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-18400 | Preordering induces Ordering | Let $\struct {S, \RR}$ be a relational structure such that $\RR$ is a preordering.
Let $\sim_\RR$ denote the equivalence on $S$ induced by $\RR$:
:$x \sim_\RR y$ {{iff}} $x \mathrel \RR y$ and $y \mathrel \RR x$
Let $\preccurlyeq_\RR$ be the relation defined on the quotient set $S / {\sim_\RR}$ by:
:$\eqclass x {\sim_\... | First we note from Ordering Induced by Preordering is Well-Defined that $\preccurlyeq_\RR$ is a well-defined relation.
Checking in turn each of the criteria for an ordering: | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]] such that $\RR$ is a [[Definition:Preordering|preordering]].
Let $\sim_\RR$ denote the [[Definition:Equivalence Relation Induced by Preordering|equivalence on $S$ induced by $\RR$]]:
:$x \sim_\RR y$ {{iff}} $x \mathrel \RR y$ and $y \... | First we note from [[Ordering Induced by Preordering is Well-Defined]] that $\preccurlyeq_\RR$ is a [[Definition:Well-Defined Relation|well-defined relation]].
Checking in turn each of the criteria for an [[Definition:Ordering|ordering]]: | Preordering induces Ordering | https://proofwiki.org/wiki/Preordering_induces_Ordering | https://proofwiki.org/wiki/Preordering_induces_Ordering | [
"Preorderings",
"Orderings"
] | [
"Definition:Relational Structure",
"Definition:Preordering",
"Definition:Equivalence Relation Induced by Preordering",
"Definition:Ordering Induced by Preordering",
"Definition:Quotient Set",
"Definition:Equivalence Class",
"Definition:Ordering"
] | [
"Ordering Induced by Preordering is Well-Defined",
"Definition:Well-Defined/Relation",
"Definition:Ordering",
"Definition:Ordering"
] |
proofwiki-18401 | Norm on Bounded Linear Transformation is Finite | <onlyinclude>
Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $A: X \to Y$ be a bounded linear transformation.
Let $\norm A$ denote the norm of $A$ defined by:
:$\norm A = \inf \set {c > 0: \forall h \in X : \norm {A... | By definition of a bounded linear transformation:
:$\exists c \in \R_{> 0}: \forall x \in X : \norm{A x}_Y \le c \norm x_X$
Hence:
:$\set {\lambda > 0: \forall x \in X : \norm {A x}_Y \le \lambda \norm x_X} \ne \O$
By definition:
:$\set {\lambda > 0: \forall x \in X : \norm {A x}_Y \le \lambda \norm x_X}$ is bounded be... | <onlyinclude>
Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $A: X \to Y$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let $\norm A$ denote t... | By definition of a [[Definition:Bounded Linear Transformation|bounded linear transformation]]:
:$\exists c \in \R_{> 0}: \forall x \in X : \norm{A x}_Y \le c \norm x_X$
Hence:
:$\set {\lambda > 0: \forall x \in X : \norm {A x}_Y \le \lambda \norm x_X} \ne \O$
By definition:
:$\set {\lambda > 0: \forall x \in X : \no... | Norm on Bounded Linear Transformation is Finite | https://proofwiki.org/wiki/Norm_on_Bounded_Linear_Transformation_is_Finite | https://proofwiki.org/wiki/Norm_on_Bounded_Linear_Transformation_is_Finite | [
"Norm on Space of Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Transformation",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Definition:Bounded Linear Transformation",
"Definition:Bounded Below Set/Real Numbers",
"Greatest Lower Bound Property",
"Category:Norm on Space of Bounded Linear Transformations"
] |
proofwiki-18402 | Mapping on Total Ordering reflects Transitivity | Let $\struct {S, \preccurlyeq}$ be a totally ordered set.
Let $f: S \to T$ be a mapping to an arbitrary set $T$.
Let $\RR$ be a relation on $T$ defined such that:
:$\RR: = \set {\tuple {\map f x, \map f y}: x \preccurlyeq y}$
That is, $a$ is related to $b$ in $T$ {{iff}} they have preimages $x$ and $y$ under $f$ such t... | Let $a, b, c \in T$ such that:
:$a \mathrel \RR b$
:$b \mathrel \RR c$
Then there exist $x, y, z \in S$ such that:
:$a = \map f x$
:$b = \map f y$
:$c = \map f z$
such that:
:$x \preccurlyeq y$
:$y \preccurlyeq z$
As $\preccurlyeq$ is a total ordering, it follows that:
:$x \preccurlyeq z$
and so by definition of $\RR$:... | Let $\struct {S, \preccurlyeq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] to an arbitrary [[Definition:Set|set]] $T$.
Let $\RR$ be a [[Definition:Relation|relation]] on $T$ defined such that:
:$\RR: = \set {\tuple {\map f x, \map f y}: x \preccur... | Let $a, b, c \in T$ such that:
:$a \mathrel \RR b$
:$b \mathrel \RR c$
Then there exist $x, y, z \in S$ such that:
:$a = \map f x$
:$b = \map f y$
:$c = \map f z$
such that:
:$x \preccurlyeq y$
:$y \preccurlyeq z$
As $\preccurlyeq$ is a [[Definition:Total Ordering|total ordering]], it follows that:
:$x \preccurlyeq... | Mapping on Total Ordering reflects Transitivity | https://proofwiki.org/wiki/Mapping_on_Total_Ordering_reflects_Transitivity | https://proofwiki.org/wiki/Mapping_on_Total_Ordering_reflects_Transitivity | [
"Total Orderings",
"Transitive Relations"
] | [
"Definition:Totally Ordered Set",
"Definition:Mapping",
"Definition:Set",
"Definition:Relation",
"Definition:Relation",
"Definition:Preimage/Mapping/Element",
"Definition:Precede",
"Definition:Transitive Relation"
] | [
"Definition:Total Ordering",
"Definition:Transitive Relation"
] |
proofwiki-18403 | Surjection on Total Ordering reflects Preordering | Let $\struct {S, \preccurlyeq}$ be a totally ordered set.
Let $f: S \to T$ be a mapping to an arbitrary set $T$.
Let $\RR$ be a relation on $T$ defined such that:
:$\RR: = \set {\tuple {\map f x, \map f y}: x \preccurlyeq y}$
That is, $a$ is related to $b$ in $T$ {{iff}} they have preimages $x$ and $y$ under $f$ such t... | === Sufficient Condition ===
Let $f: S \to T$ be a surjection.
By definition, a preordering is a relation which is both transitive and reflexive.
From Mapping on Total Ordering reflects Transitivity, $\RR$ is a transitive relation.
It remains to be shown that $\RR$ is a reflexive relation.
Let $a \in T$.
Because $f$ is... | Let $\struct {S, \preccurlyeq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] to an arbitrary [[Definition:Set|set]] $T$.
Let $\RR$ be a [[Definition:Relation|relation]] on $T$ defined such that:
:$\RR: = \set {\tuple {\map f x, \map f y}: x \preccur... | === Sufficient Condition ===
Let $f: S \to T$ be a [[Definition:Surjection|surjection]].
By definition, a [[Definition:Preordering|preordering]] is a [[Definition:Relation|relation]] which is both [[Definition:Transitive Relation|transitive]] and [[Definition:Reflexive Relation|reflexive]].
From [[Mapping on Total ... | Surjection on Total Ordering reflects Preordering | https://proofwiki.org/wiki/Surjection_on_Total_Ordering_reflects_Preordering | https://proofwiki.org/wiki/Surjection_on_Total_Ordering_reflects_Preordering | [
"Total Orderings",
"Preorderings"
] | [
"Definition:Totally Ordered Set",
"Definition:Mapping",
"Definition:Set",
"Definition:Relation",
"Definition:Relation",
"Definition:Preimage/Mapping/Element",
"Definition:Precede",
"Definition:Surjection",
"Definition:Set",
"Definition:Preordering"
] | [
"Definition:Surjection",
"Definition:Preordering",
"Definition:Relation",
"Definition:Transitive Relation",
"Definition:Reflexive Relation",
"Mapping on Total Ordering reflects Transitivity",
"Definition:Transitive Relation",
"Definition:Reflexive Relation",
"Definition:Surjection",
"Definition:To... |
proofwiki-18404 | Bijection on Total Ordering reflects Total Ordering | Let $\struct {S, \preccurlyeq}$ be a totally ordered set.
Let $f: S \to T$ be a mapping to an arbitrary set $T$.
Let $\RR$ be a relation on $T$ defined such that:
:$\RR: = \set {\tuple {\map f x, \map f y}: x \preccurlyeq y}$
That is, $a$ is related to $b$ in $T$ {{iff}} they have preimages $x$ and $y$ under $f$ such t... | === Sufficient Condition ===
Let $f$ be a bijection.
Then $f$ is {{afortiori}} a surjection.
Hence from Surjection on Total Ordering reflects Preordering it follows that $\RR$ is a preordering.
A total ordering is a preordering which is also antisymmetric and connected.
Hence it remains to be shown that:
:$\RR$ is anti... | Let $\struct {S, \preccurlyeq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] to an arbitrary [[Definition:Set|set]] $T$.
Let $\RR$ be a [[Definition:Relation|relation]] on $T$ defined such that:
:$\RR: = \set {\tuple {\map f x, \map f y}: x \preccur... | === Sufficient Condition ===
Let $f$ be a [[Definition:Bijection|bijection]].
Then $f$ is {{afortiori}} a [[Definition:Surjection|surjection]].
Hence from [[Surjection on Total Ordering reflects Preordering]] it follows that $\RR$ is a [[Definition:Preordering|preordering]].
A [[Definition:Total Ordering|total ord... | Bijection on Total Ordering reflects Total Ordering | https://proofwiki.org/wiki/Bijection_on_Total_Ordering_reflects_Total_Ordering | https://proofwiki.org/wiki/Bijection_on_Total_Ordering_reflects_Total_Ordering | [
"Total Orderings",
"Bijections"
] | [
"Definition:Totally Ordered Set",
"Definition:Mapping",
"Definition:Set",
"Definition:Relation",
"Definition:Relation",
"Definition:Preimage/Mapping/Element",
"Definition:Precede",
"Definition:Bijection",
"Definition:Total Ordering"
] | [
"Definition:Bijection",
"Definition:Surjection",
"Surjection on Total Ordering reflects Preordering",
"Definition:Preordering",
"Definition:Total Ordering",
"Definition:Preordering",
"Definition:Antisymmetric Relation",
"Definition:Connected Relation",
"Definition:Antisymmetric Relation",
"Definit... |
proofwiki-18405 | Bijection on Total Ordering is Order Isomorphism | Let $\struct {S, \preccurlyeq}$ be a totally ordered set.
Let $f: S \to T$ be a bijection to an arbitrary set $T$.
Let $\RR$ be a relation on $T$ defined such that:
:$\RR: = \set {\tuple {\map f x, \map f y}: x \preccurlyeq y}$
Then $f$ is an order isomorphism between $\struct {S, \preccurlyeq}$ and $\struct {T, \RR}$. | From Bijection on Total Ordering reflects Total Ordering, we have that $\RR$ is a total ordering.
We have that $f$ is a bijection.
Hence a fortiori $f$ is also a surjection.
Let $x, y \in S$ such that $x \preccurlyeq y$.
Then by definition of $\RR$:
:$\map f x \mathrel \RR \map f y$
Now let $\map f x \mathrel \RR \map ... | Let $\struct {S, \preccurlyeq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $f: S \to T$ be a [[Definition:Bijection|bijection]] to an arbitrary [[Definition:Set|set]] $T$.
Let $\RR$ be a [[Definition:Relation|relation]] on $T$ defined such that:
:$\RR: = \set {\tuple {\map f x, \map f y}: x \pre... | From [[Bijection on Total Ordering reflects Total Ordering]], we have that $\RR$ is a [[Definition:Total Ordering|total ordering]].
We have that $f$ is a [[Definition:Bijection|bijection]].
Hence [[Definition:A Fortiori|a fortiori]] $f$ is also a [[Definition:Surjection|surjection]].
Let $x, y \in S$ such that $x \... | Bijection on Total Ordering is Order Isomorphism | https://proofwiki.org/wiki/Bijection_on_Total_Ordering_is_Order_Isomorphism | https://proofwiki.org/wiki/Bijection_on_Total_Ordering_is_Order_Isomorphism | [
"Total Orderings",
"Bijections",
"Order Isomorphisms"
] | [
"Definition:Totally Ordered Set",
"Definition:Bijection",
"Definition:Set",
"Definition:Relation",
"Definition:Order Isomorphism"
] | [
"Bijection on Total Ordering reflects Total Ordering",
"Definition:Total Ordering",
"Definition:Bijection",
"Definition:A Fortiori",
"Definition:Surjection",
"Definition:Order Isomorphism"
] |
proofwiki-18406 | Mapping from Preordering reflects Ordering | Let $\struct {S, \RR}$ be a preordered set.
Then there exists:
:an ordered set $\struct {T, \preccurlyeq}$
:a mapping $f: S \to T$ such that:
::$\RR: = \set {\tuple {x, y}: \map f x \preccurlyeq \map f y}$ | Let $\sim_\RR$ denote the equivalence on $S$ induced by $\RR$:
:$x \sim_\RR y$ {{iff}} $x \mathrel \RR y$ and $y \mathrel \RR x$
Let $\preccurlyeq_\RR$ be the ordering on the quotient set $S / {\sim_\RR}$ by $\RR$:
:$\eqclass x {\sim_\RR} \preccurlyeq_\RR \eqclass y {\sim_\RR} \iff x \mathrel \RR y$
where $\eqclass x {... | Let $\struct {S, \RR}$ be a [[Definition:Preordered Set|preordered set]].
Then there exists:
:an [[Definition:Ordered Set|ordered set]] $\struct {T, \preccurlyeq}$
:a [[Definition:Mapping|mapping]] $f: S \to T$ such that:
::$\RR: = \set {\tuple {x, y}: \map f x \preccurlyeq \map f y}$ | Let $\sim_\RR$ denote the [[Definition:Equivalence Relation Induced by Preordering|equivalence on $S$ induced by $\RR$]]:
:$x \sim_\RR y$ {{iff}} $x \mathrel \RR y$ and $y \mathrel \RR x$
Let $\preccurlyeq_\RR$ be the [[Definition:Ordering Induced by Preordering|ordering]] on the [[Definition:Quotient Set|quotient set... | Mapping from Preordering reflects Ordering | https://proofwiki.org/wiki/Mapping_from_Preordering_reflects_Ordering | https://proofwiki.org/wiki/Mapping_from_Preordering_reflects_Ordering | [
"Orderings",
"Preorderings"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Ordered Set",
"Definition:Mapping"
] | [
"Definition:Equivalence Relation Induced by Preordering",
"Definition:Ordering Induced by Preordering",
"Definition:Quotient Set",
"Definition:Equivalence Class",
"Definition:Quotient Mapping",
"Preordering induces Ordering",
"Definition:Ordered Set"
] |
proofwiki-18407 | Set is Equivalent to Itself | Let $S$ be a set.
Then:
:$S \sim S$
where $\sim$ denotes set equivalence. | From Identity Mapping is Bijection, the identity mapping $I_S: S \to S$ is a bijection from $S$ to $S$.
Thus there exists a bijection from $S$ to itself
Hence by definition $S$ is therefore equivalent to itself. | Let $S$ be a [[Definition:Set|set]].
Then:
:$S \sim S$
where $\sim$ denotes [[Definition:Set Equivalence|set equivalence]]. | From [[Identity Mapping is Bijection]], the [[Definition:Identity Mapping|identity mapping]] $I_S: S \to S$ is a [[Definition:Bijection|bijection]] from $S$ to $S$.
Thus there exists a [[Definition:Bijection|bijection]] from $S$ to itself
Hence by definition $S$ is therefore [[Definition:Set Equivalence|equivalent]] ... | Set is Equivalent to Itself | https://proofwiki.org/wiki/Set_is_Equivalent_to_Itself | https://proofwiki.org/wiki/Set_is_Equivalent_to_Itself | [
"Set Equivalence behaves like Equivalence Relation"
] | [
"Definition:Set",
"Definition:Set Equivalence"
] | [
"Identity Mapping is Bijection",
"Definition:Identity Mapping",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Set Equivalence"
] |
proofwiki-18408 | Set Equivalence behaves like Equivalence Relation/Symmetric | Set equivalence behaves like a symmetric relation:
:$S \sim T \implies T \sim S$ | {{begin-eqn}}
{{eqn | o =
| r = S \sim T
| c =
}}
{{eqn | o = \leadsto
| r = \exists \phi: S \to T
| c = {{Defof|Set Equivalence}}, where $\phi$ is a bijection
}}
{{eqn | o = \leadsto
| r = \exists \phi^{-1}: T \to S
| c = Bijection iff Inverse is Bijection
}}
{{eqn | o = \leadsto
... | [[Definition:Set Equivalence|Set equivalence]] behaves like a [[Definition:Symmetric Relation|symmetric relation]]:
:$S \sim T \implies T \sim S$ | {{begin-eqn}}
{{eqn | o =
| r = S \sim T
| c =
}}
{{eqn | o = \leadsto
| r = \exists \phi: S \to T
| c = {{Defof|Set Equivalence}}, where $\phi$ is a [[Definition:Bijection|bijection]]
}}
{{eqn | o = \leadsto
| r = \exists \phi^{-1}: T \to S
| c = [[Bijection iff Inverse is Bijecti... | Set Equivalence behaves like Equivalence Relation/Symmetric | https://proofwiki.org/wiki/Set_Equivalence_behaves_like_Equivalence_Relation/Symmetric | https://proofwiki.org/wiki/Set_Equivalence_behaves_like_Equivalence_Relation/Symmetric | [
"Set Equivalence behaves like Equivalence Relation",
"Symmetric Relations"
] | [
"Definition:Set Equivalence",
"Definition:Symmetric Relation"
] | [
"Definition:Bijection",
"Inverse of Bijection is Bijection",
"Definition:Bijection"
] |
proofwiki-18409 | Set Equivalence behaves like Equivalence Relation/Transitive | Set equivalence behaves like a transitive relation:
:$S_1 \sim S_2 \land S_2 \sim S_3 \implies S_1 \sim S_3$ | {{begin-eqn}}
{{eqn | o =
| r = S_1 \sim S_2 \land S_2 \sim S_3
| c =
}}
{{eqn | o = \leadsto
| r = \exists \phi_1: S_1 \to S_2 \land \exists \phi_2: S_2 \to S_3
| c = {{Defof|Set Equivalence}}: $\phi_1$ and $\phi_2$ are bijections
}}
{{eqn | o = \leadsto
| r = \exists \phi_2 \circ \phi_... | [[Definition:Set Equivalence|Set equivalence]] behaves like a [[Definition:Transitive Relation|transitive relation]]:
:$S_1 \sim S_2 \land S_2 \sim S_3 \implies S_1 \sim S_3$ | {{begin-eqn}}
{{eqn | o =
| r = S_1 \sim S_2 \land S_2 \sim S_3
| c =
}}
{{eqn | o = \leadsto
| r = \exists \phi_1: S_1 \to S_2 \land \exists \phi_2: S_2 \to S_3
| c = {{Defof|Set Equivalence}}: $\phi_1$ and $\phi_2$ are [[Definition:Bijection|bijections]]
}}
{{eqn | o = \leadsto
| r = \... | Set Equivalence behaves like Equivalence Relation/Transitive | https://proofwiki.org/wiki/Set_Equivalence_behaves_like_Equivalence_Relation/Transitive | https://proofwiki.org/wiki/Set_Equivalence_behaves_like_Equivalence_Relation/Transitive | [
"Set Equivalence behaves like Equivalence Relation",
"Transitive Relations"
] | [
"Definition:Set Equivalence",
"Definition:Transitive Relation"
] | [
"Definition:Bijection",
"Composite of Bijections is Bijection",
"Definition:Bijection"
] |
proofwiki-18410 | Collection of Sets Equivalent to Set Containing Empty Set is Proper Class | Let $S = \set \O$ be the singleton whose element is the empty set.
Let $C$ be the collection of all sets which are equivalent to $S$.
$C$ is a proper class. | By definition of cardinality, $C$ is the collection of all singletons:
:$\set {x: \exists y: x = \set y}$
Define a class mapping $f: C \to V$, where $V$ is the universal class, such that $\map f {\set x} = x$.
This is a mapping on the domain $C$, as all elements of $C$ are singletons.
Take an arbitrary $y \in V$.
Then ... | Let $S = \set \O$ be the [[Definition:Singleton|singleton]] whose [[Definition:Element|element]] is the [[Definition:Empty Set|empty set]].
Let $C$ be the [[Definition:Collection|collection]] of all [[Definition:Set|sets]] which are [[Definition:Set Equivalence|equivalent]] to $S$.
$C$ is a [[Definition:Proper Class|... | By definition of [[Definition:Cardinality|cardinality]], $C$ is the [[Definition:Collection|collection]] of all [[Definition:Singleton|singletons]]:
:$\set {x: \exists y: x = \set y}$
Define a [[Definition:Class Mapping|class mapping]] $f: C \to V$, where $V$ is the [[Definition:Universal Class|universal class]], such ... | Collection of Sets Equivalent to Set Containing Empty Set is Proper Class | https://proofwiki.org/wiki/Collection_of_Sets_Equivalent_to_Set_Containing_Empty_Set_is_Proper_Class | https://proofwiki.org/wiki/Collection_of_Sets_Equivalent_to_Set_Containing_Empty_Set_is_Proper_Class | [
"Class Theory",
"Singletons",
"Empty Set"
] | [
"Definition:Singleton",
"Definition:Element",
"Definition:Empty Set",
"Definition:Collection",
"Definition:Set",
"Definition:Set Equivalence",
"Definition:Class (Class Theory)/Proper Class"
] | [
"Definition:Cardinality",
"Definition:Collection",
"Definition:Singleton",
"Definition:Mapping/Class Theory",
"Definition:Universal Class",
"Definition:Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Element",
"Definition:Singleton",
"Definition:Surjection/Class Theory",
"Univers... |
proofwiki-18411 | Finite Set of Continuous Functions between Metric Spaces is Pointwise Equicontinuous | Let $\struct {X, d}$ and $\struct {Y, d'}$ be metric spaces.
Let $\map \CC {X, Y}$ be the set of continuous functions $X \to Y$.
Let $\FF = \set {f_1, f_2, \ldots, f_n}$ be a finite subset of $\map \CC {X, Y}$.
Then $\FF$ is pointwise equicontinuous. | Let $x \in X$.
Let $\epsilon \in \R_{>0}$.
Let $i$ be a natural number with $1 \le i \le n$.
Since $f_i$ is continuous at $x$, there exists $\delta_i > 0$ such that whenever:
:$\map d {x, y} < \delta_i$
we have:
:$\map {d'} {\map {f_i} x, \map {f_i} y} < \epsilon$
Let:
:$\ds \delta = \min_i \set {\delta_i}$
so that: ... | Let $\struct {X, d}$ and $\struct {Y, d'}$ be [[Definition:Metric Space|metric spaces]].
Let $\map \CC {X, Y}$ be the set of [[Definition:Continuous Mapping (Metric Space)|continuous functions]] $X \to Y$.
Let $\FF = \set {f_1, f_2, \ldots, f_n}$ be a [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of ... | Let $x \in X$.
Let $\epsilon \in \R_{>0}$.
Let $i$ be a [[Definition:Natural Number|natural number]] with $1 \le i \le n$.
Since $f_i$ is [[Definition:Continuous Mapping (Metric Space)|continuous]] at $x$, there exists $\delta_i > 0$ such that whenever:
:$\map d {x, y} < \delta_i$
we have:
:$\map {d'} {\map {f... | Finite Set of Continuous Functions between Metric Spaces is Pointwise Equicontinuous | https://proofwiki.org/wiki/Finite_Set_of_Continuous_Functions_between_Metric_Spaces_is_Pointwise_Equicontinuous | https://proofwiki.org/wiki/Finite_Set_of_Continuous_Functions_between_Metric_Spaces_is_Pointwise_Equicontinuous | [
"Pointwise Equicontinuity"
] | [
"Definition:Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Pointwise Equicontinuous"
] | [
"Definition:Natural Numbers",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Pointwise Equicontinuous",
"Category:Pointwise Equicontinuity"
] |
proofwiki-18412 | Ordering of Cardinality of Sets is Well-Defined | The relation $\le$ in the context of cardinalities of sets is well-defined in the sense that:
:if $\card A = \card {A'}$ and $\card B = \card {B'}$, then there exists an injection from $A$ into $B$ {{iff}} there exists an injection from $A'$ into $B'$. | {{ProofWanted|I'm losing interest -- I'm going to work on something else for a bit.}} | The [[Definition:Relation|relation]] $\le$ in the context of [[Definition:Cardinality|cardinalities]] of [[Definition:Set|sets]] is [[Definition:Well-Defined Relation|well-defined]] in the sense that:
:if $\card A = \card {A'}$ and $\card B = \card {B'}$, then there exists an [[Definition:Injection|injection]] from $A... | {{ProofWanted|I'm losing interest -- I'm going to work on something else for a bit.}} | Ordering of Cardinality of Sets is Well-Defined | https://proofwiki.org/wiki/Ordering_of_Cardinality_of_Sets_is_Well-Defined | https://proofwiki.org/wiki/Ordering_of_Cardinality_of_Sets_is_Well-Defined | [
"Cardinality",
"Orderings"
] | [
"Definition:Relation",
"Definition:Cardinality",
"Definition:Set",
"Definition:Well-Defined/Relation",
"Definition:Injection",
"Definition:Injection"
] | [] |
proofwiki-18413 | Symmetric Group is Generated by Transposition and n-Cycle | Let $n \in \Z: n > 1$.
Let $S_n$ denote the symmetric group on $n$ letters.
Then the set of cyclic permutations:
:$\set {\begin {pmatrix} 1 & 2 \end{pmatrix}, \begin {pmatrix} 1 & 2 & \cdots & n \end{pmatrix} }$
is a generator for $S_n$. | Denote:
:$s = \begin {pmatrix} 1 & 2 \end{pmatrix}$
:$r = \begin {pmatrix} 1 & 2 & \cdots & n \end{pmatrix}$
By Cycle Decomposition of Conjugate,:
:$r s r^{-1} = r \begin {pmatrix} 1 & 2 \end{pmatrix} r^{-1} = \begin {pmatrix} \map r 1 & \map r 2 \end{pmatrix} = \begin {pmatrix} 2 & 3 \end{pmatrix}$.
By repeatedly usin... | Let $n \in \Z: n > 1$.
Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Then the [[Definition:Set|set]] of [[Definition:Cyclic Permutation|cyclic permutations]]:
:$\set {\begin {pmatrix} 1 & 2 \end{pmatrix}, \begin {pmatrix} 1 & 2 & \cdots & n \end{pmatrix} }$
is a [[... | Denote:
:$s = \begin {pmatrix} 1 & 2 \end{pmatrix}$
:$r = \begin {pmatrix} 1 & 2 & \cdots & n \end{pmatrix}$
By [[Cycle Decomposition of Conjugate]],:
:$r s r^{-1} = r \begin {pmatrix} 1 & 2 \end{pmatrix} r^{-1} = \begin {pmatrix} \map r 1 & \map r 2 \end{pmatrix} = \begin {pmatrix} 2 & 3 \end{pmatrix}$.
By repeatedl... | Symmetric Group is Generated by Transposition and n-Cycle | https://proofwiki.org/wiki/Symmetric_Group_is_Generated_by_Transposition_and_n-Cycle | https://proofwiki.org/wiki/Symmetric_Group_is_Generated_by_Transposition_and_n-Cycle | [
"Symmetric Groups",
"Examples of Generators of Groups"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Set",
"Definition:Cyclic Permutation",
"Definition:Generator of Group"
] | [
"Cycle Decomposition of Conjugate",
"Cycle Decomposition of Conjugate",
"Transpositions of Adjacent Elements generate Symmetric Group"
] |
proofwiki-18414 | Intersection of Nested Closed Subsets of Compact Space is Non-Empty | Let $\struct {T, \tau}$ be a compact topological space.
Let $\sequence {V_n}$ be a sequence of non-empty closed subsets of $T$ with:
:$V_{i + 1} \subseteq V_i$
for each $i$.
Then:
:$\ds \bigcap_{n \mathop = 1}^\infty V_n \ne \O$ | From Closed Subspace of Compact Space is Compact:
:$V_n$ is compact for each $n$.
{{AimForCont}} that:
:$\ds \bigcap_{n \mathop = 1}^\infty V_n = \O$
Then:
{{begin-eqn}}
{{eqn | l = V_1
| r = V_1 \setminus \paren {\bigcap_{n \mathop = 1}^\infty V_n}
}}
{{eqn | r = \bigcup_{n \mathop = 1}^\infty \paren {V_1 \setminus ... | Let $\struct {T, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $\sequence {V_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set|closed subsets]] of $T$ with:
:$V_{i + 1} \subseteq V_i$
for each $i$.
Then:
:$\ds \bigcap_... | From [[Closed Subspace of Compact Space is Compact]]:
:$V_n$ is [[Definition:Compact Topological Space|compact]] for each $n$.
{{AimForCont}} that:
:$\ds \bigcap_{n \mathop = 1}^\infty V_n = \O$
Then:
{{begin-eqn}}
{{eqn | l = V_1
| r = V_1 \setminus \paren {\bigcap_{n \mathop = 1}^\infty V_n}
}}
{{eqn | r = \bi... | Intersection of Nested Closed Subsets of Compact Space is Non-Empty | https://proofwiki.org/wiki/Intersection_of_Nested_Closed_Subsets_of_Compact_Space_is_Non-Empty | https://proofwiki.org/wiki/Intersection_of_Nested_Closed_Subsets_of_Compact_Space_is_Non-Empty | [
"Compact Topological Spaces",
"Closed Sets"
] | [
"Definition:Compact Topological Space",
"Definition:Sequence",
"Definition:Non-Empty Set",
"Definition:Closed Set"
] | [
"Closed Subspace of Compact Space is Compact",
"Definition:Compact Topological Space",
"De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection",
"Definition:Closed Set",
"Closed Set in Topological Subspace/Corollary",
"Definition:Closed Set",
"Definition:Open Set",
"Def... |
proofwiki-18415 | Family of Lipschitz Continuous Functions with same Lipschitz Constant is Uniformly Equicontinuous | Let $\struct {X, d}$ and $\struct {Y, d'}$ be metric spaces.
Let $\map \CC {X, Y}$ be the set of continuous functions $X \to Y$.
Let $\FF \subset \map \CC {X, Y}$ be a set of Lipschitz continuous functions all with Lipschitz constant $M \ge 0$.
Then $\FF$ is uniformly equicontinuous. | Note that for each $f \in \FF$, we have:
:$\map {d'} {\map f x, \map f y} \le M \map d {x, y}$
for each $x, y \in X$.
Note that if $M = 0$, we have:
:$\map {d'} {\map f x, \map f y} = 0$
for each $f \in \FF$ and $x, y \in X$.
That is:
:$\map {d'} {\map f x, \map f y} < \epsilon$
for each $\epsilon > 0$, $f \in \FF$... | Let $\struct {X, d}$ and $\struct {Y, d'}$ be [[Definition:Metric Space|metric spaces]].
Let $\map \CC {X, Y}$ be the set of [[Definition:Continuous Mapping (Metric Space)|continuous functions]] $X \to Y$.
Let $\FF \subset \map \CC {X, Y}$ be a [[Definition:Set|set]] of [[Definition:Lipschitz Continuity|Lipschitz co... | Note that for each $f \in \FF$, we have:
:$\map {d'} {\map f x, \map f y} \le M \map d {x, y}$
for each $x, y \in X$.
Note that if $M = 0$, we have:
:$\map {d'} {\map f x, \map f y} = 0$
for each $f \in \FF$ and $x, y \in X$.
That is:
:$\map {d'} {\map f x, \map f y} < \epsilon$
for each $\epsilon > 0$, $f... | Family of Lipschitz Continuous Functions with same Lipschitz Constant is Uniformly Equicontinuous | https://proofwiki.org/wiki/Family_of_Lipschitz_Continuous_Functions_with_same_Lipschitz_Constant_is_Uniformly_Equicontinuous | https://proofwiki.org/wiki/Family_of_Lipschitz_Continuous_Functions_with_same_Lipschitz_Constant_is_Uniformly_Equicontinuous | [
"Uniform Equicontinuity"
] | [
"Definition:Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Set",
"Definition:Lipschitz Continuity",
"Definition:Lipschitz Continuity/Lipschitz Constant",
"Definition:Uniformly Equicontinuous"
] | [
"Definition:Uniformly Equicontinuous",
"Definition:Uniformly Equicontinuous",
"Category:Uniform Equicontinuity"
] |
proofwiki-18416 | Continuous Function from Compact Hausdorff Space to Itself Fixes a Non-Empty Set | Let $\struct {X, \tau}$ be a compact Hausdorff space.
Let $f : X \to X$ be a continuous function.
Then there exists a non-empty subset $A \subseteq X$ such that:
:$f \sqbrk A = A$ | Define a sequence of sets $\sequence {X_i}_{i \mathop \in \N}$ by:
:$X_i = \begin{cases} X & : i = 1 \\ f \sqbrk {X_{i - 1} } & : i \ge 2 \end{cases}$
Since:
:$f \sqbrk X \subseteq X$
we have:
:$X_i \subseteq X$
for each $i \in \N$.
Define:
:$\ds A = \bigcap_{n \mathop = 1}^\infty X_n$
We have:
{{begin-eqn}}
{{eqn |... | Let $\struct {X, \tau}$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $f : X \to X$ be a [[Definition:Continuous Mapping (Topology)|continuous function]].
Then there exists a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] $A \subseteq ... | Define a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]] $\sequence {X_i}_{i \mathop \in \N}$ by:
:$X_i = \begin{cases} X & : i = 1 \\ f \sqbrk {X_{i - 1} } & : i \ge 2 \end{cases}$
Since:
:$f \sqbrk X \subseteq X$
we have:
:$X_i \subseteq X$
for each $i \in \N$.
Define:
:$\ds A = \bigcap_{n \math... | Continuous Function from Compact Hausdorff Space to Itself Fixes a Non-Empty Set | https://proofwiki.org/wiki/Continuous_Function_from_Compact_Hausdorff_Space_to_Itself_Fixes_a_Non-Empty_Set | https://proofwiki.org/wiki/Continuous_Function_from_Compact_Hausdorff_Space_to_Itself_Fixes_a_Non-Empty_Set | [
"Continuous Function from Compact Hausdorff Space to Itself Fixes a Non-Empty Set",
"Compact Topological Spaces",
"Hausdorff Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Non-Empty Set",
"Definition:Subset"
] | [
"Definition:Sequence",
"Definition:Set"
] |
proofwiki-18417 | Fundamental Property of Norm on Bounded Linear Transformation | Let $\HH, \KK$ be Hilbert spaces.
Let $A: \HH \to \KK$ be a bounded linear transformation.
Let $\norm A$ denote the norm of $A$ defined by:
:$\norm A = \inf \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$
Then:
:$\forall h \in \HH: \norm {A h}_\KK \le \norm A \norm h_\HH$ | From Norm on Bounded Linear Transformation is Finite:
:$\norm A = \inf \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$ exists
and
:$\norm A < \infty$
Let $x \in \HH \setminus \set{0_\HH}$
Let $\lambda \in \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$.
Then:
{{begin-eqn}}
{{eqn ... | Let $\HH, \KK$ be [[Definition:Hilbert Space|Hilbert spaces]].
Let $A: \HH \to \KK$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let $\norm A$ denote the [[Definition:Norm on Bounded Linear Transformation|norm]] of $A$ defined by:
:$\norm A = \inf \set {c > 0: \forall h \in \HH: \n... | From [[Norm on Bounded Linear Transformation is Finite]]:
:$\norm A = \inf \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$ exists
and
:$\norm A < \infty$
Let $x \in \HH \setminus \set{0_\HH}$
Let $\lambda \in \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$.
Then:
{{begin-eqn}... | Fundamental Property of Norm on Bounded Linear Transformation | https://proofwiki.org/wiki/Fundamental_Property_of_Norm_on_Bounded_Linear_Transformation | https://proofwiki.org/wiki/Fundamental_Property_of_Norm_on_Bounded_Linear_Transformation | [
"Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Transformation",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Norm on Bounded Linear Transformation is Finite",
"Definition:Infimum of Set"
] |
proofwiki-18418 | Equivalence of Definitions of Norm of Linear Transformation/Lemma | :$\forall \lambda > 0 : \norm{A 0_H}_K = \lambda \norm{0_H}_H$ | Let $\lambda > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \norm{A 0_H}_K
| r = \norm{0_K}_K
| c = Linear Transformation Maps Zero Vector to Zero Vector
}}
{{eqn | r = 0
| c = {{Norm-axiom-mult|1}}
}}
{{eqn | r = \lambda \cdot 0
}}
{{eqn | r = \lambda \norm{0_H}
| c = {{Norm-axiom-mult|1}}
}}
{{end-e... | :$\forall \lambda > 0 : \norm{A 0_H}_K = \lambda \norm{0_H}_H$ | Let $\lambda > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \norm{A 0_H}_K
| r = \norm{0_K}_K
| c = [[Linear Transformation Maps Zero Vector to Zero Vector]]
}}
{{eqn | r = 0
| c = {{Norm-axiom-mult|1}}
}}
{{eqn | r = \lambda \cdot 0
}}
{{eqn | r = \lambda \norm{0_H}
| c = {{Norm-axiom-mult|1}}
}}
{{... | Equivalence of Definitions of Norm of Linear Transformation/Lemma | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Lemma | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Lemma | [
"Equivalence of Definitions of Norm of Linear Transformation"
] | [] | [
"Linear Transformation Maps Zero Vector to Zero Vector",
"Category:Equivalence of Definitions of Norm of Linear Transformation"
] |
proofwiki-18419 | Quotient of Symmetric Group by Alternating Group is Parity Group | Let $n \ge 2$ be a natural number.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $A_n$ be the alternating group on $n$ letters.
Then the quotient of $S_n$ by $A_n$ is the parity group $C_2$. | From Alternating Group is Normal Subgroup of Symmetric Group, $A_n$ is a normal subgroup of $S_n$ whose index is $2$.
By definition of index, $\dfrac {S_n} {A_n}$ is a group of order $2$.
The result follows by definition of the parity group.
{{qed}} | Let $n \ge 2$ be a [[Definition:Natural Number|natural number]].
Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $A_n$ be the [[Definition:Alternating Group|alternating group on $n$ letters]].
Then the [[Definition:Quotient Group|quotient of $S_n$ by $A_n$]] is t... | From [[Alternating Group is Normal Subgroup of Symmetric Group]], $A_n$ is a [[Definition:Normal Subgroup|normal subgroup]] of $S_n$ whose [[Definition:Index of Subgroup|index]] is $2$.
By definition of [[Definition:Index of Subgroup|index]], $\dfrac {S_n} {A_n}$ is a [[Definition:Group|group]] of [[Definition:Order o... | Quotient of Symmetric Group by Alternating Group is Parity Group | https://proofwiki.org/wiki/Quotient_of_Symmetric_Group_by_Alternating_Group_is_Parity_Group | https://proofwiki.org/wiki/Quotient_of_Symmetric_Group_by_Alternating_Group_is_Parity_Group | [
"Symmetric Groups",
"Alternating Groups",
"Parity Group"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group/n Letters",
"Definition:Alternating Group",
"Definition:Quotient Group",
"Definition:Parity Group"
] | [
"Alternating Group is Normal Subgroup of Symmetric Group",
"Definition:Normal Subgroup",
"Definition:Index of Subgroup",
"Definition:Index of Subgroup",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Parity Group"
] |
proofwiki-18420 | Identity Mapping on Symmetric Group is Even Permutation | Let $S_n$ denote the symmetric group on $n$ letters.
Let $e \in S_n$ be the identity permutation on $S_n$.
Then $e$ is an even permutation on $S_n$. | By definition of the identity permutation:
:$\forall i \in \N_{>0}: \map e i = i$
Thus $e$ fixes all elements of $S_n$.
We have that a Transposition is of Odd Parity.
Hence a permutation is odd {{iff}} if is the composition of an odd number of transpositions.
The identity permutation is the composition of $0$ (zero) tr... | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $e \in S_n$ be the [[Definition:Identity Mapping|identity permutation]] on $S_n$.
Then $e$ is an [[Definition:Even Permutation|even permutation]] on $S_n$. | By definition of the [[Definition:Identity Mapping|identity permutation]]:
:$\forall i \in \N_{>0}: \map e i = i$
Thus $e$ [[Definition:Fixed Element under Permutation|fixes]] all elements of $S_n$.
We have that a [[Transposition is of Odd Parity]].
Hence a [[Definition:Permutation on n Letters|permutation]] is [[De... | Identity Mapping on Symmetric Group is Even Permutation | https://proofwiki.org/wiki/Identity_Mapping_on_Symmetric_Group_is_Even_Permutation | https://proofwiki.org/wiki/Identity_Mapping_on_Symmetric_Group_is_Even_Permutation | [
"Symmetric Groups",
"Identity Mappings",
"Even Permutations"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Identity Mapping",
"Definition:Even Permutation"
] | [
"Definition:Identity Mapping",
"Definition:Fixed Element under Permutation",
"Transposition is of Odd Parity",
"Definition:Permutation on n Letters",
"Definition:Odd Permutation",
"Definition:Composition of Mappings",
"Definition:Odd Integer",
"Definition:Transposition",
"Definition:Identity Mapping... |
proofwiki-18421 | Even Permutation is Product of 3-Cycles | Let $n \in \N$ such that $n \ge 3$.
Let $\pi: \N_n \to \N_n$ be a '''permutation on $n$ letters'''.
Let $\pi$ be an even permutation.
Then $\pi$ can be expressed as the composition of cyclic permutations of length $3$. | Let $A_n$ denote the alternating group on $n$ letters.
By Alternating Group is Set of Even Permutations, $A_n$ is the set of all even permutations of $S_n$.
$\pi$ can therefore be analysed in the context of elements of $A_n$.
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition... | Let $n \in \N$ such that $n \ge 3$.
Let $\pi: \N_n \to \N_n$ be a '''[[Definition:Permutation on n Letters|permutation on $n$ letters]]'''.
Let $\pi$ be an [[Definition:Even Permutation|even permutation]].
Then $\pi$ can be expressed as the [[Definition:Composition of Mappings|composition]] of [[Definition:Cyclic P... | Let $A_n$ denote the [[Definition:Alternating Group|alternating group on $n$ letters]].
By [[Alternating Group is Set of Even Permutations]], $A_n$ is the [[Definition:Set|set]] of all [[Definition:Even Permutation|even permutations]] of $S_n$.
$\pi$ can therefore be analysed in the context of [[Definition:Element|el... | Even Permutation is Product of 3-Cycles | https://proofwiki.org/wiki/Even_Permutation_is_Product_of_3-Cycles | https://proofwiki.org/wiki/Even_Permutation_is_Product_of_3-Cycles | [
"Even Permutations",
"Cyclic Permutations",
"Proofs by Induction"
] | [
"Definition:Permutation on n Letters",
"Definition:Even Permutation",
"Definition:Composition of Mappings",
"Definition:Cyclic Permutation",
"Definition:Cyclic Permutation"
] | [
"Definition:Alternating Group",
"Alternating Group is Set of Even Permutations",
"Definition:Set",
"Definition:Even Permutation",
"Definition:Element",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Composition of Mappings",
"Definition:Cyclic Permutation",
"Definitio... |
proofwiki-18422 | Alternating Group is Generated by 3-Cycles | Let $n \in \N$ such that $n \ge 3$.
Let $A_n$ denote the alternating group on $n$ letters.
Then $A_n$ can be generated by the set of $3$-cycles:
:$\set {\tuple {1, i, n}: i \in \set {2, 3, \ldots, n - 1} }$ | {{tidy|in due course}}
{{MissingLinks|throughout}}
Let $B_n$ be the group generated by $\set {\tuple {1, i, n}: i \in \set {2, 3, \ldots, n - 1} }$.
As every $3$-cycle is even, we have:
:$B_n \subseteq A_n$
Therefore, we want to prove $A_n \subseteq B_n$.
We shall prove the statement by induction on $n$. | Let $n \in \N$ such that $n \ge 3$.
Let $A_n$ denote the [[Definition:Alternating Group|alternating group on $n$ letters]].
Then $A_n$ can be [[Definition:Generator of Group|generated]] by the [[Definition:Set|set]] of [[Definition:Cyclic Permutation|$3$-cycles]]:
:$\set {\tuple {1, i, n}: i \in \set {2, 3, \ldots, n... | {{tidy|in due course}}
{{MissingLinks|throughout}}
Let $B_n$ be the [[Definition:Group|group]] generated by $\set {\tuple {1, i, n}: i \in \set {2, 3, \ldots, n - 1} }$.
As every [[Definition:Cyclic Permutation|$3$-cycle]] is [[Definition:Even Permutation|even]], we have:
:$B_n \subseteq A_n$
Therefore, we want to p... | Alternating Group is Generated by 3-Cycles | https://proofwiki.org/wiki/Alternating_Group_is_Generated_by_3-Cycles | https://proofwiki.org/wiki/Alternating_Group_is_Generated_by_3-Cycles | [
"Alternating Groups",
"Examples of Generators of Groups",
"Proofs by Induction"
] | [
"Definition:Alternating Group",
"Definition:Generator of Group",
"Definition:Set",
"Definition:Cyclic Permutation"
] | [
"Definition:Group",
"Definition:Cyclic Permutation",
"Definition:Even Permutation",
"Definition:Mathematical Induction",
"Definition:Even Permutation",
"Definition:Even Permutation"
] |
proofwiki-18423 | Alternating Group is Simple except on 4 Letters/Lemma 1 | Let $\alpha \in A_n$ be a permutation on $\N_n$ such that $\map \alpha 1 = 2$.
Let $\beta$ be the $3$-cycle $\tuple {3, 4, 5}$.
Then the permutation $\beta^{-1} \alpha^{-1} \beta \alpha$ fixes $1$. | We let $\beta^{-1} \alpha^{-1} \beta \alpha$ act on $1$.
By construction:
:$\map \alpha 1 = 2$
Thus:
{{begin-eqn}}
{{eqn | l = \map {\beta^{-1} \alpha^{-1} \beta \alpha} 1
| r = \map {\paren {3, 5, 4} \alpha^{-1} \tuple {3, 4, 5} } {\map \alpha 1}
| c =
}}
{{eqn | r = \map {\paren {3, 5, 4} \alpha^{-1} \tu... | Let $\alpha \in A_n$ be a [[Definition:Permutation on n Letters|permutation on $\N_n$]] such that $\map \alpha 1 = 2$.
Let $\beta$ be the [[Definition:Cyclic Permutation|$3$-cycle]] $\tuple {3, 4, 5}$.
Then the [[Definition:Permutation on n Letters|permutation]] $\beta^{-1} \alpha^{-1} \beta \alpha$ [[Definition:Fixe... | We let $\beta^{-1} \alpha^{-1} \beta \alpha$ act on $1$.
By construction:
:$\map \alpha 1 = 2$
Thus:
{{begin-eqn}}
{{eqn | l = \map {\beta^{-1} \alpha^{-1} \beta \alpha} 1
| r = \map {\paren {3, 5, 4} \alpha^{-1} \tuple {3, 4, 5} } {\map \alpha 1}
| c =
}}
{{eqn | r = \map {\paren {3, 5, 4} \alpha^{-1} ... | Alternating Group is Simple except on 4 Letters/Lemma 1 | https://proofwiki.org/wiki/Alternating_Group_is_Simple_except_on_4_Letters/Lemma_1 | https://proofwiki.org/wiki/Alternating_Group_is_Simple_except_on_4_Letters/Lemma_1 | [
"Alternating Group is Simple except on 4 Letters"
] | [
"Definition:Permutation on n Letters",
"Definition:Cyclic Permutation",
"Definition:Permutation on n Letters",
"Definition:Fixed Element under Permutation"
] | [
"Definition:Fixed Element under Permutation",
"Definition:Fixed Element under Permutation"
] |
proofwiki-18424 | Alternating Group is Simple except on 4 Letters/Lemma 2 | Let $\alpha \in A_n$ be the permutation on $\N_n$ in the form:
:$\alpha = \tuple {1, 2} \tuple {3, 4}$
Let $\beta$ be the $3$-cycle $\tuple {3, 4, 5}$.
Then:
:$\beta^{-1} \alpha^{-1} \beta \alpha = \beta$ | {{begin-eqn}}
{{eqn | l = \beta^{-1} \alpha^{-1} \beta \alpha
| r = \paren {3, 5, 4} \tuple {1, 2} \tuple {3, 4} \tuple {3, 4, 5} \tuple {1, 2} \tuple {3, 4}
| c =
}}
{{eqn | r = \paren {3, 5, 4} \tuple {1, 2} \tuple {3, 4} \tuple {1, 2} \tuple {3, 5}
| c =
}}
{{eqn | r = \paren {3, 5, 4} \tuple {3,... | Let $\alpha \in A_n$ be the [[Definition:Permutation on n Letters|permutation on $\N_n$]] in the form:
:$\alpha = \tuple {1, 2} \tuple {3, 4}$
Let $\beta$ be the [[Definition:Cyclic Permutation|$3$-cycle]] $\tuple {3, 4, 5}$.
Then:
:$\beta^{-1} \alpha^{-1} \beta \alpha = \beta$ | {{begin-eqn}}
{{eqn | l = \beta^{-1} \alpha^{-1} \beta \alpha
| r = \paren {3, 5, 4} \tuple {1, 2} \tuple {3, 4} \tuple {3, 4, 5} \tuple {1, 2} \tuple {3, 4}
| c =
}}
{{eqn | r = \paren {3, 5, 4} \tuple {1, 2} \tuple {3, 4} \tuple {1, 2} \tuple {3, 5}
| c =
}}
{{eqn | r = \paren {3, 5, 4} \tuple {3,... | Alternating Group is Simple except on 4 Letters/Lemma 2 | https://proofwiki.org/wiki/Alternating_Group_is_Simple_except_on_4_Letters/Lemma_2 | https://proofwiki.org/wiki/Alternating_Group_is_Simple_except_on_4_Letters/Lemma_2 | [
"Alternating Group is Simple except on 4 Letters"
] | [
"Definition:Permutation on n Letters",
"Definition:Cyclic Permutation"
] | [] |
proofwiki-18425 | Alternating Group has no Subgroup of Order 6 | Let $A_4$ denote the alternating group on $4$ letters.
$A_4$ has no subgroup of order $6$. | We list the subgroups of $A_4$:
{{:Alternating Group on 4 Letters/Subgroups}}
So there is no subgroup of $A_4$ of order $6$.
{{qed}} | Let $A_4$ denote the [[Alternating Group on 4 Letters|alternating group on $4$ letters]].
$A_4$ has no [[Definition:Subgroup|subgroup]] of [[Definition:Order of Group|order $6$]]. | We list the [[Alternating Group on 4 Letters/Subgroups|subgroups of $A_4$]]:
{{:Alternating Group on 4 Letters/Subgroups}}
So there is no [[Definition:Subgroup|subgroup]] of $A_4$ of [[Definition:Order of Structure|order]] $6$.
{{qed}} | Alternating Group has no Subgroup of Order 6 | https://proofwiki.org/wiki/Alternating_Group_has_no_Subgroup_of_Order_6 | https://proofwiki.org/wiki/Alternating_Group_has_no_Subgroup_of_Order_6 | [
"Alternating Group on 4 Letters"
] | [
"Alternating Group on 4 Letters",
"Definition:Subgroup",
"Definition:Order of Structure"
] | [
"Alternating Group on 4 Letters/Subgroups",
"Definition:Subgroup",
"Definition:Order of Structure"
] |
proofwiki-18426 | Alternating Group on More than 3 Letters is not Abelian | Let $n$ be an integer such that $n > 3$.
Then the $n$th alternating group $A_n$ is not abelian. | Let $\tuple {1, 2, 3}$ and $\tuple {2, 3, 4}$ be elements of $A_n$.
Then we have:
{{begin-eqn}}
{{eqn | l = \tuple {1, 2, 3} \tuple {2, 3, 4}
| r = \tuple {1 2} \tuple {3 4}
| c =
}}
{{eqn | l = \tuple {2, 3, 4} \tuple {1, 2, 3}
| r = \tuple {1 3} \tuple {2 4}
| c =
}}
{{end-eqn}}
So:
:$\tuple... | Let $n$ be an [[Definition:Integer|integer]] such that $n > 3$.
Then the $n$th [[Definition:Alternating Group|alternating group]] $A_n$ is not [[Definition:Abelian Group|abelian]]. | Let $\tuple {1, 2, 3}$ and $\tuple {2, 3, 4}$ be [[Definition:Element|elements]] of $A_n$.
Then we have:
{{begin-eqn}}
{{eqn | l = \tuple {1, 2, 3} \tuple {2, 3, 4}
| r = \tuple {1 2} \tuple {3 4}
| c =
}}
{{eqn | l = \tuple {2, 3, 4} \tuple {1, 2, 3}
| r = \tuple {1 3} \tuple {2 4}
| c =
}... | Alternating Group on More than 3 Letters is not Abelian | https://proofwiki.org/wiki/Alternating_Group_on_More_than_3_Letters_is_not_Abelian | https://proofwiki.org/wiki/Alternating_Group_on_More_than_3_Letters_is_not_Abelian | [
"Alternating Groups",
"Cyclic Groups"
] | [
"Definition:Integer",
"Definition:Alternating Group",
"Definition:Abelian Group"
] | [
"Definition:Element",
"Definition:Abelian Group"
] |
proofwiki-18427 | Symmetric Group on Greater than 4 Letters is Not Solvable | Let $n \in \N$ such that $n > 4$.
Let $S_n$ denote the symmetric group on $n$ letters.
Then $S_n$ is not a solvable group. | As stated, let $n > 4$ in the below.
Recall the definition of solvable group:
:A finite group $G$ is a '''solvable group''' {{iff}} it has a composition series in which each factor is a cyclic group.
:A '''composition series for $G$''' is a normal series for $G$ which has no proper refinement.
:A '''normal series''' fo... | Let $n \in \N$ such that $n > 4$.
Let $S_n$ denote the [[Definition:Symmetric Group|symmetric group on $n$ letters]].
Then $S_n$ is not a [[Definition:Solvable Group|solvable group]]. | As stated, let $n > 4$ in the below.
Recall the definition of [[Definition:Solvable Group|solvable group]]:
:A [[Definition:Finite Group|finite group]] $G$ is a '''[[Definition:Solvable Group|solvable group]]''' {{iff}} it has a [[Definition:Composition Series|composition series]] in which each [[Definition:Factor o... | Symmetric Group on Greater than 4 Letters is Not Solvable | https://proofwiki.org/wiki/Symmetric_Group_on_Greater_than_4_Letters_is_Not_Solvable | https://proofwiki.org/wiki/Symmetric_Group_on_Greater_than_4_Letters_is_Not_Solvable | [
"Symmetric Groups"
] | [
"Definition:Symmetric Group",
"Definition:Solvable Group"
] | [
"Definition:Solvable Group",
"Definition:Finite Group",
"Definition:Solvable Group",
"Definition:Composition Series",
"Definition:Normal Series/Factor Group",
"Definition:Cyclic Group",
"Definition:Composition Series",
"Definition:Normal Series",
"Definition:Refinement of Normal Series/Proper Refine... |
proofwiki-18428 | Isomorphism of Finite Group with Permutations of Quotient with Subgroup | Let $G$ be a finite group.
Let $H$ be a subgroup of $G$.
Let $H$ contain no non-trivial normal subgroup of $G$.
Let $G / H$ denote the left coset space of $G$ by $H$.
Then $G$ is isomorphic to a subgroup of the group of permutations $\map \Gamma {G / H}$ of $G / H$. | Let a homomorphism $\phi: G \to \map \Gamma {G / H}$ be defined as:
:$\forall x \in G: \map {\map \phi g} {x H} = \paren {g x} H$
By Kernel is Normal Subgroup of Domain, $\ker \phi$ is a normal subgroup of $G$.
So:
{{begin-eqn}}
{{eqn | l = g
| o = \in
| r = \ker \phi
| c =
}}
{{eqn | ll= \leadstoand... | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $H$ contain no [[Definition:Non-Trivial Subgroup|non-trivial]] [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $G / H$ denote the [[Definition:Left Coset Space|left coset space]] of $G$ by $H$.... | Let a [[Definition:Homomorphism|homomorphism]] $\phi: G \to \map \Gamma {G / H}$ be defined as:
:$\forall x \in G: \map {\map \phi g} {x H} = \paren {g x} H$
By [[Kernel is Normal Subgroup of Domain]], $\ker \phi$ is a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
So:
{{begin-eqn}}
{{eqn | l = g
| o ... | Isomorphism of Finite Group with Permutations of Quotient with Subgroup | https://proofwiki.org/wiki/Isomorphism_of_Finite_Group_with_Permutations_of_Quotient_with_Subgroup | https://proofwiki.org/wiki/Isomorphism_of_Finite_Group_with_Permutations_of_Quotient_with_Subgroup | [
"Group Isomorphisms",
"Symmetric Groups"
] | [
"Definition:Finite Group",
"Definition:Subgroup",
"Definition:Non-Trivial Subgroup",
"Definition:Normal Subgroup",
"Definition:Coset Space/Left Coset Space",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Subgroup",
"Definition:Symmetric Group"
] | [
"Definition:Homomorphism",
"Kernel is Normal Subgroup of Domain",
"Definition:Normal Subgroup",
"Definition:Set Intersection",
"Definition:Conjugate (Group Theory)/Subset",
"Definition:Non-Trivial Subgroup",
"Definition:Normal Subgroup",
"Definition:Trivial Subgroup",
"Kernel is Trivial iff Monomorp... |
proofwiki-18429 | Equivalence of Definitions of Norm of Linear Transformation/Definition 4 Greater or Equal Definition 2 | Let $\HH, \KK$ be Hilbert spaces.
Let $A: \HH \to \KK$ be a bounded linear transformation.
Let:
:$\lambda_2 = \sup \set {\dfrac {\norm {A h}_\KK} {\norm h_\HH}: h \in \HH, h \ne 0_\HH}$
and
:$\lambda_4 = \inf \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$
Then:
:$\lambda_4 \ge \lambda_2$ | From Fundamental Property of Norm on Bounded Linear Transformation:
:$\forall h \in \HH : \norm{A h}_\KK \le \lambda_4 \norm h_\HH$
Hence:
:$\forall h \in H, h \ne 0_\HH : \dfrac {\norm{A h}_\KK} {\norm h_\HH} \le \lambda_4$
From Continuum Property:
:$\lambda_2 = \sup \set {\dfrac {\norm {A h}_\KK} {\norm h_\HH}: h \in... | Let $\HH, \KK$ be [[Definition:Hilbert Space|Hilbert spaces]].
Let $A: \HH \to \KK$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let:
:$\lambda_2 = \sup \set {\dfrac {\norm {A h}_\KK} {\norm h_\HH}: h \in \HH, h \ne 0_\HH}$
and
:$\lambda_4 = \inf \set {c > 0: \forall h \in \HH: \no... | From [[Fundamental Property of Norm on Bounded Linear Transformation]]:
:$\forall h \in \HH : \norm{A h}_\KK \le \lambda_4 \norm h_\HH$
Hence:
:$\forall h \in H, h \ne 0_\HH : \dfrac {\norm{A h}_\KK} {\norm h_\HH} \le \lambda_4$
From [[Continuum Property]]:
:$\lambda_2 = \sup \set {\dfrac {\norm {A h}_\KK} {\norm h_\... | Equivalence of Definitions of Norm of Linear Transformation/Definition 4 Greater or Equal Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Definition_4_Greater_or_Equal_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Definition_4_Greater_or_Equal_Definition_2 | [
"Equivalence of Definitions of Norm of Linear Transformation"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Transformation"
] | [
"Fundamental Property of Norm on Bounded Linear Transformation",
"Continuum Property",
"Definition:Supremum of Set"
] |
proofwiki-18430 | Equivalence of Definitions of Norm of Linear Transformation/Definition 2 Greater or Equal Definition 1 | Let $H, K$ be Hilbert spaces.
Let $A: H \to K$ be a bounded linear transformation.
Let:
:$\lambda_1 = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
and
:$\lambda_2 = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne 0_H}$
Then:
:$\lambda_2 \ge \lambda_1$ | By definition of the supremum:
:$\forall h \in H, h \ne \mathbf 0_H: \dfrac {\norm {A h}_K} {\norm h_H} \le \lambda_2$
Hence:
:$\forall h \in H, h \ne \mathbf 0_H: \norm {A h}_K \le \lambda_2 \norm h_H$
From Lemma:
:$\norm{A 0_H}_K = \lambda_2 \norm{0_H}_H$
Hence:
:$\forall h \in H: \norm {A h}_K \le \lambda_2 \norm h_... | Let $H, K$ be [[Definition:Hilbert Space|Hilbert spaces]].
Let $A: H \to K$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let:
:$\lambda_1 = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
and
:$\lambda_2 = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne 0_H}$
Then:
:$\... | By definition of the [[Definition:Supremum of Set|supremum]]:
:$\forall h \in H, h \ne \mathbf 0_H: \dfrac {\norm {A h}_K} {\norm h_H} \le \lambda_2$
Hence:
:$\forall h \in H, h \ne \mathbf 0_H: \norm {A h}_K \le \lambda_2 \norm h_H$
From [[Equivalence of Definitions of Norm of Linear Transformation/Lemma|Lemma]]:
:$... | Equivalence of Definitions of Norm of Linear Transformation/Definition 2 Greater or Equal Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Definition_2_Greater_or_Equal_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Definition_2_Greater_or_Equal_Definition_1 | [
"Equivalence of Definitions of Norm of Linear Transformation"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Supremum of Set",
"Equivalence of Definitions of Norm of Linear Transformation/Lemma",
"Continuum Property",
"Definition:Supremum of Set"
] |
proofwiki-18431 | Equivalence of Definitions of Norm of Linear Transformation/Definition 1 Greater or Equal Definition 3 | Let $H, K$ be Hilbert spaces.
Let $A: H \to K$ be a bounded linear transformation.
Let:
:$\lambda_1 = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
and
:$\lambda_3 = \sup \set {\norm {A h}_K: \norm h_H = 1}$
Then:
:$\lambda_1 \ge \lambda_3$ | By definition of the supremum:
:$\forall h \in H, \norm h_H \le 1 : \norm{A h}_K \le \lambda_1$
In particular:
:$\forall h \in H, \norm h_H = 1 : \norm{A h}_K \le \lambda_1$
From Continuum Property:
:$\lambda_3 = \sup \set {\norm {A h}_K: \norm h_H = 1}$ exists
By definition of the supremum:
:$\lambda_3 \le \lambda_1$ | Let $H, K$ be [[Definition:Hilbert Space|Hilbert spaces]].
Let $A: H \to K$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let:
:$\lambda_1 = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
and
:$\lambda_3 = \sup \set {\norm {A h}_K: \norm h_H = 1}$
Then:
:$\lambda_1 \ge \lambda_3$ | By definition of the [[Definition:Supremum of Set|supremum]]:
:$\forall h \in H, \norm h_H \le 1 : \norm{A h}_K \le \lambda_1$
In particular:
:$\forall h \in H, \norm h_H = 1 : \norm{A h}_K \le \lambda_1$
From [[Continuum Property]]:
:$\lambda_3 = \sup \set {\norm {A h}_K: \norm h_H = 1}$ exists
By definition of the... | Equivalence of Definitions of Norm of Linear Transformation/Definition 1 Greater or Equal Definition 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Definition_1_Greater_or_Equal_Definition_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Definition_1_Greater_or_Equal_Definition_3 | [
"Equivalence of Definitions of Norm of Linear Transformation"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Supremum of Set",
"Continuum Property",
"Definition:Supremum of Set"
] |
proofwiki-18432 | Equivalence of Definitions of Norm of Linear Transformation/Definition 3 Greater or Equal Definition 4 | Let $H, K$ be Hilbert spaces.
Let $A: H \to K$ be a bounded linear transformation.
Let:
:$\lambda_3 = \sup \set {\norm {A h}_K: \norm h_H = 1}$
and
:$\lambda_4 = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$
Let:
:$\lambda_3 \ge \lambda_4$ | Let $h \in H: h \ne 0_h$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {\dfrac 1 {\norm h_H} h }_H
| r = \dfrac {\norm h_H}{\norm h_H}
| c = {{Norm-axiom-mult|2}}
}}
{{eqn | r = 1
}}
{{end-eqn}}
and
{{begin-eqn}}
{{eqn | l = \dfrac {\norm{A h}_K} {\norm h_H}
| r = \norm {\dfrac 1 {\norm h_H} A h}_K
... | Let $H, K$ be [[Definition:Hilbert Space|Hilbert spaces]].
Let $A: H \to K$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let:
:$\lambda_3 = \sup \set {\norm {A h}_K: \norm h_H = 1}$
and
:$\lambda_4 = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$
Let:
:$\lambd... | Let $h \in H: h \ne 0_h$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {\dfrac 1 {\norm h_H} h }_H
| r = \dfrac {\norm h_H}{\norm h_H}
| c = {{Norm-axiom-mult|2}}
}}
{{eqn | r = 1
}}
{{end-eqn}}
and
{{begin-eqn}}
{{eqn | l = \dfrac {\norm{A h}_K} {\norm h_H}
| r = \norm {\dfrac 1 {\norm h_H} A h}_K
... | Equivalence of Definitions of Norm of Linear Transformation/Definition 3 Greater or Equal Definition 4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Definition_3_Greater_or_Equal_Definition_4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Norm_of_Linear_Transformation/Definition_3_Greater_or_Equal_Definition_4 | [
"Equivalence of Definitions of Norm of Linear Transformation"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Transformation"
] | [
"Linear Transformation Maps Zero Vector to Zero Vector",
"Equivalence of Definitions of Norm of Linear Transformation/Lemma",
"Definition:Infimum of Set"
] |
proofwiki-18433 | Index of Proper Subgroup of Symmetric Group | Let $n \in \N$ be a natural number such that $n > 4$.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $A_n$ denote the alternating group on $n$ letters.
$A_n$ is the only proper subgroup of $S_n$ whose index is less than $n$. | From Normal Subgroup of Symmetric Group on More than 4 Letters is Alternating Group:
:$A_n$ is the only proper non-trivial normal subgroup of $S_n$.
Suppose $H$ is a subgroup of $S_n$ whose index $\index {S_n} H$ is less than $n$.
If $\index {S_n} H = 2$ then from Subgroup of Index 2 is Normal $H$ is normal.
Hence $H =... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]] such that $n > 4$.
Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $A_n$ denote the [[Definition:Alternating Group|alternating group on $n$ letters]].
$A_n$ is the only [[Definition:Proper Subgroup|... | From [[Normal Subgroup of Symmetric Group on More than 4 Letters is Alternating Group]]:
:$A_n$ is the only [[Definition:Non-Trivial Proper Subgroup|proper non-trivial]] [[Definition:Normal Subgroup|normal subgroup]] of $S_n$.
Suppose $H$ is a [[Definition:Subgroup|subgroup]] of $S_n$ whose [[Definition:Index of Sub... | Index of Proper Subgroup of Symmetric Group | https://proofwiki.org/wiki/Index_of_Proper_Subgroup_of_Symmetric_Group | https://proofwiki.org/wiki/Index_of_Proper_Subgroup_of_Symmetric_Group | [
"Symmetric Groups",
"Examples of Subgroups"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group/n Letters",
"Definition:Alternating Group",
"Definition:Proper Subgroup",
"Definition:Subgroup",
"Definition:Index of Subgroup"
] | [
"Normal Subgroup of Symmetric Group on More than 4 Letters is Alternating Group",
"Definition:Proper Subgroup/Non-Trivial",
"Definition:Normal Subgroup",
"Definition:Subgroup",
"Definition:Index of Subgroup",
"Subgroup of Index 2 is Normal",
"Definition:Normal Subgroup",
"Isomorphism of Finite Group w... |
proofwiki-18434 | Number of Distinct Functions on n Variables obtained by Permutation | Let $\map f {x_1, x_2, \ldots, x_n}$ be a function on $n$ independent variables where $n > 4$.
Let $\nu$ denote the number of distinct functions that can be obtained when $\tuple {x_1, x_2, \ldots, x_n}$ are permuted.
Then:
:$\nu > 2 \implies \nu \ge n$ | {{ProofWanted|Probably uses Index of Proper Subgroup of Symmetric Group, as this looks suggestive}} | Let $\map f {x_1, x_2, \ldots, x_n}$ be a [[Definition:Function|function]] on $n$ [[Definition:Independent Variable|independent variables]] where $n > 4$.
Let $\nu$ denote the number of [[Definition:Distinct Elements|distinct]] [[Definition:Function|functions]] that can be obtained when $\tuple {x_1, x_2, \ldots, x_n}... | {{ProofWanted|Probably uses [[Index of Proper Subgroup of Symmetric Group]], as this looks suggestive}} | Number of Distinct Functions on n Variables obtained by Permutation | https://proofwiki.org/wiki/Number_of_Distinct_Functions_on_n_Variables_obtained_by_Permutation | https://proofwiki.org/wiki/Number_of_Distinct_Functions_on_n_Variables_obtained_by_Permutation | [
"Function Theory"
] | [
"Definition:Function",
"Definition:Independent Variable",
"Definition:Distinct/Plural",
"Definition:Function",
"Definition:Permutation"
] | [
"Index of Proper Subgroup of Symmetric Group"
] |
proofwiki-18435 | Fundamental Solution to 2D Laplace's Equation | Let $\delta_{\tuple {0, 0}} \in \map {\DD'} {\R^2}$ be the Dirac delta distribution.
Let $\Delta = \dfrac {\partial^2} {\partial x^2} + \dfrac {\partial^2} {\partial y^2}$ be the Laplacian in $2$-dimensional Euclidean space.
Then in the distributional sense:
:$\ds \map \Delta {\frac {\ln r} {2 \pi}} = \delta_{\tuple {0... | Let $\norm {\,\cdot\,}_2$ be the 2-norm. | Let $\delta_{\tuple {0, 0}} \in \map {\DD'} {\R^2}$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Let $\Delta = \dfrac {\partial^2} {\partial x^2} + \dfrac {\partial^2} {\partial y^2}$ be the [[Definition:Laplacian on Scalar Field|Laplacian]] in [[Definition:Dimension of Vector Space|$2$-dim... | Let $\norm {\,\cdot\,}_2$ be the [[Definition:P-Norm|2-norm]]. | Fundamental Solution to 2D Laplace's Equation | https://proofwiki.org/wiki/Fundamental_Solution_to_2D_Laplace's_Equation | https://proofwiki.org/wiki/Fundamental_Solution_to_2D_Laplace's_Equation | [
"Laplace's Equation",
"Examples of Fundamental Solutions"
] | [
"Definition:Dirac Delta Distribution",
"Definition:Laplacian/Scalar Field",
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Definition:Distributional Derivative"
] | [
"Definition:P-Norm"
] |
proofwiki-18436 | Transitive Subgroup of Prime containing Transposition | Let $p$ be a prime number.
Let $S_p$ denote the symmetric group on $p$ letters.
Let $H$ be a transitive subgroup of $S_p$.
If $H$ contains a transposition, then $H = S_p$. | {{WLOG}}, let $\tuple {1, 2}$ be the transposition contained by $H$.
Let us define an equivalence relation $\sim$ on the set $\N_p = \set {1, 2, \ldots, p}$ as:
:$i \sim j \iff \tuple {i, j} \in H$
Because $H$ is a transitive subgroup it follows that each $\sim$-equivalence class has the same number of elements.
In fac... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $S_p$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $p$ letters]].
Let $H$ be a [[Definition:Transitive Subgroup|transitive subgroup]] of $S_p$.
If $H$ contains a [[Definition:Transposition|transposition]], then $H = S_p$. | {{WLOG}}, let $\tuple {1, 2}$ be the [[Definition:Transposition|transposition]] contained by $H$.
Let us define an [[Definition:Equivalence Relation|equivalence relation]] $\sim$ on the [[Definition:Set|set]] $\N_p = \set {1, 2, \ldots, p}$ as:
:$i \sim j \iff \tuple {i, j} \in H$
Because $H$ is a [[Definition:Transi... | Transitive Subgroup of Prime containing Transposition | https://proofwiki.org/wiki/Transitive_Subgroup_of_Prime_containing_Transposition | https://proofwiki.org/wiki/Transitive_Subgroup_of_Prime_containing_Transposition | [
"Transitive Subgroups"
] | [
"Definition:Prime Number",
"Definition:Symmetric Group/n Letters",
"Definition:Transitive Subgroup",
"Definition:Transposition"
] | [
"Definition:Transposition",
"Definition:Equivalence Relation",
"Definition:Set",
"Definition:Transitive Subgroup",
"Definition:Equivalence Class",
"Definition:Element",
"Definition:Bijection",
"Definition:Equivalence Class",
"Definition:Element",
"Definition:Equivalence Class",
"Definition:Divis... |
proofwiki-18437 | Inverse of Field Product with Inverse | Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$.
Let $a, b \in F$ such that $a \ne 0$.
Then:
:$\paren {a \times b^{-1} }^{-1} = b \times a^{-1}$
This can also be expressed using the notation of division as:
:$1 / \paren {a / b} = b / a$ | By definition, a field is a non-trivial division ring whose ring product is commutative.
By definition, a division ring is a ring with unity such that every non-zero element is a unit.
Hence we can use Inverse of Division Product:
:$\paren {\dfrac a b}^{-1} = \dfrac {1_R} {\paren {a / b}} = \dfrac b a$
which applies t... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0$ and whose [[Definition:Unity of Field|unity]] is $1$.
Let $a, b \in F$ such that $a \ne 0$.
Then:
:$\paren {a \times b^{-1} }^{-1} = b \times a^{-1}$
This can also be expressed using the notat... | By definition, a [[Definition:Field (Abstract Algebra)|field]] is a [[Definition:Non-Trivial Ring|non-trivial]] [[Definition:Division Ring|division ring]] whose [[Definition:Ring Product|ring product]] is [[Definition:Commutative Operation|commutative]].
By definition, a [[Definition:Division Ring|division ring]] is a... | Inverse of Field Product with Inverse/Proof 1 | https://proofwiki.org/wiki/Inverse_of_Field_Product_with_Inverse | https://proofwiki.org/wiki/Inverse_of_Field_Product_with_Inverse/Proof_1 | [
"Field Theory",
"Inverse of Field Product with Inverse"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Division/Field"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Non-Trivial Ring",
"Definition:Division Ring",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Commutative/Operation",
"Definition:Division Ring",
"Definition:Ring with Unity",
"Definition:Ring Zero",
"Definition:Element",
"Definition:Un... |
proofwiki-18438 | Addition of Division Products in Field | Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$.
Let $a, b, c, d \in F$ such that $b \ne 0$ and $d \ne 0$.
Then:
:$\dfrac a b + \dfrac c d = \dfrac {a d + b c} {b d}$
where $\dfrac x z$ is defined as $x \paren {z^{-1} }$. | By definition, a field is a non-trivial division ring whose ring product is commutative.
By definition, a division ring is a ring with unity such that every non-zero element is a unit.
Hence we can use Addition of Division Products:
:$\dfrac a b + \dfrac c d = \dfrac {a \circ d + b \circ c} {b \circ d}$
which applies ... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0$ and whose [[Definition:Unity of Field|unity]] is $1$.
Let $a, b, c, d \in F$ such that $b \ne 0$ and $d \ne 0$.
Then:
:$\dfrac a b + \dfrac c d = \dfrac {a d + b c} {b d}$
where $\dfrac x z$ i... | By definition, a [[Definition:Field (Abstract Algebra)|field]] is a [[Definition:Non-Trivial Ring|non-trivial]] [[Definition:Division Ring|division ring]] whose [[Definition:Ring Product|ring product]] is [[Definition:Commutative Operation|commutative]].
By definition, a [[Definition:Division Ring|division ring]] is a... | Addition of Division Products in Field | https://proofwiki.org/wiki/Addition_of_Division_Products_in_Field | https://proofwiki.org/wiki/Addition_of_Division_Products_in_Field | [
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Non-Trivial Ring",
"Definition:Division Ring",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Commutative/Operation",
"Definition:Division Ring",
"Definition:Ring with Unity",
"Definition:Ring Zero",
"Definition:Element",
"Definition:Un... |
proofwiki-18439 | Norm on Bounded Linear Functional is Finite | Let $V$ be a normed vector space.
Let $L$ be a bounded linear functional on $V$.
Let $\norm L$ denote the norm on $L$ defined as:
:$\norm L = \inf \set {c > 0: \forall v \in V: \size {L v} \le c \norm v_V}$
Then:
:$\norm L < \infty$ | By definition of a bounded linear functional:
:$\exists c \in \R_{> 0}: \forall v \in V: \size{L v} \le c \norm v_V$
Hence:
:$\set {\lambda > 0: \forall v \in V: \size {L v} \le \lambda \norm v_V} \ne \O$
By definition:
$\set {\lambda > 0: \forall v \in V: \size {L v} \le \lambda \norm v_V}$ is bounded below by $0$.
Fr... | Let $V$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $L$ be a [[Definition:Bounded Linear Functional|bounded linear functional]] on $V$.
Let $\norm L$ denote the [[Definition:Norm on Bounded Linear Transformation|norm]] on $L$ defined as:
:$\norm L = \inf \set {c > 0: \forall v \in V: \size {L v} ... | By definition of a [[Definition:Bounded Linear Functional|bounded linear functional]]:
:$\exists c \in \R_{> 0}: \forall v \in V: \size{L v} \le c \norm v_V$
Hence:
:$\set {\lambda > 0: \forall v \in V: \size {L v} \le \lambda \norm v_V} \ne \O$
By definition:
$\set {\lambda > 0: \forall v \in V: \size {L v} \le \la... | Norm on Bounded Linear Functional is Finite | https://proofwiki.org/wiki/Norm_on_Bounded_Linear_Functional_is_Finite | https://proofwiki.org/wiki/Norm_on_Bounded_Linear_Functional_is_Finite | [
"Bounded Linear Functionals"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Functional",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Definition:Bounded Linear Functional",
"Definition:Bounded Below Set/Real Numbers",
"Greatest Lower Bound Property",
"Category:Bounded Linear Functionals"
] |
proofwiki-18440 | Construction of Direct Product of Fields | Let $\struct {F, +_F, \times_F}$ be a field whose zero is $0$ and whose multiplicative identity is $1$.
Let $E = F \times F$ be the Cartesian product of $F$ with itself.
Let addition be defined on $E$ by:
:$\forall a, b, c, d \in F: \tuple {a, b} +_E \tuple {c, d} := \tuple {a +_F c, b +_F d}$
Let multiplication be def... | In order to define the structure rigorously, each of the field addition and field multiplication operations were explicitly stated in the above.
However, in order to simplify presentation, the operations will be denoted in the following as:
{{begin-eqn}}
{{eqn | q = \forall a, b, c, d \in F
| l = \tuple {a, b} + ... | Let $\struct {F, +_F, \times_F}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0$ and whose [[Definition:Multiplicative Identity|multiplicative identity]] is $1$.
Let $E = F \times F$ be the [[Definition:Cartesian Product|Cartesian product]] of $F$ with itself.
Let [[Defi... | In order to define the structure rigorously, each of the [[Definition:Field Addition|field addition]] and [[Definition:Field Product|field multiplication]] [[Definition:Binary Operation|operations]] were explicitly stated in the above.
However, in order to simplify presentation, the [[Definition:Binary Operation|opera... | Construction of Direct Product of Fields | https://proofwiki.org/wiki/Construction_of_Direct_Product_of_Fields | https://proofwiki.org/wiki/Construction_of_Direct_Product_of_Fields | [
"External Direct Products",
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Cartesian Product",
"Definition:Field (Abstract Algebra)/Addition",
"Definition:Field (Abstract Algebra)/Product",
"Definition:Field (Abstract Algebra)"
] | [
"Definition:Field (Abstract Algebra)/Addition",
"Definition:Field (Abstract Algebra)/Product",
"Definition:Operation/Binary Operation",
"Definition:Operation/Binary Operation",
"Definition:Field (Abstract Algebra)",
"Definition:Group Direct Product",
"Definition:Abelian Group",
"External Direct Produc... |
proofwiki-18441 | Field Homomorphism is either Trivial or Injection | Let $\struct {E, +_E, \times_E}$ and $\struct {F, +_F, \times_F}$ be fields.
Let $\phi: E \to F$ be a (field) homomorphism.
Then $\phi$ is either an injection or the trivial homomorphism. | This is an instance of Ring Homomorphism from Field is Monomorphism or Zero Homomorphism.
{{qed}} | Let $\struct {E, +_E, \times_E}$ and $\struct {F, +_F, \times_F}$ be [[Definition:Field (Abstract Algebra)|fields]].
Let $\phi: E \to F$ be a [[Definition:Field Homomorphism|(field) homomorphism]].
Then $\phi$ is either an [[Definition:Injection|injection]] or the [[Definition:Trivial Homomorphism|trivial homomorphi... | This is an instance of [[Ring Homomorphism from Field is Monomorphism or Zero Homomorphism]].
{{qed}} | Field Homomorphism is either Trivial or Injection/Proof 1 | https://proofwiki.org/wiki/Field_Homomorphism_is_either_Trivial_or_Injection | https://proofwiki.org/wiki/Field_Homomorphism_is_either_Trivial_or_Injection/Proof_1 | [
"Field Homomorphism is either Trivial or Injection",
"Field Homomorphisms"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Homomorphism",
"Definition:Injection",
"Definition:Zero Homomorphism"
] | [
"Ring Homomorphism from Field is Monomorphism or Zero Homomorphism"
] |
proofwiki-18442 | Surjective Field Homomorphism is Field Isomorphism | Let $E$ and $F$ be fields.
Let $\phi: E \to F$ be a (field) homomorphism.
Let $\phi$ be a surjection.
Then $\phi$ is an isomorphism. | As asserted, let $\phi$ be a surjection.
From Field Homomorphism is either Trivial or Injection, $\phi$ is either an injection or the trivial homomorphism.
If $\phi$ is an injection, then, by definition, $\phi$ is a bijection.
Hence, again by definition, $\phi$ is an isomorphism.
If $\phi$ is not an injection, then $\p... | Let $E$ and $F$ be [[Definition:Field (Abstract Algebra)|fields]].
Let $\phi: E \to F$ be a [[Definition:Field Homomorphism|(field) homomorphism]].
Let $\phi$ be a [[Definition:Surjection|surjection]].
Then $\phi$ is an [[Definition:Field Isomorphism|isomorphism]]. | As asserted, let $\phi$ be a [[Definition:Surjection|surjection]].
From [[Field Homomorphism is either Trivial or Injection]], $\phi$ is either an [[Definition:Injection|injection]] or the [[Definition:Trivial Homomorphism|trivial homomorphism]].
If $\phi$ is an [[Definition:Injection|injection]], then, by definition... | Surjective Field Homomorphism is Field Isomorphism | https://proofwiki.org/wiki/Surjective_Field_Homomorphism_is_Field_Isomorphism | https://proofwiki.org/wiki/Surjective_Field_Homomorphism_is_Field_Isomorphism | [
"Field Homomorphisms",
"Field Isomorphisms"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Homomorphism",
"Definition:Surjection",
"Definition:Isomorphism (Abstract Algebra)/Field Isomorphism"
] | [
"Definition:Surjection",
"Field Homomorphism is either Trivial or Injection",
"Definition:Injection",
"Definition:Zero Homomorphism",
"Definition:Injection",
"Definition:Bijection",
"Definition:Isomorphism (Abstract Algebra)/Field Isomorphism",
"Definition:Injection",
"Definition:Zero Homomorphism",... |
proofwiki-18443 | Field Contains at least 2 Elements | Let $\struct {F, +, \times}$ be a field.
Then $E$ contains at least $2$ elements. | By definition, $\struct {F, +, \times}$ is an algebraic structure such that:
:$\struct {F, +}$ is an abelian group whose identity element is $0 \in F$
:$\struct {F^*, \times}$ is also an abelian group, where $F^* = F \setminus \set 0$.
From Group is not Empty, $\struct {F^*, \times}$ cannot be the empty set.
So there a... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]].
Then $E$ contains at least $2$ [[Definition:Element|elements]]. | By definition, $\struct {F, +, \times}$ is an [[Definition:Algebraic Structure with Two Operations|algebraic structure]] such that:
:$\struct {F, +}$ is an [[Definition:Abelian Group|abelian group]] whose [[Definition:Identity Element|identity element]] is $0 \in F$
:$\struct {F^*, \times}$ is also an [[Definition:Abel... | Field Contains at least 2 Elements | https://proofwiki.org/wiki/Field_Contains_at_least_2_Elements | https://proofwiki.org/wiki/Field_Contains_at_least_2_Elements | [
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Element"
] | [
"Definition:Algebraic Structure/Two Operations",
"Definition:Abelian Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Abelian Group",
"Group is not Empty",
"Definition:Empty Set",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Defi... |
proofwiki-18444 | Subfield Test/Four-Step | $\struct {K, +, \circ}$ is a subfield of $\struct {F, +, \times}$ {{iff}} these all hold:
:$(1): \quad K^* \ne \O$
:$(2): \quad \forall x, y \in K: x + \paren {-y} \in K$
:$(3): \quad \forall x, y \in K: x \times y \in K$
:$(4): \quad x \in K^* \implies x^{-1} \in K^*$
where $K^*$ denotes $K \setminus \set {0_F}$. | === Necessary Condition ===
Let $\struct {K, +, \times}$ be a subfield of $\struct {F, +, \circ}$.
Then the conditions $(1)$ to $(4)$ all hold by virtue of the field axioms.
{{qed|lemma}} | $\struct {K, +, \circ}$ is a [[Definition:Subfield|subfield]] of $\struct {F, +, \times}$ {{iff}} these all hold:
:$(1): \quad K^* \ne \O$
:$(2): \quad \forall x, y \in K: x + \paren {-y} \in K$
:$(3): \quad \forall x, y \in K: x \times y \in K$
:$(4): \quad x \in K^* \implies x^{-1} \in K^*$
where $K^*$ denotes $... | === Necessary Condition ===
Let $\struct {K, +, \times}$ be a [[Definition:Subfield|subfield]] of $\struct {F, +, \circ}$.
Then the conditions $(1)$ to $(4)$ all hold by virtue of the [[Axiom:Field Axioms|field axioms]].
{{qed|lemma}} | Subfield Test/Four-Step | https://proofwiki.org/wiki/Subfield_Test/Four-Step | https://proofwiki.org/wiki/Subfield_Test/Four-Step | [
"Subfield Test"
] | [
"Definition:Subfield"
] | [
"Definition:Subfield",
"Axiom:Field Axioms"
] |
proofwiki-18445 | Subfield Test/Three-Step | $\struct {K, +, \times}$ is a subfield of $\struct {F, +, \times}$ {{iff}} these all hold:
:$(1): \quad K^* \ne \O$
:$(2): \quad \forall x, y \in K: x - y \in K$
:$(3): \quad \forall x \in K: \forall y \in K^*: \dfrac x y \in K$
where $K^*$ denotes $K \setminus \set {0_F}$. | === Necessary Condition ===
Let $\struct {K, +, \times}$ be a subfield of $\struct {F, +, \times}$.
Then the conditions $(1)$ to $(3)$ all hold by virtue of the field axioms.
{{qed|lemma}} | $\struct {K, +, \times}$ is a [[Definition:Subfield|subfield]] of $\struct {F, +, \times}$ {{iff}} these all hold:
:$(1): \quad K^* \ne \O$
:$(2): \quad \forall x, y \in K: x - y \in K$
:$(3): \quad \forall x \in K: \forall y \in K^*: \dfrac x y \in K$
where $K^*$ denotes $K \setminus \set {0_F}$. | === Necessary Condition ===
Let $\struct {K, +, \times}$ be a [[Definition:Subfield|subfield]] of $\struct {F, +, \times}$.
Then the conditions $(1)$ to $(3)$ all hold by virtue of the [[Axiom:Field Axioms|field axioms]].
{{qed|lemma}} | Subfield Test/Three-Step | https://proofwiki.org/wiki/Subfield_Test/Three-Step | https://proofwiki.org/wiki/Subfield_Test/Three-Step | [
"Subfield Test"
] | [
"Definition:Subfield"
] | [
"Definition:Subfield",
"Axiom:Field Axioms"
] |
proofwiki-18446 | Continuous Function with Sequential Limits at Infinity has Limit at Infinity | Let $f : \openint 0 \infty \to \R$ be a continuous function such that:
:for each $x \in \openint 0 \infty$, the sequence $\sequence {\map f {n x} }$ converges to $0$.
Then:
:$\ds \lim_{x \mathop \to \infty} \map f x = 0$ | Fix $\epsilon > 0$.
For each $n \in \N$, define $g_n : \openint 0 \infty \to \R$ by:
:$\map {g_n} x = \map f {n x}$
From Composite of Continuous Mappings is Continuous, we have:
:$g_n$ is continuous for each $n$.
For each $m \in \N$, define the set $X_m$ by:
:$X_m = \map { {g_m}^{-1} } {\closedint {- \epsilon} \epsil... | Let $f : \openint 0 \infty \to \R$ be a [[Definition:Continuous Function|continuous function]] such that:
:for each $x \in \openint 0 \infty$, the [[Definition:Sequence|sequence]] $\sequence {\map f {n x} }$ [[Definition:Convergent Sequence|converges]] to $0$.
Then:
:$\ds \lim_{x \mathop \to \infty} \map f x = 0$ | Fix $\epsilon > 0$.
For each $n \in \N$, define $g_n : \openint 0 \infty \to \R$ by:
:$\map {g_n} x = \map f {n x}$
From [[Composite of Continuous Mappings is Continuous]], we have:
:$g_n$ is [[Definition:Continuous Function|continuous]] for each $n$.
For each $m \in \N$, define the [[Definition:Set|set]] $X_m$ ... | Continuous Function with Sequential Limits at Infinity has Limit at Infinity | https://proofwiki.org/wiki/Continuous_Function_with_Sequential_Limits_at_Infinity_has_Limit_at_Infinity | https://proofwiki.org/wiki/Continuous_Function_with_Sequential_Limits_at_Infinity_has_Limit_at_Infinity | [
"Continuous Functions"
] | [
"Definition:Continuous Function",
"Definition:Sequence",
"Definition:Convergent Sequence"
] | [
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Function",
"Definition:Set",
"Continuity Defined from Closed Sets",
"Definition:Closed Set",
"Intersection of Closed Sets is Closed",
"Definition:Closed Set",
"Definition:Sequence",
"Definition:Convergent Sequence",
"Definiti... |
proofwiki-18447 | Complex Numbers of Type Rational a plus b root 2 form Field | Let $\Q \sqbrk {\sqrt 2}$ denote the set:
:$\Q \sqbrk {\sqrt 2} := \set {x \in \C: x = a + b \sqrt 2: a, b \in \Q}$
that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rational numbers.
Then the algebraic structure:
:$\struct {\Q \sqbrk {\sqrt 2}, +, \times}$
where $+$ and $\times$ are conventional a... | From Real Numbers of Type Rational a plus b root 2 form Field, the set:
:$\Q \sqbrk {\sqrt 2} := \set {x \in \R: x = a + b \sqrt 2: a, b \in \Q}$
forms a field.
As $\Q \sqbrk {\sqrt 2} \subseteq \R$ and $\R \subseteq \C$ it follows that $\struct {\Q \sqbrk {\sqrt 2}, +, \times}$ is a subfield of $\C$.
Hence the result ... | Let $\Q \sqbrk {\sqrt 2}$ denote the [[Definition:Set|set]]:
:$\Q \sqbrk {\sqrt 2} := \set {x \in \C: x = a + b \sqrt 2: a, b \in \Q}$
that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are [[Definition:Rational Number|rational numbers]].
Then the [[Definition:Algebraic Structure with Two Operations|a... | From [[Real Numbers of Type Rational a plus b root 2 form Field]], the [[Definition:Set|set]]:
:$\Q \sqbrk {\sqrt 2} := \set {x \in \R: x = a + b \sqrt 2: a, b \in \Q}$
forms a [[Definition:Field (Abstract Algebra)|field]].
As $\Q \sqbrk {\sqrt 2} \subseteq \R$ and $\R \subseteq \C$ it follows that $\struct {\Q \sqbr... | Complex Numbers of Type Rational a plus b root 2 form Field | https://proofwiki.org/wiki/Complex_Numbers_of_Type_Rational_a_plus_b_root_2_form_Field | https://proofwiki.org/wiki/Complex_Numbers_of_Type_Rational_a_plus_b_root_2_form_Field | [
"Number Fields"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Algebraic Structure/Two Operations",
"Definition:Addition/Real Numbers",
"Definition:Multiplication/Real Numbers",
"Definition:Real Number",
"Definition:Number Field "
] | [
"Real Numbers of Type Rational a plus b root 2 form Field",
"Definition:Set",
"Definition:Field (Abstract Algebra)",
"Definition:Subfield",
"Definition:Number Field "
] |
proofwiki-18448 | Number Field has Rational Numbers as Subfield | Let $F$ be a number field.
The field of rational numbers $\struct {\Q, +, \times}$ is a subfield of $F$. | Let $F$ be a number field.
By definition, $F$ is a subfield of the field of complex numbers $\struct {\C, +, \times}$
We have that Rational Numbers form Subfield of Complex Numbers.
Let $G = \Q \cap F$ be the intersection of $\Q$ with $F$.
By Intersection of Subfields is Subfield, $G$ is a subfield of $\C$ and so also ... | Let $F$ be a [[Definition:Number Field|number field]].
The [[Definition:Field of Rational Numbers|field of rational numbers]] $\struct {\Q, +, \times}$ is a [[Definition:Subfield|subfield]] of $F$. | Let $F$ be a [[Definition:Number Field|number field]].
By definition, $F$ is a [[Definition:Subfield|subfield]] of the [[Definition:Field of Complex Numbers|field of complex numbers]] $\struct {\C, +, \times}$
We have that [[Rational Numbers form Subfield of Complex Numbers]].
Let $G = \Q \cap F$ be the [[Definition... | Number Field has Rational Numbers as Subfield | https://proofwiki.org/wiki/Number_Field_has_Rational_Numbers_as_Subfield | https://proofwiki.org/wiki/Number_Field_has_Rational_Numbers_as_Subfield | [
"Number Fields",
"Rational Numbers"
] | [
"Definition:Number Field",
"Definition:Field of Rational Numbers",
"Definition:Subfield"
] | [
"Definition:Number Field",
"Definition:Subfield",
"Definition:Field of Complex Numbers",
"Rational Numbers form Subfield of Complex Numbers",
"Definition:Set Intersection",
"Intersection of Subfields is Subfield",
"Definition:Subfield",
"Definition:Number Field",
"Intersection is Subset",
"Definit... |
proofwiki-18449 | Element to Power of Positive Characteristic of Field is Zero | Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$.
Let the characteristic of $F$ be $n$ such that $n > 0$.
Then:
:$n \cdot a = 0$
where $n \cdot a$ denotes the power of $a$ in the context of the additive group $\struct {F, +}$:
:$n \cdot a = \begin {cases}
0 & : n = 0 \\
\paren {\paren {n... | By definition, the characteristic of $\struct {F, +, \times}$ is the order of the additive group $\struct {F, +}$.
By Element to Power of Group Order is Identity it follows directly that:
:$n \cdot a = 0$
{{qed}} | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0$ and whose [[Definition:Unity of Field|unity]] is $1$.
Let the [[Definition:Characteristic of Field|characteristic]] of $F$ be $n$ such that $n > 0$.
Then:
:$n \cdot a = 0$
where $n \cdot a$ den... | By definition, the [[Definition:Characteristic of Field|characteristic]] of $\struct {F, +, \times}$ is the [[Definition:Order of Group|order]] of the [[Definition:Additive Group of Field|additive group]] $\struct {F, +}$.
By [[Element to Power of Group Order is Identity]] it follows directly that:
:$n \cdot a = 0$
{{... | Element to Power of Positive Characteristic of Field is Zero | https://proofwiki.org/wiki/Element_to_Power_of_Positive_Characteristic_of_Field_is_Zero | https://proofwiki.org/wiki/Element_to_Power_of_Positive_Characteristic_of_Field_is_Zero | [
"Characteristics of Fields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Characteristic of Field",
"Definition:Power of Element/Group",
"Definition:Additive Group"
] | [
"Definition:Characteristic of Field",
"Definition:Order of Structure",
"Definition:Additive Group of Field",
"Element to Power of Group Order is Identity"
] |
proofwiki-18450 | Field has Characteristic of Zero iff exists Monomorphism from Rationals | Let $F$ be a field.
Then:
:there exists a field monomorphism $\phi: \Q \to F$ from the field of rational numbers $\Q$ and $F$.
{{iff}}:
:$\Char F = 0$
where $\Char F$ denotes the characteristic of $F$. | === Necessary Condition ===
Let $\Char F = 0$.
Then from Field of Characteristic Zero has Unique Prime Subfield, $F$ has a unique prime subfield $K$ such that:
:$K \cong \Q$
where $\cong$ denotes isomorphism.
Thus there exists an isomorphism from $\Q$ to a subfield of $F$.
From Injection to Image is Bijection, it follo... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Then:
:there exists a [[Definition:Field Monomorphism|field monomorphism]] $\phi: \Q \to F$ from the [[Definition:Field of Rational Numbers|field of rational numbers]] $\Q$ and $F$.
{{iff}}:
:$\Char F = 0$
where $\Char F$ denotes the [[Definition:Characterist... | === Necessary Condition ===
Let $\Char F = 0$.
Then from [[Field of Characteristic Zero has Unique Prime Subfield]], $F$ has a [[Definition:Unique|unique]] [[Definition:Prime Subfield|prime subfield]] $K$ such that:
:$K \cong \Q$
where $\cong$ denotes [[Definition:Field Isomorphism|isomorphism]].
Thus there exist... | Field has Characteristic of Zero iff exists Monomorphism from Rationals | https://proofwiki.org/wiki/Field_has_Characteristic_of_Zero_iff_exists_Monomorphism_from_Rationals | https://proofwiki.org/wiki/Field_has_Characteristic_of_Zero_iff_exists_Monomorphism_from_Rationals | [
"Characteristics of Fields",
"Rational Numbers",
"Field Monomorphisms"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Monomorphism",
"Definition:Field of Rational Numbers",
"Definition:Characteristic of Field"
] | [
"Field of Characteristic Zero has Unique Prime Subfield",
"Definition:Unique",
"Definition:Prime Subfield",
"Definition:Isomorphism (Abstract Algebra)/Field Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Field Isomorphism",
"Definition:Subfield",
"Injection to Image is Bijection",
"Definitio... |
proofwiki-18451 | Field has Prime Characteristic p iff exists Monomorphism from Field of Integers Modulo p | Let $F$ be a field.
Then:
:there exists some prime number $p$ such that $\Char F = p$
{{iff}}:
:there exists a field monomorphism $\phi: \Z_p \to F$
where:
:$\Char F$ denotes the characteristic of $F$.
:$\Z_p$ denotes the field of integers modulo $p$. | Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$. | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Then:
:there exists some [[Definition:Prime Number|prime number]] $p$ such that $\Char F = p$
{{iff}}:
:there exists a [[Definition:Field Monomorphism|field monomorphism]] $\phi: \Z_p \to F$
where:
:$\Char F$ denotes the [[Definition:Characteristic of Field|c... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0$ and whose [[Definition:Unity of Field|unity]] is $1$. | Field has Prime Characteristic p iff exists Monomorphism from Field of Integers Modulo p | https://proofwiki.org/wiki/Field_has_Prime_Characteristic_p_iff_exists_Monomorphism_from_Field_of_Integers_Modulo_p | https://proofwiki.org/wiki/Field_has_Prime_Characteristic_p_iff_exists_Monomorphism_from_Field_of_Integers_Modulo_p | [
"Characteristics of Fields",
"Field Monomorphisms",
"Field of Integers Modulo Prime"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Prime Number",
"Definition:Field Monomorphism",
"Definition:Characteristic of Field",
"Definition:Field of Integers Modulo Prime"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity"
] |
proofwiki-18452 | Group is Cancellable Monoid | Let $\struct {G, \circ}$ be a group.
Then $\struct {G, \circ}$ is a cancellable monoid. | By definition, a group is {{afortiori}} a monoid.
From Group Operation is Cancellable, $\circ$ is a cancellable operation in $G$.
Hence the result by definition of cancellable monoid.
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Then $\struct {G, \circ}$ is a [[Definition:Cancellable Monoid|cancellable monoid]]. | By definition, a [[Definition:Group|group]] is {{afortiori}} a [[Definition:Monoid|monoid]].
From [[Group Operation is Cancellable]], $\circ$ is a [[Definition:Cancellable Operation|cancellable operation]] in $G$.
Hence the result by definition of [[Definition:Cancellable Monoid|cancellable monoid]].
{{qed}} | Group is Cancellable Monoid | https://proofwiki.org/wiki/Group_is_Cancellable_Monoid | https://proofwiki.org/wiki/Group_is_Cancellable_Monoid | [
"Cancellable Monoids",
"Group Theory"
] | [
"Definition:Group",
"Definition:Cancellable Monoid"
] | [
"Definition:Group",
"Definition:Monoid",
"Cancellation Laws",
"Definition:Cancellable Operation",
"Definition:Cancellable Monoid"
] |
proofwiki-18453 | Intersection of Submonoids with Monoid Identity is Submonoid | Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.
Let $I$ be an indexing set.
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of submonoids of $S$.
For each $S_\alpha \in \family {S_\alpha}_{\alpha \mathop \in I}$, let $e_S \in S_\alpha$.
Let $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ denote ... | First we show that $\struct {\ds \bigcap_{\alpha \mathop \in I} S_\alpha, \circ}$ is a semigroup: | Let $\struct {S, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity]] is $e_S$.
Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family of Subsets|family]] of [[Definition:Submonoid|submonoids]] of $S$... | First we show that $\struct {\ds \bigcap_{\alpha \mathop \in I} S_\alpha, \circ}$ is a [[Definition:Semigroup|semigroup]]: | Intersection of Submonoids with Monoid Identity is Submonoid | https://proofwiki.org/wiki/Intersection_of_Submonoids_with_Monoid_Identity_is_Submonoid | https://proofwiki.org/wiki/Intersection_of_Submonoids_with_Monoid_Identity_is_Submonoid | [
"Submonoids"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Subsets",
"Definition:Submonoid",
"Definition:Set Intersection/Family of Sets",
"Definition:Submonoid"
] | [
"Definition:Semigroup",
"Definition:Semigroup"
] |
proofwiki-18454 | Generated Submonoid is Intersection of Submonoids containing Generator | Let $\struct {M, \circ}$ be a monoid whose identity is $e_M$.
Let $S \subseteq M$.
Let $\struct {H, \circ}$ be the '''submonoid of $\struct {M, \circ}$ generated by $S$'''.
Then $\struct {H, \circ}$ is the intersection of all submonoids of $\struct {M, \circ}$ containing $S \cup \set {e_M}$. | Let $\struct {H, \circ}$ be the '''submonoid of $\struct {M, \circ}$ generated by $S$'''.
Then by definition $H$ is the smallest (with respect to set inclusion) submonoid of $\struct {M, \circ}$ containing $S \cup \set {e_M}$.
Let $\mathbb S$ be the set of submonoids of $\struct {M, \circ}$ containing $S \cup \set {e_M... | Let $\struct {M, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity]] is $e_M$.
Let $S \subseteq M$.
Let $\struct {H, \circ}$ be the '''[[Definition:Generated Submonoid|submonoid of $\struct {M, \circ}$ generated by $S$]]'''.
Then $\struct {H, \circ}$ is the [[Definition:Set Inte... | Let $\struct {H, \circ}$ be the '''[[Definition:Generated Submonoid|submonoid of $\struct {M, \circ}$ generated by $S$]]'''.
Then by definition $H$ is the [[Definition:Smallest Set by Set Inclusion|smallest (with respect to set inclusion)]] [[Definition:Submonoid|submonoid]] of $\struct {M, \circ}$ containing $S \cup ... | Generated Submonoid is Intersection of Submonoids containing Generator | https://proofwiki.org/wiki/Generated_Submonoid_is_Intersection_of_Submonoids_containing_Generator | https://proofwiki.org/wiki/Generated_Submonoid_is_Intersection_of_Submonoids_containing_Generator | [
"Submonoids"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Generated Submonoid",
"Definition:Set Intersection",
"Definition:Submonoid"
] | [
"Definition:Generated Submonoid",
"Definition:Smallest Set by Set Inclusion",
"Definition:Submonoid",
"Definition:Submonoid",
"Definition:Submonoid",
"Intersection is Subset/General Result",
"Intersection of Submonoids with Monoid Identity is Submonoid",
"Definition:Submonoid",
"Definition:Smallest ... |
proofwiki-18455 | Fundamental Solution to Reduced Linear First Order ODE with Constant Coefficients | Let $H$ be the Heaviside step function.
Let $\lambda \in \R$.
Let $\map f x = \map H x \map \exp {\lambda x}$.
Let $T_f$ be the Schwartz distribution associated with $f$.
Then, in the distributional sense, $T_f$ is the fundamental solution of
:$\paren {\dfrac \d {\d x} - \lambda} T_f = \delta$ | $x \stackrel f {\longrightarrow} \map H x \map \exp {\lambda x}$ is a continuously differentiable real function on $\R \setminus \set 0$ and possibly has a discontinuity at $x = 0$.
By Differentiable Function as Distribution we have that:
:$T'_f = T_{f'}$
Moreover:
:$x < 0 \implies \paren {\map H x \map \exp {\lambda x... | Let $H$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $\lambda \in \R$.
Let $\map f x = \map H x \map \exp {\lambda x}$.
Let $T_f$ be the [[Definition:Schwartz Distribution|Schwartz distribution]] associated with $f$.
Then, in the [[Definition:Distributional Derivative|distributional]]... | $x \stackrel f {\longrightarrow} \map H x \map \exp {\lambda x}$ is a [[Definition:Continuously Differentiable Real Function on Open Interval|continuously differentiable real function]] on $\R \setminus \set 0$ and possibly has a [[Definition:Discontinuity|discontinuity]] at $x = 0$.
By [[Differentiable Function as Di... | Fundamental Solution to Reduced Linear First Order ODE with Constant Coefficients | https://proofwiki.org/wiki/Fundamental_Solution_to_Reduced_Linear_First_Order_ODE_with_Constant_Coefficients | https://proofwiki.org/wiki/Fundamental_Solution_to_Reduced_Linear_First_Order_ODE_with_Constant_Coefficients | [
"Examples of Fundamental Solutions",
"Distributional Derivatives",
"First Order ODEs"
] | [
"Definition:Heaviside Step Function",
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative",
"Definition:Fundamental Solution"
] | [
"Definition:Continuously Differentiable/Real Function/Open Interval",
"Definition:Discontinuity",
"Differentiable Function as Distribution",
"Jump Rule",
"Definition:Term of Expression",
"Definition:Schwartz Distribution"
] |
proofwiki-18456 | Fundamental Solution to y'' + w y | Let $H$ be the Heaviside step function.
Let $\omega \in \R : \omega \ne 0$.
Let $\map f x = \map H x \dfrac {\map \sin {\omega x} } \omega$.
Let $T_f$ be the Schwartz distribution associated with $f$.
Then, in the distributional sense, $T_f$ is the fundamental solution of
:$\paren {\dfrac {\d^2} {\d x^2} + \omega^2} T_... | $x \stackrel f {\longrightarrow} \map H x \dfrac {\map \sin {\omega x} } \omega$ is a continuously differentiable real function on $\R \setminus \set 0$ and possibly has a discontinuity at $x = 0$.
By Differentiable Function as Distribution we have that $T'_f = T_{f'}$.
Moreover:
:$x < 0 \implies \paren {\map H x \dfra... | Let $H$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $\omega \in \R : \omega \ne 0$.
Let $\map f x = \map H x \dfrac {\map \sin {\omega x} } \omega$.
Let $T_f$ be the [[Definition:Schwartz Distribution|Schwartz distribution]] associated with $f$.
Then, in the [[Definition:Distribution... | $x \stackrel f {\longrightarrow} \map H x \dfrac {\map \sin {\omega x} } \omega$ is a [[Definition:Continuously Differentiable Real Function on Open Interval|continuously differentiable real function]] on $\R \setminus \set 0$ and possibly has a [[Definition:Discontinuous|discontinuity]] at $x = 0$.
By [[Differentiabl... | Fundamental Solution to y'' + w y | https://proofwiki.org/wiki/Fundamental_Solution_to_y''_+_w_y | https://proofwiki.org/wiki/Fundamental_Solution_to_y''_+_w_y | [
"Examples of Fundamental Solutions",
"Distributional Derivatives",
"Second Order ODEs"
] | [
"Definition:Heaviside Step Function",
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative",
"Definition:Fundamental Solution"
] | [
"Definition:Continuously Differentiable/Real Function/Open Interval",
"Definition:Discontinuous",
"Differentiable Function as Distribution",
"Jump Rule",
"Definition:Continuously Differentiable/Real Function/Open Interval",
"Definition:Discontinuity",
"Differentiable Function as Distribution",
"Jump R... |
proofwiki-18457 | Order of Floor Function | Let $\floor x$ denote the floor function of $x$.
Then:
:$\floor x = x + \map \OO 1$
where $\OO$ is big-O notation. | From Floor is between Number and One Less:
:$\floor x \le x < \floor x + 1$
so:
:$0 \le x - \floor x < 1$
By the definition of the absolute value function, we have:
:$\size {\floor x - x} < 1$
so by the definition of Big-O notation, we have:
:$\floor x - x = \map \OO 1$
We can conclude that:
:$\floor x = x + \map \OO... | Let $\floor x$ denote the [[Definition:Floor Function|floor function]] of $x$.
Then:
:$\floor x = x + \map \OO 1$
where $\OO$ is [[Definition:Big-O Estimate for Real Function|big-O notation]]. | From [[Floor is between Number and One Less]]:
:$\floor x \le x < \floor x + 1$
so:
:$0 \le x - \floor x < 1$
By the definition of the [[Definition:Absolute Value|absolute value function]], we have:
:$\size {\floor x - x} < 1$
so by the definition of [[Definition:Big-O Estimate for Real Function|Big-O notation]... | Order of Floor Function | https://proofwiki.org/wiki/Order_of_Floor_Function | https://proofwiki.org/wiki/Order_of_Floor_Function | [
"Analytic Number Theory",
"Floor Function"
] | [
"Definition:Floor Function",
"Definition:Big-O Notation/Real"
] | [
"Floor is between Number and One Less",
"Definition:Absolute Value",
"Definition:Big-O Notation/Real",
"Category:Analytic Number Theory",
"Category:Floor Function"
] |
proofwiki-18458 | Fundamental Solution to Nth Derivative | Let $H$ be the Heaviside step function.
Let $n \in \N_{>0}$.
Let $\map {f_n} x = \map H x \dfrac {x^{n - 1} } {\paren {n - 1}!}$.
Let $T_{f_n}$ be the Schwartz distribution associated with $f_n$.
Then, in the distributional sense, $T_{f_n}$ is the fundamental solution of
:$\dfrac {\rd^n} {\rd x^n} T_{f_n} = \delta$ | Proof by Principle of Mathematical Induction: | Let $H$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $n \in \N_{>0}$.
Let $\map {f_n} x = \map H x \dfrac {x^{n - 1} } {\paren {n - 1}!}$.
Let $T_{f_n}$ be the [[Definition:Schwartz Distribution|Schwartz distribution]] associated with $f_n$.
Then, in the [[Definition:Distributional De... | Proof by [[Principle of Mathematical Induction]]: | Fundamental Solution to Nth Derivative | https://proofwiki.org/wiki/Fundamental_Solution_to_Nth_Derivative | https://proofwiki.org/wiki/Fundamental_Solution_to_Nth_Derivative | [
"Examples of Fundamental Solutions",
"Distributional Derivatives"
] | [
"Definition:Heaviside Step Function",
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative",
"Definition:Fundamental Solution"
] | [
"Principle of Mathematical Induction"
] |
proofwiki-18459 | Coordinate Representation of Divergence | Let $\struct {M, g}$ be a Riemannian manifold.
Let $U \subseteq M$ be an open set.
Let $\tuple {x^i}$ be local smooth coordinates.
Let $X$ be a smooth vector field on $M$.
Let $\operatorname {div}$ be the divergence operator.
Then:
:$\map {\operatorname {div} } {X^i \dfrac \partial {\partial x^i}} = \dfrac 1 {\sqrt g} ... | Let $\omega$ be the Riemannian volume form on $M$.
Recall that the divergence of $X$ is defined by the relation:
:$\map \d {\omega \rfloor X} = \map {\operatorname {div} } X \omega$, where $\rfloor$ denotes the interior product.
In local coordinates we have:
:$\omega = \sqrt g \rd x_1 \wedge \dots \wedge \d x_n$
Thus:
... | Let $\struct {M, g}$ be a [[Definition:Riemannian Manifold|Riemannian manifold]].
Let $U \subseteq M$ be an [[Definition:Open Set|open set]].
Let $\tuple {x^i}$ be [[Definition:Local Smooth Coordinates|local smooth coordinates]].
Let $X$ be a [[Definition:Smooth Vector Field|smooth vector field]] on $M$.
Let $\oper... | Let $\omega$ be the [[Definition:Riemannian Volume Form|Riemannian volume form]] on $M$.
Recall that the [[Definition:Divergence Operator on Riemannian Manifold|divergence of $X$]] is defined by the [[Definition:Relation|relation]]:
:$\map \d {\omega \rfloor X} = \map {\operatorname {div} } X \omega$, where $\rfloor$ ... | Coordinate Representation of Divergence | https://proofwiki.org/wiki/Coordinate_Representation_of_Divergence | https://proofwiki.org/wiki/Coordinate_Representation_of_Divergence | [
"Riemannian Geometry",
"Divergence Operator"
] | [
"Definition:Riemannian Manifold",
"Definition:Open Set",
"Definition:Local Smooth Coordinates",
"Definition:Smooth Vector Field",
"Definition:Divergence Operator/Riemannian Manifold"
] | [
"Definition:Riemannian Volume Form",
"Definition:Divergence Operator/Riemannian Manifold",
"Definition:Relation",
"Definition:Interior Multiplication"
] |
proofwiki-18460 | Transplanting Theorem/Corollary | Let $\struct {S, \circ}$ be an algebraic structure.
Let $f: S \to S$ be an automorphism on $\struct {S, \circ}$.
Then the transplant of $\circ$ under $f$ is $\circ$ itself. | From the Transplanting Theorem there exists one and only one operation $\circ$ such that $f: \struct {S, \circ} \to \struct {S, \circ}$ is an automorphism.
{{qed}} | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]].
Let $f: S \to S$ be an [[Definition:Automorphism (Abstract Algebra)|automorphism]] on $\struct {S, \circ}$.
Then the [[Definition:Transplant (Abstract Algebra)|transplant]] of $\circ$ under $f$ is $\circ$ itself... | From the [[Transplanting Theorem]] there exists [[Definition:Unique|one and only one]] [[Definition:Binary Operation|operation]] $\circ$ such that $f: \struct {S, \circ} \to \struct {S, \circ}$ is an [[Definition:Automorphism (Abstract Algebra)|automorphism]].
{{qed}} | Transplanting Theorem/Corollary | https://proofwiki.org/wiki/Transplanting_Theorem/Corollary | https://proofwiki.org/wiki/Transplanting_Theorem/Corollary | [
"Transplanting Theorem"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Automorphism (Abstract Algebra)",
"Definition:Transplant (Abstract Algebra)"
] | [
"Transplanting Theorem",
"Definition:Unique",
"Definition:Operation/Binary Operation",
"Definition:Automorphism (Abstract Algebra)"
] |
proofwiki-18461 | Invertible Element of Associative Structure is Cancellable/Corollary | Let $\struct {S, \circ}$ be a monoid whose identity element is $e_S$.
An element of $\struct {S, \circ}$ which is invertible is also cancellable. | By definition, a monoid is an associative algebraic structure with an identity element.
The result follows from Invertible Element of Associative Structure is Cancellable.
{{Qed}}
Category:Invertible Element of Associative Structure is Cancellable
3z54y864c0xutnlpqvr3ewiqiqcirwl | Let $\struct {S, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity element]] is $e_S$.
An [[Definition:Element|element]] of $\struct {S, \circ}$ which is [[Definition:Invertible Element|invertible]] is also [[Definition:Cancellable Element|cancellable]]. | By definition, a [[Definition:Monoid|monoid]] is an [[Definition:Associative Algebraic Structure|associative algebraic structure]] with an [[Definition:Identity Element|identity element]].
The result follows from [[Invertible Element of Associative Structure is Cancellable]].
{{Qed}}
[[Category:Invertible Element of ... | Invertible Element of Associative Structure is Cancellable/Corollary | https://proofwiki.org/wiki/Invertible_Element_of_Associative_Structure_is_Cancellable/Corollary | https://proofwiki.org/wiki/Invertible_Element_of_Associative_Structure_is_Cancellable/Corollary | [
"Invertible Element of Associative Structure is Cancellable"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Invertible Element",
"Definition:Cancellable Element"
] | [
"Definition:Monoid",
"Definition:Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Invertible Element of Associative Structure is Cancellable",
"Category:Invertible Element of Associative Structure is Cancellable"
] |
proofwiki-18462 | Smooth Real Function times Derivative of Dirac Delta Distribution | Let $\alpha \in \map {C^\infty} \R$ be a smooth real function.
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Then in the Schwartz distributional sense it holds that:
:$\alpha \cdot \delta' = \map \alpha 0 \delta' - \map {\alpha'} 0 \delta$ | Let $\phi \in \map \DD \R$ be a test function.
{{begin-eqn}}
{{eqn | l = \map {\alpha \cdot \delta'} \phi
| r = \map {\delta'} {\alpha \phi}
| c = {{Defof|Multiplication of Schwartz Distribution by Smooth Function}}
}}
{{eqn | r = - \map \delta {\paren {\alpha \phi}'}
| c = {{Defof|Distributional Deri... | Let $\alpha \in \map {C^\infty} \R$ be a [[Definition:Smooth Real Function|smooth real function]].
Let $\delta \in \map {\DD'} \R$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Then in the [[Definition:Schwartz Distribution|Schwartz distributional]] sense it holds that:
:$\alpha \cdot \de... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
{{begin-eqn}}
{{eqn | l = \map {\alpha \cdot \delta'} \phi
| r = \map {\delta'} {\alpha \phi}
| c = {{Defof|Multiplication of Schwartz Distribution by Smooth Function}}
}}
{{eqn | r = - \map \delta {\paren {\alpha \phi}'}
| c... | Smooth Real Function times Derivative of Dirac Delta Distribution | https://proofwiki.org/wiki/Smooth_Real_Function_times_Derivative_of_Dirac_Delta_Distribution | https://proofwiki.org/wiki/Smooth_Real_Function_times_Derivative_of_Dirac_Delta_Distribution | [
"Smooth Real Function times Derivative of Dirac Delta Distribution",
"Dirac Delta Function"
] | [
"Definition:Smooth Real Function",
"Definition:Dirac Delta Distribution",
"Definition:Schwartz Distribution"
] | [
"Definition:Test Function"
] |
proofwiki-18463 | Parity Group is Only Group with 2 Elements | Let $\struct {G, \circ}$ be a group with exactly $2$ elements.
Then $\struct {G, \circ}$ is isomorphic to the parity group, which can be exemplified $\struct {\Z_2, +_2}$.
That is, the additive group of integers modulo $2$. | We have that $2$ is a prime number.
Hence $\struct {\Z_2, +_2}$ is a prime group.
The result follows from Prime Groups of Same Order are Isomorphic. | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] with exactly $2$ [[Definition:Element|elements]].
Then $\struct {G, \circ}$ is [[Definition:Isomorphism|isomorphic]] to the [[Definition:Parity Group|parity group]], which can be exemplified $\struct {\Z_2, +_2}$.
That is, the [[Definition:Additive Group of In... | We have that $2$ is a [[Definition:Prime Number|prime number]].
Hence $\struct {\Z_2, +_2}$ is a [[Definition:Prime Group|prime group]].
The result follows from [[Prime Groups of Same Order are Isomorphic]]. | Parity Group is Only Group with 2 Elements | https://proofwiki.org/wiki/Parity_Group_is_Only_Group_with_2_Elements | https://proofwiki.org/wiki/Parity_Group_is_Only_Group_with_2_Elements | [
"Parity Group",
"Groups of Order 2"
] | [
"Definition:Group",
"Definition:Element",
"Definition:Isomorphism",
"Definition:Parity Group",
"Definition:Additive Group of Integers Modulo m"
] | [
"Definition:Prime Number",
"Definition:Prime Group",
"Prime Groups of Same Order are Isomorphic"
] |
proofwiki-18464 | Identity of Submagma containing Identity of Magma is Same Identity | Let $\struct {S, \circ}$ be a magma which has an identity $e$.
Let $\struct {T, \circ}$ be a submagma of $\struct {S, \circ}$ such that $e \in T$.
Then $e$ is the identity of $T$. | From Identity is Unique, there can be only one identity $e$ of $\struct {S, \circ}$.
We have that:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = T
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = S
| c =
}}
{{eqn | ll= \leadsto
| l = x \circ e
| r = x = e \circ... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]] which has an [[Definition:Identity Element|identity]] $e$.
Let $\struct {T, \circ}$ be a [[Definition:Submagma|submagma]] of $\struct {S, \circ}$ such that $e \in T$.
Then $e$ is the [[Definition:Identity Element|identity]] of $T$. | From [[Identity is Unique]], there can be [[Definition:Unique|only one]] [[Definition:Identity Element|identity]] $e$ of $\struct {S, \circ}$.
We have that:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = T
| c =
}}
{{eqn | ll= \leadsto
| l = x
| o = \in
| r = S
| c =
}}
{{eqn ... | Identity of Submagma containing Identity of Magma is Same Identity | https://proofwiki.org/wiki/Identity_of_Submagma_containing_Identity_of_Magma_is_Same_Identity | https://proofwiki.org/wiki/Identity_of_Submagma_containing_Identity_of_Magma_is_Same_Identity | [
"Identity Elements",
"Magmas"
] | [
"Definition:Magma",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Submagma",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Identity is Unique",
"Definition:Unique",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unique",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-18465 | Subsemigroup of Monoid is not necessarily Monoid | Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.
Let $\struct {T, \circ}$ be a subsemigroup of $\struct {S, \circ}$
Then it is not necessarily the case that $\struct {T, \circ}$ has an identity. | Consider the set of integers under multiplication $\struct {\Z, \times}$.
From Integers under Multiplication form Monoid, $\struct {\Z, \times}$ is a monoid.
Let $n \in \Z$ such that $n > 1$.
Let $n \Z$ be the set of integer multiples of $n$:
:$\set {x \in \Z: n \divides x}$
where $\divides$ denotes divisibility.
From ... | Let $\struct {S, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity]] is $e_S$.
Let $\struct {T, \circ}$ be a [[Definition:Subsemigroup|subsemigroup]] of $\struct {S, \circ}$
Then it is not necessarily the case that $\struct {T, \circ}$ has an [[Definition:Identity Element|identit... | Consider the [[Definition:Set|set]] of [[Definition:Integer|integers]] under [[Definition:Integer Multiplication|multiplication]] $\struct {\Z, \times}$.
From [[Integers under Multiplication form Monoid]], $\struct {\Z, \times}$ is a [[Definition:Monoid|monoid]].
Let $n \in \Z$ such that $n > 1$.
Let $n \Z$ be the ... | Subsemigroup of Monoid is not necessarily Monoid | https://proofwiki.org/wiki/Subsemigroup_of_Monoid_is_not_necessarily_Monoid | https://proofwiki.org/wiki/Subsemigroup_of_Monoid_is_not_necessarily_Monoid | [
"Identity Elements",
"Subsemigroups",
"Monoids"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subsemigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Multiplication/Integers",
"Integers under Multiplication form Monoid",
"Definition:Monoid",
"Definition:Set of Integer Multiples",
"Definition:Divisor (Algebra)/Integer",
"Integer Multiples under Multiplication form Semigroup",
"Definition:Semigrou... |
proofwiki-18466 | Image of P-adic Norm | Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.
Then the image of $\norm {\,\cdot\,}_p$ is:
:$\Img {\norm {\,\cdot\,}_p} = \set {p^n : n \in \Z} \cup \set 0$ | This follows immediately from:
:{{Defof|P-adic Norm}}
:{{Defof|P-adic Valuation}}
{{qed}}
Category:P-adic Number Theory
loyf7o0bnhfz37u556teza8xvny0slx | Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]] for some [[Definition:Prime Number|prime number]] $p$.
Then the [[Definition:Image of Mapping|image]] of $\norm {\,\cdot\,}_p$ is:
:$\Img {\norm {\,\cdot\,}_p} = \set {p^n : n \in \Z} \cup \... | This follows immediately from:
:{{Defof|P-adic Norm}}
:{{Defof|P-adic Valuation}}
{{qed}}
[[Category:P-adic Number Theory]]
loyf7o0bnhfz37u556teza8xvny0slx | Image of P-adic Norm | https://proofwiki.org/wiki/Image_of_P-adic_Norm | https://proofwiki.org/wiki/Image_of_P-adic_Norm | [
"P-adic Number Theory"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Prime Number",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Category:P-adic Number Theory"
] |
proofwiki-18467 | Left Operation is Closed for All Subsets | :$\struct {T, \leftarrow}$ is a subsemigroup of $\struct {S, \leftarrow}$. | From Structure under Left Operation is Semigroup we have that $\struct {S, \leftarrow}$ is a semigroup, whatever the nature of $S$.
Let $T \in \powerset S$.
Then:
:From Structure under Left Operation is Semigroup, $\struct {T, \leftarrow}$ is a semigroup, and therefore a subsemigroup of $\struct {S, \leftarrow}$.
This ... | :$\struct {T, \leftarrow}$ is a [[Definition:Subsemigroup|subsemigroup]] of $\struct {S, \leftarrow}$. | From [[Structure under Left Operation is Semigroup]] we have that $\struct {S, \leftarrow}$ is a [[Definition:Semigroup|semigroup]], whatever the nature of $S$.
Let $T \in \powerset S$.
Then:
:From [[Structure under Left Operation is Semigroup]], $\struct {T, \leftarrow}$ is a [[Definition:Semigroup|semigroup]], and ... | Left Operation is Closed for All Subsets | https://proofwiki.org/wiki/Left_Operation_is_Closed_for_All_Subsets | https://proofwiki.org/wiki/Left_Operation_is_Closed_for_All_Subsets | [
"Left Operation"
] | [
"Definition:Subsemigroup"
] | [
"Structure under Left Operation is Semigroup",
"Definition:Semigroup",
"Structure under Left Operation is Semigroup",
"Definition:Semigroup",
"Definition:Subsemigroup",
"Definition:Subset"
] |
proofwiki-18468 | Right Operation is Closed for All Subsets | :$\struct {T, \rightarrow}$ is a subsemigroup of $\struct {S, \rightarrow}$. | From Structure under Right Operation is Semigroup we have that $\struct {S, \rightarrow}$ is a semigroup, whatever the nature of $S$.
Let $T \in \powerset S$.
Then:
:From Structure under Right Operation is Semigroup, $\struct {T, \rightarrow}$ is a semigroup, and therefore a subsemigroup of $\struct {S, \rightarrow}$.
... | :$\struct {T, \rightarrow}$ is a [[Definition:Subsemigroup|subsemigroup]] of $\struct {S, \rightarrow}$. | From [[Structure under Right Operation is Semigroup]] we have that $\struct {S, \rightarrow}$ is a [[Definition:Semigroup|semigroup]], whatever the nature of $S$.
Let $T \in \powerset S$.
Then:
:From [[Structure under Right Operation is Semigroup]], $\struct {T, \rightarrow}$ is a [[Definition:Semigroup|semigroup]], ... | Right Operation is Closed for All Subsets | https://proofwiki.org/wiki/Right_Operation_is_Closed_for_All_Subsets | https://proofwiki.org/wiki/Right_Operation_is_Closed_for_All_Subsets | [
"Right Operation"
] | [
"Definition:Subsemigroup"
] | [
"Structure under Right Operation is Semigroup",
"Definition:Semigroup",
"Structure under Right Operation is Semigroup",
"Definition:Semigroup",
"Definition:Subsemigroup",
"Definition:Subset"
] |
proofwiki-18469 | Equivalence Class Equivalent Statements/1 iff 2 | Let $\RR$ be an equivalence relation on $S$.
Let $x, y \in S$.
{{TFAE}}
:$(1): \quad x$ and $y$ are in the same $\RR$-class
:$(2): \quad \eqclass x \RR = \eqclass y \RR$ | By Equivalence Class is Unique:
:$\eqclass x \RR$ is the unique $\RR$-class to which $x$ belongs
and:
:$\eqclass y \RR$ is the unique $\RR$-class to which $y$ belongs.
As these are unique for each, they must be the same set.
{{qed}} | Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $x, y \in S$.
{{TFAE}}
:$(1): \quad x$ and $y$ are in the same [[Definition:Equivalence Class|$\RR$-class]]
:$(2): \quad \eqclass x \RR = \eqclass y \RR$ | By [[Equivalence Class is Unique]]:
:$\eqclass x \RR$ is the [[Definition:Unique|unique]] [[Definition:Equivalence Class|$\RR$-class]] to which $x$ belongs
and:
:$\eqclass y \RR$ is the [[Definition:Unique|unique]] [[Definition:Equivalence Class|$\RR$-class]] to which $y$ belongs.
As these are [[Definition:Unique|uniq... | Equivalence Class Equivalent Statements/1 iff 2 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/1_iff_2 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/1_iff_2 | [
"Equivalence Class Equivalent Statements"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Class"
] | [
"Equivalence Class is Unique",
"Definition:Unique",
"Definition:Equivalence Class",
"Definition:Unique",
"Definition:Equivalence Class",
"Definition:Unique",
"Definition:Set"
] |
proofwiki-18470 | Equivalence Class Equivalent Statements/2 iff 3 | Let $\RR$ be an equivalence relation on $S$.
Let $x, y \in S$.
{{TFAE}}
:$\eqclass x \RR = \eqclass y \RR$
:$x \mathrel \RR y$ | By Equivalence Class holds Equivalent Elements:
{{:Equivalence Class holds Equivalent Elements}} | Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $x, y \in S$.
{{TFAE}}
:$\eqclass x \RR = \eqclass y \RR$
:$x \mathrel \RR y$ | By [[Equivalence Class holds Equivalent Elements]]:
{{:Equivalence Class holds Equivalent Elements}} | Equivalence Class Equivalent Statements/2 iff 3 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/2_iff_3 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/2_iff_3 | [
"Equivalence Class Equivalent Statements"
] | [
"Definition:Equivalence Relation"
] | [
"Equivalence Class holds Equivalent Elements"
] |
proofwiki-18471 | Equivalence Class Equivalent Statements/3 iff 4 | Let $\RR$ be an equivalence relation on $S$.
Let $x, y \in S$.
{{TFAE}}
:$x \mathrel \RR y$
:$x \in \eqclass y \RR$ | This follows directly by the definition of equivalence class.
{{qed}} | Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $x, y \in S$.
{{TFAE}}
:$x \mathrel \RR y$
:$x \in \eqclass y \RR$ | This follows directly by the definition of [[Definition:Equivalence Class|equivalence class]].
{{qed}} | Equivalence Class Equivalent Statements/3 iff 4 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/3_iff_4 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/3_iff_4 | [
"Equivalence Class Equivalent Statements"
] | [
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Class"
] |
proofwiki-18472 | Equivalence Class Equivalent Statements/3 iff 5 | Let $\RR$ be an equivalence relation on $S$.
Let $x, y \in S$.
{{TFAE}}
:$x \mathrel \RR y$
:$y \in \eqclass x \RR$ | This follows through dint of the symmetry of $\RR$ and the definition of Equivalence Class.
{{qed}} | Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $x, y \in S$.
{{TFAE}}
:$x \mathrel \RR y$
:$y \in \eqclass x \RR$ | This follows through dint of the [[Definition:Symmetric Relation|symmetry]] of $\RR$ and the definition of [[Definition:Equivalence Class|Equivalence Class]].
{{qed}} | Equivalence Class Equivalent Statements/3 iff 5 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/3_iff_5 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/3_iff_5 | [
"Equivalence Class Equivalent Statements"
] | [
"Definition:Equivalence Relation"
] | [
"Definition:Symmetric Relation",
"Definition:Equivalence Class"
] |
proofwiki-18473 | Equivalence Class Equivalent Statements/3 iff 6 | Let $\RR$ be an equivalence relation on $S$.
Let $x, y \in S$.
{{TFAE}}
:$x \mathrel \RR y$
:$\eqclass x \RR \cap \eqclass y \RR \ne \O$ | Follows directly from Equivalence Classes are Disjoint.
{{qed}} | Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $x, y \in S$.
{{TFAE}}
:$x \mathrel \RR y$
:$\eqclass x \RR \cap \eqclass y \RR \ne \O$ | Follows directly from [[Equivalence Classes are Disjoint]].
{{qed}} | Equivalence Class Equivalent Statements/3 iff 6 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/3_iff_6 | https://proofwiki.org/wiki/Equivalence_Class_Equivalent_Statements/3_iff_6 | [
"Equivalence Class Equivalent Statements"
] | [
"Definition:Equivalence Relation"
] | [
"Equivalence Classes are Disjoint"
] |
proofwiki-18474 | Excess Kurtosis of Logistic Distribution | Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
Then the excess kurtosis $\gamma_2$ of $X$ is equal to $\dfrac 6 5$. | From Kurtosis in terms of Non-Central Moments, we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Logistic Distribution we have:
:$\mu = \mu$
By V... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] which satisfies the [[Definition:Logistic Distribution|logistic distribution]]:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is equal to $\dfrac 6 5$. | From [[Kurtosis in terms of Non-Central Moments]], we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
... | Excess Kurtosis of Logistic Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Logistic_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Logistic_Distribution | [
"Kurtosis",
"Logistic Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Excess Kurtosis"
] | [
"Kurtosis in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Logistic Distribution",
"Variance of Logistic Distribution",
"Moment in terms of Moment Generating Function",
"Definition:Moment Generating Function",
"Derivatives of Moment Generatin... |
proofwiki-18475 | Characterization of P-adic Valuation on Integers | Let $p \in \N$ be a prime number.
Let $\nu_p^\Z: \Z \to \N \cup \set {+\infty}$ be the $p$-adic valuation on $\Z$.
Let $n \in \Z_{\ne 0}$.
Then $\map {\nu_p^\Z} n$ is the unique $r \in \N$ such that:
:$\exists k \in \Z: n = p^r k : p \nmid k$ | === Uniqueness of $r$ ===
Let $r, r'$ be such that:
:$\exists k \in \Z: n = p^r k : p \nmid k$
and:
:$\exists k' \in \Z: n = p^{r'} k' : p \nmid k'$
{{WLOG}}, suppose $r \ge r'$.
By subtracting the above equations:
:$0 = p^r k - p^{r'} k' = p^{r'} \paren {p^{r - r'} k - k'} = 0$
Therefore:
:$p ^{r - r'} k = k'$
Thus be... | Let $p \in \N$ be a [[Definition:Prime Number|prime number]].
Let $\nu_p^\Z: \Z \to \N \cup \set {+\infty}$ be the [[Definition:Restricted P-adic Valuation|$p$-adic valuation]] on $\Z$.
Let $n \in \Z_{\ne 0}$.
Then $\map {\nu_p^\Z} n$ is the unique $r \in \N$ such that:
:$\exists k \in \Z: n = p^r k : p \nmid k$ | === Uniqueness of $r$ ===
Let $r, r'$ be such that:
:$\exists k \in \Z: n = p^r k : p \nmid k$
and:
:$\exists k' \in \Z: n = p^{r'} k' : p \nmid k'$
{{WLOG}}, suppose $r \ge r'$.
By subtracting the above [[Definition:Equation|equations]]:
:$0 = p^r k - p^{r'} k' = p^{r'} \paren {p^{r - r'} k - k'} = 0$
Therefore:
:... | Characterization of P-adic Valuation on Integers | https://proofwiki.org/wiki/Characterization_of_P-adic_Valuation_on_Integers | https://proofwiki.org/wiki/Characterization_of_P-adic_Valuation_on_Integers | [
"P-adic Valuations"
] | [
"Definition:Prime Number",
"Definition:P-adic Valuation/Integers"
] | [
"Definition:Equation",
"Definition:P-adic Valuation/Integers",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Supremum of Set",
"Category:P-adic Valuations"
] |
proofwiki-18476 | Commutator of x and Distributional Derivative acting on Distribution | Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Let:
:$\sqbrk {x, \dfrac \d {\d x} } T := x \dfrac {\d T} {\d x} - \dfrac {\d \paren {x T} } {\d x}$
where derivatives are to be understood in the distributional sense.
Then:
:$\sqbrk {x, \dfrac \d {\d x} } T = - T$ | Let $\phi \in \map \DD \R$ be a test function.
{{begin-eqn}}
{{eqn | l = \map {\paren{x \dfrac {\d T} {\d x} - \dfrac {\d \paren {x T} } {\d x} } } \phi
| r = \map {\paren {x \dfrac {\d T} {\d x} } } \phi - \map {\paren {\dfrac {\d \paren {x T} } {\d x} } } \phi
| c = Linearity of Schwartz distribution
}}
{... | Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Let:
:$\sqbrk {x, \dfrac \d {\d x} } T := x \dfrac {\d T} {\d x} - \dfrac {\d \paren {x T} } {\d x}$
where [[Definition:Derivative|derivatives]] are to be understood in the [[Definition:Distributional Derivative|distributiona... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
{{begin-eqn}}
{{eqn | l = \map {\paren{x \dfrac {\d T} {\d x} - \dfrac {\d \paren {x T} } {\d x} } } \phi
| r = \map {\paren {x \dfrac {\d T} {\d x} } } \phi - \map {\paren {\dfrac {\d \paren {x T} } {\d x} } } \phi
| c = [[Definit... | Commutator of x and Distributional Derivative acting on Distribution | https://proofwiki.org/wiki/Commutator_of_x_and_Distributional_Derivative_acting_on_Distribution | https://proofwiki.org/wiki/Commutator_of_x_and_Distributional_Derivative_acting_on_Distribution | [
"Distributional Derivatives"
] | [
"Definition:Schwartz Distribution",
"Definition:Derivative",
"Definition:Distributional Derivative"
] | [
"Definition:Test Function",
"Definition:Linear Transformation",
"Definition:Schwartz Distribution",
"Definition:Linear Transformation",
"Definition:Schwartz Distribution",
"Definition:Linear Transformation",
"Definition:Schwartz Distribution"
] |
proofwiki-18477 | Commutative and Associative Product on Space of Distributions does not Exist | Let $\map {\DD'} \R$ be the distribution space.
Let $\alpha \in \map {C^\infty} \R$ be a smooth function.
Let $\circ$ be a product operation on $\map {\DD'} \R$.
Suppose:
:$\forall T \in \map {\DD'} \R : \forall \alpha \in \map {C^\infty} \R : \alpha \circ T := \alpha \cdot T$
where $\cdot$ stands for multiplication of... | {{AimForCont}} there is such a product.
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
We also have from Principal Value of One over x is Distribution that $\ds \PV \frac 1 x$ is a Schwartz distribution.
Let $\phi, \psi \in \map \DD \R$ be test functions.
Then:
{{begin-eqn}}
{{eqn | l = x \cdot \map {... | Let $\map {\DD'} \R$ be the [[Definition:Distribution Space|distribution space]].
Let $\alpha \in \map {C^\infty} \R$ be a [[Definition:Smooth Function|smooth function]].
Let $\circ$ be a [[Definition:General Algebraic Product|product operation]] on $\map {\DD'} \R$.
Suppose:
:$\forall T \in \map {\DD'} \R : \foral... | {{AimForCont}} there is such a [[Definition:General Algebraic Product|product]].
Let $\delta \in \map {\DD'} \R$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
We also have from [[Principal Value of One over x is Distribution]] that $\ds \PV \frac 1 x$ is a [[Definition:Schwartz Distribution... | Commutative and Associative Product on Space of Distributions does not Exist | https://proofwiki.org/wiki/Commutative_and_Associative_Product_on_Space_of_Distributions_does_not_Exist | https://proofwiki.org/wiki/Commutative_and_Associative_Product_on_Space_of_Distributions_does_not_Exist | [
"Schwartz Distributions",
"Commutativity",
"Associativity"
] | [
"Definition:Distribution Space",
"Definition:Smooth Real Function",
"Definition:Operation/Binary Operation/Product",
"Definition:Multiplication of Schwartz Distribution by Smooth Function",
"Definition:Commutative/Operation",
"Definition:Associative Operation",
"Definition:Existential Statement"
] | [
"Definition:Operation/Binary Operation/Product",
"Definition:Dirac Delta Distribution",
"Principal Value of One over x is Distribution",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Differentiable Function as Distribution",
"Definition:Product",
"Definition:Schwartz Distribution",
... |
proofwiki-18478 | Exp (-x^2) is Schwartz Test Function | {{explain|What is the domain and codomain for the exponential function defined? Is it over the Real Numbers?}}
$\map \exp {-x^2}$ is a Schwartz test function. | Let:
{{explain|What is p_n and what is p_n'? Please clarify}}
:$\dfrac {\d^n}{\d x^n} \map \exp {-x^2} = \map {p_n} x \map \exp {-x^2}$
We have that $\map {p_0} x = 1$ and $\map {p_1} x = -2 x$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\d^{n + 1} } {\d x^{n + 1} } \map \exp {-x^2}
| r = \dfrac \d {\d x} \dfrac {\d... | {{explain|What is the domain and codomain for the exponential function defined? Is it over the Real Numbers?}}
$\map \exp {-x^2}$ is a [[Definition:Schwartz Test Function|Schwartz test function]]. | Let:
{{explain|What is p_n and what is p_n'? Please clarify}}
:$\dfrac {\d^n}{\d x^n} \map \exp {-x^2} = \map {p_n} x \map \exp {-x^2}$
We have that $\map {p_0} x = 1$ and $\map {p_1} x = -2 x$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\d^{n + 1} } {\d x^{n + 1} } \map \exp {-x^2}
| r = \dfrac \d {\d x} \dfrac ... | Exp (-x^2) is Schwartz Test Function | https://proofwiki.org/wiki/Exp_(-x^2)_is_Schwartz_Test_Function | https://proofwiki.org/wiki/Exp_(-x^2)_is_Schwartz_Test_Function | [
"Examples of Schwartz Test Functions"
] | [
"Definition:Schwartz Test Function"
] | [
"Definition:Derivative/Real Function",
"Definition:Multiplication/Real Numbers",
"Definition:Subtraction/Real Numbers",
"Definition:Polynomial",
"Definition:Polynomial",
"Definition:Maclaurin Series",
"Absolute Value Function is Completely Multiplicative",
"Absolute Value Function is Completely Multip... |
proofwiki-18479 | Discrete Space iff Diagonal Set on Product is Open | Let $\struct {X, \tau}$ be a topological space.
Endow $X \times X$ with the product topology.
Let:
:$\Delta = \set {\tuple {x, x} : x \in X}$
Then:
:$\tau = \powerset X$
{{iff}}:
:$\Delta$ is open in $X$.
That is:
:the topology on $X$ is discrete {{iff}} $\Delta$ is open in $X$. | === Sufficient Condition ===
Suppose that $\tau = \powerset X$.
Then for each $x \in X$, the set:
:$\set {\tuple {x, x} }$
is open.
Note that we can write:
:$\ds \Delta = \bigcup_{x \in X} \set {\tuple {x, x} }$
This is the union of open sets, so from the definition of a topology we have that:
:$\Delta$ is open.
{{q... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Endow $X \times X$ with the [[Definition:Product Topology/Two Factor Spaces|product topology]].
Let:
:$\Delta = \set {\tuple {x, x} : x \in X}$
Then:
:$\tau = \powerset X$
{{iff}}:
:$\Delta$ is [[Definition:Open Set|open]] in $X$.
... | === Sufficient Condition ===
Suppose that $\tau = \powerset X$.
Then for each $x \in X$, the [[Definition:Set|set]]:
:$\set {\tuple {x, x} }$
is [[Definition:Open Set|open]].
Note that we can write:
:$\ds \Delta = \bigcup_{x \in X} \set {\tuple {x, x} }$
This is the union of [[Definition:Open Set|open sets]], ... | Discrete Space iff Diagonal Set on Product is Open | https://proofwiki.org/wiki/Discrete_Space_iff_Diagonal_Set_on_Product_is_Open | https://proofwiki.org/wiki/Discrete_Space_iff_Diagonal_Set_on_Product_is_Open | [
"Discrete Topologies",
"Product Topology"
] | [
"Definition:Topological Space",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Open Set",
"Definition:Topology",
"Definition:Discrete Topology",
"Definition:Open Set"
] | [
"Definition:Set",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Topology",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Set",
"Definition:Open Set"
] |
proofwiki-18480 | Order of Second Chebyshev Function | :$\ds \map \psi x = \sum_{p \mathop \le x} \ln p + \map \OO {\sqrt x \paren {\ln x}^2}$ | From the definition of the Second Chebyshev Function, we have:
:$\ds \map \psi x = \sum_{k \mathop = 1}^\infty \paren {\sum_{p^k \mathop \le x} \ln p}$
where the inner sum runs over the primes $p$ with $p^k \le x$.
That is, the primes $p$ with $p \le x^{1/k}$, so we can write:
:$\ds \sum_{p^k \mathop \le x} \ln p = \... | :$\ds \map \psi x = \sum_{p \mathop \le x} \ln p + \map \OO {\sqrt x \paren {\ln x}^2}$ | From the definition of the [[Definition:Second Chebyshev Function|Second Chebyshev Function]], we have:
:$\ds \map \psi x = \sum_{k \mathop = 1}^\infty \paren {\sum_{p^k \mathop \le x} \ln p}$
where the inner sum runs over the [[Definition:Prime Number|primes]] $p$ with $p^k \le x$.
That is, the [[Definition:Prime ... | Order of Second Chebyshev Function | https://proofwiki.org/wiki/Order_of_Second_Chebyshev_Function | https://proofwiki.org/wiki/Order_of_Second_Chebyshev_Function | [
"Second Chebyshev Function",
"Analytic Number Theory"
] | [] | [
"Definition:Second Chebyshev Function",
"Definition:Prime Number",
"Definition:Prime Number",
"Logarithm is Strictly Increasing",
"Logarithm of Power",
"Definition:Big-O Notation",
"Category:Second Chebyshev Function",
"Category:Analytic Number Theory"
] |
proofwiki-18481 | Arithmetic Average of Second Chebyshev Function | :$\ds \sum_{n \mathop \le x} \map \psi {x/n} = x \ln x - x + \map \OO {\map \ln {x + 1} }$ | We have, by the definition of the second Chebyshev function:
:$\ds \sum_{n \mathop \le x} \map \psi {x/n} = \sum_{n \mathop \le x} \sum_{m \mathop \le x/n} \map \Lambda m$
where $\Lambda$ is the Von Mangoldt function.
Consider the sum:
:$\ds \sum_{n \mathop \le x} \sum_{m \mathop \le x/n} \map \Lambda m$
The sum runs... | :$\ds \sum_{n \mathop \le x} \map \psi {x/n} = x \ln x - x + \map \OO {\map \ln {x + 1} }$ | We have, by the definition of the [[Definition:Second Chebyshev Function|second Chebyshev function]]:
:$\ds \sum_{n \mathop \le x} \map \psi {x/n} = \sum_{n \mathop \le x} \sum_{m \mathop \le x/n} \map \Lambda m$
where $\Lambda$ is the [[Definition:Von Mangoldt Function|Von Mangoldt function]].
Consider the sum:
... | Arithmetic Average of Second Chebyshev Function | https://proofwiki.org/wiki/Arithmetic_Average_of_Second_Chebyshev_Function | https://proofwiki.org/wiki/Arithmetic_Average_of_Second_Chebyshev_Function | [
"Second Chebyshev Function",
"Arithmetic Average of Second Chebyshev Function"
] | [] | [
"Definition:Second Chebyshev Function",
"Definition:Von Mangoldt Function",
"Definition:Natural Numbers",
"Definition:Divisor",
"Definition:Natural Numbers",
"Sum Over Divisors of von Mangoldt is Logarithm",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Logarithm is Strictly Incr... |
proofwiki-18482 | Sum of Big-O Estimates/Real Analysis | Let $c$ be a real number.
Let $f, g : \hointr c \infty \to \R$ be real functions.
Let $\OO$ denote big-$\OO$ notation.
Let $R_1 : \hointr c \infty \to \R$ be a real function such that $f = \map \OO {R_1}$.
Let $R_2 : \hointr c \infty \to \R$ be a real function such that $g = \map \OO {R_2}$.
Then:
:$f + g = \map \OO {... | Since:
:$f = \map \OO {R_1}$
there exists $x_1 \in \hointr c \infty$ and a real number $C_1$ such that:
:$\size {\map f x} \le C_1 \size {\map {R_1} x}$
for $x \ge x_1$.
Similarly, since:
:$g = \map \OO {R_2}$
there exists $x_2 \in \hointr c \infty$ and a real number $C_2$ such that:
:$\size {\map g x} \le C_2 \size... | Let $c$ be a [[Definition:Real Number|real number]].
Let $f, g : \hointr c \infty \to \R$ be [[Definition:Real Function|real functions]].
Let $\OO$ denote [[Definition:Big-O Notation|big-$\OO$ notation]].
Let $R_1 : \hointr c \infty \to \R$ be a [[Definition:Real Function|real function]] such that $f = \map \OO {R_1... | Since:
:$f = \map \OO {R_1}$
there exists $x_1 \in \hointr c \infty$ and a [[Definition:Real Number|real number]] $C_1$ such that:
:$\size {\map f x} \le C_1 \size {\map {R_1} x}$
for $x \ge x_1$.
Similarly, since:
:$g = \map \OO {R_2}$
there exists $x_2 \in \hointr c \infty$ and a [[Definition:Real Number|rea... | Sum of Big-O Estimates/Real Analysis | https://proofwiki.org/wiki/Sum_of_Big-O_Estimates/Real_Analysis | https://proofwiki.org/wiki/Sum_of_Big-O_Estimates/Real_Analysis | [
"Big-O Notation"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Big-O Notation",
"Definition:Real Function",
"Definition:Real Function"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Triangle Inequality",
"Definition:Big-O Notation",
"Category:Big-O Notation"
] |
proofwiki-18483 | Product of Big-O Estimates/Real Analysis | Let $c$ be a real number.
Let $f, g : \hointr c \infty \to \R$ be real functions.
Let $\OO$ denote big-$\OO$ notation.
Let $R_1 : \hointr c \infty \to \R$ be a real function such that $f = \map \OO {R_1}$.
Let $R_2 : \hointr c \infty \to \R$ be a real function such that $g = \map \OO {R_2}$.
Then:
:$f g = \map \OO {R_... | Since:
:$f = \map \OO {R_1}$
there exists $x_1 \in \hointr c \infty$ and a real number $C_1$ such that:
:$\size {\map f x} \le C_1 \size {\map {R_1} x}$
for $x \ge x_1$.
Similarly, since:
:$g = \map \OO {R_2}$
there exists $x_2 \in \hointr c \infty$ and a real number $C_2$ such that:
:$\size {\map g x} \le C_2 \size... | Let $c$ be a [[Definition:Real Number|real number]].
Let $f, g : \hointr c \infty \to \R$ be [[Definition:Real Function|real functions]].
Let $\OO$ denote [[Definition:Big-O Notation|big-$\OO$ notation]].
Let $R_1 : \hointr c \infty \to \R$ be a [[Definition:Real Function|real function]] such that $f = \map \OO {R_1... | Since:
:$f = \map \OO {R_1}$
there exists $x_1 \in \hointr c \infty$ and a [[Definition:Real Number|real number]] $C_1$ such that:
:$\size {\map f x} \le C_1 \size {\map {R_1} x}$
for $x \ge x_1$.
Similarly, since:
:$g = \map \OO {R_2}$
there exists $x_2 \in \hointr c \infty$ and a [[Definition:Real Number|rea... | Product of Big-O Estimates/Real Analysis | https://proofwiki.org/wiki/Product_of_Big-O_Estimates/Real_Analysis | https://proofwiki.org/wiki/Product_of_Big-O_Estimates/Real_Analysis | [
"Big-O Notation"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Big-O Notation",
"Definition:Real Function",
"Definition:Real Function"
] | [
"Definition:Real Number",
"Definition:Real Number",
"Definition:Big-O Notation",
"Category:Big-O Notation"
] |
proofwiki-18484 | Bounds for Prime-Counting Function in terms of Second Chebyshev Function | There exists a real function $R : \hointr 2 \infty \to \R$ such that:
:$\ds \frac {\map \psi x} {\ln x} + \map R x \le \map \pi x \le \frac {2 \map \psi x} {\ln x} + \sqrt x$
for all real numbers $x \ge 2$, where:
:$\pi$ is the prime counting function
:$\psi$ is the second Chebyshev function
:$R = \map \OO {\sqrt x \ln... | We have, from the definition of the prime counting function:
:$\ds \map \pi x = \sum_{p \le x} 1$
We can write:
:$\ds \sum_{p \le x} 1 = \sum_{p \le \sqrt x} 1 + \sum_{\sqrt x < p \le x} 1$
We have that:
{{begin-eqn}}
{{eqn | l = \sum_{p \le \sqrt x} 1
| o = \le
| r = \sum_{n \le \sqrt x} 1
}}
{{eqn | r = \floor {... | There exists a [[Definition:Real Function|real function]] $R : \hointr 2 \infty \to \R$ such that:
:$\ds \frac {\map \psi x} {\ln x} + \map R x \le \map \pi x \le \frac {2 \map \psi x} {\ln x} + \sqrt x$
for all [[Definition:Real Number|real numbers]] $x \ge 2$, where:
:$\pi$ is the [[Definition:Prime-Counting Functi... | We have, from the definition of the [[Definition:Prime-Counting Function|prime counting function]]:
:$\ds \map \pi x = \sum_{p \le x} 1$
We can write:
:$\ds \sum_{p \le x} 1 = \sum_{p \le \sqrt x} 1 + \sum_{\sqrt x < p \le x} 1$
We have that:
{{begin-eqn}}
{{eqn | l = \sum_{p \le \sqrt x} 1
| o = \le
| r = \s... | Bounds for Prime-Counting Function in terms of Second Chebyshev Function | https://proofwiki.org/wiki/Bounds_for_Prime-Counting_Function_in_terms_of_Second_Chebyshev_Function | https://proofwiki.org/wiki/Bounds_for_Prime-Counting_Function_in_terms_of_Second_Chebyshev_Function | [
"Second Chebyshev Function",
"Prime-Counting Function",
"Inequalities"
] | [
"Definition:Real Function",
"Definition:Real Number",
"Definition:Prime-Counting Function",
"Definition:Second Chebyshev Function",
"Definition:Big-O Notation"
] | [
"Definition:Prime-Counting Function",
"Logarithm is Strictly Increasing",
"Logarithm of Power",
"Logarithm is Strictly Increasing",
"Order of Second Chebyshev Function",
"Definition:Real Function",
"Logarithm is Strictly Increasing",
"Product of Big-O Estimates",
"Category:Second Chebyshev Function"... |
proofwiki-18485 | Second Chebyshev Function is Theta of x | We have:
:$\map \psi x = \map \Theta x$
where:
:$\Theta$ is $\Theta$ notation
:$\psi$ is the second Chebyshev function. | We show that:
:$\map \psi x = \map \OO x$
and:
:$x = \map \OO {\map \psi x}$
where $\OO$ denotes big-$\OO$ notation.
Note that:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop \le x} \map \psi {\frac x n} - 2 \sum_{n \mathop \le x/2} \map \psi {\frac {\frac x 2} n}
| r = x \ln x - x - 2 \paren {\frac x 2 \ln \frac x ... | We have:
:$\map \psi x = \map \Theta x$
where:
:$\Theta$ is [[Definition:Theta Notation|$\Theta$ notation]]
:$\psi$ is the [[Definition:Second Chebyshev Function|second Chebyshev function]]. | We show that:
:$\map \psi x = \map \OO x$
and:
:$x = \map \OO {\map \psi x}$
where $\OO$ denotes [[Definition:Big-O Notation|big-$\OO$ notation]].
Note that:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop \le x} \map \psi {\frac x n} - 2 \sum_{n \mathop \le x/2} \map \psi {\frac {\frac x 2} n}
| r = x \ln x -... | Second Chebyshev Function is Theta of x | https://proofwiki.org/wiki/Second_Chebyshev_Function_is_Theta_of_x | https://proofwiki.org/wiki/Second_Chebyshev_Function_is_Theta_of_x | [
"Second Chebyshev Function",
"Theta Notation"
] | [
"Definition:Theta Notation",
"Definition:Second Chebyshev Function"
] | [
"Definition:Big-O Notation",
"Order of Second Chebyshev Function",
"Sum of Big-O Estimates",
"Difference of Logarithms",
"Second Chebyshev Function is Increasing",
"Definition:Odd Integer",
"Second Chebyshev Function is Increasing",
"Definition:Even Integer",
"Second Chebyshev Function is Increasing... |
proofwiki-18486 | Order of Natural Logarithm Function | Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
Then:
:$\ln x = \map \OO {x^\epsilon}$ as $x \to \infty$
and:
:$\ln x = \map \OO {x^{-\epsilon} }$ as $x \to 0^+$
where $\OO$ is big-O notation. | We first show that:
:$\ln x = \map \OO {x^\epsilon}$ as $x \to \infty$
We show that for $x \ge 1$, we have:
:$\ds 0 \le \ln x \le \frac 1 \epsilon x^\epsilon$
We first show that for $t \ge 0$, we have:
:$\ds t \le \frac 1 \epsilon e^{\epsilon t}$
The claim then follows taking $t = \ln x$.
Define a real function $f :... | Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Then:
:$\ln x = \map \OO {x^\epsilon}$ as $x \to \infty$
and:
:$\ln x = \map \OO {x^{-\epsilon} }$ as $x \to 0^+$
where $\OO$ is [[Definition:Big-O Estimate for Real Function|big-O notation]]. | We first show that:
:$\ln x = \map \OO {x^\epsilon}$ as $x \to \infty$
We show that for $x \ge 1$, we have:
:$\ds 0 \le \ln x \le \frac 1 \epsilon x^\epsilon$
We first show that for $t \ge 0$, we have:
:$\ds t \le \frac 1 \epsilon e^{\epsilon t}$
The claim then follows taking $t = \ln x$.
Define a [[Definitio... | Order of Natural Logarithm Function | https://proofwiki.org/wiki/Order_of_Natural_Logarithm_Function | https://proofwiki.org/wiki/Order_of_Natural_Logarithm_Function | [
"Natural Logarithms"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Big-O Notation/Real"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping",
"Definition:Derivative",
"Derivative of Exponential Function",
"Real Function with Positive Derivative is Increasing",
"Definition:Increasing/Real Function",
"Definition:Big-O Notation",
"Logarithm of Power",
"Definition:Absolute Value... |
proofwiki-18487 | Extension Theorem for Distributive Operations/Existence and Uniqueness | :There exists a unique operation $\circ'$ on $T$ which distributes over $*$ in $T$ and induces on $R$ the operation $\circ$. | We have {{hypothesis}} that all the elements of $\struct {R, *}$ are cancellable.
Thus Inverse Completion of Commutative Semigroup is Abelian Group can be applied.
So $\struct {T, *}$ is an abelian group. | :There exists a [[Definition:Unique|unique]] [[Definition:Binary Operation|operation]] $\circ'$ on $T$ which [[Definition:Distributive Operation|distributes]] over $*$ in $T$ and induces on $R$ the [[Definition:Binary Operation|operation]] $\circ$. | We have {{hypothesis}} that all the [[Definition:Element|elements]] of $\struct {R, *}$ are [[Definition:Cancellable Element|cancellable]].
Thus [[Inverse Completion of Commutative Semigroup is Abelian Group]] can be applied.
So $\struct {T, *}$ is an [[Definition:Abelian Group|abelian group]]. | Extension Theorem for Distributive Operations/Existence and Uniqueness | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Existence_and_Uniqueness | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Existence_and_Uniqueness | [
"Extension Theorem for Distributive Operations"
] | [
"Definition:Unique",
"Definition:Operation/Binary Operation",
"Definition:Distributive Operation",
"Definition:Operation/Binary Operation"
] | [
"Definition:Element",
"Definition:Cancellable Element",
"Inverse Completion of Commutative Semigroup is Abelian Group",
"Definition:Abelian Group",
"Definition:Abelian Group"
] |
proofwiki-18488 | Extension Theorem for Distributive Operations/Associativity | :If $\circ$ is associative, then so is $\circ'$. | We have that $\circ'$ exists and is unique by Extension Theorem for Distributive Operations: Existence and Uniqueness.
Suppose $\circ$ is associative.
As $\circ'$ distributes over $*$, for all $n, p \in R$, the mappings:
{{begin-eqn}}
{{eqn | l = x \mapsto \paren {x \circ' n} \circ' p
| o = ,
| r = x \in T
... | :If $\circ$ is [[Definition:Associative Operation|associative]], then so is $\circ'$. | We have that $\circ'$ exists and is [[Definition:Unique|unique]] by [[Extension Theorem for Distributive Operations/Existence and Uniqueness|Extension Theorem for Distributive Operations: Existence and Uniqueness]].
Suppose $\circ$ is [[Definition:Associative Operation|associative]].
As $\circ'$ [[Definition:Distrib... | Extension Theorem for Distributive Operations/Associativity | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Associativity | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Associativity | [
"Extension Theorem for Distributive Operations"
] | [
"Definition:Associative Operation"
] | [
"Definition:Unique",
"Extension Theorem for Distributive Operations/Existence and Uniqueness",
"Definition:Associative Operation",
"Definition:Distributive Operation",
"Definition:Mapping",
"Definition:Endomorphism",
"Definition:Associative Operation",
"Definition:Mapping",
"Definition:Mapping",
"... |
proofwiki-18489 | Extension Theorem for Distributive Operations/Commutativity | :If $\circ$ is commutative, then so is $\circ'$ | We have that $\circ'$ exists and is unique by Extension Theorem for Distributive Operations: Existence and Uniqueness.
Suppose $\circ$ is commutative.
As $\circ'$ distributes over $*$, for all $n \in R$, the mappings:
{{begin-eqn}}
{{eqn | l = x \mapsto x \circ' n
| o = ,
| r = x \in T
| c =
}}
{{eqn... | :If $\circ$ is [[Definition:Commutative Operation|commutative]], then so is $\circ'$ | We have that $\circ'$ exists and is [[Definition:Unique|unique]] by [[Extension Theorem for Distributive Operations/Existence and Uniqueness|Extension Theorem for Distributive Operations: Existence and Uniqueness]].
Suppose $\circ$ is [[Definition:Commutative Operation|commutative]].
As $\circ'$ distributes over $*$... | Extension Theorem for Distributive Operations/Commutativity | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Commutativity | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Commutativity | [
"Extension Theorem for Distributive Operations"
] | [
"Definition:Commutative/Operation"
] | [
"Definition:Unique",
"Extension Theorem for Distributive Operations/Existence and Uniqueness",
"Definition:Commutative/Operation",
"Definition:Mapping",
"Definition:Endomorphism",
"Definition:Commutative/Operation",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Endomorphism",
"Definition... |
proofwiki-18490 | Extension Theorem for Distributive Operations/Identity | :If $e$ is an identity for $\circ$, then $e$ is also an identity for $\circ'$. | We have that $\circ'$ exists and is unique by Extension Theorem for Distributive Operations: Existence and Uniqueness.
Let $e$ be the identity element of $\struct {R, \circ}$.
Then the restrictions to $R$ of the endomorphisms $\lambda_e: x \mapsto e \circ' x$ and $\rho_e: x \mapsto x \circ' e$ of $\struct {T, *}$ are m... | :If $e$ is an [[Definition:Identity Element|identity]] for $\circ$, then $e$ is also an [[Definition:Identity Element|identity]] for $\circ'$. | We have that $\circ'$ exists and is [[Definition:Unique|unique]] by [[Extension Theorem for Distributive Operations/Existence and Uniqueness|Extension Theorem for Distributive Operations: Existence and Uniqueness]].
Let $e$ be the [[Definition:Identity Element|identity element]] of $\struct {R, \circ}$.
Then the [[D... | Extension Theorem for Distributive Operations/Identity | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Identity | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Identity | [
"Extension Theorem for Distributive Operations"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Unique",
"Extension Theorem for Distributive Operations/Existence and Uniqueness",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Restriction/Mapping",
"Definition:Endomorphism",
"Definition:Monomorphism (Abstract Algebra)",
"Definition:Monomorphism (Abstract Algebr... |
proofwiki-18491 | Extension Theorem for Distributive Operations/Cancellability | :Every element of $R$ cancellable for $\circ$ is also cancellable for $\circ'$. | Let $a$ be an element of $R$ cancellable for $\circ$.
Then the restrictions to $R$ of the endomorphisms:
:$\lambda_a: x \mapsto a \circ' x$
:$\rho_a: x \mapsto x \circ' a$
of $\struct {T, *}$ are monomorphisms.
But then $\lambda_a$ and $\rho_a$ are monomorphisms by the Extension Theorem for Homomorphisms.
Hence $a$ is ... | :Every [[Definition:Element|element]] of $R$ [[Definition:Cancellable Element|cancellable]] for $\circ$ is also [[Definition:Cancellable Element|cancellable]] for $\circ'$. | Let $a$ be an [[Definition:Element|element]] of $R$ [[Definition:Cancellable Element|cancellable]] for $\circ$.
Then the [[Definition:Restriction of Mapping|restrictions]] to $R$ of the [[Definition:Endomorphism|endomorphisms]]:
:$\lambda_a: x \mapsto a \circ' x$
:$\rho_a: x \mapsto x \circ' a$
of $\struct {T, *}$ ar... | Extension Theorem for Distributive Operations/Cancellability | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Cancellability | https://proofwiki.org/wiki/Extension_Theorem_for_Distributive_Operations/Cancellability | [
"Extension Theorem for Distributive Operations"
] | [
"Definition:Element",
"Definition:Cancellable Element",
"Definition:Cancellable Element"
] | [
"Definition:Element",
"Definition:Cancellable Element",
"Definition:Restriction/Mapping",
"Definition:Endomorphism",
"Definition:Monomorphism (Abstract Algebra)",
"Definition:Monomorphism (Abstract Algebra)",
"Extension Theorem for Homomorphisms",
"Definition:Cancellable Element"
] |
proofwiki-18492 | Modulus of Sine of x Less Than or Equal To Absolute Value of x | :$\size {\sin x} \le \size x$ | Clearly the inequality holds if $x = 0$.
Take $x \ne 0$.
From the Mean Value Theorem and Derivative of Sine Function, there exists $c \in \R$ such that:
:$\ds \frac {\sin x - \sin 0} {x - 0} = \cos c$
so:
:$\sin x = x \cos c$
Then we have:
{{begin-eqn}}
{{eqn | l = \size {\sin x}
| r = \size {x \cos c}
}}
{{eqn | o... | :$\size {\sin x} \le \size x$ | Clearly the inequality holds if $x = 0$.
Take $x \ne 0$.
From the [[Mean Value Theorem]] and [[Derivative of Sine Function]], there exists $c \in \R$ such that:
:$\ds \frac {\sin x - \sin 0} {x - 0} = \cos c$
so:
:$\sin x = x \cos c$
Then we have:
{{begin-eqn}}
{{eqn | l = \size {\sin x}
| r = \size {x \cos ... | Modulus of Sine of x Less Than or Equal To Absolute Value of x | https://proofwiki.org/wiki/Modulus_of_Sine_of_x_Less_Than_or_Equal_To_Absolute_Value_of_x | https://proofwiki.org/wiki/Modulus_of_Sine_of_x_Less_Than_or_Equal_To_Absolute_Value_of_x | [
"Inequalities",
"Sine Function",
"Absolute Value Function"
] | [] | [
"Mean Value Theorem",
"Derivative of Sine Function",
"Category:Inequalities",
"Category:Sine Function",
"Category:Absolute Value Function"
] |
proofwiki-18493 | Order Modulo n of Power of Integer/Corollary | Let $a$ be a primitive root of $n$.
Then:
:$a^k$ is also a primitive root of $n$
{{iff}}:
:$k \perp \map \phi n$
where $\phi$ is the Euler phi function.
Furthermore, if $n$ has a primitive root, it has exactly $\map \phi {\map \phi n}$ of them. | Let $a$ be a primitive root of $n$.
Then $R = \set {a, a^2, \ldots, a^{\map \phi n}}$ is a reduced residue system for $n$.
Hence all primitive roots are contained in $R$.
By Order Modulo n of Power of Integer, the multiplicative order $a^k$ modulo $n$ is $\dfrac {\map \phi n} {\gcd \set {\map \phi n, k} }$.
Hence $a^k$... | Let $a$ be a [[Definition:Primitive Root (Number Theory)|primitive root]] of $n$.
Then:
:$a^k$ is also a [[Definition:Primitive Root (Number Theory)|primitive root]] of $n$
{{iff}}:
:$k \perp \map \phi n$
where $\phi$ is the [[Definition:Euler Phi Function|Euler phi function]].
Furthermore, if $n$ has a [[Definitio... | Let $a$ be a [[Definition:Primitive Root (Number Theory)|primitive root]] of $n$.
Then $R = \set {a, a^2, \ldots, a^{\map \phi n}}$ is a [[Definition:Reduced Residue System|reduced residue system]] for $n$.
Hence all [[Definition:Primitive Root (Number Theory)|primitive roots]] are contained in $R$.
By [[Order Modul... | Order Modulo n of Power of Integer/Corollary | https://proofwiki.org/wiki/Order_Modulo_n_of_Power_of_Integer/Corollary | https://proofwiki.org/wiki/Order_Modulo_n_of_Power_of_Integer/Corollary | [
"Number Theory"
] | [
"Definition:Primitive Root (Number Theory)",
"Definition:Primitive Root (Number Theory)",
"Definition:Euler Phi Function",
"Definition:Primitive Root (Number Theory)"
] | [
"Definition:Primitive Root (Number Theory)",
"Definition:Reduced Residue System",
"Definition:Primitive Root (Number Theory)",
"Order Modulo n of Power of Integer",
"Definition:Multiplicative Order of Integer",
"Definition:Primitive Root (Number Theory)",
"Definition:Primitive Root (Number Theory)",
"... |
proofwiki-18494 | Second Chebyshev Function is Increasing | The second Chebyshev function $\psi$ is increasing. | Let $x \ge y$.
Then:
{{begin-eqn}}
{{eqn | l = \map \psi y
| r = \sum_{k \mathop \ge 1} \sum_{p^k \mathop \le y} \ln p
| c = {{Defof|Second Chebyshev Function}}
}}
{{eqn | r = \sum_{k \mathop \ge 1} \paren {\sum_{p^k \mathop \le x} \ln p + \sum_{x \mathop < p^k \mathop \le y} \ln p}
}}
{{eqn | r = \sum_{k \mathop \... | The [[Definition:Second Chebyshev Function|second Chebyshev function]] $\psi$ is [[Definition:Increasing Function|increasing]]. | Let $x \ge y$.
Then:
{{begin-eqn}}
{{eqn | l = \map \psi y
| r = \sum_{k \mathop \ge 1} \sum_{p^k \mathop \le y} \ln p
| c = {{Defof|Second Chebyshev Function}}
}}
{{eqn | r = \sum_{k \mathop \ge 1} \paren {\sum_{p^k \mathop \le x} \ln p + \sum_{x \mathop < p^k \mathop \le y} \ln p}
}}
{{eqn | r = \sum_{k \mathop... | Second Chebyshev Function is Increasing | https://proofwiki.org/wiki/Second_Chebyshev_Function_is_Increasing | https://proofwiki.org/wiki/Second_Chebyshev_Function_is_Increasing | [
"Second Chebyshev Function"
] | [
"Definition:Second Chebyshev Function",
"Definition:Increasing/Real Function"
] | [
"Logarithm is Strictly Increasing",
"Definition:Increasing/Real Function",
"Category:Second Chebyshev Function"
] |
proofwiki-18495 | Order of Sum of Von Mangoldt Function of n over n | :$\ds \sum_{n \le x} \frac {\map \Lambda n} n = \ln x + \map \OO 1$ | Consider the sum:
:$\ds \sum_{t \le x} \sum_{m \divides t} \map \Lambda m$
We are summing over the pairs of natural numbers $\tuple {t, m}$ such that $t \le x$ and $m$ divides $t$.
Since $m$ divides $t$ and $t \le x$, we must have $m \le x$.
So we can equivalently sum over the pairs of natural numbers $\tuple {m, t}... | :$\ds \sum_{n \le x} \frac {\map \Lambda n} n = \ln x + \map \OO 1$ | Consider the sum:
:$\ds \sum_{t \le x} \sum_{m \divides t} \map \Lambda m$
We are summing over the pairs of [[Definition:Natural Number|natural numbers]] $\tuple {t, m}$ such that $t \le x$ and $m$ [[Definition:Divisor in Natural Numbers|divides]] $t$.
Since $m$ divides $t$ and $t \le x$, we must have $m \le x$.
... | Order of Sum of Von Mangoldt Function of n over n | https://proofwiki.org/wiki/Order_of_Sum_of_Von_Mangoldt_Function_of_n_over_n | https://proofwiki.org/wiki/Order_of_Sum_of_Von_Mangoldt_Function_of_n_over_n | [
"Von Mangoldt Function"
] | [] | [
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Natural Numbers",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Natural Numbers",
"Quantity of Positive Integers Divisible by Particular Integer",
"Order of Floor Function",
"Definition:Real Function",
"Arithmetic Average of Sec... |
proofwiki-18496 | Order of Sum over Primes of Logarithm of p over p | :$\ds \sum_{p \mathop \le x} \frac {\ln p} p = \ln x + \map \OO 1$ | From the definition of the Von Mangoldt function, we have that:
:$\ds \map \Lambda m = \begin {cases}\ln p & m = p^k \text { for some prime } p \text { and } k \in \N \\ 0 & \text {otherwise} \end {cases}$
so:
{{begin-eqn}}
{{eqn | l = \sum_{m \mathop \le x} \frac {\map \Lambda m} m
| r = \sum_{k \mathop \ge 1} \sum_... | :$\ds \sum_{p \mathop \le x} \frac {\ln p} p = \ln x + \map \OO 1$ | From the definition of the [[Definition:Von Mangoldt Function|Von Mangoldt function]], we have that:
:$\ds \map \Lambda m = \begin {cases}\ln p & m = p^k \text { for some prime } p \text { and } k \in \N \\ 0 & \text {otherwise} \end {cases}$
so:
{{begin-eqn}}
{{eqn | l = \sum_{m \mathop \le x} \frac {\map \Lambda ... | Order of Sum over Primes of Logarithm of p over p | https://proofwiki.org/wiki/Order_of_Sum_over_Primes_of_Logarithm_of_p_over_p | https://proofwiki.org/wiki/Order_of_Sum_over_Primes_of_Logarithm_of_p_over_p | [
"Analytic Number Theory"
] | [] | [
"Definition:Von Mangoldt Function",
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Natural Numbers",
"Sum of Geometric Sequence",
"Order of Natural Logarithm Function",
"Convergence of P-Series",
"Definition:Convergent Series",
"Convergent Real Sequ... |
proofwiki-18497 | Limit to Infinity of Power of x by Exponential of -a x | Let $k$ and $a$ be positive real numbers.
Then:
:$\ds \lim_{x \mathop \to \infty} x^k e^{-a x} = 0$ | By Power Series Expansion for Exponential Function, we have:
:$\ds e^{a x} = \sum_{n \mathop = 0}^\infty \frac {\paren {a x}^n} {n!}$
Since for $x > 0$ each term in this sum is non-negative, we have:
:$\ds e^{a x} \ge \frac {\paren {a x}^{\floor k + 1} } {\paren {\floor k + 1}!}$
for each $k$.
So, for each $x > 0$ we... | Let $k$ and $a$ be [[Definition:Positive Real Number|positive real numbers]].
Then:
:$\ds \lim_{x \mathop \to \infty} x^k e^{-a x} = 0$ | By [[Power Series Expansion for Exponential Function]], we have:
:$\ds e^{a x} = \sum_{n \mathop = 0}^\infty \frac {\paren {a x}^n} {n!}$
Since for $x > 0$ each term in this sum is [[Definition:Non-Negative Real Number|non-negative]], we have:
:$\ds e^{a x} \ge \frac {\paren {a x}^{\floor k + 1} } {\paren {\floor ... | Limit to Infinity of Power of x by Exponential of -a x | https://proofwiki.org/wiki/Limit_to_Infinity_of_Power_of_x_by_Exponential_of_-a_x | https://proofwiki.org/wiki/Limit_to_Infinity_of_Power_of_x_by_Exponential_of_-a_x | [
"Exponential Function"
] | [
"Definition:Positive/Real Number"
] | [
"Power Series Expansion for Exponential Function",
"Definition:Positive/Real Number",
"Definition:Floor Function",
"Limit to Infinity of Power",
"Squeeze Theorem",
"Category:Exponential Function"
] |
proofwiki-18498 | Limit of Power of x by Absolute Value of Power of Logarithm of x | Let $\alpha$ and $\beta$ be positive real numbers.
Then:
:$\ds \lim_{x \mathop \to 0^+} x^\alpha \size {\ln x}^\beta = 0$ | From Order of Natural Logarithm Function, we have:
:$\ln x = \map \OO {x^{-\frac \alpha {2 \beta} } }$ as $x \to 0^+$
That is, by the definition of big-O notation there exists positive real numbers $x_0$ and $C$ such that:
:$0 \le \size {\ln x} \le C x^{-\frac \alpha {2 \beta} }$
for $0 < x \le x_0$.
So:
:$0 \le \siz... | Let $\alpha$ and $\beta$ be [[Definition:Positive Real Number|positive real numbers]].
Then:
:$\ds \lim_{x \mathop \to 0^+} x^\alpha \size {\ln x}^\beta = 0$ | From [[Order of Natural Logarithm Function]], we have:
:$\ln x = \map \OO {x^{-\frac \alpha {2 \beta} } }$ as $x \to 0^+$
That is, by the definition of [[Definition:Big-O Estimate for Real Function|big-O notation]] there exists [[Definition:Positive Real Number|positive real numbers]] $x_0$ and $C$ such that:
:$0 ... | Limit of Power of x by Absolute Value of Power of Logarithm of x | https://proofwiki.org/wiki/Limit_of_Power_of_x_by_Absolute_Value_of_Power_of_Logarithm_of_x | https://proofwiki.org/wiki/Limit_of_Power_of_x_by_Absolute_Value_of_Power_of_Logarithm_of_x | [
"Logarithms",
"Limits of Real Functions",
"Limit of Power of x by Absolute Value of Power of Logarithm of x"
] | [
"Definition:Positive/Real Number"
] | [
"Order of Natural Logarithm Function",
"Definition:Big-O Notation/Real",
"Definition:Positive/Real Number",
"Squeeze Theorem/Functions",
"Category:Logarithms",
"Category:Limits of Real Functions",
"Category:Limit of Power of x by Absolute Value of Power of Logarithm of x"
] |
proofwiki-18499 | X + y + z equals 1 implies xy + yz + zx less than Half | Let $x$, $y$ and $z$ be real numbers such that:
:$x + y + z = 1$
Then:
:$x y + y z + z x < \dfrac 1 2$ | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = \paren {x + y + z}^2
}}
{{eqn | r = \paren {\paren {x + y} + z}^2
}}
{{eqn | r = \paren {x + y}^2 + 2 z \paren {x + y} + z^2
| c = Square of Sum
}}
{{eqn | r = x^2 + 2 x y + y^2 + 2 x z + 2 y z + z^2
| c = Square of Sum
}}
{{eqn | r = 2 \paren {x y + y z + z x} + \paren {x... | Let $x$, $y$ and $z$ be [[Definition:Real Number|real numbers]] such that:
:$x + y + z = 1$
Then:
:$x y + y z + z x < \dfrac 1 2$ | We have:
{{begin-eqn}}
{{eqn | l = 1
| r = \paren {x + y + z}^2
}}
{{eqn | r = \paren {\paren {x + y} + z}^2
}}
{{eqn | r = \paren {x + y}^2 + 2 z \paren {x + y} + z^2
| c = [[Square of Sum]]
}}
{{eqn | r = x^2 + 2 x y + y^2 + 2 x z + 2 y z + z^2
| c = [[Square of Sum]]
}}
{{eqn | r = 2 \paren {x y + y z + z x} + ... | X + y + z equals 1 implies xy + yz + zx less than Half | https://proofwiki.org/wiki/X_+_y_+_z_equals_1_implies_xy_+_yz_+_zx_less_than_Half | https://proofwiki.org/wiki/X_+_y_+_z_equals_1_implies_xy_+_yz_+_zx_less_than_Half | [
"Inequalities",
"X + y + z equals 1 implies xy + yz + zx less than Half"
] | [
"Definition:Real Number"
] | [
"Square of Sum",
"Square of Sum",
"Definition:Real Number"
] |
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