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-17000 | Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1 | :$\ds \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n \alpha} 2} + i \map \sin {\theta + \frac {n \alpha} 2} } \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} }$ | First note that if $\alpha = 2 \pi k$ for $k \in \Z$, then $e^{i \alpha} = 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} }
| r = e^{i \theta} \sum_{k \mathop = 0}^n e^{i k \alpha}
| c = factorising $e^{i \theta}$
}}
{{eqn | r = e^{i \theta} \paren {\frac {e^{i \paren {... | :$\ds \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n \alpha} 2} + i \map \sin {\theta + \frac {n \alpha} 2} } \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} }$ | First note that if $\alpha = 2 \pi k$ for $k \in \Z$, then $e^{i \alpha} = 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} }
| r = e^{i \theta} \sum_{k \mathop = 0}^n e^{i k \alpha}
| c = factorising $e^{i \theta}$
}}
{{eqn | r = e^{i \theta} \paren {\frac {e^{i \paren ... | Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1 | https://proofwiki.org/wiki/Sum_of_Complex_Exponentials_of_i_times_Arithmetic_Sequence_of_Angles/Formulation_1 | https://proofwiki.org/wiki/Sum_of_Complex_Exponentials_of_i_times_Arithmetic_Sequence_of_Angles/Formulation_1 | [
"Sum of Complex Exponentials of i times Arithmetic Sequence of Angles"
] | [] | [
"Sum of Geometric Sequence",
"Exponential of Sum",
"Euler's Formula",
"Euler's Sine Identity",
"Category:Sum of Complex Exponentials of i times Arithmetic Sequence of Angles"
] |
proofwiki-17001 | Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2 | :$\ds \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \frac {n + 1} 2 \alpha} } \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$ | First note that if $\alpha = 2 \pi k$ for $k \in \Z$, then $e^{i \alpha} = 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} }
| r = e^{i \theta} e^{i \alpha} \sum_{k \mathop = 0}^{n - 1} e^{i k \alpha}
| c = factorising $e^{i \theta} e^{i \alpha}$
}}
{{eqn | r = e^{i \the... | :$\ds \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \frac {n + 1} 2 \alpha} } \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$ | First note that if $\alpha = 2 \pi k$ for $k \in \Z$, then $e^{i \alpha} = 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} }
| r = e^{i \theta} e^{i \alpha} \sum_{k \mathop = 0}^{n - 1} e^{i k \alpha}
| c = factorising $e^{i \theta} e^{i \alpha}$
}}
{{eqn | r = e^{i \th... | Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2 | https://proofwiki.org/wiki/Sum_of_Complex_Exponentials_of_i_times_Arithmetic_Sequence_of_Angles/Formulation_2 | https://proofwiki.org/wiki/Sum_of_Complex_Exponentials_of_i_times_Arithmetic_Sequence_of_Angles/Formulation_2 | [
"Sum of Complex Exponentials of i times Arithmetic Sequence of Angles"
] | [] | [
"Sum of Geometric Sequence",
"Exponential of Sum",
"Euler's Formula",
"Euler's Sine Identity"
] |
proofwiki-17002 | Sum of Sines of Arithmetic Sequence of Angles/Formulation 1 | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \map \sin {\theta + k \alpha}
| r = \sin \theta + \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb
}}
{{eqn | r = \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} } \map \sin {\theta + \frac ... | From Sum of Complex Exponentials of i times Arithmetic Sequence of Angles: Formulation 1:
:$\ds \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n \alpha} 2} + i \map \sin {\theta + \frac {n \alpha} 2} } \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2}... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \map \sin {\theta + k \alpha}
| r = \sin \theta + \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb
}}
{{eqn | r = \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} } \map \sin {\theta + \frac ... | From [[Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1|Sum of Complex Exponentials of i times Arithmetic Sequence of Angles: Formulation 1]]:
:$\ds \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n \alpha} 2} + i \map \sin {\theta + \fra... | Sum of Sines of Arithmetic Sequence of Angles/Formulation 1 | https://proofwiki.org/wiki/Sum_of_Sines_of_Arithmetic_Sequence_of_Angles/Formulation_1 | https://proofwiki.org/wiki/Sum_of_Sines_of_Arithmetic_Sequence_of_Angles/Formulation_1 | [
"Sine Function"
] | [] | [
"Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1",
"Sine of Integer Multiple of Pi",
"Euler's Formula",
"Definition:Complex Number/Imaginary Part"
] |
proofwiki-17003 | Sum of Sines of Arithmetic Sequence of Angles/Formulation 2 | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \map \sin {\theta + k \alpha}
| r = \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb
}}
{{eqn | r = \map \sin {\theta + \frac {n + 1} 2 \alpha}\frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }
}}
{{end-e... | From Sum of Complex Exponentials of i times Arithmetic Sequence of Angles: Formulation 2:
:$\ds \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \frac {n + 1} 2 \alpha} } \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$
It... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \map \sin {\theta + k \alpha}
| r = \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb
}}
{{eqn | r = \map \sin {\theta + \frac {n + 1} 2 \alpha}\frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }
}}
{{end-e... | From [[Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2|Sum of Complex Exponentials of i times Arithmetic Sequence of Angles: Formulation 2]]:
:$\ds \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \... | Sum of Sines of Arithmetic Sequence of Angles/Formulation 2 | https://proofwiki.org/wiki/Sum_of_Sines_of_Arithmetic_Sequence_of_Angles/Formulation_2 | https://proofwiki.org/wiki/Sum_of_Sines_of_Arithmetic_Sequence_of_Angles/Formulation_2 | [
"Sine Function"
] | [] | [
"Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2",
"Sine of Integer Multiple of Pi",
"Euler's Formula",
"Definition:Complex Number/Imaginary Part"
] |
proofwiki-17004 | Sum of Cosines of Arithmetic Sequence of Angles/Formulation 1 | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \map \cos {\theta + k \alpha}
| r = \cos \theta + \map \cos {\theta + \alpha} + \map \cos {\theta + 2 \alpha} + \map \cos {\theta + 3 \alpha} + \dotsb
}}
{{eqn | r = \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} } \map \cos {\theta + \frac ... | From Sum of Complex Exponentials of i times Arithmetic Sequence of Angles: Formulation 1:
:$\ds \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n \alpha} 2} + i \map \sin {\theta + \frac {n \alpha} 2} } \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2}... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \map \cos {\theta + k \alpha}
| r = \cos \theta + \map \cos {\theta + \alpha} + \map \cos {\theta + 2 \alpha} + \map \cos {\theta + 3 \alpha} + \dotsb
}}
{{eqn | r = \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} } \map \cos {\theta + \frac ... | From [[Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1|Sum of Complex Exponentials of i times Arithmetic Sequence of Angles: Formulation 1]]:
:$\ds \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n \alpha} 2} + i \map \sin {\theta + \fra... | Sum of Cosines of Arithmetic Sequence of Angles/Formulation 1 | https://proofwiki.org/wiki/Sum_of_Cosines_of_Arithmetic_Sequence_of_Angles/Formulation_1 | https://proofwiki.org/wiki/Sum_of_Cosines_of_Arithmetic_Sequence_of_Angles/Formulation_1 | [
"Sum of Cosines of Arithmetic Sequence of Angles"
] | [] | [
"Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1",
"Sine of Integer Multiple of Pi",
"Euler's Formula",
"Definition:Complex Number/Real Part"
] |
proofwiki-17005 | Sum of Cosines of Arithmetic Sequence of Angles/Formulation 2 | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \map \cos {\theta + k \alpha}
| r = \map \cos {\theta + \alpha} + \map \cos {\theta + 2 \alpha} + \map \cos {\theta + 3 \alpha} + \dotsb
}}
{{eqn | r = \map \cos {\theta + \frac {n + 1} 2 \alpha} \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }
}}
{{end-... | From Sum of Complex Exponentials of i times Arithmetic Sequence of Angles: Formulation 2:
:$\ds \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \frac {n + 1} 2 \alpha} } \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$
It... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \map \cos {\theta + k \alpha}
| r = \map \cos {\theta + \alpha} + \map \cos {\theta + 2 \alpha} + \map \cos {\theta + 3 \alpha} + \dotsb
}}
{{eqn | r = \map \cos {\theta + \frac {n + 1} 2 \alpha} \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }
}}
{{end-... | From [[Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2|Sum of Complex Exponentials of i times Arithmetic Sequence of Angles: Formulation 2]]:
:$\ds \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \... | Sum of Cosines of Arithmetic Sequence of Angles/Formulation 2 | https://proofwiki.org/wiki/Sum_of_Cosines_of_Arithmetic_Sequence_of_Angles/Formulation_2 | https://proofwiki.org/wiki/Sum_of_Cosines_of_Arithmetic_Sequence_of_Angles/Formulation_2 | [
"Sum of Cosines of Arithmetic Sequence of Angles"
] | [] | [
"Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2",
"Sine of Integer Multiple of Pi",
"Euler's Formula",
"Definition:Complex Number/Real Part"
] |
proofwiki-17006 | Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 3 | :$\ds \sum_{k \mathop = p}^q e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {\paren {p + q} \alpha} 2} + i \map \sin {\theta + \frac {\paren {p + q} \alpha} 2} } \frac {\map \sin {\paren {q - p + 1} \alpha / 2} } {\map \sin {\alpha / 2} }$ | First note that if $\alpha = 2 \pi k$ for $k \in \Z$, then $e^{i \alpha} = 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = p}^q e^{i \paren {\theta + k \alpha} }
| r = e^{i \theta} e^{i p \alpha} \sum_{k \mathop = 0}^{q - p} e^{i k \alpha}
| c = factorising $e^{i \theta} e^{i p \alpha}$
}}
{{eqn | r = e^{i ... | :$\ds \sum_{k \mathop = p}^q e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {\paren {p + q} \alpha} 2} + i \map \sin {\theta + \frac {\paren {p + q} \alpha} 2} } \frac {\map \sin {\paren {q - p + 1} \alpha / 2} } {\map \sin {\alpha / 2} }$ | First note that if $\alpha = 2 \pi k$ for $k \in \Z$, then $e^{i \alpha} = 1$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = p}^q e^{i \paren {\theta + k \alpha} }
| r = e^{i \theta} e^{i p \alpha} \sum_{k \mathop = 0}^{q - p} e^{i k \alpha}
| c = factorising $e^{i \theta} e^{i p \alpha}$
}}
{{eqn | r = e^{i... | Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 3 | https://proofwiki.org/wiki/Sum_of_Complex_Exponentials_of_i_times_Arithmetic_Sequence_of_Angles/Formulation_3 | https://proofwiki.org/wiki/Sum_of_Complex_Exponentials_of_i_times_Arithmetic_Sequence_of_Angles/Formulation_3 | [
"Sum of Complex Exponentials of i times Arithmetic Sequence of Angles"
] | [] | [
"Sum of Geometric Sequence",
"Exponential of Sum",
"Euler's Formula",
"Euler's Sine Identity",
"Category:Sum of Complex Exponentials of i times Arithmetic Sequence of Angles"
] |
proofwiki-17007 | Weierstrass Approximation Theorem/Lemma 1 | :$\ds \sum_{k \mathop = 0}^n k \map {p_{n, k} } t = n t$ | From Binomial Theorem for Integral Index:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^n
| r = \sum_{k \mathop = 0}^n \binom n k y^k x^{n - k}
}}
{{eqn | ll= \leadsto
| l = 1
| r = \sum_{k \mathop = 0}^n \binom n k t^k \paren {1 - t}^{n - k}
| c = $y = t, ~x = 1 - t$
}}
{{eqn | ll= \leadsto
| ... | :$\ds \sum_{k \mathop = 0}^n k \map {p_{n, k} } t = n t$ | From [[Binomial Theorem for Integral Index]]:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^n
| r = \sum_{k \mathop = 0}^n \binom n k y^k x^{n - k}
}}
{{eqn | ll= \leadsto
| l = 1
| r = \sum_{k \mathop = 0}^n \binom n k t^k \paren {1 - t}^{n - k}
| c = $y = t, ~x = 1 - t$
}}
{{eqn | ll= \leadsto
... | Weierstrass Approximation Theorem/Lemma 1 | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem/Lemma_1 | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem/Lemma_1 | [
"Weierstrass Approximation Theorem"
] | [] | [
"Binomial Theorem/Integral Index",
"Definition:Derivative/Real Function"
] |
proofwiki-17008 | Weierstrass Approximation Theorem/Lemma 2 | :$\ds \sum_{k \mathop = 0}^n \paren {k - n t}^2 \map {p_{n, k} } t = n t \paren {1 - t}$ | From Binomial Theorem for Integral Index:
:$\ds 1 = \sum_{k \mathop = 0}^n \binom n k t^k \paren {1 - t}^{n - k}$
From {{Lemma|Weierstrass Approximation Theorem|1}}:
{{begin-eqn}}
{{eqn | l = n t
| r = \sum_{k \mathop = 0}^n \binom n k k t^k \paren {1 - t}^{n - k}
}}
{{eqn | ll= \leadsto
| l = n
| r =... | :$\ds \sum_{k \mathop = 0}^n \paren {k - n t}^2 \map {p_{n, k} } t = n t \paren {1 - t}$ | From [[Binomial Theorem for Integral Index]]:
:$\ds 1 = \sum_{k \mathop = 0}^n \binom n k t^k \paren {1 - t}^{n - k}$
From {{Lemma|Weierstrass Approximation Theorem|1}}:
{{begin-eqn}}
{{eqn | l = n t
| r = \sum_{k \mathop = 0}^n \binom n k k t^k \paren {1 - t}^{n - k}
}}
{{eqn | ll= \leadsto
| l = n
... | Weierstrass Approximation Theorem/Lemma 2 | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem/Lemma_2 | https://proofwiki.org/wiki/Weierstrass_Approximation_Theorem/Lemma_2 | [
"Weierstrass Approximation Theorem"
] | [] | [
"Binomial Theorem/Integral Index"
] |
proofwiki-17009 | Convergent Sequence in Normed Vector Space has Unique Limit | Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.
Let $\sequence {x_n}$ be a sequence in $\struct {X, \norm {\,\cdot\,} }$.
Then $\sequence {x_n}$ can have at most one limit. | {{AimForCont}} $\ds \lim_{n \mathop \to \infty} x_n = L_1$ and $\ds \lim_{n \mathop \to \infty} x_n = L_2$ such that $L_1 \ne L_2$.
Let $\epsilon = \dfrac {\norm {L_1 - L_2} } 3$.
From the norm axioms it follows that $\epsilon > 0$.
By definition:
:$\exists N_1 \in \N : \forall n > N_1 : \norm {x_n - L_1} < \epsilon$
:... | Let $\struct {X, \norm {\,\cdot\,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $\struct {X, \norm {\,\cdot\,} }$.
Then $\sequence {x_n}$ can have at most one [[Definition:Limit of Sequence in Normed Vector Space|limit]]. | {{AimForCont}} $\ds \lim_{n \mathop \to \infty} x_n = L_1$ and $\ds \lim_{n \mathop \to \infty} x_n = L_2$ such that $L_1 \ne L_2$.
Let $\epsilon = \dfrac {\norm {L_1 - L_2} } 3$.
From the [[Axiom:Vector Space Norm Axioms|norm axioms]] it follows that $\epsilon > 0$.
By [[Definition:Convergent Sequence in Normed Vec... | Convergent Sequence in Normed Vector Space has Unique Limit | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Vector_Space_has_Unique_Limit | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Vector_Space_has_Unique_Limit | [
"Limits of Sequences",
"Normed Vector Spaces",
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Limit of Sequence/Normed Vector Space"
] | [
"Axiom:Vector Space Norm Axioms",
"Definition:Convergent Sequence/Normed Vector Space",
"Triangle Inequality",
"Definition:Contradiction"
] |
proofwiki-17010 | Product Space is T3.5 iff Factor Spaces are T3.5/Sufficient Condition | Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$.
Let $\ds T = \struct{S, \tau} = \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$.
Let $T$ be a $... | Let $T$ be a $T_{3 \frac 1 2}$ space.
Since $S_\alpha \ne \O$ we also have $S \ne \O$ by the {{Axiom-link|Choice}}.
Let $\alpha \in I$ be arbitrary.
From Subspace of Product Space is Homeomorphic to Factor Space:
:$\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to a subspace $T_\alpha$ of $T$.
From $T_{3 \frac 1 2}$ ... | Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty|non-empty]] [[Definition:Topological Space|topological spaces]] for $\alpha$ in some [[Definition:Indexing Set|indexing set]] $I$.
Let $\ds T = \struct{S, \ta... | Let $T$ be a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]].
Since $S_\alpha \ne \O$ we also have $S \ne \O$ by the {{Axiom-link|Choice}}.
Let $\alpha \in I$ be arbitrary.
From [[Subspace of Product Space is Homeomorphic to Factor Space]]:
:$\struct {S_\alpha, \tau_\alpha}$ is [[Definition:Homeomorphism (Topologi... | Product Space is T3.5 iff Factor Spaces are T3.5/Sufficient Condition | https://proofwiki.org/wiki/Product_Space_is_T3.5_iff_Factor_Spaces_are_T3.5/Sufficient_Condition | https://proofwiki.org/wiki/Product_Space_is_T3.5_iff_Factor_Spaces_are_T3.5/Sufficient_Condition | [
"Product Space is T3.5 iff Factor Spaces are T3.5"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T3.5 Space",
"Definition:T3.5 Space"
] | [
"Definition:T3.5 Space",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subspace",
"T3.5 Property is Hereditary",
"T3.5 Property is Preserved under Homeomorphism",
"Definition:T3.5 Space"
] |
proofwiki-17011 | Product Space is T3.5 iff Factor Spaces are T3.5/Necessary Condition | Let $\mathbb S = \family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$ with $S_\alpha \ne \O$ for every $\alpha \in I$.
Let $\ds T = \struct{S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the produc... | Let $\struct {S_\alpha, \tau_\alpha}$ is a $T_{3 \frac 1 2}$ space for each $\alpha \in I$.
Let $x \in S$.
Let $F$ be a closed subset of $S$ such that $x \notin F$.
By definition of a closed subset:
:$S \setminus F \in \tau$
By definition of the product topology, there exists an open set $B$ of the natural basis conta... | Let $\mathbb S = \family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] for $\alpha$ in some [[Definition:Indexing Set|indexing set]] $I$ with $S_\alpha \ne \O$ for every $\alpha \in I$.
Let $\ds T = \... | Let $\struct {S_\alpha, \tau_\alpha}$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]] for each $\alpha \in I$.
Let $x \in S$.
Let $F$ be a [[Definition:Closed Set|closed subset]] of $S$ such that $x \notin F$.
By definition of a [[Definition:Closed Set|closed subset]]:
:$S \setminus F \in \tau$
By definit... | Product Space is T3.5 iff Factor Spaces are T3.5/Necessary Condition | https://proofwiki.org/wiki/Product_Space_is_T3.5_iff_Factor_Spaces_are_T3.5/Necessary_Condition | https://proofwiki.org/wiki/Product_Space_is_T3.5_iff_Factor_Spaces_are_T3.5/Necessary_Condition | [
"Product Space is T3.5 iff Factor Spaces are T3.5"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T3.5 Space",
"Definition:T3.5 Space"
] | [
"Definition:T3.5 Space",
"Definition:Closed Set",
"Definition:Closed Set",
"Definition:Product Topology",
"Definition:Open Set",
"Definition:Product Topology/Natural Basis",
"Definition:Disjoint Sets",
"Definition:Product Topology/Natural Basis",
"Definition:Projection",
"Definition:T3.5 Space",
... |
proofwiki-17012 | Half-Range Fourier Sine Series/Sine of Non-Integer Multiple of x over 0 to Pi | Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer.
Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:
:$\map f x = \sin \lambda x$
Then its half-range Fourier sine series can be expressed as:
{{begin-eqn}}
{{eqn | l = \map f x
| o = \sim
| r = \frac {2 \sin \lambda ... | By definition of half-range Fourier sine series:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \sin n x$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$
Because $\lambda \notin \Z$ we have that $\lambda \ne n$ for all $n$.
Thus for $n > 0$:
{{begin-eqn}}... | Let $\lambda \in \R \setminus \Z$ be a [[Definition:Real Number|real number]] which is not an [[Definition:Integer|integer]].
Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 \pi$ as:
:$\map f x = \sin \lambda x$
Then its [[Definition:Half-Range Fourier Sine Series|half-range... | By definition of [[Definition:Half-Range Fourier Sine Series|half-range Fourier sine series]]:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \sin n x$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$
Because $\lambda \notin \Z$ we have that $\lambda ... | Half-Range Fourier Sine Series/Sine of Non-Integer Multiple of x over 0 to Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series/Sine_of_Non-Integer_Multiple_of_x_over_0_to_Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series/Sine_of_Non-Integer_Multiple_of_x_over_0_to_Pi | [
"Examples of Half-Range Fourier Series"
] | [
"Definition:Real Number",
"Definition:Integer",
"Definition:Real Function",
"Definition:Half-Range Fourier Sine Series"
] | [
"Definition:Half-Range Fourier Sine Series",
"Primitive of Sine of a x by Sine of b x",
"Sine of Integer Multiple of Pi",
"Sine of Sum",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi",
"Difference of Two Squares"
] |
proofwiki-17013 | Minimum Rule for Continuous Functions | Let $\struct {S, \tau}$ be a topological space.
Let $f, g: S \to \R$ be continuous real-valued functions.
Let $\min \set {f, g}: S \to \R$ denote the pointwise minimum of $f$ and $g$.
Then:
:$\min \set {f, g}$ is continuous. | Let $x \in S$.
Let $\epsilon > 0$.
{{WLOG}}, assume that $\map f x \le \map g x$. | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f, g: S \to \R$ be [[Definition:Everywhere Continuous Mapping (Topology)|continuous]] [[Definition:Real-Valued Function|real-valued functions]].
Let $\min \set {f, g}: S \to \R$ denote the [[Definition:Pointwise Minimum of Real-Valu... | Let $x \in S$.
Let $\epsilon > 0$.
{{WLOG}}, assume that $\map f x \le \map g x$. | Minimum Rule for Continuous Functions | https://proofwiki.org/wiki/Minimum_Rule_for_Continuous_Functions | https://proofwiki.org/wiki/Minimum_Rule_for_Continuous_Functions | [
"Min Operation",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Real-Valued Function",
"Definition:Pointwise Minimum of Mappings/Real-Valued Functions",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [] |
proofwiki-17014 | Min is Half of Sum Less Absolute Difference | For all numbers $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$:
:$\min \set {a, b} = \dfrac 1 2 \paren {a + b - \size {a - b} }$ | From the definition of min:
:<nowiki>$\map \min {a, b} = \begin{cases}
a: & a \le b \\
b: & b \le a
\end{cases}$</nowiki>
Let $a < b$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 2 \paren {a + b - \size {a - b} }
| r = \dfrac 1 2 \paren {a + b - \paren {b - a} }
| c = {{Defof|Absolute Value}}
}}
{{eqn | r = \d... | For all [[Definition:Number|numbers]] $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$:
:$\min \set {a, b} = \dfrac 1 2 \paren {a + b - \size {a - b} }$ | From the definition of [[Definition:Min Operation|min]]:
:<nowiki>$\map \min {a, b} = \begin{cases}
a: & a \le b \\
b: & b \le a
\end{cases}$</nowiki>
Let $a < b$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 2 \paren {a + b - \size {a - b} }
| r = \dfrac 1 2 \paren {a + b - \paren {b - a} }
| c = {{Defof|Ab... | Min is Half of Sum Less Absolute Difference | https://proofwiki.org/wiki/Min_is_Half_of_Sum_Less_Absolute_Difference | https://proofwiki.org/wiki/Min_is_Half_of_Sum_Less_Absolute_Difference | [
"Min Operation"
] | [
"Definition:Number"
] | [
"Definition:Min Operation"
] |
proofwiki-17015 | Continuity Test for Real-Valued Functions | Let $\struct{S, \tau}$ be a topological space.
Let $f: S \to \R$ be a real-valued function.
Let $x \in S$.
Then $f$ is continuous at $x$ {{iff}}:
:$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$ | === Necessary Condition ===
{{:Continuity Test for Real-Valued Functions/Necessary Condition}}{{qed|lemma}} | Let $\struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f: S \to \R$ be a [[Definition:Real-Valued Function|real-valued function]].
Let $x \in S$.
Then $f$ is [[Definition:Continuous Mapping at Point (Topology)|continuous at $x$]] {{iff}}:
:$\forall \epsilon \in \R_{>0} : \exists U \in \... | === [[Continuity Test for Real-Valued Functions/Necessary Condition|Necessary Condition]] ===
{{:Continuity Test for Real-Valued Functions/Necessary Condition}}{{qed|lemma}} | Continuity Test for Real-Valued Functions | https://proofwiki.org/wiki/Continuity_Test_for_Real-Valued_Functions | https://proofwiki.org/wiki/Continuity_Test_for_Real-Valued_Functions | [
"Continuous Mappings",
"Continuity Test for Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping (Topology)/Point"
] | [
"Continuity Test for Real-Valued Functions/Necessary Condition"
] |
proofwiki-17016 | Continuity Test for Real-Valued Functions/Everywhere Continuous | Let $\struct{S, \tau}$ be a topological space.
Let $f: S \to \R$ be a real-valued function.
Then $f$ is everywhere continuous {{iff}}:
:$\forall x \in S : \forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$ | By definition, $f$ is everywhere continuous {{iff}} $f$ is continuous at every point $x \in S$.
From Continuity Test for Real-Valued Functions, $f$ is continuous at every point $x \in S$ {{iff}}:
:$\forall x \in S : \forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f ... | Let $\struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f: S \to \R$ be a [[Definition:Real-Valued Function|real-valued function]].
Then $f$ is [[Definition:Everywhere Continuous Mapping (Topology)|everywhere continuous]] {{iff}}:
:$\forall x \in S : \forall \epsilon \in \R_{>0} : \exists... | By definition, $f$ is [[Definition:Everywhere Continuous Mapping (Topology)|everywhere continuous]] {{iff}} $f$ is [[Definition:Continuous Mapping at Point (Topology)|continuous at every point]] $x \in S$.
From [[Continuity Test for Real-Valued Functions]], $f$ is [[Definition:Continuous Mapping at Point (Topology)|co... | Continuity Test for Real-Valued Functions/Everywhere Continuous | https://proofwiki.org/wiki/Continuity_Test_for_Real-Valued_Functions/Everywhere_Continuous | https://proofwiki.org/wiki/Continuity_Test_for_Real-Valued_Functions/Everywhere_Continuous | [
"Continuity Test for Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Continuous Mapping (Topology)/Point",
"Continuity Test for Real-Valued Functions",
"Definition:Continuous Mapping (Topology)/Point",
"Category:Continuity Test for Real-Valued Functions"
] |
proofwiki-17017 | Definite Integral of Fourier Series at Ends of Interval | Let $f: \R \to \R$ be a real function defined in the open interval $\openint {-\pi} \pi$.
Let $f$ fulfil the Dirichlet conditions in $\openint {-\pi} \pi$.
Let $a_0, a_1, \dotsc; b_1, \dotsc$ be the Fourier coefficients of $f$ in $\openint {-\pi} \pi$.
Consider the real function:
:$\map F x = \ds \int_{-\pi}^x \map f t... | From Definite Integral on Zero Interval:
{{begin-eqn}}
{{eqn | l = \map F {-\pi}
| r = \int_{-\pi}^{-\pi} \map f t \rd t - \dfrac {a_0} 2 \paren {-\pi}
| c =
}}
{{eqn | r = 0 - \dfrac {a_0} 2 \paren {-\pi}
| c =
}}
{{eqn | r = \dfrac {a_0 \pi} 2
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn ... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] defined in the [[Definition:Open Real Interval|open interval]] $\openint {-\pi} \pi$.
Let $f$ fulfil the [[Definition:Dirichlet Conditions|Dirichlet conditions]] in $\openint {-\pi} \pi$.
Let $a_0, a_1, \dotsc; b_1, \dotsc$ be the [[Definition:Fourier... | From [[Definite Integral on Zero Interval]]:
{{begin-eqn}}
{{eqn | l = \map F {-\pi}
| r = \int_{-\pi}^{-\pi} \map f t \rd t - \dfrac {a_0} 2 \paren {-\pi}
| c =
}}
{{eqn | r = 0 - \dfrac {a_0} 2 \paren {-\pi}
| c =
}}
{{eqn | r = \dfrac {a_0 \pi} 2
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}... | Definite Integral of Fourier Series at Ends of Interval | https://proofwiki.org/wiki/Definite_Integral_of_Fourier_Series_at_Ends_of_Interval | https://proofwiki.org/wiki/Definite_Integral_of_Fourier_Series_at_Ends_of_Interval | [
"Definite Integrals",
"Fourier Analysis"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open",
"Definition:Dirichlet Conditions",
"Definition:Fourier Series/Fourier Coefficient",
"Definition:Real Function"
] | [
"Definite Integral on Zero Interval",
"Fourier's Theorem"
] |
proofwiki-17018 | Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 1 | Let $G := \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$.
Let $\circ: G \times G \to G$ be the binary operation defined as:
:$\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$
{{:Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers}} | To prove $G$ is isomorphic to $\struct {\R, +}$, we need to find a bijective homorphism $\phi: \openint {-1} 1 \to \R$:
:$\forall x, y \in G: \map \phi {x \circ y} = \map \phi x + \map \phi y$
From Group Examples: $\dfrac {x + y} {1 + x y}$:
:the identity element of $G$ is $0$
:the inverse of $x$ in $G$ is $-x$.
This a... | Let $G := \set {x \in \R: -1 < x < 1}$ be the [[Definition:Set|set]] of all [[Definition:Real Number|real numbers]] whose [[Definition:Absolute Value|absolute value]] is less than $1$.
Let $\circ: G \times G \to G$ be the [[Definition:Binary Operation|binary operation]] defined as:
:$\forall x, y \in G: x \circ y = \d... | To prove $G$ is [[Definition:Isomorphism|isomorphic]] to $\struct {\R, +}$, we need to find a [[Definition:Bijection|bijective]] [[Definition:Group Homomorphism|homorphism]] $\phi: \openint {-1} 1 \to \R$:
:$\forall x, y \in G: \map \phi {x \circ y} = \map \phi x + \map \phi y$
From [[Group/Examples/x+y over 1+xy|Gro... | Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 1 | https://proofwiki.org/wiki/Group/Examples/x+y_over_1+xy/Isomorphic_to_Real_Numbers/Proof_1 | https://proofwiki.org/wiki/Group/Examples/x+y_over_1+xy/Isomorphic_to_Real_Numbers/Proof_1 | [
"Examples of Groups/x+y over 1+xy"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Absolute Value",
"Definition:Operation/Binary Operation"
] | [
"Definition:Isomorphism",
"Definition:Bijection",
"Definition:Group Homomorphism",
"Group/Examples/x+y over 1+xy",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Odd Function",
"Definition:Real Interval/Open",
"Definition:Inv... |
proofwiki-17019 | Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 2 | Let $G := \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$.
Let $\circ: G \times G \to G$ be the binary operation defined as:
:$\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$
{{:Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers}} | To prove $G$ is isomorphic to $\struct {\R, +}$, it is sufficient to find a bijective homorphism $\phi: \to \R \to G$:
:$\forall x, y \in G: \map \phi {x + y} = \map \phi x \circ \map \phi y$
From Group Examples: $\dfrac {x + y} {1 + x y}$:
:the identity element of $G$ is $0$
:the inverse of $x$ in $G$ is $-x$.
This al... | Let $G := \set {x \in \R: -1 < x < 1}$ be the [[Definition:Set|set]] of all [[Definition:Real Number|real numbers]] whose [[Definition:Absolute Value|absolute value]] is less than $1$.
Let $\circ: G \times G \to G$ be the [[Definition:Binary Operation|binary operation]] defined as:
:$\forall x, y \in G: x \circ y = \d... | To prove $G$ is [[Definition:Isomorphism|isomorphic]] to $\struct {\R, +}$, it is sufficient to find a [[Definition:Bijection|bijective]] [[Definition:Group Homomorphism|homorphism]] $\phi: \to \R \to G$:
:$\forall x, y \in G: \map \phi {x + y} = \map \phi x \circ \map \phi y$
From [[Group/Examples/x+y over 1+xy|Grou... | Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 2 | https://proofwiki.org/wiki/Group/Examples/x+y_over_1+xy/Isomorphic_to_Real_Numbers/Proof_2 | https://proofwiki.org/wiki/Group/Examples/x+y_over_1+xy/Isomorphic_to_Real_Numbers/Proof_2 | [
"Examples of Groups/x+y over 1+xy"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Absolute Value",
"Definition:Operation/Binary Operation"
] | [
"Definition:Isomorphism",
"Definition:Bijection",
"Definition:Group Homomorphism",
"Group/Examples/x+y over 1+xy",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Odd Function",
"Definition:Image (Set Theory)/Mapping/Mapping",
... |
proofwiki-17020 | Triangle Inequality for Complex Numbers/Corollary 2 | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\cmod z$ be the modulus of $z$.
Then:
:$\cmod {z_1 + z_2} \ge \cmod {\cmod {z_1} - \cmod {z_2} }$ | {{begin-eqn}}
{{eqn | l = \cmod {z_1 + z_2}
| o = \ge
| r = \cmod {z_1} - \cmod {z_2}
| c = {{Corollary|Triangle Inequality for Complex Numbers|1}}
}}
{{eqn | l = \cmod {z_1 + z_2}
| o = \ge
| r = \cmod {z_2} - \cmod {z_1}
| c = {{Corollary|Triangle Inequality for Complex Numbers|1}}... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\cmod z$ be the [[Definition:Modulus of Complex Number|modulus]] of $z$.
Then:
:$\cmod {z_1 + z_2} \ge \cmod {\cmod {z_1} - \cmod {z_2} }$ | {{begin-eqn}}
{{eqn | l = \cmod {z_1 + z_2}
| o = \ge
| r = \cmod {z_1} - \cmod {z_2}
| c = {{Corollary|Triangle Inequality for Complex Numbers|1}}
}}
{{eqn | l = \cmod {z_1 + z_2}
| o = \ge
| r = \cmod {z_2} - \cmod {z_1}
| c = {{Corollary|Triangle Inequality for Complex Numbers|1}}... | Triangle Inequality for Complex Numbers/Corollary 2 | https://proofwiki.org/wiki/Triangle_Inequality_for_Complex_Numbers/Corollary_2 | https://proofwiki.org/wiki/Triangle_Inequality_for_Complex_Numbers/Corollary_2 | [
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [] |
proofwiki-17021 | Continuity Test for Real-Valued Functions/Necessary Condition | Let $\struct {S, \tau}$ be a topological space.
Let $f: S \to \R$ be a real-valued function.
Let $x \in S$.
Let $f$ be continuous at $x$
Then:
:$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$ | Let $f$ be continuous at $x$.
Then by definition:
:For every neighborhood $N$ of $\map f x$ in $\R$, there exists a neighborhood $M$ of $x$ in $S$ such that $\map {f^\to} M \subseteq N$.
From:
:Open Ball in Real Number Line is Open Interval
:Open Ball of Metric Space is Open Set
:Set is Open iff Neighborhood of all its... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f: S \to \R$ be a [[Definition:Real-Valued Function|real-valued function]].
Let $x \in S$.
Let $f$ be [[Definition:Continuous Mapping at Point (Topology)|continuous at $x$]]
Then:
:$\forall \epsilon \in \R_{>0} : \exists U \in \t... | Let $f$ be [[Definition:Continuous Mapping at Point (Topology)|continuous at $x$]].
Then by definition:
:For every [[Definition:Neighborhood of Point|neighborhood]] $N$ of $\map f x$ in $\R$, there exists a [[Definition:Neighborhood of Point|neighborhood]] $M$ of $x$ in $S$ such that $\map {f^\to} M \subseteq N$.
Fro... | Continuity Test for Real-Valued Functions/Necessary Condition | https://proofwiki.org/wiki/Continuity_Test_for_Real-Valued_Functions/Necessary_Condition | https://proofwiki.org/wiki/Continuity_Test_for_Real-Valued_Functions/Necessary_Condition | [
"Continuity Test for Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping (Topology)/Point"
] | [
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Open Ball in Real Number Line is Open Interval",
"Open Ball is Open Set/Pseudometric Space",
"Set is Open iff Neighborhood of all its Points",
"Definition:Neighborho... |
proofwiki-17022 | Continuity Test for Real-Valued Functions/Sufficient Condition | Let $\struct {S, \tau}$ be a topological space.
Let $f: S \to \R$ be a real-valued function.
Let $x \in S$.
Let $f$ satisfy:
:$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$
Then $f$ is continuous at $x$ | Let $f$ satisfy:
:$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$
Let $N$ be a neighborhood of $\map f x$.
By definition of a neighborhood of $\map f x$:
:there exists $U$ such that $z \in U \subseteq N$
By definition of the t... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f: S \to \R$ be a [[Definition:Real-Valued Function|real-valued function]].
Let $x \in S$.
Let $f$ satisfy:
:$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\m... | Let $f$ satisfy:
:$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$
Let $N$ be a [[Definition:Neighborhood of Point|neighborhood]] of $\map f x$.
By definition of a [[Definition:Neighborhood of Point|neighborhood]] of $\map ... | Continuity Test for Real-Valued Functions/Sufficient Condition | https://proofwiki.org/wiki/Continuity_Test_for_Real-Valued_Functions/Sufficient_Condition | https://proofwiki.org/wiki/Continuity_Test_for_Real-Valued_Functions/Sufficient_Condition | [
"Continuity Test for Real-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Topology Induced by Metric",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Open Ball",
"Open Ball in Real Number Line is Open Interval",
"Set is Open iff Neighborhood of all its Points",
"... |
proofwiki-17023 | Max is Half of Sum Plus Absolute Difference | For all numbers $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$:
:$\max \set {a, b} = \dfrac 1 2 \paren {a + b + \size {a - b} }$ | From the Trichotomy Law for Real Numbers exactly one of the following holds:
{{begin-itemize}}
{{item||$x < y$ and so $\max \set {x, y} {{=}} y$}}
{{item||$x {{=}} y$ and so $\max \set {x, y} {{=}} x {{=}} y$}}
{{item||$y < x$ and so $\max \set {x, y} {{=}} x$}}
{{end-itemize}}
By the definition of the absolute value f... | For all [[Definition:Number|numbers]] $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$:
:$\max \set {a, b} = \dfrac 1 2 \paren {a + b + \size {a - b} }$ | From the [[Trichotomy Law for Real Numbers]] exactly one of the following holds:
{{begin-itemize}}
{{item||$x < y$ and so $\max \set {x, y} {{=}} y$}}
{{item||$x {{=}} y$ and so $\max \set {x, y} {{=}} x {{=}} y$}}
{{item||$y < x$ and so $\max \set {x, y} {{=}} x$}}
{{end-itemize}}
By the definition of the [[Definit... | Max is Half of Sum Plus Absolute Difference/Proof 2 | https://proofwiki.org/wiki/Max_is_Half_of_Sum_Plus_Absolute_Difference | https://proofwiki.org/wiki/Max_is_Half_of_Sum_Plus_Absolute_Difference/Proof_2 | [
"Max is Half of Sum Plus Absolute Difference",
"Max Operation"
] | [
"Definition:Number"
] | [
"Trichotomy Law for Real Numbers",
"Definition:Absolute Value",
"Definition:Commutative/Operation",
"Definition:Associative Operation"
] |
proofwiki-17024 | Maximum Rule for Continuous Functions | Let $\struct {S, \tau}$ be a topological space.
Let $f, g: S \to \R$ be continuous real-valued functions.
Let $\max \set {f, g}: S \to \R$ denote the pointwise maximum of $f$ and $g$.
Then:
:$\max \set {f, g}$ is continuous. | From Sum Less Minimum is Maximum:
:$\forall x \in S : \max \set {\map f x, \map g x} = \map f x + \map g x - \min \set {\map f x, \map g x}$
Thus:
:$\max \set {f, g} = f + g - \min \set{f, g}$
From Minimum Rule for Continuous Functions:
:$\min \set {f, g}$ is continuous
From Multiple Rule for Continuous Mappings into ... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f, g: S \to \R$ be [[Definition:Everywhere Continuous Mapping (Topology)|continuous]] [[Definition:Real-Valued Function|real-valued functions]].
Let $\max \set {f, g}: S \to \R$ denote the [[Definition:Pointwise Maximum of Real-Valu... | From [[Sum Less Minimum is Maximum]]:
:$\forall x \in S : \max \set {\map f x, \map g x} = \map f x + \map g x - \min \set {\map f x, \map g x}$
Thus:
:$\max \set {f, g} = f + g - \min \set{f, g}$
From [[Minimum Rule for Continuous Functions]]:
:$\min \set {f, g}$ is [[Definition:Everywhere Continuous Mapping (Topo... | Maximum Rule for Continuous Functions | https://proofwiki.org/wiki/Maximum_Rule_for_Continuous_Functions | https://proofwiki.org/wiki/Maximum_Rule_for_Continuous_Functions | [
"Max Operation",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Real-Valued Function",
"Definition:Pointwise Maximum of Mappings/Real-Valued Functions",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Sum Less Minimum is Maximum",
"Minimum Rule for Continuous Functions",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Combination Theorem for Continuous Mappings/Topological Ring/Multiple Rule",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Combination Theorem for Continuous Mappings/T... |
proofwiki-17025 | Equivalence of Definitions of T2.5 Space | {{TFAE|def = T2.5 Space|view = a $T_{2 \frac 1 2}$ space}}
Let $T = \struct {S, \tau}$ be a topological space. | === Definition 1 implies Definition 2 ===
Let $\struct {S, \tau}$ satisfy:
:$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$
Let $x, y \subseteq S , x \ne y $ be arbitrary.
Then:
:$\exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$
Let $N_x = U$ and $N_y = V$.
From Se... | {{TFAE|def = T2.5 Space|view = a $T_{2 \frac 1 2}$ space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]]. | === Definition 1 implies Definition 2 ===
Let $\struct {S, \tau}$ satisfy:
:$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$
Let $x, y \subseteq S , x \ne y $ be arbitrary.
Then:
:$\exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$
Let $N_x = U$ and $N_y = V$.
F... | Equivalence of Definitions of T2.5 Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T2.5_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T2.5_Space | [
"T2.5 Spaces"
] | [
"Definition:Topological Space"
] | [
"Set is Subset of Itself"
] |
proofwiki-17026 | Product Space is T2.5 iff Factor Spaces are T2.5/Sufficient Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$.
Let $\ds T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a $T_{2 \frac... | Let $T$ be a $T_{2 \frac 1 2}$ space.
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the axiom of choice.
Let $\alpha \in I$.
From Subspace of Product Space is Homeomorphic to Factor Space, $\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to a subspace $T_\alpha$ of $T$.
From $T_{2 \frac 1 2}$ property is hereditary,... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] for $\alpha$ in some [[Definition:Indexing Set|indexing set]] $I$.
Let $\ds T = \struct {S, \ta... | Let $T$ be a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the [[Axiom:Axiom of Choice|axiom of choice]].
Let $\alpha \in I$.
From [[Subspace of Product Space is Homeomorphic to Factor Space]], $\struct {S_\alpha, \tau_\alpha}$ is [[Definition:Homeomorphism (Topol... | Product Space is T2.5 iff Factor Spaces are T2.5/Sufficient Condition | https://proofwiki.org/wiki/Product_Space_is_T2.5_iff_Factor_Spaces_are_T2.5/Sufficient_Condition | https://proofwiki.org/wiki/Product_Space_is_T2.5_iff_Factor_Spaces_are_T2.5/Sufficient_Condition | [
"Product Space is T2.5 iff Factor Spaces are T2.5"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T2.5 Space",
"Definition:T2.5 Space"
] | [
"Definition:T2.5 Space",
"Axiom:Axiom of Choice",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subspace",
"T2.5 Property is Hereditary",
"Definition:T2.5 Space",
"T2.5 Property is Preserved under Homeomorphism",
"Definition:T... |
proofwiki-17027 | Metric Space is T4 | A metric space $M = \struct {A, d}$ is a $T_4$ space. | Let $H$ and $K$ be disjoint closed sets of $M$.
Let $g: A \to \R$ be defined as:
:$g = f_K - f_H$
where:
:$\forall x \in A: \map {f_K} x = \map d {x, K}$
:$\forall x \in A: \map {f_H} x = \map d {x, H}$
where $\map d {x, K}$, $\map d {x, H}$ denotes the distance from $x$ to $K$ and from $x$ to $H$ respectively.
From Di... | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:T4 Space|$T_4$ space]]. | Let $H$ and $K$ be [[Definition:Disjoint Sets|disjoint]] [[Definition:Closed Set (Topology)|closed sets]] of $M$.
Let $g: A \to \R$ be defined as:
:$g = f_K - f_H$
where:
:$\forall x \in A: \map {f_K} x = \map d {x, K}$
:$\forall x \in A: \map {f_H} x = \map d {x, H}$
where $\map d {x, K}$, $\map d {x, H}$ denotes the... | Metric Space is T4/Proof 1 | https://proofwiki.org/wiki/Metric_Space_is_T4 | https://proofwiki.org/wiki/Metric_Space_is_T4/Proof_1 | [
"Metric Space is T4",
"Metric Space fulfils all Separation Axioms",
"Examples of T4 Spaces"
] | [
"Definition:Metric Space",
"Definition:T4 Space"
] | [
"Definition:Disjoint Sets",
"Definition:Closed Set/Topology",
"Definition:Distance/Sets/Metric Spaces",
"Distance from Point to Subset is Continuous Function",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Disjoint Sets",
"Definition:Real I... |
proofwiki-17028 | Metric Space is T4 | A metric space $M = \struct {A, d}$ is a $T_4$ space. | We have that a metric space is fully $T_4$.
Then we have that a fully $T_4$ space is $T_4$.
{{qed}} | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:T4 Space|$T_4$ space]]. | We have that a [[Metric Space is Fully T4|metric space is fully $T_4$]].
Then we have that a [[Fully T4 Space is T4|fully $T_4$ space is $T_4$]].
{{qed}} | Metric Space is T4/Proof 2 | https://proofwiki.org/wiki/Metric_Space_is_T4 | https://proofwiki.org/wiki/Metric_Space_is_T4/Proof_2 | [
"Metric Space is T4",
"Metric Space fulfils all Separation Axioms",
"Examples of T4 Spaces"
] | [
"Definition:Metric Space",
"Definition:T4 Space"
] | [
"Metric Space is Fully T4",
"Fully T4 Space is T4"
] |
proofwiki-17029 | Product Space is T2.5 iff Factor Spaces are T2.5/Necessary Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$.
Let $\ds T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let each of $\struct {S_\alpha, \tau_\alpha}$ f... | Let each of $\struct {S_\alpha, \tau_\alpha}$ for $\alpha \in I$ be a $T_{2 \frac 1 2}$ space.
Let $x, y \in S : x \ne y$ be arbitrary.
Then $x_\alpha \ne y_\alpha$ for some $\alpha \in I$.
Since $\struct {S_\alpha, \tau_\alpha}$ is a $T_{2 \frac 1 2}$ space then:
:$\exists U, V \in \tau_\alpha: x_\alpha \in U, y_\alph... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] for $\alpha$ in some [[Definition:Indexing Set|indexing set]] $I$.
Let $\ds T = \struct {S, \tau} = \prod_{\alpha \math... | Let each of $\struct {S_\alpha, \tau_\alpha}$ for $\alpha \in I$ be a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
Let $x, y \in S : x \ne y$ be arbitrary.
Then $x_\alpha \ne y_\alpha$ for some $\alpha \in I$.
Since $\struct {S_\alpha, \tau_\alpha}$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]] then:
... | Product Space is T2.5 iff Factor Spaces are T2.5/Necessary Condition | https://proofwiki.org/wiki/Product_Space_is_T2.5_iff_Factor_Spaces_are_T2.5/Necessary_Condition | https://proofwiki.org/wiki/Product_Space_is_T2.5_iff_Factor_Spaces_are_T2.5/Necessary_Condition | [
"Product Space is T2.5 iff Factor Spaces are T2.5"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T2.5 Space",
"Definition:T2.5 Space"
] | [
"Definition:T2.5 Space",
"Definition:T2.5 Space",
"Definition:Projection (Mapping Theory)",
"Preimage of Intersection under Mapping",
"Preimage of Subset is Subset of Preimage",
"Projection from Product Topology is Open and Continuous/General Result",
"Definition:Continuous Mapping (Topology)/Everywhere... |
proofwiki-17030 | P-adic Expansion Less Initial Zero Terms Represents Same P-adic Number | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.
Let $\ds \sum_{i \mathop = m}^\infty d_i p^i$ be a $p$-adic expansion that represents $a$.
Let $l$ be the first index $i \ge m$ such that $d_i \ne 0$
Then the series... | Let $\sequence {\alpha_n}$ be the sequence of partial sums:
:$\ds \forall n \in \N: \alpha _n = \sum_{i \mathop = 0}^n d_{n + m} p^{n + m}$
Let $\sequence {\beta_n}$ be the sequence of partial sums:
:$\ds \forall n \in \N: \beta _n = \sum_{i \mathop = 0}^n d_{n + l} p^{n + l}$
Then:
{{begin-eqn}}
{{eqn | l = \beta_n
... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $a$ be a [[Definition:P-adic Number|$p$-adic number]], that is [[Definition:Left Coset|left coset]], in $\Q_p$.
Let $\ds \sum_{i \mathop = m}^\in... | Let $\sequence {\alpha_n}$ be the sequence of [[Definition:Partial Sum|partial sums]]:
:$\ds \forall n \in \N: \alpha _n = \sum_{i \mathop = 0}^n d_{n + m} p^{n + m}$
Let $\sequence {\beta_n}$ be the sequence of [[Definition:Partial Sum|partial sums]]:
:$\ds \forall n \in \N: \beta _n = \sum_{i \mathop = 0}^n d_{n + l... | P-adic Expansion Less Initial Zero Terms Represents Same P-adic Number | https://proofwiki.org/wiki/P-adic_Expansion_Less_Initial_Zero_Terms_Represents_Same_P-adic_Number | https://proofwiki.org/wiki/P-adic_Expansion_Less_Initial_Zero_Terms_Represents_Same_P-adic_Number | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Number",
"Definition:Coset/Left Coset",
"Definition:P-adic Expansion",
"Definition:Equivalence Class/Representative",
"Definition:Smallest Element",
"Definition:Index",
"Definition:Series",
"Definition:P-adi... | [
"Definition:Series/Sequence of Partial Sums",
"Definition:Series/Sequence of Partial Sums",
"Definition:Subsequence",
"Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Subsequence is Equivalent to Cauchy Sequence",
"Definitio... |
proofwiki-17031 | Subsequence is Equivalent to Cauchy Sequence | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a Cauchy sequence in $R$.
Let $\sequence {x_{m_n} }$ be a subsequence of $\sequence {x_n}$.
Then:
:$\ds \lim_{n \mathop \to \infty} {x_n - x_{m_n} } = 0$ | From Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence:
:$\sequence {x_{m_n} }$ is a Cauchy sequence
Let $\epsilon > 0$.
By definition of a Cauchy sequence:
:$\exists N: \forall n, m > N: \norm {x_n - x_m } < \epsilon$
From Index of Subsequence not Less than its Index:
$\forall n \in \N : m_n \g... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] in $R$.
Let $\sequence {x_{m_n} }$ be a [[Definition:Subsequence|subsequence]] of $\sequence {x_n}$.
Then:
:$\ds \li... | From [[Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence]]:
:$\sequence {x_{m_n} }$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]
Let $\epsilon > 0$.
By definition of a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]:
:$\exists N: \forall n,... | Subsequence is Equivalent to Cauchy Sequence | https://proofwiki.org/wiki/Subsequence_is_Equivalent_to_Cauchy_Sequence | https://proofwiki.org/wiki/Subsequence_is_Equivalent_to_Cauchy_Sequence | [
"Cauchy Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Subsequence"
] | [
"Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Index of Subsequence not Less than its Index",
"Definition:Convergent Sequence/Normed Division Ring",
"Category:Cauchy Sequenc... |
proofwiki-17032 | Numbers with Square-Free Binomial Coefficients/Lemma | Let $n$ be a (strictly) positive integer.
Let $p$ be a prime number.
By Basis Representation Theorem, there is a unique sequence $\sequence {a_j}_{0 \mathop \le j \mathop \le r}$ such that:
:$(1): \quad \ds n = \sum_{k \mathop = 0}^r a_k p^k$
:$(2): \quad \ds \forall k \in \closedint 0 r: a_k \in \N_b$
:$(3): \quad r_... | Suppose $\forall i: 0 \le i \le r - 2: a_i = p - 1$.
Then:
{{begin-eqn}}
{{eqn | l = n + 1
| r = 1 + \sum_{i \mathop = 0}^r a_ip^i
| c = from $\ds n = \sum_{k \mathop = 0}^r a_k p^k$
}}
{{eqn | r = 1 + a_r p^r + a_{r - 1} p^{r - 1} + \sum_{i \mathop = 0}^{r - 2} (p - 1) p^i
| c = from $a_i = p - 1$ f... | Let $n$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $p$ be a [[Definition:Prime Number|prime number]].
By [[Basis Representation Theorem]], there is a [[Definition:Exactly One|unique]] [[Definition:Sequence|sequence]] $\sequence {a_j}_{0 \mathop \le j \mathop \le r}$ such that:
:$... | Suppose $\forall i: 0 \le i \le r - 2: a_i = p - 1$.
Then:
{{begin-eqn}}
{{eqn | l = n + 1
| r = 1 + \sum_{i \mathop = 0}^r a_ip^i
| c = from $\ds n = \sum_{k \mathop = 0}^r a_k p^k$
}}
{{eqn | r = 1 + a_r p^r + a_{r - 1} p^{r - 1} + \sum_{i \mathop = 0}^{r - 2} (p - 1) p^i
| c = from $a_i = p - 1$... | Numbers with Square-Free Binomial Coefficients/Lemma | https://proofwiki.org/wiki/Numbers_with_Square-Free_Binomial_Coefficients/Lemma | https://proofwiki.org/wiki/Numbers_with_Square-Free_Binomial_Coefficients/Lemma | [
"Square-Free Integers",
"Binomial Coefficients"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number",
"Basis Representation Theorem",
"Definition:Unique",
"Definition:Sequence",
"Definition:Divisor (Algebra)/Integer"
] | [
"Difference of Two Powers",
"Definition:Integer",
"Definition:Contrapositive Statement",
"Definition:Contrapositive Statement",
"Definition:Integer",
"Basis Representation Theorem",
"Difference of Two Powers",
"Basis Representation Theorem",
"Definition:Carry Digit",
"Kummer's Theorem",
"Definit... |
proofwiki-17033 | Derivative of Generating Function/General Result/Corollary | Let the coefficient of $z^n$ extracted from $\map G z$ be denoted:
:$\sqbrk {z^n} \map G z := a_n$
Then:
:$\sqbrk {z^m} \map G z = \dfrac 1 {m!} \map {G^{\paren m} } 0$
where $G^{\paren m}$ denotes the $m$th derivative of $G$. | {{begin-eqn}}
{{eqn | l = \dfrac {\d^m} {\d z^m} \map G z
| r = \sum_{k \mathop \ge 0} \dfrac {\paren {k + m}!} {k!} a_{k + m} z^k
| c = Derivative of Generating Function: General Result
}}
{{eqn | r = \dfrac {m!} {0!} a_m + \sum_{k \mathop \ge 1} \dfrac {\paren {k + m}!} {k!} a_{k + m} z^k
| c =
}}... | Let the [[Definition:Generating Function/Extraction of Coefficient|coefficient of $z^n$ extracted from $\map G z$]] be denoted:
:$\sqbrk {z^n} \map G z := a_n$
Then:
:$\sqbrk {z^m} \map G z = \dfrac 1 {m!} \map {G^{\paren m} } 0$
where $G^{\paren m}$ denotes the [[Definition:Higher Derivative|$m$th derivative]] of $G$... | {{begin-eqn}}
{{eqn | l = \dfrac {\d^m} {\d z^m} \map G z
| r = \sum_{k \mathop \ge 0} \dfrac {\paren {k + m}!} {k!} a_{k + m} z^k
| c = [[Derivative of Generating Function/General Result|Derivative of Generating Function: General Result]]
}}
{{eqn | r = \dfrac {m!} {0!} a_m + \sum_{k \mathop \ge 1} \dfrac... | Derivative of Generating Function/General Result/Corollary | https://proofwiki.org/wiki/Derivative_of_Generating_Function/General_Result/Corollary | https://proofwiki.org/wiki/Derivative_of_Generating_Function/General_Result/Corollary | [
"Generating Functions"
] | [
"Definition:Generating Function/Extraction of Coefficient",
"Definition:Derivative/Higher Derivatives/Higher Order"
] | [
"Derivative of Generating Function/General Result",
"Factorial/Examples/0",
"Category:Generating Functions"
] |
proofwiki-17034 | Derivative of Generating Function/General Result | Let $m$ be a positive integer.
Then:
:$\ds \dfrac {\d^m} {\d z^m} \map G z = \sum_{k \mathop \ge 0} \dfrac {\paren {k + m}!} {k!} a_{k + m} z^k$
=== Corollary ===
{{:Derivative of Generating Function/General Result/Corollary}} | Proof by induction: | Let $m$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$\ds \dfrac {\d^m} {\d z^m} \map G z = \sum_{k \mathop \ge 0} \dfrac {\paren {k + m}!} {k!} a_{k + m} z^k$
=== [[Derivative of Generating Function/General Result/Corollary|Corollary]] ===
{{:Derivative of Generating Function/General Result/Coroll... | Proof by [[Principle of Mathematical Induction|induction]]: | Derivative of Generating Function/General Result | https://proofwiki.org/wiki/Derivative_of_Generating_Function/General_Result | https://proofwiki.org/wiki/Derivative_of_Generating_Function/General_Result | [
"Generating Functions"
] | [
"Definition:Positive/Integer",
"Derivative of Generating Function/General Result/Corollary"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-17035 | D'Alembert's Formula | Let $t$ be time.
Let $x$ be position.
Let $\tuple {t, x} \stackrel u {\longrightarrow} \map u {t, x}: \R^2 \to \R$ be a twice-differentiable function in both variables.
Let $x \stackrel \phi {\longrightarrow} \map \phi x: \R \to \R$ be a differentiable function.
Let $x \stackrel \psi {\longrightarrow} \map \psi x: \R... | The general solution to the $1$-D wave equation:
:$\dfrac {\partial^2} {\partial t^2} \map u {x, t} = c^2 \dfrac {\partial^2} {\partial x^2} \map u {x, t}$
is given by:
:$\map u {x, t} = \map f {x + c t} + \map g {x - c t}$
where $f, g$ are arbitrary twice-differentiable functions.
From initial conditions we have:
{{be... | Let $t$ be [[Definition:Time|time]].
Let $x$ be [[Definition:Position|position]].
Let $\tuple {t, x} \stackrel u {\longrightarrow} \map u {t, x}: \R^2 \to \R$ be a [[Definition:Differentiability Class|twice-differentiable function]] in [[Definition:Real Function of Two Variables|both variables]].
Let $x \stackrel \... | The general [[Definition:Solution to Partial Differential Equation|solution]] to the [[Definition:1-Dimensional Wave Equation|$1$-D wave equation]]:
:$\dfrac {\partial^2} {\partial t^2} \map u {x, t} = c^2 \dfrac {\partial^2} {\partial x^2} \map u {x, t}$
is given by:
:$\map u {x, t} = \map f {x + c t} + \map g {x -... | D'Alembert's Formula | https://proofwiki.org/wiki/D'Alembert's_Formula | https://proofwiki.org/wiki/D'Alembert's_Formula | [
"Partial Differential Equations"
] | [
"Definition:Time",
"Definition:Position",
"Definition:Differentiability Class",
"Definition:Real Function/Two Variables",
"Definition:Differentiable Mapping/Real Function/Real Number Line",
"Definition:Definite Integral/Riemann",
"Definition:Constant",
"Definition:Solution to Partial Differential Equa... | [
"Definition:Solution to Partial Differential Equation",
"Definition:1-Dimensional Wave Equation",
"Definition:Differentiability Class",
"Definition:Constant",
"Chain Rule for Partial Derivatives",
"Sum Rule for Derivatives",
"Fundamental Theorem of Calculus",
"Definition:Primitive (Calculus)/Constant ... |
proofwiki-17036 | Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Banach Space | Let $I = \closedint a b$ be a closed real interval.
Let $\map C I$ be the space of real-valued functions, continuous on $I$.
Let $\norm {\,\cdot\,}_\infty$ be the supremum norm on real-valued functions, continuous on $I$.
Then $\struct {\map C I, \norm {\,\cdot\,}_\infty}$ is a Banach space. | A Banach space is a normed vector space, where a Cauchy sequence converges {{WRT}} the supplied norm.
To prove the theorem, we need to show that a Cauchy sequence in $\struct {\map C I, \norm {\,\cdot\,}_\infty}$ converges.
We take a Cauchy sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\struct {\map C I, \norm {\,\... | Let $I = \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\map C I$ be the [[Definition:Space of Real-Valued Functions Continuous on Closed Interval|space of real-valued functions, continuous on $I$]].
Let $\norm {\,\cdot\,}_\infty$ be the [[Definition:Supremum Norm/Continuous on Cl... | A [[Definition:Banach Space|Banach space]] is a [[Definition:Normed Vector Space|normed vector space]], where a [[Definition:Cauchy Sequence|Cauchy sequence]] [[Definition:Convergent Sequence in Normed Vector Space|converges]] {{WRT}} the supplied [[Definition:Norm on Vector Space|norm]].
To prove the theorem, we need... | Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm forms Banach Space | https://proofwiki.org/wiki/Space_of_Continuous_on_Closed_Interval_Real-Valued_Functions_with_Supremum_Norm_forms_Banach_Space | https://proofwiki.org/wiki/Space_of_Continuous_on_Closed_Interval_Real-Valued_Functions_with_Supremum_Norm_forms_Banach_Space | [
"Functional Analysis",
"Banach Spaces"
] | [
"Definition:Real Interval/Closed",
"Definition:Space of Real-Valued Functions Continuous on Closed Interval",
"Definition:Supremum Norm/Continuous on Closed Interval Real-Valued Function",
"Definition:Banach Space"
] | [
"Definition:Banach Space",
"Definition:Normed Vector Space",
"Definition:Cauchy Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Cauchy Se... |
proofwiki-17037 | Inductive Set under Mapping has Minimally Inductive Subset | Let $A$ be an inductive class under a mapping $g$.
Let $A$ be a set.
Then there exists some subset $S$ of $A$ such that $S$ is minimally inductive under $g$. | Let $S$ be the set defined as:
:$S = \set {s: \exists t \in A: t = \map g s} \cup \set \O$
By definition, we have that:
:$\O \in S$
:$s \in S \implies \map g s \in S$
So, by definition, $S$ is inductive under $g$.
Because $A$ is inductive under $g$:
:$\O \in A$
and so by definition of subset:
:$S \subseteq A$
It remain... | Let $A$ be an [[Definition:Inductive Class under General Mapping|inductive class]] under a [[Definition:Mapping|mapping]] $g$.
Let $A$ be a [[Definition:Set|set]].
Then there exists some [[Definition:Subset|subset]] $S$ of $A$ such that $S$ is [[Definition:Minimally Inductive Class under General Mapping|minimally in... | Let $S$ be the set defined as:
:$S = \set {s: \exists t \in A: t = \map g s} \cup \set \O$
By definition, we have that:
:$\O \in S$
:$s \in S \implies \map g s \in S$
So, by definition, $S$ is [[Definition:Inductive Class under General Mapping|inductive]] under $g$.
Because $A$ is [[Definition:Inductive Class under... | Inductive Set under Mapping has Minimally Inductive Subset | https://proofwiki.org/wiki/Inductive_Set_under_Mapping_has_Minimally_Inductive_Subset | https://proofwiki.org/wiki/Inductive_Set_under_Mapping_has_Minimally_Inductive_Subset | [
"Inductive Classes",
"Proofs by Induction",
"Minimally Inductive Classes"
] | [
"Definition:Inductive Class/General",
"Definition:Mapping",
"Definition:Set",
"Definition:Subset",
"Definition:Minimally Inductive Class under General Mapping"
] | [
"Definition:Inductive Class/General",
"Definition:Inductive Class/General",
"Definition:Subset",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Inductive Class/General",
"Principle of General Induction",
"Definition:Propositional Function",
"Definition:Inductive Class/Genera... |
proofwiki-17038 | Recurrence Relation for Bell Numbers | Let $B_n$ be the Bell number for $n \in \Z_{\ge 0}$.
Then:
:$B_{n + 1} = \ds \sum_{k \mathop = 0}^n \dbinom n k B_k$
where $\dbinom n k$ are binomial coefficients. | By definition of Bell numbers:
:$B_{n + 1}$ is the number of partitions of a (finite) set whose cardinality is $n + 1$.
Let $k \in \set {k \in \Z: 0 \le k \le n}$.
Let us form a partition of a (finite) set $S$ with cardinality $n + 1$ such that one component has $n + 1 - k > 0$ elements.
We can do this by first choosin... | Let $B_n$ be the [[Definition:Bell Number|Bell number]] for $n \in \Z_{\ge 0}$.
Then:
:$B_{n + 1} = \ds \sum_{k \mathop = 0}^n \dbinom n k B_k$
where $\dbinom n k$ are [[Definition:Binomial Coefficient|binomial coefficients]]. | By definition of [[Definition:Bell Number|Bell numbers]]:
:$B_{n + 1}$ is the number of [[Definition:Set Partition|partitions]] of a [[Definition:Finite Set|(finite) set]] whose [[Definition:Cardinality|cardinality]] is $n + 1$.
Let $k \in \set {k \in \Z: 0 \le k \le n}$.
Let us form a [[Definition:Set Partition|pa... | Recurrence Relation for Bell Numbers | https://proofwiki.org/wiki/Recurrence_Relation_for_Bell_Numbers | https://proofwiki.org/wiki/Recurrence_Relation_for_Bell_Numbers | [
"Recurrence Relation for Bell Numbers",
"Bell Numbers",
"Recurrence Relations"
] | [
"Definition:Bell Number",
"Definition:Binomial Coefficient"
] | [
"Definition:Bell Number",
"Definition:Set Partition",
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Set Partition",
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Set Partition/Component",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Defin... |
proofwiki-17039 | Natural Numbers are Comparable/Strong Result | Let $\N$ be the natural numbers.
Let $m, n \in \N$.
Then either:
:$(1): \quad m + 1 \le n$
or:
:$(2): \quad n \le m$ | Let $\N$ be defined as the von Neumann construction $\omega$.
By definition of the ordering on von Neumann construction:
:$m \le n \iff m \subseteq n$
From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.
Then from Minimally Inductive... | Let $\N$ be the [[Definition:Natural Numbers|natural numbers]].
Let $m, n \in \N$.
Then either:
:$(1): \quad m + 1 \le n$
or:
:$(2): \quad n \le m$ | Let $\N$ be defined as the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]] $\omega$.
By definition of the [[Definition:Ordering on Von Neumann Construction of Natural Numbers|ordering on von Neumann construction]]:
:$m \le n \iff m \subseteq n$
From [[Von Neumann Construction of N... | Natural Numbers are Comparable/Strong Result/Proof 1 | https://proofwiki.org/wiki/Natural_Numbers_are_Comparable/Strong_Result | https://proofwiki.org/wiki/Natural_Numbers_are_Comparable/Strong_Result/Proof_1 | [
"Natural Numbers are Comparable"
] | [
"Definition:Natural Numbers"
] | [
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Ordering on Natural Numbers/Von Neumann Construction",
"Von Neumann Construction of Natural Numbers is Minimally Inductive",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Natural Numbers/Von Neumann Construction/S... |
proofwiki-17040 | Natural Numbers are Comparable/Strong Result | Let $\N$ be the natural numbers.
Let $m, n \in \N$.
Then either:
:$(1): \quad m + 1 \le n$
or:
:$(2): \quad n \le m$ | {{ProofWanted|Proof using Minimally Inductive Class under Slowly Progressing Mapping is Nest by exploiting Successor Mapping is Slowly Progressing.}} | Let $\N$ be the [[Definition:Natural Numbers|natural numbers]].
Let $m, n \in \N$.
Then either:
:$(1): \quad m + 1 \le n$
or:
:$(2): \quad n \le m$ | {{ProofWanted|Proof using [[Minimally Inductive Class under Slowly Progressing Mapping is Nest]] by exploiting [[Successor Mapping is Slowly Progressing]].}} | Natural Numbers are Comparable/Strong Result/Proof 2 | https://proofwiki.org/wiki/Natural_Numbers_are_Comparable/Strong_Result | https://proofwiki.org/wiki/Natural_Numbers_are_Comparable/Strong_Result/Proof_2 | [
"Natural Numbers are Comparable"
] | [
"Definition:Natural Numbers"
] | [
"Minimally Inductive Class under Slowly Progressing Mapping is Nest",
"Successor Mapping is Slowly Progressing"
] |
proofwiki-17041 | Natural Number m is Less than n implies n is not Greater than Successor of n | Let $\N$ be the natural numbers.
Let $m, n \in \N$.
Then:
:$m < n \implies m + 1 \le n$ | Let $\N$ be considered as the naturally ordered semigroup:
:$\struct {\N, +, \le}$
The result follows from Sum with One is Immediate Successor in Naturally Ordered Semigroup. | Let $\N$ be the [[Definition:Natural Numbers|natural numbers]].
Let $m, n \in \N$.
Then:
:$m < n \implies m + 1 \le n$ | Let $\N$ be considered as the [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]]:
:$\struct {\N, +, \le}$
The result follows from [[Sum with One is Immediate Successor in Naturally Ordered Semigroup]]. | Natural Number m is Less than n implies n is not Greater than Successor of n/Proof using Naturally Ordered Semigroup | https://proofwiki.org/wiki/Natural_Number_m_is_Less_than_n_implies_n_is_not_Greater_than_Successor_of_n | https://proofwiki.org/wiki/Natural_Number_m_is_Less_than_n_implies_n_is_not_Greater_than_Successor_of_n/Proof_using_Naturally_Ordered_Semigroup | [
"Ordering on Natural Numbers",
"Natural Number m is Less than n implies n is not Greater than Successor of n"
] | [
"Definition:Natural Numbers"
] | [
"Definition:Naturally Ordered Semigroup",
"Sum with One is Immediate Successor in Naturally Ordered Semigroup"
] |
proofwiki-17042 | Natural Number m is Less than n implies n is not Greater than Successor of n | Let $\N$ be the natural numbers.
Let $m, n \in \N$.
Then:
:$m < n \implies m + 1 \le n$ | Let $\N$ be defined as the von Neumann construction $\omega$.
By definition of the ordering on von Neumann construction:
:$m \le n \iff m \subseteq n$
From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.
The result is then a direct a... | Let $\N$ be the [[Definition:Natural Numbers|natural numbers]].
Let $m, n \in \N$.
Then:
:$m < n \implies m + 1 \le n$ | Let $\N$ be defined as the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]] $\omega$.
By definition of the [[Definition:Ordering on Von Neumann Construction of Natural Numbers|ordering on von Neumann construction]]:
:$m \le n \iff m \subseteq n$
From [[Von Neumann Construction of N... | Natural Number m is Less than n implies n is not Greater than Successor of n/Proof using Von Neumann Construction | https://proofwiki.org/wiki/Natural_Number_m_is_Less_than_n_implies_n_is_not_Greater_than_Successor_of_n | https://proofwiki.org/wiki/Natural_Number_m_is_Less_than_n_implies_n_is_not_Greater_than_Successor_of_n/Proof_using_Von_Neumann_Construction | [
"Ordering on Natural Numbers",
"Natural Number m is Less than n implies n is not Greater than Successor of n"
] | [
"Definition:Natural Numbers"
] | [
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Ordering on Natural Numbers/Von Neumann Construction",
"Von Neumann Construction of Natural Numbers is Minimally Inductive",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Natural Numbers/Von Neumann Construction/S... |
proofwiki-17043 | Natural Number Ordering is Preserved by Successor Mapping | Let $\N$ be the natural numbers.
Let $m, n \in \N$.
Then:
:$n \le m \implies n^+ \le m^+$ | Let $\N$ be defined as the von Neumann construction $\omega$.
By definition of the ordering on von Neumann construction:
:$m \le n \iff m \subseteq n$
From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.
From Successor Mapping on Nat... | Let $\N$ be the [[Definition:Natural Numbers|natural numbers]].
Let $m, n \in \N$.
Then:
:$n \le m \implies n^+ \le m^+$ | Let $\N$ be defined as the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]] $\omega$.
By definition of the [[Definition:Ordering on Von Neumann Construction of Natural Numbers|ordering on von Neumann construction]]:
:$m \le n \iff m \subseteq n$
From [[Von Neumann Construction of N... | Natural Number Ordering is Preserved by Successor Mapping | https://proofwiki.org/wiki/Natural_Number_Ordering_is_Preserved_by_Successor_Mapping | https://proofwiki.org/wiki/Natural_Number_Ordering_is_Preserved_by_Successor_Mapping | [
"Ordering on Natural Numbers"
] | [
"Definition:Natural Numbers"
] | [
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Ordering on Natural Numbers/Von Neumann Construction",
"Von Neumann Construction of Natural Numbers is Minimally Inductive",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Natural Numbers/Von Neumann Construction/S... |
proofwiki-17044 | Non-Empty Bounded Subset of Natural Numbers has Greatest Element | Let $\omega$ be the set of natural numbers defined as the von Neumann construction.
Then every non-empty bounded subset of $\omega$ has a greatest element. | From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.
From Successor Mapping on Natural Numbers is Progressing, this successor mapping is a progressing mapping.
The result is a direct application of Non-Empty Bounded Subset of Minimal... | Let $\omega$ be the [[Definition:Natural Numbers|set of natural numbers]] defined as the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]].
Then every [[Definition:Non-Empty Set|non-empty]] [[Definition:Bounded Subset of Class|bounded subset]] of $\omega$ has a [[Definition:Greatest S... | From [[Von Neumann Construction of Natural Numbers is Minimally Inductive]], $\omega$ is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class]] under the [[Definition:Successor Mapping on Von Neumann Construction|successor mapping]].
From [[Successor Mapping on Natural Numbers is Prog... | Non-Empty Bounded Subset of Natural Numbers has Greatest Element | https://proofwiki.org/wiki/Non-Empty_Bounded_Subset_of_Natural_Numbers_has_Greatest_Element | https://proofwiki.org/wiki/Non-Empty_Bounded_Subset_of_Natural_Numbers_has_Greatest_Element | [
"Ordering on Natural Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Non-Empty Set",
"Definition:Bounded Class/Bounded Subset of Class",
"Definition:Greatest Set by Set Inclusion/Class Theory"
] | [
"Von Neumann Construction of Natural Numbers is Minimally Inductive",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Natural Numbers/Von Neumann Construction/Successor Mapping",
"Successor Mapping on Natural Numbers is Progressing",
"Definition:Natural Numbers/Von Neumann Construc... |
proofwiki-17045 | Natural Number m is Less than n iff m is an Element of n | Let $\omega$ be the set of natural numbers defined as the von Neumann construction.
Let $m, n \in \omega$.
Then:
:$m < n \iff m \in n$
That is, every natural number is the set of all smaller natural numbers. | By definition of the ordering on von Neumann construction:
:$m \le n \iff m \subseteq n$ | Let $\omega$ be the [[Definition:Natural Numbers|set of natural numbers]] defined as the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]].
Let $m, n \in \omega$.
Then:
:$m < n \iff m \in n$
That is, every [[Definition:Natural Numbers|natural number]] is the [[Definition:Set|set]] ... | By definition of the [[Definition:Ordering on Von Neumann Construction of Natural Numbers|ordering on von Neumann construction]]:
:$m \le n \iff m \subseteq n$ | Natural Number m is Less than n iff m is an Element of n | https://proofwiki.org/wiki/Natural_Number_m_is_Less_than_n_iff_m_is_an_Element_of_n | https://proofwiki.org/wiki/Natural_Number_m_is_Less_than_n_iff_m_is_an_Element_of_n | [
"Ordering on Natural Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers"
] | [
"Definition:Ordering on Natural Numbers/Von Neumann Construction"
] |
proofwiki-17046 | Trichotomy Law for Natural Numbers | Let $\omega$ be the set of natural numbers defined as the von Neumann construction.
Let $m, n \in \omega$.
Then one of the following cases holds:
:$m \in n$
:$m = n$
:$n \in m$ | By definition of the ordering on von Neumann construction:
:$m \le n \iff m \subseteq n$
From Natural Number m is Less than n iff m is an Element of n, we have:
:$m < n \iff m \in n$
Hence the theorem is equivalent to the statement that for every $m, n \in \omega$, one of the following holds:
:$m \subsetneq n$
:$m = n$... | Let $\omega$ be the [[Definition:Natural Numbers|set of natural numbers]] defined as the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]].
Let $m, n \in \omega$.
Then one of the following cases holds:
:$m \in n$
:$m = n$
:$n \in m$ | By definition of the [[Definition:Ordering on Von Neumann Construction of Natural Numbers|ordering on von Neumann construction]]:
:$m \le n \iff m \subseteq n$
From [[Natural Number m is Less than n iff m is an Element of n]], we have:
:$m < n \iff m \in n$
Hence the theorem is equivalent to the statement that for ... | Trichotomy Law for Natural Numbers | https://proofwiki.org/wiki/Trichotomy_Law_for_Natural_Numbers | https://proofwiki.org/wiki/Trichotomy_Law_for_Natural_Numbers | [
"Ordering on Natural Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Natural Numbers/Von Neumann Construction"
] | [
"Definition:Ordering on Natural Numbers/Von Neumann Construction",
"Natural Number m is Less than n iff m is an Element of n",
"Natural Numbers are Comparable/Strong Result"
] |
proofwiki-17047 | Natural Number Less than or Equal to Successor of Another | Let $\N$ be the natural numbers.
Let $m, n \in \N$ such that $m \le n^+$.
Then either:
:$(1): \quad m \le n$
or:
:$(2): \quad m = n^+$ | Let $m \le n^+$.
Suppose $m \le n$ is false.
Then:
:$n^+ \le m$
and because $m \le n^+$:
:$m = n^+$
{{qed}} | Let $\N$ be the [[Definition:Natural Numbers|natural numbers]].
Let $m, n \in \N$ such that $m \le n^+$.
Then either:
:$(1): \quad m \le n$
or:
:$(2): \quad m = n^+$ | Let $m \le n^+$.
Suppose $m \le n$ is false.
Then:
:$n^+ \le m$
and because $m \le n^+$:
:$m = n^+$
{{qed}} | Natural Number Less than or Equal to Successor of Another | https://proofwiki.org/wiki/Natural_Number_Less_than_or_Equal_to_Successor_of_Another | https://proofwiki.org/wiki/Natural_Number_Less_than_or_Equal_to_Successor_of_Another | [
"Ordering on Natural Numbers"
] | [
"Definition:Natural Numbers"
] | [] |
proofwiki-17048 | Mapping whose Image of Natural Number n is Subset of Image of Successor | Let $f: \N \to A$ be a mapping from the set of natural numbers $\N$ to a class $A$.
Let $f$ have the property that:
:$\forall n \in \N: \map f n \subseteq \map f {n^+}$
where $n^+$ is the successor of $n$.
Then:
:$\forall n, m \in N: n \le m \implies \map f n \subseteq \map f m$ | Let us establish that:
:$n = m \implies \map f n = \map f m$
and so:
:$n = m \implies \map f n \subseteq \map f m$
Hence it is sufficient to demonstrate that:
:$\forall n < m: \map f n \subseteq \map f m$
The proof will proceed by the Principle of Finite Induction on $\N$.
Let $S$ be the set defined as:
:$S := \set {m ... | Let $f: \N \to A$ be a [[Definition:Mapping|mapping]] from the [[Definition:Natural Numbers|set of natural numbers]] $\N$ to a [[Definition:Class (Class Theory)|class]] $A$.
Let $f$ have the property that:
:$\forall n \in \N: \map f n \subseteq \map f {n^+}$
where $n^+$ is the [[Definition:Successor Mapping on Natur... | Let us establish that:
:$n = m \implies \map f n = \map f m$
and so:
:$n = m \implies \map f n \subseteq \map f m$
Hence it is sufficient to demonstrate that:
:$\forall n < m: \map f n \subseteq \map f m$
The proof will proceed by the [[Principle of Finite Induction]] on $\N$.
Let $S$ be the [[Definition:Set|set]]... | Mapping whose Image of Natural Number n is Subset of Image of Successor | https://proofwiki.org/wiki/Mapping_whose_Image_of_Natural_Number_n_is_Subset_of_Image_of_Successor | https://proofwiki.org/wiki/Mapping_whose_Image_of_Natural_Number_n_is_Subset_of_Image_of_Successor | [
"Ordering on Natural Numbers"
] | [
"Definition:Mapping",
"Definition:Natural Numbers",
"Definition:Class (Class Theory)",
"Definition:Successor Mapping on Natural Numbers"
] | [
"Principle of Finite Induction",
"Definition:Set",
"Definition:Set",
"Definition:Vacuous Truth",
"Principle of Finite Induction"
] |
proofwiki-17049 | Mapping whose Image of Natural Number n is Subset of Image of Successor/Corollary | Let $f$ have the property that:
:$\forall n \in \N: \map f n \subsetneqq \map f {n^+}$
where $n^+$ is the successor of $n$.
Then:
:$\forall n, m \in N: n < m \implies \map f n \subsetneqq \map f m$ | The proof will proceed by the Principle of Finite Induction on $\N$.
Let $S$ be the set defined as:
:$S := \set {m \in \N: \forall n < m: \map f n \subsetneqq \map f m}$
That is, $S$ is to be the set of all $n$ such that:
:$\forall n < m: \map f n \subsetneqq \map f m$
First we note that:
:$\not \exists n \in \N: n < 0... | Let $f$ have the property that:
:$\forall n \in \N: \map f n \subsetneqq \map f {n^+}$
where $n^+$ is the [[Definition:Successor Mapping on Natural Numbers|successor]] of $n$.
Then:
:$\forall n, m \in N: n < m \implies \map f n \subsetneqq \map f m$ | The proof will proceed by the [[Principle of Finite Induction]] on $\N$.
Let $S$ be the [[Definition:Set|set]] defined as:
:$S := \set {m \in \N: \forall n < m: \map f n \subsetneqq \map f m}$
That is, $S$ is to be the [[Definition:Set|set]] of all $n$ such that:
:$\forall n < m: \map f n \subsetneqq \map f m$
Firs... | Mapping whose Image of Natural Number n is Subset of Image of Successor/Corollary | https://proofwiki.org/wiki/Mapping_whose_Image_of_Natural_Number_n_is_Subset_of_Image_of_Successor/Corollary | https://proofwiki.org/wiki/Mapping_whose_Image_of_Natural_Number_n_is_Subset_of_Image_of_Successor/Corollary | [
"Ordering on Natural Numbers"
] | [
"Definition:Successor Mapping on Natural Numbers"
] | [
"Principle of Finite Induction",
"Definition:Set",
"Definition:Set",
"Definition:Vacuous Truth",
"Principle of Finite Induction"
] |
proofwiki-17050 | Finite Class is Set | Let $A$ be a finite class.
Then $A$ is a set. | Let it be assumed that all classes are subclasses of a basic universe $V$.
The proof proceeds by induction.
For all $n \in \N$, let $\map P n$ be the proposition:
:If $A$ is a finite class with $n$ elements, then $A$ is a set.
The {{axiom-link|the Empty Set|Class Theory}} gives that the empty class $\O$ is a set.
From ... | Let $A$ be a [[Definition:Finite Class|finite class]].
Then $A$ is a [[Definition:Set|set]]. | Let it be assumed that all [[Definition:Class (Class Theory)|classes]] are [[Definition:Subclass|subclasses]] of a [[Definition:Basic Universe|basic universe]] $V$.
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]... | Finite Class is Set | https://proofwiki.org/wiki/Finite_Class_is_Set | https://proofwiki.org/wiki/Finite_Class_is_Set | [
"Finite Classes",
"Finite Sets"
] | [
"Definition:Finite Class",
"Definition:Set"
] | [
"Definition:Class (Class Theory)",
"Definition:Subclass",
"Definition:Basic Universe",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Finite Class",
"Definition:Element/Class",
"Definition:Set",
"Definition:Empty Class (Class Theory)",
"Definition:Set",
"Empty Set i... |
proofwiki-17051 | Non-Empty Finite Set of Natural Numbers has Greatest Element | Let $A$ be a non-empty finite set of natural numbers.
Then $A$ has a greatest element. | The proof proceeds by induction.
By definition, if $A$ is a non-empty finite set, then $A$ has $m$ elements for some $m \in \N$ such that $m > 0$.
So, for all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:If $A$ has $n$ elements, then $A$ has a greatest element. | Let $A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|finite set]] of [[Definition:Natural Number|natural numbers]].
Then $A$ has a [[Definition:Greatest Element|greatest element]]. | The proof proceeds by [[Principle of Mathematical Induction|induction]].
By definition, if $A$ is a [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|finite set]], then $A$ has $m$ [[Definition:Element|elements]] for some $m \in \N$ such that $m > 0$.
So, for all $n \in \N_{>0}$, let $\map P n$ be the [[... | Non-Empty Finite Set of Natural Numbers has Greatest Element | https://proofwiki.org/wiki/Non-Empty_Finite_Set_of_Natural_Numbers_has_Greatest_Element | https://proofwiki.org/wiki/Non-Empty_Finite_Set_of_Natural_Numbers_has_Greatest_Element | [
"Ordering on Natural Numbers",
"Finite Sets"
] | [
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Natural Numbers",
"Definition:Greatest Element"
] | [
"Principle of Mathematical Induction",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Element",
"Definition:Proposition",
"Definition:Element",
"Definition:Greatest Element",
"Definition:Element",
"Definition:Greatest Element",
"Definition:Element",
"Definition:Greatest Element... |
proofwiki-17052 | Set of Subsets of Element of Minimally Inductive Class under Progressing Mapping is Finite | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Let $x \in M$.
Let $S$ be the set of all $y \in M$ such that $y \subseteq x$.
Then $S$ is finite. | The proof proceeds by general induction.
For all $x \in M$, let $\map P x$ be the proposition:
:$\set {y \in M: y \subseteq x}$ is finite. | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let $x \in M$.
Let $S$ be the [[Definition:Set|set]] of all $y \in M$ such that $y \subseteq x$.
Then... | The proof proceeds by [[Principle of General Induction|general induction]].
For all $x \in M$, let $\map P x$ be the [[Definition:Proposition|proposition]]:
:$\set {y \in M: y \subseteq x}$ is [[Definition:Finite Set|finite]]. | Set of Subsets of Element of Minimally Inductive Class under Progressing Mapping is Finite | https://proofwiki.org/wiki/Set_of_Subsets_of_Element_of_Minimally_Inductive_Class_under_Progressing_Mapping_is_Finite | https://proofwiki.org/wiki/Set_of_Subsets_of_Element_of_Minimally_Inductive_Class_under_Progressing_Mapping_is_Finite | [
"Minimally Inductive Classes",
"Progressing Mappings",
"Finite Sets"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping",
"Definition:Set",
"Definition:Finite Set"
] | [
"Principle of General Induction",
"Definition:Proposition",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Principle of General Induction",
"Definition:Finite Set"
] |
proofwiki-17053 | Minimally Inductive Class under Progressing Mapping with Fixed Element is Finite | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Let there exist an element $x \in M$ such that $x = \map g x$.
Then $M$ is a finite class. | By Set of Subsets of Element of Minimally Inductive Class under Progressing Mapping is Finite, the set:
:$\set {y \in M: y \subseteq x}$
is finite.
From Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element, $x$ is the greatest element of $M$.
Thus:
:$M = \set {y \in M: y \subseteq x}$
Hen... | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let there exist an [[Definition:Element of Class|element]] $x \in M$ such that $x = \map g x$.
Then $M... | By [[Set of Subsets of Element of Minimally Inductive Class under Progressing Mapping is Finite]], the [[Definition:Set|set]]:
:$\set {y \in M: y \subseteq x}$
is [[Definition:Finite Set|finite]].
From [[Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element]], $x$ is the [[Definition:Grea... | Minimally Inductive Class under Progressing Mapping with Fixed Element is Finite | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Progressing_Mapping_with_Fixed_Element_is_Finite | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Progressing_Mapping_with_Fixed_Element_is_Finite | [
"Minimally Inductive Classes",
"Progressing Mappings",
"Finite Classes"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping",
"Definition:Element/Class",
"Definition:Finite Class"
] | [
"Set of Subsets of Element of Minimally Inductive Class under Progressing Mapping is Finite",
"Definition:Set",
"Definition:Finite Set",
"Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element",
"Definition:Greatest Element"
] |
proofwiki-17054 | Subset of Natural Numbers is either Finite or Denumerable | Let $S$ be a subset of the natural numbers $\N$.
Then $S$ is either finite or denumerable. | Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction.
By the Well-Ordering Principle, $\omega$ is well-ordered by the $\le$ relation.
Thus from the Well-Ordering Principle, $S$ has a smallest element.
Let this smallest element of $S$ be denoted $s_0$.
Also from the Well-Ordering Pri... | Let $S$ be a [[Definition:Subset|subset]] of the [[Definition:Natural Number|natural numbers]] $\N$.
Then $S$ is either [[Definition:Finite Set|finite]] or [[Definition:Countably Infinite Class|denumerable]]. | Let $\omega$ denote the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] as defined by the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]].
By the [[Well-Ordering Principle]], $\omega$ is [[Definition:Well-Ordered Class under Subset Relation|well-ordered]] by... | Subset of Natural Numbers is either Finite or Denumerable | https://proofwiki.org/wiki/Subset_of_Natural_Numbers_is_either_Finite_or_Denumerable | https://proofwiki.org/wiki/Subset_of_Natural_Numbers_is_either_Finite_or_Denumerable | [
"Natural Numbers"
] | [
"Definition:Subset",
"Definition:Natural Numbers",
"Definition:Finite Set",
"Definition:Countably Infinite/Class"
] | [
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Natural Numbers/Von Neumann Construction",
"Well-Ordering Principle",
"Definition:Well-Ordered Class under Subset Relation",
"Definition:Relation",
"Well-Ordering Principle",
"Definition:Smallest Element",
"Definition:Smallest Element",
"... |
proofwiki-17055 | Non-Empty Set of Natural Numbers with no Greatest Element is Denumerable | Let $A$ be a non-empty set of natural numbers.
Let $A$ have no greatest element.
Then $A$ is a countably infinite set. | {{AimForCont}} $A$ is finite.
From Subset of Naturals is Finite iff Bounded, it follows that $A$ is bounded.
Hence $A$ has a greatest element.
This contradicts the fact that $A$ has no greatest element.
Hence by Proof by Contradiction it follows that $A$ is not finite.
From Subset of Natural Numbers is either Finite or... | Let $A$ be a [[Definition:Non-Empty Set|non-empty set]] of [[Definition:Natural Number|natural numbers]].
Let $A$ have no [[Definition:Greatest Element|greatest element]].
Then $A$ is a [[Definition:Countably Infinite Set|countably infinite set]]. | {{AimForCont}} $A$ is [[Definition:Finite Set|finite]].
From [[Subset of Naturals is Finite iff Bounded]], it follows that $A$ is [[Definition:Bounded Set|bounded]].
Hence $A$ has a [[Definition:Greatest Element|greatest element]].
This [[Definition:Contradiction|contradicts]] the fact that $A$ has no [[Definition:G... | Non-Empty Set of Natural Numbers with no Greatest Element is Denumerable | https://proofwiki.org/wiki/Non-Empty_Set_of_Natural_Numbers_with_no_Greatest_Element_is_Denumerable | https://proofwiki.org/wiki/Non-Empty_Set_of_Natural_Numbers_with_no_Greatest_Element_is_Denumerable | [
"Natural Numbers",
"Countably Infinite Sets"
] | [
"Definition:Non-Empty Set",
"Definition:Natural Numbers",
"Definition:Greatest Element",
"Definition:Countably Infinite/Set"
] | [
"Definition:Finite Set",
"Subset of Naturals is Finite iff Bounded",
"Definition:Bounded Set",
"Definition:Greatest Element",
"Definition:Contradiction",
"Definition:Greatest Element",
"Proof by Contradiction",
"Definition:Finite Set",
"Subset of Natural Numbers is either Finite or Denumerable",
"... |
proofwiki-17056 | Like Electric Charges Repel | Let $a$ and $b$ be stationary particles, each carrying an electric charge of $q_a$ and $q_b$ respectively.
Let $q_a$ and $q_b$ be of the same polarity.
That is, let $q_a$ and $q_b$ be like charges.
Then the forces exerted by $a$ on $b$, and by $b$ on $a$, are such as to cause $a$ and $b$ to repel each other. | By Coulomb's Law of Electrostatics:
:$\mathbf F_{a b} \propto \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$
where:
:$\mathbf F_{a b}$ is the force exerted on $b$ by the electric charge on $a$
:$\mathbf r_{a b}$ is the displacement vector from $a$ to $b$
:$r$ is the distance between $a$ and $b$.
Let $q_a$ and $q_b$ both b... | Let $a$ and $b$ be [[Definition:Stationary|stationary]] [[Definition:Particle|particles]], each carrying an [[Definition:Electric Charge|electric charge]] of $q_a$ and $q_b$ respectively.
Let $q_a$ and $q_b$ be of the same [[Definition:Polarity of Electric Charge|polarity]].
That is, let $q_a$ and $q_b$ be [[Definiti... | By [[Coulomb's Law of Electrostatics]]:
:$\mathbf F_{a b} \propto \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$
where:
:$\mathbf F_{a b}$ is the [[Definition:Force|force]] exerted on $b$ by the [[Definition:Electric Charge|electric charge]] on $a$
:$\mathbf r_{a b}$ is the [[Definition:Displacement|displacement vector]]... | Like Electric Charges Repel | https://proofwiki.org/wiki/Like_Electric_Charges_Repel | https://proofwiki.org/wiki/Like_Electric_Charges_Repel | [
"Electric Charge"
] | [
"Definition:Stationary",
"Definition:Particle",
"Definition:Electric Charge",
"Definition:Electric Charge/Polarity",
"Definition:Electric Charge/Polarity/Like",
"Definition:Force"
] | [
"Coulomb's Law of Electrostatics",
"Definition:Force",
"Definition:Electric Charge",
"Definition:Displacement",
"Definition:Distance between Points",
"Definition:Electric Charge/Polarity/Positive",
"Definition:Positive/Real Number",
"Definition:Multiplication/Real Numbers",
"Definition:Positive/Real... |
proofwiki-17057 | Unlike Electric Charges Attract | Let $a$ and $b$ be stationary particles, each carrying an electric charge of $q_a$ and $q_b$ respectively.
Let $q_a$ and $q_b$ be of the opposite polarity.
That is, let $q_a$ and $q_b$ be unlike charges.
Then the forces exerted by $a$ on $b$, and by $b$ on $a$, are such as to cause $a$ and $b$ to attract each other. | By Coulomb's Law of Electrostatics:
:$\mathbf F_{a b} \propto \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$
where:
:$\mathbf F_{a b}$ is the force exerted on $b$ by the electric charge on $a$
:$\mathbf r_{a b}$ is the displacement vector from $a$ to $b$
:$r$ is the distance between $a$ and $b$.
{{WLOG}}, let $q_a$ be pos... | Let $a$ and $b$ be [[Definition:Stationary|stationary]] [[Definition:Particle|particles]], each carrying an [[Definition:Electric Charge|electric charge]] of $q_a$ and $q_b$ respectively.
Let $q_a$ and $q_b$ be of the opposite [[Definition:Polarity of Electric Charge|polarity]].
That is, let $q_a$ and $q_b$ be [[Defi... | By [[Coulomb's Law of Electrostatics]]:
:$\mathbf F_{a b} \propto \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$
where:
:$\mathbf F_{a b}$ is the [[Definition:Force|force]] exerted on $b$ by the [[Definition:Electric Charge|electric charge]] on $a$
:$\mathbf r_{a b}$ is the [[Definition:Displacement|displacement vector]]... | Unlike Electric Charges Attract | https://proofwiki.org/wiki/Unlike_Electric_Charges_Attract | https://proofwiki.org/wiki/Unlike_Electric_Charges_Attract | [
"Electric Charge"
] | [
"Definition:Stationary",
"Definition:Particle",
"Definition:Electric Charge",
"Definition:Electric Charge/Polarity",
"Definition:Electric Charge/Polarity/Unlike",
"Definition:Force"
] | [
"Coulomb's Law of Electrostatics",
"Definition:Force",
"Definition:Electric Charge",
"Definition:Displacement",
"Definition:Distance between Points",
"Definition:Electric Charge/Polarity/Positive",
"Definition:Electric Charge/Polarity/Negative",
"Definition:Positive/Real Number",
"Definition:Multipl... |
proofwiki-17058 | Value of Vacuum Permittivity | The value of the '''vacuum permittivity''' is calculated as:
:$\varepsilon_0 = 8 \cdotp 85418 \, 78128 (13) \times 10^{-12} \, \mathrm F \, \mathrm m^{-1}$ (farads per metre)
with a relative uncertainty of $1 \cdotp 5 \times 10^{-10}$. | The '''vacuum permittivity''' is the physical constant denoted $\varepsilon_0$ defined as:
:$\varepsilon_0 := \dfrac {e^2} {2 \alpha h c}$
where:
:$e$ is the elementary charge
:$\alpha$ is the fine-structure constant
:$h$ is Planck's constant
:$c$ is the speed of light defined in $\mathrm m \, \mathrm s^{-1}$
$e$ is de... | The value of the '''[[Definition:Vacuum Permittivity|vacuum permittivity]]''' is calculated as:
:$\varepsilon_0 = 8 \cdotp 85418 \, 78128 (13) \times 10^{-12} \, \mathrm F \, \mathrm m^{-1}$ ([[Definition:Farad|farads]] per [[Definition:Metre|metre]])
with a [[Definition:Relative Uncertainty|relative uncertainty]] of... | The '''[[Definition:Vacuum Permittivity|vacuum permittivity]]''' is the [[Definition:Physical Constant|physical constant]] denoted $\varepsilon_0$ defined as:
:$\varepsilon_0 := \dfrac {e^2} {2 \alpha h c}$
where:
:$e$ is the [[Definition:Elementary Charge|elementary charge]]
:$\alpha$ is the [[Definition:Fine-Structu... | Value of Vacuum Permittivity/Proof 1 | https://proofwiki.org/wiki/Value_of_Vacuum_Permittivity | https://proofwiki.org/wiki/Value_of_Vacuum_Permittivity/Proof_1 | [
"Value of Vacuum Permittivity",
"Vacuum Permittivity"
] | [
"Definition:Vacuum Permittivity",
"Definition:Farad",
"Definition:Metric System/Length/Metre",
"Definition:Relative Uncertainty"
] | [
"Definition:Vacuum Permittivity",
"Definition:Physical Constant",
"Definition:Electric Charge/Quantum",
"Definition:Fine-Structure Constant",
"Definition:Planck's Constant",
"Definition:Speed of Light",
"Definition:Coulomb",
"Definition:SI/Energy/Joule",
"Definition:Time/Unit/Second",
"Definition:... |
proofwiki-17059 | Value of Vacuum Permittivity | The value of the '''vacuum permittivity''' is calculated as:
:$\varepsilon_0 = 8 \cdotp 85418 \, 78128 (13) \times 10^{-12} \, \mathrm F \, \mathrm m^{-1}$ (farads per metre)
with a relative uncertainty of $1 \cdotp 5 \times 10^{-10}$. | The '''vacuum permittivity''' is the physical constant denoted $\varepsilon_0$ defined as:
:$\varepsilon_0 := \dfrac 1 {\mu_0 c^2}$
where:
:$\mu_0$ is the vacuum permeability defined in $\mathrm H \, \mathrm m^{-1}$ (henries per metre)
:$c$ is the speed of light defined in $\mathrm m \, \mathrm s^{-1}$
$\mu_0$ has the ... | The value of the '''[[Definition:Vacuum Permittivity|vacuum permittivity]]''' is calculated as:
:$\varepsilon_0 = 8 \cdotp 85418 \, 78128 (13) \times 10^{-12} \, \mathrm F \, \mathrm m^{-1}$ ([[Definition:Farad|farads]] per [[Definition:Metre|metre]])
with a [[Definition:Relative Uncertainty|relative uncertainty]] of... | The '''[[Definition:Vacuum Permittivity|vacuum permittivity]]''' is the [[Definition:Physical Constant|physical constant]] denoted $\varepsilon_0$ defined as:
:$\varepsilon_0 := \dfrac 1 {\mu_0 c^2}$
where:
:$\mu_0$ is the [[Definition:Vacuum Permeability|vacuum permeability]] defined in $\mathrm H \, \mathrm m^{-1}$ ... | Value of Vacuum Permittivity/Proof 2 | https://proofwiki.org/wiki/Value_of_Vacuum_Permittivity | https://proofwiki.org/wiki/Value_of_Vacuum_Permittivity/Proof_2 | [
"Value of Vacuum Permittivity",
"Vacuum Permittivity"
] | [
"Definition:Vacuum Permittivity",
"Definition:Farad",
"Definition:Metric System/Length/Metre",
"Definition:Relative Uncertainty"
] | [
"Definition:Vacuum Permittivity",
"Definition:Physical Constant",
"Definition:Vacuum Permeability",
"Definition:Henry",
"Definition:Metric System/Length/Metre",
"Definition:Speed of Light",
"Definition:Henry/Base Units",
"Definition:Farad/Base Units"
] |
proofwiki-17060 | Successor Mapping on Natural Numbers has no Fixed Element | Let $\N$ denote the set of natural numbers.
Then:
:$\forall n \in \N: n + 1 \ne n$ | Consider the set of natural numbers as defined by the von Neumann construction.
From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\N$ is a minimally inductive class under the successor mapping.
Let $s: \N \to \N$ denote this successor mapping:
:$\forall x \in \N: \map s x := x + 1$
{{AimForCont}... | Let $\N$ denote the [[Definition:Natural Numbers|set of natural numbers]].
Then:
:$\forall n \in \N: n + 1 \ne n$ | Consider the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] as defined by the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]].
From [[Von Neumann Construction of Natural Numbers is Minimally Inductive]], $\N$ is a [[Definition:Minimally Inductive Class unde... | Successor Mapping on Natural Numbers has no Fixed Element | https://proofwiki.org/wiki/Successor_Mapping_on_Natural_Numbers_has_no_Fixed_Element | https://proofwiki.org/wiki/Successor_Mapping_on_Natural_Numbers_has_no_Fixed_Element | [
"Natural Numbers"
] | [
"Definition:Natural Numbers"
] | [
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Natural Numbers/Von Neumann Construction",
"Von Neumann Construction of Natural Numbers is Minimally Inductive",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Natural Numbers/Von Neumann Construction/Successor Mapping"... |
proofwiki-17061 | Residue of Fibonacci Number Modulo Fibonacci Number/Lemma | :<nowiki>$F_{m n + 1} \equiv \paren {\begin{cases} F_1 & : m \bmod 4 = 0 \\
F_{n - 1} & : m \bmod 4 = 1 \\
\paren {-1}^n F_1 & : m \bmod 4 = 2 \\
\paren {-1}^n F_{n - 1} & : m \bmod 4 = 3 \end{cases} } \pmod {F_n}$</nowiki> | We prove this by induction on $m$.
For all $m \in \N$, let $\map P m$ be the proposition:
:<nowiki>$F_{m n + 1} \equiv \paren {\begin{cases} F_1 & : m \bmod 4 = 0 \\
F_{n - 1} & : m \bmod 4 = 1 \\
\paren {-1}^n F_1 & : m \bmod 4 = 2 \\
\paren {-1}^n F_{n - 1} & : m \bmod 4 = 3 \end{cases} } \pmod {F_n}$</nowiki> | :<nowiki>$F_{m n + 1} \equiv \paren {\begin{cases} F_1 & : m \bmod 4 = 0 \\
F_{n - 1} & : m \bmod 4 = 1 \\
\paren {-1}^n F_1 & : m \bmod 4 = 2 \\
\paren {-1}^n F_{n - 1} & : m \bmod 4 = 3 \end{cases} } \pmod {F_n}$</nowiki> | We prove this by [[Principle of Mathematical Induction|induction]] on $m$.
For all $m \in \N$, let $\map P m$ be the [[Definition:Proposition|proposition]]:
:<nowiki>$F_{m n + 1} \equiv \paren {\begin{cases} F_1 & : m \bmod 4 = 0 \\
F_{n - 1} & : m \bmod 4 = 1 \\
\paren {-1}^n F_1 & : m \bmod 4 = 2 \\
\paren {-1}^n F_... | Residue of Fibonacci Number Modulo Fibonacci Number/Lemma | https://proofwiki.org/wiki/Residue_of_Fibonacci_Number_Modulo_Fibonacci_Number/Lemma | https://proofwiki.org/wiki/Residue_of_Fibonacci_Number_Modulo_Fibonacci_Number/Lemma | [
"Proofs by Induction",
"Residue of Fibonacci Number Modulo Fibonacci Number"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-17062 | Complement of Open Interval Defined by Absolute Value | :$\set {x \in \R: \size {\xi - x} \ge \delta} = \R \setminus \openint {\xi - \delta} {\xi + \delta}$
where:
:$\openint {\xi - \delta} {\xi + \delta}$ is the open real interval between $\xi - \delta$ and $\xi + \delta$
:$\setminus$ denotes the set difference operator. | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \set {x \in \R: \size {\xi - x} \ge \delta}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \notin
| r = \set {x \in \R: \size {\xi - x} < \delta}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \notin
| r = \... | :$\set {x \in \R: \size {\xi - x} \ge \delta} = \R \setminus \openint {\xi - \delta} {\xi + \delta}$
where:
:$\openint {\xi - \delta} {\xi + \delta}$ is the [[Definition:Open Real Interval|open real interval]] between $\xi - \delta$ and $\xi + \delta$
:$\setminus$ denotes the [[Definition:Set Difference|set difference ... | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \set {x \in \R: \size {\xi - x} \ge \delta}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \notin
| r = \set {x \in \R: \size {\xi - x} < \delta}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \notin
| r = \... | Complement of Open Interval Defined by Absolute Value | https://proofwiki.org/wiki/Complement_of_Open_Interval_Defined_by_Absolute_Value | https://proofwiki.org/wiki/Complement_of_Open_Interval_Defined_by_Absolute_Value | [
"Real Intervals",
"Absolute Value Function"
] | [
"Definition:Real Interval/Open",
"Definition:Set Difference"
] | [
"Open Interval Defined by Absolute Value"
] |
proofwiki-17063 | Complement of Closed Interval Defined by Absolute Value | :$\set {x \in \R: \size {\xi - x} > \delta} = \R \setminus \closedint {\xi - \delta} {\xi + \delta}$
where:
:$\closedint {\xi - \delta} {\xi + \delta}$ is the closed real interval between $\xi - \delta$ and $\xi + \delta$
:$\setminus$ denotes the set difference operator. | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \set {x \in \R: \size {\xi - x} > \delta}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \notin
| r = \set {x \in \R: \size {\xi - x} \le \delta}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \notin
| r = \... | :$\set {x \in \R: \size {\xi - x} > \delta} = \R \setminus \closedint {\xi - \delta} {\xi + \delta}$
where:
:$\closedint {\xi - \delta} {\xi + \delta}$ is the [[Definition:Closed Real Interval|closed real interval]] between $\xi - \delta$ and $\xi + \delta$
:$\setminus$ denotes the [[Definition:Set Difference|set diffe... | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \set {x \in \R: \size {\xi - x} > \delta}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \notin
| r = \set {x \in \R: \size {\xi - x} \le \delta}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \notin
| r = \... | Complement of Closed Interval Defined by Absolute Value | https://proofwiki.org/wiki/Complement_of_Closed_Interval_Defined_by_Absolute_Value | https://proofwiki.org/wiki/Complement_of_Closed_Interval_Defined_by_Absolute_Value | [
"Real Intervals",
"Absolute Value Function"
] | [
"Definition:Real Interval/Closed",
"Definition:Set Difference"
] | [
"Closed Interval Defined by Absolute Value"
] |
proofwiki-17064 | P-Sequence Space with P-Norm forms Banach Space | Let $\ell^p$ be a p-sequence space.
Let $\norm {\, \cdot \,}_p$ be a p-norm.
Then $\struct {\ell^p, \norm {\, \cdot \,}_p}$ is a Banach space. | A Banach space is a normed vector space, where a Cauchy sequence converges {{WRT}} the supplied norm.
To prove the theorem, we need to show that a Cauchy sequence in $\struct {\ell^p, \norm {\,\cdot\,}_p}$ converges.
We take a Cauchy sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\struct {\ell^p, \norm {\,\cdot\,}_p... | Let $\ell^p$ be a [[Definition:P-Sequence Space|p-sequence space]].
Let $\norm {\, \cdot \,}_p$ be a [[Definition:P-Norm|p-norm]].
Then $\struct {\ell^p, \norm {\, \cdot \,}_p}$ is a [[Definition:Banach Space|Banach space]]. | A [[Definition:Banach Space|Banach space]] is a [[Definition:Normed Vector Space|normed vector space]], where a [[Definition:Cauchy Sequence|Cauchy sequence]] [[Definition:Convergent Sequence in Normed Vector Space|converges]] {{WRT}} the supplied [[Definition:Norm on Vector Space|norm]].
To prove the theorem, we need... | P-Sequence Space with P-Norm forms Banach Space | https://proofwiki.org/wiki/P-Sequence_Space_with_P-Norm_forms_Banach_Space | https://proofwiki.org/wiki/P-Sequence_Space_with_P-Norm_forms_Banach_Space | [
"Functional Analysis",
"Banach Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:P-Norm",
"Definition:Banach Space"
] | [
"Definition:Banach Space",
"Definition:Normed Vector Space",
"Definition:Cauchy Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Cauchy Se... |
proofwiki-17065 | Size of y-1 lt n and Size of y+1 gt 1 over n | Let $T_n \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:
:$T_n = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}$
Then:
:$T_n = \openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n}$ | :500pxthumbright
First note that:
{{begin-eqn}}
{{eqn | l = T_n
| r = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}
| c =
}}
{{eqn | r = \set {y: \size {y - 1} < n} \cap \set {y: \size {y + 1} > \dfrac 1 n}
| c =
}}
{{end-eqn}}
We have:
{{begin-eqn}}
{{eqn | l = \set {y: \size {y - 1}... | Let $T_n \subseteq \R$ be the [[Definition:Subset|subset]] of the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ defined as:
:$T_n = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}$
Then:
:$T_n = \openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n}$ | :[[File:Union-of-Family-Example-1.png|500px|thumb|right]]
First note that:
{{begin-eqn}}
{{eqn | l = T_n
| r = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}
| c =
}}
{{eqn | r = \set {y: \size {y - 1} < n} \cap \set {y: \size {y + 1} > \dfrac 1 n}
| c =
}}
{{end-eqn}}
We have:
{{... | Size of y-1 lt n and Size of y+1 gt 1 over n | https://proofwiki.org/wiki/Size_of_y-1_lt_n_and_Size_of_y+1_gt_1_over_n | https://proofwiki.org/wiki/Size_of_y-1_lt_n_and_Size_of_y+1_gt_1_over_n | [
"Unions of Families",
"Intersections of Families"
] | [
"Definition:Subset",
"Definition:Set",
"Definition:Real Number"
] | [
"File:Union-of-Family-Example-1.png",
"Open Interval Defined by Absolute Value",
"Complement of Closed Interval Defined by Absolute Value",
"Category:Unions of Families",
"Category:Intersections of Families"
] |
proofwiki-17066 | Real Part as Mapping is Surjection | Let $f: \C \to \R$ be the projection from the complex numbers to the real numbers defined as:
:$\forall z \in \C: \map f z = \map \Re z$
where $\map \Re z$ denotes the real part of $z$.
Then $f$ is a surjection. | Let $x \in \R$ be a real number.
Let $y \in \R$ be an arbitrary real number.
Let $z \in \C$ be the complex number defined as:
:$z = x + i y$
Then we have:
:$\map \Re z = x$
That is:
:$\exists z \in \C: \map f z = x$
The result follows by definition of surjection.
{{qed}} | Let $f: \C \to \R$ be the [[Definition:Projection (Mapping Theory)|projection]] from the [[Definition:Complex Number|complex numbers]] to the [[Definition:Real Number|real numbers]] defined as:
:$\forall z \in \C: \map f z = \map \Re z$
where $\map \Re z$ denotes the [[Definition:Real Part|real part]] of $z$.
Then $f$... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $y \in \R$ be an arbitrary [[Definition:Real Number|real number]].
Let $z \in \C$ be the [[Definition:Complex Number|complex number]] defined as:
:$z = x + i y$
Then we have:
:$\map \Re z = x$
That is:
:$\exists z \in \C: \map f z = x$
The result foll... | Real Part as Mapping is Surjection | https://proofwiki.org/wiki/Real_Part_as_Mapping_is_Surjection | https://proofwiki.org/wiki/Real_Part_as_Mapping_is_Surjection | [
"Real Parts",
"Examples of Surjections"
] | [
"Definition:Projection (Mapping Theory)",
"Definition:Complex Number",
"Definition:Real Number",
"Definition:Complex Number/Real Part",
"Definition:Surjection"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Surjection"
] |
proofwiki-17067 | Imaginary Part as Mapping is Surjection | Let $f: \C \to \R$ be the projection from the complex numbers to the real numbers defined as:
:$\forall z \in \C: \map f z = \map \Im z$
where $\map \Im z$ denotes the imaginary part of $z$.
Then $f$ is a surjection. | Let $y \in \R$ be a real number.
Let $x \in \R$ be an arbitrary real number.
Let $z \in \C$ be the complex number defined as:
:$z = x + i y$
Then we have:
:$\map \Im z = y$
That is:
:$\exists z \in \C: \map f z = y$
The result follows by definition of surjection.
{{qed}} | Let $f: \C \to \R$ be the [[Definition:Projection (Mapping Theory)|projection]] from the [[Definition:Complex Number|complex numbers]] to the [[Definition:Real Number|real numbers]] defined as:
:$\forall z \in \C: \map f z = \map \Im z$
where $\map \Im z$ denotes the [[Definition:Imaginary Part|imaginary part]] of $z$.... | Let $y \in \R$ be a [[Definition:Real Number|real number]].
Let $x \in \R$ be an arbitrary [[Definition:Real Number|real number]].
Let $z \in \C$ be the [[Definition:Complex Number|complex number]] defined as:
:$z = x + i y$
Then we have:
:$\map \Im z = y$
That is:
:$\exists z \in \C: \map f z = y$
The result foll... | Imaginary Part as Mapping is Surjection | https://proofwiki.org/wiki/Imaginary_Part_as_Mapping_is_Surjection | https://proofwiki.org/wiki/Imaginary_Part_as_Mapping_is_Surjection | [
"Imaginary Parts",
"Examples of Surjections"
] | [
"Definition:Projection (Mapping Theory)",
"Definition:Complex Number",
"Definition:Real Number",
"Definition:Complex Number/Imaginary Part",
"Definition:Surjection"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Surjection"
] |
proofwiki-17068 | Condition for Mapping from Quotient Set to be Well-Defined | Let $S$ and $T$ be sets.
Let $\RR$ be an equivalence relation on $S$.
Let $f: S \to T$ be a mapping from $S$ to $T$.
Let $S / \RR$ be the quotient set of $S$ induced by $\RR$.
Let $q_\RR: S \to S / \RR$ be the quotient mapping induced by $\RR$.
Then:
:there exists a mapping $\phi: S / \RR \to T$ such that $\phi \circ q... | From Condition for Composite Mapping on Left, we have:
:$\exists \phi: S / \RR \to T$ such that $\phi$ is a mapping and $\phi \circ q_\RR = f$
{{iff}}:
:$\forall x, y \in S: \map {q_\RR} x = \map {q_\RR} y \implies \map f x = \map f y$
But by definition of the quotient mapping induced by $\RR$:
:$\map {q_\RR} x = \map ... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] from $S$ to $T$.
Let $S / \RR$ be the [[Definition:Quotient Set|quotient set of $S$ induced by $\RR$]].
Let $q_\RR: S \to S / \RR$ be the... | From [[Condition for Composite Mapping on Left]], we have:
:$\exists \phi: S / \RR \to T$ such that $\phi$ is a [[Definition:Mapping|mapping]] and $\phi \circ q_\RR = f$
{{iff}}:
:$\forall x, y \in S: \map {q_\RR} x = \map {q_\RR} y \implies \map f x = \map f y$
But by definition of the [[Definition:Quotient Mapping|... | Condition for Mapping from Quotient Set to be Well-Defined | https://proofwiki.org/wiki/Condition_for_Mapping_from_Quotient_Set_to_be_Well-Defined | https://proofwiki.org/wiki/Condition_for_Mapping_from_Quotient_Set_to_be_Well-Defined | [
"Quotient Mappings",
"Quotient Sets"
] | [
"Definition:Set",
"Definition:Equivalence Relation",
"Definition:Mapping",
"Definition:Quotient Set",
"Definition:Quotient Mapping",
"Definition:Mapping"
] | [
"Condition for Composite Mapping on Left",
"Definition:Mapping",
"Definition:Quotient Mapping"
] |
proofwiki-17069 | Automorphism Maps Generator to Generator | Let $G$ be a cyclic group.
Let $g$ be a generator of $G$.
Let $\phi$ be an automorphism on $G$.
Then $\map \phi g$ is also a generator of $G$. | By definition of automorphism, $\phi$ is a homomorphism
It follows that this result is a specific instance of Homomorphic Image of Cyclic Group is Cyclic Group.
{{qed}}
Category:Cyclic Groups
Category:Group Automorphisms
kwnpb54bt14so7j5unset7a5q7styla | Let $G$ be a [[Definition:Cyclic Group|cyclic group]].
Let $g$ be a [[Definition:Generator of Cyclic Group|generator]] of $G$.
Let $\phi$ be an [[Definition:Group Automorphism|automorphism]] on $G$.
Then $\map \phi g$ is also a [[Definition:Generator of Cyclic Group|generator]] of $G$. | By definition of [[Definition:Group Automorphism|automorphism]], $\phi$ is a [[Definition:Group Homomorphism|homomorphism]]
It follows that this result is a specific instance of [[Homomorphic Image of Cyclic Group is Cyclic Group]].
{{qed}}
[[Category:Cyclic Groups]]
[[Category:Group Automorphisms]]
kwnpb54bt14so7j5u... | Automorphism Maps Generator to Generator | https://proofwiki.org/wiki/Automorphism_Maps_Generator_to_Generator | https://proofwiki.org/wiki/Automorphism_Maps_Generator_to_Generator | [
"Cyclic Groups",
"Group Automorphisms"
] | [
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator",
"Definition:Group Automorphism",
"Definition:Cyclic Group/Generator"
] | [
"Definition:Group Automorphism",
"Definition:Group Homomorphism",
"Homomorphic Image of Cyclic Group is Cyclic Group",
"Category:Cyclic Groups",
"Category:Group Automorphisms"
] |
proofwiki-17070 | Condition for Mapping from Quotient Set to be Surjection | Let the mapping $\phi: S / \RR \to T$ defined as:
:$\phi \circ q_\RR = f$
be well-defined.
Then:
:$\phi$ is a surjection
{{iff}}:
:$f$ is a surjection. | We are given that:
:$\phi \circ q_\RR = f$
is well-defined.
Note that from Quotient Mapping is Surjection, $q_\RR$ is a surjection. | Let the [[Definition:Mapping|mapping]] $\phi: S / \RR \to T$ defined as:
:$\phi \circ q_\RR = f$
be [[Definition:Well-Defined Mapping|well-defined]].
Then:
:$\phi$ is a [[Definition:Surjection|surjection]]
{{iff}}:
:$f$ is a [[Definition:Surjection|surjection]]. | We are given that:
:$\phi \circ q_\RR = f$
is [[Definition:Well-Defined Mapping|well-defined]].
Note that from [[Quotient Mapping is Surjection]], $q_\RR$ is a [[Definition:Surjection|surjection]]. | Condition for Mapping from Quotient Set to be Surjection | https://proofwiki.org/wiki/Condition_for_Mapping_from_Quotient_Set_to_be_Surjection | https://proofwiki.org/wiki/Condition_for_Mapping_from_Quotient_Set_to_be_Surjection | [
"Quotient Mappings",
"Quotient Sets",
"Surjections"
] | [
"Definition:Mapping",
"Definition:Well-Defined/Mapping",
"Definition:Surjection",
"Definition:Surjection"
] | [
"Definition:Well-Defined/Mapping",
"Quotient Mapping is Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Surjection"
] |
proofwiki-17071 | Condition for Mapping from Quotient Set to be Injection | Let the mapping $\phi: S / \RR \to T$ defined as:
:$\phi \circ q_\RR = f$
be well-defined.
Then:
:$\phi$ is an injection
{{iff}}:
:$\forall x, y \in S: \tuple {x, y} \in \RR \iff \map f x = \map f y$ | From Condition for Mapping from Quotient Set to be Well-Defined, $\phi$ is well-defined {{iff}}:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$
By definition of injection, $\phi$ is injective {{iff}}:
:$\map \phi {\eqclass x \RR} = \map \phi {\eqclass y \RR} \implies \eqclass x \RR = \eqclass... | Let the [[Definition:Mapping|mapping]] $\phi: S / \RR \to T$ defined as:
:$\phi \circ q_\RR = f$
be [[Definition:Well-Defined Mapping|well-defined]].
Then:
:$\phi$ is an [[Definition:Injection|injection]]
{{iff}}:
:$\forall x, y \in S: \tuple {x, y} \in \RR \iff \map f x = \map f y$ | From [[Condition for Mapping from Quotient Set to be Well-Defined]], $\phi$ is [[Definition:Well-Defined Mapping|well-defined]] {{iff}}:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$
By definition of [[Definition:Injection|injection]], $\phi$ is [[Definition:Injection|injective]] {{iff}}:
... | Condition for Mapping from Quotient Set to be Injection | https://proofwiki.org/wiki/Condition_for_Mapping_from_Quotient_Set_to_be_Injection | https://proofwiki.org/wiki/Condition_for_Mapping_from_Quotient_Set_to_be_Injection | [
"Quotient Mappings",
"Quotient Sets",
"Injections"
] | [
"Definition:Mapping",
"Definition:Well-Defined/Mapping",
"Definition:Injection"
] | [
"Condition for Mapping from Quotient Set to be Well-Defined",
"Definition:Well-Defined/Mapping",
"Definition:Injection",
"Definition:Injection"
] |
proofwiki-17072 | Mapping from Quotient Set when Defined is Unique | Let the mapping $\phi: S / \RR \to T$ defined as:
:$\phi \circ q_\RR = f$
be well-defined.
Then $\phi$ is unique. | From Condition for Mapping from Quotient Set to be Well-Defined, $\phi$ is well-defined {{iff}}:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$
From Quotient Mapping is Surjection, $q_\RR$ is a surjection.
Suppose $\psi: S / \RR \to T$ is another well-defined mapping defined as:
:$\psi \circ ... | Let the [[Definition:Mapping|mapping]] $\phi: S / \RR \to T$ defined as:
:$\phi \circ q_\RR = f$
be [[Definition:Well-Defined Mapping|well-defined]].
Then $\phi$ is [[Definition:Unique|unique]]. | From [[Condition for Mapping from Quotient Set to be Well-Defined]], $\phi$ is [[Definition:Well-Defined Mapping|well-defined]] {{iff}}:
:$\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$
From [[Quotient Mapping is Surjection]], $q_\RR$ is a [[Definition:Surjection|surjection]].
Suppose $\psi:... | Mapping from Quotient Set when Defined is Unique | https://proofwiki.org/wiki/Mapping_from_Quotient_Set_when_Defined_is_Unique | https://proofwiki.org/wiki/Mapping_from_Quotient_Set_when_Defined_is_Unique | [
"Quotient Mappings",
"Quotient Sets"
] | [
"Definition:Mapping",
"Definition:Well-Defined/Mapping",
"Definition:Unique"
] | [
"Condition for Mapping from Quotient Set to be Well-Defined",
"Definition:Well-Defined/Mapping",
"Quotient Mapping is Surjection",
"Definition:Surjection",
"Definition:Well-Defined/Mapping",
"Definition:Mapping",
"Surjection iff Right Cancellable"
] |
proofwiki-17073 | Limit of Subsequence equals Limit of Sequence/Normed Vector Space | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\sequence {x_n}$ be a sequence in $X$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$.
Then:
:... | Let $\epsilon > 0$.
Since $\ds \lim_{n \mathop \to \infty} x_n = l$, it follows from the definition of limit that:
:$\exists N \in \N : \forall n \in \N : n > N \implies \norm {x_n - l} < \epsilon$
Now let $R = N$.
Then from Strictly Increasing Sequence of Natural Numbers:
:$\forall r > R: n_r \ge r$
Thus $n_r > N$ and... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $X$]].
Let $\sequence {x_n}$ be [[Definition:Convergent Sequence in Normed Vector Space|convergent in the norm]] $\norm {\, \cdot \,}$ to the following [[De... | Let $\epsilon > 0$.
Since $\ds \lim_{n \mathop \to \infty} x_n = l$, it follows from the definition of [[Definition:Limit of Sequence in Normed Vector Space|limit]] that:
:$\exists N \in \N : \forall n \in \N : n > N \implies \norm {x_n - l} < \epsilon$
Now let $R = N$.
Then from [[Strictly Increasing Sequence of N... | Limit of Subsequence equals Limit of Sequence/Normed Vector Space | https://proofwiki.org/wiki/Limit_of_Subsequence_equals_Limit_of_Sequence/Normed_Vector_Space | https://proofwiki.org/wiki/Limit_of_Subsequence_equals_Limit_of_Sequence/Normed_Vector_Space | [
"Normed Vector Spaces",
"Convergence",
"Limits of Sequences"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Limit of Sequence/Normed Vector Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Definition:Limit of Sequence/Normed Vector Space",
"Strictly Increasing Sequence of Natural Numbers",
"Category:Normed Vector Spaces",
"Category:Convergence",
"Category:Limits of Sequences"
] |
proofwiki-17074 | Conditions for Commutative Diagram on Quotient Mappings between Mappings | Let $A$ and $B$ be sets.
Let $\RR_S$ and $\RR_T$ be equivalence relations on $S$ and $T$ respectively.
Let $f: S \to T$ be a mapping from $S$ to $T$.
Let $S / \RR_S$ and $T / \RR_T$ be the quotient sets of $S$ and $T$ induced by $\RR_S$ and $\RR_T$ respectively.
Let $q_S: S \to S / \RR_S$ and $q_T: T \to T / \RR_T$ be ... | Consider the commutative diagram:
::<nowiki>$\begin {xy} \xymatrix@L + 2mu@ + 1em {
S \ar[rr]^*{f}
\ar[dd]_*{q_S}
\ar[ddrr]^*{q_T \circ f}
& & T \ar[dd]^*{q_T} \\
& & \\
S / \RR_S \ar@{-->}[rr]_*{g} & & T / \RR_T } \end {xy}$</nowiki>
We consider the mapping $q_T \circ f: S \to T / \RR_T$.
From Condition for Map... | Let $A$ and $B$ be [[Definition:Set|sets]].
Let $\RR_S$ and $\RR_T$ be [[Definition:Equivalence Relation|equivalence relations]] on $S$ and $T$ respectively.
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] from $S$ to $T$.
Let $S / \RR_S$ and $T / \RR_T$ be the [[Definition:Quotient Set|quotient sets]] of $S$ ... | Consider the [[Definition:Commutative Diagram|commutative diagram]]:
::<nowiki>$\begin {xy} \xymatrix@L + 2mu@ + 1em {
S \ar[rr]^*{f}
\ar[dd]_*{q_S}
\ar[ddrr]^*{q_T \circ f}
& & T \ar[dd]^*{q_T} \\
& & \\
S / \RR_S \ar@{-->}[rr]_*{g} & & T / \RR_T } \end {xy}$</nowiki>
We consider the [[Definition:Mapping|map... | Conditions for Commutative Diagram on Quotient Mappings between Mappings | https://proofwiki.org/wiki/Conditions_for_Commutative_Diagram_on_Quotient_Mappings_between_Mappings | https://proofwiki.org/wiki/Conditions_for_Commutative_Diagram_on_Quotient_Mappings_between_Mappings | [
"Quotient Mappings"
] | [
"Definition:Set",
"Definition:Equivalence Relation",
"Definition:Mapping",
"Definition:Quotient Set",
"Definition:Quotient Mapping",
"Definition:Mapping"
] | [
"Definition:Commutative Diagram",
"Definition:Mapping",
"Condition for Mapping from Quotient Set to be Well-Defined",
"Definition:Mapping"
] |
proofwiki-17075 | Subset of Finite Dimensional Normed Vector Space is Compact iff Closed and Bounded/Sufficient Condition | Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space.
Let $K \subset X$ be a compact subset.
Then $K$ is closed and bounded. | === Closedness ===
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $K$.
Suppose, $\sequence {x_n}_{n \mathop \in \N}$ converges to $L \in K$.
Then there is a subsequence $\sequence {x_{n_k}}_{k \mathop \in \N}$ convergent to $L' \in K$.
But $\sequence {x_{n_k}}_{k \mathop \in \N}$ is a subsequence $\sequenc... | Let $\struct {X, \norm {\,\cdot\,}}$ be a [[Definition:Finite Dimensional Vector Space|finite-dimensional]] [[Definition:Normed Vector Space|normed vector space]].
Let $K \subset X$ be a [[Definition:Compact Space/Normed Vector Space/Subspace|compact]] [[Definition:Subset|subset]].
Then $K$ is [[Definition:Closed Se... | === Closedness ===
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence in $K$]].
Suppose, $\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence in Normed Vector Space|converges]] to $L \in K$.
Then there is a [[Definition:Subsequence|subsequence]] $\sequence {x_{n_k}}_{k \ma... | Subset of Finite Dimensional Normed Vector Space is Compact iff Closed and Bounded/Sufficient Condition | https://proofwiki.org/wiki/Subset_of_Finite_Dimensional_Normed_Vector_Space_is_Compact_iff_Closed_and_Bounded/Sufficient_Condition | https://proofwiki.org/wiki/Subset_of_Finite_Dimensional_Normed_Vector_Space_is_Compact_iff_Closed_and_Bounded/Sufficient_Condition | [
"Subset of Finite Dimensional Normed Vector Space is Compact iff Closed and Bounded"
] | [
"Definition:Dimension of Vector Space/Finite",
"Definition:Normed Vector Space",
"Definition:Compact Space/Normed Vector Space/Subspace",
"Definition:Subset",
"Definition:Closed Set/Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Subsequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Subsequence",
"Limit of Subsequence equals Limit of Sequence/Normed Vector Space",
"Convergent Sequence in Normed Vector Space has Uniq... |
proofwiki-17076 | Geometrical Interpretation of Complex Modulus | Let $z \in \C$ be a complex number expressed in the complex plane.
Then the modulus of $z$ can be interpreted as the distance of $z$ from the origin. | Let $z = x + i y$.
By definition of the complex plane, it can be represented by the point $\tuple {x, y}$.
By the Distance Formula, the distance $d$ of $z$ from the origin is:
{{begin-eqn}}
{{eqn | l = d
| r = \sqrt {\paren {x - 0}^2 + \paren {y - 0}^2}
| c =
}}
{{eqn | r = \sqrt {x^2 + y^2}
| c =
}... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in the [[Definition:Complex Plane|complex plane]].
Then the [[Definition:Complex Modulus|modulus]] of $z$ can be interpreted as the [[Definition:Distance between Points|distance]] of $z$ from the [[Definition:Origin|origin]]. | Let $z = x + i y$.
By definition of the [[Definition:Complex Plane|complex plane]], it can be represented by the [[Definition:Point|point]] $\tuple {x, y}$.
By the [[Distance Formula]], the [[Definition:Distance between Points|distance]] $d$ of $z$ from the [[Definition:Origin|origin]] is:
{{begin-eqn}}
{{eqn | l = ... | Geometrical Interpretation of Complex Modulus | https://proofwiki.org/wiki/Geometrical_Interpretation_of_Complex_Modulus | https://proofwiki.org/wiki/Geometrical_Interpretation_of_Complex_Modulus | [
"Complex Modulus",
"Geometry of Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Modulus",
"Definition:Distance between Points",
"Definition:Coordinate System/Origin"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Point",
"Distance Formula",
"Definition:Distance between Points",
"Definition:Coordinate System/Origin",
"Definition:Complex Modulus"
] |
proofwiki-17077 | Condition for Factoring of Quotient Mapping between Modulo Addition Groups | Let $m, n \in \Z_{>0}$ be strictly positive integers.
Let $\struct {\Z, +}$ denote the additive group of integers.
Let $\struct {\Z_m, +_m}$ and $\struct {\Z_n, +_n}$ denote the additive groups of integers modulo $m$ and $n$ respectively.
Let $f: \Z \to \Z_n$ be the quotient epimorphism from $\struct {\Z, +}$ to $\stru... | An example of the use of Third Isomorphism Theorem/Groups/Corollary.
{{ProofWanted|details}} | Let $m, n \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Let $\struct {\Z, +}$ denote the [[Definition:Additive Group of Integers|additive group of integers]].
Let $\struct {\Z_m, +_m}$ and $\struct {\Z_n, +_n}$ denote the [[Definition:Additive Group of Integers Modulo m|additive... | An example of the use of [[Third Isomorphism Theorem/Groups/Corollary]].
{{ProofWanted|details}} | Condition for Factoring of Quotient Mapping between Modulo Addition Groups | https://proofwiki.org/wiki/Condition_for_Factoring_of_Quotient_Mapping_between_Modulo_Addition_Groups | https://proofwiki.org/wiki/Condition_for_Factoring_of_Quotient_Mapping_between_Modulo_Addition_Groups | [
"Additive Groups of Integer Multiples"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Additive Group of Integers",
"Definition:Additive Group of Integers Modulo m",
"Definition:Quotient Epimorphism",
"Definition:Quotient Epimorphism",
"Definition:Group Homomorphism",
"Definition:Divisor (Algebra)/Integer"
] | [
"Third Isomorphism Theorem/Groups/Corollary"
] |
proofwiki-17078 | Factors of Group Direct Product are not Subgroups | Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups.
Let $\struct {G \times H, \circ}$ be the group direct product of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$.
Then neither $\struct {G, \circ_1}$ nor $\struct {H, \circ_2}$ is a subgroup of $\struct {G \times H, \circ}$. | A subgroup is by definition a subset which is a group.
But neither $G$ nor $H$ are actually subsets of their cartesian product $G \times H$.
Hence the result.
{{qed}}
Category:Group Direct Products
Category:Subgroups
ninzsg4wxsaaotzi11ntmic8a0nz9sn | Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be [[Definition:Group|groups]].
Let $\struct {G \times H, \circ}$ be the [[Definition:Group Direct Product|group direct product]] of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$.
Then neither $\struct {G, \circ_1}$ nor $\struct {H, \circ_2}$ is a [[Definitio... | A [[Definition:Subgroup|subgroup]] is by definition a [[Definition:Subset|subset]] which is a [[Definition:Group|group]].
But neither $G$ nor $H$ are actually [[Definition:Subset|subsets]] of their [[Definition:Cartesian Product|cartesian product]] $G \times H$.
Hence the result.
{{qed}}
[[Category:Group Direct Prod... | Factors of Group Direct Product are not Subgroups | https://proofwiki.org/wiki/Factors_of_Group_Direct_Product_are_not_Subgroups | https://proofwiki.org/wiki/Factors_of_Group_Direct_Product_are_not_Subgroups | [
"Group Direct Products",
"Subgroups"
] | [
"Definition:Group",
"Definition:Group Direct Product",
"Definition:Subgroup"
] | [
"Definition:Subgroup",
"Definition:Subset",
"Definition:Group",
"Definition:Subset",
"Definition:Cartesian Product",
"Category:Group Direct Products",
"Category:Subgroups"
] |
proofwiki-17079 | Canonical Injection of Real Number Line into Complex Plane | Let $\struct {\C, +}$ be the additive group of complex numbers.
Let $\struct {\R, +}$ be the additive group of real numbers.
Let $f: \R \to \C$ be the mapping from the real numbers to the complex numbers defined as:
:$\forall x \in \R: \map f z = x + 0 y$
Then $f: \struct {\R, +} \to \struct {\C, +}$ is a monomorphism. | Consider the mapping $g: \C \to \R$ defined as:
:$\forall z \in \C: \map f z = \map \Re z$
where $\map \Re z$ denotes the real part of $z$.
From Real Part as Mapping is Endomorphism for Complex Addition, this is a projection from $\C$ to $\R$.
The result follows from Canonical Injection is Right Inverse of Projection.
... | Let $\struct {\C, +}$ be the [[Definition:Additive Group of Complex Numbers|additive group of complex numbers]].
Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $f: \R \to \C$ be the [[Definition:Mapping|mapping]] from the [[Definition:Real Number|real nu... | Consider the [[Definition:Mapping|mapping]] $g: \C \to \R$ defined as:
:$\forall z \in \C: \map f z = \map \Re z$
where $\map \Re z$ denotes the [[Definition:Real Part|real part]] of $z$.
From [[Real Part as Mapping is Endomorphism for Complex Addition]], this is a [[Definition:Projection (Mapping Theory)|projection]]... | Canonical Injection of Real Number Line into Complex Plane | https://proofwiki.org/wiki/Canonical_Injection_of_Real_Number_Line_into_Complex_Plane | https://proofwiki.org/wiki/Canonical_Injection_of_Real_Number_Line_into_Complex_Plane | [
"Real Numbers",
"Complex Numbers",
"Group Direct Products"
] | [
"Definition:Additive Group of Complex Numbers",
"Definition:Additive Group of Real Numbers",
"Definition:Mapping",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Monomorphism"
] | [
"Definition:Mapping",
"Definition:Complex Number/Real Part",
"Real Part as Mapping is Endomorphism for Complex Addition",
"Definition:Projection (Mapping Theory)",
"Canonical Injection is Right Inverse of Projection"
] |
proofwiki-17080 | Additive Group of Complex Numbers is Direct Product of Reals with Reals | Let $\struct {\C, +}$ be the additive group of complex numbers.
Let $\struct {\R, +}$ be the additive group of real numbers.
Then the direct product $\struct {\R, +} \times \struct {\R, +}$ is isomorphic with $\struct {\C, +}$. | Let us define the mapping $\phi: \R^2 \to \C$ as:
:$\forall \tuple {x, y} \in \R^2: \phi: \tuple {x, y} = x + y i$
We will show that $\phi$ is a group isomorphism. | Let $\struct {\C, +}$ be the [[Definition:Additive Group of Complex Numbers|additive group of complex numbers]].
Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Then the [[Definition:Group Direct Product|direct product]] $\struct {\R, +} \times \struct {\R, ... | Let us define the [[Definition:Mapping|mapping]] $\phi: \R^2 \to \C$ as:
:$\forall \tuple {x, y} \in \R^2: \phi: \tuple {x, y} = x + y i$
We will show that $\phi$ is a [[Definition:Group Isomorphism|group isomorphism]]. | Additive Group of Complex Numbers is Direct Product of Reals with Reals | https://proofwiki.org/wiki/Additive_Group_of_Complex_Numbers_is_Direct_Product_of_Reals_with_Reals | https://proofwiki.org/wiki/Additive_Group_of_Complex_Numbers_is_Direct_Product_of_Reals_with_Reals | [
"Additive Group of Complex Numbers",
"Additive Group of Real Numbers",
"Group Direct Products"
] | [
"Definition:Additive Group of Complex Numbers",
"Definition:Additive Group of Real Numbers",
"Definition:Group Direct Product",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Mapping",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] |
proofwiki-17081 | Multiplicative Group of Complex Numbers is not Direct Product of Reals with Reals | Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.
Let $\struct {\R_{\ne 0}, \times}$ be the multiplicative group of real numbers.
Then the direct product $\struct {\R_{\ne 0}, \times} \times \struct {\R_{\ne 0}, \times}$ is not isomorphic with $\struct {\C_{\ne 0}, \times}$. | Let $\tuple {a, b}$ and $\tuple {c, d}$ be pairs of non-zero real numbers:
:$\tuple {a, b} \in \R_{\ne 0} \times \R_{\ne 0}$
:$\tuple {c, d} \in \R_{\ne 0} \times \R_{\ne 0}$
Then by definition of group direct product:
:$\tuple {a, b} \times \tuple {c, d} = \tuple {a \times c, b \times d}$
However, by interpreting $\tu... | Let $\struct {\C_{\ne 0}, \times}$ be the [[Definition:Multiplicative Group of Complex Numbers|multiplicative group of complex numbers]].
Let $\struct {\R_{\ne 0}, \times}$ be the [[Definition:Multiplicative Group of Real Numbers|multiplicative group of real numbers]].
Then the [[Definition:Group Direct Product|dire... | Let $\tuple {a, b}$ and $\tuple {c, d}$ be [[Definition:Ordered Pair|pairs]] of non-[[Definition:Zero (Number)|zero]] [[Definition:Real Number|real numbers]]:
:$\tuple {a, b} \in \R_{\ne 0} \times \R_{\ne 0}$
:$\tuple {c, d} \in \R_{\ne 0} \times \R_{\ne 0}$
Then by definition of [[Definition:Group Direct Product|grou... | Multiplicative Group of Complex Numbers is not Direct Product of Reals with Reals | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Numbers_is_not_Direct_Product_of_Reals_with_Reals | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Numbers_is_not_Direct_Product_of_Reals_with_Reals | [
"Multiplicative Group of Real Numbers",
"Multiplicative Group of Complex Numbers",
"Group Direct Products"
] | [
"Definition:Multiplicative Group of Complex Numbers",
"Definition:Multiplicative Group of Real Numbers",
"Definition:Group Direct Product",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Ordered Pair",
"Definition:Zero (Number)",
"Definition:Real Number",
"Definition:Group Direct Product",
"Definition:Complex Number/Definition 2",
"Definition:Multiplication/Complex Numbers"
] |
proofwiki-17082 | Imaginary Numbers under Addition form Group | Let $\II$ denote the set of complex numbers of the form $0 + i y$
That is, let $\II$ be the set of all wholly imaginary numbers.
Then the algebraic structure $\struct {\II, +}$ is a group. | We have that $\II$ is a non-empty subset of the complex numbers $\C$.
Indeed, for example:
:$0 + 0 i \in \II$
Now, let $0 + i x, 0 + i y \in \II$.
Then we have:
{{begin-eqn}}
{{eqn | l = \paren {0 + i x} + \paren {-\paren {0 + i y} }
| r = \paren {0 + i x} - \paren {0 + i y}
| c = Inverse for Complex Additi... | Let $\II$ denote the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] of the form $0 + i y$
That is, let $\II$ be the [[Definition:Set|set]] of all [[Definition:Wholly Imaginary|wholly imaginary]] numbers.
Then the [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\str... | We have that $\II$ is a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Complex Number|complex numbers]] $\C$.
Indeed, for example:
:$0 + 0 i \in \II$
Now, let $0 + i x, 0 + i y \in \II$.
Then we have:
{{begin-eqn}}
{{eqn | l = \paren {0 + i x} + \paren {-\paren {0 + i y} }
... | Imaginary Numbers under Addition form Group | https://proofwiki.org/wiki/Imaginary_Numbers_under_Addition_form_Group | https://proofwiki.org/wiki/Imaginary_Numbers_under_Addition_form_Group | [
"Complex Addition",
"Examples of Groups"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Set",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Algebraic Structure/One Operation",
"Definition:Group"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Complex Number",
"Inverse for Complex Addition",
"One-Step Subgroup Test"
] |
proofwiki-17083 | Imaginary Numbers under Multiplication do not form Group | Let $\II$ denote the set of complex numbers of the form $0 + i y$ for $y \in \R_{\ne 0}$.
That is, let $\II$ be the set of all wholly imaginary non-zero numbers.
Then the algebraic structure $\struct {\II, \times}$ is not a group. | Let $0 + i x \in \II$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {0 + i x} \times \paren {0 + i x}
| r = \paren {0 - x^2} + i \paren {0 \times x + 0 \times x}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = -x^2
| c =
}}
{{eqn | o = \notin
| r = \II
| c =
}}
{{end-eqn}}
So $\struct... | Let $\II$ denote the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] of the form $0 + i y$ for $y \in \R_{\ne 0}$.
That is, let $\II$ be the [[Definition:Set|set]] of all [[Definition:Wholly Imaginary|wholly imaginary]] non-[[Definition:Zero (Number)|zero]] numbers.
Then the [[Definition:Alge... | Let $0 + i x \in \II$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {0 + i x} \times \paren {0 + i x}
| r = \paren {0 - x^2} + i \paren {0 \times x + 0 \times x}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = -x^2
| c =
}}
{{eqn | o = \notin
| r = \II
| c =
}}
{{end-eqn}}
So $\str... | Imaginary Numbers under Multiplication do not form Group | https://proofwiki.org/wiki/Imaginary_Numbers_under_Multiplication_do_not_form_Group | https://proofwiki.org/wiki/Imaginary_Numbers_under_Multiplication_do_not_form_Group | [
"Complex Multiplication"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Set",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Zero (Number)",
"Definition:Algebraic Structure/One Operation",
"Definition:Group"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Group"
] |
proofwiki-17084 | Set of Isometries in Complex Plane under Composition forms Group | Let $S$ be the set of all bijective complex functions $f: \C \to \C$ which preserve distance when embedded in the complex plane.
That is:
:$\size {\map f a - \map f b} = \size {a - b}$
where $\size z$ denotes the complex modulus of $z \in \C$.
Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the ... | From Complex Plane is Metric Space, $\C$ can be treated as a metric space.
We have {{hypothesis}} that $f$ is:
:a distance-preserving mapping.
:a bijection.
Hence $f$ is an isometry on $\C$.
Taking the group axioms in turn: | Let $S$ be the [[Definition:Set|set]] of all [[Definition:Bijection|bijective]] [[Definition:Complex Function|complex functions]] $f: \C \to \C$ which preserve [[Definition:Distance between Points|distance]] when embedded in the [[Definition:Complex Plane|complex plane]].
That is:
:$\size {\map f a - \map f b} = \size... | From [[Complex Plane is Metric Space]], $\C$ can be treated as a [[Definition:Metric Space|metric space]].
We have {{hypothesis}} that $f$ is:
:a [[Definition:Distance-Preserving Mapping|distance-preserving mapping]].
:a [[Definition:Bijection|bijection]].
Hence $f$ is an [[Definition:Isometry (Metric Spaces)|isometr... | Set of Isometries in Complex Plane under Composition forms Group | https://proofwiki.org/wiki/Set_of_Isometries_in_Complex_Plane_under_Composition_forms_Group | https://proofwiki.org/wiki/Set_of_Isometries_in_Complex_Plane_under_Composition_forms_Group | [
"Complex Numbers",
"Isometries (Euclidean Geometry)",
"Examples of Groups"
] | [
"Definition:Set",
"Definition:Bijection",
"Definition:Complex Function",
"Definition:Distance between Points",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Modulus",
"Definition:Algebraic Structure/One Operation",
"Definition:Composition of Mappings",
"Definition:Group"
] | [
"Complex Plane is Metric Space",
"Definition:Metric Space",
"Definition:Distance-Preserving Mapping",
"Definition:Bijection",
"Definition:Isometry (Metric Spaces)",
"Axiom:Group Axioms",
"Definition:Isometry (Metric Spaces)",
"Definition:Isometry (Metric Spaces)",
"Definition:Isometry (Metric Spaces... |
proofwiki-17085 | Set of Affine Mappings on Real Line under Composition forms Group | Let $S$ be the set of all real functions $f: \R \to \R$ of the form:
:$\forall x \in \R: \map f x = r x + s$
where $r \in \R_{\ne 0}$ and $s \in \R$
Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$.
Then $\struct {S, \circ}$ is a group. | We note that $S$ is a subset of the set of all real functions on $\R$.
From Set of all Self-Maps under Composition forms Semigroup, we have that $\circ$ is associative.
Consider the real function $I: \R \to \R$ defined as:
:$\forall x \in \R: \map I x = 1 \times x + 0$
We have that:
:$I \in S$
:$I$ is the identity mapp... | Let $S$ be the [[Definition:Set|set]] of all [[Definition:Real Function|real functions]] $f: \R \to \R$ of the form:
:$\forall x \in \R: \map f x = r x + s$
where $r \in \R_{\ne 0}$ and $s \in \R$
Let $\struct {S, \circ}$ be the [[Definition:Algebraic Structure with One Operation|algebraic structure]] formed from $S... | We note that $S$ is a [[Definition:Subset|subset]] of the [[Definition:Set|set]] of all [[Definition:Real Function|real functions]] on $\R$.
From [[Set of all Self-Maps under Composition forms Semigroup]], we have that $\circ$ is [[Definition:Associative Operation|associative]].
Consider the [[Definition:Real Functi... | Set of Affine Mappings on Real Line under Composition forms Group | https://proofwiki.org/wiki/Set_of_Affine_Mappings_on_Real_Line_under_Composition_forms_Group | https://proofwiki.org/wiki/Set_of_Affine_Mappings_on_Real_Line_under_Composition_forms_Group | [
"Affine Geometry"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Algebraic Structure/One Operation",
"Definition:Composition of Mappings",
"Definition:Group"
] | [
"Definition:Subset",
"Definition:Set",
"Definition:Real Function",
"Set of all Self-Maps under Composition forms Semigroup",
"Definition:Associative Operation",
"Definition:Real Function",
"Definition:Identity Mapping",
"Definition:Empty Set",
"Definition:Closure (Abstract Algebra)/Algebraic Structu... |
proofwiki-17086 | Arbitrary Cyclic Group of Order 4 | Let $S = \set {1, 2, 3, 4}$.
Consider the algebraic structure $\struct {S, \circ}$ given by the Cayley table:
:$\begin{array}{r|rrrr}
\circ & 2 & 3 & 4 & 1 \\
\hline
2 & 2 & 3 & 4 & 1
\\
3 & 3 & 4 & 1 & 2
\\
4 & 4 & 1 & 2 & 3
\\
1 & 1 & 2 & 3 & 4
\\
\end{array}$
Then $\struct {S, \circ}$ is a group.
Specifically, $\st... | Let $S' = \set {0, 1, 2, 3}$.
Let $\phi: S \to S'$ be the bijection:
{{begin-eqn}}
{{eqn | l = \map \phi 2
| r = 0
}}
{{eqn | l = \map \phi 3
| r = 1
}}
{{eqn | l = \map \phi 4
| r = 2
}}
{{eqn | l = \map \phi 1
| r = 3
}}
{{end-eqn}}
By inspection, the Cayley table presented above is in the sam... | Let $S = \set {1, 2, 3, 4}$.
Consider the [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\struct {S, \circ}$ given by the [[Definition:Cayley Table|Cayley table]]:
:$\begin{array}{r|rrrr}
\circ & 2 & 3 & 4 & 1 \\
\hline
2 & 2 & 3 & 4 & 1
\\
3 & 3 & 4 & 1 & 2
\\
4 & 4 & 1 & 2 & 3
\\
1 & 1 ... | Let $S' = \set {0, 1, 2, 3}$.
Let $\phi: S \to S'$ be the [[Definition:Bijection|bijection]]:
{{begin-eqn}}
{{eqn | l = \map \phi 2
| r = 0
}}
{{eqn | l = \map \phi 3
| r = 1
}}
{{eqn | l = \map \phi 4
| r = 2
}}
{{eqn | l = \map \phi 1
| r = 3
}}
{{end-eqn}}
By inspection, the [[Definition:Ca... | Arbitrary Cyclic Group of Order 4 | https://proofwiki.org/wiki/Arbitrary_Cyclic_Group_of_Order_4 | https://proofwiki.org/wiki/Arbitrary_Cyclic_Group_of_Order_4 | [
"Cyclic Group of Order 4"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Cayley Table",
"Definition:Group",
"Definition:Cyclic Group",
"Definition:Order of Structure"
] | [
"Definition:Bijection",
"Definition:Cayley Table",
"Definition:Cayley Table",
"Definition:Cyclic Group",
"Definition:Order of Structure"
] |
proofwiki-17087 | Group Product Identity therefore Inverses/Part 1 | :$g h = e \implies h = g^{-1}$ and $g = h^{-1}$ | From the Division Laws for Groups:
:$g h = e \implies g = e h^{-1} = h^{-1}$
Also by the Division Laws for Groups:
:$g h = e \implies h = g^{-1} e = g^{-1}$
{{qed}} | :$g h = e \implies h = g^{-1}$ and $g = h^{-1}$ | From the [[Division Laws for Groups]]:
:$g h = e \implies g = e h^{-1} = h^{-1}$
Also by the [[Division Laws for Groups]]:
:$g h = e \implies h = g^{-1} e = g^{-1}$
{{qed}} | Group Product Identity therefore Inverses/Part 1/Proof 1 | https://proofwiki.org/wiki/Group_Product_Identity_therefore_Inverses/Part_1 | https://proofwiki.org/wiki/Group_Product_Identity_therefore_Inverses/Part_1/Proof_1 | [
"Group Product Identity therefore Inverses"
] | [] | [
"Division Laws for Groups",
"Division Laws for Groups"
] |
proofwiki-17088 | Group Product Identity therefore Inverses/Part 1 | :$g h = e \implies h = g^{-1}$ and $g = h^{-1}$ | Let $g h = e$.
Then:
{{begin-eqn}}
{{eqn | l = h
| r = e h
| c = {{Group-axiom|2}}
}}
{{eqn | r = \paren {g^{-1} g} h
| c = {{Group-axiom|3}}
}}
{{eqn | r = g^{-1} \paren {g h}
| c = {{Group-axiom|1}}
}}
{{eqn | r = g^{-1} e
| c = {{hypothesis}}
}}
{{eqn | r = g^{-1}
| c = {{Group-ax... | :$g h = e \implies h = g^{-1}$ and $g = h^{-1}$ | Let $g h = e$.
Then:
{{begin-eqn}}
{{eqn | l = h
| r = e h
| c = {{Group-axiom|2}}
}}
{{eqn | r = \paren {g^{-1} g} h
| c = {{Group-axiom|3}}
}}
{{eqn | r = g^{-1} \paren {g h}
| c = {{Group-axiom|1}}
}}
{{eqn | r = g^{-1} e
| c = {{hypothesis}}
}}
{{eqn | r = g^{-1}
| c = {{Group-a... | Group Product Identity therefore Inverses/Part 1/Proof 2 | https://proofwiki.org/wiki/Group_Product_Identity_therefore_Inverses/Part_1 | https://proofwiki.org/wiki/Group_Product_Identity_therefore_Inverses/Part_1/Proof_2 | [
"Group Product Identity therefore Inverses"
] | [] | [] |
proofwiki-17089 | Group Product Identity therefore Inverses/Part 2 | :$h g = e \implies h = g^{-1}$ and $g = h^{-1}$ | From the Division Laws for Groups:
:$h g = e \implies g = e h^{-1} = h^{-1}$
Also by the Division Laws for Groups:
:$h g = e \implies h = g^{-1} e = g^{-1}$
{{qed}} | :$h g = e \implies h = g^{-1}$ and $g = h^{-1}$ | From the [[Division Laws for Groups]]:
:$h g = e \implies g = e h^{-1} = h^{-1}$
Also by the [[Division Laws for Groups]]:
:$h g = e \implies h = g^{-1} e = g^{-1}$
{{qed}} | Group Product Identity therefore Inverses/Part 2/Proof 1 | https://proofwiki.org/wiki/Group_Product_Identity_therefore_Inverses/Part_2 | https://proofwiki.org/wiki/Group_Product_Identity_therefore_Inverses/Part_2/Proof_1 | [
"Group Product Identity therefore Inverses"
] | [] | [
"Division Laws for Groups",
"Division Laws for Groups"
] |
proofwiki-17090 | Group Product Identity therefore Inverses/Part 2 | :$h g = e \implies h = g^{-1}$ and $g = h^{-1}$ | Let $h g = e$.
Then:
{{begin-eqn}}
{{eqn | l = g
| r = e g
| c = {{Group-axiom|2}}
}}
{{eqn | r = \paren {h^{-1} h} g
| c = {{Group-axiom|3}}
}}
{{eqn | r = h^{-1} \paren {h g}
| c = {{Group-axiom|1}}
}}
{{eqn | r = h^{-1} e
| c = {{hypothesis}}
}}
{{eqn | r = h^{-1}
| c = {{Group-ax... | :$h g = e \implies h = g^{-1}$ and $g = h^{-1}$ | Let $h g = e$.
Then:
{{begin-eqn}}
{{eqn | l = g
| r = e g
| c = {{Group-axiom|2}}
}}
{{eqn | r = \paren {h^{-1} h} g
| c = {{Group-axiom|3}}
}}
{{eqn | r = h^{-1} \paren {h g}
| c = {{Group-axiom|1}}
}}
{{eqn | r = h^{-1} e
| c = {{hypothesis}}
}}
{{eqn | r = h^{-1}
| c = {{Group-a... | Group Product Identity therefore Inverses/Part 2/Proof 2 | https://proofwiki.org/wiki/Group_Product_Identity_therefore_Inverses/Part_2 | https://proofwiki.org/wiki/Group_Product_Identity_therefore_Inverses/Part_2/Proof_2 | [
"Group Product Identity therefore Inverses"
] | [] | [] |
proofwiki-17091 | Factors of Sums of Powers of 100,000/General Result | All integers $n$ of the form:
:$n = \ds \sum_{k \mathop = 0}^m 10^{r k}$ for $m \in \Z_{> 0}$
are composite for $r \ge 2$.
The only exceptions are:
:$r = 2^k, m = 1$ for some $k \in \N$
:$r = m + 1 =$ some odd prime
in which cases $n$ may be prime. | === Case $1$: $m + 1$ is Composite ===
Suppose $m + 1$ is composite.
Then:
:$\exists p, q > 1: m + 1 = p q$
By Division Theorem, for each $k$ with $0 \le z \le m$:
:$\exists i, j \in \N: 0 \le i \le q - 1, \, 0 \le j \le p - 1: k = i + q j$
Thus:
{{begin-eqn}}
{{eqn | l = n
| r = \sum_{k \mathop = 0}^m 10^{r k}
}... | All [[Definition:Integer|integers]] $n$ of the form:
:$n = \ds \sum_{k \mathop = 0}^m 10^{r k}$ for $m \in \Z_{> 0}$
are [[Definition:Composite Number|composite]] for $r \ge 2$.
The only exceptions are:
:$r = 2^k, m = 1$ for some $k \in \N$
:$r = m + 1 =$ some [[Definition:Odd Prime|odd prime]]
in which cases $n$ may... | === Case $1$: $m + 1$ is [[Definition:Composite Number|Composite]] ===
Suppose $m + 1$ is [[Definition:Composite Number|composite]].
Then:
:$\exists p, q > 1: m + 1 = p q$
By [[Division Theorem]], for each $k$ with $0 \le z \le m$:
:$\exists i, j \in \N: 0 \le i \le q - 1, \, 0 \le j \le p - 1: k = i + q j$
Thus:
... | Factors of Sums of Powers of 100,000/General Result | https://proofwiki.org/wiki/Factors_of_Sums_of_Powers_of_100,000/General_Result | https://proofwiki.org/wiki/Factors_of_Sums_of_Powers_of_100,000/General_Result | [
"Factors of Sums of Powers of 100,000"
] | [
"Definition:Integer",
"Definition:Composite Number",
"Definition:Odd Prime",
"Definition:Prime Number"
] | [
"Definition:Composite Number",
"Definition:Composite Number",
"Division Theorem",
"Definition:Addition/Integers",
"Definition:Composite Number"
] |
proofwiki-17092 | Odd Integers under Addition do not form Subgroup of Integers | Let $S$ denote the set of odd integers.
Then $\struct {S, +}$ is not a subgroup of the additive group of integers $\struct {\Z, +}$. | Consider the odd integers $1$ and $3$.
We have that $1 + 3 = 4$.
But $4$ is not odd.
Thus addition on $\struct {S, +}$ is not closed.
Hence $\struct {S, +}$ is not a group, let alone a subgroup of $\struct {\Z, +}$
{{qed}} | Let $S$ denote the [[Definition:Odd Integer|set of odd integers]].
Then $\struct {S, +}$ is not a [[Definition:Subgroup|subgroup]] of the [[Definition:Additive Group of Integers|additive group of integers]] $\struct {\Z, +}$. | Consider the [[Definition:Odd Integer|odd integers]] $1$ and $3$.
We have that $1 + 3 = 4$.
But $4$ is not [[Definition:Odd Integer|odd]].
Thus [[Definition:Integer Addition|addition]] on $\struct {S, +}$ is not [[Definition:Closed Operation|closed]].
Hence $\struct {S, +}$ is not a [[Definition:Group|group]], let ... | Odd Integers under Addition do not form Subgroup of Integers | https://proofwiki.org/wiki/Odd_Integers_under_Addition_do_not_form_Subgroup_of_Integers | https://proofwiki.org/wiki/Odd_Integers_under_Addition_do_not_form_Subgroup_of_Integers | [
"Integer Addition",
"Integral Domains"
] | [
"Definition:Odd Integer",
"Definition:Subgroup",
"Definition:Additive Group of Integers"
] | [
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Addition/Integers",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Group",
"Definition:Subgroup"
] |
proofwiki-17093 | Set of Transpositions is not Subgroup of Symmetric Group | Let $S$ be a finite set with $n$ elements such that $n > 2$.
Let $G = \struct {\map \Gamma S, \circ}$ denote the symmetric group on $S$.
Let $H \subseteq G$ denote the set of all transpositions of $S$ along with the identity mapping which moves no elements of $S$.
Then $H$ does not form a subgroup of $G$. | First it is noted that $H \subseteq G$, and that the identity mapping is an element of $H$.
Hence to demonstrate that $H$ is a subgroup of $G$, one may use the Two-Step Subgroup Test.
Let $\phi \in H$ be a transposition.
Then from Transposition is Self-Inverse, $\phi^{-1} \in H$.
So $H$ is closed under inversions.
Let ... | Let $S$ be a [[Definition:Finite Set|finite set]] with $n$ [[Definition:Element|elements]] such that $n > 2$.
Let $G = \struct {\map \Gamma S, \circ}$ denote the [[Definition:Symmetric Group|symmetric group]] on $S$.
Let $H \subseteq G$ denote the [[Definition:Set|set]] of all [[Definition:Transposition|transpositio... | First it is noted that $H \subseteq G$, and that the [[Definition:Identity Mapping|identity mapping]] is an [[Definition:Element|element]] of $H$.
Hence to demonstrate that $H$ is a [[Definition:Subgroup|subgroup]] of $G$, one may use the [[Two-Step Subgroup Test]].
Let $\phi \in H$ be a [[Definition:Transposition|tr... | Set of Transpositions is not Subgroup of Symmetric Group | https://proofwiki.org/wiki/Set_of_Transpositions_is_not_Subgroup_of_Symmetric_Group | https://proofwiki.org/wiki/Set_of_Transpositions_is_not_Subgroup_of_Symmetric_Group | [
"Symmetric Groups"
] | [
"Definition:Finite Set",
"Definition:Element",
"Definition:Symmetric Group",
"Definition:Set",
"Definition:Transposition",
"Definition:Identity Mapping",
"Definition:Element",
"Definition:Subgroup"
] | [
"Definition:Identity Mapping",
"Definition:Element",
"Definition:Subgroup",
"Two-Step Subgroup Test",
"Definition:Transposition",
"Transposition is Self-Inverse",
"Definition:Closed under Inversion",
"Definition:Cyclic Permutation",
"Definition:Transposition",
"Definition:Closure (Abstract Algebra... |
proofwiki-17094 | Positive Real Axis forms Subgroup of Complex Numbers under Multiplication | Let $S$ be the subset of the set of complex numbers $\C$ defined as:
:$S = \set {z \in \C: z = x + 0 i, x > 0}$
That is, let $S$ be the positive real axis of the complex plane.
Then the algebraic structure $\struct {S, \times}$ is a subgroup of the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$. | We have that $S$ is the same thing as $\R_{>0}$, the set of strictly positive real numbers:
:$\R_{>0} = \set {x \in \R: x > 0}$
From Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group, $\struct {S, \times}$ is a group.
Hence as $S$ is a group which is a subset of $\struct {\C_{\ne 0}, \t... | Let $S$ be the [[Definition:Subset|subset]] of the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ defined as:
:$S = \set {z \in \C: z = x + 0 i, x > 0}$
That is, let $S$ be the [[Definition:Positive Real Number|positive]] [[Definition:Real Axis|real axis]] of the [[Definition:Complex Pla... | We have that $S$ is the same thing as $\R_{>0}$, the set of [[Definition:Strictly Positive Real Number|strictly positive real numbers]]:
:$\R_{>0} = \set {x \in \R: x > 0}$
From [[Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group]], $\struct {S, \times}$ is a [[Definition:Group|group]]... | Positive Real Axis forms Subgroup of Complex Numbers under Multiplication | https://proofwiki.org/wiki/Positive_Real_Axis_forms_Subgroup_of_Complex_Numbers_under_Multiplication | https://proofwiki.org/wiki/Positive_Real_Axis_forms_Subgroup_of_Complex_Numbers_under_Multiplication | [
"Real Multiplication",
"Multiplicative Group of Complex Numbers",
"Examples of Subgroups"
] | [
"Definition:Subset",
"Definition:Set",
"Definition:Complex Number",
"Definition:Positive/Real Number",
"Definition:Complex Number/Complex Plane/Real Axis",
"Definition:Complex Number/Complex Plane",
"Definition:Algebraic Structure/One Operation",
"Definition:Subgroup",
"Definition:Multiplicative Gro... | [
"Definition:Strictly Positive/Real Number",
"Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group",
"Definition:Group",
"Definition:Group",
"Definition:Subset",
"Definition:Subgroup"
] |
proofwiki-17095 | Homomorphism from Reals to Circle Group/Corollary | Let $\struct {\R, +}$ be the additive group of real numbers.
Let $\struct {C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.
Let $\phi: \struct {\R, +} \to \struct {C_{\ne 0}, \times}$ be the mapping defined as:
:$\forall x \in \R: \map \phi x = \cos x + i \sin x$
Then $\phi$ is a (group) homomorphism... | By Euler's Identity, $\phi$ can also be expressed as:
:$\forall x \in \R: \map \phi x = e^{i x}$
From Homomorphism from Reals to Circle Group, $\phi$ is a homomorphism from $\struct {\R, +}$ to the circle group $\struct {K, \times}$.
From Circle Group is Infinite Abelian Group, we note that $\struct {K, \times}$ is a s... | Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $\struct {C_{\ne 0}, \times}$ be the [[Definition:Multiplicative Group of Complex Numbers|multiplicative group of complex numbers]].
Let $\phi: \struct {\R, +} \to \struct {C_{\ne 0}, \times}$ be the [[Defin... | By [[Euler's Identity]], $\phi$ can also be expressed as:
:$\forall x \in \R: \map \phi x = e^{i x}$
From [[Homomorphism from Reals to Circle Group]], $\phi$ is a [[Definition:Homomorphism|homomorphism]] from $\struct {\R, +}$ to the [[Definition:Circle Group|circle group]] $\struct {K, \times}$.
From [[Circle Group... | Homomorphism from Reals to Circle Group/Corollary | https://proofwiki.org/wiki/Homomorphism_from_Reals_to_Circle_Group/Corollary | https://proofwiki.org/wiki/Homomorphism_from_Reals_to_Circle_Group/Corollary | [
"Circle Group",
"Examples of Group Homomorphisms"
] | [
"Definition:Additive Group of Real Numbers",
"Definition:Multiplicative Group of Complex Numbers",
"Definition:Mapping",
"Definition:Group Homomorphism"
] | [
"Euler's Identity",
"Homomorphism from Reals to Circle Group",
"Definition:Homomorphism",
"Definition:Circle Group",
"Circle Group is Infinite Abelian Group",
"Definition:Subgroup",
"Definition:Multiplicative Group of Complex Numbers"
] |
proofwiki-17096 | Equivalence of Definitions of Closed Set in Normed Vector Space | {{TFAE|def = Closed Set in Normed Vector Space|view = Closed Set|context = Normed Vector Space|contextview = Normed Vector Spaces}}
Let $V = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Let $F \subseteq X$. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $F$ with a limit point $x$ in $X$. | {{TFAE|def = Closed Set in Normed Vector Space|view = Closed Set|context = Normed Vector Space|contextview = Normed Vector Spaces}}
Let $V = \struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $F \subseteq X$. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $F$ with a [[Definition:Limit Point (Normed Vector Space)|limit point]] $x$ in $X$. | Equivalence of Definitions of Closed Set in Normed Vector Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Closed_Set_in_Normed_Vector_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Closed_Set_in_Normed_Vector_Space | [
"Closed Sets"
] | [
"Definition:Normed Vector Space"
] | [
"Definition:Sequence",
"Definition:Limit Point/Normed Vector Space",
"Definition:Limit Point/Normed Vector Space",
"Definition:Limit Point/Normed Vector Space",
"Definition:Limit Point/Normed Vector Space",
"Definition:Sequence"
] |
proofwiki-17097 | Cosets of Positive Reals in Multiplicative Group of Complex Numbers | Let $S$ be the positive real axis of the complex plane:
:$S = \set {z \in \C: z = x + 0 i, x \in \R_{>0} }$
Consider the algebraic structure $\struct {S, \times}$ as a subgroup of the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$.
The cosets of $\struct {S, \times}$ are the sets of the form:
:$... | Let $z_0 \in \C_{\ne 0}$.
Write $z_0 = r_0 e^{i \theta}$, where $r_0 > 0$ and $\theta \in \hointr 0 {2 \pi}$.
We will show that:
:$z_0 S = \set {z \in \C: \exists r \in \R_{>0}: z = r e^{i \theta}}$
Pick any $w \in z_0 S$.
Then there exists some $x \in S$ such that $w = z_0 x$.
Note that $x \in \R_{>0}$ and $r_0 x \in ... | Let $S$ be the [[Definition:Positive Real Number|positive]] [[Definition:Real Axis|real axis]] of the [[Definition:Complex Plane|complex plane]]:
:$S = \set {z \in \C: z = x + 0 i, x \in \R_{>0} }$
Consider the [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\struct {S, \times}$ as a [[Defi... | Let $z_0 \in \C_{\ne 0}$.
Write $z_0 = r_0 e^{i \theta}$, where $r_0 > 0$ and $\theta \in \hointr 0 {2 \pi}$.
We will show that:
:$z_0 S = \set {z \in \C: \exists r \in \R_{>0}: z = r e^{i \theta}}$
Pick any $w \in z_0 S$.
Then there exists some $x \in S$ such that $w = z_0 x$.
Note that $x \in \R_{>0}$ and $r_0 ... | Cosets of Positive Reals in Multiplicative Group of Complex Numbers | https://proofwiki.org/wiki/Cosets_of_Positive_Reals_in_Multiplicative_Group_of_Complex_Numbers | https://proofwiki.org/wiki/Cosets_of_Positive_Reals_in_Multiplicative_Group_of_Complex_Numbers | [
"Real Multiplication",
"Multiplicative Group of Complex Numbers",
"Examples of Cosets"
] | [
"Definition:Positive/Real Number",
"Definition:Complex Number/Complex Plane/Real Axis",
"Definition:Complex Number/Complex Plane",
"Definition:Algebraic Structure/One Operation",
"Definition:Subgroup",
"Definition:Multiplicative Group of Complex Numbers",
"Definition:Coset",
"Definition:Set",
"Defin... | [
"Definition:Set Equality"
] |
proofwiki-17098 | Morphism from Multiplicative Group of Complex Numbers to Unit Circle | Let $\struct {\C_{\ne 0}, \times}$ denote the multiplicative group of complex numbers.
Let $f: \C_{\ne 0} \to \C_{\ne 0}$ be the mapping defined as:
:$\forall z \in \C_{\ne 0}: \map f z = \dfrac z {\cmod z}$
where $\cmod z$ denotes the modulus of $z$.
Then $f$ is an endomorphism on $\struct {\C_{\ne 0}, \times}$ whose ... | {{ProofWanted|Straightforward but tedious}} | Let $\struct {\C_{\ne 0}, \times}$ denote the [[Definition:Multiplicative Group of Complex Numbers|multiplicative group of complex numbers]].
Let $f: \C_{\ne 0} \to \C_{\ne 0}$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall z \in \C_{\ne 0}: \map f z = \dfrac z {\cmod z}$
where $\cmod z$ denotes the [[D... | {{ProofWanted|Straightforward but tedious}} | Morphism from Multiplicative Group of Complex Numbers to Unit Circle | https://proofwiki.org/wiki/Morphism_from_Multiplicative_Group_of_Complex_Numbers_to_Unit_Circle | https://proofwiki.org/wiki/Morphism_from_Multiplicative_Group_of_Complex_Numbers_to_Unit_Circle | [
"Multiplicative Group of Complex Numbers",
"Examples of Group Homomorphisms"
] | [
"Definition:Multiplicative Group of Complex Numbers",
"Definition:Mapping",
"Definition:Complex Modulus",
"Definition:Endomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Positive/Real Number",
"Definition:Complex Number/Complex Plane/Real Axis",
"Definition:Image (Set Theory)/Mapping... | [] |
proofwiki-17099 | Rational Numbers are Dense Subfield of P-adic Numbers | Let $p$ be a prime number.
Let $\norm {\,\cdot\,}^{\Q}_p$ be the p-adic norm on the rational numbers $\Q$.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\phi: \Q \to \Q_p$ be the mapping defined by:
:$\map \phi r = \eqclass {r, r, r, \dotsc} {}$
where $\eqclass {r, r, r, \dotsc} {}$ is the lef... | From P-adic Numbers form Completion of Rational Numbers with P-adic Norm:
:$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a completion of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$
From Embedding Division Ring into Quotient Ring of Cauchy Sequences:
:the mapping $\phi: \Q \to \Q_p$ is a distance-preserving monomorphism.
From Nor... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\norm {\,\cdot\,}^{\Q}_p$ be the [[Definition:P-adic Norm|p-adic norm]] on the [[Definition:Rational Numbers|rational numbers $\Q$]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\phi:... | From [[P-adic Numbers form Completion of Rational Numbers with P-adic Norm]]:
:$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a [[Definition:Completion (Normed Division Ring)|completion]] of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$
From [[Embedding Division Ring into Quotient Ring of Cauchy Sequences]]:
:the [[Definition:Mapp... | Rational Numbers are Dense Subfield of P-adic Numbers | https://proofwiki.org/wiki/Rational_Numbers_are_Dense_Subfield_of_P-adic_Numbers | https://proofwiki.org/wiki/Rational_Numbers_are_Dense_Subfield_of_P-adic_Numbers | [
"Rational Numbers",
"Everywhere Dense",
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Mapping",
"Definition:Coset/Left Coset",
"Definition:Sequence",
"Definition:Isometric Isomorphism/Normed Division Ring",
"Definition:Everywhere Dense",
"Defin... | [
"P-adic Numbers form Completion of Rational Numbers with P-adic Norm",
"Definition:Completion (Normed Division Ring)",
"Embedding Division Ring into Quotient Ring of Cauchy Sequences",
"Definition:Mapping",
"Definition:Distance-Preserving Mapping",
"Definition:Ring Monomorphism",
"Normed Division Ring i... |
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