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-15100 | Ring with Multiplicative Norm has No Proper Zero Divisors | Let $\struct {R, +, \circ}$ be a ring.
Let its zero be denoted by $0_R$.
Let $\norm {\,\cdot\,}$ be a multiplicative norm on $R$.
Then $R$ has no proper zero divisors.
That is:
:$\forall x, y \in R^*: x \circ y \ne 0_R$
where $R^*$ is defined as $R \setminus \set {0_R}$. | {{AimForCont}}:
:$\exists x, y \in {R^*} : x \circ y = 0_R$
By positive definiteness:
:$x, y \ne 0_R \iff \norm x, \norm y \ne 0$
Thus:
:$\norm x \norm y \ne 0$
But we also have:
{{begin-eqn}}
{{eqn | l = \norm x \norm y
| r = \norm {x \circ y}
| c = Multiplicativity
}}
{{eqn | r = \norm {0_R}
| c = {... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let its [[Definition:Ring Zero|zero]] be denoted by $0_R$.
Let $\norm {\,\cdot\,}$ be a [[Definition:Multiplicative Norm on Ring|multiplicative norm]] on $R$.
Then $R$ has no [[Definition:Proper Zero Divisor|proper zero divisors]].
That... | {{AimForCont}}:
:$\exists x, y \in {R^*} : x \circ y = 0_R$
By [[Definition:Multiplicative Norm on Ring|positive definiteness]]:
:$x, y \ne 0_R \iff \norm x, \norm y \ne 0$
Thus:
:$\norm x \norm y \ne 0$
But we also have:
{{begin-eqn}}
{{eqn | l = \norm x \norm y
| r = \norm {x \circ y}
| c = [[Defin... | Ring with Multiplicative Norm has No Proper Zero Divisors | https://proofwiki.org/wiki/Ring_with_Multiplicative_Norm_has_No_Proper_Zero_Divisors | https://proofwiki.org/wiki/Ring_with_Multiplicative_Norm_has_No_Proper_Zero_Divisors | [
"Norm Theory",
"Zero Divisors"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Zero",
"Definition:Norm/Ring/Multiplicative",
"Definition:Proper Zero Divisor",
"Definition:Ring (Abstract Algebra)/Ring Less Zero"
] | [
"Definition:Norm/Ring/Multiplicative",
"Definition:Norm/Ring/Multiplicative",
"Definition:Norm/Ring/Multiplicative",
"Category:Norm Theory",
"Category:Zero Divisors"
] |
proofwiki-15101 | Finite Ring with Multiplicative Norm is Field | Let $R$ be a finite ring with a multiplicative norm.
Then $R$ is a field. | From Ring with Multiplicative Norm has No Proper Zero Divisors, $R$ has no proper zero divisors.
From Finite Ring with No Proper Zero Divisors is Field, $R$ is a field.
{{qed}}
Category:Ring Theory
Category:Field Theory
Category:Norm Theory
enxnwlyyj6tvyk47644xl7ho37i4chw | Let $R$ be a [[Definition:Finite Set|finite]] [[Definition:Ring (Abstract Algebra)|ring]] with a [[Definition:Multiplicative Norm on Ring|multiplicative norm]].
Then $R$ is a [[Definition:Field (Abstract Algebra)|field]]. | From [[Ring with Multiplicative Norm has No Proper Zero Divisors]], $R$ has no [[Definition:Proper Zero Divisor|proper zero divisors]].
From [[Finite Ring with No Proper Zero Divisors is Field]], $R$ is a [[Definition:Field (Abstract Algebra)|field]].
{{qed}}
[[Category:Ring Theory]]
[[Category:Field Theory]]
[[Categ... | Finite Ring with Multiplicative Norm is Field | https://proofwiki.org/wiki/Finite_Ring_with_Multiplicative_Norm_is_Field | https://proofwiki.org/wiki/Finite_Ring_with_Multiplicative_Norm_is_Field | [
"Ring Theory",
"Field Theory",
"Norm Theory"
] | [
"Definition:Finite Set",
"Definition:Ring (Abstract Algebra)",
"Definition:Norm/Ring/Multiplicative",
"Definition:Field (Abstract Algebra)"
] | [
"Ring with Multiplicative Norm has No Proper Zero Divisors",
"Definition:Proper Zero Divisor",
"Finite Ring with No Proper Zero Divisors is Field",
"Definition:Field (Abstract Algebra)",
"Category:Ring Theory",
"Category:Field Theory",
"Category:Norm Theory"
] |
proofwiki-15102 | Composition of Distance-Preserving Mappings is Distance-Preserving | Let:
:$\struct {X_1, d_1}$
:$\struct {X_2, d_2}$
:$\struct {X_3, d_3}$
be metric spaces.
Let:
:$\phi: \struct {X_1, d_1} \to \struct {X_2, d_2}$
:$\psi: \struct {X_2, d_2} \to \struct {X_3, d_3}$
be distance-preserving mappings.
Then the composite of $\phi$ and $\psi$ is also a distance-preserving mapping. | Let $x,y \in X_1$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {x,y}
| r = \map {d_2} {\map \phi x, \map \phi y}
| c = $\phi$ is a distance-preserving mapping
}}
{{eqn | r = \map {d_3} {\map \psi {\map \phi x}, \map \psi {\map \phi y} }
| c = $\psi$ is a distance-preserving mapping
}}
{{eqn | r = \map ... | Let:
:$\struct {X_1, d_1}$
:$\struct {X_2, d_2}$
:$\struct {X_3, d_3}$
be [[Definition:Metric Space|metric spaces]].
Let:
:$\phi: \struct {X_1, d_1} \to \struct {X_2, d_2}$
:$\psi: \struct {X_2, d_2} \to \struct {X_3, d_3}$
be [[Definition:Distance-Preserving Mapping|distance-preserving mappings]].
Then the [[Defini... | Let $x,y \in X_1$.
Then:
{{begin-eqn}}
{{eqn | l = \map {d_1} {x,y}
| r = \map {d_2} {\map \phi x, \map \phi y}
| c = $\phi$ is a [[Definition:Distance-Preserving Mapping|distance-preserving mapping]]
}}
{{eqn | r = \map {d_3} {\map \psi {\map \phi x}, \map \psi {\map \phi y} }
| c = $\psi$ is a [[De... | Composition of Distance-Preserving Mappings is Distance-Preserving | https://proofwiki.org/wiki/Composition_of_Distance-Preserving_Mappings_is_Distance-Preserving | https://proofwiki.org/wiki/Composition_of_Distance-Preserving_Mappings_is_Distance-Preserving | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Distance-Preserving Mapping",
"Definition:Composition of Mappings",
"Definition:Distance-Preserving Mapping"
] | [
"Definition:Distance-Preserving Mapping",
"Definition:Distance-Preserving Mapping",
"Definition:Distance-Preserving Mapping",
"Definition:Distance-Preserving Mapping",
"Category:Metric Spaces"
] |
proofwiki-15103 | Composition of Isometries is Isometry | Let:
:$\struct {X_1, d_1}$
:$\struct {X_2, d_2}$
:$\struct {X_3, d_3}$
be metric spaces.
Let:
:$\phi: \struct {X_1, d_1} \to \struct {X_2, d_2}$
:$\psi: \struct {X_2, d_2} \to \struct {X_3, d_3}$
be isometries.
Then the composite of $\phi$ and $\psi$ is also an isometry. | An isometry is a distance-preserving mapping which is also a bijection.
From Composition of Distance-Preserving Mappings is Distance-Preserving, $\psi \circ \phi$ is a distance-preserving mapping.
From Composite of Bijections is Bijection, $\psi \circ \phi$ is a bijection.
{{qed}}
Category:Metric Spaces
m8cjqegtg9dd5z4... | Let:
:$\struct {X_1, d_1}$
:$\struct {X_2, d_2}$
:$\struct {X_3, d_3}$
be [[Definition:Metric Space|metric spaces]].
Let:
:$\phi: \struct {X_1, d_1} \to \struct {X_2, d_2}$
:$\psi: \struct {X_2, d_2} \to \struct {X_3, d_3}$
be [[Definition:Isometry (Metric Spaces)|isometries]].
Then the [[Definition:Composition of Ma... | An [[Definition:Isometry (Metric Spaces)|isometry]] is a [[Definition:Distance-Preserving Mapping|distance-preserving mapping]] which is also a [[Definition:Bijection|bijection]].
From [[Composition of Distance-Preserving Mappings is Distance-Preserving]], $\psi \circ \phi$ is a [[Definition:Distance-Preserving Mappi... | Composition of Isometries is Isometry | https://proofwiki.org/wiki/Composition_of_Isometries_is_Isometry | https://proofwiki.org/wiki/Composition_of_Isometries_is_Isometry | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Isometry (Metric Spaces)",
"Definition:Composition of Mappings",
"Definition:Isometry (Metric Spaces)"
] | [
"Definition:Isometry (Metric Spaces)",
"Definition:Distance-Preserving Mapping",
"Definition:Bijection",
"Composition of Distance-Preserving Mappings is Distance-Preserving",
"Definition:Distance-Preserving Mapping",
"Composite of Bijections is Bijection",
"Definition:Bijection",
"Category:Metric Spac... |
proofwiki-15104 | Subgroup Action is Group Action | Let $\struct {G, \circ}$ be a group.
Let $\struct {H, \circ}$ be a subgroup of $G$.
Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as:
:$\forall h \in H, g \in G: h * g := h \circ g$
Then $*$ is a group action. | Let $g \in G$.
First we note that since $G$ is closed, and $h \circ g \in G$, it follows that $h * g \in G$.
Next we note:
:$e * g = e \circ g = g$
and so {{GroupActionAxiom|2}} is satisfied.
Now let $h_1, h_2 \in G$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {h_1 \circ h_2} * g
| r = \paren {h_1 \circ h_2} \circ... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $\struct {H, \circ}$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $*: H \times G \to G$ be the [[Definition:Subgroup Action|subgroup action]] defined for all $h \in H, g \in G$ as:
:$\forall h \in H, g \in G: h * g := h \circ g$
Then $*$ is a [[Defi... | Let $g \in G$.
First we note that since $G$ is [[Definition:Closed Algebraic Structure|closed]], and $h \circ g \in G$, it follows that $h * g \in G$.
Next we note:
:$e * g = e \circ g = g$
and so {{GroupActionAxiom|2}} is satisfied.
Now let $h_1, h_2 \in G$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {h_1 \circ... | Subgroup Action is Group Action | https://proofwiki.org/wiki/Subgroup_Action_is_Group_Action | https://proofwiki.org/wiki/Subgroup_Action_is_Group_Action | [
"Subgroup Action"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Subgroup Action",
"Definition:Group Action"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-15105 | Orbit of Subgroup Action is Coset | Let $\struct {G, \circ}$ be a group.
Let $\struct {H, \circ}$ be a subgroup of $G$.
Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as:
:$\forall h \in H, g \in G: h * g := h \circ g$
The orbit of $x \in G$ is the right coset by $x$ of $H$:
:$\Orb x = H x$ | From Subgroup Action is Group Action we have that $*$ is a group action.
Let $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \Orb x
| r = \set {g \in G: \exists h \in H: g = h * x}
| c = {{Defof|Orbit (Group Theory)|Orbit}}
}}
{{eqn | r = \set {g \in G: \exists h \in H: g = h \circ x}
| c = Definition of $... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $\struct {H, \circ}$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $*: H \times G \to G$ be the [[Definition:Subgroup Action|subgroup action]] defined for all $h \in H, g \in G$ as:
:$\forall h \in H, g \in G: h * g := h \circ g$
The [[Definition:Orb... | From [[Subgroup Action is Group Action]] we have that $*$ is a [[Definition:Group Action|group action]].
Let $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \Orb x
| r = \set {g \in G: \exists h \in H: g = h * x}
| c = {{Defof|Orbit (Group Theory)|Orbit}}
}}
{{eqn | r = \set {g \in G: \exists h \in H: g = h \c... | Orbit of Subgroup Action is Coset | https://proofwiki.org/wiki/Orbit_of_Subgroup_Action_is_Coset | https://proofwiki.org/wiki/Orbit_of_Subgroup_Action_is_Coset | [
"Subgroup Action"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Subgroup Action",
"Definition:Orbit (Group Theory)",
"Definition:Coset/Right Coset"
] | [
"Subgroup Action is Group Action",
"Definition:Group Action",
"Definition:Coset/Right Coset"
] |
proofwiki-15106 | Removable Singularity at Infinity implies Constant Function | Let $f : \C \to \C$ be an entire function.
Let $f$ have an removable singularity at $\infty$.
Then $f$ is constant. | We are given that $f$ has a removable singularity at $\infty$.
By Riemann Removable Singularities Theorem, $f$ must be bounded in a neighborhood of $\infty$.
That is, there exists a real number $M > 0$ such that:
:$\forall z \in \set {z : \cmod z > r}: \cmod {\map f z} \le M$
for some real $r \ge 0$.
However, by Contin... | Let $f : \C \to \C$ be an [[Definition:Entire Function|entire function]].
Let $f$ have an [[Definition:Removable Singularity (Complex Plane)|removable singularity]] at $\infty$.
Then $f$ is [[Definition:Constant Function|constant]]. | We are given that $f$ has a [[Definition:Removable Singularity (Complex Plane)|removable singularity]] at $\infty$.
By [[Riemann Removable Singularities Theorem]], $f$ must be [[Definition:Bounded Complex-Valued Function|bounded]] in a [[Definition:Neighborhood of Infinity (Complex Analysis)|neighborhood of $\infty$]]... | Removable Singularity at Infinity implies Constant Function | https://proofwiki.org/wiki/Removable_Singularity_at_Infinity_implies_Constant_Function | https://proofwiki.org/wiki/Removable_Singularity_at_Infinity_implies_Constant_Function | [
"Removable Singularities",
"Entire Functions",
"Constant Mappings"
] | [
"Definition:Entire Function",
"Definition:Removable Singularity/Complex Function",
"Definition:Constant Mapping"
] | [
"Definition:Removable Singularity/Complex Function",
"Riemann Removable Singularities Theorem",
"Definition:Bounded Mapping/Complex-Valued",
"Definition:Neighborhood of Infinity (Complex Analysis)",
"Definition:Real Number",
"Definition:Real Number",
"Continuous Function on Compact Space is Bounded",
... |
proofwiki-15107 | Stabilizer of Subgroup Action is Identity | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $\struct {H, \circ}$ be a subgroup of $G$.
Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as:
:$\forall h \in H, g \in G: h * g := h \circ g$
The stabilizer of $x \in G$ is $\set e$:
:$\Stab x = \set e$ | From Subgroup Action is Group Action we have that $*$ is a group action.
Let $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \Stab x
| r = \set {h \in H: h * x = x}
| c = {{Defof|Stabilizer}}
}}
{{eqn | r = \set {h \in H: h \circ x = x}
| c = Definition of $*$
}}
{{eqn | r = \set {h \in H: h = x \circ x^{-... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\struct {H, \circ}$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $*: H \times G \to G$ be the [[Definition:Subgroup Action|subgroup action]] defined for all $h \in H, g \in G$ as:
:$\forall h \in ... | From [[Subgroup Action is Group Action]] we have that $*$ is a [[Definition:Group Action|group action]].
Let $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \Stab x
| r = \set {h \in H: h * x = x}
| c = {{Defof|Stabilizer}}
}}
{{eqn | r = \set {h \in H: h \circ x = x}
| c = Definition of $*$
}}
{{eqn | r... | Stabilizer of Subgroup Action is Identity | https://proofwiki.org/wiki/Stabilizer_of_Subgroup_Action_is_Identity | https://proofwiki.org/wiki/Stabilizer_of_Subgroup_Action_is_Identity | [
"Subgroup Action",
"Stabilizers"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Subgroup Action",
"Definition:Stabilizer"
] | [
"Subgroup Action is Group Action",
"Definition:Group Action",
"Definition:Coset/Right Coset"
] |
proofwiki-15108 | Stabilizers of Elements in Same Orbit are Conjugate Subgroups | Let $G$ be a group acting on a set $X$.
Let:
:$y, z \in \Orb x$
where $\Orb x$ denotes the orbit of some $x \in X$.
Then their stabilizers $\Stab y$ and $\Stab z$ are conjugate subgroups. | From Stabilizer is Subgroup we have that both $\Stab y$ and $\Stab z$ are subgroups of $G$.
From definition of orbits:
:$\exists h_1, h_2 \in G: y = h_1 * x, z = h_2 * x$
Then $y = h_1 * \paren {h_2^{-1} * z} = h_1 h_2^{-1} * z$.
Thus:
{{begin-eqn}}
{{eqn | l = \Stab y
| r = \set {g \in G: g * y = y}
| c = ... | Let $G$ be a [[Definition:Group Action|group acting]] on a set $X$.
Let:
:$y, z \in \Orb x$
where $\Orb x$ denotes the [[Definition:Orbit (Group Theory)|orbit]] of some $x \in X$.
Then their [[Definition:Stabilizer|stabilizers]] $\Stab y$ and $\Stab z$ are [[Definition:Conjugate of Group Subset|conjugate]] [[Definit... | From [[Stabilizer is Subgroup]] we have that both $\Stab y$ and $\Stab z$ are [[Definition:Subgroup|subgroups]] of $G$.
From [[Definition:Orbit (Group Theory)|definition of orbits]]:
:$\exists h_1, h_2 \in G: y = h_1 * x, z = h_2 * x$
Then $y = h_1 * \paren {h_2^{-1} * z} = h_1 h_2^{-1} * z$.
Thus:
{{begin-eqn}}
{{... | Stabilizers of Elements in Same Orbit are Conjugate Subgroups | https://proofwiki.org/wiki/Stabilizers_of_Elements_in_Same_Orbit_are_Conjugate_Subgroups | https://proofwiki.org/wiki/Stabilizers_of_Elements_in_Same_Orbit_are_Conjugate_Subgroups | [
"Group Actions",
"Stabilizers"
] | [
"Definition:Group Action",
"Definition:Orbit (Group Theory)",
"Definition:Stabilizer",
"Definition:Conjugate (Group Theory)/Subset",
"Definition:Subgroup"
] | [
"Stabilizer is Subgroup",
"Definition:Subgroup",
"Definition:Orbit (Group Theory)",
"Definition:Conjugate (Group Theory)/Subset"
] |
proofwiki-15109 | Stabilizer of Subgroup Action on Left Coset Space | Let $G$ be a group.
Let $H$ and $K$ be subgroups of $G$.
Let $K$ act on the left coset space $G / H^l$ by:
:$\forall \tuple {k, g H} \in K \times G / H^l: k * \paren {g H} := \paren {k g} H$
The stabilizer of $g H$ is $K \cap H^g$, where $H^g$ denotes the $G$-conjugate of $H$ by $g$. | {{Proofread| Check the proof for the well-definedness of the action}}
Let $\RR^l_H$ be the equivalence defined as left congruence modulo $H$.
By Left Congruence Class Modulo Subgroup is Left Coset, $\RR^l_H$ is an equivalence relation.
We check that $*$ is a well-defined mapping over $\RR^l_H$.
Suppose:
:$\eqclass g {... | Let $G$ be a [[Definition:Group|group]].
Let $H$ and $K$ be [[Definition:Subgroup|subgroups]] of $G$.
Let $K$ [[Definition:Group Action|act on]] the [[Definition:Left Coset Space|left coset space]] $G / H^l$ by:
:$\forall \tuple {k, g H} \in K \times G / H^l: k * \paren {g H} := \paren {k g} H$
The [[Definition:St... | {{Proofread| Check the proof for the well-definedness of the action}}
Let $\RR^l_H$ be the [[Definition:Equivalence Relation|equivalence]] defined as [[Definition:Left Congruence Modulo Subgroup|left congruence modulo $H$]].
By [[Left Congruence Class Modulo Subgroup is Left Coset]], $\RR^l_H$ is an [[Definition:Equi... | Stabilizer of Subgroup Action on Left Coset Space | https://proofwiki.org/wiki/Stabilizer_of_Subgroup_Action_on_Left_Coset_Space | https://proofwiki.org/wiki/Stabilizer_of_Subgroup_Action_on_Left_Coset_Space | [
"Stabilizers",
"Group Action on Coset Space"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Group Action",
"Definition:Coset Space/Left Coset Space",
"Definition:Stabilizer",
"Definition:Conjugate (Group Theory)/Subset"
] | [
"Definition:Equivalence Relation",
"Definition:Congruence Modulo Subgroup/Left Congruence",
"Left Congruence Class Modulo Subgroup is Left Coset",
"Definition:Equivalence Relation",
"Definition:Well-Defined/Mapping",
"Definition:Equivalence Class",
"Left Congruence Class Modulo Subgroup is Left Coset",
... |
proofwiki-15110 | Length of Orbit of Subgroup Action on Left Coset Space | Let $G$ be a group.
Let $H$ and $K$ be subgroups of $G$.
Let $K$ act on the left coset space $G / H^l$ by:
:$\forall \tuple {k, g H} \in K \times G / H^l: k * g H := \paren {k g} H$
The length of the orbit of $g H$ is $\index K {K \cap H^g}$. | {{begin-eqn}}
{{eqn | l = \card {\Orb {g H} }
| r = \index K {\Stab {g H} }
| c = Orbit-Stabilizer Theorem
}}
{{eqn | r = \index K {K \cap H^g}
| c = Stabilizer of Subgroup Action on Left Coset Space
}}
{{end-eqn}}
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $H$ and $K$ be [[Definition:Subgroup|subgroups]] of $G$.
Let $K$ [[Definition:Group Action|act on]] the [[Definition:Left Coset Space|left coset space]] $G / H^l$ by:
:$\forall \tuple {k, g H} \in K \times G / H^l: k * g H := \paren {k g} H$
The [[Definition:Length of O... | {{begin-eqn}}
{{eqn | l = \card {\Orb {g H} }
| r = \index K {\Stab {g H} }
| c = [[Orbit-Stabilizer Theorem]]
}}
{{eqn | r = \index K {K \cap H^g}
| c = [[Stabilizer of Subgroup Action on Left Coset Space]]
}}
{{end-eqn}}
{{qed}} | Length of Orbit of Subgroup Action on Left Coset Space | https://proofwiki.org/wiki/Length_of_Orbit_of_Subgroup_Action_on_Left_Coset_Space | https://proofwiki.org/wiki/Length_of_Orbit_of_Subgroup_Action_on_Left_Coset_Space | [
"Group Action on Coset Space"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Group Action",
"Definition:Coset Space/Left Coset Space",
"Definition:Orbit (Group Theory)/Length",
"Definition:Orbit (Group Theory)"
] | [
"Orbit-Stabilizer Theorem",
"Stabilizer of Subgroup Action on Left Coset Space"
] |
proofwiki-15111 | Pole at Infinity implies Polynomial Function | Let $f : \C \to \C$ be an entire function.
Let $f$ have a pole of order $N$ at $\infty$.
Then $f$ is a polynomial of degree $N$. | By Complex Function is Entire iff it has Everywhere Convergent Power Series, there exists a power series:
:$\ds \map f z = \sum_{n \mathop = 0}^\infty a_n z^n$
convergent for all $z \in \C$, where $\sequence {a_n}$ is a sequence of complex coefficients.
This gives:
:$\ds \map f {\frac 1 z} = \sum_{n \mathop = 0}^\i... | Let $f : \C \to \C$ be an [[Definition:Entire Function|entire function]].
Let $f$ have a [[Definition:Pole (Complex Analysis)|pole]] of [[Definition:Order of Pole|order]] $N$ at $\infty$.
Then $f$ is a [[Definition:Complex Polynomial Function|polynomial]] of [[Definition:Degree of Polynomial|degree]] $N$. | By [[Complex Function is Entire iff it has Everywhere Convergent Power Series]], there exists a [[Definition:Complex Power Series|power series]]:
:$\ds \map f z = \sum_{n \mathop = 0}^\infty a_n z^n$
convergent for all $z \in \C$, where $\sequence {a_n}$ is a [[Definition:Sequence|sequence]] of complex coefficients... | Pole at Infinity implies Polynomial Function | https://proofwiki.org/wiki/Pole_at_Infinity_implies_Polynomial_Function | https://proofwiki.org/wiki/Pole_at_Infinity_implies_Polynomial_Function | [
"Complex Analysis"
] | [
"Definition:Entire Function",
"Definition:Isolated Singularity/Pole",
"Definition:Order of Pole",
"Definition:Polynomial Function/Complex",
"Definition:Degree of Polynomial"
] | [
"Complex Function is Entire iff it has Everywhere Convergent Power Series",
"Definition:Power Series/Complex Domain",
"Definition:Sequence",
"Definition:Holomorphic Function",
"Definition:Polynomial Function/Complex",
"Definition:Degree of Polynomial",
"Category:Complex Analysis"
] |
proofwiki-15112 | Cardinality of Set Difference with Subset | Let $S$ and $T$ be sets such that $T$ is finite.
Let $T \subseteq S$.
Then:
:$\card {S \setminus T} = \card S - \card T$
where $\card S$ denotes the cardinality of $S$. | From Set Difference with Superset is Empty Set:
:$T \subseteq S \iff T \setminus S = \O$
From Set Difference and Intersection form Partition:
:$S = \paren {S \setminus T} \cup T$
Thus from Cardinality of Set Union:
:$\card S = \card T + \card {S \setminus T} - \card {T \cap \paren {S \setminus T} }$
But from Set Differ... | Let $S$ and $T$ be [[Definition:Set|sets]] such that $T$ is [[Definition:Finite Set|finite]].
Let $T \subseteq S$.
Then:
:$\card {S \setminus T} = \card S - \card T$
where $\card S$ denotes the [[Definition:Cardinality|cardinality]] of $S$. | From [[Set Difference with Superset is Empty Set]]:
:$T \subseteq S \iff T \setminus S = \O$
From [[Set Difference and Intersection form Partition]]:
:$S = \paren {S \setminus T} \cup T$
Thus from [[Cardinality of Set Union]]:
:$\card S = \card T + \card {S \setminus T} - \card {T \cap \paren {S \setminus T} }$
But ... | Cardinality of Set Difference with Subset | https://proofwiki.org/wiki/Cardinality_of_Set_Difference_with_Subset | https://proofwiki.org/wiki/Cardinality_of_Set_Difference_with_Subset | [
"Set Difference",
"Cardinality"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Cardinality"
] | [
"Set Difference with Superset is Empty Set",
"Set Difference and Intersection form Partition",
"Cardinality of Set Union",
"Set Difference Intersection with Second Set is Empty Set"
] |
proofwiki-15113 | Riemann Zeta Function at Non-Positive Integers | Let $n \ge 0$ be a integer.
Then:
:$\map \zeta {-n} = \paren {-1}^n \dfrac {B_{n + 1} } {n + 1}$
where:
:$B_n$ is the $n$th Bernoulli number
:$\zeta$ is the Riemann Zeta function | By Hankel Representation of Riemann Zeta Function:
:$\ds \map \zeta {-n} = \frac {i \map \Gamma {1 + n} } {2 \pi} \oint_C \frac 1 {z^{n + 1} \paren {e^z - 1} } \rd z$
where $C$ is the Hankel contour.
Note that the integrand is meromorphic, with exactly one pole at $z = 0$ lying inside the contour.
So:
{{begin-eqn}}
... | Let $n \ge 0$ be a [[Definition:Integer|integer]].
Then:
:$\map \zeta {-n} = \paren {-1}^n \dfrac {B_{n + 1} } {n + 1}$
where:
:$B_n$ is the [[Definition:Bernoulli Numbers|$n$th Bernoulli number]]
:$\zeta$ is the [[Definition:Riemann Zeta Function|Riemann Zeta function]] | By [[Hankel Representation of Riemann Zeta Function]]:
:$\ds \map \zeta {-n} = \frac {i \map \Gamma {1 + n} } {2 \pi} \oint_C \frac 1 {z^{n + 1} \paren {e^z - 1} } \rd z$
where $C$ is the [[Definition:Hankel Contour|Hankel contour]].
Note that the integrand is [[Definition:Meromorphic Function|meromorphic]], with ... | Riemann Zeta Function at Non-Positive Integers | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Non-Positive_Integers | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Non-Positive_Integers | [
"Bernoulli Numbers",
"Riemann Zeta Function"
] | [
"Definition:Integer",
"Definition:Bernoulli Numbers",
"Definition:Riemann Zeta Function"
] | [
"Hankel Representation of Riemann Zeta Function",
"Definition:Hankel Contour",
"Definition:Meromorphic Function",
"Gamma Function Extends Factorial",
"Cauchy's Residue Theorem",
"Definition:Residue (Complex Analysis)"
] |
proofwiki-15114 | Remainder on Division is Least Positive Residue | Let $a, b \in \Z$ be integers such that $a \ge 0$ and $b \ne 0$.
Let $r$ be the remainder resulting from the operation of integer division of $a$ by $b$:
$a = q b + r, 0 \le r < \size b$
Then $r$ is equal to the least positive residue of $a \pmod b$. | By definition of least positive residue:
:$a = q b + r \iff r \equiv a \pmod b$
for some $q \in \Z$.
By the Division Theorem, there exists a $q$ such that:
:$0 \le r < \size b$
which is precisely the definition of the least positive residue of $a \pmod b$.
{{qed}} | Let $a, b \in \Z$ be [[Definition:Integer|integers]] such that $a \ge 0$ and $b \ne 0$.
Let $r$ be the [[Definition:Remainder|remainder]] resulting from the operation of [[Definition:Integer Division|integer division]] of $a$ by $b$:
$a = q b + r, 0 \le r < \size b$
Then $r$ is equal to the [[Definition:Least Posit... | By definition of [[Definition:Least Positive Residue|least positive residue]]:
:$a = q b + r \iff r \equiv a \pmod b$
for some $q \in \Z$.
By the [[Division Theorem]], there exists a $q$ such that:
:$0 \le r < \size b$
which is precisely the definition of the [[Definition:Least Positive Residue|least positive residue... | Remainder on Division is Least Positive Residue | https://proofwiki.org/wiki/Remainder_on_Division_is_Least_Positive_Residue | https://proofwiki.org/wiki/Remainder_on_Division_is_Least_Positive_Residue | [
"Modulo Arithmetic"
] | [
"Definition:Integer",
"Definition:Remainder",
"Definition:Integer Division",
"Definition:Set of Residue Classes/Least Positive"
] | [
"Definition:Set of Residue Classes/Least Positive",
"Division Theorem",
"Definition:Set of Residue Classes/Least Positive"
] |
proofwiki-15115 | Floor Function/Examples/Floor of Root 5 | :$\floor {\sqrt 5} = 2$ | The decimal expansion of $\sqrt 5$ is:
:$\sqrt 5 \approx 2 \cdotp 23606 \, 79774 \, 99789 \, 6964 \ldots$
Thus:
:$2 \le \sqrt 5 < 3$
Hence $2$ is the floor of $\sqrt 5$ by definition.
{{qed}} | :$\floor {\sqrt 5} = 2$ | The [[Square Root of 5|decimal expansion of $\sqrt 5$]] is:
:$\sqrt 5 \approx 2 \cdotp 23606 \, 79774 \, 99789 \, 6964 \ldots$
Thus:
:$2 \le \sqrt 5 < 3$
Hence $2$ is the [[Definition:Floor Function|floor]] of $\sqrt 5$ by definition.
{{qed}} | Floor Function/Examples/Floor of Root 5 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_Root_5 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_Root_5 | [
"Examples of Floor Function"
] | [] | [
"Square Root/Examples/5",
"Definition:Floor Function"
] |
proofwiki-15116 | Floor Function/Examples/Floor of 3 | :$\floor 3 = 3$ | We have that $3$ is an integer.
Thus this is a specific example of Real Number is Integer iff equals Floor:
$\floor x = x \iff x \in \Z$
{{qed}} | :$\floor 3 = 3$ | We have that $3$ is an [[Definition:Integer|integer]].
Thus this is a specific example of [[Real Number is Integer iff equals Floor]]:
$\floor x = x \iff x \in \Z$
{{qed}} | Floor Function/Examples/Floor of 3 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_3 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_3 | [
"Examples of Floor Function"
] | [] | [
"Definition:Integer",
"Real Number is Integer iff equals Floor"
] |
proofwiki-15117 | Divisibility by 12 | Let $N \in \N$ be expressed as:
:$N = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$
Then $N$ is divisible by $12$ {{iff}} $a_0 - 2 a_1 + 4 \paren {\ds \sum_{r \mathop = 2}^n a_r}$ is divisible by $12$. | We first prove that $100 \times 10^n = 4 \pmod {12}$, where $n \in \N$.
Proof by induction:
For all $n \in \N$, let $P \paren n$ be the proposition:
:$100 \times 10^n = 4 \pmod {12}$ | Let $N \in \N$ be expressed as:
:$N = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$
Then $N$ is [[Definition:Divisor of Integer|divisible]] by $12$ {{iff}} $a_0 - 2 a_1 + 4 \paren {\ds \sum_{r \mathop = 2}^n a_r}$ is [[Definition:Divisor of Integer|divisible]] by $12$. | We first prove that $100 \times 10^n = 4 \pmod {12}$, where $n \in \N$.
Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N$, let $P \paren n$ be the [[Definition:Proposition|proposition]]:
:$100 \times 10^n = 4 \pmod {12}$ | Divisibility by 12 | https://proofwiki.org/wiki/Divisibility_by_12 | https://proofwiki.org/wiki/Divisibility_by_12 | [
"Divisibility Tests",
"12"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-15118 | Exponential on Real Numbers is Injection | Let $\exp: \R \to \R$ be the exponential function:
:$\map \exp x = e^x$
Then $\exp$ is an injection. | <onlyinclude>
From Exponential is Strictly Increasing:
:$\exp$ is strictly increasing on $\R$.
From Strictly Monotone Mapping with Totally Ordered Domain is Injective:
:$\exp$ is an injection.
{{qed}} | Let $\exp: \R \to \R$ be the [[Definition:Real Exponential Function|exponential function]]:
:$\map \exp x = e^x$
Then $\exp$ is an [[Definition:Injection|injection]]. | <onlyinclude>
From [[Exponential is Strictly Increasing]]:
:$\exp$ is [[Definition:Strictly Increasing Real Function|strictly increasing]] on $\R$.
From [[Strictly Monotone Mapping with Totally Ordered Domain is Injective]]:
:$\exp$ is an [[Definition:Injection|injection]].
{{qed}} | Exponential on Real Numbers is Injection | https://proofwiki.org/wiki/Exponential_on_Real_Numbers_is_Injection | https://proofwiki.org/wiki/Exponential_on_Real_Numbers_is_Injection | [
"Exponential Function",
"Injections"
] | [
"Definition:Exponential Function/Real",
"Definition:Injection"
] | [
"Exponential is Strictly Increasing",
"Definition:Strictly Increasing/Real Function",
"Strictly Monotone Mapping with Totally Ordered Domain is Injective",
"Definition:Injection"
] |
proofwiki-15119 | Surjective Restriction of Real Exponential Function | Let $\exp: \R \to \R$ be the exponential function:
:$\map \exp x = e^x$
Then the restriction of the codomain of $\exp$ to the strictly positive real numbers:
:$\exp: \R \to \R_{>0}$
is a surjective restriction.
Hence:
:$\exp: \R \to \R_{>0}$
is a bijection. | We have Exponential on Real Numbers is Injection.
Let $y \in \R_{> 0}$.
Then $\exists x \in \R: x = \map \ln y$
That is:
:$\exp x = y$
and so $\exp: \R \to \R_{>0}$ is a surjection.
Hence the result.
{{qed}} | Let $\exp: \R \to \R$ be the [[Definition:Real Exponential Function|exponential function]]:
:$\map \exp x = e^x$
Then the [[Definition:Restriction of Mapping|restriction]] of the [[Definition:Codomain|codomain]] of $\exp$ to the [[Definition:Strictly Positive Real Number|strictly positive real numbers]]:
:$\exp: \R \... | We have [[Exponential on Real Numbers is Injection]].
Let $y \in \R_{> 0}$.
Then $\exists x \in \R: x = \map \ln y$
That is:
:$\exp x = y$
and so $\exp: \R \to \R_{>0}$ is a [[Definition:Surjection|surjection]].
Hence the result.
{{qed}} | Surjective Restriction of Real Exponential Function | https://proofwiki.org/wiki/Surjective_Restriction_of_Real_Exponential_Function | https://proofwiki.org/wiki/Surjective_Restriction_of_Real_Exponential_Function | [
"Exponential Function",
"Surjections"
] | [
"Definition:Exponential Function/Real",
"Definition:Restriction/Mapping",
"Definition:Codomain",
"Definition:Strictly Positive/Real Number",
"Definition:Surjective Restriction",
"Definition:Bijection"
] | [
"Exponential on Real Numbers is Injection",
"Definition:Surjection"
] |
proofwiki-15120 | Maximum Rule for Real Sequences | :$\ds \lim_{n \mathop \to \infty} \max \set {x_n, y_n} = \max \set {l, m}$ | === Case $1$: $l = m$ ===
Let $l = m$.
Then:
:$\max \set {l, m} = l = m$
Let $\epsilon > 0$ be given.
By definition of the limit of a real sequence, we can find $N_1$ such that:
:$\forall n > N_1: \size {x_n - l} < \epsilon$
Similarly we can find $N_2$ such that:
:$\forall n > N_2: \size {y_n - m} < \epsilon$
Let $N = ... | :$\ds \lim_{n \mathop \to \infty} \max \set {x_n, y_n} = \max \set {l, m}$ | === Case $1$: $l = m$ ===
Let $l = m$.
Then:
:$\max \set {l, m} = l = m$
Let $\epsilon > 0$ be given.
By definition of the [[Definition:Limit of Real Sequence|limit of a real sequence]], we can find $N_1$ such that:
:$\forall n > N_1: \size {x_n - l} < \epsilon$
Similarly we can find $N_2$ such that:
:$\forall n ... | Maximum Rule for Real Sequences | https://proofwiki.org/wiki/Maximum_Rule_for_Real_Sequences | https://proofwiki.org/wiki/Maximum_Rule_for_Real_Sequences | [
"Real Sequences"
] | [] | [
"Definition:Limit of Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers"
] |
proofwiki-15121 | Minimum Rule for Real Sequences | :$\ds \lim_{n \mathop \to \infty} \min \set {x_n, y_n} = \min \set {l, m}$ | By Sum Less Maximum is Minimum:
:$\forall n \in \R: \min \set {x_n, y_n} = x_n + y_n - \max \set {x_n, y_n}$
and
:$\min \set {m, l} = m + l - \max \set {m, l}$
By Maximum Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \max \set {x_n, y_n} = \max \set {m, l}$
By the Multiple Rule for Real Sequences:
:$\ds \l... | :$\ds \lim_{n \mathop \to \infty} \min \set {x_n, y_n} = \min \set {l, m}$ | By [[Sum Less Maximum is Minimum]]:
:$\forall n \in \R: \min \set {x_n, y_n} = x_n + y_n - \max \set {x_n, y_n}$
and
:$\min \set {m, l} = m + l - \max \set {m, l}$
By [[Maximum Rule for Real Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \max \set {x_n, y_n} = \max \set {m, l}$
By the [[Multiple Rule for Real Sequen... | Minimum Rule for Real Sequences | https://proofwiki.org/wiki/Minimum_Rule_for_Real_Sequences | https://proofwiki.org/wiki/Minimum_Rule_for_Real_Sequences | [
"Real Sequences"
] | [] | [
"Sum Less Maximum is Minimum",
"Maximum Rule for Real Sequences",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Combination Theorem for Sequences/Real/Sum Rule"
] |
proofwiki-15122 | Division Subring of Normed Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $S$ be a division subring of $R$.
Then:
:$\struct {S, \norm {\, \cdot \,}_S}$ is a normed division subring of $\struct {R, \norm {\, \cdot \,} }$
where $\norm {\, \cdot \,}_S$ is the norm $\norm {\,\cdot\,}$ restricted to $S$. | === {{Norm-axiom-mult|1|nolink}} ===
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \norm x_S
| r = 0
}}
{{eqn | ll= \leadstoandfrom
| l = \norm x
| r = 0
| c = Definition of $\norm x_S$
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = 0
| c = {{Norm-axiom-mult|1}}
}}
{{end-... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $S$ be a [[Definition:Division Subring|division subring]] of $R$.
Then:
:$\struct {S, \norm {\, \cdot \,}_S}$ is a [[Definition:Normed Division Subring|normed division subring]] of $\struct {R, \norm {\, \cdot \... | === {{Norm-axiom-mult|1|nolink}} ===
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \norm x_S
| r = 0
}}
{{eqn | ll= \leadstoandfrom
| l = \norm x
| r = 0
| c = Definition of $\norm x_S$
}}
{{eqn | ll= \leadstoandfrom
| l = x
| r = 0
| c = {{Norm-axiom-mult|1}}
}}
{{end... | Division Subring of Normed Division Ring | https://proofwiki.org/wiki/Division_Subring_of_Normed_Division_Ring | https://proofwiki.org/wiki/Division_Subring_of_Normed_Division_Ring | [
"Division Rings",
"Norm Theory"
] | [
"Definition:Normed Division Ring",
"Definition:Division Subring",
"Definition:Normed Division Subring",
"Definition:Norm/Division Ring",
"Definition:Restriction/Mapping"
] | [] |
proofwiki-15123 | Normed Division Ring is Dense Subring of Completion | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$
Then:
:$\struct {R, \norm {\, \cdot \,} }$ is isometrically isomorphic to a dense normed division subring of $\struct {R', \norm {\, \c... | By the definition of a normed division ring completion then:
:$(1): \quad$ there exists a distance-preserving ring monomorphism $\phi: R \to R'$.
:$(2): \quad \struct {R', \norm {\, \cdot \,}' }$ is a complete metric space.
:$(3): \quad \phi \sqbrk R$ is a dense subspace in $\struct {R', \norm {\, \cdot \,}' }$.
By ima... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {R', \norm {\, \cdot \,}' }$ be a [[Definition:Completion (Normed Division Ring)|normed division ring completion]] of $\struct {R, \norm {\, \cdot \,} }$
Then:
:$\struct {R, \norm {\, \cdot \,} }$ is [[... | By the definition of a [[Definition:Completion (Normed Division Ring)|normed division ring completion]] then:
:$(1): \quad$ there exists a [[Definition:Distance-Preserving Mapping|distance-preserving]] [[Definition:Ring Monomorphism|ring monomorphism]] $\phi: R \to R'$.
:$(2): \quad \struct {R', \norm {\, \cdot \,}' }$... | Normed Division Ring is Dense Subring of Completion | https://proofwiki.org/wiki/Normed_Division_Ring_is_Dense_Subring_of_Completion | https://proofwiki.org/wiki/Normed_Division_Ring_is_Dense_Subring_of_Completion | [
"Normed Division Rings",
"Complete Metric Spaces",
"Completion of Normed Division Ring"
] | [
"Definition:Normed Division Ring",
"Definition:Completion (Normed Division Ring)",
"Definition:Isometric Isomorphism/Normed Division Ring",
"Definition:Everywhere Dense",
"Definition:Normed Division Subring"
] | [
"Definition:Completion (Normed Division Ring)",
"Definition:Distance-Preserving Mapping",
"Definition:Ring Monomorphism",
"Definition:Complete Metric Space",
"Definition:Everywhere Dense",
"Definition:Topological Subspace",
"Ring Homomorphism Preserves Subrings/Corollary",
"Definition:Subring",
"Def... |
proofwiki-15124 | Inverse of Isometric Isomorphism | Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings.
Let $\phi:R \to S$ be a mapping.
Then $\phi:R \to S$ is an isometric isomorphism {{iff}} $\phi^{-1}: S \to R$ is also an isometric isomorphism. | By Inverse of Algebraic Structure Isomorphism is Isomorphism then:
:$\phi: R \to S$ is an ring isomorphism {{iff}} $\phi^{-1}: S \to R$ is also an ring isomorphism.
Let $d_R$ and $d_S$ be the metric induced by the norms $\norm {\,\cdot\,}_R$ and $\norm {\,\cdot\,}_S$ respectively.
By Inverse of Isometry of Metric Space... | Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be [[Definition:Normed Division Ring|normed division rings]].
Let $\phi:R \to S$ be a mapping.
Then $\phi:R \to S$ is an [[Definition:Isometric Isomorphism on Normed Division Ring|isometric isomorphism]] {{iff}} $\phi^{-1}: S \to R$ is als... | By [[Inverse of Algebraic Structure Isomorphism is Isomorphism]] then:
:$\phi: R \to S$ is an [[Definition:Ring Isomorphism|ring isomorphism]] {{iff}} $\phi^{-1}: S \to R$ is also an [[Definition:Ring Isomorphism|ring isomorphism]].
Let $d_R$ and $d_S$ be the [[Definition:Metric Induced by Norm on Division Ring|metri... | Inverse of Isometric Isomorphism | https://proofwiki.org/wiki/Inverse_of_Isometric_Isomorphism | https://proofwiki.org/wiki/Inverse_of_Isometric_Isomorphism | [
"Isometric Isomorphisms (Normed Division Rings)",
"Normed Division Rings",
"Isometric Isomorphisms (Normed Division Rings)"
] | [
"Definition:Normed Division Ring",
"Definition:Isometric Isomorphism/Normed Division Ring",
"Definition:Isometric Isomorphism/Normed Division Ring"
] | [
"Inverse of Algebraic Structure Isomorphism is Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Equivalence of Definitions of Iso... |
proofwiki-15125 | Isometric Isomorphism is Norm-Preserving | Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings.
Let $\phi: R \to S$ be a ring isomorphism.
Then $\phi: R \to S$ is an isometric isomorphism {{iff}} $\phi$ satisfies:
:$\forall x \in R: \norm {\map \phi x}_S = \norm x_R $ | Let $d_R$ and $d_S$ be the metric induced by the norms $\norm {\,\cdot\,}_R$ and $\norm {\,\cdot\,}_S$ respectively. | Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be [[Definition:Normed Division Ring|normed division rings]].
Let $\phi: R \to S$ be a [[Definition:Ring Isomorphism|ring isomorphism]].
Then $\phi: R \to S$ is an [[Definition:Isometric Isomorphism on Normed Division Ring|isometric isomor... | Let $d_R$ and $d_S$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] by the [[Definition:Norm on Division Ring|norms]] $\norm {\,\cdot\,}_R$ and $\norm {\,\cdot\,}_S$ respectively. | Isometric Isomorphism is Norm-Preserving | https://proofwiki.org/wiki/Isometric_Isomorphism_is_Norm-Preserving | https://proofwiki.org/wiki/Isometric_Isomorphism_is_Norm-Preserving | [
"Isometric Isomorphisms (Normed Division Rings)",
"Normed Division Rings",
"Isometric Isomorphisms (Normed Division Rings)"
] | [
"Definition:Normed Division Ring",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"Definition:Isometric Isomorphism/Normed Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring"
] |
proofwiki-15126 | Taylor Series of Holomorphic Function | :$\ds \map f z = \sum_{n \mathop = 0}^\infty \frac {\map {f^{\paren n} } a} {n!} \paren {z - a}^n$ | In Holomorphic Function is Analytic, it is shown that:
:$\ds \map f z = \sum_{n \mathop = 0}^\infty \paren {\frac 1 {2 \pi i} \int_{\partial D} \frac {\map f t} {\paren {t - a}^{n + 1} } \rd t} \paren {z - a}^n$
for all $z \in D$.
From Cauchy's Integral Formula for Derivatives, we have:
:$\ds \frac 1 {2 \pi i} \int_... | :$\ds \map f z = \sum_{n \mathop = 0}^\infty \frac {\map {f^{\paren n} } a} {n!} \paren {z - a}^n$ | In [[Holomorphic Function is Analytic]], it is shown that:
:$\ds \map f z = \sum_{n \mathop = 0}^\infty \paren {\frac 1 {2 \pi i} \int_{\partial D} \frac {\map f t} {\paren {t - a}^{n + 1} } \rd t} \paren {z - a}^n$
for all $z \in D$.
From [[Cauchy's Integral Formula for Derivatives]], we have:
:$\ds \frac 1 {2 ... | Taylor Series of Holomorphic Function | https://proofwiki.org/wiki/Taylor_Series_of_Holomorphic_Function | https://proofwiki.org/wiki/Taylor_Series_of_Holomorphic_Function | [
"Holomorphic Functions",
"Taylor Series"
] | [] | [
"Holomorphic Function is Analytic",
"Cauchy's Integral Formula/General Result",
"Category:Holomorphic Functions",
"Category:Taylor Series"
] |
proofwiki-15127 | Equivalence of Definitions of Square Function | Let $\F$ denote one of the standard classes of numbers: $\N$, $\Z$, $\Q$, $\R$, $\C$.
{{TFAE|def = Square Function}} | By definition of $n$th power (for positive $n$):
:$x^n = \begin {cases} 1 & : n = 0 \\ x \times x^{n - 1} & : n > 0 \end {cases}$
Thus:
{{begin-eqn}}
{{eqn | l = x^2
| r = x \times x^1
| c =
}}
{{eqn | r = x \times x \times x^0
| c =
}}
{{eqn | r = x \times x \times 1
| c =
}}
{{eqn | r = x \... | Let $\F$ denote one of the [[Definition:Number|standard classes of numbers]]: $\N$, $\Z$, $\Q$, $\R$, $\C$.
{{TFAE|def = Square Function}} | By definition of [[Definition:Integer Power|$n$th power]] (for [[Definition:Positive Integer|positive]] $n$):
:$x^n = \begin {cases} 1 & : n = 0 \\ x \times x^{n - 1} & : n > 0 \end {cases}$
Thus:
{{begin-eqn}}
{{eqn | l = x^2
| r = x \times x^1
| c =
}}
{{eqn | r = x \times x \times x^0
| c =
}}
... | Equivalence of Definitions of Square Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Square_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Square_Function | [
"Square Function"
] | [
"Definition:Number"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Positive/Integer",
"Category:Square Function"
] |
proofwiki-15128 | Subring of Non-Archimedean Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with non-archimedean norm $\norm {\, \cdot \,}$.
Let $\struct {S, \norm {\, \cdot \,}_S }$ be a normed division subring of $R$.
Then:
:$\norm {\, \cdot \,}_S$ is a non-archimedean norm. | $\forall x, y \in S$:
{{begin-eqn}}
{{eqn | l = \norm {x + y}_S
| r = \norm {x + y}
| c = Definition of $\norm {\,\cdot\,}_S$
}}
{{eqn | o = \le
| r = \max \set {\norm x, \norm y}
| c = $(\text N 4)$: Ultrametric Inequality
}}
{{eqn | r = \max \set {\norm x_S, \norm y_S}
| c = Definition o... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Non-Archimedean Division Ring Norm|non-archimedean norm]] $\norm {\, \cdot \,}$.
Let $\struct {S, \norm {\, \cdot \,}_S }$ be a [[Definition:Normed Division Subring|normed division subring]] of $R$.
... | $\forall x, y \in S$:
{{begin-eqn}}
{{eqn | l = \norm {x + y}_S
| r = \norm {x + y}
| c = Definition of $\norm {\,\cdot\,}_S$
}}
{{eqn | o = \le
| r = \max \set {\norm x, \norm y}
| c = [[Definition:Non-Archimedean Division Ring Norm|$(\text N 4)$: Ultrametric Inequality]]
}}
{{eqn | r = \max \... | Subring of Non-Archimedean Division Ring | https://proofwiki.org/wiki/Subring_of_Non-Archimedean_Division_Ring | https://proofwiki.org/wiki/Subring_of_Non-Archimedean_Division_Ring | [
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Normed Division Subring",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)",
"Category:Normed Division Rings"
] |
proofwiki-15129 | Isometrically Isomorphic Non-Archimedean Division Rings | Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings.
Let $\phi:R \to S$ be an isometric isomorphism.
Then:
:$\norm {\,\cdot\,}_R$ is a non-archimedean norm {{iff}} $\norm {\,\cdot\,}_S$ is a non-archimedean norm. | === Necessary Condition ===
Let $\norm {\,\cdot\,}_R$ be a non-archimedean norm.
Then for all $x,y \in R$:
{{begin-eqn}}
{{eqn | l = \norm {x + y}_S
| r = \norm {\map \phi {\map {\phi^{-1} } x} + \map \phi {\map {\phi^{-1} } y} }_S
| c = $\phi$ is a bijection
}}
{{eqn | r = \norm {\map {\phi^{-1} } x + \ma... | Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be [[Definition:Normed Division Ring|normed division rings]].
Let $\phi:R \to S$ be an [[Definition:Isometric Isomorphism on Normed Division Ring|isometric isomorphism]].
Then:
:$\norm {\,\cdot\,}_R$ is a [[Definition:Non-Archimedean Divis... | === Necessary Condition ===
Let $\norm {\,\cdot\,}_R$ be a [[Definition:Non-Archimedean Division Ring Norm|non-archimedean norm]].
Then for all $x,y \in R$:
{{begin-eqn}}
{{eqn | l = \norm {x + y}_S
| r = \norm {\map \phi {\map {\phi^{-1} } x} + \map \phi {\map {\phi^{-1} } y} }_S
| c = $\phi$ is a [[Defi... | Isometrically Isomorphic Non-Archimedean Division Rings | https://proofwiki.org/wiki/Isometrically_Isomorphic_Non-Archimedean_Division_Rings | https://proofwiki.org/wiki/Isometrically_Isomorphic_Non-Archimedean_Division_Rings | [
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Isometric Isomorphism/Normed Division Ring",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Bijection",
"Definition:Isometry (Metric Spaces)",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Isometry (Metric Spaces)",
"Definition:Bijection",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archime... |
proofwiki-15130 | Non-Archimedean Division Ring iff Non-Archimedean Completion | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$
Then:
:$\norm {\, \cdot \,}$ is non-archimedean {{iff}} $\norm {\, \cdot \,}'$ is non-archimedean. | By the definition of a normed division ring completion then:
:$(1): \quad$ there exists a distance-preserving ring monomorphism $\phi: R \to R'$.
:$(2): \quad \struct {R', \norm {\, \cdot \,}' }$ is a complete metric space.
:$(3): \quad \phi \sqbrk R$ is a dense subspace in $\struct {R', \norm {\, \cdot \,}' }$.
By Nor... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {R', \norm {\, \cdot \,}' }$ be a [[Definition:Completion (Normed Division Ring)|normed division ring completion]] of $\struct {R, \norm {\, \cdot \,} }$
Then:
:$\norm {\, \cdot \,}$ is [[Definition:Non... | By the definition of a [[Definition:Completion (Normed Division Ring)|normed division ring completion]] then:
:$(1): \quad$ there exists a [[Definition:Distance-Preserving Mapping|distance-preserving]] [[Definition:Ring Monomorphism|ring monomorphism]] $\phi: R \to R'$.
:$(2): \quad \struct {R', \norm {\, \cdot \,}' }$... | Non-Archimedean Division Ring iff Non-Archimedean Completion | https://proofwiki.org/wiki/Non-Archimedean_Division_Ring_iff_Non-Archimedean_Completion | https://proofwiki.org/wiki/Non-Archimedean_Division_Ring_iff_Non-Archimedean_Completion | [
"Normed Division Rings",
"Non-Archimedean Norms"
] | [
"Definition:Normed Division Ring",
"Definition:Completion (Normed Division Ring)",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Definition:Completion (Normed Division Ring)",
"Definition:Distance-Preserving Mapping",
"Definition:Ring Monomorphism",
"Definition:Complete Metric Space",
"Definition:Everywhere Dense",
"Definition:Topological Subspace",
"Normed Division Ring is Dense Subring of Completion",
"Definition:Dense",
"... |
proofwiki-15131 | Domain of Integer Square Function | The domain of the integer square function is the entire set of integers $\Z$. | The operation of integer multiplication is defined on all integers.
Thus:
:$\forall x \in \Z: \exists y \in \Z: x^2 = y$
Hence the result by definition of domain.
{{qed}}
Category:Square Function
thaqmabzebfpu0jntuszq9f2o226uxb | The [[Definition:Domain of Mapping|domain]] of the [[Definition:Integer Square Function|integer square function]] is the entire [[Definition:Integer|set of integers]] $\Z$. | The operation of [[Definition:Integer Multiplication|integer multiplication]] is defined on all [[Definition:Integer|integers]].
Thus:
:$\forall x \in \Z: \exists y \in \Z: x^2 = y$
Hence the result by definition of [[Definition:Domain of Mapping|domain]].
{{qed}}
[[Category:Square Function]]
thaqmabzebfpu0jntuszq9f... | Domain of Integer Square Function | https://proofwiki.org/wiki/Domain_of_Integer_Square_Function | https://proofwiki.org/wiki/Domain_of_Integer_Square_Function | [
"Square Function"
] | [
"Definition:Domain (Set Theory)/Mapping",
"Definition:Square Function/Integer",
"Definition:Integer"
] | [
"Definition:Multiplication/Integers",
"Definition:Integer",
"Definition:Domain (Set Theory)/Mapping",
"Category:Square Function"
] |
proofwiki-15132 | Image of Integer Square Function | The image of the integer square function is the set of square numbers. | Follows directly by definition of the integer square function.
{{qed}}
Category:Square Function
crigqa9nrja5iymme5m6dh111j0ao06 | The [[Definition:Image of Mapping|image]] of the [[Definition:Integer Square Function|integer square function]] is the [[Definition:Set|set]] of [[Definition:Square Number|square numbers]]. | Follows directly by definition of the [[Definition:Integer Square Function|integer square function]].
{{qed}}
[[Category:Square Function]]
crigqa9nrja5iymme5m6dh111j0ao06 | Image of Integer Square Function | https://proofwiki.org/wiki/Image_of_Integer_Square_Function | https://proofwiki.org/wiki/Image_of_Integer_Square_Function | [
"Square Function"
] | [
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Square Function/Integer",
"Definition:Set",
"Definition:Square Number"
] | [
"Definition:Square Function/Integer",
"Category:Square Function"
] |
proofwiki-15133 | Restriction of Real Square Mapping to Positive Reals is Bijection | Let $f: \R \to \R$ be the real square function:
:$\forall x \in \R: \map f x = x^2$
Let $g: \R_{\ge 0} \to R_{\ge 0} := f {\restriction_{\R_{\ge 0} \times R_{\ge 0} } }$ be the restriction of $f$ to the positive real numbers $\R_{\ge 0}$.
Then $g$ is a bijective restriction of $f$. | From Order is Preserved on Positive Reals by Squaring, $f$ is strictly increasing on $\R_{\ge 0}$.
By definition, a strictly increasing real function is strictly monotone.
The result follows from Strictly Monotone Real Function is Bijective.
{{qed}} | Let $f: \R \to \R$ be the [[Definition:Real Square Function|real square function]]:
:$\forall x \in \R: \map f x = x^2$
Let $g: \R_{\ge 0} \to R_{\ge 0} := f {\restriction_{\R_{\ge 0} \times R_{\ge 0} } }$ be the [[Definition:Restriction of Mapping|restriction]] of $f$ to the [[Definition:Positive Real Number|positive... | From [[Order is Preserved on Positive Reals by Squaring]], $f$ is [[Definition:Strictly Increasing Real Function|strictly increasing]] on $\R_{\ge 0}$.
By definition, a [[Definition:Strictly Increasing Real Function|strictly increasing real function]] is [[Definition:Strictly Monotone Real Function|strictly monotone]]... | Restriction of Real Square Mapping to Positive Reals is Bijection | https://proofwiki.org/wiki/Restriction_of_Real_Square_Mapping_to_Positive_Reals_is_Bijection | https://proofwiki.org/wiki/Restriction_of_Real_Square_Mapping_to_Positive_Reals_is_Bijection | [
"Square Function",
"Examples of Bijections"
] | [
"Definition:Square Function/Real",
"Definition:Restriction/Mapping",
"Definition:Positive/Real Number",
"Definition:Bijective Restriction"
] | [
"Order is Preserved on Positive Reals by Squaring",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Monotone/Real Function",
"Strictly Monotone Real Function is Bijective"
] |
proofwiki-15134 | Inverse of Real Square Function on Positive Reals | Let $f: \R_{\ge 0} \to R_{\ge 0}$ be the restriction of the real square function to the positive real numbers $\R_{\ge 0}$.
The inverse of $f$ is $f^{-1}: \R_{\ge 0} \times R_{\ge 0}$ defined as:
:$\forall x \in \R_{\ge 0}: \map {f^{-1} } x = \sqrt x$
where $\sqrt x$ is the positive square root of $x$. | From Restriction of Real Square Mapping to Positive Reals is Bijection, $f$ is a bijection.
By definition of the positive square root:
:$y = \sqrt x \iff x = y^2$
for $x, y \in \R_{\ge 0}$.
Hence the result.
{{qed}} | Let $f: \R_{\ge 0} \to R_{\ge 0}$ be the [[Definition:Restriction of Mapping|restriction]] of the [[Definition:Real Square Function|real square function]] to the [[Definition:Positive Real Number|positive real numbers]] $\R_{\ge 0}$.
The [[Definition:Inverse Mapping|inverse]] of $f$ is $f^{-1}: \R_{\ge 0} \times R_{\g... | From [[Restriction of Real Square Mapping to Positive Reals is Bijection]], $f$ is a [[Definition:Bijection|bijection]].
By definition of the [[Definition:Positive Square Root|positive square root]]:
:$y = \sqrt x \iff x = y^2$
for $x, y \in \R_{\ge 0}$.
Hence the result.
{{qed}} | Inverse of Real Square Function on Positive Reals | https://proofwiki.org/wiki/Inverse_of_Real_Square_Function_on_Positive_Reals | https://proofwiki.org/wiki/Inverse_of_Real_Square_Function_on_Positive_Reals | [
"Square Function"
] | [
"Definition:Restriction/Mapping",
"Definition:Square Function/Real",
"Definition:Positive/Real Number",
"Definition:Inverse Mapping",
"Definition:Square Root/Positive"
] | [
"Restriction of Real Square Mapping to Positive Reals is Bijection",
"Definition:Bijection",
"Definition:Square Root/Positive"
] |
proofwiki-15135 | Real Square Function is not Bijective | Let $f: \R \to \R$ be the real square function:
:$\forall x \in \R: \map f x = x^2$
Then $f$ is not a bijection. | From Real Square Function is not Injective, $f$ is not an injection.
From Real Square Function is not Surjective, $f$ is not a surjection.
The result follows by definition of bijection.
{{qed}} | Let $f: \R \to \R$ be the [[Definition:Real Square Function|real square function]]:
:$\forall x \in \R: \map f x = x^2$
Then $f$ is not a [[Definition:Bijection|bijection]]. | From [[Real Square Function is not Injective]], $f$ is not an [[Definition:Injection|injection]].
From [[Real Square Function is not Surjective]], $f$ is not a [[Definition:Surjection|surjection]].
The result follows by definition of [[Definition:Bijection|bijection]].
{{qed}} | Real Square Function is not Bijective | https://proofwiki.org/wiki/Real_Square_Function_is_not_Bijective | https://proofwiki.org/wiki/Real_Square_Function_is_not_Bijective | [
"Square Function"
] | [
"Definition:Square Function/Real",
"Definition:Bijection"
] | [
"Real Square Function is not Injective",
"Definition:Injection",
"Real Square Function is not Surjective",
"Definition:Surjection",
"Definition:Bijection"
] |
proofwiki-15136 | Inverse of Linear Function on Real Numbers | Let $a, b \in \R$ be real numbers such that $a \ne 0$.
Let $f: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map f x = a x + b$
Then the inverse of $f$ is given by:
:$\forall y \in \R: \map {f^{-1} } y = \dfrac {y - b} a$ | We have that Linear Function on Real Numbers is Bijection.
Let $y = \map f x$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = \map f x
| c =
}}
{{eqn | r = a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \dfrac {y - b} a
| c =
}}
{{end-eqn}}
and so:
:$\forall y \in \R: \map {f^{-1} }... | Let $a, b \in \R$ be [[Definition:Real Number|real numbers]] such that $a \ne 0$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\forall x \in \R: \map f x = a x + b$
Then the [[Definition:Inverse Mapping|inverse]] of $f$ is given by:
:$\forall y \in \R: \map {f^{-1} } y = \dfrac ... | We have that [[Linear Function on Real Numbers is Bijection]].
Let $y = \map f x$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = \map f x
| c =
}}
{{eqn | r = a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \dfrac {y - b} a
| c =
}}
{{end-eqn}}
and so:
:$\forall y \in \R: \map {f... | Inverse of Linear Function on Real Numbers | https://proofwiki.org/wiki/Inverse_of_Linear_Function_on_Real_Numbers | https://proofwiki.org/wiki/Inverse_of_Linear_Function_on_Real_Numbers | [
"Linear Algebra"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Inverse Mapping"
] | [
"Linear Function on Real Numbers is Bijection"
] |
proofwiki-15137 | Linear Function on Real Numbers is Bijection | Let $a, b \in \R$ be real numbers.
Let $f: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map f x = a x + b$
Then $f$ is a bijection {{iff}} $a \ne 0$. | Let $a \ne 0$.
Let $y = \map f x$.
{{begin-eqn}}
{{eqn | l = y
| r = \map f x
| c =
}}
{{eqn | r = a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \dfrac {y - b} a
| c =
}}
{{end-eqn}}
and so:
:$\forall y \in \R: \exists x \in \R; y = \map f x$
demonstrating that $f$ is surjec... | Let $a, b \in \R$ be [[Definition:Real Number|real numbers]].
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\forall x \in \R: \map f x = a x + b$
Then $f$ is a [[Definition:Bijection|bijection]] {{iff}} $a \ne 0$. | Let $a \ne 0$.
Let $y = \map f x$.
{{begin-eqn}}
{{eqn | l = y
| r = \map f x
| c =
}}
{{eqn | r = a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \dfrac {y - b} a
| c =
}}
{{end-eqn}}
and so:
:$\forall y \in \R: \exists x \in \R; y = \map f x$
demonstrating that $f$ is [[... | Linear Function on Real Numbers is Bijection | https://proofwiki.org/wiki/Linear_Function_on_Real_Numbers_is_Bijection | https://proofwiki.org/wiki/Linear_Function_on_Real_Numbers_is_Bijection | [
"Linear Algebra"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Bijection"
] | [
"Definition:Surjection",
"Definition:Injection",
"Definition:Bijection",
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection"
] |
proofwiki-15138 | Composition of Linear Real Functions | Let $a, b, c, d \in \R$ be real numbers.
Let $\theta_{a, b}: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$
Let $\theta_{c, d} \circ \theta_{a, b}$ denote the composition of $\theta_{c, d}$ with $\theta_{a, b}$.
Then:
:$\theta_{c, d} \circ \theta_{a, b} = \theta_{a c... | {{begin-eqn}}
{{eqn | l = \map {\paren {\theta_{c, d} \circ \theta_{a, b} } } x
| r = \map {\theta_{c, d} } {\map {\theta_{a, b} } x}
| c =
}}
{{eqn | r = \map {\theta_{c, d} } {a x + b}
| c =
}}
{{eqn | r = c \paren {a x + b} + d
| c =
}}
{{eqn | r = \paren {a c} x + \paren {b c + d}
|... | Let $a, b, c, d \in \R$ be [[Definition:Real Number|real numbers]].
Let $\theta_{a, b}: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$
Let $\theta_{c, d} \circ \theta_{a, b}$ denote the [[Definition:Composition of Mappings|composition]]... | {{begin-eqn}}
{{eqn | l = \map {\paren {\theta_{c, d} \circ \theta_{a, b} } } x
| r = \map {\theta_{c, d} } {\map {\theta_{a, b} } x}
| c =
}}
{{eqn | r = \map {\theta_{c, d} } {a x + b}
| c =
}}
{{eqn | r = c \paren {a x + b} + d
| c =
}}
{{eqn | r = \paren {a c} x + \paren {b c + d}
|... | Composition of Linear Real Functions | https://proofwiki.org/wiki/Composition_of_Linear_Real_Functions | https://proofwiki.org/wiki/Composition_of_Linear_Real_Functions | [
"Linear Algebra"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Composition of Mappings"
] | [] |
proofwiki-15139 | Condition for Composition of Linear Real Functions to be Commutative | Let $a, b, c, d \in \R$ be real numbers.
Let $\theta_{a, b}: \R \to \R$ be the real function defined as:
:$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$
Let $\theta_{c, d} \circ \theta_{a, b}$ denote the composition of $\theta_{c, d}$ with $\theta_{a, b}$.
Then:
:$\theta_{c, d} \circ \theta_{a, b} = \theta_{a, ... | {{begin-eqn}}
{{eqn | l = \map {\theta_{c, d} \circ \theta_{a, b} } x
| r = \map {\theta_{a, b} \circ \theta_{c, d} } x
| c =
}}
{{eqn | ll= \leadsto
| l = \theta_{a c, b c + d}
| r = \theta_{c a, a d + b}
| c =
}}
{{eqn | ll= \leadsto
| l = b c + d
| r = a d + b
| c = ... | Let $a, b, c, d \in \R$ be [[Definition:Real Number|real numbers]].
Let $\theta_{a, b}: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$
Let $\theta_{c, d} \circ \theta_{a, b}$ denote the [[Definition:Composition of Mappings|composition]]... | {{begin-eqn}}
{{eqn | l = \map {\theta_{c, d} \circ \theta_{a, b} } x
| r = \map {\theta_{a, b} \circ \theta_{c, d} } x
| c =
}}
{{eqn | ll= \leadsto
| l = \theta_{a c, b c + d}
| r = \theta_{c a, a d + b}
| c =
}}
{{eqn | ll= \leadsto
| l = b c + d
| r = a d + b
| c = ... | Condition for Composition of Linear Real Functions to be Commutative | https://proofwiki.org/wiki/Condition_for_Composition_of_Linear_Real_Functions_to_be_Commutative | https://proofwiki.org/wiki/Condition_for_Composition_of_Linear_Real_Functions_to_be_Commutative | [
"Linear Algebra"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Composition of Mappings"
] | [] |
proofwiki-15140 | Composition of Right Inverse with Mapping is Idempotent | Let $f: S \to T$ be a mapping.
Let $g: T \to S$ be a right inverse mapping of $f$.
Then:
:$\paren {g \circ f} \circ \paren {g \circ f} = g \circ f$ | {{begin-eqn}}
{{eqn | l = \paren {g \circ f} \circ \paren {g \circ f}
| r = g \circ \paren {f \circ g} \circ f
| c = Composition of Mappings is Associative
}}
{{eqn | r = g \circ I_T \circ f
| c = {{Defof|Right Inverse Mapping}}
}}
{{eqn | r = g \circ f
| c = {{Defof|Identity Mapping}}
}}
{{end-... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $g: T \to S$ be a [[Definition:Right Inverse Mapping|right inverse mapping]] of $f$.
Then:
:$\paren {g \circ f} \circ \paren {g \circ f} = g \circ f$ | {{begin-eqn}}
{{eqn | l = \paren {g \circ f} \circ \paren {g \circ f}
| r = g \circ \paren {f \circ g} \circ f
| c = [[Composition of Mappings is Associative]]
}}
{{eqn | r = g \circ I_T \circ f
| c = {{Defof|Right Inverse Mapping}}
}}
{{eqn | r = g \circ f
| c = {{Defof|Identity Mapping}}
}}
{{... | Composition of Right Inverse with Mapping is Idempotent | https://proofwiki.org/wiki/Composition_of_Right_Inverse_with_Mapping_is_Idempotent | https://proofwiki.org/wiki/Composition_of_Right_Inverse_with_Mapping_is_Idempotent | [
"Composite Mappings"
] | [
"Definition:Mapping",
"Definition:Right Inverse Mapping"
] | [
"Composition of Mappings is Associative"
] |
proofwiki-15141 | Set of Even Integers is Equivalent to Set of Integers | Let $\Z$ denote the set of integers.
Let $2 \Z$ denote the set of even integers.
Then:
:$2 \Z \sim \Z$
where $\sim$ denotes set equivalence. | To demonstrate set equivalence, it is sufficient to construct a bijection between the two sets.
Let $f: \Z \to 2 \Z$ defined as:
:$\forall x \in \Z: \map f x = 2 x$
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
| c =
}}
{{eqn | ll= \leadsto
| l = 2 x
| r = 2 y
| c =
}}
{{eqn | ll= \l... | Let $\Z$ denote the [[Definition:Set|set]] of [[Definition:Integer|integers]].
Let $2 \Z$ denote the [[Definition:Set|set]] of [[Definition:Even Integer|even integers]].
Then:
:$2 \Z \sim \Z$
where $\sim$ denotes [[Definition:Set Equivalence|set equivalence]]. | To demonstrate [[Definition:Set Equivalence|set equivalence]], it is sufficient to construct a [[Definition:Bijection|bijection]] between the two [[Definition:Set|sets]].
Let $f: \Z \to 2 \Z$ defined as:
:$\forall x \in \Z: \map f x = 2 x$
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
| c =
}}
{{eqn... | Set of Even Integers is Equivalent to Set of Integers | https://proofwiki.org/wiki/Set_of_Even_Integers_is_Equivalent_to_Set_of_Integers | https://proofwiki.org/wiki/Set_of_Even_Integers_is_Equivalent_to_Set_of_Integers | [
"Set Equivalence",
"Integers"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Set",
"Definition:Even Integer",
"Definition:Set Equivalence"
] | [
"Definition:Set Equivalence",
"Definition:Bijection",
"Definition:Set",
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection"
] |
proofwiki-15142 | Sets of Permutations of Equivalent Sets are Equivalent | Let $A$ and $B$ be sets such that:
:$A \sim B$
where $\sim$ denotes set equivalence.
Let $\map \Gamma A$ denote the set of permutations on $A$.
Then:
:$\map \Gamma A \sim \map \Gamma B$ | By definition of set equivalence, let $f: A \to B$ be a bijection.
Define $\Phi : \map \Gamma A \to \map \Gamma B$ by:
:$\map \Phi \gamma := f \circ \gamma \circ f^{-1}$
By definition of permutation, each $\gamma \in \map \Gamma A$ is a bijection.
By Composite of Bijections is Bijection, each $f \circ \gamma$ is a bije... | Let $A$ and $B$ be [[Definition:Set|sets]] such that:
:$A \sim B$
where $\sim$ denotes [[Definition:Set Equivalence|set equivalence]].
Let $\map \Gamma A$ denote the [[Definition:Set|set]] of [[Definition:Permutation|permutations]] on $A$.
Then:
:$\map \Gamma A \sim \map \Gamma B$ | By definition of [[Definition:Set Equivalence|set equivalence]], let $f: A \to B$ be a [[Definition:Bijection|bijection]].
Define $\Phi : \map \Gamma A \to \map \Gamma B$ by:
:$\map \Phi \gamma := f \circ \gamma \circ f^{-1}$
By definition of [[Definition:Permutation|permutation]], each $\gamma \in \map \Gamma A$ is ... | Sets of Permutations of Equivalent Sets are Equivalent | https://proofwiki.org/wiki/Sets_of_Permutations_of_Equivalent_Sets_are_Equivalent | https://proofwiki.org/wiki/Sets_of_Permutations_of_Equivalent_Sets_are_Equivalent | [
"Permutations",
"Set Equivalence"
] | [
"Definition:Set",
"Definition:Set Equivalence",
"Definition:Set",
"Definition:Permutation"
] | [
"Definition:Set Equivalence",
"Definition:Bijection",
"Definition:Permutation",
"Definition:Bijection",
"Composite of Bijections is Bijection",
"Definition:Bijection",
"Inverse of Bijection is Bijection",
"Definition:Inverse Mapping",
"Definition:Bijection",
"Composite of Bijections is Bijection",... |
proofwiki-15143 | Composition of Permutations is not Commutative | Let $S$ be a set.
Let $\map \Gamma S$ denote the set of permutations on $S$.
Let $\pi, \rho$ be elements of $\map \Gamma S$
Then it is not necessarily the case that:
:$\pi \circ \rho = \rho \circ \pi$
where $\circ$ denotes composition. | Proof by Counterexample:
Let $S := \set {1, 2, 3}$.
Let:
{{begin-eqn}}
{{eqn | l = \pi
| o = :=
| r = \dbinom {1 \ 2 \ 3} {2 \ 3 \ 1}
}}
{{eqn | l = \rho
| o = :=
| r = \dbinom {1 \ 2 \ 3} {1 \ 3 \ 2}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \pi \circ \rho
| r = \dbinom {1 \ 2 \ 3} {... | Let $S$ be a [[Definition:Set|set]].
Let $\map \Gamma S$ denote the [[Definition:Set|set]] of [[Definition:Permutation|permutations]] on $S$.
Let $\pi, \rho$ be [[Definition:Element|elements]] of $\map \Gamma S$
Then it is not necessarily the case that:
:$\pi \circ \rho = \rho \circ \pi$
where $\circ$ denotes [[Defi... | [[Proof by Counterexample]]:
Let $S := \set {1, 2, 3}$.
Let:
{{begin-eqn}}
{{eqn | l = \pi
| o = :=
| r = \dbinom {1 \ 2 \ 3} {2 \ 3 \ 1}
}}
{{eqn | l = \rho
| o = :=
| r = \dbinom {1 \ 2 \ 3} {1 \ 3 \ 2}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \pi \circ \rho
| r = \dbinom {1 ... | Composition of Permutations is not Commutative | https://proofwiki.org/wiki/Composition_of_Permutations_is_not_Commutative | https://proofwiki.org/wiki/Composition_of_Permutations_is_not_Commutative | [
"Permutations",
"Composite Mappings",
"Commutativity"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Permutation",
"Definition:Element",
"Definition:Composition of Mappings"
] | [
"Proof by Counterexample"
] |
proofwiki-15144 | Three Points in Ultrametric Space have Two Equal Distances/Corollary 2 | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,
Let $x, y \in R$ and $\norm x \ne \norm y$.
Then:
:$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y}$ | Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
By Non-Archimedean Norm iff Non-Archimedean Metric then $d$ is a non-Archimedean metric and $\struct {R, d}$ is an ultrametric space.
Let $x, y \in R$ and $\norm x \ne \norm y$.
By the definition of the non-Archimedean metric $d$ then:
:$\norm x = \norm {x ... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm{\,\cdot\,}$,
Let $x, y \in R$ and $\norm x \ne \norm y$.
Then:
:$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \n... | Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by the norm]] $\norm {\,\cdot\,}$.
By [[Non-Archimedean Norm iff Non-Archimedean Metric]] then $d$ is a [[Definition:Non-Archimedean Metric|non-Archimedean metric]] and $\struct {R, d}$ is an [[Definition:Ultrametric Space|ultrametric s... | Three Points in Ultrametric Space have Two Equal Distances/Corollary 2 | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary_2 | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary_2 | [
"Three Points in Ultrametric Space have Two Equal Distances"
] | [
"Definition:Normed Division Ring",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Non-Archimedean Norm iff Non-Archimedean Metric",
"Definition:Non-Archimedean/Metric",
"Definition:Ultrametric Space",
"Definition:Non-Archimedean/Metric",
"Three Points in Ultrametric Space have Two Equal Distances",
"Properties of Norm on Division... |
proofwiki-15145 | Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls | :$y \in \map {B_r} x \implies \map {B_r} y = \map {B_r} x$ | Let $y \in \map {B_r} x$.
Let $a \in \map {B_r} y$.
By the definition of an open ball, then:
:$\norm {a - y} < r$
:$\norm {y - x} < r$
Hence:
{{begin-eqn}}
{{eqn | l = \norm {a - x}
| r = \norm {a - y + y - x}
}}
{{eqn | o = \le
| r = \max \set {\norm {a - y}, \norm{y - x} }
| c = {{Defof|Non-Archime... | :$y \in \map {B_r} x \implies \map {B_r} y = \map {B_r} x$ | Let $y \in \map {B_r} x$.
Let $a \in \map {B_r} y$.
By the definition of an [[Definition:Open Ball of Normed Division Ring|open ball]], then:
:$\norm {a - y} < r$
:$\norm {y - x} < r$
Hence:
{{begin-eqn}}
{{eqn | l = \norm {a - x}
| r = \norm {a - y + y - x}
}}
{{eqn | o = \le
| r = \max \set {\norm {a ... | Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Centers_of_Open_Balls | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Centers_of_Open_Balls | [
"Topological Properties of Non-Archimedean Division Rings"
] | [] | [
"Definition:Open Ball/Normed Division Ring",
"Definition:Open Ball/Normed Division Ring",
"Properties of Norm on Division Ring/Norm of Negative",
"Definition:Open Ball/Normed Division Ring",
"Definition:Set Equality/Definition 2"
] |
proofwiki-15146 | Topological Properties of Non-Archimedean Division Rings/Centers of Closed Balls | :$y \in \map { {B_r}^-} x \implies \map { {B_r}^-} y = \map { {B_r}^-} x$ | Let $y \in \map { {B_r}^-} x$.
Let $a \in \map { {B_r}^-} y$.
By the definition of an closed ball, then:
:$\norm {a - y} \le r$
:$\norm {y - x} \le r$
Hence:
{{begin-eqn}}
{{eqn | l = \norm {a - x}
| r = \norm {a - y + y - x}
}}
{{eqn | r = \max \set {\norm {a - y}, \norm {y - x} }
| o = \le
| c = {{... | :$y \in \map { {B_r}^-} x \implies \map { {B_r}^-} y = \map { {B_r}^-} x$ | Let $y \in \map { {B_r}^-} x$.
Let $a \in \map { {B_r}^-} y$.
By the definition of an [[Definition:Closed Ball of Normed Division Ring|closed ball]], then:
:$\norm {a - y} \le r$
:$\norm {y - x} \le r$
Hence:
{{begin-eqn}}
{{eqn | l = \norm {a - x}
| r = \norm {a - y + y - x}
}}
{{eqn | r = \max \set {\norm {... | Topological Properties of Non-Archimedean Division Rings/Centers of Closed Balls | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Centers_of_Closed_Balls | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Centers_of_Closed_Balls | [
"Topological Properties of Non-Archimedean Division Rings"
] | [] | [
"Definition:Closed Ball/Normed Division Ring",
"Definition:Closed Ball/Normed Division Ring",
"Properties of Norm on Division Ring/Norm of Negative",
"Definition:Closed Ball/Normed Division Ring",
"Definition:Set Equality/Definition 2"
] |
proofwiki-15147 | Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen | :The open $r$-ball of $x$, $\map {B_r} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$. | Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
By the definition of an open ball in $\norm {\,\cdot\,}$:
:$\map {B_r} x$ is an open ball in the metric space $\struct {R, d}$.
By Open Ball of Metric Space is Open Set then $\map {B_r} x$ is open in $\struct {R, d}$.
So it remains to show that $\map {B_r} ... | :The [[Definition:Open Ball of Normed Division Ring|open $r$-ball of $x$]], $\map {B_r} x$, is both [[Definition:Open Set of Metric Space|open]] and [[Definition:Closed Set of Metric Space|closed]] in the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] by $\norm {\,\cdot\,}$. | Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by the norm]] $\norm {\,\cdot\,}$.
By the definition of an [[Definition:Open Ball of Normed Division Ring|open ball]] in $\norm {\,\cdot\,}$:
:$\map {B_r} x$ is an [[Definition:Open Ball|open ball]] in the [[Definition:Metric Space|metr... | Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Open_Balls_are_Clopen | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Open_Balls_are_Clopen | [
"Topological Properties of Non-Archimedean Division Rings"
] | [
"Definition:Open Ball/Normed Division Ring",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Metric Induced by Norm on Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Open Ball/Normed Division Ring",
"Definition:Open Ball",
"Definition:Metric Space",
"Open Ball is Open Set/Pseudometric Space",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Closure (Topology)/Met... |
proofwiki-15148 | Topological Properties of Non-Archimedean Division Rings/Closed Balls are Clopen | :The closed $r$-ball of $x$, $\map { {B_r}^-} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$. | Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$.
By the definition of a closed ball in $\norm {\,\cdot\,}$ then:
:$\map { {B_r}^-} x$ is a closed ball in the metric space $\struct {R, d}$.
By Closed Ball is Closed in Metric Space then $\map { {B_r}^-} c$ is closed in $d$.
So it remains to show that $\map ... | :The [[Definition:Closed Ball of Normed Division Ring|closed $r$-ball of $x$]], $\map { {B_r}^-} x$, is both [[Definition:Open Set of Metric Space|open]] and [[Definition:Closed Set of Metric Space|closed]] in the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] by $\norm {\,\cdot\,}$. | Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced by the norm]] $\norm {\,\cdot\,}$.
By the definition of a [[Definition:Closed Ball of Normed Division Ring|closed ball]] in $\norm {\,\cdot\,}$ then:
:$\map { {B_r}^-} x$ is a [[Definition:Closed Ball|closed ball]] in the [[Definition:M... | Topological Properties of Non-Archimedean Division Rings/Closed Balls are Clopen | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Closed_Balls_are_Clopen | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Closed_Balls_are_Clopen | [
"Topological Properties of Non-Archimedean Division Rings"
] | [
"Definition:Closed Ball/Normed Division Ring",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Metric Induced by Norm on Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Closed Ball/Normed Division Ring",
"Definition:Closed Ball",
"Definition:Metric Space",
"Closed Ball is Closed/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Open Set/Metric Space",
"Topological Properties of Non-Archi... |
proofwiki-15149 | Topological Properties of Non-Archimedean Division Rings/Intersection of Open Balls | :$\map {B_r} x \cap \map {B_s} y \ne \O \iff \map {B_r} x \subseteq \map {B_s} y$ or $\map {B_s} y \subseteq \map {B_r} x$ | === Necessary Condition ===
Let $z \in \map {B_r} x \cap \map {B_s} y$.
If $r \le s$ then:
{{begin-eqn}}
{{eqn| l = \map {B_r} x
| r = \map {B_r} z
| c = Every element in an open ball is the center
}}
{{eqn| o = \subseteq
| r = \map {B_s} z
| c = as $r \le s$
}}
{{eqn| r = \map {B_s} y
| c = E... | :$\map {B_r} x \cap \map {B_s} y \ne \O \iff \map {B_r} x \subseteq \map {B_s} y$ or $\map {B_s} y \subseteq \map {B_r} x$ | === Necessary Condition ===
Let $z \in \map {B_r} x \cap \map {B_s} y$.
If $r \le s$ then:
{{begin-eqn}}
{{eqn| l = \map {B_r} x
| r = \map {B_r} z
| c = [[Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls|Every element in an open ball is the center]]
}}
{{eqn| o = \subseteq
... | Topological Properties of Non-Archimedean Division Rings/Intersection of Open Balls | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Intersection_of_Open_Balls | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Intersection_of_Open_Balls | [
"Topological Properties of Non-Archimedean Division Rings"
] | [] | [
"Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls",
"Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls"
] |
proofwiki-15150 | Topological Properties of Non-Archimedean Division Rings/Intersection of Closed Balls | :$\map { {B_r}^-} x \cap \map { {B_s}^-} y \ne \O \iff \map { {B_r}^-} x \subseteq \map { {B_s}^-} y$ or $\map { {B_s}^-} y \subseteq \map { {B_r}^-} x$ | === Necessary Condition ===
Let $z \in \map { {B_r}^-} x \cap \map { {B_s}^-} y$.
If $r \le s$ then:
{{begin-eqn}}
{{eqn| l = \map { {B_r}^-} x
| r = \map { {B_r}^-} z
| c = Every element in an open ball is the center
}}
{{eqn| o = \subseteq
| r = \map { {B_s}^-} z
| c = as $r \le s$
}}
{{eqn| r = ... | :$\map { {B_r}^-} x \cap \map { {B_s}^-} y \ne \O \iff \map { {B_r}^-} x \subseteq \map { {B_s}^-} y$ or $\map { {B_s}^-} y \subseteq \map { {B_r}^-} x$ | === Necessary Condition ===
Let $z \in \map { {B_r}^-} x \cap \map { {B_s}^-} y$.
If $r \le s$ then:
{{begin-eqn}}
{{eqn| l = \map { {B_r}^-} x
| r = \map { {B_r}^-} z
| c = [[Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls|Every element in an open ball is the center]]
}}
{{... | Topological Properties of Non-Archimedean Division Rings/Intersection of Closed Balls | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Intersection_of_Closed_Balls | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Intersection_of_Closed_Balls | [
"Topological Properties of Non-Archimedean Division Rings"
] | [] | [
"Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls",
"Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls"
] |
proofwiki-15151 | Mittag-Leffler Expansion for Hyperbolic Cotangent Function | :$\ds \pi \map \coth {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}$
where:
:$z \in \C$ is not an integer multiple of $i$
:$\coth$ is the hyperbolic cotangent function. | {{begin-eqn}}
{{eqn | l = \pi \map \coth {\pi z}
| r = \pi i \map \cot {\pi i z}
| c = Hyperbolic Cotangent in terms of Cotangent
}}
{{eqn | r = i \paren {\frac 1 {i z} + 2 i \sum_{n \mathop = 1}^\infty \frac z {\paren {i z}^2 - n^2} }
| c = Mittag-Leffler Expansion for Cotangent Function
}}
{{eqn | r = \frac 1 z - ... | :$\ds \pi \map \coth {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]] multiple of $i$
:$\coth$ is the [[Definition:Hyperbolic Cotangent|hyperbolic cotangent function]]. | {{begin-eqn}}
{{eqn | l = \pi \map \coth {\pi z}
| r = \pi i \map \cot {\pi i z}
| c = [[Hyperbolic Cotangent in terms of Cotangent]]
}}
{{eqn | r = i \paren {\frac 1 {i z} + 2 i \sum_{n \mathop = 1}^\infty \frac z {\paren {i z}^2 - n^2} }
| c = [[Mittag-Leffler Expansion for Cotangent Function]]
}}
{{eqn | r = \fra... | Mittag-Leffler Expansion for Hyperbolic Cotangent Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Hyperbolic_Cotangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Hyperbolic_Cotangent_Function | [
"Mittag-Leffler Expansions",
"Hyperbolic Cotangent Function"
] | [
"Definition:Integer",
"Definition:Hyperbolic Cotangent"
] | [
"Hyperbolic Cotangent in terms of Cotangent",
"Mittag-Leffler Expansion for Cotangent Function"
] |
proofwiki-15152 | Mittag-Leffler Expansion for Secant Function | :$\ds \pi \map \sec {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 z^2}$
where:
:$z \in \C$ is not a half-integer
:$\sec$ is the secant function. | {{begin-eqn}}
{{eqn | l = \pi \map \sec {\pi z}
| r = \pi \map \csc {\frac \pi 2 - \pi z}
| c = Secant and Cosecant are Cofunctions in radians
}}
{{eqn | r = \frac 1 {1/2 - z} + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {1/2 - z} {\paren {1/2 - z}^2 - n^2}
| c = Mittag-Leffler Expansion for Co... | :$\ds \pi \map \sec {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 z^2}$
where:
:$z \in \C$ is not a [[Definition:Half-Integer|half-integer]]
:$\sec$ is the [[Definition:Secant Function|secant function]]. | {{begin-eqn}}
{{eqn | l = \pi \map \sec {\pi z}
| r = \pi \map \csc {\frac \pi 2 - \pi z}
| c = [[Secant and Cosecant are Cofunctions]] in radians
}}
{{eqn | r = \frac 1 {1/2 - z} + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {1/2 - z} {\paren {1/2 - z}^2 - n^2}
| c = [[Mittag-Leffler Expansion ... | Mittag-Leffler Expansion for Secant Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Secant_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Secant_Function | [
"Mittag-Leffler Expansions",
"Secant Function"
] | [
"Definition:Half-Integer",
"Definition:Secant Function"
] | [
"Secant and Cosecant are Cofunctions",
"Mittag-Leffler Expansion for Cosecant Function",
"Definition:Partial Fractions Expansion",
"Difference of Two Squares"
] |
proofwiki-15153 | Pringsheim's Theorem | Let $f$ be a holomorphic function defined on a unit disc centered at the origin of the complex plane and is denoted by its Taylor series:
:$\map f z = \ds \sum_{n \mathop = 0}^{\infty} c_n z^n$
Let:
:$(1): \quad \forall n \ge 0: c_n \ge 0$
:$(2): \quad$ the radius of convergence of the Taylor series of function $f$ is ... | {{ProofWanted}}
{{Namedfor|Alfred Pringsheim|cat = Pringsheim}}
ljnkhn1tkbziuxpfgxvxwj0cle7jp03 | Let $f$ be a [[Definition:Holomorphic Function|holomorphic function]] defined on a unit disc centered at the origin of the complex plane and is denoted by its [[Definition:Taylor Series|Taylor series]]:
:$\map f z = \ds \sum_{n \mathop = 0}^{\infty} c_n z^n$
Let:
:$(1): \quad \forall n \ge 0: c_n \ge 0$
:$(2): \quad$ ... | {{ProofWanted}}
{{Namedfor|Alfred Pringsheim|cat = Pringsheim}}
ljnkhn1tkbziuxpfgxvxwj0cle7jp03 | Pringsheim's Theorem | https://proofwiki.org/wiki/Pringsheim's_Theorem | https://proofwiki.org/wiki/Pringsheim's_Theorem | [] | [
"Definition:Holomorphic Function",
"Definition:Taylor Series",
"Definition:Radius of Convergence/Complex Domain",
"Definition:Taylor Series",
"Definition:Isolated Singularity"
] | [] |
proofwiki-15154 | Mittag-Leffler Expansion for Tangent Function | :$\ds \pi \map \tan {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {\paren {2 n + 1}^2 - 4 z^2}$
where:
:$z \in \C$ is not a half-integer
:$\tan$ is the tangent function. | From {{Corollary|Mittag-Leffler Expansion for Tangent Function}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi {2 n} \map \tan {\frac {\pi m} {2 n} }
| r = \sum_{k \mathop = 0}^\infty \paren {\frac 1 {\paren {2 k + 1} n - m} - \frac 1 {\paren {2 k + 1} n + m} }
| c =
}}
{{eqn | r = \paren {\frac 1 {n - m} ... | :$\ds \pi \map \tan {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {\paren {2 n + 1}^2 - 4 z^2}$
where:
:$z \in \C$ is not a [[Definition:Half-Integer|half-integer]]
:$\tan$ is the [[Definition:Tangent Function|tangent function]]. | From {{Corollary|Mittag-Leffler Expansion for Tangent Function}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi {2 n} \map \tan {\frac {\pi m} {2 n} }
| r = \sum_{k \mathop = 0}^\infty \paren {\frac 1 {\paren {2 k + 1} n - m} - \frac 1 {\paren {2 k + 1} n + m} }
| c =
}}
{{eqn | r = \paren {\frac 1 {n - m}... | Leibniz's Formula for Pi/Proof by Mittag-Leffler Expansion for Tangent Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Tangent_Function | https://proofwiki.org/wiki/Leibniz's_Formula_for_Pi/Proof_by_Mittag-Leffler_Expansion_for_Tangent_Function | [
"Mittag-Leffler Expansion for Tangent Function",
"Tangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Half-Integer",
"Definition:Tangent Function"
] | [] |
proofwiki-15155 | Mittag-Leffler Expansion for Tangent Function | :$\ds \pi \map \tan {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {\paren {2 n + 1}^2 - 4 z^2}$
where:
:$z \in \C$ is not a half-integer
:$\tan$ is the tangent function. | {{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}} | :$\ds \pi \map \tan {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {\paren {2 n + 1}^2 - 4 z^2}$
where:
:$z \in \C$ is not a [[Definition:Half-Integer|half-integer]]
:$\tan$ is the [[Definition:Tangent Function|tangent function]]. | {{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}} | Mittag-Leffler Expansion for Tangent Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Tangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Tangent_Function | [
"Mittag-Leffler Expansion for Tangent Function",
"Tangent Function",
"Mittag-Leffler Expansions"
] | [
"Definition:Half-Integer",
"Definition:Tangent Function"
] | [] |
proofwiki-15156 | Mittag-Leffler Expansion for Hyperbolic Tangent Function | :$\ds \pi \map \tanh {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {4 z^2 + \paren {2 n + 1}^2}$
where:
:$z \in \C$ is not a half-integer multiple of $i$
:$\tanh$ is the hyperbolic tangent function. | {{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}} | :$\ds \pi \map \tanh {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {4 z^2 + \paren {2 n + 1}^2}$
where:
:$z \in \C$ is not a [[Definition:Half-Integer|half-integer]] multiple of $i$
:$\tanh$ is the [[Definition:Hyperbolic Tangent|hyperbolic tangent function]]. | {{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}} | Mittag-Leffler Expansion for Hyperbolic Tangent Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Hyperbolic_Tangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Hyperbolic_Tangent_Function | [
"Mittag-Leffler Expansions",
"Hyperbolic Tangent Function"
] | [
"Definition:Half-Integer",
"Definition:Hyperbolic Tangent"
] | [] |
proofwiki-15157 | Mittag-Leffler Expansion for Hyperbolic Secant Function | :$\ds \pi \map \sech {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 + 4 z^2}$
where:
:$z \in \C$ is not a half-integer multiple of $i$
:$\sech$ is the hyperbolic secant function. | {{begin-eqn}}
{{eqn | l = \pi \map \sech {\pi z}
| r = \pi \map \sec {i \pi z}
| c = Hyperbolic Secant in terms of Secant
}}
{{eqn | r = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 \paren {i z}^2}
| c = Mittag-Leffler Expansion for Secant Function
}}
{{eqn | r =... | :$\ds \pi \map \sech {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 + 4 z^2}$
where:
:$z \in \C$ is not a [[Definition:Half-Integer|half-integer]] multiple of $i$
:$\sech$ is the [[Definition:Hyperbolic Secant|hyperbolic secant function]]. | {{begin-eqn}}
{{eqn | l = \pi \map \sech {\pi z}
| r = \pi \map \sec {i \pi z}
| c = [[Hyperbolic Secant in terms of Secant]]
}}
{{eqn | r = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 \paren {i z}^2}
| c = [[Mittag-Leffler Expansion for Secant Function]]
}}
{{e... | Mittag-Leffler Expansion for Hyperbolic Secant Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Hyperbolic_Secant_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Hyperbolic_Secant_Function | [
"Mittag-Leffler Expansions",
"Hyperbolic Secant Function"
] | [
"Definition:Half-Integer",
"Definition:Hyperbolic Secant"
] | [
"Hyperbolic Secant in terms of Secant",
"Mittag-Leffler Expansion for Secant Function"
] |
proofwiki-15158 | Mittag-Leffler Expansion for Hyperbolic Cosecant Function | :$\ds \pi \map \csch {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 + n^2}$
where:
:$z \in \C$ is not an integer multiple of $i$
:$\csch$ is the hyperbolic cosecant function. | {{begin-eqn}}
{{eqn | l = \pi \map \csch {\pi z}
| r = i \pi \map \csc {i \pi z}
| c = Hyperbolic Cosecant in terms of Cosecant
}}
{{eqn | r = i \paren {\dfrac 1 {i z} + 2 i \sum _{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {\paren {i z}^2 - n^2} }
| c = Mittag-Leffler Expansion for Cosecant Functio... | :$\ds \pi \map \csch {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 + n^2}$
where:
:$z \in \C$ is not an [[Definition:Integer|integer]] multiple of $i$
:$\csch$ is the [[Definition:Hyperbolic Cosecant|hyperbolic cosecant function]]. | {{begin-eqn}}
{{eqn | l = \pi \map \csch {\pi z}
| r = i \pi \map \csc {i \pi z}
| c = [[Hyperbolic Cosecant in terms of Cosecant]]
}}
{{eqn | r = i \paren {\dfrac 1 {i z} + 2 i \sum _{n \mathop = 1}^\infty \paren {-1}^n \dfrac z {\paren {i z}^2 - n^2} }
| c = [[Mittag-Leffler Expansion for Cosecant F... | Mittag-Leffler Expansion for Hyperbolic Cosecant Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Hyperbolic_Cosecant_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Hyperbolic_Cosecant_Function | [
"Mittag-Leffler Expansions",
"Hyperbolic Cosecant Function"
] | [
"Definition:Integer",
"Definition:Hyperbolic Cosecant"
] | [
"Hyperbolic Cosecant in terms of Cosecant",
"Mittag-Leffler Expansion for Cosecant Function"
] |
proofwiki-15159 | Subtraction has no Identity Element | The operation of subtraction on numbers of any kind has no identity. | {{AimForCont}} there exists an identity $e$ in one of the standard number systems $\GF$.
{{begin-eqn}}
{{eqn | q = \forall x \in \GF
| l = x
| r = x - e
| c =
}}
{{eqn | r = e - x
| c =
}}
{{eqn | ll= \leadsto
| l = x + \paren {-e}
| r = e + \paren {-x}
| c =
}}
{{eqn | ll= ... | The [[Definition:Binary Operation|operation]] of [[Definition:Subtraction|subtraction]] on [[Definition:Standard Number System|numbers]] of any kind has no [[Definition:Identity Element|identity]]. | {{AimForCont}} there exists an [[Definition:Identity Element|identity]] $e$ in one of the [[Definition:Standard Number System|standard number systems]] $\GF$.
{{begin-eqn}}
{{eqn | q = \forall x \in \GF
| l = x
| r = x - e
| c =
}}
{{eqn | r = e - x
| c =
}}
{{eqn | ll= \leadsto
| l = x... | Subtraction has no Identity Element | https://proofwiki.org/wiki/Subtraction_has_no_Identity_Element | https://proofwiki.org/wiki/Subtraction_has_no_Identity_Element | [
"Numbers",
"Subtraction",
"Examples of Identity Elements"
] | [
"Definition:Operation/Binary Operation",
"Definition:Subtraction",
"Definition:Number",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Number",
"Identity is Unique",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Proof by Contradiction"
] |
proofwiki-15160 | Zero Element of Multiplication on Numbers | On all the number systems:
* natural numbers $\N$
* integers $\Z$
* rational numbers $\Q$
* real numbers $\R$
* complex numbers $\C$
the zero element of multiplication is zero ($0$). | This is demonstrated by showing that:
:$n \times 0 = 0 = 0 \times n$
for all $n$ in all standard number systems.
{{qed}} | On all the number systems:
* [[Definition:Natural Numbers|natural numbers]] $\N$
* [[Definition:Integer|integers]] $\Z$
* [[Definition:Rational Number|rational numbers]] $\Q$
* [[Definition:Real Number|real numbers]] $\R$
* [[Definition:Complex Number|complex numbers]] $\C$
the [[Definition:Zero Element|zero element]] ... | This is demonstrated by showing that:
:$n \times 0 = 0 = 0 \times n$
for all $n$ in all [[Definition:Standard Number System|standard number systems]].
{{qed}} | Zero Element of Multiplication on Numbers | https://proofwiki.org/wiki/Zero_Element_of_Multiplication_on_Numbers | https://proofwiki.org/wiki/Zero_Element_of_Multiplication_on_Numbers | [
"Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Integer",
"Definition:Rational Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Zero Element",
"Definition:Multiplication",
"Definition:Zero (Number)"
] | [
"Definition:Number"
] |
proofwiki-15161 | Addition on Numbers has no Zero Element | On all the number systems:
* natural numbers $\N$
* integers $\Z$
* rational numbers $\Q$
* real numbers $\R$
* complex numbers $\C$
there exists no zero element for addition. | Suppose $z$ is a zero element for addition in a standard number system $\F$.
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \F
| l = n + z
| r = z
| c =
}}
{{eqn | ll= \leadsto
| l = n
| r = 0
| c = subtracting $z$ from both sides
}}
{{end-eqn}}
As $n$ is arbitrary, and therefore not... | On all the number systems:
* [[Definition:Natural Numbers|natural numbers]] $\N$
* [[Definition:Integer|integers]] $\Z$
* [[Definition:Rational Number|rational numbers]] $\Q$
* [[Definition:Real Number|real numbers]] $\R$
* [[Definition:Complex Number|complex numbers]] $\C$
there exists no [[Definition:Zero Element|zer... | Suppose $z$ is a [[Definition:Zero Element|zero element]] for [[Definition:Addition|addition]] in a [[Definition:Standard Number System|standard number system]] $\F$.
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \F
| l = n + z
| r = z
| c =
}}
{{eqn | ll= \leadsto
| l = n
| r = 0
... | Addition on Numbers has no Zero Element | https://proofwiki.org/wiki/Addition_on_Numbers_has_no_Zero_Element | https://proofwiki.org/wiki/Addition_on_Numbers_has_no_Zero_Element | [
"Numbers",
"Addition",
"Zero Elements"
] | [
"Definition:Natural Numbers",
"Definition:Integer",
"Definition:Rational Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Zero Element",
"Definition:Addition"
] | [
"Definition:Zero Element",
"Definition:Addition",
"Definition:Number"
] |
proofwiki-15162 | Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y') | Let $F$ and its partial derivatives $F_y, F_{y'}$ be real functions, defined on the closed interval $I = \closedint a b$.
Let $F, F_y, F_{y'} $ be continuous at every point $\tuple {x, y}$ for all finite $y'$.
Suppose there exists a constant $k > 0$ such that:
:$\map {F_y} {x, y, y'} > k$
Suppose there exist real funct... | === Lemma 1 (Uniqueness) ===
{{:Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')/Lemma 1}}{{qed|lemma}} | Let $F$ and its [[Definition:Partial Derivative|partial derivatives]] $F_y, F_{y'}$ be [[Definition:Real Function|real functions]], defined on the [[Definition:Closed Interval|closed interval]] $I = \closedint a b$.
Let $F, F_y, F_{y'} $ be [[Definition:Continuous on Interval|continuous]] at every [[Definition:Point|p... | === [[Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')/Lemma 1|Lemma 1 (Uniqueness)]] ===
{{:Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')/Lemma 1}}{{qed|lemma}} | Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y') | https://proofwiki.org/wiki/Bernstein's_Theorem_on_Unique_Global_Solution_to_y''=F(x,y,y') | https://proofwiki.org/wiki/Bernstein's_Theorem_on_Unique_Global_Solution_to_y''=F(x,y,y') | [
"Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')",
"Calculus of Variations"
] | [
"Definition:Partial Derivative",
"Definition:Real Function",
"Definition:Interval/Ordered Set/Closed",
"Definition:Continuous Real Function/Interval",
"Definition:Point",
"Definition:Finite",
"Definition:Constant Mapping",
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Defi... | [
"Bernstein's Theorem on Unique Global Solution to y''=F(x,y,y')/Lemma 1"
] |
proofwiki-15163 | Real Numbers under Subtraction do not form Semigroup | The set of real numbers under subtraction $\struct {\R, -}$ does not form a semigroup. | We have that Subtraction on Numbers is Not Associative.
Hence $\struct {\R, -}$ is not a semigroup by definition.
{{qed}} | The [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] under [[Definition:Real Subtraction|subtraction]] $\struct {\R, -}$ does not form a [[Definition:Semigroup|semigroup]]. | We have that [[Subtraction on Numbers is Not Associative]].
Hence $\struct {\R, -}$ is not a [[Definition:Semigroup|semigroup]] by definition.
{{qed}} | Real Numbers under Subtraction do not form Semigroup | https://proofwiki.org/wiki/Real_Numbers_under_Subtraction_do_not_form_Semigroup | https://proofwiki.org/wiki/Real_Numbers_under_Subtraction_do_not_form_Semigroup | [
"Real Subtraction",
"Semigroups"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Subtraction/Real Numbers",
"Definition:Semigroup"
] | [
"Subtraction on Numbers is Not Associative",
"Definition:Semigroup"
] |
proofwiki-15164 | Group has Latin Square Property/Additive Notation | Let $\struct {G, +}$ be a group.
Then $G$ satisfies the Latin square property.
That is, for all $a, b \in G$, there exists a unique $g \in G$ such that $a + g = b$.
Similarly, there exists a unique $h \in G$ such that $h + a = b$. | From Group has Latin Square Property, we have that:
{{begin-eqn}}
{{eqn | l = a + g
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = g
| r = \paren {-a} + b
| c =
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = h + a
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = h
| r = b... | Let $\struct {G, +}$ be a [[Definition:Group|group]].
Then $G$ satisfies the [[Definition:Latin Square Property|Latin square property]].
That is, for all $a, b \in G$, there exists a [[Definition:Unique|unique]] $g \in G$ such that $a + g = b$.
Similarly, there exists a [[Definition:Unique|unique]] $h \in G$ such th... | From [[Group has Latin Square Property]], we have that:
{{begin-eqn}}
{{eqn | l = a + g
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = g
| r = \paren {-a} + b
| c =
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = h + a
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = h
... | Group has Latin Square Property/Additive Notation | https://proofwiki.org/wiki/Group_has_Latin_Square_Property/Additive_Notation | https://proofwiki.org/wiki/Group_has_Latin_Square_Property/Additive_Notation | [
"Group has Latin Square Property"
] | [
"Definition:Group",
"Definition:Latin Square Property",
"Definition:Unique",
"Definition:Unique"
] | [
"Group has Latin Square Property"
] |
proofwiki-15165 | Mittag-Leffler's Expansion Theorem | Let $f$ be a meromorphic function that:
:has only simple poles
:is continuous, or has a removable singularity, at $0$.
Let $X$ be the set of poles of $f$.
For $N \in \N$, let $C_N$ be a disk, centred at the origin, of radius $R_N$ where:
:$R_N \to \infty$ as $N \to \infty$
:$\partial C_N$ contains no poles of $f$... | Let $\zeta \in \C \setminus X$.
Then:
:$\ds \frac {\map f z} {z - \zeta}$
has simple poles for $z \in X \cup \set \zeta$.
Let $X_N$ be the set of poles contained within $C_N$.
Then:
{{begin-eqn}}
{{eqn | l = \frac 1 {2 \pi i} \oint_{\partial C_N} \frac {\map f z} {z - \zeta} \rd z
| r = \Res {\frac {\map f ... | Let $f$ be a [[Definition:Meromorphic Function|meromorphic function]] that:
:has only [[Definition:Simple Pole|simple poles]]
:is continuous, or has a [[Definition:Removable Singularity (Complex Plane)|removable singularity]], at $0$.
Let $X$ be the set of [[Definition:Pole|poles]] of $f$.
For $N \in \N$, let $C_... | Let $\zeta \in \C \setminus X$.
Then:
:$\ds \frac {\map f z} {z - \zeta}$
has simple poles for $z \in X \cup \set \zeta$.
Let $X_N$ be the set of poles contained within $C_N$.
Then:
{{begin-eqn}}
{{eqn | l = \frac 1 {2 \pi i} \oint_{\partial C_N} \frac {\map f z} {z - \zeta} \rd z
| r = \Res {\frac {\... | Mittag-Leffler's Expansion Theorem | https://proofwiki.org/wiki/Mittag-Leffler's_Expansion_Theorem | https://proofwiki.org/wiki/Mittag-Leffler's_Expansion_Theorem | [
"Mittag-Leffler Expansions"
] | [
"Definition:Meromorphic Function",
"Definition:Order of Pole/Simple Pole",
"Definition:Removable Singularity/Complex Function",
"Definition:Pole",
"Definition:Real Number",
"Definition:Residue",
"Definition:Removable Singularity/Complex Function"
] | [
"Cauchy's Residue Theorem",
"Residue at Simple Pole",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Residue at Simple Pole",
"Estimation Lemma for Contour Integrals",
"Reverse Triangle Inequality/Real and Complex Fields"
] |
proofwiki-15166 | Normed Division Ring Operations are Continuous/Addition | :$+ : \struct {R \times R, d_p} \to \struct{R,d}$
is continuous. | By $p$-Product Metric Induces Product Topology and Continuous Mapping is Continuous on Induced Topological Spaces, it suffices to consider the case $p = \infty$.
Let $\tuple {x_0, y_0} \in R \times R$.
Let $\epsilon > 0$ be given.
Let $\tuple {x, y} \in R \times R$ such that:
:$\map {d_\infty} {\tuple {x, y}, \tuple{x_... | :$+ : \struct {R \times R, d_p} \to \struct{R,d}$
is [[Definition:Continuous Mapping (Metric Space)|continuous]]. | By [[P-Product Metric Induces Product Topology|$p$-Product Metric Induces Product Topology]] and [[Continuous Mapping is Continuous on Induced Topological Spaces]], it suffices to consider the case $p = \infty$.
Let $\tuple {x_0, y_0} \in R \times R$.
Let $\epsilon > 0$ be given.
Let $\tuple {x, y} \in R \times R$... | Normed Division Ring Operations are Continuous/Addition | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Addition | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Addition | [
"Normed Division Rings",
"Topological Division Rings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"P-Product Metric Induces Product Topology",
"Continuous Mapping is Continuous on Induced Topological Spaces",
"Definition:P-Product Metric",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-15167 | Normed Division Ring Operations are Continuous/Negation | :$\eta: \struct {R, d} \to \struct {R, d}: \map \eta x = -x$
is continuous. | Let $x_0 \in R$.
Let $\epsilon > 0$ be given.
Let $x \in R$ such that:
:$\map d {x, x_0} < \epsilon$
Then:
{{begin-eqn}}
{{eqn | l = \map d {-x, -x_0}
| r = \norm {-x - \paren {-x_0} }
| c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | r = \norm {-x + x_0}
}}
{{eqn | r = \norm {x_0 - x}
... | :$\eta: \struct {R, d} \to \struct {R, d}: \map \eta x = -x$
is [[Definition:Continuous Mapping (Metric Space)|continuous]]. | Let $x_0 \in R$.
Let $\epsilon > 0$ be given.
Let $x \in R$ such that:
:$\map d {x, x_0} < \epsilon$
Then:
{{begin-eqn}}
{{eqn | l = \map d {-x, -x_0}
| r = \norm {-x - \paren {-x_0} }
| c = {{Defof|Metric Induced by Norm on Division Ring}}
}}
{{eqn | r = \norm {-x + x_0}
}}
{{eqn | r = \norm {x_0 - x}
... | Normed Division Ring Operations are Continuous/Negation | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Negation | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Negation | [
"Normed Division Rings",
"Topological Division Rings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Commutative/Operation",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-15168 | Normed Division Ring Operations are Continuous/Multiplication | :$* : \struct {R \times R, d_p} \to \struct {R, d}$
is continuous. | By $p$-Product Metric Induces Product Topology and Continuous Mapping is Continuous on Induced Topological Spaces, it suffices to consider the case $p = \infty$.
Let $\tuple {x_0, y_0} \in R \times R$.
Let $\epsilon > 0$ be given.
Let $\delta = \min \set {\dfrac \epsilon {1 + \norm {y_0} + \norm {x_0} }, 1}$
Since $1 +... | :$* : \struct {R \times R, d_p} \to \struct {R, d}$
is [[Definition:Continuous Mapping (Metric Space)|continuous]]. | By [[P-Product Metric Induces Product Topology|$p$-Product Metric Induces Product Topology]] and [[Continuous Mapping is Continuous on Induced Topological Spaces]], it suffices to consider the case $p = \infty$.
Let $\tuple {x_0, y_0} \in R \times R$.
Let $\epsilon > 0$ be given.
Let $\delta = \min \set {\dfrac \e... | Normed Division Ring Operations are Continuous/Multiplication | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Multiplication | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Multiplication | [
"Normed Division Rings",
"Topological Division Rings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"P-Product Metric Induces Product Topology",
"Continuous Mapping is Continuous on Induced Topological Spaces",
"Definition:P-Product Metric",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-15169 | Normed Division Ring Operations are Continuous/Inversion | :$\iota : \struct {R^* ,d^*} \to \struct {R, d} : \map \iota x = x^{-1}$
is continuous. | Let $x_0 \in R^*$.
Let $\epsilon > 0$ be given.
Let $\delta = \min \set {\dfrac {\norm {x_0} } 2, \dfrac {\norm {x_0}^2 \epsilon} 2 }$
Let $x \in R^*$ such that:
:$\map {d^*} {x, x_0} < \delta$
By the definition of the subspace metric on $R^*$ and the definition of the metric induced by the norm on $R$:
:$\map {d^*} {x... | :$\iota : \struct {R^* ,d^*} \to \struct {R, d} : \map \iota x = x^{-1}$
is [[Definition:Continuous Mapping (Metric Space)|continuous]]. | Let $x_0 \in R^*$.
Let $\epsilon > 0$ be given.
Let $\delta = \min \set {\dfrac {\norm {x_0} } 2, \dfrac {\norm {x_0}^2 \epsilon} 2 }$
Let $x \in R^*$ such that:
:$\map {d^*} {x, x_0} < \delta$
By the definition of the [[Definition:Metric Subspace|subspace metric]] on $R^*$ and the definition of the [[Definition:M... | Normed Division Ring Operations are Continuous/Inversion | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Inversion | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Inversion | [
"Normed Division Rings",
"Topological Division Rings"
] | [
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Metric Subspace",
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)"
] |
proofwiki-15170 | Normed Division Ring Operations are Continuous/Corollary | Let $\tau$ be the topology induced by the metric $d$.
Then:
:$\struct {R, \tau}$ is a topological division ring. | Let $d_\infty$ be the Chebyshev distance metric on $R \times R$.
Let $\tau^\times$ be the product topology on $R \times R$.
By $p$-Product Metric Induces Product Topology, $\tau^\times$ is the topology induced by the metric $d_\infty$.
Let $R^* = R \setminus \set 0$.
Let $d^*$ be the restriction of $d$ to $R^*$.
Let $\... | Let $\tau$ be the [[Definition:Topology Induced by Metric|topology induced by the metric]] $d$.
Then:
:$\struct {R, \tau}$ is a [[Definition:Topological Division Ring|topological division ring]]. | Let $d_\infty$ be the [[Definition:Chebyshev Distance|Chebyshev distance metric]] on $R \times R$.
Let $\tau^\times$ be the [[Definition:Product Topology|product topology]] on $R \times R$.
By [[P-Product Metric Induces Product Topology|$p$-Product Metric Induces Product Topology]], $\tau^\times$ is the [[Definition:... | Normed Division Ring Operations are Continuous/Corollary | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Corollary | https://proofwiki.org/wiki/Normed_Division_Ring_Operations_are_Continuous/Corollary | [
"Normed Division Rings",
"Topological Division Rings"
] | [
"Definition:Topology Induced by Metric",
"Definition:Topological Division Ring"
] | [
"Definition:Chebyshev Distance",
"Definition:Product Topology",
"P-Product Metric Induces Product Topology",
"Definition:Topology Induced by Metric",
"Definition:Restriction",
"Definition:Topological Subspace",
"Metric Subspace Induces Subspace Topology",
"Definition:Topology Induced by Metric",
"No... |
proofwiki-15171 | Structure with Element both Identity and Zero has One Element | Let $\struct {S, \circ}$ be an algebraic structure.
Let $z \in S$ such that $z$ is both an identity element and a zero element.
Then:
:$S = \set z$ | Let $x \in S$.
Then
{{begin-eqn}}
{{eqn | l = x
| r = x \circ z
| c = {{Defof|Identity Element}}
}}
{{eqn | r = z
| c = {{Defof|Zero Element}}
}}
{{end-eqn}}
and so there is no other element of $S$ but $z$.
{{qed}} | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]].
Let $z \in S$ such that $z$ is both an [[Definition:Identity Element|identity element]] and a [[Definition:Zero Element|zero element]].
Then:
:$S = \set z$ | Let $x \in S$.
Then
{{begin-eqn}}
{{eqn | l = x
| r = x \circ z
| c = {{Defof|Identity Element}}
}}
{{eqn | r = z
| c = {{Defof|Zero Element}}
}}
{{end-eqn}}
and so there is no other [[Definition:Element|element]] of $S$ but $z$.
{{qed}} | Structure with Element both Identity and Zero has One Element | https://proofwiki.org/wiki/Structure_with_Element_both_Identity_and_Zero_has_One_Element | https://proofwiki.org/wiki/Structure_with_Element_both_Identity_and_Zero_has_One_Element | [
"Identity Elements",
"Zero Elements"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Zero Element"
] | [
"Definition:Element"
] |
proofwiki-15172 | Group/Examples/Linear Functions | Let $G$ be the set of all real functions $\theta_{a, b}: \R \to \R$ defined as:
:$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$
where $a, b \in \R$ such that $a \ne 0$.
The algebraic structure $\struct {G, \circ}$, where $\circ$ denotes composition of mappings, is a group.
$\struct {G, \circ}$ is specifically no... | We verify the group axioms, in the following order (for convenience): | Let $G$ be the [[Definition:Set|set]] of all [[Definition:Real Function|real functions]] $\theta_{a, b}: \R \to \R$ defined as:
:$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$
where $a, b \in \R$ such that $a \ne 0$.
The [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\struct {G, \ci... | We verify the [[Axiom:Group Axioms|group axioms]], in the following order (for convenience): | Group/Examples/Linear Functions | https://proofwiki.org/wiki/Group/Examples/Linear_Functions | https://proofwiki.org/wiki/Group/Examples/Linear_Functions | [
"Examples of Groups"
] | [
"Definition:Set",
"Definition:Real Function",
"Definition:Algebraic Structure/One Operation",
"Definition:Composition of Mappings",
"Definition:Group",
"Definition:Abelian Group"
] | [
"Axiom:Group Axioms"
] |
proofwiki-15173 | Integer Multiples under Addition form Subgroup of Integers | Let $\struct {\Z, +}$ denote the additive group of integers.
Let $n \Z$ be the set of integer multiples of $n$.
Then $\struct {n \Z, +}$ is a subgroup of $\struct {\Z, +}$.
Hence $\struct {n \Z, +}$ can be justifiably referred to as the additive group of integer multiples. | Clearly $0 \in n \Z$ so $n \Z \ne \O$.
Now suppose $x, y \in n \Z$.
Then $\exists r, s \in \Z: x = n r, y = n s$.
Also, $-y = - n s$.
So $x - y = n \paren {r - s}$.
As $r - s \in \Z$ it follows that $x - y \in n \Z$.
So by the One-Step Subgroup Test it follows that $\struct {n \Z, +}$ is a subgroup of the additive grou... | Let $\struct {\Z, +}$ denote the [[Definition:Additive Group of Integers|additive group of integers]].
Let $n \Z$ be the [[Definition:Set of Integer Multiples|set of integer multiples]] of $n$.
Then $\struct {n \Z, +}$ is a [[Definition:Subgroup|subgroup]] of $\struct {\Z, +}$.
Hence $\struct {n \Z, +}$ can be jus... | Clearly $0 \in n \Z$ so $n \Z \ne \O$.
Now suppose $x, y \in n \Z$.
Then $\exists r, s \in \Z: x = n r, y = n s$.
Also, $-y = - n s$.
So $x - y = n \paren {r - s}$.
As $r - s \in \Z$ it follows that $x - y \in n \Z$.
So by the [[One-Step Subgroup Test]] it follows that $\struct {n \Z, +}$ is a [[Definition:Subgro... | Integer Multiples under Addition form Subgroup of Integers | https://proofwiki.org/wiki/Integer_Multiples_under_Addition_form_Subgroup_of_Integers | https://proofwiki.org/wiki/Integer_Multiples_under_Addition_form_Subgroup_of_Integers | [
"Additive Groups of Integer Multiples",
"Additive Group of Integers"
] | [
"Definition:Additive Group of Integers",
"Definition:Set of Integer Multiples",
"Definition:Subgroup",
"Definition:Additive Group of Integer Multiples"
] | [
"One-Step Subgroup Test",
"Definition:Subgroup",
"Definition:Additive Group of Integers"
] |
proofwiki-15174 | Subgroup Generated by Subgroup | Let $G$ be a group.
Let $H \le G$ be a subgroup of $G$.
Then:
:$H = \gen H$
where $\gen H$ denotes the subgroup generated by $H$. | By definition of generated subgroup, $\gen H$ is the smallest subgroup of $H$ containing $H$.
Hence the result.
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $H \le G$ be a [[Definition:Subgroup|subgroup]] of $G$.
Then:
:$H = \gen H$
where $\gen H$ denotes the [[Definition:Generated Subgroup|subgroup generated]] by $H$. | By definition of [[Definition:Generated Subgroup|generated subgroup]], $\gen H$ is the [[Definition:Smallest Set by Set Inclusion|smallest]] [[Definition:Subgroup|subgroup]] of $H$ containing $H$.
Hence the result.
{{qed}} | Subgroup Generated by Subgroup | https://proofwiki.org/wiki/Subgroup_Generated_by_Subgroup | https://proofwiki.org/wiki/Subgroup_Generated_by_Subgroup | [
"Generated Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Generated Subgroup"
] | [
"Definition:Generated Subgroup",
"Definition:Smallest Set by Set Inclusion",
"Definition:Subgroup"
] |
proofwiki-15175 | Group is Generated by Itself | Let $G$ be a group.
Then:
:$G = \gen G$
where $\gen G$ denotes the group generated by $G$. | By definition of generated subgroup, $\gen G$ is the smallest subgroup of $G$ containing $G$.
Hence the result by Group is Subgroup of Itself.
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Then:
:$G = \gen G$
where $\gen G$ denotes the [[Definition:Generator of Group|group generated]] by $G$. | By definition of [[Definition:Generated Subgroup|generated subgroup]], $\gen G$ is the [[Definition:Smallest Set by Set Inclusion|smallest]] [[Definition:Subgroup|subgroup]] of $G$ containing $G$.
Hence the result by [[Group is Subgroup of Itself]].
{{qed}} | Group is Generated by Itself | https://proofwiki.org/wiki/Group_is_Generated_by_Itself | https://proofwiki.org/wiki/Group_is_Generated_by_Itself | [
"Generators of Groups"
] | [
"Definition:Group",
"Definition:Generator of Group"
] | [
"Definition:Generated Subgroup",
"Definition:Smallest Set by Set Inclusion",
"Definition:Subgroup",
"Group is Subgroup of Itself"
] |
proofwiki-15176 | Action of Inverse of Group Element | Let $\struct {G, \circ}$ be a group.
Let $S$ be a sets.
Let $*: G \times S \to S$ be a group action.
Then:
:$g * a = b \iff g^{-1} * b = a$ | {{begin-eqn}}
{{eqn | l = g * a
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = g^{-1} * \paren {g * a}
| r = g^{-1} * b
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {g^{-1} \circ g} * a
| r = g^{-1} * b
| c = {{GroupActionAxiom|2}}
}}
{{eqn | ll= \leadsto
| l = e * a
... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $S$ be a [[Definition:Set|sets]].
Let $*: G \times S \to S$ be a [[Definition:Group Action|group action]].
Then:
:$g * a = b \iff g^{-1} * b = a$ | {{begin-eqn}}
{{eqn | l = g * a
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = g^{-1} * \paren {g * a}
| r = g^{-1} * b
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {g^{-1} \circ g} * a
| r = g^{-1} * b
| c = {{GroupActionAxiom|2}}
}}
{{eqn | ll= \leadsto
| l = e * a
... | Action of Inverse of Group Element | https://proofwiki.org/wiki/Action_of_Inverse_of_Group_Element | https://proofwiki.org/wiki/Action_of_Inverse_of_Group_Element | [
"Group Actions"
] | [
"Definition:Group",
"Definition:Set",
"Definition:Group Action"
] | [] |
proofwiki-15177 | Union Operation on Supersets of Subset is Closed | Let $S$ be a set.
Let $T \subseteq S$ be a given subset of $S$.
Let $\powerset S$ denote the power set of $S$
Let $\mathscr S$ be the subset of $\powerset S$ defined as:
:$\mathscr S = \set {Y \in \powerset S: T \subseteq Y}$
Then the algebraic structure $\struct {\mathscr S, \cup}$ is closed. | Let $A, B \in \mathscr S$.
We have that:
{{begin-eqn}}
{{eqn | l = T
| o = \subseteq
| r = A
| c = Definition of $\mathscr S$
}}
{{eqn | l = T
| o = \subseteq
| r = B
| c = Definition of $\mathscr S$
}}
{{eqn | n = 1
| ll= \leadsto
| l = T
| o = \subseteq
| r ... | Let $S$ be a [[Definition:Set|set]].
Let $T \subseteq S$ be a given [[Definition:Subset|subset]] of $S$.
Let $\powerset S$ denote the [[Definition:Power Set|power set]] of $S$
Let $\mathscr S$ be the [[Definition:Subset|subset]] of $\powerset S$ defined as:
:$\mathscr S = \set {Y \in \powerset S: T \subseteq Y}$
... | Let $A, B \in \mathscr S$.
We have that:
{{begin-eqn}}
{{eqn | l = T
| o = \subseteq
| r = A
| c = Definition of $\mathscr S$
}}
{{eqn | l = T
| o = \subseteq
| r = B
| c = Definition of $\mathscr S$
}}
{{eqn | n = 1
| ll= \leadsto
| l = T
| o = \subseteq
| ... | Union Operation on Supersets of Subset is Closed | https://proofwiki.org/wiki/Union_Operation_on_Supersets_of_Subset_is_Closed | https://proofwiki.org/wiki/Union_Operation_on_Supersets_of_Subset_is_Closed | [
"Set Union"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Subset",
"Definition:Algebraic Structure/One Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Set is Subset of Union",
"Union is Smallest Superset",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-15178 | Intersection Operation on Supersets of Subset is Closed | Let $S$ be a set.
Let $T \subseteq S$ be a given subset of $S$.
Let $\powerset S$ denote the power set of $S$
Let $\mathscr S$ be the subset of $\powerset S$ defined as:
:$\mathscr S = \set {Y \in \powerset S: T \subseteq Y}$
Then the algebraic structure $\struct {\mathscr S, \cap}$ is closed. | Let $A, B \in \mathscr S$.
We have that:
{{begin-eqn}}
{{eqn | l = T
| o = \subseteq
| r = A
| c = Definition of $\mathscr S$
}}
{{eqn | l = T
| o = \subseteq
| r = B
| c = Definition of $\mathscr S$
}}
{{eqn | n = 1
| ll= \leadsto
| l = T
| o = \subseteq
| r ... | Let $S$ be a [[Definition:Set|set]].
Let $T \subseteq S$ be a given [[Definition:Subset|subset]] of $S$.
Let $\powerset S$ denote the [[Definition:Power Set|power set]] of $S$
Let $\mathscr S$ be the [[Definition:Subset|subset]] of $\powerset S$ defined as:
:$\mathscr S = \set {Y \in \powerset S: T \subseteq Y}$
... | Let $A, B \in \mathscr S$.
We have that:
{{begin-eqn}}
{{eqn | l = T
| o = \subseteq
| r = A
| c = Definition of $\mathscr S$
}}
{{eqn | l = T
| o = \subseteq
| r = B
| c = Definition of $\mathscr S$
}}
{{eqn | n = 1
| ll= \leadsto
| l = T
| o = \subseteq
| ... | Intersection Operation on Supersets of Subset is Closed | https://proofwiki.org/wiki/Intersection_Operation_on_Supersets_of_Subset_is_Closed | https://proofwiki.org/wiki/Intersection_Operation_on_Supersets_of_Subset_is_Closed | [
"Set Union"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Subset",
"Definition:Algebraic Structure/One Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Intersection is Largest Subset",
"Intersection is Subset",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-15179 | Existence of Magma with no Proper Submagma | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $S$ be a set of cardinality $n$:
:$\card S = n$
Then there exists an operation $\circ$ on $S$ such that:
:$\struct {S, \circ}$ is a magma
:$\struct {S, \circ}$ has no submagma $\struct {T, \circ}$ such that $T$ is a non-empty proper subset of $S$. | For $n = 1$ the result follows trivially: there are no non-empty proper subsets of a singleton.
Let $S = \set {s_1, s_2, \ldots, s_n}$.
Let $\circ$ be defined on $S$ such that:
:$\forall s_a \in S: s_a \circ s_a = \begin{cases} s_{a + 1} & : a < n \\ s_1 & : a = n \end{cases}$
For $a \ne b$ the operation $s_a \circ s_b... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $S$ be a [[Definition:Set|set]] of [[Definition:Cardinality|cardinality]] $n$:
:$\card S = n$
Then there exists an [[Definition:Binary Operation|operation]] $\circ$ on $S$ such that:
:$\struct {S, \circ}$ is a [[Definiti... | For $n = 1$ the result follows trivially: there are no [[Definition:Non-Empty Set|non-empty]] [[Definition:Proper Subset|proper subsets]] of a [[Definition:Singleton|singleton]].
Let $S = \set {s_1, s_2, \ldots, s_n}$.
Let $\circ$ be defined on $S$ such that:
:$\forall s_a \in S: s_a \circ s_a = \begin{cases} s_{a +... | Existence of Magma with no Proper Submagma | https://proofwiki.org/wiki/Existence_of_Magma_with_no_Proper_Submagma | https://proofwiki.org/wiki/Existence_of_Magma_with_no_Proper_Submagma | [
"Magmas"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Cardinality",
"Definition:Operation/Binary Operation",
"Definition:Magma",
"Definition:Submagma",
"Definition:Non-Empty Set",
"Definition:Proper Subset"
] | [
"Definition:Non-Empty Set",
"Definition:Proper Subset",
"Definition:Singleton",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Submagma"
] |
proofwiki-15180 | Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup | Let $\struct {G, \circ}$ be an abelian group.
Let $S \subset G$ be a non-empty subset of $G$ such that $\struct {S, \circ}$ is closed.
Let $H$ be the set defined as:
:$H := \set {x \circ y^{-1}: x, y \in S}$
Then $\struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$. | Let $x \in S$.
Then:
:$x \circ x^{-1} \in H$
and so $H \ne \O$.
Now let $a, b \in H$.
Then:
:$a = x_a \circ y_a^{-1}$
and:
:$b = x_b \circ y_b^{-1}$
for some $x_a, y_a, x_b, y_b \in S$.
Thus:
{{begin-eqn}}
{{eqn | l = a \circ b^{-1}
| r = \paren {x_a \circ y_a^{-1} } \circ \paren {x_b \circ y_b^{-1} }^{-1}
... | Let $\struct {G, \circ}$ be an [[Definition:Abelian Group|abelian group]].
Let $S \subset G$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $G$ such that $\struct {S, \circ}$ is [[Definition:Closed Algebraic Structure|closed]].
Let $H$ be the [[Definition:Set|set]] defined as:
:$H := \set... | Let $x \in S$.
Then:
:$x \circ x^{-1} \in H$
and so $H \ne \O$.
Now let $a, b \in H$.
Then:
:$a = x_a \circ y_a^{-1}$
and:
:$b = x_b \circ y_b^{-1}$
for some $x_a, y_a, x_b, y_b \in S$.
Thus:
{{begin-eqn}}
{{eqn | l = a \circ b^{-1}
| r = \paren {x_a \circ y_a^{-1} } \circ \paren {x_b \circ y_b^{-1} }^{-1}
... | Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup | https://proofwiki.org/wiki/Subset_of_Abelian_Group_Generated_by_Product_of_Element_with_Inverse_Element_is_Subgroup | https://proofwiki.org/wiki/Subset_of_Abelian_Group_Generated_by_Product_of_Element_with_Inverse_Element_is_Subgroup | [
"Abelian Groups",
"Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup"
] | [
"Definition:Abelian Group",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set",
"Definition:Subgroup"
] | [
"Inverse of Group Product",
"Inverse of Group Inverse",
"Inverse of Group Product",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"One-Step Subgroup Test",
"Definition:Subgroup"
] |
proofwiki-15181 | Local Basis Test | Let $\struct {S, \tau}$ be a topological space.
Let $x \in S$.
Let $\BB$ be a local basis for $x$ in $\struct {S, \tau}$.
Let $\CC$ be a set of open neighborhoods of $x$.
Then:
:$\CC$ is a local basis {{iff}}:
::$\forall B \in \BB \implies \exists C \in \CC: C \subseteq B$ | === Necessary Condition ===
Let $\CC$ be a local basis.
Let $B \in \BB$.
Since $\BB$ is a local basis, by the definition of a local basis then $B$ is open.
Since $\CC$ is a local basis, by the definition of a local basis then:
:$\exists C \in \CC : C\subseteq B$.
{{qed|lemma}} | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $\BB$ be a [[Definition:Local Basis|local basis]] for $x$ in $\struct {S, \tau}$.
Let $\CC$ be a [[Definition:Set|set]] of [[Definition:Open Neighborhood of Point|open neighborhoods]] of $x$.
Then:
:$\CC$ is a [[Defi... | === Necessary Condition ===
Let $\CC$ be a [[Definition:Local Basis|local basis]].
Let $B \in \BB$.
Since $\BB$ is a [[Definition:Local Basis|local basis]], by the definition of a [[Definition:Local Basis|local basis]] then $B$ is [[Definition:Open Set|open]].
Since $\CC$ is a [[Definition:Local Basis|local basis]]... | Local Basis Test | https://proofwiki.org/wiki/Local_Basis_Test | https://proofwiki.org/wiki/Local_Basis_Test | [
"Local Bases"
] | [
"Definition:Topological Space",
"Definition:Local Basis",
"Definition:Set",
"Definition:Open Neighborhood/Point",
"Definition:Local Basis"
] | [
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Open Set",
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Local Basis"
] |
proofwiki-15182 | Non-Zero Integers under Multiplication are not Subgroup of Reals | Let $\struct {\Z_{\ne 0}, \times}$ denote the algebraic structure formed by the set of non-zero integers under multiplication.
Let $\struct {\R_{\ne 0}, \times}$ denote the algebraic structure formed by the set of non-zero real numbers under multiplication.
Then, while $\struct {\Z_{\ne 0}, \times}$ is closed, it is no... | We have that Non-Zero Real Numbers under Multiplication form Group.
We also have that the set of non-zero integers $\Z_{\ne 0}$ form a subset of $\R_{\ne 0}$.
From Non-Zero Integers Closed under Multiplication:
:$\forall a, b \in \Z_{\ne 0}: a \times b \in \Z_{\ne 0}$
We have that:
:$\forall x \in \Z_{\ne 0}: 1 \times ... | Let $\struct {\Z_{\ne 0}, \times}$ denote the [[Definition:Algebraic Structure with One Operation|algebraic structure]] formed by the [[Definition:Set|set]] of [[Definition:Zero (Number)|non-zero]] [[Definition:Integer|integers]] under [[Definition:Integer Multiplication|multiplication]].
Let $\struct {\R_{\ne 0}, \ti... | We have that [[Non-Zero Real Numbers under Multiplication form Group]].
We also have that the [[Definition:Set|set]] of [[Definition:Zero (Number)|non-zero]] [[Definition:Integer|integers]] $\Z_{\ne 0}$ form a [[Definition:Subset|subset]] of $\R_{\ne 0}$.
From [[Non-Zero Integers Closed under Multiplication]]:
:$\fo... | Non-Zero Integers under Multiplication are not Subgroup of Reals | https://proofwiki.org/wiki/Non-Zero_Integers_under_Multiplication_are_not_Subgroup_of_Reals | https://proofwiki.org/wiki/Non-Zero_Integers_under_Multiplication_are_not_Subgroup_of_Reals | [
"Integer Multiplication",
"Real Multiplication"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Set",
"Definition:Zero (Number)",
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Algebraic Structure/One Operation",
"Definition:Set",
"Definition:Zero (Number)",
"Definition:Real Number",
"Definition:Multiplicati... | [
"Non-Zero Real Numbers under Multiplication form Group",
"Definition:Set",
"Definition:Zero (Number)",
"Definition:Integer",
"Definition:Subset",
"Non-Zero Integers Closed under Multiplication",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inverse... |
proofwiki-15183 | Condition for Elements of Group to be in Subgroup | Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Let $x, y \in G$ be such that $2$ elements of $\set {x, y, x y}$ are elements of $h$.
Then ''all'' the elements of $\set {x, y, x y}$ are in $H$. | As $H$ is a subgroup of $G$, it is a group in its own right.
Thus the group axioms all apply to $H$.
Let $x, y \in H$.
Then by {{Group-axiom|0}}, $x y \in H$.
Let $x, x y \in H$.
As $x \in H$, it follows that $x^{-1} \in H$ by {{Group-axiom|3}}.
Thus by {{Group-axiom|0}}, $x^{-1} \paren {x y} = y \in H$.
Let $y, x y \i... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $x, y \in G$ be such that $2$ [[Definition:Element|elements]] of $\set {x, y, x y}$ are [[Definition:Element|elements]] of $h$.
Then ''all'' the [[Definition:Element|elements]] of $\set {x, y, x y}$ are in $H$. | As $H$ is a [[Definition:Subgroup|subgroup]] of $G$, it is a [[Definition:Group|group]] in its own right.
Thus the [[Axiom:Group Axioms|group axioms]] all apply to $H$.
Let $x, y \in H$.
Then by {{Group-axiom|0}}, $x y \in H$.
Let $x, x y \in H$.
As $x \in H$, it follows that $x^{-1} \in H$ by {{Group-axiom|3}}.... | Condition for Elements of Group to be in Subgroup | https://proofwiki.org/wiki/Condition_for_Elements_of_Group_to_be_in_Subgroup | https://proofwiki.org/wiki/Condition_for_Elements_of_Group_to_be_in_Subgroup | [
"Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Element",
"Definition:Element",
"Definition:Element"
] | [
"Definition:Subgroup",
"Definition:Group",
"Axiom:Group Axioms"
] |
proofwiki-15184 | Union of Subgroups/Corollary 1 | Let $H \cup K$ be a subgroup of $G$.
Then either $H \subseteq K$ or $K \subseteq H$. | {{AimForCont}} neither $H \subseteq K$ nor $K \subseteq H$.
Then from Union of Subgroups it follows that $H \cup K$ is not a subgroup of $G$.
The result follows by Proof by Contradiction.
{{qed}} | Let $H \cup K$ be a [[Definition:Subgroup|subgroup]] of $G$.
Then either $H \subseteq K$ or $K \subseteq H$. | {{AimForCont}} neither $H \subseteq K$ nor $K \subseteq H$.
Then from [[Union of Subgroups]] it follows that $H \cup K$ is not a [[Definition:Subgroup|subgroup]] of $G$.
The result follows by [[Proof by Contradiction]].
{{qed}} | Union of Subgroups/Corollary 1 | https://proofwiki.org/wiki/Union_of_Subgroups/Corollary_1 | https://proofwiki.org/wiki/Union_of_Subgroups/Corollary_1 | [
"Union of Subgroups"
] | [
"Definition:Subgroup"
] | [
"Union of Subgroups",
"Definition:Subgroup",
"Proof by Contradiction"
] |
proofwiki-15185 | Subgroup Generated by Commuting Elements is Abelian | Let $\struct {G, \circ}$ be a group.
Let $S \subseteq G$ such that:
:$\forall x, y \in S: x \circ y = y \circ x$
Then the subgroup generated by $S$ is abelian. | Let $H = \gen S$ denote the subgroup generated by $S$.
Let $a, b \in H$.
Then:
:$a = s_1$
:$b = s_2$
for some words $s_1, s_2$ of the set of words $\map W S$ of $S$.
Then:
{{begin-eqn}}
{{eqn | l = a \circ b
| r = s_1 \circ s_2
| c =
}}
{{eqn | r = s_2 \circ s_1
| c = as all elements of $S$ commute w... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $S \subseteq G$ such that:
:$\forall x, y \in S: x \circ y = y \circ x$
Then the [[Definition:Generated Subgroup|subgroup generated]] by $S$ is [[Definition:Abelian Group|abelian]]. | Let $H = \gen S$ denote the [[Definition:Generated Subgroup|subgroup generated]] by $S$.
Let $a, b \in H$.
Then:
:$a = s_1$
:$b = s_2$
for some [[Definition:Word (Group Theory)|words]] $s_1, s_2$ of the [[Definition:Set of Words|set of words]] $\map W S$ of $S$.
Then:
{{begin-eqn}}
{{eqn | l = a \circ b
| r ... | Subgroup Generated by Commuting Elements is Abelian | https://proofwiki.org/wiki/Subgroup_Generated_by_Commuting_Elements_is_Abelian | https://proofwiki.org/wiki/Subgroup_Generated_by_Commuting_Elements_is_Abelian | [
"Abelian Groups"
] | [
"Definition:Group",
"Definition:Generated Subgroup",
"Definition:Abelian Group"
] | [
"Definition:Generated Subgroup",
"Definition:Word (Abstract Algebra)",
"Definition:Word (Abstract Algebra)",
"Definition:Element",
"Definition:Commutative/Elements",
"Definition:Abelian Group"
] |
proofwiki-15186 | Equivalence of Definitions of Generated Subgroup | {{TFAE|def = Generated Subgroup}}
Let $G$ be a group.
Let $S \subset G$ be a subset. | === $(1)$ is equivalent to $(2)$ ===
Let $H$ be the smallest subgroup containing $S$.
Let $\mathbb S$ be the set of subgroups containing $S$.
To show the equivalence of the two definitions, we need to show that $H = \bigcap \mathbb S$.
Since $H$ is a subgroup containing $S$:
:$H \in \mathbb S$
By Intersection is Subset... | {{TFAE|def = Generated Subgroup}}
Let $G$ be a [[Definition:Group|group]].
Let $S \subset G$ be a [[Definition:Subset|subset]]. | === $(1)$ is equivalent to $(2)$ ===
Let $H$ be the [[Definition:Smallest Set by Set Inclusion|smallest]] [[Definition:Subgroup|subgroup]] containing $S$.
Let $\mathbb S$ be the set of [[Definition:Subgroup|subgroups]] containing $S$.
To show the equivalence of the two definitions, we need to show that $H = \bigcap ... | Equivalence of Definitions of Generated Subgroup | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generated_Subgroup | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generated_Subgroup | [
"Generated Subgroups"
] | [
"Definition:Group",
"Definition:Subset"
] | [
"Definition:Smallest Set by Set Inclusion",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Subgroup",
"Intersection is Subset/General Result",
"Intersection of Subgroups is Subgroup/General Result",
"Definition:Subgroup",
"Definition:Smallest Set by Set Inclusion",
"Definition:Subgroup",
... |
proofwiki-15187 | Stabilizer of Element after Group Action | Let $\struct {G, \circ}$ be a group.
Let $S$ be a set.
Let $*_S: G \times S \to S$ be a group action.
Let $x \in S, a \in G$.
Then:
:$\Stab {a * x} = a^{-1} \circ \Stab x \circ a$ | {{begin-eqn}}
{{eqn | l = \Stab {a * x}
| r = \set {g \in G: g * \paren {a * x} = a * x}
| c = {{Defof|Stabilizer}}
}}
{{eqn | r = \set {g \in G: \paren {g \circ a} * x = a * x}
| c = {{GroupActionAxiom|2}}
}}
{{eqn | r = \set {g \in G: a^{-1} * \paren {g \circ a} * x = a^{-1} * \paren {a * x} }
... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $S$ be a [[Definition:Set|set]].
Let $*_S: G \times S \to S$ be a [[Definition:Group Action|group action]].
Let $x \in S, a \in G$.
Then:
:$\Stab {a * x} = a^{-1} \circ \Stab x \circ a$ | {{begin-eqn}}
{{eqn | l = \Stab {a * x}
| r = \set {g \in G: g * \paren {a * x} = a * x}
| c = {{Defof|Stabilizer}}
}}
{{eqn | r = \set {g \in G: \paren {g \circ a} * x = a * x}
| c = {{GroupActionAxiom|2}}
}}
{{eqn | r = \set {g \in G: a^{-1} * \paren {g \circ a} * x = a^{-1} * \paren {a * x} }
... | Stabilizer of Element after Group Action | https://proofwiki.org/wiki/Stabilizer_of_Element_after_Group_Action | https://proofwiki.org/wiki/Stabilizer_of_Element_after_Group_Action | [
"Stabilizers"
] | [
"Definition:Group",
"Definition:Set",
"Definition:Group Action"
] | [] |
proofwiki-15188 | Group Action of Symmetric Group/Subset | Let $r \in \N: 0 < r \le n$.
Let $B_r$ denote the set of all subsets of $\N_n$ of cardinality $r$:
:$B_r := \set {S \subseteq \N_n: \card S = r}$
Let $*$ be the mapping $*: S_n \times B_r \to B_r$ defined as:
:$\forall \pi \in S_n, \forall S \in B_r: \pi * S = \pi \sqbrk S$
where $\pi \sqbrk S$ denotes the image of $S... | The group action axioms are investigated in turn.
Let $\pi, \rho \in S_n$.
Let $S \in B_r$.
Thus:
{{begin-eqn}}
{{eqn | l = \pi * \paren {\rho * S}
| r = \pi * \rho \sqbrk S
| c = Definition of $*$
}}
{{eqn | r = \pi \sqbrk {\rho \sqbrk S}
| c = Definition of $*$
}}
{{eqn | r = \paren {\pi \circ \rho}... | Let $r \in \N: 0 < r \le n$.
Let $B_r$ denote the [[Definition:Set|set]] of all [[Definition:Subset|subsets]] of $\N_n$ of [[Definition:Cardinality|cardinality]] $r$:
:$B_r := \set {S \subseteq \N_n: \card S = r}$
Let $*$ be the [[Definition:Mapping|mapping]] $*: S_n \times B_r \to B_r$ defined as:
:$\forall \pi \i... | The [[Axiom:Group Action Axioms|group action axioms]] are investigated in turn.
Let $\pi, \rho \in S_n$.
Let $S \in B_r$.
Thus:
{{begin-eqn}}
{{eqn | l = \pi * \paren {\rho * S}
| r = \pi * \rho \sqbrk S
| c = Definition of $*$
}}
{{eqn | r = \pi \sqbrk {\rho \sqbrk S}
| c = Definition of $*$
}}
... | Group Action of Symmetric Group/Subset | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group/Subset | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group/Subset | [
"Group Action of Symmetric Group"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Cardinality",
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Group Action"
] | [
"Axiom:Group Action Axioms",
"Definition:Identity Mapping"
] |
proofwiki-15189 | Group Action of Symmetric Group on Subset is Transitive | Let $r \in \N: 0 < r \le n$.
Let $B_r$ denote the set of all subsets of $\N_n$ of cardinality $r$:
:$B_r := \set {S \subseteq \N_n: \card S = r}$
Let $*$ be the mapping $*: S_n \times B_r \to B_r$ defined as:
:$\forall \pi \in S_n, \forall S \in B_r: \pi * B_r = \pi \sqbrk S$
where $\pi \sqbrk S$ denotes the image of ... | From Group Action of Symmetric Group on Subset it is established that $*$ is a group action.
Let $U = \set {u_1, u_2, \ldots, u_r}$ and $V = \set {v_1, v_2, \ldots, v_r}$ be elements of $B_r$.
Then there exists a permutation $\rho \in S_n$ such that:
:$\map \rho {u_k} = v_k$
for all $k \in \set {1, 2, \ldots, r}$.
Thus... | Let $r \in \N: 0 < r \le n$.
Let $B_r$ denote the [[Definition:Set|set]] of all [[Definition:Subset|subsets]] of $\N_n$ of [[Definition:Cardinality|cardinality]] $r$:
:$B_r := \set {S \subseteq \N_n: \card S = r}$
Let $*$ be the [[Definition:Mapping|mapping]] $*: S_n \times B_r \to B_r$ defined as:
:$\forall \pi \i... | From [[Group Action of Symmetric Group on Subset]] it is established that $*$ is a [[Definition:Group Action|group action]].
Let $U = \set {u_1, u_2, \ldots, u_r}$ and $V = \set {v_1, v_2, \ldots, v_r}$ be [[Definition:Element|elements]] of $B_r$.
Then there exists a [[Definition:Permutation|permutation]] $\rho \in S... | Group Action of Symmetric Group on Subset is Transitive | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_on_Subset_is_Transitive | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_on_Subset_is_Transitive | [
"Group Action of Symmetric Group",
"Transitive Group Actions"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Cardinality",
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Transitive Group Action"
] | [
"Group Action of Symmetric Group/Subset",
"Definition:Group Action",
"Definition:Element",
"Definition:Permutation",
"Definition:Orbit of Group Action",
"Definition:Transitive Group Action"
] |
proofwiki-15190 | Coset Product on Non-Normal Subgroup is not Well-Defined | Let $\struct {G, \circ}$ be a group.
Let $H$ be a subgroup of $G$ which is not normal.
Let $a, b \in G$.
Then it is not necessarily the case that the coset product:
:$\paren {a \circ H} \circ \paren {b \circ H} = \paren {a \circ b} \circ H$
is well-defined. | Proof by Counterexample:
Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as:
{{:Symmetric Group on 3 Letters/Cayley Table}}
Consider the subgroups of $S_3$:
{{:Symmetric Group on 3 Letters/Subgroups}}
Let $H = \set {e, \tuple {12} }$.
From Normal Subgroups in Symmetric Group on 3 Letters,... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$ which is not [[Definition:Normal Subgroup|normal]].
Let $a, b \in G$.
Then it is not necessarily the case that the [[Definition:Coset Product|coset product]]:
:$\paren {a \circ H} \circ \paren {b \circ H} ... | [[Proof by Counterexample]]:
Let $S_3$ denote the [[Symmetric Group on 3 Letters]], whose [[Symmetric Group on 3 Letters/Cayley Table|Cayley table]] is given as:
{{:Symmetric Group on 3 Letters/Cayley Table}}
Consider the [[Symmetric Group on 3 Letters/Subgroups|subgroups]] of $S_3$:
{{:Symmetric Group on 3 Letters/S... | Coset Product on Non-Normal Subgroup is not Well-Defined | https://proofwiki.org/wiki/Coset_Product_on_Non-Normal_Subgroup_is_not_Well-Defined | https://proofwiki.org/wiki/Coset_Product_on_Non-Normal_Subgroup_is_not_Well-Defined | [
"Coset Product"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Normal Subgroup",
"Definition:Coset Product",
"Definition:Well-Defined/Operation"
] | [
"Proof by Counterexample",
"Symmetric Group on 3 Letters",
"Symmetric Group on 3 Letters/Cayley Table",
"Symmetric Group on 3 Letters/Subgroups",
"Symmetric Group on 3 Letters/Normal Subgroups",
"Definition:Normal Subgroup",
"Definition:Coset/Left Coset",
"Definition:Coset/Left Coset",
"Definition:C... |
proofwiki-15191 | Klein Four-Group is Normal in A4 | Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as:
{{:Alternating Group on 4 Letters/Cayley Table}}
The subsets of $A_4$ which form subgroups of $A_4$ are as follows:
Consider the order $4$ subgroup $V$ of $A_4$, presented by Cayley table:
:<nowiki>$\begin{array}{c|cccc}
\circ & e & ... | {{ProofWanted|Straightforward but tedious, unless someone has a short cut better than testing all the products}}
:$\index {A_4} V = 3$ follows from Lagrange's Theorem. | Let $A_4$ denote the [[Alternating Group on 4 Letters|alternating group on $4$ letters]], whose [[Alternating Group on 4 Letters/Cayley Table|Cayley table]] is given as:
{{:Alternating Group on 4 Letters/Cayley Table}}
The [[Definition:Subset|subsets]] of $A_4$ which form [[Definition:Subgroup|subgroups]] of $A_4$ are... | {{ProofWanted|Straightforward but tedious, unless someone has a short cut better than testing all the products}}
:$\index {A_4} V = 3$ follows from [[Lagrange's Theorem (Group Theory)|Lagrange's Theorem]]. | Klein Four-Group is Normal in A4/Proof 1 | https://proofwiki.org/wiki/Klein_Four-Group_is_Normal_in_A4 | https://proofwiki.org/wiki/Klein_Four-Group_is_Normal_in_A4/Proof_1 | [
"Klein Four-Group is Normal in A4",
"Alternating Group on 4 Letters",
"Examples of Normal Subgroups"
] | [
"Alternating Group on 4 Letters",
"Alternating Group on 4 Letters/Cayley Table",
"Definition:Subset",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Subgroup",
"Klein Four-Group/Cayley Table",
"Definition:Normal Subgroup",
"Definition:Index of Subgroup",
"Definition:Coset/Left... | [
"Lagrange's Theorem (Group Theory)"
] |
proofwiki-15192 | Klein Four-Group is Normal in A4 | Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as:
{{:Alternating Group on 4 Letters/Cayley Table}}
The subsets of $A_4$ which form subgroups of $A_4$ are as follows:
Consider the order $4$ subgroup $V$ of $A_4$, presented by Cayley table:
:<nowiki>$\begin{array}{c|cccc}
\circ & e & ... | By Order of Conjugate of Subgroup, conjugate subgroups have the same order.
Therefore, any conjugation of $V$ has $4$ elements.
By Subgroups of $A_4$, the only order-$4$ subgroup of $A_4$ is $V$.
Hence, any conjugation of $V$ in $A_4$ is $V$ itself.
By Subgroup equals Conjugate iff Normal, $V$ is normal in $A_4$.
{{qed... | Let $A_4$ denote the [[Alternating Group on 4 Letters|alternating group on $4$ letters]], whose [[Alternating Group on 4 Letters/Cayley Table|Cayley table]] is given as:
{{:Alternating Group on 4 Letters/Cayley Table}}
The [[Definition:Subset|subsets]] of $A_4$ which form [[Definition:Subgroup|subgroups]] of $A_4$ are... | By [[Order of Conjugate of Subgroup]], conjugate subgroups have the same order.
Therefore, any [[Definition:Conjugate of Group Subset|conjugation]] of $V$ has $4$ [[Definition:Element|elements]].
By [[Alternating Group on 4 Letters/Subgroups|Subgroups of $A_4$]], the only [[Definition:Order of Group|order-$4$]] [[Def... | Klein Four-Group is Normal in A4/Proof 2 | https://proofwiki.org/wiki/Klein_Four-Group_is_Normal_in_A4 | https://proofwiki.org/wiki/Klein_Four-Group_is_Normal_in_A4/Proof_2 | [
"Klein Four-Group is Normal in A4",
"Alternating Group on 4 Letters",
"Examples of Normal Subgroups"
] | [
"Alternating Group on 4 Letters",
"Alternating Group on 4 Letters/Cayley Table",
"Definition:Subset",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Subgroup",
"Klein Four-Group/Cayley Table",
"Definition:Normal Subgroup",
"Definition:Index of Subgroup",
"Definition:Coset/Left... | [
"Order of Conjugate of Subgroup",
"Definition:Conjugate (Group Theory)/Subset",
"Definition:Element",
"Alternating Group on 4 Letters/Subgroups",
"Definition:Order of Structure",
"Definition:Subgroup",
"Definition:Conjugate (Group Theory)/Subset",
"Subgroup equals Conjugate iff Normal",
"Definition:... |
proofwiki-15193 | Coset of Trivial Subgroup is Singleton | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $E := \struct {\set e, \circ}$ denote the trivial subgroup of $\struct {G, \circ}$.
Let $g \in G$.
Then the left coset and right coset of $E$ by $g$ is $\set g$. | {{begin-eqn}}
{{eqn | l = g \circ \set e
| r = \set {g \circ x: x \in \set e}
| c = {{Defof|Left Coset}}
}}
{{eqn | r = \set {g \circ e}
| c =
}}
{{eqn | r = \set g
| c =
}}
{{end-eqn}}
Similarly:
{{begin-eqn}}
{{eqn | l = \set e \circ g
| r = \set {x \circ g: x \in \set e}
| c = {... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $E := \struct {\set e, \circ}$ denote the [[Definition:Trivial Subgroup|trivial subgroup]] of $\struct {G, \circ}$.
Let $g \in G$.
Then the [[Definition:Left Coset|left coset]] and [[Definition:Right ... | {{begin-eqn}}
{{eqn | l = g \circ \set e
| r = \set {g \circ x: x \in \set e}
| c = {{Defof|Left Coset}}
}}
{{eqn | r = \set {g \circ e}
| c =
}}
{{eqn | r = \set g
| c =
}}
{{end-eqn}}
Similarly:
{{begin-eqn}}
{{eqn | l = \set e \circ g
| r = \set {x \circ g: x \in \set e}
| c ... | Coset of Trivial Subgroup is Singleton | https://proofwiki.org/wiki/Coset_of_Trivial_Subgroup_is_Singleton | https://proofwiki.org/wiki/Coset_of_Trivial_Subgroup_is_Singleton | [
"Cosets",
"Trivial Group"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Trivial Subgroup",
"Definition:Coset/Left Coset",
"Definition:Coset/Right Coset"
] | [] |
proofwiki-15194 | Inverse Elements of Right Transversal is Left Transversal | Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Let $S \subseteq G$ be a right transversal for $H$ in $G$.
Let $T$ be the set defined as:
:$T := \set {x^{-1}: x \in S}$
where $x^{-1}$ is the inverse of $x$ in $G$.
Then $T$ is a left transversal for $H$ in $G$. | Let $g \in G$.
We show that $g H$ contains exactly $1$ element of $T$.
Since $S$ is a right transversal:
:$\exists ! x \in S: x \in H g^{-1}$
By Right Cosets are Equal iff Element in Other Right Coset:
:$H x = H g^{-1}$
By Right Cosets are Equal iff Left Cosets by Inverse are Equal:
:$x^{-1} H = g H$
We have from defin... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $S \subseteq G$ be a [[Definition:Right Transversal|right transversal]] for $H$ in $G$.
Let $T$ be the [[Definition:Set|set]] defined as:
:$T := \set {x^{-1}: x \in S}$
where $x^{-1}$ is the [[Definition:Inverse Elem... | Let $g \in G$.
We show that $g H$ contains exactly $1$ element of $T$.
Since $S$ is a [[Definition:Right Transversal|right transversal]]:
:$\exists ! x \in S: x \in H g^{-1}$
By [[Right Cosets are Equal iff Element in Other Right Coset]]:
:$H x = H g^{-1}$
By [[Right Cosets are Equal iff Left Cosets by Inverse a... | Inverse Elements of Right Transversal is Left Transversal | https://proofwiki.org/wiki/Inverse_Elements_of_Right_Transversal_is_Left_Transversal | https://proofwiki.org/wiki/Inverse_Elements_of_Right_Transversal_is_Left_Transversal | [
"Transversals (Group Theory)"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Transversal (Group Theory)/Right Transversal",
"Definition:Set",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Transversal (Group Theory)/Left Transversal"
] | [
"Definition:Transversal (Group Theory)/Right Transversal",
"Right Cosets are Equal iff Element in Other Right Coset",
"Right Cosets are Equal iff Left Cosets by Inverse are Equal",
"Definition:Unique"
] |
proofwiki-15195 | Condition for Subset of Group to be Right Transversal | Let $G$ be a group.
Let $H$ be a subgroup of $G$ whose index in $G$ is $n$:
:$\index G H = n$
Let $S \subseteq G$ be a subset of $G$ of cardinality $n$.
Then $S$ is a right transversal for $H$ in $G$ {{iff}}:
:$\forall x, y \in S: x \ne y \implies x y^{-1} \notin H$ | From {{Defof|Right Transversal}}, $S$ is a right transversal for $H$ in $G$ {{iff}} every right coset of $H$ contains exactly one element of $S$.
Since there are $n$ right cosets of $H$ and $S$ has cardinality $n$, if $S$ is not a right transversal for $H$ in $G$, at least one right coset of $H$ contain more than one ... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$ whose [[Definition:Index of Subgroup|index]] in $G$ is $n$:
:$\index G H = n$
Let $S \subseteq G$ be a [[Definition:Subset|subset]] of $G$ of [[Definition:Cardinality|cardinality]] $n$.
Then $S$ is a [[Definition:Right Tra... | From {{Defof|Right Transversal}}, $S$ is a [[Definition:Right Transversal|right transversal]] for $H$ in $G$ {{iff}} every [[Definition:Right Coset|right coset]] of $H$ contains exactly one [[Definition:Element|element]] of $S$.
Since there are $n$ [[Definition:Right Coset|right cosets]] of $H$ and $S$ has [[Definiti... | Condition for Subset of Group to be Right Transversal | https://proofwiki.org/wiki/Condition_for_Subset_of_Group_to_be_Right_Transversal | https://proofwiki.org/wiki/Condition_for_Subset_of_Group_to_be_Right_Transversal | [
"Transversals (Group Theory)"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Index of Subgroup",
"Definition:Subset",
"Definition:Cardinality",
"Definition:Transversal (Group Theory)/Right Transversal"
] | [
"Definition:Transversal (Group Theory)/Right Transversal",
"Definition:Coset/Right Coset",
"Definition:Element",
"Definition:Coset/Right Coset",
"Definition:Cardinality",
"Definition:Transversal (Group Theory)/Right Transversal",
"Definition:Coset/Right Coset",
"Definition:Element",
"Definition:Cont... |
proofwiki-15196 | Group Action on Coset Space is Transitive | Let $G$ be a group whose identity is $e$.
Let $H$ be a subgroup of $G$.
Let $*: G \times G / H \to G / H$ be the action on the (left) coset space:
:$\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$
Then $G$ is a transitive group action. | It is established in Action of Group on Coset Space is Group Action that $*$ is a group action.
It remains to be shown that:
:$\forall g' H \in G / H: \Orb {g' H} = G / H$
where $\Orb {g' H}$ denotes the orbit of $g' H \in G / H$ under $*$.
Let $a H, b H \in G / H$ such that $a H \ne b H$.
We have that:
:$\exists x \in... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $*: G \times G / H \to G / H$ be the [[Definition:Group Action on Coset Space|action on the (left) coset space]]:
:$\forall g \in G, \forall g' H \in G / H: g * \pa... | It is established in [[Action of Group on Coset Space is Group Action]] that $*$ is a [[Definition:Group Action|group action]].
It remains to be shown that:
:$\forall g' H \in G / H: \Orb {g' H} = G / H$
where $\Orb {g' H}$ denotes the [[Definition:Orbit (Group Theory)|orbit]] of $g' H \in G / H$ under $*$.
Let $a H... | Group Action on Coset Space is Transitive | https://proofwiki.org/wiki/Group_Action_on_Coset_Space_is_Transitive | https://proofwiki.org/wiki/Group_Action_on_Coset_Space_is_Transitive | [
"Group Action on Coset Space",
"Transitive Group Actions"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Group Action on Coset Space",
"Definition:Transitive Group Action"
] | [
"Action of Group on Coset Space is Group Action",
"Definition:Group Action",
"Definition:Orbit (Group Theory)"
] |
proofwiki-15197 | Stabilizer of Coset under Group Action on Coset Space | Let $G$ be a group whose identity is $e$.
Let $H$ be a subgroup of $G$.
Let $*: G \times G / H \to G / H$ be the action on the (left) coset space:
:$\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$
Then the stabilizer of $a H$ under $*$ is given by:
:$\Stab {a H} = a H a^{-1}$ | It is established in Action of Group on Coset Space is Group Action that $*$ is a group action.
Then:
{{begin-eqn}}
{{eqn | l = \Stab {a H}
| r = \set {g \in G: g * a H = a H}
| c = {{Defof|Stabilizer}}
}}
{{eqn | r = \set {g \in G: \paren {g a} H = a H}
| c =
}}
{{eqn | r = \set {g \in G: g H = a H ... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $*: G \times G / H \to G / H$ be the [[Definition:Group Action on Coset Space|action on the (left) coset space]]:
:$\forall g \in G, \forall g' H \in G / H: g * \par... | It is established in [[Action of Group on Coset Space is Group Action]] that $*$ is a [[Definition:Group Action|group action]].
Then:
{{begin-eqn}}
{{eqn | l = \Stab {a H}
| r = \set {g \in G: g * a H = a H}
| c = {{Defof|Stabilizer}}
}}
{{eqn | r = \set {g \in G: \paren {g a} H = a H}
| c =
}}
{{e... | Stabilizer of Coset under Group Action on Coset Space | https://proofwiki.org/wiki/Stabilizer_of_Coset_under_Group_Action_on_Coset_Space | https://proofwiki.org/wiki/Stabilizer_of_Coset_under_Group_Action_on_Coset_Space | [
"Group Action on Coset Space",
"Stabilizers"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Group Action on Coset Space",
"Definition:Stabilizer"
] | [
"Action of Group on Coset Space is Group Action",
"Definition:Group Action"
] |
proofwiki-15198 | Index of Subgroup equals Index of Conjugate | Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Then:
:$\index G H = \index G {a H a^{-1} }$
where $\index G H$ denotes the index of $H$ in $G$. | {{ProofWanted|tired}} | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Then:
:$\index G H = \index G {a H a^{-1} }$
where $\index G H$ denotes the [[Definition:Index of Subgroup|index]] of $H$ in $G$. | {{ProofWanted|tired}} | Index of Subgroup equals Index of Conjugate | https://proofwiki.org/wiki/Index_of_Subgroup_equals_Index_of_Conjugate | https://proofwiki.org/wiki/Index_of_Subgroup_equals_Index_of_Conjugate | [
"Subgroups",
"Conjugacy"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Index of Subgroup"
] | [] |
proofwiki-15199 | Normality Relation is not Transitive | Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $K$ be a normal subgroup of $N$.
Then it is not necessarily the case that $K$ is a normal subgroup of $G$. | Proof by Counterexample:
Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as:
{{:Alternating Group on 4 Letters/Cayley Table}}
From Normality of Subgroups of Alternating Group on 4 Letters:
:$K := \set {e, t, u, v}$ is a normal subgroup of $A_4$
:$T := \set {e, t}$ is not a normal subg... | Let $G$ be a [[Definition:Group|group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $K$ be a [[Definition:Normal Subgroup|normal subgroup]] of $N$.
Then it is not necessarily the case that $K$ is a [[Definition:Normal Subgroup|normal subgroup]] of $G$. | [[Proof by Counterexample]]:
Let $A_4$ denote the [[Alternating Group on 4 Letters|alternating group on $4$ letters]], whose [[Alternating Group on 4 Letters/Cayley Table|Cayley table]] is given as:
{{:Alternating Group on 4 Letters/Cayley Table}}
From [[Normality of Subgroups of Alternating Group on 4 Letters]]:
:$K... | Normality Relation is not Transitive/Proof 1 | https://proofwiki.org/wiki/Normality_Relation_is_not_Transitive | https://proofwiki.org/wiki/Normality_Relation_is_not_Transitive/Proof_1 | [
"Normal Subgroups",
"Normality Relation is not Transitive"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup"
] | [
"Proof by Counterexample",
"Alternating Group on 4 Letters",
"Alternating Group on 4 Letters/Cayley Table",
"Alternating Group on 4 Letters/Normality of Subgroups",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Subgroup of Abelian Group is Normal",
"Definition:Normal Subgroup"
] |
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