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-16100 | Sum of Sequence of n Choose 2 | Let $n \in \Z$ be an integer such that $n \ge 2$.
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 2}^n \dbinom j 2
| r = \dbinom 2 2 + \dbinom 3 2 + \dbinom 4 2 + \dotsb + \dbinom n 2
| c =
}}
{{eqn | r = \dbinom {n + 1} 3
| c =
}}
{{end-eqn}}
where $\dbinom n j$ denotes a binomial coefficient. | The proof proceeds by induction.
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 2}^n \dbinom j 2 = \dbinom {n + 1} 3$
=== Basis for the Induction ===
$\map P 2$ is the case:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 2}^2 \dbinom j 2
| r = \dbinom 2 2
| c =
}}
{{eqn... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 2}^n \dbinom j 2
| r = \dbinom 2 2 + \dbinom 3 2 + \dbinom 4 2 + \dotsb + \dbinom n 2
| c =
}}
{{eqn | r = \dbinom {n + 1} 3
| c =
}}
{{end-eqn}}
where $\dbinom n j$ denotes a [[Def... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 2}^n \dbinom j 2 = \dbinom {n + 1} 3$
=== Basis for the Induction ===
$\map P 2$ is the case:
{{begin-eqn}}
{{eqn | l = \sum_{j ... | Sum of Sequence of n Choose 2/Proof 2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_n_Choose_2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_n_Choose_2/Proof_2 | [
"Binomial Coefficients",
"Sum of Sequence of n Choose 2"
] | [
"Definition:Integer",
"Definition:Binomial Coefficient"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Sum of Sequence of n Choose 2/Proof 2",
"Pascal's Rule",
"Principle of Mathematical Induction"
] |
proofwiki-16101 | If n is Triangular then so is 25n + 3 | Let $n$ be a triangular number.
Then $25 n + 3$ is also triangular. | Let $n$ be triangular.
Then:
:$\exists k \in \Z: n = \dfrac {k \paren {k + 1} } 2$
So:
{{begin-eqn}}
{{eqn | l = 25 n + 3
| r = 25 \frac {k \paren {k + 1} } 2 + 3
| c =
}}
{{eqn | r = \frac {25 k^2 + 25 k + 6} 2
| c =
}}
{{eqn | r = \frac {\paren {5 k + 2} \paren {5 k + 3} } 2
| c =
}}
{{end-... | Let $n$ be a [[Definition:Triangular Number|triangular number]].
Then $25 n + 3$ is also [[Definition:Triangular Number|triangular]]. | Let $n$ be [[Definition:Triangular Number|triangular]].
Then:
:$\exists k \in \Z: n = \dfrac {k \paren {k + 1} } 2$
So:
{{begin-eqn}}
{{eqn | l = 25 n + 3
| r = 25 \frac {k \paren {k + 1} } 2 + 3
| c =
}}
{{eqn | r = \frac {25 k^2 + 25 k + 6} 2
| c =
}}
{{eqn | r = \frac {\paren {5 k + 2} \paren ... | If n is Triangular then so is 25n + 3 | https://proofwiki.org/wiki/If_n_is_Triangular_then_so_is_25n_+_3 | https://proofwiki.org/wiki/If_n_is_Triangular_then_so_is_25n_+_3 | [
"Triangular Numbers"
] | [
"Definition:Triangular Number",
"Definition:Triangular Number"
] | [
"Definition:Triangular Number",
"Definition:Triangular Number"
] |
proofwiki-16102 | If n is Triangular then so is 49n + 6 | Let $n$ be a triangular number.
Then $49 n + 6$ is also triangular. | Let $n$ be triangular.
Then:
:$\exists k \in \Z: n = \dfrac {k \paren {k + 1} } 2$
So:
{{begin-eqn}}
{{eqn | l = 49 n + 6
| r = 49 \frac {k \paren {k + 1} } 2 + 6
| c =
}}
{{eqn | r = \frac {49 k^2 + 49 k + 12} 2
| c =
}}
{{eqn | r = \frac {\paren {7 k + 3} \paren {7 k + 4} } 2
| c =
}}
{{end... | Let $n$ be a [[Definition:Triangular Number|triangular number]].
Then $49 n + 6$ is also [[Definition:Triangular Number|triangular]]. | Let $n$ be [[Definition:Triangular Number|triangular]].
Then:
:$\exists k \in \Z: n = \dfrac {k \paren {k + 1} } 2$
So:
{{begin-eqn}}
{{eqn | l = 49 n + 6
| r = 49 \frac {k \paren {k + 1} } 2 + 6
| c =
}}
{{eqn | r = \frac {49 k^2 + 49 k + 12} 2
| c =
}}
{{eqn | r = \frac {\paren {7 k + 3} \paren... | If n is Triangular then so is 49n + 6 | https://proofwiki.org/wiki/If_n_is_Triangular_then_so_is_49n_+_6 | https://proofwiki.org/wiki/If_n_is_Triangular_then_so_is_49n_+_6 | [
"Triangular Numbers"
] | [
"Definition:Triangular Number",
"Definition:Triangular Number"
] | [
"Definition:Triangular Number",
"Definition:Triangular Number"
] |
proofwiki-16103 | Sum of Sequence of Triangular Numbers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $T_n$ denote the $n$th triangular number.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n T_j
| r = T_1 + T_2 + T_3 + \dotsb + T_n
| c =
}}
{{eqn | r = \dfrac {n \paren {n + 1} \paren {n + 2} } 6
| c =
}}
{{end-eqn}} | From Sum of Sequence of n Choose 2 we have:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 2}^n \dbinom j 2
| r = \dbinom 2 2 + \dbinom 3 2 + \dbinom 4 2 + \dotsb + \dbinom n 2
| c =
}}
{{eqn | r = \dbinom {n + 1} 3
| c =
}}
{{end-eqn}}
and so:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 2}^{n + 1} \db... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $T_n$ denote the $n$th [[Definition:Triangular Number|triangular number]].
Then:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n T_j
| r = T_1 + T_2 + T_3 + \dotsb + T_n
| c =
}}
{{eqn | r = \dfrac {n \p... | From [[Sum of Sequence of n Choose 2]] we have:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 2}^n \dbinom j 2
| r = \dbinom 2 2 + \dbinom 3 2 + \dbinom 4 2 + \dotsb + \dbinom n 2
| c =
}}
{{eqn | r = \dbinom {n + 1} 3
| c =
}}
{{end-eqn}}
and so:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 2}^{n +... | Sum of Sequence of Triangular Numbers/Proof 1 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Triangular_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Triangular_Numbers/Proof_1 | [
"Sum of Sequence of Triangular Numbers",
"Triangular Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Triangular Number"
] | [
"Sum of Sequence of n Choose 2",
"Binomial Coefficient with Two"
] |
proofwiki-16104 | Sum of Sequence of Triangular Numbers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $T_n$ denote the $n$th triangular number.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n T_j
| r = T_1 + T_2 + T_3 + \dotsb + T_n
| c =
}}
{{eqn | r = \dfrac {n \paren {n + 1} \paren {n + 2} } 6
| c =
}}
{{end-eqn}} | First let $n$ be even.
Thus we have:
:$n = 2 m$
Then:
{{begin-eqn}}
{{eqn | l = T_1 + T_2 + T_3 + \dotsb + T_{2 m}
| r = \paren {T_1 + T_2} + \paren {T_3 + T_4} + \dotsb + \paren {T_{2 m - 1} + T_{2 m} }
| c =
}}
{{eqn | r = 2^2 + 4^2 + \dotsb + \paren {2 m}^2
| c = Sum of Consecutive Triangular Numb... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $T_n$ denote the $n$th [[Definition:Triangular Number|triangular number]].
Then:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n T_j
| r = T_1 + T_2 + T_3 + \dotsb + T_n
| c =
}}
{{eqn | r = \dfrac {n \p... | First let $n$ be [[Definition:Even Integer|even]].
Thus we have:
:$n = 2 m$
Then:
{{begin-eqn}}
{{eqn | l = T_1 + T_2 + T_3 + \dotsb + T_{2 m}
| r = \paren {T_1 + T_2} + \paren {T_3 + T_4} + \dotsb + \paren {T_{2 m - 1} + T_{2 m} }
| c =
}}
{{eqn | r = 2^2 + 4^2 + \dotsb + \paren {2 m}^2
| c = [[S... | Sum of Sequence of Triangular Numbers/Proof 2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Triangular_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Triangular_Numbers/Proof_2 | [
"Sum of Sequence of Triangular Numbers",
"Triangular Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Triangular Number"
] | [
"Definition:Even Integer",
"Sum of Consecutive Triangular Numbers is Square",
"Sum of Consecutive Triangular Numbers is Square",
"Definition:Odd Integer",
"Closed Form for Triangular Numbers"
] |
proofwiki-16105 | Square of Odd Multiple of 3 is Difference between Triangular Numbers | Let $n \in \Z_{\ge 0}$ be a positive integer.
Let $T_n$ denote the $n$th triangular number.
Let $m = 2 n + 1$ be an odd integer
Then:
:$\paren {3 m}^2 = T_{9 n + 4} - T_{3 n + 1}$ | {{begin-eqn}}
{{eqn | l = T_{9 n + 4} - T_{3 n + 1}
| r = \dfrac {\paren {9 n + 4} \paren {9 n + 5} } 2 - \dfrac {\paren {3 n + 1} \paren {3 n + 2} } 2
| c = Closed Form for Triangular Numbers
}}
{{eqn | r = \dfrac {\paren {81 n^2 + 81 n + 20} - \paren {9 n^2 + 9 n + 2} } 2
| c =
}}
{{eqn | r = \dfra... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Let $T_n$ denote the $n$th [[Definition:Triangular Number|triangular number]].
Let $m = 2 n + 1$ be an [[Definition:Odd Integer|odd integer]]
Then:
:$\paren {3 m}^2 = T_{9 n + 4} - T_{3 n + 1}$ | {{begin-eqn}}
{{eqn | l = T_{9 n + 4} - T_{3 n + 1}
| r = \dfrac {\paren {9 n + 4} \paren {9 n + 5} } 2 - \dfrac {\paren {3 n + 1} \paren {3 n + 2} } 2
| c = [[Closed Form for Triangular Numbers]]
}}
{{eqn | r = \dfrac {\paren {81 n^2 + 81 n + 20} - \paren {9 n^2 + 9 n + 2} } 2
| c =
}}
{{eqn | r = \... | Square of Odd Multiple of 3 is Difference between Triangular Numbers | https://proofwiki.org/wiki/Square_of_Odd_Multiple_of_3_is_Difference_between_Triangular_Numbers | https://proofwiki.org/wiki/Square_of_Odd_Multiple_of_3_is_Difference_between_Triangular_Numbers | [
"Triangular Numbers",
"Square Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Triangular Number",
"Definition:Odd Integer"
] | [
"Closed Form for Triangular Numbers"
] |
proofwiki-16106 | Square Sum of Three Consecutive Triangular Numbers | Let $T_n$ denote the $n$th triangular number for $n \in \Z_{>0}$ a (strictly) positive integer.
Let $T_n + T_{n + 1} + T_{n + 2}$ be a square number.
Then at least one value of $n$ fulfils this condition:
:$n = 5$ | Let $T_n + T_{n + 1} + T_{n + 2} = m^2$ for some $m \in \Z_{>0}$.
We have:
{{begin-eqn}}
{{eqn | l = T_n + T_{n + 1} + T_{n + 2}
| r = \dfrac {n \paren {n + 1} } 2 + \dfrac {\paren {n + 1} \paren {n + 2} } 2 + \dfrac {\paren {n + 2} \paren {n + 3} } 2
| c =
}}
{{eqn | r = \dfrac {n \paren {n + 1} + \paren ... | Let $T_n$ denote the $n$th [[Definition:Triangular Number|triangular number]] for $n \in \Z_{>0}$ a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $T_n + T_{n + 1} + T_{n + 2}$ be a [[Definition:Square Number|square number]].
Then at least one value of $n$ fulfils this condition:
:$n = 5$ | Let $T_n + T_{n + 1} + T_{n + 2} = m^2$ for some $m \in \Z_{>0}$.
We have:
{{begin-eqn}}
{{eqn | l = T_n + T_{n + 1} + T_{n + 2}
| r = \dfrac {n \paren {n + 1} } 2 + \dfrac {\paren {n + 1} \paren {n + 2} } 2 + \dfrac {\paren {n + 2} \paren {n + 3} } 2
| c =
}}
{{eqn | r = \dfrac {n \paren {n + 1} + \pare... | Square Sum of Three Consecutive Triangular Numbers | https://proofwiki.org/wiki/Square_Sum_of_Three_Consecutive_Triangular_Numbers | https://proofwiki.org/wiki/Square_Sum_of_Three_Consecutive_Triangular_Numbers | [
"Triangular Numbers",
"Square Numbers"
] | [
"Definition:Triangular Number",
"Definition:Strictly Positive/Integer",
"Definition:Square Number"
] | [
"Definition:Square Number"
] |
proofwiki-16107 | Square Product of Three Consecutive Triangular Numbers | Let $T_n$ denote the $n$th triangular number for $n \in \Z_{>0}$ a (strictly) positive integer.
Let $T_n \times T_{n + 1} \times T_{n + 2}$ be a square number.
Then at least one value of $n$ fulfils this condition:
:$n = 3$ | Let $T_n \times T_{n + 1} \times T_{n + 2} = m^2$ for some $m \in \Z_{>0}$.
We have:
{{begin-eqn}}
{{eqn | l = T_n \times T_{n + 1} \times T_{n + 2}
| r = \dfrac {n \paren {n + 1} } 2 \dfrac {\paren {n + 1} \paren {n + 2} } 2 \dfrac {\paren {n + 2} \paren {n + 3} } 2
| c =
}}
{{eqn | r = \dfrac {n \paren {... | Let $T_n$ denote the $n$th [[Definition:Triangular Number|triangular number]] for $n \in \Z_{>0}$ a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $T_n \times T_{n + 1} \times T_{n + 2}$ be a [[Definition:Square Number|square number]].
Then at least one value of $n$ fulfils this condition:... | Let $T_n \times T_{n + 1} \times T_{n + 2} = m^2$ for some $m \in \Z_{>0}$.
We have:
{{begin-eqn}}
{{eqn | l = T_n \times T_{n + 1} \times T_{n + 2}
| r = \dfrac {n \paren {n + 1} } 2 \dfrac {\paren {n + 1} \paren {n + 2} } 2 \dfrac {\paren {n + 2} \paren {n + 3} } 2
| c =
}}
{{eqn | r = \dfrac {n \paren... | Square Product of Three Consecutive Triangular Numbers | https://proofwiki.org/wiki/Square_Product_of_Three_Consecutive_Triangular_Numbers | https://proofwiki.org/wiki/Square_Product_of_Three_Consecutive_Triangular_Numbers | [
"Triangular Numbers",
"Square Numbers"
] | [
"Definition:Triangular Number",
"Definition:Strictly Positive/Integer",
"Definition:Square Number"
] | [
"Definition:Square Number"
] |
proofwiki-16108 | Sufficient Condition for Square of Product to be Triangular | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $2 n^2 \pm 1 = m^2$ be a square number.
Then $\paren {m n}^2$ is a triangular number. | {{begin-eqn}}
{{eqn | l = \paren {m n}^2
| r = \paren {2 n^2 \pm 1} \times n^2
| c =
}}
{{eqn | r = \dfrac {\paren {2 n^2 \pm 1} \paren {2 n^2} } 2
| c =
}}
{{end-eqn}}
That is, either:
:$\paren {m n}^2 = \dfrac {\paren {2 n^2 - 1} \paren {2 n^2} } 2$
and so:
:$\paren {m n}^2 = T_{2 n^2 - 1}$
or:
:$... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $2 n^2 \pm 1 = m^2$ be a [[Definition:Square Number|square number]].
Then $\paren {m n}^2$ is a [[Definition:Triangular Number|triangular number]]. | {{begin-eqn}}
{{eqn | l = \paren {m n}^2
| r = \paren {2 n^2 \pm 1} \times n^2
| c =
}}
{{eqn | r = \dfrac {\paren {2 n^2 \pm 1} \paren {2 n^2} } 2
| c =
}}
{{end-eqn}}
That is, either:
:$\paren {m n}^2 = \dfrac {\paren {2 n^2 - 1} \paren {2 n^2} } 2$
and so:
:$\paren {m n}^2 = T_{2 n^2 - 1}$
o... | Sufficient Condition for Square of Product to be Triangular | https://proofwiki.org/wiki/Sufficient_Condition_for_Square_of_Product_to_be_Triangular | https://proofwiki.org/wiki/Sufficient_Condition_for_Square_of_Product_to_be_Triangular | [
"Triangular Numbers",
"Square Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Square Number",
"Definition:Triangular Number"
] | [] |
proofwiki-16109 | Trivial Topological Space is Indiscrete | Let $T = \struct {S, \tau}$ be a trivial topological space.
Then $\tau$ is an indiscrete topology. | By definition of trivial topological space, $S$ is a singleton.
That is, $S$ is a set containing exactly one element.
Suppose $S = \set x$ for some object $x$.
Then the power set of $S$ is the set:
:$\powerset S = \set {\O, \set x}$
That is:
:$\powerset S = \set {\O, S}$
Let $\tau$ be a topology on $S$.
We have that $... | Let $T = \struct {S, \tau}$ be a [[Definition:Trivial Topological Space|trivial topological space]].
Then $\tau$ is an [[Definition:Indiscrete Topology|indiscrete topology]]. | By definition of [[Definition:Trivial Topological Space|trivial topological space]], $S$ is a [[Definition:Singleton|singleton]].
That is, $S$ is a [[Definition:Set|set]] containing exactly one [[Definition:Element|element]].
Suppose $S = \set x$ for some [[Definition:Object|object]] $x$.
Then the [[Definition:Powe... | Trivial Topological Space is Indiscrete | https://proofwiki.org/wiki/Trivial_Topological_Space_is_Indiscrete | https://proofwiki.org/wiki/Trivial_Topological_Space_is_Indiscrete | [
"Trivial Topological Spaces",
"Indiscrete Topology"
] | [
"Definition:Trivial Topological Space",
"Definition:Indiscrete Topology"
] | [
"Definition:Trivial Topological Space",
"Definition:Singleton",
"Definition:Set",
"Definition:Element",
"Definition:Object",
"Definition:Power Set",
"Definition:Set",
"Definition:Topology",
"Definition:Subset",
"Definition:Set",
"Definition:Topology",
"Empty Set is Element of Topology",
"Ind... |
proofwiki-16110 | Existence of Divisor with Remainder between 2b and 3b | For every pair of integers $a, b$ where $b > 0$, there exist unique integers $q$ and $r$ where $2 b \le r < 3 b$ such that:
:$a = q b + r$ | From the Division Theorem, we have that:
:$\forall a, b \in \Z, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$
So, with a view to where this is going, let $q$ and $r$ be renamed such that $a = q' b + r'$ with $0 \le r' < b$.
Then let $q' = q + 2$.
We have:
{{begin-eqn}}
{{eqn | l = a
| r = q' b + r'
... | For every pair of [[Definition:Integer|integers]] $a, b$ where $b > 0$, there exist [[Definition:Unique|unique]] [[Definition:Integer|integers]] $q$ and $r$ where $2 b \le r < 3 b$ such that:
:$a = q b + r$ | From the [[Division Theorem]], we have that:
:$\forall a, b \in \Z, b > 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < b$
So, with a view to where this is going, let $q$ and $r$ be renamed such that $a = q' b + r'$ with $0 \le r' < b$.
Then let $q' = q + 2$.
We have:
{{begin-eqn}}
{{eqn | l = a
| r = q' b... | Existence of Divisor with Remainder between 2b and 3b | https://proofwiki.org/wiki/Existence_of_Divisor_with_Remainder_between_2b_and_3b | https://proofwiki.org/wiki/Existence_of_Divisor_with_Remainder_between_2b_and_3b | [
"Division Theorem"
] | [
"Definition:Integer",
"Definition:Unique",
"Definition:Integer"
] | [
"Division Theorem"
] |
proofwiki-16111 | Integer of form 6k + 5 is of form 3k + 2 but not Conversely | Let $n \in \Z$ be an integer of the form:
:$n = 6 k + 5$
where $k \in \Z$.
Then $n$ can also be expressed in the form:
:$n = 3 k + 2$
for some other $k \in \Z$.
However it is not necessarily the case that if $n$ can be expressed in the form:
:$n = 3 k + 2$
then it can also be expressed in the form:
:$n = 6 k + 5$ | {{begin-eqn}}
{{eqn | l = n
| r = 6 k + 5
| c =
}}
{{eqn | r = 3 \paren {2 k} + 3 + 2
| c =
}}
{{eqn | r = 3 \paren {2 k + 1} + 2
| c = for some $2 k + 1 \in \Z$
}}
{{end-eqn}}
Replacing $2 k + 1$ with $k$ gives the result.
{{qed|lemma}}
Now consider $n = 8$.
We have that:
:$8 = 3 \times 2 + 2... | Let $n \in \Z$ be an [[Definition:Integer|integer]] of the form:
:$n = 6 k + 5$
where $k \in \Z$.
Then $n$ can also be expressed in the form:
:$n = 3 k + 2$
for some other $k \in \Z$.
However it is not necessarily the case that if $n$ can be expressed in the form:
:$n = 3 k + 2$
then it can also be expressed in the... | {{begin-eqn}}
{{eqn | l = n
| r = 6 k + 5
| c =
}}
{{eqn | r = 3 \paren {2 k} + 3 + 2
| c =
}}
{{eqn | r = 3 \paren {2 k + 1} + 2
| c = for some $2 k + 1 \in \Z$
}}
{{end-eqn}}
Replacing $2 k + 1$ with $k$ gives the result.
{{qed|lemma}}
Now consider $n = 8$.
We have that:
:$8 = 3 \times 2... | Integer of form 6k + 5 is of form 3k + 2 but not Conversely | https://proofwiki.org/wiki/Integer_of_form_6k_+_5_is_of_form_3k_+_2_but_not_Conversely | https://proofwiki.org/wiki/Integer_of_form_6k_+_5_is_of_form_3k_+_2_but_not_Conversely | [
"Modulo Arithmetic"
] | [
"Definition:Integer"
] | [] |
proofwiki-16112 | Odd Integer Modulo 4 | Let $n$ be an odd integer.
Then $n$ can be expressed either as:
:$n = 4 k + 1$
or as:
:$n = 4 k + 3$ | By the Division Theorem, $n$ can be expressed as:
:$n = 4 k + r$
where:
:$k, r \in \Z$
:$0 \le r < 4$
That is, one of the following holds:
{{begin-eqn}}
{{eqn | l = n
| r = 4 k
}}
{{eqn | l = n
| r = 4 k + 1
}}
{{eqn | l = n
| r = 4 k + 2
}}
{{eqn | l = n
| r = 4 k + 3
}}
{{end-eqn}}
Of these:
{... | Let $n$ be an [[Definition:Odd Integer|odd integer]].
Then $n$ can be expressed either as:
:$n = 4 k + 1$
or as:
:$n = 4 k + 3$ | By the [[Division Theorem]], $n$ can be expressed as:
:$n = 4 k + r$
where:
:$k, r \in \Z$
:$0 \le r < 4$
That is, one of the following holds:
{{begin-eqn}}
{{eqn | l = n
| r = 4 k
}}
{{eqn | l = n
| r = 4 k + 1
}}
{{eqn | l = n
| r = 4 k + 2
}}
{{eqn | l = n
| r = 4 k + 3
}}
{{end-eqn}}
Of... | Odd Integer Modulo 4 | https://proofwiki.org/wiki/Odd_Integer_Modulo_4 | https://proofwiki.org/wiki/Odd_Integer_Modulo_4 | [
"Odd Integers"
] | [
"Definition:Odd Integer"
] | [
"Division Theorem",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-16113 | Cube Modulo 9 | Let $x \in \Z$ be an integer.
Then one of the following holds:
{{begin-eqn}}
{{eqn | l = x^3
| o = \equiv
| r = 0 \pmod 9
| c =
}}
{{eqn | l = x^3
| o = \equiv
| r = 1 \pmod 9
| c =
}}
{{eqn | l = x^3
| o = \equiv
| r = 8 \pmod 9
| c =
}}
{{end-eqn}} | Let $x$ be an integer.
There are three cases to consider:
:$(1): \quad x \equiv 0 \pmod 3$: we have $x = 3 k$
:$(2): \quad x \equiv 1 \pmod 3$: we have $x = 3 k + 1$
:$(3): \quad x \equiv 2 \pmod 3$: we have $x = 3 k + 2$
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| r = 3 k
| c =
}}
{{eqn | ll= \leads... | Let $x \in \Z$ be an [[Definition:Integer|integer]].
Then one of the following holds:
{{begin-eqn}}
{{eqn | l = x^3
| o = \equiv
| r = 0 \pmod 9
| c =
}}
{{eqn | l = x^3
| o = \equiv
| r = 1 \pmod 9
| c =
}}
{{eqn | l = x^3
| o = \equiv
| r = 8 \pmod 9
| c =
}}... | Let $x$ be an [[Definition:Integer|integer]].
There are three cases to consider:
:$(1): \quad x \equiv 0 \pmod 3$: we have $x = 3 k$
:$(2): \quad x \equiv 1 \pmod 3$: we have $x = 3 k + 1$
:$(3): \quad x \equiv 2 \pmod 3$: we have $x = 3 k + 2$
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| r = 3 k
... | Cube Modulo 9 | https://proofwiki.org/wiki/Cube_Modulo_9 | https://proofwiki.org/wiki/Cube_Modulo_9 | [
"Modulo Arithmetic",
"Cube Numbers"
] | [
"Definition:Integer"
] | [
"Definition:Integer"
] |
proofwiki-16114 | N (n + 1) (2n + 1) over 6 is Integer | Let $n \in \Z$ be an integer.
Then $\dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$ is also an integer. | This is equivalent to proving that $n \paren {n + 1} \paren {2 n + 1}$ is a multiple of $6$.
There are $6$ cases to consider:
:$(1): \quad n \equiv 0 \pmod 6$: we have $n = 6 k$
:$(2): \quad n \equiv 1 \pmod 6$: we have $n = 6 k + 1$
:$(3): \quad n \equiv 2 \pmod 6$: we have $n = 6 k + 2$
:$(4): \quad n \equiv 3 \pmod ... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then $\dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$ is also an [[Definition:Integer|integer]]. | This is equivalent to proving that $n \paren {n + 1} \paren {2 n + 1}$ is a [[Definition:Multiple of Integer|multiple]] of $6$.
There are $6$ cases to consider:
:$(1): \quad n \equiv 0 \pmod 6$: we have $n = 6 k$
:$(2): \quad n \equiv 1 \pmod 6$: we have $n = 6 k + 1$
:$(3): \quad n \equiv 2 \pmod 6$: we have $n =... | N (n + 1) (2n + 1) over 6 is Integer | https://proofwiki.org/wiki/N_(n_+_1)_(2n_+_1)_over_6_is_Integer | https://proofwiki.org/wiki/N_(n_+_1)_(2n_+_1)_over_6_is_Integer | [
"Divisibility"
] | [
"Definition:Integer",
"Definition:Integer"
] | [
"Definition:Multiple/Integer"
] |
proofwiki-16115 | Number which is Square and Cube Modulo 7 | Let $n \in \Z$ be an integer.
Let $n$ be both a square and a cube at the same time.
Then either:
:$n \equiv 0 \pmod 7$
or:
:$n \equiv 1 \pmod 7$ | Let $n = r^2 = s^3$ for some $r, s \in \Z$.
Then:
:$n = \paren {m^2}^3 = \paren {m^3}^2 = m^6$
for some $m \in \Z$
There are $7$ cases to consider:
:$(0): \quad m \equiv 0 \pmod 7$: we have $m = 7 k$
:$(1): \quad m \equiv 1 \pmod 7$: we have $m = 7 k + 1$
:$(2): \quad m \equiv 2 \pmod 7$: we have $m = 7 k + 2$
:$(3): \... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Let $n$ be both a [[Definition:Square Number|square]] and a [[Definition:Cube Number|cube]] at the same time.
Then either:
:$n \equiv 0 \pmod 7$
or:
:$n \equiv 1 \pmod 7$ | Let $n = r^2 = s^3$ for some $r, s \in \Z$.
Then:
:$n = \paren {m^2}^3 = \paren {m^3}^2 = m^6$
for some $m \in \Z$
There are $7$ cases to consider:
:$(0): \quad m \equiv 0 \pmod 7$: we have $m = 7 k$
:$(1): \quad m \equiv 1 \pmod 7$: we have $m = 7 k + 1$
:$(2): \quad m \equiv 2 \pmod 7$: we have $m = 7 k + 2$
:... | Number which is Square and Cube Modulo 7/Proof 1 | https://proofwiki.org/wiki/Number_which_is_Square_and_Cube_Modulo_7 | https://proofwiki.org/wiki/Number_which_is_Square_and_Cube_Modulo_7/Proof_1 | [
"Number which is Square and Cube Modulo 7",
"Cube Numbers",
"Square Numbers",
"Modulo Arithmetic"
] | [
"Definition:Integer",
"Definition:Square Number",
"Definition:Cube Number"
] | [
"Congruence of Powers",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Integer"
] |
proofwiki-16116 | Division Theorem/Half Remainder Version | For every pair of integers $a, b$ where $b \ne 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $-\dfrac {\size b} 2 \le r < \dfrac {\size b} 2$:
:$\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, -\dfrac {\size b} 2 \le r < \dfrac {\size b} 2$ | {{ProofWanted|boring}} | For every pair of [[Definition:Integer|integers]] $a, b$ where $b \ne 0$, there exist [[Definition:Unique|unique]] [[Definition:Integer|integers]] $q, r$ such that $a = q b + r$ and $-\dfrac {\size b} 2 \le r < \dfrac {\size b} 2$:
:$\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, -\dfrac {\size b} 2 ... | {{ProofWanted|boring}} | Division Theorem/Half Remainder Version | https://proofwiki.org/wiki/Division_Theorem/Half_Remainder_Version | https://proofwiki.org/wiki/Division_Theorem/Half_Remainder_Version | [
"Division Theorem"
] | [
"Definition:Integer",
"Definition:Unique",
"Definition:Integer"
] | [] |
proofwiki-16117 | Khinchin's Law | Let $P$ be a population.
Let $P$ have mean $\mu$ and finite variance.
Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of random variables forming a random sample from $P$.
Let:
:$\ds {\overline X}_n = \frac 1 n \sum_{i \mathop = 1}^n X_i$
Then:
:${\overline X}_n \xrightarrow p \mu$
where $\xrightarrow p$ denote... | Let $\sigma$ be the standard deviation of $P$.
By the definition of convergence in probability, we aim to show that:
:$\ds \lim_{n \mathop \to \infty} \map \Pr {\size { {\overline X}_n - \mu} < \epsilon} = 1$
for all real $\epsilon > 0$.
Let $\epsilon > 0$ be a real number.
By Variance of Sample Mean:
:$\var {{\overli... | Let $P$ be a [[Definition:Population|population]].
Let $P$ have [[Definition:Expectation|mean]] $\mu$ and finite [[Definition:Variance|variance]].
Let $\sequence {X_n}_{n \mathop \ge 1}$ be a [[Definition:Sequence|sequence]] of [[Definition:Random Variable|random variables]] forming a [[Definition:Random Sample (Sta... | Let $\sigma$ be the [[Definition:Standard Deviation|standard deviation]] of $P$.
By the definition of [[Definition:Convergence in Probability|convergence in probability]], we aim to show that:
:$\ds \lim_{n \mathop \to \infty} \map \Pr {\size { {\overline X}_n - \mu} < \epsilon} = 1$
for all [[Definition:Real Numbe... | Khinchin's Law | https://proofwiki.org/wiki/Khinchin's_Law | https://proofwiki.org/wiki/Khinchin's_Law | [
"Khinchin's Law",
"Laws of Large Numbers",
"Probability Theory"
] | [
"Definition:Population",
"Definition:Expectation",
"Definition:Variance",
"Definition:Sequence",
"Definition:Random Variable",
"Definition:Random Sample (Statistics)",
"Definition:Convergence in Probability"
] | [
"Definition:Standard Deviation",
"Definition:Convergence in Probability",
"Definition:Real Number",
"Definition:Real Number",
"Variance of Sample Mean",
"Bienaymé-Chebyshev Inequality",
"Definition:Real Number",
"Squeeze Theorem",
"Definition:Real Number"
] |
proofwiki-16118 | Variance of Linear Combination of Random Variables/Corollary | Let $X$ and $Y$ be independent random variables. | From Variance of Linear Combination of Random Variables, we have:
:$\var {a X + b Y} = a^2 \, \var X + b^2 \, \var Y + 2 a b \, \cov {X, Y}$
where $\cov {X, Y}$ is the covariance of $X$ and $Y$.
From Covariance of Independent Random Variables is Zero:
:$2 a b \, \cov {X, Y} = 0$
The result follows.
{{qed}}
Category:... | Let $X$ and $Y$ be [[Definition:Independent Random Variables|independent random variables]]. | From [[Variance of Linear Combination of Random Variables]], we have:
:$\var {a X + b Y} = a^2 \, \var X + b^2 \, \var Y + 2 a b \, \cov {X, Y}$
where $\cov {X, Y}$ is the [[Definition:Covariance|covariance]] of $X$ and $Y$.
From [[Covariance of Independent Random Variables is Zero]]:
:$2 a b \, \cov {X, Y} = 0$... | Variance of Linear Combination of Random Variables/Corollary | https://proofwiki.org/wiki/Variance_of_Linear_Combination_of_Random_Variables/Corollary | https://proofwiki.org/wiki/Variance_of_Linear_Combination_of_Random_Variables/Corollary | [
"Variance"
] | [
"Definition:Independent Random Variables"
] | [
"Variance of Linear Combination of Random Variables",
"Definition:Covariance",
"Covariance of Independent Random Variables is Zero",
"Category:Variance"
] |
proofwiki-16119 | Singleton is Connected in Topological Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Then the singleton $\set x$ is connected. | Let $A = \set x$.
From definition $3$ of a connected set, $A$ is connected in $T$ {{iff}} the subspace $\struct {A, \tau_A}$ is a connected space.
From Topology on Singleton is Indiscrete Topology, $\tau_A$ is the indiscrete topology.
From Indiscrete Space is Connected, $\struct {A, \tau_A}$ is a connected space.
{{qed... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Then the [[Definition:Singleton|singleton]] $\set x$ is [[Definition:Connected Set (Topology)|connected]]. | Let $A = \set x$.
From [[Definition:Connected Set (Topology)/Definition 3|definition $3$ of a connected set]], $A$ is [[Definition:Connected Set (Topology)/Definition 3|connected]] in $T$ {{iff}} the [[Definition:Topological Subspace|subspace]] $\struct {A, \tau_A}$ is a [[Definition:Connected Topological Space|conne... | Singleton is Connected in Topological Space | https://proofwiki.org/wiki/Singleton_is_Connected_in_Topological_Space | https://proofwiki.org/wiki/Singleton_is_Connected_in_Topological_Space | [
"Connected Sets (Topology)",
"Singletons"
] | [
"Definition:Topological Space",
"Definition:Singleton",
"Definition:Connected Set (Topology)"
] | [
"Definition:Connected Set (Topology)/Definition 3",
"Definition:Connected Set (Topology)/Definition 3",
"Definition:Topological Subspace",
"Definition:Connected Topological Space",
"Trivial Topological Space is Indiscrete",
"Definition:Indiscrete Topology",
"Indiscrete Space is Connected",
"Definition... |
proofwiki-16120 | Equivalence of Definitions of Component/Lemma 1 | :$C$ is connected in $T$ and $C \in \CC_x$. | From Singleton is Connected in Topological Space, $\set{x}$ is a connected set of $T$ containing $x$.
It follows that $x \in C$.
From Union of Connected Sets with Common Point is Connected, $C$ is a connected set of $T$.
Hence $C \in \CC_x$.
{{qed}}
Category:Equivalence of Definitions of Component
g3treh4at5mzazjpq27g... | :$C$ is [[Definition:Connected Set (Topology)|connected]] in $T$ and $C \in \CC_x$. | From [[Singleton is Connected in Topological Space]], $\set{x}$ is a [[Definition:Connected Set (Topology)|connected set]] of $T$ containing $x$.
It follows that $x \in C$.
From [[Union of Connected Sets with Common Point is Connected]], $C$ is a [[Definition:Connected Set (Topology)|connected set]] of $T$.
Hence $... | Equivalence of Definitions of Component/Lemma 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component/Lemma_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Component/Lemma_1 | [
"Equivalence of Definitions of Component"
] | [
"Definition:Connected Set (Topology)"
] | [
"Singleton is Connected in Topological Space",
"Definition:Connected Set (Topology)",
"Union of Connected Sets with Common Point is Connected",
"Definition:Connected Set (Topology)",
"Category:Equivalence of Definitions of Component"
] |
proofwiki-16121 | Variance of Sample Mean | Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$.
Let:
:$\ds \overline X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
Then:
:$\var {\overline X} = \dfrac {\sigma^2} n$ | {{begin-eqn}}
{{eqn | l = \var {\overline X}
| r = \var {\frac 1 n \sum_{i \mathop = 1}^n X_i}
}}
{{eqn | r = \frac 1 {n^2} \sum_{i \mathop = 1}^n \var {X_i}
| c = repeated application of {{Corollary|Variance of Linear Combination of Random Variables}}
}}
{{eqn | r = \frac 1 {n^2} \sum_{i \mathop = 1}^n \si... | Let $X_1, X_2, \ldots, X_n$ form a [[Definition:Random Sample (Statistics)|random sample]] from a population with [[Definition:Expectation|mean]] $\mu$ and [[Definition:Variance|variance]] $\sigma^2$.
Let:
:$\ds \overline X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
Then:
:$\var {\overline X} = \dfrac {\sigma^2} n$ | {{begin-eqn}}
{{eqn | l = \var {\overline X}
| r = \var {\frac 1 n \sum_{i \mathop = 1}^n X_i}
}}
{{eqn | r = \frac 1 {n^2} \sum_{i \mathop = 1}^n \var {X_i}
| c = repeated application of {{Corollary|Variance of Linear Combination of Random Variables}}
}}
{{eqn | r = \frac 1 {n^2} \sum_{i \mathop = 1}^n \si... | Variance of Sample Mean | https://proofwiki.org/wiki/Variance_of_Sample_Mean | https://proofwiki.org/wiki/Variance_of_Sample_Mean | [
"Variance",
"Inductive Statistics"
] | [
"Definition:Random Sample (Statistics)",
"Definition:Expectation",
"Definition:Variance"
] | [
"Category:Variance",
"Category:Inductive Statistics"
] |
proofwiki-16122 | Product of Divisors is Divisor of Product | Let $a, b, c, d \in \Z$ be integers such that $a, c \ne 0$.
Let $a \divides b$ and $c \divides d$, where $\divides$ denotes divisibility.
Then:
:$a c \divides b d$ | By definition of divisibility:
{{begin-eqn}}
{{eqn | q = \exists k_1 \in \Z
| l = b
| r = a k_1
}}
{{eqn | q = \exists k_2 \in \Z
| l = d
| r = c k_2
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = b d
| r = \paren {a k_1} \paren {c k_2}
| c =
}}
{{eqn | r = k_1 k_2 \paren {a c}
... | Let $a, b, c, d \in \Z$ be [[Definition:Integer|integers]] such that $a, c \ne 0$.
Let $a \divides b$ and $c \divides d$, where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Then:
:$a c \divides b d$ | By definition of [[Definition:Divisor of Integer|divisibility]]:
{{begin-eqn}}
{{eqn | q = \exists k_1 \in \Z
| l = b
| r = a k_1
}}
{{eqn | q = \exists k_2 \in \Z
| l = d
| r = c k_2
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = b d
| r = \paren {a k_1} \paren {c k_2}
| c =
}}
{... | Product of Divisors is Divisor of Product | https://proofwiki.org/wiki/Product_of_Divisors_is_Divisor_of_Product | https://proofwiki.org/wiki/Product_of_Divisors_is_Divisor_of_Product | [
"Divisors"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-16123 | Common Divisor Divides Integer Combination/General Result | Let $c$ be a common divisor of a set of integers $A := \set {a_1, a_2, \dotsc, a_n}$.
That is:
:$\forall x \in A: c \divides x$
Then $c$ divides any integer combination of elements of $A$:
:$\forall x_1, x_2, \dotsc, x_n \in \Z: c \divides \paren {a_1 x_2 + a_2 x_2 + \dotsb + a_n x_n}$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
:$\forall x \in \set {a_1, a_2, \dotsc, a_n}: c \divides x \implies \forall x_1, x_2, \dotsc, x_n \in \Z: c \divides \paren {a_1 x_2 + a_2 x_2 + \dotsb + a_n x_n}$ | Let $c$ be a [[Definition:Common Divisor of Integers|common divisor]] of a [[Definition:Set|set]] of [[Definition:Integer|integers]] $A := \set {a_1, a_2, \dotsc, a_n}$.
That is:
:$\forall x \in A: c \divides x$
Then $c$ divides any [[Definition:Integer Combination|integer combination]] of [[Definition:Element|eleme... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\forall x \in \set {a_1, a_2, \dotsc, a_n}: c \divides x \implies \forall x_1, x_2, \dotsc, x_n \in \Z: c \divides \paren {a_1 x_2 + a_2 x_2 + \dotsb + a_... | Common Divisor Divides Integer Combination/General Result | https://proofwiki.org/wiki/Common_Divisor_Divides_Integer_Combination/General_Result | https://proofwiki.org/wiki/Common_Divisor_Divides_Integer_Combination/General_Result | [
"Common Divisor Divides Integer Combination"
] | [
"Definition:Common Divisor/Integers",
"Definition:Set",
"Definition:Integer",
"Definition:Integer Combination",
"Definition:Element"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-16124 | One is Common Divisor of Integers | Let $a, b \in \Z$ be integers.
Then $1$ is a common divisor of $a$ and $b$. | From One Divides all Integers:
:$1 \divides a$
and:
:$1 \divides b$
where $\divides$ denotes divisibility.
The result follows by definition of common divisor.
{{Qed}} | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Then $1$ is a [[Definition:Common Divisor of Integers|common divisor]] of $a$ and $b$. | From [[One Divides all Integers]]:
:$1 \divides a$
and:
:$1 \divides b$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
The result follows by definition of [[Definition:Common Divisor of Integers|common divisor]].
{{Qed}} | One is Common Divisor of Integers | https://proofwiki.org/wiki/One_is_Common_Divisor_of_Integers | https://proofwiki.org/wiki/One_is_Common_Divisor_of_Integers | [
"Divisors"
] | [
"Definition:Integer",
"Definition:Common Divisor/Integers"
] | [
"Integer Divisor Results/One Divides all Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Common Divisor/Integers"
] |
proofwiki-16125 | Set of Common Divisors of Integers is not Empty | Let $a, b \in \Z$ be integers.
Let $S$ be the set of common divisors of $a$ and $b$.
Then $S$ is not empty. | From One is Common Divisor of Integers:
:$1$ is a common divisor of $a$ and $b$.
Thus, whatever $a$ and $b$ are:
:$1 \in S$
The result follows by definition of empty set.
{{Qed}} | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Let $S$ be the [[Definition:Set|set]] of [[Definition:Common Divisor of Integers|common divisors]] of $a$ and $b$.
Then $S$ is not [[Definition:Empty Set|empty]]. | From [[One is Common Divisor of Integers]]:
:$1$ is a [[Definition:Common Divisor of Integers|common divisor]] of $a$ and $b$.
Thus, whatever $a$ and $b$ are:
:$1 \in S$
The result follows by definition of [[Definition:Empty Set|empty set]].
{{Qed}} | Set of Common Divisors of Integers is not Empty | https://proofwiki.org/wiki/Set_of_Common_Divisors_of_Integers_is_not_Empty | https://proofwiki.org/wiki/Set_of_Common_Divisors_of_Integers_is_not_Empty | [
"Divisors"
] | [
"Definition:Integer",
"Definition:Set",
"Definition:Common Divisor/Integers",
"Definition:Empty Set"
] | [
"One is Common Divisor of Integers",
"Definition:Common Divisor/Integers",
"Definition:Empty Set"
] |
proofwiki-16126 | Kolmogorov's Law | Let $P$ be a population.
Let $P$ have mean $\mu$ and finite variance.
Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of random variables forming a random sample from $P$.
Let:
:$\ds {\overline X}_n = \frac 1 n \sum_{i \mathop = 1}^n X_i$
Then:
:$\ds {\overline X}_n \xrightarrow {\text {a.s.} } \mu$
where $\xri... | We may assume that $X_n \ge 0$ for all $n \ge 1$.
Indeed, otherwise consider:
:${X_n}^+ := \max \set {X_n, 0}$
and:
:${X_n}^- := \max \set {- X_n, 0}$
instead of $X_n$.
Let $\varepsilon \in \R_{>0}$.
For $k \ge 1$ let:
:$\ell_k := \floor {\paren {1 + \epsilon}^k }$
be the floor of $\paren {1 + \epsilon}^k$.
Then:
{{beg... | Let $P$ be a [[Definition:Population|population]].
Let $P$ have [[Definition:Expectation|mean]] $\mu$ and finite [[Definition:Variance|variance]].
Let $\sequence {X_n}_{n \mathop \ge 1}$ be a [[Definition:Sequence|sequence]] of [[Definition:Random Variable|random variables]] forming a [[Definition:Random Sample (Sta... | We may assume that $X_n \ge 0$ for all $n \ge 1$.
Indeed, otherwise consider:
:${X_n}^+ := \max \set {X_n, 0}$
and:
:${X_n}^- := \max \set {- X_n, 0}$
instead of $X_n$.
Let $\varepsilon \in \R_{>0}$.
For $k \ge 1$ let:
:$\ell_k := \floor {\paren {1 + \epsilon}^k }$
be the [[Definition:Floor Function|floor]] of $\pa... | Kolmogorov's Law | https://proofwiki.org/wiki/Kolmogorov's_Law | https://proofwiki.org/wiki/Kolmogorov's_Law | [
"Kolmogorov's Law",
"Laws of Large Numbers",
"Probability Theory"
] | [
"Definition:Population",
"Definition:Expectation",
"Definition:Variance",
"Definition:Sequence",
"Definition:Random Variable",
"Definition:Random Sample (Statistics)",
"Definition:Almost Sure Convergence"
] | [
"Definition:Floor Function",
"Borel-Cantelli Lemma"
] |
proofwiki-16127 | Equivalence of Definitions of Path Component | {{TFAE|def = Path Component|view = path component|context = Topology (Mathematical Branch)|contextview = topology}}
Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in T$.
=== Equivalence Class ===
{{:Definition:Path Component/Equivalence Class}}
=== Union of Path-Connected Sets ===
{{:Definition:Path Compon... | Let $\CC_x = \set {A \subseteq S : x \in A \land A \text { is path-connected in } T}$
Let $C = \bigcup \CC_x$. | {{TFAE|def = Path Component|view = path component|context = Topology (Mathematical Branch)|contextview = topology}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in T$.
=== [[Definition:Path Component/Equivalence Class|Equivalence Class]] ===
{{:Definition:Path Compon... | Let $\CC_x = \set {A \subseteq S : x \in A \land A \text { is path-connected in } T}$
Let $C = \bigcup \CC_x$. | Equivalence of Definitions of Path Component | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component | [
"Equivalence of Definitions of Path Component",
"Path Components",
"Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Path Component/Equivalence Class",
"Definition:Path Component/Union of Path-Connected Sets",
"Definition:Path Component/Maximal Path-Connected Set"
] | [] |
proofwiki-16128 | GCD of Integer and its Negative | Let $a \in \Z$ be an integer.
Then:
:$\gcd \set {a, -a} = \size a$
where:
:$\gcd$ denotes greatest common divisor
:$\size a$ denotes the absolute value of $a$. | From Integer Divisor Results, the divisors of $a$ include $a$ itself.
From Integer Divides its Negative, $a \divides \paren {-a}$.
Thus we have:
:$a \divides a$
and:
:$a \divides -a$
and so:
:$\gcd \set {a, -a} \ge \size a$
From Absolute Value of Integer is not less than Divisors, there is no divisor of $a$ which is gr... | Let $a \in \Z$ be an [[Definition:Integer|integer]].
Then:
:$\gcd \set {a, -a} = \size a$
where:
:$\gcd$ denotes [[Definition:Greatest Common Divisor of Integers|greatest common divisor]]
:$\size a$ denotes the [[Definition:Absolute Value|absolute value]] of $a$. | From [[Integer Divisor Results]], the [[Definition:Divisor of Integer|divisors]] of $a$ include $a$ itself.
From [[Integer Divides its Negative]], $a \divides \paren {-a}$.
Thus we have:
:$a \divides a$
and:
:$a \divides -a$
and so:
:$\gcd \set {a, -a} \ge \size a$
From [[Absolute Value of Integer is not less than... | GCD of Integer and its Negative | https://proofwiki.org/wiki/GCD_of_Integer_and_its_Negative | https://proofwiki.org/wiki/GCD_of_Integer_and_its_Negative | [
"Greatest Common Divisor"
] | [
"Definition:Integer",
"Definition:Greatest Common Divisor/Integers",
"Definition:Absolute Value"
] | [
"Integer Divisor Results",
"Definition:Divisor (Algebra)/Integer",
"Integer Divisor Results/Integer Divides its Negative",
"Absolute Value of Integer is not less than Divisors",
"Definition:Divisor (Algebra)/Integer",
"Category:Greatest Common Divisor"
] |
proofwiki-16129 | Equivalence of Definitions of Path Component/Lemma | :$C$ is path-connected in $T$ and $C \in \CC_x$. | From Point is Path-Connected to Itself, $\set x$ is a path-connected subset of $T$ containing $x$.
It follows that $x \in C$.
From Union of Path-Connected Sets with Common Point is Path-Connected, $C$ is a path-connected subset of $T$.
Hence $C \in \CC_x$.
{{qed}}
Category:Equivalence of Definitions of Path Component
... | :$C$ is [[Definition:Path-Connected Set|path-connected]] in $T$ and $C \in \CC_x$. | From [[Point is Path-Connected to Itself]], $\set x$ is a [[Definition:Path-Connected Set|path-connected subset]] of $T$ containing $x$.
It follows that $x \in C$.
From [[Union of Path-Connected Sets with Common Point is Path-Connected]], $C$ is a [[Definition:Path-Connected Set|path-connected subset]] of $T$.
Henc... | Equivalence of Definitions of Path Component/Lemma | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component/Lemma | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component/Lemma | [
"Equivalence of Definitions of Path Component"
] | [
"Definition:Path-Connected/Set"
] | [
"Point is Path-Connected to Itself",
"Definition:Path-Connected/Set",
"Union of Path-Connected Sets with Common Point is Path-Connected",
"Definition:Path-Connected/Set",
"Category:Equivalence of Definitions of Path Component"
] |
proofwiki-16130 | Square Divides Product of Multiples | Let $a, b, c, \in \Z$ be integers.
Let:
:$a \divides b, a \divides c$
where $\divides$ denotes divisibility.
Then:
:$a^2 \divides b c$ | We have that:
{{begin-eqn}}
{{eqn | l = a
| o = \divides
| r = b
| c =
}}
{{eqn | ll= \leadsto
| q = \exists k_1 \in \Z
| l = k_1 a
| r = b
| c = {{Defof|Divisor of Integer}}
}}
{{eqn | l = a
| o = \divides
| r = c
| c =
}}
{{eqn | ll= \leadsto
| q = \e... | Let $a, b, c, \in \Z$ be [[Definition:Integer|integers]].
Let:
:$a \divides b, a \divides c$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Then:
:$a^2 \divides b c$ | We have that:
{{begin-eqn}}
{{eqn | l = a
| o = \divides
| r = b
| c =
}}
{{eqn | ll= \leadsto
| q = \exists k_1 \in \Z
| l = k_1 a
| r = b
| c = {{Defof|Divisor of Integer}}
}}
{{eqn | l = a
| o = \divides
| r = c
| c =
}}
{{eqn | ll= \leadsto
| q = \... | Square Divides Product of Multiples | https://proofwiki.org/wiki/Square_Divides_Product_of_Multiples | https://proofwiki.org/wiki/Square_Divides_Product_of_Multiples | [
"Divisors"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-16131 | Equivalence of Definitions of Path Component/Equivalence Class equals Union of Path-Connected Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in T$.
Let $\CC_x = \leftset {A \subseteq S : x \in A \land A}$ is path-connected in $\rightset T$.
Let $C = \bigcup \CC_x$
Let $\sim$ be the equivalence relation defined by:
:$y \sim z$ {{iff}} $y$ and $z$ are path-connected in $T$.
Let $C'$ be the equivalenc... | {{begin-eqn}}
{{eqn | l = y \in C'
| o = \leadstoandfrom
| r = x \text { is path-connected to } y \text { in } T
| c = Definition of $\sim$
}}
{{eqn | o = \leadstoandfrom
| r = \exists B \text{ a connected set of } T, x \in B, y \in B
| c = Points are Path-Connected iff Contained in Path-C... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in T$.
Let $\CC_x = \leftset {A \subseteq S : x \in A \land A}$ is [[Definition:Path-Connected Set|path-connected]] in $\rightset T$.
Let $C = \bigcup \CC_x$
Let $\sim$ be the [[Definition:Equivalence Relation|equivalence... | {{begin-eqn}}
{{eqn | l = y \in C'
| o = \leadstoandfrom
| r = x \text { is path-connected to } y \text { in } T
| c = Definition of $\sim$
}}
{{eqn | o = \leadstoandfrom
| r = \exists B \text{ a connected set of } T, x \in B, y \in B
| c = [[Points are Path-Connected iff Contained in Path... | Equivalence of Definitions of Path Component/Equivalence Class equals Union of Path-Connected Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component/Equivalence_Class_equals_Union_of_Path-Connected_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component/Equivalence_Class_equals_Union_of_Path-Connected_Sets | [
"Equivalence of Definitions of Path Component"
] | [
"Definition:Topological Space",
"Definition:Path-Connected/Set",
"Definition:Equivalence Relation",
"Definition:Path-Connected/Points",
"Definition:Equivalence Class"
] | [
"Points are Path-Connected iff Contained in Path-Connected Set"
] |
proofwiki-16132 | Image of Path is Path-Connected Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $I \subset \R$ be the closed real interval $\closedint a b$.
Let $\gamma: I \to S$ be a path.
Then:
:$\map \gamma I$ is a path-connected set of $T$. | From Path-Connected iff Path-Connected to Point, $\map \gamma I$ is a path-connected set {{iff}} every point of $\map \gamma I$ is path-connected to a common point.
It is shown that every point of $\map \gamma I$ is path-connected to $\map \gamma a$.
Let $H = \map \gamma I$.
Let $x = \map \gamma a$.
From Point is Path-... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $I \subset \R$ be the [[Definition:Closed Real Interval|closed real interval]] $\closedint a b$.
Let $\gamma: I \to S$ be a [[Definition:Path (Topology)|path]].
Then:
:$\map \gamma I$ is a [[Definition:Path-Connected Set|path-c... | From [[Path-Connected iff Path-Connected to Point]], $\map \gamma I$ is a [[Definition:Path-Connected Set|path-connected set]] {{iff}} every point of $\map \gamma I$ is [[Definition:Path-Connected Points|path-connected]] to a common point.
It is shown that every point of $\map \gamma I$ is [[Definition:Path-Connected ... | Image of Path is Path-Connected Set | https://proofwiki.org/wiki/Image_of_Path_is_Path-Connected_Set | https://proofwiki.org/wiki/Image_of_Path_is_Path-Connected_Set | [
"Path-Connected Sets"
] | [
"Definition:Topological Space",
"Definition:Real Interval/Closed",
"Definition:Path (Topology)",
"Definition:Path-Connected/Set"
] | [
"Path-Connected iff Path-Connected to Point",
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Points",
"Definition:Path-Connected/Points",
"Point is Path-Connected to Itself",
"Definition:Path-Connected/Points",
"Continuity of Composite with Inclusion",
"Definition:Continuous Mapping (Topo... |
proofwiki-16133 | Acceleration of Particle moving in Circle at Constant Speed | Let $P$ be a particle moving in a circular path $C$ at a constant speed.
Then the acceleration of $P$, denoted by $\mathbf a \in \R$, is given as:
:$\mathbf a = -\dfrac {\size {\mathbf v}^2 \mathbf r} {\size {\mathbf r}^2}$
where:
:$\mathbf v$ is the instantaneous velocity of $P$
:$\mathbf r$ is the vector whose magnit... | Let the center of $C$ be placed at the origin of a two-dimensional coordinate system where the circular path is in the $xy$ plane.
Using the parametric equation of a circle, we can write the position of $P$ as the vector-valued function
:$\ds \map {\mathbf r} t = \begin{bmatrix} r \map \cos {\omega t} \\ r \map \sin {\... | Let $P$ be a [[Definition:Particle|particle]] moving in a [[Definition:Circle|circular path]] $C$ at a [[Definition:Constant Speed|constant speed]].
Then the [[Definition:Acceleration|acceleration]] of $P$, denoted by $\mathbf a \in \R$, is given as:
:$\mathbf a = -\dfrac {\size {\mathbf v}^2 \mathbf r} {\size {\math... | Let the [[Definition:Center of Circle|center]] of $C$ be placed at the [[Definition:Origin|origin]] of a two-dimensional [[Definition:Coordinate System|coordinate system]] where the [[Definition:Circle|circular path]] is in the [[Definition:XY Plane|$xy$ plane]].
Using the [[Equation of Circle/Parametric|parametric eq... | Acceleration of Particle moving in Circle at Constant Speed/Proof 1 | https://proofwiki.org/wiki/Acceleration_of_Particle_moving_in_Circle_at_Constant_Speed | https://proofwiki.org/wiki/Acceleration_of_Particle_moving_in_Circle_at_Constant_Speed/Proof_1 | [
"Acceleration of Particle moving in Circle at Constant Speed",
"Acceleration",
"Circles",
"Physics"
] | [
"Definition:Particle",
"Definition:Circle",
"Definition:Constant Speed",
"Definition:Acceleration",
"Definition:Velocity",
"Definition:Vector",
"Definition:Magnitude",
"Definition:Linear Measure/Length",
"Definition:Circle/Radius",
"Definition:Direction",
"Definition:Circle/Center",
"Definitio... | [
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Coordinate System",
"Definition:Circle",
"Definition:Cartesian Plane",
"Equation of Circle/Parametric",
"Definition:Position Vector",
"Definition:Vector-Valued Function",
"Definition:Constant",
"Definition:Time",
"Def... |
proofwiki-16134 | Closure of Hadamard Product | Let $\struct {S, \cdot}$ be an algebraic structure.
Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$.
For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$.
The operation $\circ$ is closed on $\map {\MM_S} {... | === Necessary Condition ===
Let the operation $\cdot$ be closed on $\struct {S, \cdot}$.
Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be elements of $\map {\MM_S} {m, n}$.
Let $\sqbrk c_{m n} = \sqbrk a_{m n} \cdot \sqbrk b_{m n}$.
Then:
:$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j}... | Let $\struct {S, \cdot}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]].
Let $\map {\MM_S} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $S$.
For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the [[Definition:Ha... | === Necessary Condition ===
Let the [[Definition:Binary Operation|operation]] $\cdot$ be [[Definition:Closure (Abstract Algebra)|closed]] on $\struct {S, \cdot}$.
Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be [[Definition:Element|elements]] of $\map {\MM_S} {m, n}$.
Let $\sqbrk c_{m n} = \sqbr... | Closure of Hadamard Product | https://proofwiki.org/wiki/Closure_of_Hadamard_Product | https://proofwiki.org/wiki/Closure_of_Hadamard_Product | [
"Hadamard Product"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Matrix Space",
"Definition:Hadamard Product",
"Definition:Closure (Abstract Algebra)",
"Definition:Closure (Abstract Algebra)"
] | [
"Definition:Operation/Binary Operation",
"Definition:Closure (Abstract Algebra)",
"Definition:Element",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Hadamard Product",
"Definition:Matrix/Order",
"Definition:Closure (Abstract Algebra)",
"Definition:Closure (Abstract Algebra)... |
proofwiki-16135 | Associativity of Hadamard Product | Let $\struct {S, \cdot}$ be an algebraic structure.
Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$.
For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$.
The operation $\circ$ is associative on $\map {\MM... | === Necessary Condition ===
Let the operation $\cdot$ be associative on $\struct {S, \cdot}$.
Let $\mathbf A = \sqbrk a_{m n}$, $\mathbf B = \sqbrk b_{m n}$ and $\mathbf C = \sqbrk c_{m n}$ be elements of the $m \times n$ matrix space over $S$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf A \circ \mathbf B} \circ \m... | Let $\struct {S, \cdot}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]].
Let $\map {\MM_S} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $S$.
For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the [[Definition:Ha... | === Necessary Condition ===
Let the [[Definition:Binary Operation|operation]] $\cdot$ be [[Definition:Associative Operation|associative]] on $\struct {S, \cdot}$.
Let $\mathbf A = \sqbrk a_{m n}$, $\mathbf B = \sqbrk b_{m n}$ and $\mathbf C = \sqbrk c_{m n}$ be [[Definition:Element|elements]] of the [[Definition:Matr... | Associativity of Hadamard Product | https://proofwiki.org/wiki/Associativity_of_Hadamard_Product | https://proofwiki.org/wiki/Associativity_of_Hadamard_Product | [
"Hadamard Product",
"Examples of Associative Operations"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Matrix Space",
"Definition:Hadamard Product",
"Definition:Associative Operation",
"Definition:Associative Operation"
] | [
"Definition:Operation/Binary Operation",
"Definition:Associative Operation",
"Definition:Element",
"Definition:Matrix Space",
"Definition:Associative Operation",
"Definition:Associative Operation",
"Definition:Associative Operation",
"Definition:Element",
"Definition:Associative Operation"
] |
proofwiki-16136 | Commutativity of Hadamard Product | Let $\struct {S, \cdot}$ be an algebraic structure.
Let $\map {\MM_S} {m, n}$ be a $m \times n$ matrix space over $S$.
For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the Hadamard product of $\mathbf A$ and $\mathbf B$.
The operation $\circ$ is commutative on $\map {\MM... | === Necessary Condition ===
Let the operation $\cdot$ be commutative on $\struct {S, \cdot}$.
Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be elements of the $m \times n$ matrix space over $S$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A \circ \mathbf B
| r = \sqbrk a_{m n} \circ \sqbrk b_{m n}... | Let $\struct {S, \cdot}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]].
Let $\map {\MM_S} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $S$.
For $\mathbf A, \mathbf B \in \map {\MM_S} {m, n}$, let $\mathbf A \circ \mathbf B$ be defined as the [[Definition:Ha... | === Necessary Condition ===
Let the [[Definition:Binary Operation|operation]] $\cdot$ be [[Definition:Commutative Operation|commutative]] on $\struct {S, \cdot}$.
Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be [[Definition:Element|elements]] of the [[Definition:Matrix Space|$m \times n$ matrix s... | Commutativity of Hadamard Product | https://proofwiki.org/wiki/Commutativity_of_Hadamard_Product | https://proofwiki.org/wiki/Commutativity_of_Hadamard_Product | [
"Hadamard Product",
"Examples of Commutative Operations"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Matrix Space",
"Definition:Hadamard Product",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation"
] | [
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation",
"Definition:Element",
"Definition:Matrix Space",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation",
"Definition:Element",
"Definition:Commutative/Operation"
] |
proofwiki-16137 | Matrix Entrywise Addition over Ring is Closed | Let $\struct {R, +, \circ}$ be a ring.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.
For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation $+$ is closed on $\map {\MM_R} {m, n}$. | Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be elements of $\map {\MM_R} {m, n}$.
Let $\sqbrk c_{m n} = \sqbrk a_{m n} + \sqbrk b_{m n}$.
By definition of matrix entrywise addition:
:$\forall i \in \closedint 1 m, j \in \closedint 1 n: a_{i j} + b_{i j} = c_{i j}$
By {{Ring-axiom|A0}}, $R$ is clos... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\map {\MM_R} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $R$.
For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the [[Definition:Matrix Entrywise Addition over Rin... | Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be [[Definition:Element|elements]] of $\map {\MM_R} {m, n}$.
Let $\sqbrk c_{m n} = \sqbrk a_{m n} + \sqbrk b_{m n}$.
By definition of [[Definition:Matrix Entrywise Addition|matrix entrywise addition]]:
:$\forall i \in \closedint 1 m, j \in \closedint 1... | Matrix Entrywise Addition over Ring is Closed/Proof 1 | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_over_Ring_is_Closed | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_over_Ring_is_Closed/Proof_1 | [
"Matrix Entrywise Addition",
"Algebraic Closure",
"Matrix Entrywise Addition is Closed"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix Space",
"Definition:Matrix Entrywise Addition/Ring",
"Definition:Closure (Abstract Algebra)"
] | [
"Definition:Element",
"Definition:Matrix Entrywise Addition",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Matrix Entrywise Addition",
"Definition:Matrix/Order",
"Definition:Closure (Abstract Algebra)"
] |
proofwiki-16138 | Matrix Entrywise Addition over Ring is Closed | Let $\struct {R, +, \circ}$ be a ring.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.
For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation $+$ is closed on $\map {\MM_R} {m, n}$. | By definition, matrix entrywise addition is the '''Hadamard product''' of $\mathbf A$ and $\mathbf B$ with respect to ring addition.
We have from {{Ring-axiom|A0}} that ring addition is closed.
The result then follows directly from Closure of Hadamard Product.
{{qed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\map {\MM_R} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $R$.
For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the [[Definition:Matrix Entrywise Addition over Rin... | By definition, [[Definition:Matrix Entrywise Addition|matrix entrywise addition]] is the '''[[Definition:Hadamard Product|Hadamard product]]''' of $\mathbf A$ and $\mathbf B$ with respect to [[Definition:Ring Addition|ring addition]].
We have from {{Ring-axiom|A0}} that [[Definition:Ring Addition|ring addition]] is [[... | Matrix Entrywise Addition over Ring is Closed/Proof 2 | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_over_Ring_is_Closed | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_over_Ring_is_Closed/Proof_2 | [
"Matrix Entrywise Addition",
"Algebraic Closure",
"Matrix Entrywise Addition is Closed"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix Space",
"Definition:Matrix Entrywise Addition/Ring",
"Definition:Closure (Abstract Algebra)"
] | [
"Definition:Matrix Entrywise Addition",
"Definition:Hadamard Product",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Closure of Hadamard Product"
] |
proofwiki-16139 | Points are Path-Connected iff Contained in Path-Connected Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $x, y \in S$
Then:
:$x, y$ are path-connected points in $T$ {{iff}} there exists a path-connected set of $T$ containing $x$ and $y$. | === Necessary Condition ===
Let $x, y$ be path-connected points in $T$.
Let $\gamma: \closedint 0 1 \to T$ be a path from $x$ to $y$.
From Image of Path is Path-Connected Set, $\Img \gamma$ is a path-connected set of $T$ containing $x$ and $y$.
The result follows.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x, y \in S$
Then:
:$x, y$ are [[Definition:Path-Connected Points|path-connected points]] in $T$ {{iff}} there exists a [[Definition:Path-Connected Set|path-connected set]] of $T$ containing $x$ and $y$. | === Necessary Condition ===
Let $x, y$ be [[Definition:Path-Connected Points|path-connected points]] in $T$.
Let $\gamma: \closedint 0 1 \to T$ be a [[Definition:Path (Topology)|path]] from $x$ to $y$.
From [[Image of Path is Path-Connected Set]], $\Img \gamma$ is a [[Definition:Path-Connected Set|path-connected set... | Points are Path-Connected iff Contained in Path-Connected Set | https://proofwiki.org/wiki/Points_are_Path-Connected_iff_Contained_in_Path-Connected_Set | https://proofwiki.org/wiki/Points_are_Path-Connected_iff_Contained_in_Path-Connected_Set | [
"Path-Connected Points",
"Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Path-Connected/Points",
"Definition:Path-Connected/Set"
] | [
"Definition:Path-Connected/Points",
"Definition:Path (Topology)",
"Image of Path is Path-Connected Set",
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Set",
"Definition:Path (Topology)",
"Definition:Path (Topology)"
] |
proofwiki-16140 | Equivalence of Definitions of Path Component/Union of Path-Connected Sets is Maximal Path-Connected Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in T$.
Let $\CC_x = \leftset {A \subseteq S : x \in A \land A}$ is path-connected in $\rightset T$
Let $C = \bigcup \CC_x$
Then $C$ is a maximal path-connected set of $T$. | Let $\tilde C$ be an arbitrary path-connected set such that:
:$C \subseteq \tilde C$
Then $x \in \tilde C$.
Hence $\tilde C \in \CC_x$.
From Set is Subset of Union,
:$\tilde C \subseteq C$.
Hence $\tilde C = C$.
It follows that $C$ is a maximal path-connected set of $T$ by definition. | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in T$.
Let $\CC_x = \leftset {A \subseteq S : x \in A \land A}$ is [[Definition:Path-Connected Set|path-connected]] in $\rightset T$
Let $C = \bigcup \CC_x$
Then $C$ is a [[Definition:Maximal Set|maximal]] [[Definition:Pat... | Let $\tilde C$ be an arbitrary [[Definition:Path-Connected Set|path-connected set]] such that:
:$C \subseteq \tilde C$
Then $x \in \tilde C$.
Hence $\tilde C \in \CC_x$.
From [[Set is Subset of Union]],
:$\tilde C \subseteq C$.
Hence $\tilde C = C$.
It follows that $C$ is a [[Definition:Maximal Set|maximal]] [[De... | Equivalence of Definitions of Path Component/Union of Path-Connected Sets is Maximal Path-Connected Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component/Union_of_Path-Connected_Sets_is_Maximal_Path-Connected_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component/Union_of_Path-Connected_Sets_is_Maximal_Path-Connected_Set | [
"Equivalence of Definitions of Path Component"
] | [
"Definition:Topological Space",
"Definition:Path-Connected/Set",
"Definition:Maximal/Set",
"Definition:Path-Connected/Set"
] | [
"Definition:Path-Connected/Set",
"Set is Subset of Union",
"Definition:Maximal/Set",
"Definition:Path-Connected/Set"
] |
proofwiki-16141 | Solutions to Diophantine Equation 16x^2+32x+20 = y^2+y | The indeterminate Diophantine equation:
:$16x^2 + 32x + 20 = y^2 + y$
has exactly $4$ solutions:
:$\tuple {0, 4}, \tuple {-2, 4}, \tuple {0, -5}, \tuple {-2, -5}$ | {{begin-eqn}}
{{eqn | l = 16 x^2 + 32 x + 20
| r = y^2 + y
| c =
}}
{{eqn | ll= \leadsto
| l = 16 x^2 + 32 x + 16 + 4
| r =
| c =
}}
{{eqn | l = 16 \paren {x^2 + 2 x + 1} + 4
| r =
| c =
}}
{{eqn | l = 16 \paren {x + 1}^2 + 4
| r = y^2 + y
| c =
}}
{{eqn | ll= \lea... | The [[Definition:Indeterminate Equation|indeterminate]] [[Definition:Diophantine Equation|Diophantine equation]]:
:$16x^2 + 32x + 20 = y^2 + y$
has exactly $4$ solutions:
:$\tuple {0, 4}, \tuple {-2, 4}, \tuple {0, -5}, \tuple {-2, -5}$ | {{begin-eqn}}
{{eqn | l = 16 x^2 + 32 x + 20
| r = y^2 + y
| c =
}}
{{eqn | ll= \leadsto
| l = 16 x^2 + 32 x + 16 + 4
| r =
| c =
}}
{{eqn | l = 16 \paren {x^2 + 2 x + 1} + 4
| r =
| c =
}}
{{eqn | l = 16 \paren {x + 1}^2 + 4
| r = y^2 + y
| c =
}}
{{eqn | ll= \lea... | Solutions to Diophantine Equation 16x^2+32x+20 = y^2+y | https://proofwiki.org/wiki/Solutions_to_Diophantine_Equation_16x^2+32x+20_=_y^2+y | https://proofwiki.org/wiki/Solutions_to_Diophantine_Equation_16x^2+32x+20_=_y^2+y | [
"Diophantine Equations"
] | [
"Definition:Indeterminate Equation",
"Definition:Diophantine Equation"
] | [
"Definition:Prime Number"
] |
proofwiki-16142 | Equivalence of Definitions of Path Component/Maximal Path-Connected Set is Union of Path-Connected Sets | Let $\tilde C$ be a maximal path-connected set of $T$ that contains $x$. | By definition:
:$\tilde C \in \CC_x$
From Set is Subset of Union:
:$\tilde C \subseteq C$
By maximality of $\tilde C$:
:$\tilde C = C$ | Let $\tilde C$ be a [[Definition:Maximal Set|maximal]] [[Definition:Path-Connected Set|path-connected set]] of $T$ that contains $x$. | By definition:
:$\tilde C \in \CC_x$
From [[Set is Subset of Union]]:
:$\tilde C \subseteq C$
By [[Definition:Maximal Set|maximality]] of $\tilde C$:
:$\tilde C = C$ | Equivalence of Definitions of Path Component/Maximal Path-Connected Set is Union of Path-Connected Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component/Maximal_Path-Connected_Set_is_Union_of_Path-Connected_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Path_Component/Maximal_Path-Connected_Set_is_Union_of_Path-Connected_Sets | [
"Equivalence of Definitions of Path Component"
] | [
"Definition:Maximal/Set",
"Definition:Path-Connected/Set"
] | [
"Set is Subset of Union",
"Definition:Maximal/Set"
] |
proofwiki-16143 | Power Set is Closed under Set Complement | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then:
:$\forall A \in \powerset S: \relcomp S A \in \powerset S$ | Let $A \in \powerset S$.
Then by the definition of power set, $A \subseteq S$.
By definition of relative complement:
:$\relcomp S A = \set {x \in S: x \notin A}$
Hence $\relcomp S A$ is a subset of $S$.
That is:
:$\relcomp S A \in \powerset S$
and closure is proved.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Then:
:$\forall A \in \powerset S: \relcomp S A \in \powerset S$ | Let $A \in \powerset S$.
Then by the definition of [[Definition:Power Set|power set]], $A \subseteq S$.
By definition of [[Definition:Relative Complement|relative complement]]:
:$\relcomp S A = \set {x \in S: x \notin A}$
Hence $\relcomp S A$ is a [[Definition:Subset|subset]] of $S$.
That is:
:$\relcomp S A \in \po... | Power Set is Closed under Set Complement | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Set_Complement | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Set_Complement | [
"Power Set",
"Set Complement"
] | [
"Definition:Set",
"Definition:Power Set"
] | [
"Definition:Power Set",
"Definition:Relative Complement",
"Definition:Subset",
"Definition:Closure (Abstract Algebra)"
] |
proofwiki-16144 | Normal Subgroup of Symmetric Group on More than 4 Letters is Alternating Group | Let $n \in \N$ be a natural number such that $n > 4$.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $A_n$ denote the alternating group on $n$ letters.
$A_n$ is the only proper non-trivial normal subgroup of $S_n$. | From Alternating Group is Normal Subgroup of Symmetric Group, $A_n$ is seen to be normal in $S_n$.
It remains to be shown that $A_n$ is the only such normal subgroup of $S_n$.
{{AimForCont}} $N$ is a proper non-trivial normal subgroup of $S_n$ such that $N$ is a proper subset of $A_n$.
From Intersection with Normal Sub... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]] such that $n > 4$.
Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $A_n$ denote the [[Definition:Alternating Group|alternating group on $n$ letters]].
$A_n$ is the only [[Definition:Non-Trivial Prop... | From [[Alternating Group is Normal Subgroup of Symmetric Group]], $A_n$ is seen to be [[Definition:Normal Subgroup|normal]] in $S_n$.
It remains to be shown that $A_n$ is the only such [[Definition:Normal Subgroup|normal subgroup]] of $S_n$.
{{AimForCont}} $N$ is a [[Definition:Non-Trivial Proper Subgroup|proper non... | Normal Subgroup of Symmetric Group on More than 4 Letters is Alternating Group | https://proofwiki.org/wiki/Normal_Subgroup_of_Symmetric_Group_on_More_than_4_Letters_is_Alternating_Group | https://proofwiki.org/wiki/Normal_Subgroup_of_Symmetric_Group_on_More_than_4_Letters_is_Alternating_Group | [
"Alternating Groups",
"Symmetric Groups",
"Examples of Normal Subgroups"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group/n Letters",
"Definition:Alternating Group",
"Definition:Proper Subgroup/Non-Trivial",
"Definition:Normal Subgroup"
] | [
"Alternating Group is Normal Subgroup of Symmetric Group",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Definition:Proper Subgroup/Non-Trivial",
"Definition:Normal Subgroup",
"Definition:Proper Subset",
"Intersection with Normal Subgroup is Normal",
"Definition:Normal Subgroup",
"Int... |
proofwiki-16145 | Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets | Let the path components of $T$ be open sets. | By definition, the path components of $T$ are a partition of $S$.
Hence $S$ is the union of the open path components of $T$.
Since a path component is a maximal path-connected set by definition, then $S$ is a union of open path-connected sets of $T$. | Let the [[Definition:Path Component|path components]] of $T$ be [[Definition:Open Set (Topology)|open sets]]. | By definition, the [[Definition:Path Component|path components]] of $T$ are a [[Definition:Set Partition|partition]] of $S$.
Hence $S$ is the [[Definition:Set Union|union]] of the [[Definition:Open Set (Topology)|open]] [[Definition:Path Component|path components]] of $T$.
Since a [[Definition:Path Component|path com... | Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets | https://proofwiki.org/wiki/Path_Components_are_Open_iff_Union_of_Open_Path-Connected_Sets/Path_Components_are_Open_implies_Space_is_Union_of_Open_Path-Connected_Sets | https://proofwiki.org/wiki/Path_Components_are_Open_iff_Union_of_Open_Path-Connected_Sets/Path_Components_are_Open_implies_Space_is_Union_of_Open_Path-Connected_Sets | [
"Path Components are Open iff Union of Open Path-Connected Sets"
] | [
"Definition:Path Component",
"Definition:Open Set/Topology"
] | [
"Definition:Path Component",
"Definition:Set Partition",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Path Component",
"Definition:Path Component",
"Definition:Path Component/Maximal Path-Connected Set",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Pat... |
proofwiki-16146 | Path Components are Open iff Union of Open Path-Connected Sets/Space is Union of Open Path-Connected Sets implies Path Components are Open | Let $T = \struct {S, \tau}$ be a topological space.
Let $S$ be the union of open path-connected sets of $T$.
Then:
:The path components of $T$ are open sets. | Let $S = \ds \bigcup \set {U \subseteq S : U \in \tau \text { and } U \text { is path-connected} }$.
Let $C$ be a path component of $T$.
=== Lemma ===
{{:Path Components are Open iff Union of Open Path-Connected Sets/Lemma}}{{qed|lemma}}
Then:
{{begin-eqn}}
{{eqn | l = C
| r = C \cap S
| c = Intersection wi... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $S$ be the [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definition:Path-Connected Set|path-connected sets]] of $T$.
Then:
:The [[Definition:Path Component|path components]] of $T$ are [[Definition:... | Let $S = \ds \bigcup \set {U \subseteq S : U \in \tau \text { and } U \text { is path-connected} }$.
Let $C$ be a [[Definition:Path Component|path component]] of $T$.
=== [[Path Components are Open iff Union of Open Path-Connected Sets/Lemma|Lemma]] ===
{{:Path Components are Open iff Union of Open Path-Connected Se... | Path Components are Open iff Union of Open Path-Connected Sets/Space is Union of Open Path-Connected Sets implies Path Components are Open | https://proofwiki.org/wiki/Path_Components_are_Open_iff_Union_of_Open_Path-Connected_Sets/Space_is_Union_of_Open_Path-Connected_Sets_implies_Path_Components_are_Open | https://proofwiki.org/wiki/Path_Components_are_Open_iff_Union_of_Open_Path-Connected_Sets/Space_is_Union_of_Open_Path-Connected_Sets_implies_Path_Components_are_Open | [
"Path Components are Open iff Union of Open Path-Connected Sets"
] | [
"Definition:Topological Space",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Path-Connected/Set",
"Definition:Path Component",
"Definition:Open Set/Topology"
] | [
"Definition:Path Component",
"Path Components are Open iff Union of Open Path-Connected Sets/Lemma",
"Intersection with Subset is Subset",
"Intersection Distributes over Union",
"Union with Empty Set",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Topological Space",
"Definitio... |
proofwiki-16147 | Path Components are Open iff Union of Open Path-Connected Sets/Lemma | Let $U$ be a path-connected set of $T$.
Then:
:$U \cap C \ne \O$ {{iff}} $U \ne \O$ and $U \subseteq C$ | === Necessary Condition ===
Let $U \cap C \ne \O$.
From Union of Path-Connected Sets with Common Point is Path-Connected, $U \cup C$ is a path-connected set of $T$.
From Set is Subset of Union, $C \subseteq U \cup C$.
By definition of a path component, $C$ is a maximal path-connected set.
Hence $C = U \cup C$.
From Uni... | Let $U$ be a [[Definition:Path-Connected Set|path-connected set]] of $T$.
Then:
:$U \cap C \ne \O$ {{iff}} $U \ne \O$ and $U \subseteq C$ | === Necessary Condition ===
Let $U \cap C \ne \O$.
From [[Union of Path-Connected Sets with Common Point is Path-Connected]], $U \cup C$ is a [[Definition:Path-Connected Set|path-connected set]] of $T$.
From [[Set is Subset of Union]], $C \subseteq U \cup C$.
By definition of a [[Definition:Path Component/Maximal P... | Path Components are Open iff Union of Open Path-Connected Sets/Lemma | https://proofwiki.org/wiki/Path_Components_are_Open_iff_Union_of_Open_Path-Connected_Sets/Lemma | https://proofwiki.org/wiki/Path_Components_are_Open_iff_Union_of_Open_Path-Connected_Sets/Lemma | [
"Path Components are Open iff Union of Open Path-Connected Sets"
] | [
"Definition:Path-Connected/Set"
] | [
"Union of Path-Connected Sets with Common Point is Path-Connected",
"Definition:Path-Connected/Set",
"Set is Subset of Union",
"Definition:Path Component/Maximal Path-Connected Set",
"Definition:Maximal",
"Definition:Path-Connected/Set",
"Union with Superset is Superset",
"Intersection with Subset is ... |
proofwiki-16148 | Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let the components of $T$ be open sets.
Then:
:$S$ is a union of open connected sets of $T$. | Let the components of $T$ be open.
By definition, the components of $T$ are a partition of $S$.
Hence $S$ is the union of the open components of $T$.
Since a component is a maximal connected set by definition, then $S$ is a union of open connected sets of $T$. | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let the [[Definition:Component (Topology)|components]] of $T$ be [[Definition:Open Set (Topology)|open sets]].
Then:
:$S$ is a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definition:Connected Set (Top... | Let the [[Definition:Component (Topology)|components]] of $T$ be [[Definition:Open Set (Topology)|open]].
By definition, the [[Definition:Component (Topology)|components]] of $T$ are a [[Definition:Set Partition|partition]] of $S$.
Hence $S$ is the [[Definition:Set Union|union]] of the [[Definition:Open Set (Topology... | Components are Open iff Union of Open Connected Sets/Components are Open implies Space is Union of Open Connected Sets | https://proofwiki.org/wiki/Components_are_Open_iff_Union_of_Open_Connected_Sets/Components_are_Open_implies_Space_is_Union_of_Open_Connected_Sets | https://proofwiki.org/wiki/Components_are_Open_iff_Union_of_Open_Connected_Sets/Components_are_Open_implies_Space_is_Union_of_Open_Connected_Sets | [
"Components are Open iff Union of Open Connected Sets"
] | [
"Definition:Topological Space",
"Definition:Component (Topology)",
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Connected Set (Topology)"
] | [
"Definition:Component (Topology)",
"Definition:Open Set/Topology",
"Definition:Component (Topology)",
"Definition:Set Partition",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Component (Topology)",
"Definition:Component (Topology)",
"Definition:Maximal/Set",
"Definition:Conn... |
proofwiki-16149 | Components are Open iff Union of Open Connected Sets/Space is Union of Open Connected Sets implies Components are Open | Let $T = \struct {S, \tau}$ be a topological space.
Let $S$ be a union of open connected sets of $T$.
Then:
:The components of $T$ are open sets. | Let $S = \ds \bigcup \set {U \subseteq S : U \in \tau \text { and } U \text { is connected} }$.
Let $C$ be a component of $T$.
=== Lemma ===
:For any connected set $U$ then:
{{:Components are Open iff Union of Open Connected Sets/Lemma 1}}{{qed|lemma}}
Then:
{{begin-eqn}}
{{eqn | l = C
| r = C \cap S
| c = ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $S$ be a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open]] [[Definition:Connected Set (Topology)|connected sets]] of $T$.
Then:
:The [[Definition:Component (Topology)|components]] of $T$ are [[Definition:... | Let $S = \ds \bigcup \set {U \subseteq S : U \in \tau \text { and } U \text { is connected} }$.
Let $C$ be a [[Definition:Component (Topology)|component]] of $T$.
=== [[Components are Open iff Union of Open Connected Sets/Lemma 1|Lemma]] ===
:For any [[Definition:Connected Set (Topology)|connected set]] $U$ then:
{... | Components are Open iff Union of Open Connected Sets/Space is Union of Open Connected Sets implies Components are Open | https://proofwiki.org/wiki/Components_are_Open_iff_Union_of_Open_Connected_Sets/Space_is_Union_of_Open_Connected_Sets_implies_Components_are_Open | https://proofwiki.org/wiki/Components_are_Open_iff_Union_of_Open_Connected_Sets/Space_is_Union_of_Open_Connected_Sets_implies_Components_are_Open | [
"Components are Open iff Union of Open Connected Sets"
] | [
"Definition:Topological Space",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Connected Set (Topology)",
"Definition:Component (Topology)",
"Definition:Open Set/Topology"
] | [
"Definition:Component (Topology)",
"Components are Open iff Union of Open Connected Sets/Lemma 1",
"Definition:Connected Set (Topology)",
"Intersection with Subset is Subset",
"Intersection Distributes over Union",
"Union with Empty Set",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Defi... |
proofwiki-16150 | Components are Open iff Union of Open Connected Sets/Lemma 1 | ::$U \cap C \ne \O$ {{iff}} $U \ne \O$ and $U \subseteq C$ | === Necessary Condition ===
Let $U \cap C \ne \O$.
From Union of Connected Sets with Common Point is Connected, $U \cup C$ is a connected set of $T$.
From Set is Subset of Union, $C \subseteq U \cup C$.
By definition of a component, $C$ is a maximal connected set.
Hence $C = U \cup C$.
From Union with Superset is Super... | ::$U \cap C \ne \O$ {{iff}} $U \ne \O$ and $U \subseteq C$ | === Necessary Condition ===
Let $U \cap C \ne \O$.
From [[Union of Connected Sets with Common Point is Connected]], $U \cup C$ is a [[Definition:Connected Set (Topology)|connected set]] of $T$.
From [[Set is Subset of Union]], $C \subseteq U \cup C$.
By definition of a [[Definition:Component (Topology)/Definition 3... | Components are Open iff Union of Open Connected Sets/Lemma 1 | https://proofwiki.org/wiki/Components_are_Open_iff_Union_of_Open_Connected_Sets/Lemma_1 | https://proofwiki.org/wiki/Components_are_Open_iff_Union_of_Open_Connected_Sets/Lemma_1 | [
"Components are Open iff Union of Open Connected Sets"
] | [] | [
"Union of Connected Sets with Common Point is Connected",
"Definition:Connected Set (Topology)",
"Set is Subset of Union",
"Definition:Component (Topology)/Definition 3",
"Definition:Maximal",
"Definition:Connected Set (Topology)",
"Union with Superset is Superset",
"Intersection with Subset is Subset... |
proofwiki-16151 | Integral to Infinity of Square of Sine p x over x Squared | :$\ds \int_0^\infty \paren {\frac {\sin p x} x}^2 \rd x = \frac {\pi \size p} 2$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sin^2 p x}
| r = \map {\frac \d {\d x} } {\sin p x} \map {\frac \d {\map \d {\sin p x} } } {\sin^2 p x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = 2 p \cos p x \sin p x
| c = Derivative of Sine of a x, Derivative of Power
}}
{{eqn | r ... | :$\ds \int_0^\infty \paren {\frac {\sin p x} x}^2 \rd x = \frac {\pi \size p} 2$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sin^2 p x}
| r = \map {\frac \d {\d x} } {\sin p x} \map {\frac \d {\map \d {\sin p x} } } {\sin^2 p x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = 2 p \cos p x \sin p x
| c = [[Derivative of Sine of a x]], [[Derivative of Power]]
... | Integral to Infinity of Square of Sine p x over x Squared | https://proofwiki.org/wiki/Integral_to_Infinity_of_Square_of_Sine_p_x_over_x_Squared | https://proofwiki.org/wiki/Integral_to_Infinity_of_Square_of_Sine_p_x_over_x_Squared | [
"Definite Integrals involving Sine Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Sine Function/Corollary",
"Power Rule for Derivatives",
"Double Angle Formulas/Sine",
"Primitive of Power",
"Integration by Parts",
"Real Sine Function is Bounded",
"Definition:Real Number",
"Definition:Strictly Positive/Real Number",
"Squeeze The... |
proofwiki-16152 | Integral to Infinity of One minus Cosine p x over x Squared | :$\ds \int_0^\infty \frac {1 - \cos p x} {x^2} \rd x = \frac {\pi \size p} 2$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {1 - \cos p x} {x^2} \rd x
| r = \int_0^\infty \frac {2 \sin^2 \paren {\frac {p x} 2} } {x^2} \rd x
| c = Square of Sine
}}
{{eqn | r = 2 \times \frac {\pi \size p} {2 \times 2}
| c = Integral to Infinity of $\paren {\dfrac {\sin p x} x}^2$
}}
{{eqn | r = \frac {\pi \siz... | :$\ds \int_0^\infty \frac {1 - \cos p x} {x^2} \rd x = \frac {\pi \size p} 2$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {1 - \cos p x} {x^2} \rd x
| r = \int_0^\infty \frac {2 \sin^2 \paren {\frac {p x} 2} } {x^2} \rd x
| c = [[Square of Sine]]
}}
{{eqn | r = 2 \times \frac {\pi \size p} {2 \times 2}
| c = [[Integral to Infinity of Square of Sine p x over x Squared|Integral to Infinity of... | Integral to Infinity of One minus Cosine p x over x Squared | https://proofwiki.org/wiki/Integral_to_Infinity_of_One_minus_Cosine_p_x_over_x_Squared | https://proofwiki.org/wiki/Integral_to_Infinity_of_One_minus_Cosine_p_x_over_x_Squared | [
"Definite Integrals involving Cosine Function"
] | [] | [
"Power Reduction Formulas/Sine Squared",
"Integral to Infinity of Square of Sine p x over x Squared"
] |
proofwiki-16153 | 1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways | The number $1$ can be expressed as the sum of $4$ distinct unit fractions in $6$ different ways:
{{begin-eqn}}
{{eqn | l = 1
| r = \frac 1 2 + \frac 1 3 + \frac 1 7 + \frac 1 {42}
}}
{{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 8 + \frac 1 {24}
}}
{{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 9 + \frac 1 {18}
}}
{{e... | Let:
:$1 = \dfrac 1 v + \dfrac 1 w + \dfrac 1 x + \dfrac 1 y$
where $ 1 < v < w < x < y$
Suppose $v = 3$ and take the largest potential solution that can be generated:
:$1 \stackrel {?} {=} \dfrac 1 3 + \dfrac 1 4 + \dfrac 1 5 + \dfrac 1 6$
But we find:
:$1 > \dfrac 1 3 + \dfrac 1 4 + \dfrac 1 5 + \dfrac 1 6$
Therefore... | The number $1$ can be expressed as the [[Definition:Sum (Addition)|sum]] of $4$ [[Definition:Distinct|distinct]] [[Definition:Unit Fraction|unit fractions]] in $6$ different ways:
{{begin-eqn}}
{{eqn | l = 1
| r = \frac 1 2 + \frac 1 3 + \frac 1 7 + \frac 1 {42}
}}
{{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 8 +... | Let:
:$1 = \dfrac 1 v + \dfrac 1 w + \dfrac 1 x + \dfrac 1 y$
where $ 1 < v < w < x < y$
Suppose $v = 3$ and take the largest potential solution that can be generated:
:$1 \stackrel {?} {=} \dfrac 1 3 + \dfrac 1 4 + \dfrac 1 5 + \dfrac 1 6$
But we find:
:$1 > \dfrac 1 3 + \dfrac 1 4 + \dfrac 1 5 + \dfrac 1 6$
Ther... | 1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways/Proof 1 | https://proofwiki.org/wiki/1_can_be_Expressed_as_Sum_of_4_Distinct_Unit_Fractions_in_6_Ways | https://proofwiki.org/wiki/1_can_be_Expressed_as_Sum_of_4_Distinct_Unit_Fractions_in_6_Ways/Proof_1 | [
"1",
"Unit Fractions",
"Recreational Mathematics",
"1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways"
] | [
"Definition:Addition/Sum",
"Definition:Distinct",
"Definition:Unit Fraction"
] | [] |
proofwiki-16154 | 1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways | The number $1$ can be expressed as the sum of $4$ distinct unit fractions in $6$ different ways:
{{begin-eqn}}
{{eqn | l = 1
| r = \frac 1 2 + \frac 1 3 + \frac 1 7 + \frac 1 {42}
}}
{{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 8 + \frac 1 {24}
}}
{{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 9 + \frac 1 {18}
}}
{{e... | From Sum of 4 Unit Fractions that equals 1:
{{:Sum of 4 Unit Fractions that equals 1}}
This includes repeated unit fractions.
The full list is:
{{begin-eqn}}
{{eqn | n = 1
| r = \dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {42}
| o =
}}
{{eqn | n = 2
| r = \dfrac 1 2 + \dfrac 1 3 + \dfrac 1 8 + \d... | The number $1$ can be expressed as the [[Definition:Sum (Addition)|sum]] of $4$ [[Definition:Distinct|distinct]] [[Definition:Unit Fraction|unit fractions]] in $6$ different ways:
{{begin-eqn}}
{{eqn | l = 1
| r = \frac 1 2 + \frac 1 3 + \frac 1 7 + \frac 1 {42}
}}
{{eqn | r = \frac 1 2 + \frac 1 3 + \frac 1 8 +... | From [[Sum of 4 Unit Fractions that equals 1]]:
{{:Sum of 4 Unit Fractions that equals 1}}
This includes repeated [[Definition:Unit Fraction|unit fractions]].
The full list is:
{{begin-eqn}}
{{eqn | n = 1
| r = \dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {42}
| o =
}}
{{eqn | n = 2
| r = \dfr... | 1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways/Proof 2 | https://proofwiki.org/wiki/1_can_be_Expressed_as_Sum_of_4_Distinct_Unit_Fractions_in_6_Ways | https://proofwiki.org/wiki/1_can_be_Expressed_as_Sum_of_4_Distinct_Unit_Fractions_in_6_Ways/Proof_2 | [
"1",
"Unit Fractions",
"Recreational Mathematics",
"1 can be Expressed as Sum of 4 Distinct Unit Fractions in 6 Ways"
] | [
"Definition:Addition/Sum",
"Definition:Distinct",
"Definition:Unit Fraction"
] | [
"Sum of 4 Unit Fractions that equals 1",
"Definition:Unit Fraction",
"Definition:Fraction/Denominator",
"Definition:Addition/Summand",
"Definition:Distinct"
] |
proofwiki-16155 | Equivalence of Definitions of Weakly Locally Connected at Point | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
{{TFAE|def = Weakly Locally Connected at Point}} | === Definition 1 implies Definition 2 ===
{{:Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2}}{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
{{TFAE|def = Weakly Locally Connected at Point}} | === [[Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2|Definition 1 implies Definition 2]] ===
{{:Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2}}{{qed|lemma}} | Equivalence of Definitions of Weakly Locally Connected at Point | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weakly_Locally_Connected_at_Point | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weakly_Locally_Connected_at_Point | [
"Locally Connected Spaces",
"Equivalence of Definitions of Weakly Locally Connected at Point"
] | [
"Definition:Topological Space"
] | [
"Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2"
] |
proofwiki-16156 | Integral to Infinity of Cosine p x minus Cosine q x over x | :$\ds \int_0^\infty \frac {\cos p x - \cos q x} x \rd x = \ln \frac q p$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\cos p x - \cos q x} x \rd x
| r = \int_0^\infty \map {\laptrans {\cos p t - \cos q t} } x \rd x
| c = Integral to Infinity of Function over Argument, assuming both integrals converge
}}
{{eqn | r = \int_0^\infty \frac x {x^2 + p^2} - \frac x {x^2 + q^2} \rd x
... | :$\ds \int_0^\infty \frac {\cos p x - \cos q x} x \rd x = \ln \frac q p$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\cos p x - \cos q x} x \rd x
| r = \int_0^\infty \map {\laptrans {\cos p t - \cos q t} } x \rd x
| c = [[Integral to Infinity of Function over Argument]], assuming both [[Definition:Definite Integral|integrals]] converge
}}
{{eqn | r = \int_0^\infty \frac x {x^... | Integral to Infinity of Cosine p x minus Cosine q x over x | https://proofwiki.org/wiki/Integral_to_Infinity_of_Cosine_p_x_minus_Cosine_q_x_over_x | https://proofwiki.org/wiki/Integral_to_Infinity_of_Cosine_p_x_minus_Cosine_q_x_over_x | [
"Definite Integrals involving Cosine Function"
] | [] | [
"Integral to Infinity of Function over Argument",
"Definition:Definite Integral",
"Laplace Transform of Cosine",
"Primitive of x over x squared plus a squared",
"Limit of Real Function/Examples/Reciprocal of x at Infinity"
] |
proofwiki-16157 | Empty Set is Subset of Power Set | The empty set is a subset of all power sets:
:$\forall S: \O \subseteq \powerset S$ | Follows directly from Empty Set is Subset of All Sets.
{{qed}} | The [[Definition:Empty Set|empty set]] is a [[Definition:Subset|subset]] of all [[Definition:Power Set|power sets]]:
:$\forall S: \O \subseteq \powerset S$ | Follows directly from [[Empty Set is Subset of All Sets]].
{{qed}} | Empty Set is Subset of Power Set | https://proofwiki.org/wiki/Empty_Set_is_Subset_of_Power_Set | https://proofwiki.org/wiki/Empty_Set_is_Subset_of_Power_Set | [
"Empty Set",
"Power Set",
"Subsets"
] | [
"Definition:Empty Set",
"Definition:Subset",
"Definition:Power Set"
] | [
"Empty Set is Subset of All Sets"
] |
proofwiki-16158 | Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2 | Let $x$ have a neighborhood basis consisting of connected sets. | Let $U$ be an open neighborhood of $x$.
By assumption there exists a connected neighborhood $C$ of $x$ such that $C \subseteq U$.
By definition of a neighborhood, there exists an open neighborhood $V$ of $x$ such that $V \subseteq C$.
From Subset Relation is Transitive, $V \subseteq U$.
By definition of a subset:
:$\fo... | Let $x$ have a [[Definition:Neighborhood Basis|neighborhood basis]] consisting of [[Definition:Connected Set (Topology)|connected sets]]. | Let $U$ be an [[Definition:Open Neighborhood of Point|open neighborhood]] of $x$.
By assumption there exists a [[Definition:Connected Set (Topology)|connected]] [[Definition:Neighborhood of Point|neighborhood]] $C$ of $x$ such that $C \subseteq U$.
By definition of a [[Definition:Neighborhood of Point|neighborhood]],... | Equivalence of Definitions of Weakly Locally Connected at Point/Definition 1 implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weakly_Locally_Connected_at_Point/Definition_1_implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weakly_Locally_Connected_at_Point/Definition_1_implies_Definition_2 | [
"Equivalence of Definitions of Weakly Locally Connected at Point"
] | [
"Definition:Neighborhood Basis",
"Definition:Connected Set (Topology)"
] | [
"Definition:Open Neighborhood/Point",
"Definition:Connected Set (Topology)",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Neighborhood/Point",
"Subset Relation is Transitive",
"Definition:Subset"
] |
proofwiki-16159 | Equivalence of Definitions of Weakly Locally Connected at Point/Definition 2 implies Definition 1 | Let every open neighborhood $U$ of $x$ contain an open neighborhood $V$ of $x$ such that every two points of $V$ lie in some connected subset of $U$. | Let $\BB = \set {B \subseteq S: B \text{ is a connected neighborhood of } x}$
It will be shown that $\BB$ is a neighborhood basis consisting of connected sets.
Let $N$ be any neighborhood of $x$.
By definition of a neighborhood there exists an open neighborhood $U$ of $x$ such that $U \subseteq N$
By assumption there e... | Let every [[Definition:Open Neighborhood of Point|open neighborhood]] $U$ of $x$ contain an [[Definition:Open Neighborhood of Point|open neighborhood]] $V$ of $x$ such that every two points of $V$ lie in some [[Definition:Connected Set (Topology)|connected subset]] of $U$. | Let $\BB = \set {B \subseteq S: B \text{ is a connected neighborhood of } x}$
It will be shown that $\BB$ is a [[Definition:Neighborhood Basis|neighborhood basis]] consisting of [[Definition:Connected Set (Topology)|connected sets]].
Let $N$ be any [[Definition:Neighborhood of Point|neighborhood]] of $x$.
By defini... | Equivalence of Definitions of Weakly Locally Connected at Point/Definition 2 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weakly_Locally_Connected_at_Point/Definition_2_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weakly_Locally_Connected_at_Point/Definition_2_implies_Definition_1 | [
"Equivalence of Definitions of Weakly Locally Connected at Point"
] | [
"Definition:Open Neighborhood/Point",
"Definition:Open Neighborhood/Point",
"Definition:Connected Set (Topology)"
] | [
"Definition:Neighborhood Basis",
"Definition:Connected Set (Topology)",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Neighborhood/Point",
"Definition:Open Neighborhood/Point",
"Definition:Connected Set (Topology)",
"Definition:Connected Set (... |
proofwiki-16160 | Cycle of Subsets implies Set Equality | Let $A_1, A_2, \dotsc, A_n$ be sets.
Let:
:$\forall k \in \set {2, 3, \dotsc, n}: A_{k - 1} \subseteq A_k$
and:
:$A_n \subseteq A_1$
Then:
:$\forall j, k \in \set {1, 2, \dotsc, n}: A_j = A_k$ | Consider the set of sets $\mathbb A = \set {A_1, A_2, \dotsc, A_n}$
Consider the relational structure $S = \struct {\mathbb A, \subseteq}$.
We have from Subset Relation is Ordering that $S$ is an ordered structure.
The result follows from Ordering Cycle implies Equality.
{{qed}} | Let $A_1, A_2, \dotsc, A_n$ be [[Definition:Set|sets]].
Let:
:$\forall k \in \set {2, 3, \dotsc, n}: A_{k - 1} \subseteq A_k$
and:
:$A_n \subseteq A_1$
Then:
:$\forall j, k \in \set {1, 2, \dotsc, n}: A_j = A_k$ | Consider the [[Definition:Set of Sets|set of sets]] $\mathbb A = \set {A_1, A_2, \dotsc, A_n}$
Consider the [[Definition:Relational Structure|relational structure]] $S = \struct {\mathbb A, \subseteq}$.
We have from [[Subset Relation is Ordering]] that $S$ is an [[Definition:Ordered Structure|ordered structure]].
T... | Cycle of Subsets implies Set Equality | https://proofwiki.org/wiki/Cycle_of_Subsets_implies_Set_Equality | https://proofwiki.org/wiki/Cycle_of_Subsets_implies_Set_Equality | [
"Subsets"
] | [
"Definition:Set"
] | [
"Definition:Set of Sets",
"Definition:Relational Structure",
"Subset Relation is Ordering",
"Definition:Ordered Structure",
"Ordering Cycle implies Equality/General Case"
] |
proofwiki-16161 | Integral Representation of Dirichlet Beta Function in terms of Gamma Function | :$\ds \map \beta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 + e^{-2 x} } \rd x$ | We have, by Laplace Transform of Power:
:$\ds \frac {\paren {-1}^n \map \Gamma s} {\paren {2 n + 1}^s} = \paren {-1}^n \int_0^\infty x^{s - 1} e^{-\paren {2 n + 1} x} \rd x$
for $\map \Re s > 0$.
Summing, we have:
{{begin-eqn}}
{{eqn | l = \map \Gamma s \sum_{n \mathop = 0}^N \frac {\paren {-1}^n} {\paren {2 n + 1}^... | :$\ds \map \beta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 + e^{-2 x} } \rd x$ | We have, by [[Laplace Transform of Power]]:
:$\ds \frac {\paren {-1}^n \map \Gamma s} {\paren {2 n + 1}^s} = \paren {-1}^n \int_0^\infty x^{s - 1} e^{-\paren {2 n + 1} x} \rd x$
for $\map \Re s > 0$.
Summing, we have:
{{begin-eqn}}
{{eqn | l = \map \Gamma s \sum_{n \mathop = 0}^N \frac {\paren {-1}^n} {\paren {... | Integral Representation of Dirichlet Beta Function in terms of Gamma Function | https://proofwiki.org/wiki/Integral_Representation_of_Dirichlet_Beta_Function_in_terms_of_Gamma_Function | https://proofwiki.org/wiki/Integral_Representation_of_Dirichlet_Beta_Function_in_terms_of_Gamma_Function | [
"Dirichlet Beta Function",
"Gamma Function"
] | [] | [
"Laplace Transform of Power",
"Linear Combination of Integrals/Definite",
"Combination Theorem for Limits of Functions/Complex/Multiple Rule",
"Lebesgue's Dominated Convergence Theorem",
"Sum of Infinite Geometric Sequence",
"Category:Dirichlet Beta Function",
"Category:Gamma Function"
] |
proofwiki-16162 | Dirichlet Beta Function at Odd Positive Integers | {{begin-eqn}}
{{eqn | l = \map \beta {2 n + 1}
| r = \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k} {\paren {2 k + 1}^{2 n + 1} }
| c =
}}
{{eqn | r = \frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots
| c =
}}
{{eqn | r = \paren {-1}^n \dfra... | {{begin-eqn}}
{{eqn | l = \map \beta {2 n + 1}
| r = \dfrac 1 {4^{2 n + 1} } \paren {\map \zeta {2 n + 1, \frac 1 4} - \map \zeta {2 n + 1, \frac 3 4} }
| c = Dirichlet Beta Function in terms of Hurwitz Zeta Function
}}
{{eqn | r = \dfrac 1 {4^{2 n + 1} } \paren { \dfrac {\map {\psi_{2 n} } {\dfrac 1 4} - \... | {{begin-eqn}}
{{eqn | l = \map \beta {2 n + 1}
| r = \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k} {\paren {2 k + 1}^{2 n + 1} }
| c =
}}
{{eqn | r = \frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots
| c =
}}
{{eqn | r = \paren {-1}^n \dfra... | {{begin-eqn}}
{{eqn | l = \map \beta {2 n + 1}
| r = \dfrac 1 {4^{2 n + 1} } \paren {\map \zeta {2 n + 1, \frac 1 4} - \map \zeta {2 n + 1, \frac 3 4} }
| c = [[Dirichlet Beta Function in terms of Hurwitz Zeta Function]]
}}
{{eqn | r = \dfrac 1 {4^{2 n + 1} } \paren { \dfrac {\map {\psi_{2 n} } {\dfrac 1 4}... | Dirichlet Beta Function at Odd Positive Integers | https://proofwiki.org/wiki/Dirichlet_Beta_Function_at_Odd_Positive_Integers | https://proofwiki.org/wiki/Dirichlet_Beta_Function_at_Odd_Positive_Integers | [
"Dirichlet Beta Function at Odd Positive Integers",
"Dirichlet Beta Function",
"Euler Numbers",
"Sums of Sequences"
] | [
"Definition:Dirichlet Beta Function",
"Definition:Euler Numbers",
"Definition:Positive/Integer"
] | [
"Dirichlet Beta Function in terms of Hurwitz Zeta Function",
"Polygamma Function in terms of Hurwitz Zeta Function",
"Polygamma Reflection Formula",
"Even Derivatives of Cotangent of Pi Z at One Fourth"
] |
proofwiki-16163 | Correspondence between Set and Ordinate of Cartesian Product is Mapping | Let $S$ and $T$ be sets such that $T \ne \O$.
Let $S \times T$ denote their cartesian product.
Let $t \in T$ be given.
Let $j_t \subseteq S \times \paren {S \times T}$ be the relation on $S \times {S \times T}$ defined as:
:$\forall s \in S: \map {j_t} s = \tuple {s, t}$
Then $j_t$ is a mapping. | First it is to be shown that $j_t$ is left-total.
This follows from the fact that $j_t$ is defined for all $s$:
:$\map {j_t} s = \tuple {s, t}$
{{qed|lemma}}
Next it is to be shown that $j_t$ is many-to-one, that is:
:$\forall s_1, s_2 \in S: \map {j_t} {s_1} \ne \map {j_t} {s_2} \implies s_1 \ne s_2$
We have that:
{{b... | Let $S$ and $T$ be [[Definition:Set|sets]] such that $T \ne \O$.
Let $S \times T$ denote their [[Definition:Cartesian Product|cartesian product]].
Let $t \in T$ be given.
Let $j_t \subseteq S \times \paren {S \times T}$ be the [[Definition:Relation|relation]] on $S \times {S \times T}$ defined as:
:$\forall s \in ... | First it is to be shown that $j_t$ is [[Definition:Left-Total Relation|left-total]].
This follows from the fact that $j_t$ is defined for all $s$:
:$\map {j_t} s = \tuple {s, t}$
{{qed|lemma}}
Next it is to be shown that $j_t$ is [[Definition:Many-to-One Relation|many-to-one]], that is:
:$\forall s_1, s_2 \in S: \ma... | Correspondence between Set and Ordinate of Cartesian Product is Mapping | https://proofwiki.org/wiki/Correspondence_between_Set_and_Ordinate_of_Cartesian_Product_is_Mapping | https://proofwiki.org/wiki/Correspondence_between_Set_and_Ordinate_of_Cartesian_Product_is_Mapping | [
"Cartesian Product"
] | [
"Definition:Set",
"Definition:Cartesian Product",
"Definition:Relation",
"Definition:Mapping"
] | [
"Definition:Left-Total Relation",
"Definition:Many-to-One Relation"
] |
proofwiki-16164 | Mapping from Set to Ordinate of Cartesian Product is Injection | Let $S$ and $T$ be sets such that $T \ne \O$.
Let $S \times T$ denote their cartesian product.
Let $t \in T$ be given.
Let $j_t: S \to S \times T$ be the mapping from $S$ to $S \times T$ defined as:
:$\forall s \in S: \map {j_t} s = \tuple {s, t}$
Then $j_t$ is an injection. | It has been shown in Correspondence between Set and Ordinate of Cartesian Product is Mapping that $j_t$ is a mapping.
Now it is to be shown that $j_t$ is injective, that is:
:$\forall s_1, s_2 \in S: \map {j_t} {s_1} = \map {j_t} {s_2} \implies s_1 = s_2$
We have that:
{{begin-eqn}}
{{eqn | l = \map {j_t} {s_1}
|... | Let $S$ and $T$ be [[Definition:Set|sets]] such that $T \ne \O$.
Let $S \times T$ denote their [[Definition:Cartesian Product|cartesian product]].
Let $t \in T$ be given.
Let $j_t: S \to S \times T$ be the [[Definition:Mapping|mapping]] from $S$ to $S \times T$ defined as:
:$\forall s \in S: \map {j_t} s = \tuple ... | It has been shown in [[Correspondence between Set and Ordinate of Cartesian Product is Mapping]] that $j_t$ is a [[Definition:Mapping|mapping]].
Now it is to be shown that $j_t$ is [[Definition:Injection|injective]], that is:
:$\forall s_1, s_2 \in S: \map {j_t} {s_1} = \map {j_t} {s_2} \implies s_1 = s_2$
We have t... | Mapping from Set to Ordinate of Cartesian Product is Injection | https://proofwiki.org/wiki/Mapping_from_Set_to_Ordinate_of_Cartesian_Product_is_Injection | https://proofwiki.org/wiki/Mapping_from_Set_to_Ordinate_of_Cartesian_Product_is_Injection | [
"Cartesian Product",
"Injections"
] | [
"Definition:Set",
"Definition:Cartesian Product",
"Definition:Mapping",
"Definition:Injection"
] | [
"Correspondence between Set and Ordinate of Cartesian Product is Mapping",
"Definition:Mapping",
"Definition:Injection"
] |
proofwiki-16165 | Primitive of Sine Integral Function | :$\ds \int \map \Si x \rd x = x \map \Si x + \cos x + C$ | By Derivative of Sine Integral Function, we have:
:$\map {\dfrac \d {\d x} } {\map \Si x} = \dfrac {\sin x} x$
So:
{{begin-eqn}}
{{eqn | l = \int \map \Si x \rd x
| r = \int 1 \times \map \Si x \rd x
}}
{{eqn | r = x \map \Si x - \int x \frac {\sin x} x \rd x
| c = Integration by Parts
}}
{{eqn | r = x \map \Si x -... | :$\ds \int \map \Si x \rd x = x \map \Si x + \cos x + C$ | By [[Derivative of Sine Integral Function]], we have:
:$\map {\dfrac \d {\d x} } {\map \Si x} = \dfrac {\sin x} x$
So:
{{begin-eqn}}
{{eqn | l = \int \map \Si x \rd x
| r = \int 1 \times \map \Si x \rd x
}}
{{eqn | r = x \map \Si x - \int x \frac {\sin x} x \rd x
| c = [[Integration by Parts]]
}}
{{eqn | r = x \... | Primitive of Sine Integral Function | https://proofwiki.org/wiki/Primitive_of_Sine_Integral_Function | https://proofwiki.org/wiki/Primitive_of_Sine_Integral_Function | [
"Primitives",
"Sine Integral Function"
] | [] | [
"Derivative of Sine Integral Function",
"Integration by Parts",
"Primitive of Sine Function",
"Category:Primitives",
"Category:Sine Integral Function"
] |
proofwiki-16166 | Preimage of Subset of Cartesian Product under Injection from Factor | Let $S$ and $T$ be sets such that $T \ne \O$.
Let $S \times T$ denote their cartesian product.
Let $t \in T$ be given.
Let $j_t: S \to S \times T$ be the injection from $S$ to $S \times T$ defined as:
:$\forall s \in S: \map {j_t} s = \tuple {s, t}$
Let $W \subseteq S \times T$.
Let $V = {j_t}^{-1} \sqbrk W$ denote the... | That $j_t$ is actually an injection is demonstrated in Mapping from Set to Ordinate of Cartesian Product is Injection.
Then:
{{begin-eqn}}
{{eqn | l = V
| r = {j_t}^{-1} \sqbrk W
| c =
}}
{{eqn | r = \set {s \in S : \map {j_t} s \in W}
| c =
}}
{{eqn | r = \set {s \in S : \tuple {s, t} \in W}
... | Let $S$ and $T$ be [[Definition:Set|sets]] such that $T \ne \O$.
Let $S \times T$ denote their [[Definition:Cartesian Product|cartesian product]].
Let $t \in T$ be given.
Let $j_t: S \to S \times T$ be the [[Definition:Injection|injection]] from $S$ to $S \times T$ defined as:
:$\forall s \in S: \map {j_t} s = \tu... | That $j_t$ is actually an [[Definition:Injection|injection]] is demonstrated in [[Mapping from Set to Ordinate of Cartesian Product is Injection]].
Then:
{{begin-eqn}}
{{eqn | l = V
| r = {j_t}^{-1} \sqbrk W
| c =
}}
{{eqn | r = \set {s \in S : \map {j_t} s \in W}
| c =
}}
{{eqn | r = \set {s \in ... | Preimage of Subset of Cartesian Product under Injection from Factor | https://proofwiki.org/wiki/Preimage_of_Subset_of_Cartesian_Product_under_Injection_from_Factor | https://proofwiki.org/wiki/Preimage_of_Subset_of_Cartesian_Product_under_Injection_from_Factor | [
"Cartesian Product",
"Injections"
] | [
"Definition:Set",
"Definition:Cartesian Product",
"Definition:Injection",
"Definition:Preimage/Mapping/Subset"
] | [
"Definition:Injection",
"Mapping from Set to Ordinate of Cartesian Product is Injection"
] |
proofwiki-16167 | Primitive of Gauss Error Function | :$\ds \int \map \erf x \rd x = x \map \erf x + \frac 1 {\sqrt \pi} e^{-x^2} + C$ | By Derivative of Gauss Error Function, we have:
:$\dfrac \d {\d x} \paren {\map \erf x} = \dfrac 2 {\sqrt \pi} e^{-x^2}$
So:
{{begin-eqn}}
{{eqn | l = \int \map \erf x \rd x
| r = \int 1 \times \map \erf x \rd x
}}
{{eqn | r = x \map \erf x - \frac 2 {\sqrt \pi} \int x e^{-x^2} \rd x
| c = Integration by Pa... | :$\ds \int \map \erf x \rd x = x \map \erf x + \frac 1 {\sqrt \pi} e^{-x^2} + C$ | By [[Derivative of Gauss Error Function]], we have:
:$\dfrac \d {\d x} \paren {\map \erf x} = \dfrac 2 {\sqrt \pi} e^{-x^2}$
So:
{{begin-eqn}}
{{eqn | l = \int \map \erf x \rd x
| r = \int 1 \times \map \erf x \rd x
}}
{{eqn | r = x \map \erf x - \frac 2 {\sqrt \pi} \int x e^{-x^2} \rd x
| c = [[Integrat... | Primitive of Gauss Error Function | https://proofwiki.org/wiki/Primitive_of_Gauss_Error_Function | https://proofwiki.org/wiki/Primitive_of_Gauss_Error_Function | [
"Gauss Error Function",
"Primitives"
] | [] | [
"Derivative of Gauss Error Function",
"Integration by Parts",
"Integration by Substitution",
"Primitive of Exponential Function",
"Category:Gauss Error Function",
"Category:Primitives"
] |
proofwiki-16168 | Derivative of Gauss Error Function | :$\map {\dfrac \d {\d x} } {\map \erf x} = \dfrac 2 {\sqrt \pi} e^{-x^2}$ | We have, by the definition of the Gauss error function:
:$\ds \map \erf x = \frac 2 {\sqrt \pi} \int_0^x e^{-t^2} \rd t$
By Fundamental Theorem of Calculus (First Part): Corollary, we therefore have:
:$\map {\dfrac \d {\d x} } {\map \erf x} = \dfrac 2 {\sqrt \pi} e^{-x^2}$
{{qed}}
Category:Gauss Error Function
Category... | :$\map {\dfrac \d {\d x} } {\map \erf x} = \dfrac 2 {\sqrt \pi} e^{-x^2}$ | We have, by the definition of the [[Definition:Gauss Error Function|Gauss error function]]:
:$\ds \map \erf x = \frac 2 {\sqrt \pi} \int_0^x e^{-t^2} \rd t$
By [[Fundamental Theorem of Calculus/First Part/Corollary|Fundamental Theorem of Calculus (First Part): Corollary]], we therefore have:
:$\map {\dfrac \d {\d x}... | Derivative of Gauss Error Function | https://proofwiki.org/wiki/Derivative_of_Gauss_Error_Function | https://proofwiki.org/wiki/Derivative_of_Gauss_Error_Function | [
"Gauss Error Function",
"Derivatives"
] | [] | [
"Definition:Gauss Error Function",
"Fundamental Theorem of Calculus/First Part/Corollary",
"Category:Gauss Error Function",
"Category:Derivatives"
] |
proofwiki-16169 | Limit to Infinity of Gauss Error Function | :$\ds \lim_{x \mathop \to \infty} \map \erf x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map \erf x
| r = \lim_{x \mathop \to \infty} \paren {\frac 2 {\sqrt \pi} \int_0^x e^{-t^2} \rd t}
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = \frac 2 {\sqrt \pi} \int_0^\infty e^{-t^2} \rd t
| c = Multiple Rule for Limits of Real Functions... | :$\ds \lim_{x \mathop \to \infty} \map \erf x = 1$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map \erf x
| r = \lim_{x \mathop \to \infty} \paren {\frac 2 {\sqrt \pi} \int_0^x e^{-t^2} \rd t}
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = \frac 2 {\sqrt \pi} \int_0^\infty e^{-t^2} \rd t
| c = [[Multiple Rule for Limits of Real Functio... | Limit to Infinity of Gauss Error Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Gauss_Error_Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Gauss_Error_Function | [
"Gauss Error Function",
"Examples of Limits of Real Functions"
] | [] | [
"Combination Theorem for Limits of Functions/Real/Multiple Rule",
"Integral to Infinity of Exponential of -t^2"
] |
proofwiki-16170 | Dirichlet Beta Function in terms of Hurwitz Zeta Function | :$\map \beta s = \dfrac 1 {4^s} \paren {\map \zeta {s, \dfrac 1 4} - \map \zeta {s, \dfrac 3 4} }$ | {{begin-eqn}}
{{eqn | l = \map \beta s
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}
| c = {{Defof|Dirichlet Beta Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {4 n + 1}^s} - \sum_{n \mathop = 0}^\infty \frac 1 {\paren {4 n + 3}^s}
| c = splitting... | :$\map \beta s = \dfrac 1 {4^s} \paren {\map \zeta {s, \dfrac 1 4} - \map \zeta {s, \dfrac 3 4} }$ | {{begin-eqn}}
{{eqn | l = \map \beta s
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}
| c = {{Defof|Dirichlet Beta Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {4 n + 1}^s} - \sum_{n \mathop = 0}^\infty \frac 1 {\paren {4 n + 3}^s}
| c = splitting... | Dirichlet Beta Function in terms of Hurwitz Zeta Function | https://proofwiki.org/wiki/Dirichlet_Beta_Function_in_terms_of_Hurwitz_Zeta_Function | https://proofwiki.org/wiki/Dirichlet_Beta_Function_in_terms_of_Hurwitz_Zeta_Function | [
"Dirichlet Beta Function",
"Hurwitz Zeta Function"
] | [] | [
"Definition:Summation",
"Definition:Positive/Real Number",
"Definition:Negative/Real Number",
"Definition:Hurwitz Zeta Function",
"Category:Dirichlet Beta Function",
"Category:Hurwitz Zeta Function"
] |
proofwiki-16171 | Inverse of Right-Total Relation is Left-Total | :$\RR$ is right-total {{iff}} $\RR^{-1}$ is left-total. | === Sufficient Condition ===
Let $\RR$ be right-total.
Then by definition:
:$\forall t \in T: \exists s \in S: \tuple {s, t} \in \RR$
By definition of the inverse of $\RR$, it follows that:
:$\forall t \in T: \exists s \in S: \tuple {t, s} \in \RR^{-1}$
So by definition $\RR^{-1}$ is left-total.
{{qed|lemma}} | :$\RR$ is [[Definition:Right-Total Relation|right-total]] {{iff}} $\RR^{-1}$ is [[Definition:Left-Total Relation|left-total]]. | === Sufficient Condition ===
Let $\RR$ be [[Definition:Right-Total Relation|right-total]].
Then by definition:
:$\forall t \in T: \exists s \in S: \tuple {s, t} \in \RR$
By definition of the [[Definition:Inverse Relation|inverse]] of $\RR$, it follows that:
:$\forall t \in T: \exists s \in S: \tuple {t, s} \in \RR^{... | Inverse of Right-Total Relation is Left-Total | https://proofwiki.org/wiki/Inverse_of_Right-Total_Relation_is_Left-Total | https://proofwiki.org/wiki/Inverse_of_Right-Total_Relation_is_Left-Total | [
"Inverse Relations",
"Right-Total Relations",
"Left-Total Relations"
] | [
"Definition:Right-Total Relation",
"Definition:Left-Total Relation"
] | [
"Definition:Right-Total Relation",
"Definition:Inverse Relation",
"Definition:Left-Total Relation",
"Definition:Left-Total Relation",
"Definition:Right-Total Relation"
] |
proofwiki-16172 | Inverse of Left-Total Relation is Right-Total | :$\RR$ is left-total {{iff}} $\RR^{-1}$ is right-total. | From Inverse of Inverse Relation, the inverse of $\RR^{-1}$ is $\RR$.
From Inverse of Right-Total Relation is Left-Total:
:$\RR^{-1}$ is right-total {{iff}} $\RR$ is left-total.
Hence the result.
{{qed}}
Category:Inverse Relations
Category:Left-Total Relations
Category:Right-Total Relations
p7et19gof1t0i11wuqbu8ugwwstx... | :$\RR$ is [[Definition:Left-Total Relation|left-total]] {{iff}} $\RR^{-1}$ is [[Definition:Right-Total Relation|right-total]]. | From [[Inverse of Inverse Relation]], the [[Definition:Inverse Relation|inverse]] of $\RR^{-1}$ is $\RR$.
From [[Inverse of Right-Total Relation is Left-Total]]:
:$\RR^{-1}$ is [[Definition:Right-Total Relation|right-total]] {{iff}} $\RR$ is [[Definition:Left-Total Relation|left-total]].
Hence the result.
{{qed}}
[[... | Inverse of Left-Total Relation is Right-Total | https://proofwiki.org/wiki/Inverse_of_Left-Total_Relation_is_Right-Total | https://proofwiki.org/wiki/Inverse_of_Left-Total_Relation_is_Right-Total | [
"Inverse Relations",
"Left-Total Relations",
"Right-Total Relations"
] | [
"Definition:Left-Total Relation",
"Definition:Right-Total Relation"
] | [
"Inverse of Inverse Relation",
"Definition:Inverse Relation",
"Inverse of Right-Total Relation is Left-Total",
"Definition:Right-Total Relation",
"Definition:Left-Total Relation",
"Category:Inverse Relations",
"Category:Left-Total Relations",
"Category:Right-Total Relations"
] |
proofwiki-16173 | Inverse of Mapping is Right-Total Relation | Let $f$ be a mapping.
Then its inverse $f^{-1}$ is a right-total relation. | We have that $f$ is a mapping.
Hence $f$ is {{afortiori}} a left-total relation.
Then from Inverse of Left-Total Relation is Right-Total, $f^{-1}$ is right-total.
{{Qed}}
Category:Inverse Relations
Category:Inverse Mappings
Category:Right-Total Relations
rbbo0ijqmjzkhouyjoifa8s3qq0i868 | Let $f$ be a [[Definition:Mapping|mapping]].
Then its [[Definition:Inverse of Mapping|inverse]] $f^{-1}$ is a [[Definition:Right-Total Relation|right-total relation]]. | We have that $f$ is a [[Definition:Mapping|mapping]].
Hence $f$ is {{afortiori}} a [[Definition:Left-Total Relation|left-total relation]].
Then from [[Inverse of Left-Total Relation is Right-Total]], $f^{-1}$ is [[Definition:Right-Total Relation|right-total]].
{{Qed}}
[[Category:Inverse Relations]]
[[Category:Invers... | Inverse of Mapping is Right-Total Relation | https://proofwiki.org/wiki/Inverse_of_Mapping_is_Right-Total_Relation | https://proofwiki.org/wiki/Inverse_of_Mapping_is_Right-Total_Relation | [
"Inverse Relations",
"Inverse Mappings",
"Right-Total Relations"
] | [
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Right-Total Relation"
] | [
"Definition:Mapping",
"Definition:Left-Total Relation",
"Inverse of Left-Total Relation is Right-Total",
"Definition:Right-Total Relation",
"Category:Inverse Relations",
"Category:Inverse Mappings",
"Category:Right-Total Relations"
] |
proofwiki-16174 | Inverse of One-to-One Relation is One-to-One | The inverse of a one-to-one relation is a one-to-one relation. | Let $\RR$ be a one-to-one relation.
Let $\RR^{-1}$ denote its inverse
By definition, $\RR$ is a relation which is both many-to-one and one-to-many.
From Inverse of Many-to-One Relation is One-to-Many:
:$\RR^{-1}$ is both one-to-many and many-to-one.
Hence the result by definition of one-to-one relation.
{{qed}}
Categor... | The [[Definition:Inverse Relation|inverse]] of a [[Definition:One-to-One Relation|one-to-one relation]] is a [[Definition:One-to-One Relation|one-to-one relation]]. | Let $\RR$ be a [[Definition:One-to-One Relation|one-to-one relation]].
Let $\RR^{-1}$ denote its [[Definition:Inverse Relation|inverse]]
By definition, $\RR$ is a [[Definition:Relation|relation]] which is both [[Definition:Many-to-One Relation|many-to-one]] and [[Definition:One-to-Many Relation|one-to-many]].
From ... | Inverse of One-to-One Relation is One-to-One | https://proofwiki.org/wiki/Inverse_of_One-to-One_Relation_is_One-to-One | https://proofwiki.org/wiki/Inverse_of_One-to-One_Relation_is_One-to-One | [
"Inverse Relations"
] | [
"Definition:Inverse Relation",
"Definition:One-to-One Relation",
"Definition:One-to-One Relation"
] | [
"Definition:One-to-One Relation",
"Definition:Inverse Relation",
"Definition:Relation",
"Definition:Many-to-One Relation",
"Definition:One-to-Many Relation",
"Inverse of Many-to-One Relation is One-to-Many",
"Definition:One-to-Many Relation",
"Definition:Many-to-One Relation",
"Definition:One-to-One... |
proofwiki-16175 | Inverse of Injection is One-to-One Relation | Let $f$ be an injective mapping.
Then its inverse $f^{-1}$ is a one-to-one relation. | We are given that $f$ is an injective mapping.
Hence by definition $f$ is a one-to-one relation.
The result follows from from Inverse of One-to-One Relation is One-to-One.
{{Qed}} | Let $f$ be an [[Definition:Injection|injective mapping]].
Then its [[Definition:Inverse of Mapping|inverse]] $f^{-1}$ is a [[Definition:One-to-One Relation|one-to-one relation]]. | We are given that $f$ is an [[Definition:Injection|injective mapping]].
Hence by definition $f$ is a [[Definition:One-to-One Relation|one-to-one relation]].
The result follows from from [[Inverse of One-to-One Relation is One-to-One]].
{{Qed}} | Inverse of Injection is One-to-One Relation | https://proofwiki.org/wiki/Inverse_of_Injection_is_One-to-One_Relation | https://proofwiki.org/wiki/Inverse_of_Injection_is_One-to-One_Relation | [
"Inverse Relations",
"Injections",
"Inverse Mappings"
] | [
"Definition:Injection",
"Definition:Inverse of Mapping",
"Definition:One-to-One Relation"
] | [
"Definition:Injection",
"Definition:One-to-One Relation",
"Inverse of One-to-One Relation is One-to-One"
] |
proofwiki-16176 | Inverse of Surjection is Relation both Left-Total and Right-Total | Let $f$ be an surjective mapping.
Then its inverse $f^{-1}$ is a relation which is both left-total and right-total. | We are given that $f$ is a surjective mapping.
By Inverse of Mapping is Right-Total Relation, $f^{-1}$ is a right-total relation.
By definition of surjection, $f$ is itself a right-total relation.
From Inverse of Right-Total Relation is Left-Total, $f^{-1}$ is a left-total relation.
Hence the result.
{{Qed}}
Category:I... | Let $f$ be an [[Definition:Surjection|surjective mapping]].
Then its [[Definition:Inverse of Mapping|inverse]] $f^{-1}$ is a [[Definition:Relation|relation]] which is both [[Definition:Left-Total Relation|left-total]] and [[Definition:Right-Total Relation|right-total]]. | We are given that $f$ is a [[Definition:Surjection|surjective mapping]].
By [[Inverse of Mapping is Right-Total Relation]], $f^{-1}$ is a [[Definition:Right-Total Relation|right-total relation]].
By definition of [[Definition:Surjection|surjection]], $f$ is itself a [[Definition:Right-Total Relation|right-total relat... | Inverse of Surjection is Relation both Left-Total and Right-Total | https://proofwiki.org/wiki/Inverse_of_Surjection_is_Relation_both_Left-Total_and_Right-Total | https://proofwiki.org/wiki/Inverse_of_Surjection_is_Relation_both_Left-Total_and_Right-Total | [
"Inverse Relations",
"Surjections",
"Inverse Mappings",
"Right-Total Relations",
"Left-Total Relations"
] | [
"Definition:Surjection",
"Definition:Inverse of Mapping",
"Definition:Relation",
"Definition:Left-Total Relation",
"Definition:Right-Total Relation"
] | [
"Definition:Surjection",
"Inverse of Mapping is Right-Total Relation",
"Definition:Right-Total Relation",
"Definition:Surjection",
"Definition:Right-Total Relation",
"Inverse of Right-Total Relation is Left-Total",
"Definition:Left-Total Relation",
"Category:Inverse Relations",
"Category:Surjections... |
proofwiki-16177 | Inverse is Mapping implies Mapping is Injection | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping.
Then $f$ is an injection. | Let $f^{-1}: T \to S$ be a mapping.
Let $\map f {x_a} = y$ and $\map f {x_b} = y$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x_a, y}
| o = \in
| r = f
| c = {{Defof|Mapping}}
}}
{{eqn | lo= \land
| l = \tuple {x_b, y}
| o = \in
| r = f
| c =
}}
{{eqn | ll= \leadsto
| l = \... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let the [[Definition:Inverse of Mapping|inverse]] $f^{-1} \subseteq T \times S$ itself be a [[Definition:Mapping|mapping]].
Then $f$ is an [[Definition:Injection|injection]]. | Let $f^{-1}: T \to S$ be a [[Definition:Mapping|mapping]].
Let $\map f {x_a} = y$ and $\map f {x_b} = y$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x_a, y}
| o = \in
| r = f
| c = {{Defof|Mapping}}
}}
{{eqn | lo= \land
| l = \tuple {x_b, y}
| o = \in
| r = f
| c =
}}
{{eqn | ... | Inverse is Mapping implies Mapping is Injection/Proof 1 | https://proofwiki.org/wiki/Inverse_is_Mapping_implies_Mapping_is_Injection | https://proofwiki.org/wiki/Inverse_is_Mapping_implies_Mapping_is_Injection/Proof_1 | [
"Mapping is Injection and Surjection iff Inverse is Mapping"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Mapping",
"Definition:Injection"
] | [
"Definition:Mapping",
"Definition:Many-to-One Relation",
"Definition:Injection"
] |
proofwiki-16178 | Inverse is Mapping implies Mapping is Injection | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping.
Then $f$ is an injection. | Let $f^{-1}: T \to S$ be a mapping.
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
| c =
}}
{{eqn | ll= \leadsto
| l = \map {f^{-1} } {\map f x}
| r = \map {f^{-1} } {\map f y}
| c = as $f^{-1}$ is a mapping
}}
{{eqn | ll= \leadsto
| l = x
| r = y
| c = {{Defof|Invers... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let the [[Definition:Inverse of Mapping|inverse]] $f^{-1} \subseteq T \times S$ itself be a [[Definition:Mapping|mapping]].
Then $f$ is an [[Definition:Injection|injection]]. | Let $f^{-1}: T \to S$ be a [[Definition:Mapping|mapping]].
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
| c =
}}
{{eqn | ll= \leadsto
| l = \map {f^{-1} } {\map f x}
| r = \map {f^{-1} } {\map f y}
| c = as $f^{-1}$ is a [[Definition:Mapping|mapping]]
}}
{{eqn | ll= \leadsto
|... | Inverse is Mapping implies Mapping is Injection/Proof 2 | https://proofwiki.org/wiki/Inverse_is_Mapping_implies_Mapping_is_Injection | https://proofwiki.org/wiki/Inverse_is_Mapping_implies_Mapping_is_Injection/Proof_2 | [
"Mapping is Injection and Surjection iff Inverse is Mapping"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Mapping",
"Definition:Injection"
] | [
"Definition:Mapping",
"Definition:Mapping",
"Definition:Injection"
] |
proofwiki-16179 | Inverse is Mapping implies Mapping is Surjection | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping.
Then $f$ is a surjection. | Let $f^{-1}: T \to S$ be a mapping.
We have:
{{begin-eqn}}
{{eqn | l = t
| o = \in
| r = T
| c =
}}
{{eqn | ll= \leadsto
| l = \map {f^{-1} } t
| o = \in
| r = S
| c = as $f^{-1}$ is a mapping
}}
{{eqn | ll= \leadsto
| l = \map f {\map {f^{-1} } t}
| r = t
| ... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let the [[Definition:Inverse of Mapping|inverse]] $f^{-1} \subseteq T \times S$ itself be a [[Definition:Mapping|mapping]].
Then $f$ is a [[Definition:Surjection|surjection]]. | Let $f^{-1}: T \to S$ be a [[Definition:Mapping|mapping]].
We have:
{{begin-eqn}}
{{eqn | l = t
| o = \in
| r = T
| c =
}}
{{eqn | ll= \leadsto
| l = \map {f^{-1} } t
| o = \in
| r = S
| c = as $f^{-1}$ is a [[Definition:Mapping|mapping]]
}}
{{eqn | ll= \leadsto
| l =... | Inverse is Mapping implies Mapping is Surjection/Proof 2 | https://proofwiki.org/wiki/Inverse_is_Mapping_implies_Mapping_is_Surjection | https://proofwiki.org/wiki/Inverse_is_Mapping_implies_Mapping_is_Surjection/Proof_2 | [
"Mapping is Injection and Surjection iff Inverse is Mapping"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Mapping",
"Definition:Surjection"
] | [
"Definition:Mapping",
"Definition:Mapping",
"Definition:Surjection"
] |
proofwiki-16180 | Derivative Function is not Invertible | Let $\Bbb I = \closedint a b$ be a closed interval on the set of real numbers $\R$ such that $a < b$.
Let $A$ denote the set of all continuous real functions $f: \Bbb I \to \R$.
Let $B \subseteq A$ denote the set of all functions differentiable on $\Bbb I$ whose derivative is continuous on $\Bbb I$.
Let $d: B \to A$ de... | By definition, $d$ is invertible {{iff}} $d$ is a bijection.
It is sufficient to demonstrate that $d$ is not an injection.
Hence a fortiori $d$ is shown to not be a bijection.
Consider a differentiable function $f \in B$.
Then consider the function $g \in B$ defined as:
:$\forall x \in \Bbb I: \map g x = \map f x + C$
... | Let $\Bbb I = \closedint a b$ be a [[Definition:Closed Real Interval|closed interval]] on the [[Definition:Real Number|set of real numbers]] $\R$ such that $a < b$.
Let $A$ denote the [[Definition:Set|set]] of all [[Definition:Continuous Real Function|continuous real functions]] $f: \Bbb I \to \R$.
Let $B \subseteq A... | By definition, $d$ is [[Definition:Invertible Mapping|invertible]] {{iff}} $d$ is a [[Definition:Bijection|bijection]].
It is sufficient to demonstrate that $d$ is not an [[Definition:Injection|injection]].
Hence [[Definition:A Fortiori|a fortiori]] $d$ is shown to not be a [[Definition:Bijection|bijection]].
Consi... | Derivative Function is not Invertible | https://proofwiki.org/wiki/Derivative_Function_is_not_Invertible | https://proofwiki.org/wiki/Derivative_Function_is_not_Invertible | [
"Differential Calculus"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Number",
"Definition:Set",
"Definition:Continuous Real Function",
"Definition:Set",
"Definition:Differentiable Mapping/Real Function/Interval/Closed Interval",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Continuous Real ... | [
"Definition:Inverse Mapping",
"Definition:Bijection",
"Definition:Injection",
"Definition:A Fortiori",
"Definition:Bijection",
"Definition:Differentiable Mapping/Real Function/Interval/Closed Interval",
"Definition:Real Function",
"Definition:Constant",
"Derivative of Function plus Constant",
"Def... |
proofwiki-16181 | Equivalence of Definitions of Convergent Sequence in Metric Space | Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $\sequence {x_k}$ be a sequence in $A$.
{{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}} | === Definition 1 iff Definition 2 ===
{{:Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2}}{{qed|lemma}} | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]] or a [[Definition:Pseudometric Space|pseudometric space]].
Let $\sequence {x_k}$ be a [[Definition:Sequence|sequence in $A$]].
{{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}} | === [[Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2|Definition 1 iff Definition 2]] ===
{{:Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2}}{{qed|lemma}} | Equivalence of Definitions of Convergent Sequence in Metric Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space | [
"Metric Spaces",
"Convergence",
"Sequences",
"Convergent Sequences (Metric Space)",
"Equivalence of Definitions of Convergent Sequence in Metric Space"
] | [
"Definition:Metric Space",
"Definition:Pseudometric/Pseudometric Space",
"Definition:Sequence"
] | [
"Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2"
] |
proofwiki-16182 | Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 2 implies Definition 4 | Let $\sequence {x_k}$ satisfy:
:$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$
where $\map {B_\epsilon} l$ is the open $\epsilon$-ball of $l$. | Let a fixed $\epsilon \in \R{>0}$ be selected.
Then:
:$\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$
Hence the only $x_k$ that cannot be in the open $\epsilon$-ball $\map {B_\epsilon} l$ of $l$ are those for which $n \le N$.
There are finitely many of these. | Let $\sequence {x_k}$ satisfy:
:$\forall \epsilon > 0: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$
where $\map {B_\epsilon} l$ is the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] of $l$. | Let a fixed $\epsilon \in \R{>0}$ be selected.
Then:
:$\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in \map {B_\epsilon} l$
Hence the only $x_k$ that cannot be in the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] $\map {B_\epsilon} l$ of $l$ are those for which $n \le N$.
There are [[... | Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 2 implies Definition 4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space/Definition_2_implies_Definition_4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space/Definition_2_implies_Definition_4 | [
"Equivalence of Definitions of Convergent Sequence in Metric Space"
] | [
"Definition:Open Ball"
] | [
"Definition:Open Ball",
"Definition:Finite Set"
] |
proofwiki-16183 | Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 4 implies Definition 2 | Let $\sequence {x_k}$ satisfy:
:for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $p_n$. | Let $\map {B_\epsilon} l$ be any open $\epsilon$-ball of $l$.
Let $A = \set {n: x_n \notin \map {B_\epsilon} l}$.
By assumption $A$ is finite.
From Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements, any finite subset of $\N$ has a maximum.
Let $N$ be the maximum of $A$.
Then for every $n... | Let $\sequence {x_k}$ satisfy:
:for every $\epsilon \in \R{>0}$, the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] about $l$ contains all but [[Definition:Finite Set|finitely many]] of the $p_n$. | Let $\map {B_\epsilon} l$ be any [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] of $l$.
Let $A = \set {n: x_n \notin \map {B_\epsilon} l}$.
By assumption $A$ is [[Definition:Finite Set|finite]].
From [[Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements]], any [[Definitio... | Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 4 implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space/Definition_4_implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space/Definition_4_implies_Definition_2 | [
"Equivalence of Definitions of Convergent Sequence in Metric Space"
] | [
"Definition:Open Ball",
"Definition:Finite Set"
] | [
"Definition:Open Ball",
"Definition:Finite Set",
"Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Greatest Element",
"Definition:Greatest Element",
"Definition:Open Ball"
] |
proofwiki-16184 | Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x | :$\ds \int_0^{2 \pi} \frac {\d x} {a + b \cos x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$ | Let $C$ be the unit open disk in the complex plane centred at $0$.
The boundary of $C$, $\partial C$, can be parameterized by:
:$\map z x = e^{i x}$
for $0 \le x \le 2 \pi$.
We have:
{{begin-eqn}}
{{eqn | l = \int_0^{2 \pi} \frac {\d x} {a + b \cos x}
| r = \int_0^{2 \pi} \frac {\d x} {a + \frac b 2 \paren {... | :$\ds \int_0^{2 \pi} \frac {\d x} {a + b \cos x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$ | Let $C$ be the [[Definition:Unit Disk|unit]] [[Definition:Open Complex Disk|open disk]] in the [[Definition:Complex Plane|complex plane]] centred at $0$.
The [[Definition:Boundary|boundary]] of $C$, $\partial C$, can be [[Definition:Parameterization|parameterized]] by:
:$\map z x = e^{i x}$
for $0 \le x \le 2 \pi$... | Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_a_plus_b_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_a_plus_b_Cosine_x/Proof_1 | [
"Definite Integrals involving Cosine Function",
"Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x"
] | [] | [
"Definition:Unit Disk",
"Definition:Complex Disk/Open",
"Definition:Complex Number/Complex Plane",
"Definition:Boundary",
"Definition:Parameterization",
"Euler's Cosine Identity",
"Derivative of Exponential Function",
"Completing the Square",
"Definition:Integration/Integrand",
"Definition:Isolate... |
proofwiki-16185 | Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x | :$\ds \int_0^{2 \pi} \frac {\d x} {a + b \cos x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$ | {{begin-eqn}}
{{eqn | l = \int_0^{2 \pi} \frac {\d x} {a + b \cos x}
| r = \int_0^\pi \frac {\d x} {a + b \cos x} + \int_\pi^{2 \pi} \frac {\d x} {a + b \cos x}
| c = Sum of Integrals on Adjacent Intervals for Integrable Functions
}}
{{eqn | r = \intlimits {\frac 2 {\sqrt {a^2 - b^2} } \map \arctan {\sqrt {\frac {a -... | :$\ds \int_0^{2 \pi} \frac {\d x} {a + b \cos x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$ | {{begin-eqn}}
{{eqn | l = \int_0^{2 \pi} \frac {\d x} {a + b \cos x}
| r = \int_0^\pi \frac {\d x} {a + b \cos x} + \int_\pi^{2 \pi} \frac {\d x} {a + b \cos x}
| c = [[Sum of Integrals on Adjacent Intervals for Integrable Functions]]
}}
{{eqn | r = \intlimits {\frac 2 {\sqrt {a^2 - b^2} } \map \arctan {\sqrt {\frac ... | Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_a_plus_b_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_a_plus_b_Cosine_x/Proof_2 | [
"Definite Integrals involving Cosine Function",
"Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x"
] | [] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Primitive of Reciprocal of p plus q by Cosine of a x",
"Tangent Function is Periodic on Reals",
"Tangent Function Tends to Positive and Negative Infinity",
"Limit to Positive and Negative Infinity of Arctangent Function"
] |
proofwiki-16186 | Definite Integral from 0 to 2 Pi of Reciprocal of Square of a plus b Cosine x | :$\ds \int_0^{2 \pi} \frac {\d x} {\paren {a + b \cos x}^2} = \frac {2 \pi a} {\paren {a^2 - b^2}^{3/2} }$ | From Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {a + b \cos x}$, we have:
:$\ds \int_0^{2 \pi} \frac {\d x} {a + b \cos x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$
We have:
{{begin-eqn}}
{{eqn | l = \frac \partial {\partial a} \int_0^{2 \pi} \frac {\d x} {a + b \cos x}
| r = \int_0^{2 \pi} \frac \partial {\part... | :$\ds \int_0^{2 \pi} \frac {\d x} {\paren {a + b \cos x}^2} = \frac {2 \pi a} {\paren {a^2 - b^2}^{3/2} }$ | From [[Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x|Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {a + b \cos x}$]], we have:
:$\ds \int_0^{2 \pi} \frac {\d x} {a + b \cos x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$
We have:
{{begin-eqn}}
{{eqn | l = \frac \partial {\partial a} \int_0^{2 \... | Definite Integral from 0 to 2 Pi of Reciprocal of Square of a plus b Cosine x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_Square_of_a_plus_b_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_Square_of_a_plus_b_Cosine_x/Proof_1 | [
"Definite Integrals involving Cosine Function",
"Definite Integral from 0 to 2 Pi of Reciprocal of Square of a plus b Cosine x"
] | [] | [
"Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x",
"Definite Integral of Partial Derivative",
"Quotient Rule for Derivatives",
"Quotient Rule for Derivatives"
] |
proofwiki-16187 | Derivative of Function plus Constant | Let $f$ be a real function which is differentiable on $\R$.
Let $c \in \R$ be a constant.
Then:
:$\map {\dfrac \d {\d x} } {\map f x + c} = \map {\dfrac \d {\d x} } {\map f x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\map f x + c}
| r = \map {\dfrac \d {\d x} } {\map f x} + \map f x \, c
| c = Sum Rule for Derivatives
}}
{{eqn | r = \map {\dfrac \d {\d x} } {\map f x} + 0
| c = Derivative of Constant
}}
{{eqn | r = \map {\dfrac \d {\d x} } {\map f x}
| c =
... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Differentiable on Interval|differentiable]] on $\R$.
Let $c \in \R$ be a [[Definition:Constant|constant]].
Then:
:$\map {\dfrac \d {\d x} } {\map f x + c} = \map {\dfrac \d {\d x} } {\map f x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\map f x + c}
| r = \map {\dfrac \d {\d x} } {\map f x} + \map f x \, c
| c = [[Sum Rule for Derivatives]]
}}
{{eqn | r = \map {\dfrac \d {\d x} } {\map f x} + 0
| c = [[Derivative of Constant]]
}}
{{eqn | r = \map {\dfrac \d {\d x} } {\map f x}
... | Derivative of Function plus Constant | https://proofwiki.org/wiki/Derivative_of_Function_plus_Constant | https://proofwiki.org/wiki/Derivative_of_Function_plus_Constant | [
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Constant"
] | [
"Sum Rule for Derivatives",
"Derivative of Constant",
"Category:Differential Calculus"
] |
proofwiki-16188 | Map from Set of Continuous Functions on Interval to Set of their Integrals | Let $\Bbb I = \closedint a b$ be a closed interval on the set of real numbers $\R$ such that $a < b$.
Let $A$ denote the set of all continuous real functions $f: \Bbb I \to \R$.
Let $B \subseteq A$ denote the set of all functions differentiable on $\Bbb I$ whose derivative is continuous on $\Bbb I$.
Let $C \subseteq B$... | Let $f \in A$ be an arbitrary continuous real function $f: \Bbb I \to \R$.
From Continuous Real Function is Darboux Integrable, $\map {\paren {\map h f} } x$ exists and is continuous on $\Bbb I$.
Let $x = a$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\map h f} } x
| r = \int_a^x \map f t \rd t
|... | Let $\Bbb I = \closedint a b$ be a [[Definition:Closed Real Interval|closed interval]] on the [[Definition:Real Number|set of real numbers]] $\R$ such that $a < b$.
Let $A$ denote the [[Definition:Set|set]] of all [[Definition:Continuous Real Function|continuous real functions]] $f: \Bbb I \to \R$.
Let $B \subseteq A... | Let $f \in A$ be an arbitrary [[Definition:Continuous Real Function|continuous real function]] $f: \Bbb I \to \R$.
From [[Continuous Real Function is Darboux Integrable]], $\map {\paren {\map h f} } x$ exists and is [[Definition:Continuous Real Function|continuous]] on $\Bbb I$.
Let $x = a$.
Then we have:
{{begin-... | Map from Set of Continuous Functions on Interval to Set of their Integrals | https://proofwiki.org/wiki/Map_from_Set_of_Continuous_Functions_on_Interval_to_Set_of_their_Integrals | https://proofwiki.org/wiki/Map_from_Set_of_Continuous_Functions_on_Interval_to_Set_of_their_Integrals | [
"Integral Calculus",
"Continuous Real Functions"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Number",
"Definition:Set",
"Definition:Continuous Real Function",
"Definition:Set",
"Definition:Differentiable Mapping/Real Function/Interval/Closed Interval",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Continuous Real ... | [
"Definition:Continuous Real Function",
"Continuous Real Function is Darboux Integrable",
"Definition:Continuous Real Function",
"Definite Integral on Zero Interval",
"Definition:Continuous Real Function",
"Definition:Primitive",
"Fundamental Theorem of Calculus",
"Definition:Differentiable Mapping/Rea... |
proofwiki-16189 | Definite Integral from 0 to Half Pi of Reciprocal of One plus Power of Tan x | :$\ds \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x} = \frac \pi 4$ | {{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x}
| r = \int_0^{\pi/2} \frac {\cos^m x} {\cos^m x + \sin^m x} \rd x
| c = multiplying by $\dfrac {\cos^m x} {\cos^m x}$
}}
{{eqn | r = \int_0^{\pi/2} \frac {\map {\cos^m} {\frac \pi 2 - x} } {\map {\cos^m} {\frac \pi 2 - x} + \map {\sin^m} {\frac \pi... | :$\ds \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x} = \frac \pi 4$ | {{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x}
| r = \int_0^{\pi/2} \frac {\cos^m x} {\cos^m x + \sin^m x} \rd x
| c = multiplying by $\dfrac {\cos^m x} {\cos^m x}$
}}
{{eqn | r = \int_0^{\pi/2} \frac {\map {\cos^m} {\frac \pi 2 - x} } {\map {\cos^m} {\frac \pi 2 - x} + \map {\sin^m} {\frac \pi... | Definite Integral from 0 to Half Pi of Reciprocal of One plus Power of Tan x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Reciprocal_of_One_plus_Power_of_Tan_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Reciprocal_of_One_plus_Power_of_Tan_x | [
"Definite Integrals involving Tangent Function"
] | [] | [
"Integral between Limits is Independent of Direction",
"Cosine of Complement equals Sine",
"Sine of Complement equals Cosine",
"Primitive of Constant"
] |
proofwiki-16190 | Möbius Transformation is Bijection | Let $a, b, c, d \in \C$ be complex numbers.
Let $f: \overline \C \to \overline \C$ be the Möbius transformation:
:<nowiki>$\map f z = \begin {cases} \dfrac {a z + b} {c z + d} & : z \ne -\dfrac d c \\
\infty & : z = -\dfrac d c \\
\dfrac a c & : z = \infty \\
\infty & : z = \infty \text { and } c = 0 \end {cases}$</now... | We demonstrate that $f$ is injective {{iff}} $b c - a d \ne 0$.
{{begin-eqn}}
{{eqn | l = \map f {z_1}
| r = \map f {z_2}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac {a z_1 + b} {c z_1 + d}
| r = \dfrac {a z_2 + b} {c z_2 + d}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \paren... | Let $a, b, c, d \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $f: \overline \C \to \overline \C$ be the [[Definition:Möbius Transformation|Möbius transformation]]:
:<nowiki>$\map f z = \begin {cases} \dfrac {a z + b} {c z + d} & : z \ne -\dfrac d c \\
\infty & : z = -\dfrac d c \\
\dfrac a c & : z = \... | We demonstrate that $f$ is [[Definition:Injection|injective]] {{iff}} $b c - a d \ne 0$.
{{begin-eqn}}
{{eqn | l = \map f {z_1}
| r = \map f {z_2}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac {a z_1 + b} {c z_1 + d}
| r = \dfrac {a z_2 + b} {c z_2 + d}
| c =
}}
{{eqn | ll= \leadsto... | Möbius Transformation is Bijection | https://proofwiki.org/wiki/Möbius_Transformation_is_Bijection | https://proofwiki.org/wiki/Möbius_Transformation_is_Bijection | [
"Möbius Transformation is Bijection",
"Möbius Transformations",
"Examples of Bijections"
] | [
"Definition:Complex Number",
"Definition:Möbius Transformation",
"Definition:Extended Complex Plane",
"Definition:Bijection"
] | [
"Definition:Injection",
"Definition:Injection",
"Definition:Inverse of Mapping",
"Inverse Element of Injection",
"Definition:Möbius Transformation",
"Definition:Inverse of Mapping",
"Definition:Möbius Transformation",
"Definition:Möbius Transformation",
"Definition:Injection",
"Injection is Biject... |
proofwiki-16191 | Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x | :$\ds \int_0^1 x^m \paren {\ln x}^n \rd x = \frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {m + 1}^{n + 1} }$ | Let:
:$x = \map \exp {-\dfrac u {m + 1} }$
Then, by Derivative of Exponential Function:
:$\dfrac {\d x} {\d u} = -\dfrac 1 {m + 1} \map \exp {-\dfrac u {m + 1} }$
We have by Exponential of Zero:
:as $x \to 1$, $u \to 0$
We also have, by Exponential Tends to Zero and Infinity:
:as $x \to 0$, $u \to \infty$
So:
{{be... | :$\ds \int_0^1 x^m \paren {\ln x}^n \rd x = \frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {m + 1}^{n + 1} }$ | Let:
:$x = \map \exp {-\dfrac u {m + 1} }$
Then, by [[Derivative of Exponential Function]]:
:$\dfrac {\d x} {\d u} = -\dfrac 1 {m + 1} \map \exp {-\dfrac u {m + 1} }$
We have by [[Exponential of Zero]]:
:as $x \to 1$, $u \to 0$
We also have, by [[Exponential Tends to Zero and Infinity]]:
:as $x \to 0$, $u \t... | Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Power_of_x_by_Power_of_Logarithm_of_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Power_of_x_by_Power_of_Logarithm_of_x/Proof_1 | [
"Definite Integrals involving Logarithm Function",
"Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x"
] | [] | [
"Derivative of Exponential Function",
"Exponential of Zero",
"Exponential Tends to Zero and Infinity",
"Integration by Substitution",
"Reversal of Limits of Definite Integral",
"Exponential of Sum",
"Exponent Combination Laws/Power of Power"
] |
proofwiki-16192 | Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x | :$\ds \int_0^1 x^m \paren {\ln x}^n \rd x = \frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {m + 1}^{n + 1} }$ | From Primitive of Power, we have:
:$\ds \int_0^1 x^m \rd x = \frac 1 {m + 1}$
We have:
{{begin-eqn}}
{{eqn | l = \frac {\d^n} {\d m^n} \int_0^1 x^m \rd x
| r = \int_0^1 \frac {\partial^n} {\partial m^n} x^m \rd x
| c = Definite Integral of Partial Derivative
}}
{{eqn | r = \int_0^1 x^m \paren {\ln x}^n \r... | :$\ds \int_0^1 x^m \paren {\ln x}^n \rd x = \frac {\paren {-1}^n \map \Gamma {n + 1} } {\paren {m + 1}^{n + 1} }$ | From [[Primitive of Power]], we have:
:$\ds \int_0^1 x^m \rd x = \frac 1 {m + 1}$
We have:
{{begin-eqn}}
{{eqn | l = \frac {\d^n} {\d m^n} \int_0^1 x^m \rd x
| r = \int_0^1 \frac {\partial^n} {\partial m^n} x^m \rd x
| c = [[Definite Integral of Partial Derivative]]
}}
{{eqn | r = \int_0^1 x^m \paren {... | Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Power_of_x_by_Power_of_Logarithm_of_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Power_of_x_by_Power_of_Logarithm_of_x/Proof_2 | [
"Definite Integrals involving Logarithm Function",
"Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x"
] | [] | [
"Primitive of Power",
"Definite Integral of Partial Derivative",
"Derivative of General Exponential Function",
"Derivative of Natural Logarithm Function",
"Nth Derivative of Natural Logarithm"
] |
proofwiki-16193 | Definite Integral to Infinity of Exponential of -a x by Sine of b x | :$\ds \int_0^\infty e^{-a x} \sin b x \rd x = \frac b {a^2 + b^2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-a x} \sin b x \rd x
| r = \intlimits {\frac {e^{-a x} \paren {-a \sin b x - b \cos b x} } {a^2 + b^2} } 0 \infty
| c = Primitive of $e^{a x} \sin b x$
}}
{{eqn | r = -\lim_{x \mathop \to \infty} \paren {\frac {e^{-a x} \paren {a \sin b x + b \cos b x} } {a^2 + b^2} } + \fra... | :$\ds \int_0^\infty e^{-a x} \sin b x \rd x = \frac b {a^2 + b^2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-a x} \sin b x \rd x
| r = \intlimits {\frac {e^{-a x} \paren {-a \sin b x - b \cos b x} } {a^2 + b^2} } 0 \infty
| c = [[Primitive of Exponential of a x by Sine of b x|Primitive of $e^{a x} \sin b x$]]
}}
{{eqn | r = -\lim_{x \mathop \to \infty} \paren {\frac {e^{-a x} \par... | Definite Integral to Infinity of Exponential of -a x by Sine of b x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_by_Sine_of_b_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_by_Sine_of_b_x | [
"Definite Integrals involving Sine Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Primitive of Exponential of a x by Sine of b x",
"Exponential of Zero",
"Sine of Zero is Zero",
"Cosine of Zero is One",
"Linear Combination of Sine and Cosine",
"Exponential Tends to Zero and Infinity",
"Squeeze Theorem"
] |
proofwiki-16194 | Sum of Exponential of i k x | :$\ds \sum_{k \mathop = 0}^n \map \exp {i k x} = \paren {i \sin \frac {n x} 2 + \cos \frac {n x} 2} \frac {\map \sin {\frac {\paren {n + 1} x} 2} } {\sin \frac x 2}$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \map \exp {i k x}
| r = \frac {\map \exp {i x \paren {n + 1} } - 1} {\map \exp {i x} - 1}
| c = Sum of Geometric Sequence
}}
{{eqn | r = \frac {\map \exp {\frac {i x \paren {n + 1} } 2} \paren {\map \exp {\frac {i x \paren {n + 1} } 2} - \map \exp {\frac {-i... | :$\ds \sum_{k \mathop = 0}^n \map \exp {i k x} = \paren {i \sin \frac {n x} 2 + \cos \frac {n x} 2} \frac {\map \sin {\frac {\paren {n + 1} x} 2} } {\sin \frac x 2}$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \map \exp {i k x}
| r = \frac {\map \exp {i x \paren {n + 1} } - 1} {\map \exp {i x} - 1}
| c = [[Sum of Geometric Sequence]]
}}
{{eqn | r = \frac {\map \exp {\frac {i x \paren {n + 1} } 2} \paren {\map \exp {\frac {i x \paren {n + 1} } 2} - \map \exp {\frac... | Sum of Exponential of i k x | https://proofwiki.org/wiki/Sum_of_Exponential_of_i_k_x | https://proofwiki.org/wiki/Sum_of_Exponential_of_i_k_x | [
"Exponential Function"
] | [] | [
"Sum of Geometric Sequence",
"Euler's Sine Identity",
"Euler's Formula",
"Category:Exponential Function"
] |
proofwiki-16195 | Möbius Transformation is Bijection/Restriction to Reals | Let $a, b, c, d \in \R$ be real numbers.
Let $f: \R^* \to \R^*$ be the Möbius transformation restricted to the real numbers:
:<nowiki>$\map f x = \begin {cases} \dfrac {a x + b} {c x + d} & : x \ne -\dfrac d c \\
\infty & : x = -\dfrac d c \\
\dfrac a c & : x = \infty \\
\infty & : x = \infty \text { and } c = 0 \end {... | First we note that as Real Addition is Closed and Real Multiplication is Closed:
:$\Dom {\R^*} \subseteq \R^*$
{{explain|What is $\Dom {\R^*}$?}}
Recall from Möbius Transformation is Bijection that the Möbius transformation on the extended complex plane is a bijection {{iff}} $a c - b d \ne 0$.
From Restriction of Inje... | Let $a, b, c, d \in \R$ be [[Definition:Real Number|real numbers]].
Let $f: \R^* \to \R^*$ be the [[Definition:Möbius Transformation on Real Numbers|Möbius transformation]] [[Definition:Restriction of Mapping|restricted]] to the [[Definition:Real Number|real numbers]]:
:<nowiki>$\map f x = \begin {cases} \dfrac {a x ... | First we note that as [[Real Addition is Closed]] and [[Real Multiplication is Closed]]:
:$\Dom {\R^*} \subseteq \R^*$
{{explain|What is $\Dom {\R^*}$?}}
Recall from [[Möbius Transformation is Bijection]] that the [[Definition:Möbius Transformation|Möbius transformation]] on the [[Definition:Extended Complex Plane|ext... | Möbius Transformation is Bijection/Restriction to Reals | https://proofwiki.org/wiki/Möbius_Transformation_is_Bijection/Restriction_to_Reals | https://proofwiki.org/wiki/Möbius_Transformation_is_Bijection/Restriction_to_Reals | [
"Möbius Transformation is Bijection"
] | [
"Definition:Real Number",
"Definition:Möbius Transformation/Real Numbers",
"Definition:Restriction/Mapping",
"Definition:Real Number",
"Definition:Bijection"
] | [
"Real Addition is Closed",
"Real Multiplication is Closed",
"Möbius Transformation is Bijection",
"Definition:Möbius Transformation",
"Definition:Extended Complex Plane",
"Definition:Bijection",
"Restriction of Injection is Injection",
"Definition:Injection",
"Definition:Inverse Mapping",
"Definit... |
proofwiki-16196 | Extension of Extension of Mapping is Extension | Let $A, B, C, S$ be sets such that $A \subseteq B \subseteq C$.
Let $f: A \to S$, $g: B \to S$ and $h: C \to S$ be mappings such that:
:$g$ is an extension of $f$ to $B$
:$h$ is an extension of $g$ to $C$.
Then $h$ is an extension of $f$ to $C$. | By definition of extension:
:$\forall x \in A: \map f x = \map g x$
and:
:$\forall x \in B: \map g x = \map h x$
and so:
:$\forall x \in A: \map g x = \map h x$
from which it follows that:
:$\forall x \in A: \map f x = \map h x$
{{qed}} | Let $A, B, C, S$ be [[Definition:Set|sets]] such that $A \subseteq B \subseteq C$.
Let $f: A \to S$, $g: B \to S$ and $h: C \to S$ be [[Definition:Mapping|mappings]] such that:
:$g$ is an [[Definition:Extension of Mapping|extension]] of $f$ to $B$
:$h$ is an [[Definition:Extension of Mapping|extension]] of $g$ to $C$... | By definition of [[Definition:Extension of Mapping|extension]]:
:$\forall x \in A: \map f x = \map g x$
and:
:$\forall x \in B: \map g x = \map h x$
and so:
:$\forall x \in A: \map g x = \map h x$
from which it follows that:
:$\forall x \in A: \map f x = \map h x$
{{qed}} | Extension of Extension of Mapping is Extension | https://proofwiki.org/wiki/Extension_of_Extension_of_Mapping_is_Extension | https://proofwiki.org/wiki/Extension_of_Extension_of_Mapping_is_Extension | [
"Restrictions"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Extension of Mapping",
"Definition:Extension of Mapping",
"Definition:Extension of Mapping"
] | [
"Definition:Extension of Mapping"
] |
proofwiki-16197 | Indexed Cartesian Space is Set of all Mappings | Let $I$ be an indexing set.
Let $\ds \prod_{i \mathop \in I} S$ denote the cartesian space of $S$ indexed by $I$.
Then $\ds \prod_{i \mathop \in I} S$ is the set of all mappings from $I$ to $S$, and hence the notation:
:$S^I := \ds \prod_{i \mathop \in I} S$ | Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.
By definition of the Cartesian product of $\family {S_i}_{i \mathop \in I}$:
:$(1): \quad \ds \prod_{i \mathop \in I} S_i := \set {f: \paren {f: I \to \bigcup_{i \mathop \in I} S_i} \land \paren {\forall i \in I: \paren {\map f i \in S_i} } }$
where $f... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\ds \prod_{i \mathop \in I} S$ denote the [[Definition:Indexed Cartesian Space|cartesian space of $S$ indexed by $I$]].
Then $\ds \prod_{i \mathop \in I} S$ is the [[Definition:Set of All Mappings|set of all mappings]] from $I$ to $S$, and hence the notati... | Let $\family {S_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family of Sets|indexed family of sets]].
By definition of the [[Definition:Cartesian Product of Family/Definition 2|Cartesian product of $\family {S_i}_{i \mathop \in I}$]]:
:$(1): \quad \ds \prod_{i \mathop \in I} S_i := \set {f: \paren {f: I \to \big... | Indexed Cartesian Space is Set of all Mappings | https://proofwiki.org/wiki/Indexed_Cartesian_Space_is_Set_of_all_Mappings | https://proofwiki.org/wiki/Indexed_Cartesian_Space_is_Set_of_all_Mappings | [
"Cartesian Product",
"Indexed Families"
] | [
"Definition:Indexing Set",
"Definition:Cartesian Product/Cartesian Space/Family of Sets",
"Definition:Set of All Mappings"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Cartesian Product of Family/Definition 2",
"Definition:Mapping",
"Set Union is Idempotent"
] |
proofwiki-16198 | Definite Integral from 0 to 1 of Logarithm of x over One plus x | :$\ds \int_0^1 \frac {\ln x} {1 + x} \rd x = -\frac {\pi^2} {12}$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\ln x} {1 + x} \rd x
| r = \int_0^1 \frac {\ln x} {1 - \paren {-x} } \rd x
}}
{{eqn | r = \int_0^1 \ln x \paren {\sum_{n \mathop = 0}^\infty \paren {-x}^n} \rd x
| c = Sum of Infinite Geometric Sequence
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \int_0^1 x^n \ln... | :$\ds \int_0^1 \frac {\ln x} {1 + x} \rd x = -\frac {\pi^2} {12}$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\ln x} {1 + x} \rd x
| r = \int_0^1 \frac {\ln x} {1 - \paren {-x} } \rd x
}}
{{eqn | r = \int_0^1 \ln x \paren {\sum_{n \mathop = 0}^\infty \paren {-x}^n} \rd x
| c = [[Sum of Infinite Geometric Sequence]]
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \int_0^1 x^n... | Definite Integral from 0 to 1 of Logarithm of x over One plus x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_over_One_plus_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_over_One_plus_x | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x",
"Gamma Function Extends Factorial",
"Sum of Reciprocals of Squares Alternating in Sign"
] |
proofwiki-16199 | Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2 | Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $\sequence {x_k}$ be a sequence in $A$.
{{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}} | By definition of an open ball:
:$\forall n \in \N: \map d {x_n, l} < \epsilon \iff x_n \in \map {B_\epsilon} l$
The result follows. | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]] or a [[Definition:Pseudometric Space|pseudometric space]].
Let $\sequence {x_k}$ be a [[Definition:Sequence|sequence in $A$]].
{{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}} | By definition of an [[Definition:Open Ball of Metric Space|open ball]]:
:$\forall n \in \N: \map d {x_n, l} < \epsilon \iff x_n \in \map {B_\epsilon} l$
The result follows. | Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space/Definition_1_iff_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space/Definition_1_iff_Definition_2 | [
"Equivalence of Definitions of Convergent Sequence in Metric Space"
] | [
"Definition:Metric Space",
"Definition:Pseudometric/Pseudometric Space",
"Definition:Sequence"
] | [
"Definition:Open Ball"
] |
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