraction. | {{begin-eqn}}
{{eqn | l = x - x
| r = x + \paren {-x}
| c = {{Defof|Real Subtraction}}
}}
{{eqn | r = 0
| c = Inverse for Real Addition
}}
{{end-eqn}}
{{qed}} | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$x - x = 0$
where $x - x$ denotes the operation of [[Definition:Real Subtraction|real subtraction]]. | {{begin-eqn}}
{{eqn | l = x - x
| r = x + \paren {-x}
| c = {{Defof|Real Subtraction}}
}}
{{eqn | r = 0
| c = [[Inverse for Real Addition]]
}}
{{end-eqn}}
{{qed}} | Real Number Subtracted from Itself leaves Zero | https://proofwiki.org/wiki/Real_Number_Subtracted_from_Itself_leaves_Zero | https://proofwiki.org/wiki/Real_Number_Subtracted_from_Itself_leaves_Zero | [
"Subtraction"
] | [
"Definition:Real Number",
"Definition:Subtraction/Real Numbers"
] | [
"Inverse for Real Addition"
] |
proofwiki-17596 | Real Number Ordering is Compatible with Multiplication/Positive Factor/Corollary | :$\forall a, b, c, d \in \R: 0 < a < b \land 0 < c < d \implies a c < b d$ | {{begin-eqn}}
{{eqn | l = a < b
| o = \implies
| r = a \times c < b \times c
| c = Real Number Ordering is Compatible with Multiplication: Positive Factor as $c > 0$
}}
{{eqn | l = c < d
| o = \implies
| r = b \times c < b \times d
| c = Real Number Ordering is Compatible with Multip... | :$\forall a, b, c, d \in \R: 0 < a < b \land 0 < c < d \implies a c < b d$ | {{begin-eqn}}
{{eqn | l = a < b
| o = \implies
| r = a \times c < b \times c
| c = [[Real Number Ordering is Compatible with Multiplication/Positive Factor|Real Number Ordering is Compatible with Multiplication: Positive Factor]] as $c > 0$
}}
{{eqn | l = c < d
| o = \implies
| r = b \time... | Real Number Ordering is Compatible with Multiplication/Positive Factor/Corollary | https://proofwiki.org/wiki/Real_Number_Ordering_is_Compatible_with_Multiplication/Positive_Factor/Corollary | https://proofwiki.org/wiki/Real_Number_Ordering_is_Compatible_with_Multiplication/Positive_Factor/Corollary | [
"Real Number Ordering is Compatible with Multiplication"
] | [] | [
"Real Number Ordering is Compatible with Multiplication/Positive Factor",
"Real Number Ordering is Compatible with Multiplication/Positive Factor",
"Transitive Law"
] |
proofwiki-17597 | Equivalence of Definitions of Matroid Base Axioms | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
{{TFAE|axiom = Base Axiom (Matroid)|view = matroid base axiom}}
{{:Axiom:Base Axiom (Matroid)}} | === Formulation $1$ iff Formulation $2$ ===
Formulation $1$ holds {{iff}} formulation $2$ holds follows immediately from {{Corollary|Set Difference Then Union Equals Union Then Set Difference}}.
{{qed|lemma}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
{{TFAE|axiom = Base Axiom (Matroid)|view = matroid base axiom}}
{{:Axiom:Base Axiom (Matroid)}} | === Formulation $1$ iff Formulation $2$ ===
[[Axiom:Base Axiom (Matroid)/Formulation 1|Formulation $1$]] holds {{iff}} [[Axiom:Base Axiom (Matroid)/Formulation 2|formulation $2$]] holds follows immediately from {{Corollary|Set Difference Then Union Equals Union Then Set Difference}}.
{{qed|lemma}} | Equivalence of Definitions of Matroid Base Axioms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms | [
"Matroid Bases",
"Equivalence of Definitions of Matroid Base Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty",
"Definition:Set",
"Definition:Subset"
] | [
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Axiom:Base Axiom (Matroid)/Formulation 2"
] |
proofwiki-17598 | Number of Parameters of Moving Average Model | Let $S$ be a stochastic process based on an equispaced time series.
Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t$ be the deviation from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of ... | By definition of the parameters of $M$:
{{:Definition:Parameter of Moving Average Model}}
Thus:
:there are $q$ parameters of the form $\theta_j$
:$1$ parameter $\mu$
:$1$ parameter $\sigma_a^2$.
That is: $q + 1 + 1 = q + 2$ parameters.
{{qed}} | Let $S$ be a [[Definition:Stochastic Process|stochastic process]] based on an [[Definition:Equispaced Time Series|equispaced time series]].
Let the values of $S$ at [[Definition:Timestamp of Time Series Observation|timestamps]] $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t$ be the [... | By definition of the [[Definition:Parameter of Moving Average Model|parameters]] of $M$:
{{:Definition:Parameter of Moving Average Model}}
Thus:
:there are $q$ [[Definition:Parameter of Moving Average Model|parameters]] of the form $\theta_j$
:$1$ [[Definition:Parameter of Moving Average Model|parameter]] $\mu$
:$1$ ... | Number of Parameters of Moving Average Model | https://proofwiki.org/wiki/Number_of_Parameters_of_Moving_Average_Model | https://proofwiki.org/wiki/Number_of_Parameters_of_Moving_Average_Model | [
"Moving Average Models"
] | [
"Definition:Stochastic Process",
"Definition:Time Series/Equispaced",
"Definition:Time Series/Timestamp",
"Definition:Deviation from Mean",
"Definition:Constant Mean Level",
"Definition:Sequence",
"Definition:Independent Shocks",
"Definition:Time Series/Timestamp",
"Definition:Moving Average Model",... | [
"Definition:Moving Average Model/Parameter",
"Definition:Moving Average Model/Parameter",
"Definition:Moving Average Model/Parameter",
"Definition:Moving Average Model/Parameter",
"Definition:Moving Average Model/Parameter"
] |
proofwiki-17599 | Convergent Sequence is Cauchy Sequence/Metric Space | Let $M = \struct {A, d}$ be a metric space.
Every convergent sequence in $A$ is a Cauchy sequence. | Let $\sequence {x_n}$ be a sequence in $A$ that converges to the limit $l \in A$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ converges to $l$ in $A$, we have:
:$\exists N_1 \in \R_{>0}: \forall n > N_1: \map d {x_n, l} < \dfrac \epsilon 2$
Because $\sequence {x_n}$ converges to $l... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Every [[Definition:Convergent Sequence in Metric Space|convergent sequence]] in $A$ is a [[Definition:Cauchy Sequence (Metric Space)|Cauchy sequence]]. | Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $A$ that [[Definition:Convergent Sequence in Metric Space|converges]] to the [[Definition:Limit of Sequence (Metric Space)|limit]] $l \in A$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ [[Definition:Convergent Sequen... | Convergent Sequence is Cauchy Sequence/Metric Space | https://proofwiki.org/wiki/Convergent_Sequence_is_Cauchy_Sequence/Metric_Space | https://proofwiki.org/wiki/Convergent_Sequence_is_Cauchy_Sequence/Metric_Space | [
"Convergent Sequence is Cauchy Sequence",
"Convergent Sequences (Metric Space)"
] | [
"Definition:Metric Space",
"Definition:Convergent Sequence/Metric Space",
"Definition:Cauchy Sequence/Metric Space"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Limit of Sequence/Metric Space",
"Definition:Convergent Sequence/Metric Space",
"Definition:Convergent Sequence/Metric Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Limit of Se... |
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