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-21700 | Existence of Ordinal with no Surjection from Set | Let $S$ be a set.
Then, there exists a non-empty ordinal $\alpha$ such that there is no surjection from $S$ to $\alpha$. | By Hartogs' Lemma, let $\alpha$ be an ordinal such that there is no injection from $\alpha$ to $\powerset S$.
From Empty Mapping is Injective, it follows that $\alpha$ is non-empty.
{{AimForCont}} there is a surjection $\phi : S \to \alpha$.
Define $\psi : \alpha \to \powerset S$ as:
:$\map \psi \gamma = \set {x \in S ... | Let $S$ be a [[Definition:Set|set]].
Then, there exists a [[Definition:Non-Empty Set|non-empty]] [[Definition:Ordinal|ordinal]] $\alpha$ such that there is no [[Definition:Surjection|surjection]] from $S$ to $\alpha$. | By [[Hartogs' Lemma (Set Theory)|Hartogs' Lemma]], let $\alpha$ be an [[Definition:Ordinal|ordinal]] such that there is no [[Definition:Injection|injection]] from $\alpha$ to $\powerset S$.
From [[Empty Mapping is Injective]], it follows that $\alpha$ is [[Definition:Non-Empty Set|non-empty]].
{{AimForCont}} there i... | Existence of Ordinal with no Surjection from Set | https://proofwiki.org/wiki/Existence_of_Ordinal_with_no_Surjection_from_Set | https://proofwiki.org/wiki/Existence_of_Ordinal_with_no_Surjection_from_Set | [
"Ordinals",
"Surjections"
] | [
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Ordinal",
"Definition:Surjection"
] | [
"Hartogs' Lemma (Set Theory)",
"Definition:Ordinal",
"Definition:Injection",
"Empty Mapping is Injective",
"Definition:Non-Empty Set",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Mapping",
"Definition:Injection",
"Category:Ordinals",
"Category:Surjections"
] |
proofwiki-21701 | Cartesian Product of Infinite Set Equivalent to Itself implies Axiom of Choice | Suppose that, for every Dedekind-infinite set $A$, it holds that:
:$A \sim A \times A$
where $\sim$ denotes set equivalence, and $\times$ denotes the cartesian product.
Then, the {{Axiom-link|Choice}} holds. | By Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice, it suffices to show that the hypothesis implies Zermelo's Well-Ordering Theorem.
Since every finite set is well-orderable, it suffices to consider only infinite sets.
Let $B$ be an arbitrary infinite set.
{{WLOG}}, assume that $B \cap \On = \O$.
By Ex... | Suppose that, for every [[Definition:Dedekind-Infinite Set|Dedekind-infinite set]] $A$, it holds that:
:$A \sim A \times A$
where $\sim$ denotes [[Definition:Set Equivalence|set equivalence]], and $\times$ denotes the [[Definition:Cartesian Product|cartesian product]].
Then, the {{Axiom-link|Choice}} holds. | By [[Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice]], it suffices to show that the hypothesis implies [[Zermelo's Well-Ordering Theorem]].
Since every [[Definition:Finite Set|finite set]] is [[Definition:Well-Orderable Set|well-orderable]], it suffices to consider only [[Definition:Infinite Set|infi... | Cartesian Product of Infinite Set Equivalent to Itself implies Axiom of Choice | https://proofwiki.org/wiki/Cartesian_Product_of_Infinite_Set_Equivalent_to_Itself_implies_Axiom_of_Choice | https://proofwiki.org/wiki/Cartesian_Product_of_Infinite_Set_Equivalent_to_Itself_implies_Axiom_of_Choice | [
"Cartesian Product",
"Equivalents of Axiom of Choice",
"Set Equivalence"
] | [
"Definition:Dedekind-Infinite",
"Definition:Set Equivalence",
"Definition:Cartesian Product"
] | [
"Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice",
"Zermelo's Well-Ordering Theorem",
"Definition:Finite Set",
"Definition:Well-Orderable Set",
"Definition:Infinite Set",
"Definition:Infinite Set",
"Existence of Ordinal with no Surjection from Set",
"Definition:Surjection",
"Set is ... |
proofwiki-21702 | Frame of Topological Space is Frame | :$\map \Omega T$ is a frame | Recall that the frame of topological space $T$ is defined to be:
:$\map \Omega T = \struct{\tau, \subseteq}$
where $\subseteq$ denotes the subset relation.
From Topology forms Complete Lattice:
:$\map \Omega T$ is a complete lattice
Furthermore:
:$\forall \TT \subseteq \tau : \sup \TT = \bigcup \TT, \inf \TT = \paren{\... | :$\map \Omega T$ is a [[Definition:Frame (Lattice Theory)|frame]] | Recall that the [[Definition:Frame of Topological Space|frame of topological space]] $T$ is defined to be:
:$\map \Omega T = \struct{\tau, \subseteq}$
where $\subseteq$ denotes the [[Definition:Subset|subset relation]].
From [[Topology forms Complete Lattice]]:
:$\map \Omega T$ is a [[Definition:Complete Lattice|comp... | Frame of Topological Space is Frame | https://proofwiki.org/wiki/Frame_of_Topological_Space_is_Frame | https://proofwiki.org/wiki/Frame_of_Topological_Space_is_Frame | [
"Topological Spaces",
"Frames"
] | [
"Definition:Frame (Lattice Theory)"
] | [
"Definition:Frame of Topological Space",
"Definition:Subset",
"Topology forms Complete Lattice",
"Definition:Complete Lattice",
"Definition:Interior (Topology)",
"Interior of Open Set",
"Intersection Distributes over Union/General Result",
"Axiom:Infinite Join Distributive Law",
"Definition:Frame (L... |
proofwiki-21703 | Locale of Topological Space is Locale | :$\map \Omega T$ is a locale | By definition of frame of topological space:
:the frame of $T$ is $\map \Omega T$
From Frame of Topological Space is Frame:
:$\map \Omega T$ is a frame
By definition of locale:
:$\map \Omega T$ is a locale
{{qed}} | :$\map \Omega T$ is a [[Definition:Locale (Lattice Theory)|locale]] | By definition of [[Definition:Frame of Topological Space|frame of topological space]]:
:the [[Definition:Frame of Topological Space|frame]] of $T$ is $\map \Omega T$
From [[Frame of Topological Space is Frame]]:
:$\map \Omega T$ is a [[Definition:Frame (Lattice Theory)|frame]]
By definition of [[Definition:Locale (La... | Locale of Topological Space is Locale | https://proofwiki.org/wiki/Locale_of_Topological_Space_is_Locale | https://proofwiki.org/wiki/Locale_of_Topological_Space_is_Locale | [
"Topological Spaces",
"Locales"
] | [
"Definition:Locale (Lattice Theory)"
] | [
"Definition:Frame of Topological Space",
"Definition:Frame of Topological Space",
"Frame of Topological Space is Frame",
"Definition:Frame (Lattice Theory)",
"Definition:Locale (Lattice Theory)",
"Definition:Locale (Lattice Theory)"
] |
proofwiki-21704 | Dilogarithm of One | :$\map {\Li_2} 1 = \map \zeta 2$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}
| c = Power Series Expansion for Spence's Function
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {1^n} {n^2}
| c = $z := 1$
}}
{{eqn | r = \map \zeta 2
| c = Basel Problem
}}
{{end-eqn}}
{{qed}}
Categ... | :$\map {\Li_2} 1 = \map \zeta 2$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}
| c = [[Power Series Expansion for Spence's Function]]
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {1^n} {n^2}
| c = $z := 1$
}}
{{eqn | r = \map \zeta 2
| c = [[Basel Problem]]
}}
{{end-eqn}}
{{qed... | Dilogarithm of One | https://proofwiki.org/wiki/Dilogarithm_of_One | https://proofwiki.org/wiki/Dilogarithm_of_One | [
"Examples of Dilogarithm Function",
"Spence's Function"
] | [] | [
"Power Series Expansion for Spence's Function",
"Basel Problem",
"Category:Examples of Dilogarithm Function",
"Category:Spence's Function"
] |
proofwiki-21705 | Dilogarithm Reflection Formula | :$\map {\Li_2} z + \map {\Li_2} {1 - z} = \map \zeta 2 - \map \ln z \map \ln {1 - z}$ | From the definition of the dilogarithm function:
:$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {1 - x} } x \rd x$
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \map \ln {1 - x}
... | :$\map {\Li_2} z + \map {\Li_2} {1 - z} = \map \zeta 2 - \map \ln z \map \ln {1 - z}$ | From the definition of the [[Definition:Dilogarithm Function|dilogarithm function]]:
:$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {1 - x} } x \rd x$
With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
... | Dilogarithm Reflection Formula | https://proofwiki.org/wiki/Dilogarithm_Reflection_Formula | https://proofwiki.org/wiki/Dilogarithm_Reflection_Formula | [
"Spence's Function",
"Reflection Formulas"
] | [] | [
"Definition:Spence's Function",
"Definition:Primitive",
"Derivative of Natural Logarithm Function",
"Chain Rule",
"Primitive of Reciprocal",
"Integration by Parts",
"Dilogarithm of One"
] |
proofwiki-21706 | Dilogarithm of Minus One | :$\map {\Li_2} {-1} = -\dfrac 1 2 \map \zeta 2$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}
| c = Power Series Expansion for Spence's Function
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} {n^2}
| c = $z := -1$
}}
{{eqn | r = - \map \eta 2
| c = {{Defof|Dirichlet Eta Function... | :$\map {\Li_2} {-1} = -\dfrac 1 2 \map \zeta 2$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}
| c = [[Power Series Expansion for Spence's Function]]
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} {n^2}
| c = $z := -1$
}}
{{eqn | r = - \map \eta 2
| c = {{Defof|Dirichlet Eta Func... | Dilogarithm of Minus One | https://proofwiki.org/wiki/Dilogarithm_of_Minus_One | https://proofwiki.org/wiki/Dilogarithm_of_Minus_One | [
"Examples of Dilogarithm Function",
"Spence's Function"
] | [] | [
"Power Series Expansion for Spence's Function",
"Riemann Zeta Function in terms of Dirichlet Eta Function",
"Category:Examples of Dilogarithm Function",
"Category:Spence's Function"
] |
proofwiki-21707 | Dilogarithm of One Half | :$\map {\Li_2} {\dfrac 1 2} = \dfrac 1 2 \paren {\map \zeta 2 - \paren {\map \ln 2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z + \map {\Li_2} {1 - z}
| r = \map \zeta 2 - \map \ln z \map \ln {1 - z}
| c = Dilogarithm Reflection Formula
}}
{{eqn | ll = \leadsto
| l = \map {\Li_2} {\frac 1 2} + \map {\Li_2} {\frac 1 2}
| r = \map \zeta 2 - \map \ln {\frac 1 2} \map \ln {\frac 1 2}
... | :$\map {\Li_2} {\dfrac 1 2} = \dfrac 1 2 \paren {\map \zeta 2 - \paren {\map \ln 2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z + \map {\Li_2} {1 - z}
| r = \map \zeta 2 - \map \ln z \map \ln {1 - z}
| c = [[Dilogarithm Reflection Formula]]
}}
{{eqn | ll = \leadsto
| l = \map {\Li_2} {\frac 1 2} + \map {\Li_2} {\frac 1 2}
| r = \map \zeta 2 - \map \ln {\frac 1 2} \map \ln {\frac 1... | Dilogarithm of One Half | https://proofwiki.org/wiki/Dilogarithm_of_One_Half | https://proofwiki.org/wiki/Dilogarithm_of_One_Half | [
"Examples of Dilogarithm Function",
"Spence's Function"
] | [] | [
"Dilogarithm Reflection Formula",
"Logarithm of Reciprocal",
"Category:Examples of Dilogarithm Function",
"Category:Spence's Function"
] |
proofwiki-21708 | Dilogarithm of Minus Z Plus Dilogarithm of Minus Reciprocal of Z | :$\map {\Li_2} {-z} + \map {\Li_2} {-\dfrac 1 z} = -\map \zeta 2 - \dfrac 1 2 \map {\ln^2} z$ | From the definition of the dilogarithm function:
:$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {1 - x} } x \rd x$
Taking the derivative of both sides at $-\dfrac 1 z$
{{begin-eqn}}
{{eqn | l = \frac \d {\d z} \map {\Li_2} {-\dfrac 1 z}
| r = -\paren {\dfrac {\map \ln {1 - \paren {-\dfrac 1 z} } } {\paren {-\d... | :$\map {\Li_2} {-z} + \map {\Li_2} {-\dfrac 1 z} = -\map \zeta 2 - \dfrac 1 2 \map {\ln^2} z$ | From the definition of the [[Definition:Dilogarithm Function|dilogarithm function]]:
:$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {1 - x} } x \rd x$
Taking the [[Definition:Derivative|derivative]] of both sides at $-\dfrac 1 z$
{{begin-eqn}}
{{eqn | l = \frac \d {\d z} \map {\Li_2} {-\dfrac 1 z}
| r = -\par... | Dilogarithm of Minus Z Plus Dilogarithm of Minus Reciprocal of Z | https://proofwiki.org/wiki/Dilogarithm_of_Minus_Z_Plus_Dilogarithm_of_Minus_Reciprocal_of_Z | https://proofwiki.org/wiki/Dilogarithm_of_Minus_Z_Plus_Dilogarithm_of_Minus_Reciprocal_of_Z | [
"Spence's Function"
] | [] | [
"Definition:Spence's Function",
"Definition:Derivative",
"Derivative of Reciprocal",
"Difference of Logarithms",
"Definition:Primitive (Calculus)/Integration",
"Fundamental Theorem of Calculus",
"Linear Combination of Integrals/Definite",
"Primitive of Logarithm of x over x",
"Dilogarithm of Minus O... |
proofwiki-21709 | Dilogarithm of Zero | :$\map {\Li_2} 0 = 0$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}
| c = Power Series Expansion for Spence's Function
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {0^n} {n^2}
| c = $z := 0$
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}}
Category:Examples of Dilogari... | :$\map {\Li_2} 0 = 0$ | {{begin-eqn}}
{{eqn | l = \map {\Li_2} z
| r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}
| c = [[Power Series Expansion for Spence's Function]]
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {0^n} {n^2}
| c = $z := 0$
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed}}
[[Category:Examples of D... | Dilogarithm of Zero | https://proofwiki.org/wiki/Dilogarithm_of_Zero | https://proofwiki.org/wiki/Dilogarithm_of_Zero | [
"Examples of Dilogarithm Function",
"Spence's Function"
] | [] | [
"Power Series Expansion for Spence's Function",
"Category:Examples of Dilogarithm Function",
"Category:Spence's Function"
] |
proofwiki-21710 | Dilogarithm of One Minus Z Plus Dilogarithm of One Minus Reciprocal of Z | :$\map {\Li_2} {1 - z} + \map {\Li_2} {1 - \dfrac 1 z} = -\dfrac 1 2 \map {\ln^2} z$ | From the definition of the dilogarithm function:
:$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {1 - x} } x \rd x$
Taking the derivative of both sides at $-\dfrac z {1 - z}$
{{begin-eqn}}
{{eqn | l = \frac {\d } {\d z} \map {\Li_2} {-\dfrac z {1 - z} }
| r = -\paren {\dfrac {\map \ln {1 - \paren {-\dfrac z {1 ... | :$\map {\Li_2} {1 - z} + \map {\Li_2} {1 - \dfrac 1 z} = -\dfrac 1 2 \map {\ln^2} z$ | From the definition of the [[Definition:Dilogarithm Function|dilogarithm function]]:
:$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {1 - x} } x \rd x$
Taking the [[Definition:Derivative|derivative]] of both sides at $-\dfrac z {1 - z}$
{{begin-eqn}}
{{eqn | l = \frac {\d } {\d z} \map {\Li_2} {-\dfrac z {1 - z} }
... | Dilogarithm of One Minus Z Plus Dilogarithm of One Minus Reciprocal of Z | https://proofwiki.org/wiki/Dilogarithm_of_One_Minus_Z_Plus_Dilogarithm_of_One_Minus_Reciprocal_of_Z | https://proofwiki.org/wiki/Dilogarithm_of_One_Minus_Z_Plus_Dilogarithm_of_One_Minus_Reciprocal_of_Z | [
"Spence's Function"
] | [] | [
"Definition:Spence's Function",
"Definition:Derivative",
"Difference of Logarithms",
"Natural Logarithm of 1 is 0",
"Definition:Primitive (Calculus)/Integration",
"Fundamental Theorem of Calculus",
"Linear Combination of Integrals/Definite",
"Integration by Substitution",
"Primitive of Logarithm of ... |
proofwiki-21711 | Complement of Singleton Closure is Meet-Irreducible | Let $\struct {S, \tau}$ be a topological space.
Let $x \in S$.
Then:
:$S \setminus \set x^-$ is a meet-irreducible open set
where $\set x^-$ denotes the closure of $\set x$ | From Characterization of Meet-Irreducible Open Set:
$S \setminus \set x^-$ is a '''meet-irreducible open set''' {{iff}}
:$\forall U, V \in \tau : \paren {U \cap V \subseteq S \setminus \set x^- \implies U \subseteq S \setminus \set x^- \text { or } V \subseteq S \setminus \set x^-}$
Let:
:$U, V \in \tau: U \cap V \subs... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Then:
:$S \setminus \set x^-$ is a [[Definition:Meet-Irreducible Open Set|meet-irreducible open set]]
where $\set x^-$ denotes the [[Definition:Closure (Topology)|closure]] of $\set x$ | From [[Characterization of Meet-Irreducible Open Set]]:
$S \setminus \set x^-$ is a '''[[Definition:Meet-Irreducible Open Set|meet-irreducible open set]]''' {{iff}}
:$\forall U, V \in \tau : \paren {U \cap V \subseteq S \setminus \set x^- \implies U \subseteq S \setminus \set x^- \text { or } V \subseteq S \setminus \... | Complement of Singleton Closure is Meet-Irreducible | https://proofwiki.org/wiki/Complement_of_Singleton_Closure_is_Meet-Irreducible | https://proofwiki.org/wiki/Complement_of_Singleton_Closure_is_Meet-Irreducible | [
"Meet-Irreducible Open Sets",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Meet-Irreducible Open Set",
"Definition:Closure (Topology)"
] | [
"Characterization of Meet-Irreducible Open Set",
"Definition:Meet-Irreducible Open Set",
"Set is Subset of its Topological Closure",
"Definition:Set Difference",
"Definition:Subset",
"Definition:Set Intersection",
"Definition:Set Complement",
"Definition:Closed Set",
"Closure of Subset of Closed Set... |
proofwiki-21712 | Existence of Class Intersection | For any classes $X, Y$, the intersection $X \cap Y$ exists and is unique.
That is:
:$\forall X, Y: \exists! Z: \forall u: u \in Z \iff u \in X \land u \in Y$
where $X \cap Y := Z$. | Let $X, Y$ be arbitrary.
By Axiom $\text B 2$, there is some class $Z$ such that:
:$\forall u: u \in Z \iff u \in X \land u \in Y$
satisfying the existence portion of the theorem.
{{qed|lemma}}
Now, let some $Z'$ satisfy:
:$\forall u: u \in Z' \iff u \in X \land u \in Y$
For each set $u$:
:$u \in Z \iff u \in X \land u... | For any [[Definition:Class|classes]] $X, Y$, the [[Definition:Class Intersection|intersection]] $X \cap Y$ exists and is [[Definition:Unique|unique]].
That is:
:$\forall X, Y: \exists! Z: \forall u: u \in Z \iff u \in X \land u \in Y$
where $X \cap Y := Z$. | Let $X, Y$ be arbitrary.
By [[Axiom:Axioms of Class Existence|Axiom $\text B 2$]], there is some [[Definition:Class|class]] $Z$ such that:
:$\forall u: u \in Z \iff u \in X \land u \in Y$
satisfying the existence portion of the theorem.
{{qed|lemma}}
Now, let some $Z'$ satisfy:
:$\forall u: u \in Z' \iff u \in X \lan... | Existence of Class Intersection | https://proofwiki.org/wiki/Existence_of_Class_Intersection | https://proofwiki.org/wiki/Existence_of_Class_Intersection | [
"Class Theory",
"Von Neumann-Bernays-Gödel Set Theory"
] | [
"Definition:Class",
"Definition:Class Intersection",
"Definition:Unique"
] | [
"Axiom:Axioms of Class Existence",
"Definition:Class",
"Definition:Set/Class Theory",
"Axiom:Axiom of Extension/Class Theory",
"Category:Class Theory",
"Category:Von Neumann-Bernays-Gödel Set Theory"
] |
proofwiki-21713 | Existence of Class Complement | Let $X$ be a class.
Then there is a unique class $Z$ such that, for every set $u$:
:$u \in Z \iff u \notin X$ | Let $X$ be arbitrary.
By Axiom $\text B 3$, there is a class $Z$ such that:
:$\forall u: u \in Z \iff u \notin X$
This satisfies the existence portion of the theorem.
{{qed|lemma}}
Let $Z'$ be a class such that:
:$\forall u: u \in Z' \iff u \notin X$
Then, for every set $u$:
:$u \in Z \iff u \notin X \iff u \in Z'$
The... | Let $X$ be a [[Definition:Class|class]].
Then there is a [[Definition:Unique|unique]] [[Definition:Class|class]] $Z$ such that, for every [[Definition:Set|set]] $u$:
:$u \in Z \iff u \notin X$ | Let $X$ be arbitrary.
By [[Axiom:Axioms of Class Existence|Axiom $\text B 3$]], there is a [[Definition:Class|class]] $Z$ such that:
:$\forall u: u \in Z \iff u \notin X$
This satisfies the existence portion of the theorem.
{{qed|lemma}}
Let $Z'$ be a [[Definition:Class|class]] such that:
:$\forall u: u \in Z' \iff... | Existence of Class Complement | https://proofwiki.org/wiki/Existence_of_Class_Complement | https://proofwiki.org/wiki/Existence_of_Class_Complement | [
"Class Theory",
"Von Neumann-Bernays-Gödel Set Theory"
] | [
"Definition:Class",
"Definition:Unique",
"Definition:Class",
"Definition:Set"
] | [
"Axiom:Axioms of Class Existence",
"Definition:Class",
"Definition:Class",
"Definition:Set",
"Axiom:Axiom of Extension/Class Theory",
"Category:Class Theory",
"Category:Von Neumann-Bernays-Gödel Set Theory"
] |
proofwiki-21714 | Sober Space is T0 | Let $\struct {S, \tau}$ be a sober space.
Then:
:$\struct {S, \tau}$ is a $T_0$ (Kolmogorov) space. | Let $x, y \in S : x \ne y$.
From Complement of Singleton Closure is Meet-Irreducible:
:$S \setminus \set x^-$ is a meet-irreducible open set
By definition of a sober space:
:$S \setminus \set x^-$ is uniquely defined by $x$
Hence:
:$S \setminus \set x^- \ne S \setminus \set y^-$
From Equal Relative Complements iff Equa... | Let $\struct {S, \tau}$ be a [[Definition:Sober Space|sober space]].
Then:
:$\struct {S, \tau}$ is a [[Definition:T0 Space|$T_0$ (Kolmogorov) space]]. | Let $x, y \in S : x \ne y$.
From [[Complement of Singleton Closure is Meet-Irreducible]]:
:$S \setminus \set x^-$ is a [[Definition:Meet-Irreducible Open Set|meet-irreducible open set]]
By definition of a [[Definition:Sober Space|sober space]]:
:$S \setminus \set x^-$ is uniquely defined by $x$
Hence:
:$S \setminu... | Sober Space is T0 | https://proofwiki.org/wiki/Sober_Space_is_T0 | https://proofwiki.org/wiki/Sober_Space_is_T0 | [
"Sober Spaces",
"T0 Spaces"
] | [
"Definition:Sober Space",
"Definition:T0 Space"
] | [
"Complement of Singleton Closure is Meet-Irreducible",
"Definition:Meet-Irreducible Open Set",
"Definition:Sober Space",
"Equal Relative Complements iff Equal Subsets",
"Characterization of T0 Space by Distinct Closures of Singletons",
"Definition:T0 Space"
] |
proofwiki-21715 | T2 Space is Sober Space | Let $\struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
Then:
:$\struct {S, \tau}$ is a sober space. | Let $W \in \tau$ be meet-irreducible open proper subset of $S$.
Let $x \in S \setminus W$.
We have $S \setminus W$ is a closed set by definition.
From Closure of Subset of Closed Set of Topological Space is Subset:
:$\set x^- \subseteq S \setminus W$
From Relative Complement inverts Subsets of Relative Complement:
:$W ... | Let $\struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Then:
:$\struct {S, \tau}$ is a [[Definition:Sober Space|sober space]]. | Let $W \in \tau$ be [[Definition:Meet-Irreducible Open Set|meet-irreducible open]] [[Definition:Proper Subset|proper subset]] of $S$.
Let $x \in S \setminus W$.
We have $S \setminus W$ is a [[Definition:Closed Set (Topology)|closed set]] by definition.
From [[Closure of Subset of Closed Set of Topological Space is ... | T2 Space is Sober Space | https://proofwiki.org/wiki/T2_Space_is_Sober_Space | https://proofwiki.org/wiki/T2_Space_is_Sober_Space | [
"Sober Spaces",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Sober Space"
] | [
"Definition:Meet-Irreducible Open Set",
"Definition:Proper Subset",
"Definition:Closed Set/Topology",
"Closure of Subset of Closed Set of Topological Space is Subset",
"Relative Complement inverts Subsets of Relative Complement",
"Definition:T2 Space",
"Intersection Distributes over Union",
"Set Inter... |
proofwiki-21716 | Dilogarithm of One Minus Reciprocal of Golden Mean | :$\map {\Li_2} {1 - \dfrac 1 \phi} = \dfrac 2 5 \map \zeta 2 - \paren {\map \ln \phi}^2$ | We first note the following:
{{begin-eqn}}
{{eqn | n = 1
| l = -\frac 1 \phi
| r = 1 - \phi
| c = Reciprocal Form of One Minus Golden Mean
}}
{{eqn | ll= \leadsto
| n = 2
| l = \frac 1 {\phi^2}
| r = 1 - \dfrac 1 \phi
| c = dividing through by $-\phi$ and rearranging
}}
{{end-e... | :$\map {\Li_2} {1 - \dfrac 1 \phi} = \dfrac 2 5 \map \zeta 2 - \paren {\map \ln \phi}^2$ | We first note the following:
{{begin-eqn}}
{{eqn | n = 1
| l = -\frac 1 \phi
| r = 1 - \phi
| c = [[Reciprocal Form of One Minus Golden Mean]]
}}
{{eqn | ll= \leadsto
| n = 2
| l = \frac 1 {\phi^2}
| r = 1 - \dfrac 1 \phi
| c = dividing through by $-\phi$ and rearranging
}}
{{... | Dilogarithm of One Minus Reciprocal of Golden Mean | https://proofwiki.org/wiki/Dilogarithm_of_One_Minus_Reciprocal_of_Golden_Mean | https://proofwiki.org/wiki/Dilogarithm_of_One_Minus_Reciprocal_of_Golden_Mean | [
"Examples of Dilogarithm Function",
"Spence's Function",
"Golden Mean"
] | [] | [
"Reciprocal Form of One Minus Golden Mean",
"Dilogarithm of Square",
"Dilogarithm of One Minus Z Plus Dilogarithm of One Minus Reciprocal of Z",
"Difference of Logarithms",
"Natural Logarithm of 1 is 0",
"Dilogarithm Reflection Formula",
"Difference of Logarithms",
"Natural Logarithm of 1 is 0"
] |
proofwiki-21717 | Dilogarithm of Minus Reciprocal of Golden Mean | :$\map {\Li_2} {-\dfrac 1 \phi} = -\dfrac 2 5 \map \zeta 2 + \dfrac 1 2 \paren {\map \ln \phi}^2$ | We now note:
{{begin-eqn}}
{{eqn | l = \map {\Li_2} {1 - z} + \map {\Li_2} {1 - \dfrac 1 z}
| r = -\dfrac 1 2 \map {\ln^2} z
| c = Dilogarithm of One Minus Z Plus Dilogarithm of One Minus Reciprocal of Z
}}
{{eqn | ll= \leadsto
| l = \map {\Li_2} {1 - \frac 1 \phi} + \map {\Li_2} {1 - \dfrac 1 {\frac ... | :$\map {\Li_2} {-\dfrac 1 \phi} = -\dfrac 2 5 \map \zeta 2 + \dfrac 1 2 \paren {\map \ln \phi}^2$ | We now note:
{{begin-eqn}}
{{eqn | l = \map {\Li_2} {1 - z} + \map {\Li_2} {1 - \dfrac 1 z}
| r = -\dfrac 1 2 \map {\ln^2} z
| c = [[Dilogarithm of One Minus Z Plus Dilogarithm of One Minus Reciprocal of Z]]
}}
{{eqn | ll= \leadsto
| l = \map {\Li_2} {1 - \frac 1 \phi} + \map {\Li_2} {1 - \dfrac 1 {\f... | Dilogarithm of Minus Reciprocal of Golden Mean | https://proofwiki.org/wiki/Dilogarithm_of_Minus_Reciprocal_of_Golden_Mean | https://proofwiki.org/wiki/Dilogarithm_of_Minus_Reciprocal_of_Golden_Mean | [
"Examples of Dilogarithm Function",
"Spence's Function",
"Golden Mean"
] | [] | [
"Dilogarithm of One Minus Z Plus Dilogarithm of One Minus Reciprocal of Z",
"Difference of Logarithms",
"Natural Logarithm of 1 is 0",
"Reciprocal Form of One Minus Golden Mean",
"Dilogarithm of One Minus Reciprocal of Golden Mean"
] |
proofwiki-21718 | Dilogarithm of Reciprocal of Golden Mean | :$\map {\Li_2} {\dfrac 1 \phi} = \dfrac 3 5 \map \zeta 2 - \paren {\map \ln \phi}^2$ | We first note the following:
{{begin-eqn}}
{{eqn | l = -\frac 1 \phi
| r = 1 - \phi
| c = Reciprocal Form of One Minus Golden Mean
}}
{{eqn | ll= \leadsto
| n = 1
| l = \frac 1 {\phi^2}
| r = 1 - \dfrac 1 \phi
| c = dividing through by $-\phi$ and rearranging
}}
{{end-eqn}}
We now no... | :$\map {\Li_2} {\dfrac 1 \phi} = \dfrac 3 5 \map \zeta 2 - \paren {\map \ln \phi}^2$ | We first note the following:
{{begin-eqn}}
{{eqn | l = -\frac 1 \phi
| r = 1 - \phi
| c = [[Reciprocal Form of One Minus Golden Mean]]
}}
{{eqn | ll= \leadsto
| n = 1
| l = \frac 1 {\phi^2}
| r = 1 - \dfrac 1 \phi
| c = dividing through by $-\phi$ and rearranging
}}
{{end-eqn}}
We... | Dilogarithm of Reciprocal of Golden Mean | https://proofwiki.org/wiki/Dilogarithm_of_Reciprocal_of_Golden_Mean | https://proofwiki.org/wiki/Dilogarithm_of_Reciprocal_of_Golden_Mean | [
"Examples of Dilogarithm Function",
"Spence's Function",
"Golden Mean"
] | [] | [
"Reciprocal Form of One Minus Golden Mean",
"Dilogarithm of Square",
"Dilogarithm of One Minus Reciprocal of Golden Mean",
"Dilogarithm of Minus Reciprocal of Golden Mean"
] |
proofwiki-21719 | Dilogarithm of Minus Golden Mean | :$\map {\Li_2} {-\phi} = -\dfrac 3 5 \map \zeta 2 - \paren {\map \ln \phi}^2$ | We now note:
{{begin-eqn}}
{{eqn | l = \map {\Li_2} {-z} + \map {\Li_2} {-\dfrac 1 z}
| r = -\map \zeta 2 - \dfrac 1 2 \map {\ln^2} z
| c = Dilogarithm of Minus Z Plus Dilogarithm of Minus Reciprocal of Z
}}
{{eqn | ll= \leadsto
| l = \map {\Li_2} {-\frac 1 \phi} + \map {\Li_2} {-\dfrac 1 {\frac 1 \ph... | :$\map {\Li_2} {-\phi} = -\dfrac 3 5 \map \zeta 2 - \paren {\map \ln \phi}^2$ | We now note:
{{begin-eqn}}
{{eqn | l = \map {\Li_2} {-z} + \map {\Li_2} {-\dfrac 1 z}
| r = -\map \zeta 2 - \dfrac 1 2 \map {\ln^2} z
| c = [[Dilogarithm of Minus Z Plus Dilogarithm of Minus Reciprocal of Z]]
}}
{{eqn | ll= \leadsto
| l = \map {\Li_2} {-\frac 1 \phi} + \map {\Li_2} {-\dfrac 1 {\frac 1... | Dilogarithm of Minus Golden Mean | https://proofwiki.org/wiki/Dilogarithm_of_Minus_Golden_Mean | https://proofwiki.org/wiki/Dilogarithm_of_Minus_Golden_Mean | [
"Examples of Dilogarithm Function",
"Spence's Function",
"Golden Mean"
] | [] | [
"Dilogarithm of Minus Z Plus Dilogarithm of Minus Reciprocal of Z",
"Difference of Logarithms",
"Natural Logarithm of 1 is 0",
"Dilogarithm of Minus Reciprocal of Golden Mean"
] |
proofwiki-21720 | Erdős-Moser Equation | The '''Erdős-Moser Equation''' is the Diophantine equation:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^k j^p
| r = 1 + 2^p + 3^p + \cdots + k^p
| c =
}}
{{eqn | r = \paren {k + 1}^p
| c =
}}
{{end-eqn}}
where $k, p \in \N$.
Its only solution is $k = 2$ and $p = 1$:
:$1^1 + 2^1 = 3^1$
{{questiona... | Let $\map {B_n} x$ denote the $n$th Bernoulli polynomial:
:$\ds \map {B_n} x = \sum_{k \mathop = 0}^n \binom n k b_{n - k} x^k$
where $b_n$ denotes the $n$th Bernoulli number. | The '''[[Erdős-Moser Equation]]''' is the [[Definition:Diophantine Equation|Diophantine equation]]:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^k j^p
| r = 1 + 2^p + 3^p + \cdots + k^p
| c =
}}
{{eqn | r = \paren {k + 1}^p
| c =
}}
{{end-eqn}}
where $k, p \in \N$.
Its only solution is $k = 2$... | Let $\map {B_n} x$ denote the $n$th [[Definition:Bernoulli Polynomial|Bernoulli polynomial]]:
:$\ds \map {B_n} x = \sum_{k \mathop = 0}^n \binom n k b_{n - k} x^k$
where $b_n$ denotes the $n$th [[Definition:Bernoulli Numbers|Bernoulli number]]. | Erdős-Moser Equation | https://proofwiki.org/wiki/Erdős-Moser_Equation | https://proofwiki.org/wiki/Erdős-Moser_Equation | [] | [
"Erdős-Moser Equation",
"Definition:Diophantine Equation"
] | [
"Definition:Bernoulli Polynomial",
"Definition:Bernoulli Numbers"
] |
proofwiki-21721 | Existence of Domain of Class Relation | Let $X$ be a class.
Then there is a unique class $Z$ such that:
:$\forall u: u \in Z \iff \exists v: \tuple {u, v} \in X$
where $\tuple {\cdot, \cdot}$ represents the Kuratowski formalization of ordered pairs.
That is, $Z$ is the class of all $z$ which appear as the first coordinate of an ordered pair in $X$. | Let $X$ be arbitrary.
By Axiom $\text B 4$, there is a class $Z$ such that:
:$\forall u: u \in Z \iff \exists v: \tuple {u, v} \in X$
This satisfies the existence portion of the theorem.
{{qed|lemma}}
Let $Z'$ be a class such that:
:$\forall u: u \in Z' \iff \exists v: \tuple {u, v} \in X$
Then, for every set $u$:
:$u ... | Let $X$ be a [[Definition:Class|class]].
Then there is a [[Definition:Unique|unique]] [[Definition:Class|class]] $Z$ such that:
:$\forall u: u \in Z \iff \exists v: \tuple {u, v} \in X$
where $\tuple {\cdot, \cdot}$ represents the [[Definition:Kuratowski Formalization of Ordered Pair|Kuratowski formalization of ordere... | Let $X$ be arbitrary.
By [[Axiom:Axioms of Class Existence|Axiom $\text B 4$]], there is a [[Definition:Class|class]] $Z$ such that:
:$\forall u: u \in Z \iff \exists v: \tuple {u, v} \in X$
This satisfies the existence portion of the theorem.
{{qed|lemma}}
Let $Z'$ be a [[Definition:Class|class]] such that:
:$\fora... | Existence of Domain of Class Relation | https://proofwiki.org/wiki/Existence_of_Domain_of_Class_Relation | https://proofwiki.org/wiki/Existence_of_Domain_of_Class_Relation | [
"Class Theory",
"Von Neumann-Bernays-Gödel Set Theory"
] | [
"Definition:Class",
"Definition:Unique",
"Definition:Class",
"Definition:Ordered Pair/Kuratowski Formalization",
"Definition:Class",
"Definition:Coordinate System/Coordinate/Element of Ordered Pair",
"Definition:Ordered Pair"
] | [
"Axiom:Axioms of Class Existence",
"Definition:Class",
"Definition:Class",
"Definition:Set",
"Axiom:Axiom of Extension/Class Theory",
"Category:Class Theory",
"Category:Von Neumann-Bernays-Gödel Set Theory"
] |
proofwiki-21722 | Existence of Class Union | For any classes $X, Y$, the union $X \cup Y$ exists and is unique.
That is:
:$\forall X, Y: \exists! Z: \forall u: u \in Z \iff u \in X \lor u \in Y$
where $X \cup Y := Z$. | By Existence of Class Complement, there exist unique classes $\overline X, \overline Y$ such that:
:$u \in \overline X \iff u \notin X$
:$u \in \overline Y \iff u \notin Y$
for all sets $u$.
Then, by Existence of Class Intersection, there is a unique class $\overline X \cap \overline Y$ such that:
:$u \in \overline X \... | For any [[Definition:Class|classes]] $X, Y$, the [[Definition:Class Union|union]] $X \cup Y$ exists and is [[Definition:Unique|unique]].
That is:
:$\forall X, Y: \exists! Z: \forall u: u \in Z \iff u \in X \lor u \in Y$
where $X \cup Y := Z$. | By [[Existence of Class Complement]], there exist [[Definition:Unique|unique]] [[Definition:Class|classes]] $\overline X, \overline Y$ such that:
:$u \in \overline X \iff u \notin X$
:$u \in \overline Y \iff u \notin Y$
for all [[Definition:Set|sets]] $u$.
Then, by [[Existence of Class Intersection]], there is a [[Def... | Existence of Class Union | https://proofwiki.org/wiki/Existence_of_Class_Union | https://proofwiki.org/wiki/Existence_of_Class_Union | [
"Class Theory",
"Von Neumann-Bernays-Gödel Set Theory"
] | [
"Definition:Class",
"Definition:Class Union",
"Definition:Unique"
] | [
"Existence of Class Complement",
"Definition:Unique",
"Definition:Class",
"Definition:Set",
"Existence of Class Intersection",
"Definition:Unique",
"Definition:Class",
"Existence of Class Complement",
"Definition:Unique",
"Definition:Class",
"De Morgan's Laws (Logic)/Disjunction",
"Definition:... |
proofwiki-21723 | Existence of Universal Class | There exists a unique class $V$ such that:
:$\forall u: u \in V$
That is, the universal class exists and is unique. | By Existence of Empty Class, there is a unique class $\O$ such that:
:$\forall u: u \notin \O$
for every set $u$.
Thus, by Existence of Class Complement, there is a unique class $\overline \O$ such that:
:$u \in \overline \O \iff u \notin \O$
But then, by the definition of $\O$:
:$\forall u: u \in \overline \O$
Then, $... | There exists a [[Definition:Unique|unique]] [[Definition:Class|class]] $V$ such that:
:$\forall u: u \in V$
That is, the [[Definition:Universal Class|universal class]] exists and is [[Definition:Unique|unique]]. | By [[Existence of Empty Class]], there is a [[Definition:Unique|unique]] [[Definition:Class|class]] $\O$ such that:
:$\forall u: u \notin \O$
for every [[Definition:Set|set]] $u$.
Thus, by [[Existence of Class Complement]], there is a [[Definition:Unique|unique]] [[Definition:Class|class]] $\overline \O$ such that:
:$... | Existence of Universal Class | https://proofwiki.org/wiki/Existence_of_Universal_Class | https://proofwiki.org/wiki/Existence_of_Universal_Class | [
"Class Theory",
"Von Neumann-Bernays-Gödel Set Theory"
] | [
"Definition:Unique",
"Definition:Class",
"Definition:Universal Class",
"Definition:Unique"
] | [
"Existence of Empty Class",
"Definition:Unique",
"Definition:Class",
"Definition:Set",
"Existence of Class Complement",
"Definition:Unique",
"Definition:Class",
"Definition:Unique",
"Category:Class Theory",
"Category:Von Neumann-Bernays-Gödel Set Theory"
] |
proofwiki-21724 | Existence of Inverse of Class Relation | Let $X$ be a class.
Then there is a class $Z$ such that, for all sets $u, v$:
:$\tuple {u, v} \in Z \iff \tuple {v, u} \in X$
where $\tuple {\cdot, \cdot}$ denotes the Kuratowski formalization of ordered pairs. | Let $X$ be arbitrary.
By Axiom $\text B 5$, there is a class $Z_1$ such that:
:$\paren 1 \quad \forall a, w: \tuple {a, w} \in Z_1 \iff a \in X$
Now, by Axiom $\text B 7$, there is a class $Z_2$ such that:
:$\paren 2 \quad \forall u, v, w: \tuple {\tuple {v, w}, u} \in Z_2 \iff \tuple {\tuple {v, u}, w} \in Z_1$
Next, ... | Let $X$ be a [[Definition:Class|class]].
Then there is a [[Definition:Class|class]] $Z$ such that, for all [[Definition:Set|sets]] $u, v$:
:$\tuple {u, v} \in Z \iff \tuple {v, u} \in X$
where $\tuple {\cdot, \cdot}$ denotes the [[Definition:Kuratowski Formalization of Ordered Pair|Kuratowski formalization of ordered ... | Let $X$ be arbitrary.
By [[Axiom:Axioms of Class Existence|Axiom $\text B 5$]], there is a [[Definition:Class|class]] $Z_1$ such that:
:$\paren 1 \quad \forall a, w: \tuple {a, w} \in Z_1 \iff a \in X$
Now, by [[Axiom:Axioms of Class Existence|Axiom $\text B 7$]], there is a [[Definition:Class|class]] $Z_2$ such that... | Existence of Inverse of Class Relation | https://proofwiki.org/wiki/Existence_of_Inverse_of_Class_Relation | https://proofwiki.org/wiki/Existence_of_Inverse_of_Class_Relation | [
"Class Theory",
"Von Neumann-Bernays-Gödel Set Theory"
] | [
"Definition:Class",
"Definition:Class",
"Definition:Set",
"Definition:Ordered Pair/Kuratowski Formalization"
] | [
"Axiom:Axioms of Class Existence",
"Definition:Class",
"Axiom:Axioms of Class Existence",
"Definition:Class",
"Axiom:Axioms of Class Existence",
"Definition:Class",
"Axiom:Axioms of Class Existence",
"Definition:Class",
"Biconditional is Transitive",
"Definition:Set",
"Axiom:Axiom of Infinity/Se... |
proofwiki-21725 | Existence of Right Universal Product | For each $m \in \N$, the following is a theorem of von Neumann-Bernays-Gödel set theory:
:$\forall X: \exists Z: \forall v_1, \dotsc, v_m: \forall x: \tuple {\tuple {\dots \tuple {x, v_1}, \dots}, v_m} \in Z \iff x \in X$
Informally, for any class $X$, there is some class $Z$ such that:
:$Z = \paren {\dotsm \paren {X \... | We will proceed by induction on $m$.
Importantly, this induction is ''not'' done in the context of NBG.
Rather, it is performed in the metamathematical system in which NBG is formalized. | For each $m \in \N$, the following is a [[Definition:Theorem|theorem]] of [[Definition:Von Neumann-Bernays-Gödel Set Theory|von Neumann-Bernays-Gödel set theory]]:
:$\forall X: \exists Z: \forall v_1, \dotsc, v_m: \forall x: \tuple {\tuple {\dots \tuple {x, v_1}, \dots}, v_m} \in Z \iff x \in X$
Informally, for any [... | We will proceed by [[Definition:Mathematical Induction|induction]] on $m$.
Importantly, this induction is ''not'' done in the context of [[Definition:Von Neumann-Bernays-Gödel Set Theory|NBG]].
Rather, it is performed in the metamathematical system in which [[Definition:Von Neumann-Bernays-Gödel Set Theory|NBG]] is f... | Existence of Right Universal Product | https://proofwiki.org/wiki/Existence_of_Right_Universal_Product | https://proofwiki.org/wiki/Existence_of_Right_Universal_Product | [
"Class Theory",
"Von Neumann-Bernays-Gödel Set Theory"
] | [
"Definition:Theorem",
"Definition:Von Neumann-Bernays-Gödel Set Theory",
"Definition:Class",
"Definition:Class",
"Definition:Universal Class",
"Definition:Cartesian Product/Class Theory"
] | [
"Definition:Mathematical Induction",
"Definition:Von Neumann-Bernays-Gödel Set Theory",
"Definition:Von Neumann-Bernays-Gödel Set Theory"
] |
proofwiki-21726 | Existence of Left Universal Product | For each $m \in \N_{> 0}$, the following is a theorem of von Neumann-Bernays-Gödel set theory:
:$\forall X: \exists Z: \forall x: \forall v_1, \dotsc, v_m: \sequence {v_1, \dotsc, v_m, x} \in Z \iff x \in X$
where the notation $\sequence {\cdot, \dotsc, \cdot}$ is defined inductively as:
:$\sequence {y_1} := y_1$
:$\se... | Let $X$ be arbitrary.
By Axiom $\text B 5$, there is a class $Z$ such that:
:$\forall x, v: \tuple {x, v} \in Z \iff x \in X$
By Existence of Inverse of Class Relation, there is a class $Z'$ such that:
:$\forall v, x: \tuple {v, x} \in Z' \iff \tuple {x, v} \in Z$
Thus, by Biconditional is Transitive:
:$\paren 1 \quad ... | For each $m \in \N_{> 0}$, the following is a [[Definition:Theorem|theorem]] of [[Definition:Von Neumann-Bernays-Gödel Set Theory|von Neumann-Bernays-Gödel set theory]]:
:$\forall X: \exists Z: \forall x: \forall v_1, \dotsc, v_m: \sequence {v_1, \dotsc, v_m, x} \in Z \iff x \in X$
where the notation $\sequence {\cdot,... | Let $X$ be arbitrary.
By [[Axiom:Axioms of Class Existence|Axiom $\text B 5$]], there is a [[Definition:Class|class]] $Z$ such that:
:$\forall x, v: \tuple {x, v} \in Z \iff x \in X$
By [[Existence of Inverse of Class Relation]], there is a [[Definition:Class|class]] $Z'$ such that:
:$\forall v, x: \tuple {v, x} \in ... | Existence of Left Universal Product | https://proofwiki.org/wiki/Existence_of_Left_Universal_Product | https://proofwiki.org/wiki/Existence_of_Left_Universal_Product | [
"Class Theory",
"Von Neumann-Bernays-Gödel Set Theory"
] | [
"Definition:Theorem",
"Definition:Von Neumann-Bernays-Gödel Set Theory",
"Definition:Ordered Pair/Kuratowski Formalization"
] | [
"Axiom:Axioms of Class Existence",
"Definition:Class",
"Existence of Inverse of Class Relation",
"Definition:Class",
"Biconditional is Transitive",
"Category:Class Theory",
"Category:Von Neumann-Bernays-Gödel Set Theory"
] |
proofwiki-21727 | Existence of Middle Universal Product | For each $m \in \N$, the following is a theorem of von Neumann-Bernays-Gödel set theory:
:$\forall X: \exists Z: \forall v_1, \dotsc, v_m: \forall x, y: \sequence {x, v_1, \dotsc, v_m, y} \in Z \iff \tuple {x, y} \in X$
where the notation $\sequence {\cdot, \dotsc, \cdot}$ is defined inductively as:
:$\sequence {y_1} :... | In the metatheory in which NBG is formalized, we proceed by induction on $m$. | For each $m \in \N$, the following is a [[Definition:Theorem|theorem]] of [[Definition:Von Neumann-Bernays-Gödel Set Theory|von Neumann-Bernays-Gödel set theory]]:
:$\forall X: \exists Z: \forall v_1, \dotsc, v_m: \forall x, y: \sequence {x, v_1, \dotsc, v_m, y} \in Z \iff \tuple {x, y} \in X$
where the notation $\sequ... | In the metatheory in which [[Definition:Von Neumann-Bernays-Gödel Set Theory|NBG]] is formalized, we proceed by [[Definition:Mathematical Induction|induction]] on $m$. | Existence of Middle Universal Product | https://proofwiki.org/wiki/Existence_of_Middle_Universal_Product | https://proofwiki.org/wiki/Existence_of_Middle_Universal_Product | [
"Class Theory",
"Von Neumann-Bernays-Gödel Set Theory"
] | [
"Definition:Theorem",
"Definition:Von Neumann-Bernays-Gödel Set Theory",
"Definition:Ordered Pair/Kuratowski Formalization"
] | [
"Definition:Von Neumann-Bernays-Gödel Set Theory",
"Definition:Mathematical Induction"
] |
proofwiki-21728 | Equal Relative Complements iff Equal Subsets | Let $S$ be a set.
Let $A, B \subseteq S$ be subsets of $S$.
Then:
:$\relcomp S B = \relcomp S A \iff A = B$
where $\complement_S$ denotes the complement relative to $S$. | We have:
{{begin-eqn}}
{{eqn | l = \relcomp S B = \relcomp S A
| o = \iff
| r = \relcomp S B \subseteq \relcomp S A \land \relcomp S A \subseteq \relcomp S B
| c = {{Defof|Set Equivalence}}
}}
{{eqn | o = \iff
| r = A \subseteq B \land B \subseteq A
| c = Relative Complement inverts Subset... | Let $S$ be a [[Definition:Set|set]].
Let $A, B \subseteq S$ be [[Definition:Subset|subsets]] of $S$.
Then:
:$\relcomp S B = \relcomp S A \iff A = B$
where $\complement_S$ denotes the [[Definition:Relative Complement|complement relative to $S$]]. | We have:
{{begin-eqn}}
{{eqn | l = \relcomp S B = \relcomp S A
| o = \iff
| r = \relcomp S B \subseteq \relcomp S A \land \relcomp S A \subseteq \relcomp S B
| c = {{Defof|Set Equivalence}}
}}
{{eqn | o = \iff
| r = A \subseteq B \land B \subseteq A
| c = [[Relative Complement inverts Subs... | Equal Relative Complements iff Equal Subsets | https://proofwiki.org/wiki/Equal_Relative_Complements_iff_Equal_Subsets | https://proofwiki.org/wiki/Equal_Relative_Complements_iff_Equal_Subsets | [
"Subsets",
"Relative Complement"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Relative Complement"
] | [
"Relative Complement inverts Subsets",
"Category:Subsets",
"Category:Relative Complement"
] |
proofwiki-21729 | Relative Complement inverts Subsets of Relative Complement | Let $S$ be a set.
Let $A \subseteq S, B \subseteq S$ be subsets of $S$.
Then:
:$A \subseteq \relcomp S B \iff B \subseteq \relcomp S A$
where $\complement_S$ denotes the complement relative to $S$. | We have:
{{begin-eqn}}
{{eqn | l = A \subseteq \relcomp S B
| o = \iff
| r = \relcomp S {\relcomp S B} \subseteq \relcomp S A
| c = Relative Complement inverts Subsets
}}
{{eqn | o = \iff
| r = B \subseteq \relcomp S A
| c = Relative Complement of Relative Complement
}}
{{end-eqn}}
{{qed}}... | Let $S$ be a [[Definition:Set|set]].
Let $A \subseteq S, B \subseteq S$ be [[Definition:Subset|subsets]] of $S$.
Then:
:$A \subseteq \relcomp S B \iff B \subseteq \relcomp S A$
where $\complement_S$ denotes the [[Definition:Relative Complement|complement relative to $S$]]. | We have:
{{begin-eqn}}
{{eqn | l = A \subseteq \relcomp S B
| o = \iff
| r = \relcomp S {\relcomp S B} \subseteq \relcomp S A
| c = [[Relative Complement inverts Subsets]]
}}
{{eqn | o = \iff
| r = B \subseteq \relcomp S A
| c = [[Relative Complement of Relative Complement]]
}}
{{end-eqn}}... | Relative Complement inverts Subsets of Relative Complement | https://proofwiki.org/wiki/Relative_Complement_inverts_Subsets_of_Relative_Complement | https://proofwiki.org/wiki/Relative_Complement_inverts_Subsets_of_Relative_Complement | [
"Subsets",
"Relative Complement"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Relative Complement"
] | [
"Relative Complement inverts Subsets",
"Relative Complement of Relative Complement",
"Category:Subsets",
"Category:Relative Complement"
] |
proofwiki-21730 | Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods | Let $\struct{S, \tau}$ be a topological space.
For each $x \in S$, let:
:$\map \UU x$ denote the system of open neighborhoods of $x$
Then $\struct{S, \tau}$ is a sober space {{Iff}}:
:for each completely prime filter $\FF$ in the complete lattice $\struct{\tau, \subseteq}$:
::$\exists ! x \in S : \FF = \map \UU x$ | Recall by definition of system of open neighborhoods:
:$\forall x \in S : \map \UU x = \set{U \in \tau : x \in U}$ | Let $\struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
For each $x \in S$, let:
:$\map \UU x$ denote the [[Definition:System of Open Neighborhoods|system of open neighborhoods]] of $x$
Then $\struct{S, \tau}$ is a [[Definition:Sober Space|sober space]] {{Iff}}:
:for each [[Definition:Comple... | Recall by definition of [[Definition:System of Open Neighborhoods|system of open neighborhoods]]:
:$\forall x \in S : \map \UU x = \set{U \in \tau : x \in U}$ | Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods | https://proofwiki.org/wiki/Sober_Space_iff_Completely_Prime_Filter_is_Unique_System_of_Open_Neighborhoods | https://proofwiki.org/wiki/Sober_Space_iff_Completely_Prime_Filter_is_Unique_System_of_Open_Neighborhoods | [
"Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods",
"Sober Spaces",
"Systems of Open Neighborhoods",
"Completely Prime Filters"
] | [
"Definition:Topological Space",
"Definition:System of Open Neighborhoods",
"Definition:Sober Space",
"Definition:Completely Prime Filter",
"Definition:Complete Lattice"
] | [
"Definition:System of Open Neighborhoods"
] |
proofwiki-21731 | Lower Bound of Pell Number | For all $n \in \N$:
:$P_n \ge 2 \paren {1 + \sqrt 2}^{n - 2}$
where:
:$P_n$ is the $n$th Pell number | The proof proceeds by induction.
For all $n \in \N$, let $\map P n$ be the proposition:
:$P_n \ge 2 \paren {1 + \sqrt 2}^{n - 2}$ | For all $n \in \N$:
:$P_n \ge 2 \paren {1 + \sqrt 2}^{n - 2}$
where:
:$P_n$ is the $n$th [[Definition:Pell Numbers|Pell number]] | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$P_n \ge 2 \paren {1 + \sqrt 2}^{n - 2}$ | Lower Bound of Pell Number | https://proofwiki.org/wiki/Lower_Bound_of_Pell_Number | https://proofwiki.org/wiki/Lower_Bound_of_Pell_Number | [
"Pell Numbers"
] | [
"Definition:Pell Numbers"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-21732 | Upper Bound of Pell Number | For all $n \in \N$:
:$P_n \le \paren {1 + \sqrt 2}^{n - 1}$
where:
:$P_n$ is the $n$th Pell number | The proof proceeds by induction.
For all $n \in \N$, let $\map P n$ be the proposition:
:$P_n \le \paren {1 + \sqrt 2}^{n - 1}$ | For all $n \in \N$:
:$P_n \le \paren {1 + \sqrt 2}^{n - 1}$
where:
:$P_n$ is the $n$th [[Definition:Pell Numbers|Pell number]] | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$P_n \le \paren {1 + \sqrt 2}^{n - 1}$ | Upper Bound of Pell Number | https://proofwiki.org/wiki/Upper_Bound_of_Pell_Number | https://proofwiki.org/wiki/Upper_Bound_of_Pell_Number | [
"Pell Numbers"
] | [
"Definition:Pell Numbers"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-21733 | Generalization to Dudeney's Distribution Problem | Suppose $n > 1$ people, each with $a_1, a_2, \dots, a_n$ dollars, redistribute their wealth as follows:
Going in order of their index, each person doubles everyone else's money by giving out their own money.
If they all have the same amount of money after the process, the least (strictly) positive integral solution is:... | Let us define the total amount of money $S = \ds \sum_{j \mathop = 1}^n a_j$.
Consider the wealth of the $j$th person.
They would see their wealth doubled $j - 1$ times:
:$a_j \to 2^{j - 1} a_j$
Then they would have to give out money to double everyone else's money:
:$2^{j - 1} a_j \to 2^{j - 1} a_j - \paren {S - 2^{j ... | Suppose $n > 1$ people, each with $a_1, a_2, \dots, a_n$ dollars, redistribute their wealth as follows:
Going in order of their index, each person doubles everyone else's money by giving out their own money.
If they all have the same amount of money after the process, the least [[Definition:Strictly Positive Integer|... | Let us define the total amount of money $S = \ds \sum_{j \mathop = 1}^n a_j$.
Consider the wealth of the $j$th person.
They would see their wealth doubled $j - 1$ times:
:$a_j \to 2^{j - 1} a_j$
Then they would have to give out money to double everyone else's money:
:$2^{j - 1} a_j \to 2^{j - 1} a_j - \paren {S - 2^... | Generalization to Dudeney's Distribution Problem | https://proofwiki.org/wiki/Generalization_to_Dudeney's_Distribution_Problem | https://proofwiki.org/wiki/Generalization_to_Dudeney's_Distribution_Problem | [] | [
"Definition:Strictly Positive/Integer"
] | [
"Definition:Coprime/Integers",
"Euclid's Lemma",
"Definition:Integral Multiple/Real Numbers"
] |
proofwiki-21734 | Floating-Point Operation may not result in Floating-Point Number | Let a floating-point operation be performed on two floating-point numbers.
Then it is not necessarily the case that the result of that operation is itself a number in floating-point representation. | {{ProofWanted|More background needed on how those floating-point operations combine}} | Let a [[Definition:Floating-Point Operation|floating-point operation]] be performed on two [[Definition:Floating-Point Representation|floating-point numbers]].
Then it is not necessarily the case that the result of that [[Definition:Floating-Point Operation|operation]] is itself a [[Definition:Number|number]] in [[Def... | {{ProofWanted|More background needed on how those floating-point operations combine}} | Floating-Point Operation may not result in Floating-Point Number | https://proofwiki.org/wiki/Floating-Point_Operation_may_not_result_in_Floating-Point_Number | https://proofwiki.org/wiki/Floating-Point_Operation_may_not_result_in_Floating-Point_Number | [
"Floating-Point Operations"
] | [
"Definition:Floating-Point Operation",
"Definition:Floating-Point Representation",
"Definition:Floating-Point Operation",
"Definition:Number",
"Definition:Floating-Point Representation"
] | [] |
proofwiki-21735 | Filter Contains Greatest Element | Let $\struct{S, \preceq}$ be an ordered set with greatest element $\top$.
Let $\FF$ be a filter in $\struct{S, \preceq}$.
Then:
:$\top \in \FF$ | By {{Ordered-set-filter-axiom|1}}:
:$\exists x \in \FF$
By definition of greatest element:
$x \preceq \top$
By {{Ordered-set-filter-axiom|3}}:
:$\top \in \FF$
{{qed}}
Category:Ordered Sets
Category:Filter Theory
Category:Greatest Elements
jyj5woqcv5g0kyliaouddutsw30sdx6 | Let $\struct{S, \preceq}$ be an [[Definition:Ordered Set|ordered set]] with [[Definition:Greatest Element|greatest element]] $\top$.
Let $\FF$ be a [[Definition:Filter|filter]] in $\struct{S, \preceq}$.
Then:
:$\top \in \FF$ | By {{Ordered-set-filter-axiom|1}}:
:$\exists x \in \FF$
By definition of [[Definition:Greatest Element|greatest element]]:
$x \preceq \top$
By {{Ordered-set-filter-axiom|3}}:
:$\top \in \FF$
{{qed}}
[[Category:Ordered Sets]]
[[Category:Filter Theory]]
[[Category:Greatest Elements]]
jyj5woqcv5g0kyliaouddutsw30sdx6 | Filter Contains Greatest Element | https://proofwiki.org/wiki/Filter_Contains_Greatest_Element | https://proofwiki.org/wiki/Filter_Contains_Greatest_Element | [
"Ordered Sets",
"Filter Theory",
"Greatest Elements"
] | [
"Definition:Ordered Set",
"Definition:Greatest Element",
"Definition:Filter"
] | [
"Definition:Greatest Element",
"Category:Ordered Sets",
"Category:Filter Theory",
"Category:Greatest Elements"
] |
proofwiki-21736 | System of Open Neighborhoods is a Completely Prime Filter | Let $\struct{S, \tau}$ be a topological space.
Let $x \in S$.
Let $\map \UU x$ denote the system of open neighborhoods of $x$ in $\struct{S, \tau}$.
Then:
:$\map \UU x$ is a completely prime filter in the complete lattice $\struct{\tau, \subseteq}$ | === $\map \UU x$ satisfies {{Ordered-set-filter-axiom|1}} ===
We have $x \in S$.
By {{Open-set-axiom|3}}:
:$S \in \tau$
Hence:
:$S \in \map \UU x$
It follows that $\map \UU x$ satisfies {{Ordered-set-filter-axiom|1}}.
{{qed|lemma}} | Let $\struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $\map \UU x$ denote the [[Definition:System of Open Neighborhoods|system of open neighborhoods]] of $x$ in $\struct{S, \tau}$.
Then:
:$\map \UU x$ is a [[Definition:Completely Prime Filter|completely prime filter]] in... | === $\map \UU x$ satisfies {{Ordered-set-filter-axiom|1}} ===
We have $x \in S$.
By {{Open-set-axiom|3}}:
:$S \in \tau$
Hence:
:$S \in \map \UU x$
It follows that $\map \UU x$ satisfies {{Ordered-set-filter-axiom|1}}.
{{qed|lemma}} | System of Open Neighborhoods is a Completely Prime Filter | https://proofwiki.org/wiki/System_of_Open_Neighborhoods_is_a_Completely_Prime_Filter | https://proofwiki.org/wiki/System_of_Open_Neighborhoods_is_a_Completely_Prime_Filter | [
"Systems of Open Neighborhoods",
"Completely Prime Filters"
] | [
"Definition:Topological Space",
"Definition:System of Open Neighborhoods",
"Definition:Completely Prime Filter",
"Definition:Complete Lattice"
] | [] |
proofwiki-21737 | Condition for Equilibrium using Force Polygon | Let $\SS$ be a system of forces $\SS$ act at a point.
Let $\SS$ give rise to a force polygon $A_0 A_1 \ldots A_n A_0$.
Then $\SS$ is in equilibrium {{iff}} $A_n$ and $A_0$ coincide. | {{MissingLinks}} | Let $\SS$ be a system of [[Definition:Force|forces]] $\SS$ act at a [[Definition:Point|point]].
Let $\SS$ give rise to a [[Definition:Force Polygon|force polygon]] $A_0 A_1 \ldots A_n A_0$.
Then $\SS$ is in [[Definition:Equilibrium|equilibrium]] {{iff}} $A_n$ and $A_0$ coincide. | {{MissingLinks}} | Condition for Equilibrium using Force Polygon | https://proofwiki.org/wiki/Condition_for_Equilibrium_using_Force_Polygon | https://proofwiki.org/wiki/Condition_for_Equilibrium_using_Force_Polygon | [
"Force Polygons"
] | [
"Definition:Force",
"Definition:Point",
"Definition:Force Polygon",
"Definition:Equilibrium"
] | [] |
proofwiki-21738 | Quadratic Form in Two Variables represents Conic Section | A quadratic form in $2$ variables, when put equal to a constant
:$a x^2 + b x y + c y^2 = r$
represents a conic section. | It can be seen directly that:
:$a x^2 + b x y + c y^2 = r$
is an instance of an equation for a conic section in Cartesian form:
:$a x^2 + b x y + c y^2 + d x + e y + f = 0$
by setting $f \gets -r$ and equating $d$ and $e$ to zero.
{{qed}} | A [[Definition:Quadratic Form (Polynomial Theory)|quadratic form]] in $2$ [[Definition:Variable|variables]], when put equal to a [[Definition:Constant|constant]]
:$a x^2 + b x y + c y^2 = r$
represents a [[Definition:Conic Section|conic section]]. | It can be seen directly that:
:$a x^2 + b x y + c y^2 = r$
is an instance of an equation for a [[Equation of Conic Section/Cartesian Form|conic section in Cartesian form]]:
:$a x^2 + b x y + c y^2 + d x + e y + f = 0$
by setting $f \gets -r$ and equating $d$ and $e$ to [[Definition:Zero (Number)|zero]].
{{qed}} | Quadratic Form in Two Variables represents Conic Section | https://proofwiki.org/wiki/Quadratic_Form_in_Two_Variables_represents_Conic_Section | https://proofwiki.org/wiki/Quadratic_Form_in_Two_Variables_represents_Conic_Section | [
"Quadratic Forms (Polynomial Theory)",
"Conic Sections"
] | [
"Definition:Quadratic Form (Polynomial Theory)",
"Definition:Variable",
"Definition:Constant",
"Definition:Conic Section"
] | [
"Equation of Conic Section/Cartesian Form",
"Definition:Zero (Number)"
] |
proofwiki-21739 | Maximum Speed of Rotation of Plane of Oscillation of Foucault's Pendulum | Let $P$ be a Foucault pendulum.
The maximum angular speed of the plane of oscillation of $P$ occurs at Earth's poles. | From Angular Speed of Rotation of Plane of Oscillation of Foucault's Pendulum, the angular speed $\alpha$ of the plane of oscillation of $P$ is given by:
:$\alpha = \omega \sin \lambda$
where:
:$\lambda$ denotes the latitude on Earth at which $P$ is located
:$\omega$ denotes the angular speed of rotation of Earth.
So ... | Let $P$ be a [[Definition:Foucault Pendulum|Foucault pendulum]].
The [[Definition:Maximum|maximum]] [[Definition:Angular Speed|angular speed]] of the [[Definition:Plane|plane]] of oscillation of $P$ occurs at [[Definition:Earth's Poles|Earth's poles]]. | From [[Angular Speed of Rotation of Plane of Oscillation of Foucault's Pendulum]], the [[Definition:Angular Speed|angular speed]] $\alpha$ of the [[Definition:Plane|plane]] of oscillation of $P$ is given by:
:$\alpha = \omega \sin \lambda$
where:
:$\lambda$ denotes the [[Definition:Terrestrial Latitude|latitude]] on ... | Maximum Speed of Rotation of Plane of Oscillation of Foucault's Pendulum | https://proofwiki.org/wiki/Maximum_Speed_of_Rotation_of_Plane_of_Oscillation_of_Foucault's_Pendulum | https://proofwiki.org/wiki/Maximum_Speed_of_Rotation_of_Plane_of_Oscillation_of_Foucault's_Pendulum | [
"Foucault's Pendulum"
] | [
"Definition:Foucault's Pendulum",
"Definition:Maximum Value of Real Function",
"Definition:Angular Speed",
"Definition:Plane Surface",
"Definition:Earth's Poles"
] | [
"Angular Speed of Rotation of Plane of Oscillation of Foucault's Pendulum",
"Definition:Angular Speed",
"Definition:Plane Surface",
"Definition:Latitude/Terrestrial",
"Definition:Earth",
"Definition:Angular Speed",
"Definition:Rotation (Geometry)/Space",
"Definition:Earth",
"Definition:Maximum Value... |
proofwiki-21740 | Period of Rotation of Plane of Oscillation of Foucault's Pendulum | Let $P$ be a Foucault pendulum.
The plane of oscillation of $P$ rotates through a full circle in time $T$, where:
:$T = \dfrac D {\sin \lambda}$
where:
:$\lambda$ denotes the latitude on Earth at which $P$ is located
:$D$ denotes the length of a sidereal day. | From Angular Speed of Rotation of Plane of Oscillation of Foucault's Pendulum, the angular speed of rotation $\alpha$ of the plane of oscillation of $P$ is given by:
:$\alpha = \omega \sin \lambda$
where:
:$\lambda$ denotes the latitude on Earth at which $P$ is located
:$\omega$ denotes the angular speed of rotation of... | Let $P$ be a [[Definition:Foucault Pendulum|Foucault pendulum]].
The [[Definition:Plane|plane]] of oscillation of $P$ [[Definition:Space Rotation|rotates]] through a full [[Definition:Circle|circle]] in [[Definition:Length of Time|time]] $T$, where:
:$T = \dfrac D {\sin \lambda}$
where:
:$\lambda$ denotes the [[Defin... | From [[Angular Speed of Rotation of Plane of Oscillation of Foucault's Pendulum]], the [[Definition:Angular Speed|angular speed]] of [[Definition:Space Rotation|rotation]] $\alpha$ of the [[Definition:Plane|plane]] of oscillation of $P$ is given by:
:$\alpha = \omega \sin \lambda$
where:
:$\lambda$ denotes the [[Defini... | Period of Rotation of Plane of Oscillation of Foucault's Pendulum | https://proofwiki.org/wiki/Period_of_Rotation_of_Plane_of_Oscillation_of_Foucault's_Pendulum | https://proofwiki.org/wiki/Period_of_Rotation_of_Plane_of_Oscillation_of_Foucault's_Pendulum | [
"Foucault's Pendulum"
] | [
"Definition:Foucault's Pendulum",
"Definition:Plane Surface",
"Definition:Rotation (Geometry)/Space",
"Definition:Circle",
"Definition:Time/Length",
"Definition:Latitude/Terrestrial",
"Definition:Earth",
"Definition:Time/Length",
"Definition:Sidereal Day"
] | [
"Angular Speed of Rotation of Plane of Oscillation of Foucault's Pendulum",
"Definition:Angular Speed",
"Definition:Rotation (Geometry)/Space",
"Definition:Plane Surface",
"Definition:Latitude/Terrestrial",
"Definition:Earth",
"Definition:Angular Speed",
"Definition:Rotation (Geometry)/Space",
"Defi... |
proofwiki-21741 | Subtraction of Fractions | Let $a, b, c, d \in \Z$ such that $b d \ne 0$.
Then:
:$\dfrac a b - \dfrac c d = \dfrac {a D - B c} {\lcm \set {b, d} }$
where:
:$B = \dfrac b {\gcd \set {b, d} }$
:$D = \dfrac d {\gcd \set {b, d} }$
:$\lcm$ denotes lowest common multiple
:$\gcd$ denotes greatest common divisor. | {{begin-eqn}}
{{eqn | l = \dfrac a b - \dfrac c d
| r = \dfrac a b + \dfrac {\paren {-c} } d
| c =
}}
{{eqn | r = \dfrac {a D + B \paren {-c} } {\lcm \set {b, d} }
| c = Addition of Fractions
}}
{{eqn | r = \dfrac {a D - B c} {\lcm \set {b, d} }
| c =
}}
{{end-eqn}}
{{qed}} | Let $a, b, c, d \in \Z$ such that $b d \ne 0$.
Then:
:$\dfrac a b - \dfrac c d = \dfrac {a D - B c} {\lcm \set {b, d} }$
where:
:$B = \dfrac b {\gcd \set {b, d} }$
:$D = \dfrac d {\gcd \set {b, d} }$
:$\lcm$ denotes [[Definition:Lowest Common Multiple of Integers|lowest common multiple]]
:$\gcd$ denotes [[Definit... | {{begin-eqn}}
{{eqn | l = \dfrac a b - \dfrac c d
| r = \dfrac a b + \dfrac {\paren {-c} } d
| c =
}}
{{eqn | r = \dfrac {a D + B \paren {-c} } {\lcm \set {b, d} }
| c = [[Addition of Fractions]]
}}
{{eqn | r = \dfrac {a D - B c} {\lcm \set {b, d} }
| c =
}}
{{end-eqn}}
{{qed}} | Subtraction of Fractions | https://proofwiki.org/wiki/Subtraction_of_Fractions | https://proofwiki.org/wiki/Subtraction_of_Fractions | [
"Fractions",
"Subtraction",
"Lowest Common Multiple",
"Greatest Common Divisor"
] | [
"Definition:Lowest Common Multiple/Integers",
"Definition:Greatest Common Divisor/Integers"
] | [
"Addition of Fractions"
] |
proofwiki-21742 | Multiplication of Fractions | Let $a, b, c, d \in \Z$ such that $b d \ne 0$.
Then:
:$\dfrac a b \times \dfrac c d = \dfrac {a c} {b d}$ | {{begin-eqn}}
{{eqn | l = \dfrac a b \times \dfrac c d
| r = a \times \dfrac 1 b \times c \times \dfrac 1 d
| c =
}}
{{eqn | r = a \times c \times \dfrac 1 b \times \dfrac 1 d
| c =
}}
{{eqn | r = a c \times \dfrac 1 {b d}
| c =
}}
{{eqn | r = \dfrac {a c} {b d}
| c =
}}
{{end-eqn}}
{{... | Let $a, b, c, d \in \Z$ such that $b d \ne 0$.
Then:
:$\dfrac a b \times \dfrac c d = \dfrac {a c} {b d}$ | {{begin-eqn}}
{{eqn | l = \dfrac a b \times \dfrac c d
| r = a \times \dfrac 1 b \times c \times \dfrac 1 d
| c =
}}
{{eqn | r = a \times c \times \dfrac 1 b \times \dfrac 1 d
| c =
}}
{{eqn | r = a c \times \dfrac 1 {b d}
| c =
}}
{{eqn | r = \dfrac {a c} {b d}
| c =
}}
{{end-eqn}}
{{... | Multiplication of Fractions | https://proofwiki.org/wiki/Multiplication_of_Fractions | https://proofwiki.org/wiki/Multiplication_of_Fractions | [
"Multiplication of Fractions",
"Fractions",
"Multiplication"
] | [] | [] |
proofwiki-21743 | Division of Fractions | Let $a, b, c, d \in \Z$ such that $b c d \ne 0$.
Then:
:$\dfrac a b \div \dfrac c d = \dfrac {a d} {b c}$ | {{begin-eqn}}
{{eqn | l = \dfrac a b \div \dfrac c d
| r = \dfrac a b \times \dfrac 1 {c / d}
| c =
}}
{{eqn | r = \dfrac a b \times \dfrac d c
| c =
}}
{{eqn | r = \dfrac {a d} {b c}
| c = Multiplication of Fractions
}}
{{end-eqn}}
{{qed}} | Let $a, b, c, d \in \Z$ such that $b c d \ne 0$.
Then:
:$\dfrac a b \div \dfrac c d = \dfrac {a d} {b c}$ | {{begin-eqn}}
{{eqn | l = \dfrac a b \div \dfrac c d
| r = \dfrac a b \times \dfrac 1 {c / d}
| c =
}}
{{eqn | r = \dfrac a b \times \dfrac d c
| c =
}}
{{eqn | r = \dfrac {a d} {b c}
| c = [[Multiplication of Fractions]]
}}
{{end-eqn}}
{{qed}} | Division of Fractions | https://proofwiki.org/wiki/Division_of_Fractions | https://proofwiki.org/wiki/Division_of_Fractions | [
"Division of Fractions",
"Fractions",
"Division"
] | [] | [
"Multiplication of Fractions"
] |
proofwiki-21744 | Subgroup of Free Group is Free Group | Let $G$ be a free group.
Let $H \le G$ be a subgroup of $G$ such that $H$ is not the trivial group.
Then $H$ is also a free group. | Let $x \in H$ such that $x$ is not the identity element.
Because $H$ is not the trivial group, such an $x$ always exists.
As $H$ is a subgroup of $G$:
:$x \in G$
where, {{hypothesis}}, $G$ is a free group.
By Element of Free Group can be Expressed Uniquely as Finite Product, $x$ can be expressed uniquely in the form:
... | Let $G$ be a [[Definition:Free Group|free group]].
Let $H \le G$ be a [[Definition:Subgroup|subgroup]] of $G$ such that $H$ is not the [[Definition:Trivial Group|trivial group]].
Then $H$ is also a [[Definition:Free Group|free group]]. | Let $x \in H$ such that $x$ is not the [[Definition:Identity Element|identity element]].
Because $H$ is not the [[Definition:Trivial Group|trivial group]], such an $x$ always exists.
As $H$ is a [[Definition:Subgroup|subgroup]] of $G$:
:$x \in G$
where, {{hypothesis}}, $G$ is a [[Definition:Free Group|free group]].
... | Subgroup of Free Group is Free Group | https://proofwiki.org/wiki/Subgroup_of_Free_Group_is_Free_Group | https://proofwiki.org/wiki/Subgroup_of_Free_Group_is_Free_Group | [
"Free Groups"
] | [
"Definition:Free Group",
"Definition:Subgroup",
"Definition:Trivial Group",
"Definition:Free Group"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Trivial Group",
"Definition:Subgroup",
"Definition:Free Group",
"Element of Free Group can be Expressed Uniquely as Finite Product",
"Definition:Unique",
"Definition:Element",
"Definition:Distinct/Plural",
"Definition:Zero (Num... |
proofwiki-21745 | Simpson's Dissection | Let $\omega = e^{2 i \pi / q}$ be a primitive $q$th root of unity.
Let $p \not \equiv 0 \pmod q$.
Let:
:$\ds \map f x = \sum_{n \mathop = 0}^\infty a_n x^n$
Then:
:$\ds \sum_{n \mathop = 0}^\infty a_{n q + p} x^{n q + p} = \dfrac 1 q \sum_{j \mathop = 0}^{q - 1} \omega^{- j p} \map f {\omega^j x}$ | Expanding the sum on the {{RHS}}, we obtain:
{{begin-eqn}}
{{eqn | l = \dfrac 1 q \sum_{j \mathop = 0}^{q - 1} \omega^{- j p} \map f {\omega^j x}
| r = \frac 1 q \times \omega^0 \times \paren {a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots}
}}
{{eqn | o =
| ro= +
| r = \frac 1 q \times \omega^{-p} \times \... | Let $\omega = e^{2 i \pi / q}$ be a [[Definition:Primitive Complex Root of Unity|primitive $q$th root of unity]].
Let $p \not \equiv 0 \pmod q$.
Let:
:$\ds \map f x = \sum_{n \mathop = 0}^\infty a_n x^n$
Then:
:$\ds \sum_{n \mathop = 0}^\infty a_{n q + p} x^{n q + p} = \dfrac 1 q \sum_{j \mathop = 0}^{q - 1} \omega^... | Expanding the sum on the {{RHS}}, we obtain:
{{begin-eqn}}
{{eqn | l = \dfrac 1 q \sum_{j \mathop = 0}^{q - 1} \omega^{- j p} \map f {\omega^j x}
| r = \frac 1 q \times \omega^0 \times \paren {a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots}
}}
{{eqn | o =
| ro= +
| r = \frac 1 q \times \omega^{-p} \times ... | Simpson's Dissection | https://proofwiki.org/wiki/Simpson's_Dissection | https://proofwiki.org/wiki/Simpson's_Dissection | [
"Simpson's Dissection",
"Series"
] | [
"Definition:Root of Unity/Complex/Primitive"
] | [
"Definition:Term of Expression",
"Definition:Power (Algebra)/Exponent",
"Definition:Array",
"Sum of Powers of Primitive Complex Roots of Unity"
] |
proofwiki-21746 | Characterization of Locale/Statement 3 Implies Statement 4 | Let $L = \struct{S, \preceq}$ be an complete lattice satisfying the infinite join distributive law.
Then:
:$L$ is a Heyting algebra. | Let $a, b \in S$.
Let $a \to b = \sup \set{c \in S : a \wedge c \preceq b}$
We have:
{{begin-eqn}}
{{eqn | l = a \wedge \paren{a \to b}
| r = a \wedge \sup \set{c : a \wedge c \preceq b}
| c = Definition of $a \to b$
}}
{{eqn | r = \sup \set{a \wedge c : a \wedge c \preceq b}
| c = Infinite join distr... | Let $L = \struct{S, \preceq}$ be an [[Definition:Complete Lattice|complete lattice]] satisfying the [[Axiom:Infinite Join Distributive Law|infinite join distributive law]].
Then:
:$L$ is a [[Definition:Heyting Algebra|Heyting algebra]]. | Let $a, b \in S$.
Let $a \to b = \sup \set{c \in S : a \wedge c \preceq b}$
We have:
{{begin-eqn}}
{{eqn | l = a \wedge \paren{a \to b}
| r = a \wedge \sup \set{c : a \wedge c \preceq b}
| c = Definition of $a \to b$
}}
{{eqn | r = \sup \set{a \wedge c : a \wedge c \preceq b}
| c = [[Axiom:Infinite... | Characterization of Locale/Statement 3 Implies Statement 4 | https://proofwiki.org/wiki/Characterization_of_Locale/Statement_3_Implies_Statement_4 | https://proofwiki.org/wiki/Characterization_of_Locale/Statement_3_Implies_Statement_4 | [
"Characterization of Locale"
] | [
"Definition:Complete Lattice",
"Axiom:Infinite Join Distributive Law",
"Definition:Heyting Algebra"
] | [
"Axiom:Infinite Join Distributive Law",
"Definition:Greatest Element",
"Definition:Relative Pseudocomplement",
"Definition:Greatest Element",
"Definition:Heyting Algebra"
] |
proofwiki-21747 | Characterization of Locale/Statement 5 Implies Statement 3 | Let $L = \struct{S, \preceq}$ be a complete Brouwerian lattice.
Then:
:$L$ satisfies the infinite join distributive law | Let $A \subseteq S$.
Let $a \in S$.
==== Lemma 1 ====
:$\sup \set{a \wedge b : b \in A} \preceq a \wedge \sup A$
==== Proof of Lemma 1 ====
By definition of meet:
:$\forall b \in A : a \wedge b \preceq a, a \wedge b \preceq b$
By definition of upper bound:
:$a$ is an upper bound for $\set{a \wedge b : b \in A}$
By defi... | Let $L = \struct{S, \preceq}$ be a [[Definition:Complete Lattice|complete]] [[Definition:Brouwerian Lattice|Brouwerian lattice]].
Then:
:$L$ satisfies the [[Axiom:Infinite Join Distributive Law|infinite join distributive law]] | Let $A \subseteq S$.
Let $a \in S$.
==== Lemma 1 ====
:$\sup \set{a \wedge b : b \in A} \preceq a \wedge \sup A$
==== Proof of Lemma 1 ====
By definition of [[Definition:Meet|meet]]:
:$\forall b \in A : a \wedge b \preceq a, a \wedge b \preceq b$
By definition of [[Definition:Upper Bound|upper bound]]:
:$a$ is an... | Characterization of Locale/Statement 5 Implies Statement 3 | https://proofwiki.org/wiki/Characterization_of_Locale/Statement_5_Implies_Statement_3 | https://proofwiki.org/wiki/Characterization_of_Locale/Statement_5_Implies_Statement_3 | [
"Characterization of Locale"
] | [
"Definition:Complete Lattice",
"Definition:Brouwerian Lattice",
"Axiom:Infinite Join Distributive Law"
] | [
"Definition:Meet",
"Definition:Upper Bound",
"Definition:Upper Bound",
"Definition:Supremum of Set",
"Finer Supremum Precedes Supremum",
"Definition:Meet",
"Definition:Supremum of Set",
"Relative Pseudocomplement Preserves Order",
"Definition:Supremum of Set",
"Inequality with Meet Operation is Eq... |
proofwiki-21748 | Equivalence of Definitions of Sober Space | Let $T = \struct{S, \tau}$ be a topological space.
{{TFAE|def=Sober Space}}
=== Definition 1 ===
{{:Definition:Sober Space/Definition 1}} | === Definition 1 implies Definition 2 ===
Let each closed irreducible subspace of $T$ have a unique generic point.
Let $U \ne S$ be a meet-irreducible open set.
Let $F = S \setminus U$.
From Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace:
:$F$ is closed irreducible subspace
We have {{hypothesis... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
{{TFAE|def=Sober Space}}
=== [[Definition:Sober Space/Definition 1|Definition 1]] ===
{{:Definition:Sober Space/Definition 1}} | === Definition 1 implies Definition 2 ===
Let each [[Definition:Closed Set (Topology)|closed]] [[Definition:Irreducible Space|irreducible]] [[Definition:Topological Subspace|subspace]] of $T$ have a [[Definition:Unique|unique]] [[Definition:Generic Point of Topological Space|generic point]].
Let $U \ne S$ be a [[Def... | Equivalence of Definitions of Sober Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Sober_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Sober_Space | [
"Sober Spaces"
] | [
"Definition:Topological Space",
"Definition:Sober Space/Definition 1"
] | [
"Definition:Closed Set/Topology",
"Definition:Irreducible Space",
"Definition:Topological Subspace",
"Definition:Unique",
"Definition:Generic Point of Topological Space",
"Definition:Meet-Irreducible Open Set",
"Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace",
"Definition:Clos... |
proofwiki-21749 | Gaussian Integers are Closed under Addition | The set of Gaussian integers $\Z \sqbrk i$ is closed under addition:
:$\forall x, y \in \Z \sqbrk i: x + y \in \Z \sqbrk i$ | Let $x$ and $y$ be Gaussian integers.
Then:
{{begin-eqn}}
{{eqn | q = \exists a, b \in \Z
| l = x
| r = a + b i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | q = \exists c, d \in \Z
| l = y
| r = c + d i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | ll= \leadsto
| l = x + y
|... | The [[Definition:Set|set]] of [[Definition:Gaussian Integer|Gaussian integers]] $\Z \sqbrk i$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Addition|addition]]:
:$\forall x, y \in \Z \sqbrk i: x + y \in \Z \sqbrk i$ | Let $x$ and $y$ be [[Definition:Gaussian Integer|Gaussian integers]].
Then:
{{begin-eqn}}
{{eqn | q = \exists a, b \in \Z
| l = x
| r = a + b i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | q = \exists c, d \in \Z
| l = y
| r = c + d i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | ll= \... | Gaussian Integers are Closed under Addition | https://proofwiki.org/wiki/Gaussian_Integers_are_Closed_under_Addition | https://proofwiki.org/wiki/Gaussian_Integers_are_Closed_under_Addition | [
"Gaussian Integers",
"Complex Addition",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Gaussian Integer",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Addition/Complex Numbers"
] | [
"Definition:Gaussian Integer",
"Integer Addition is Closed"
] |
proofwiki-21750 | Gaussian Integers are Closed under Negation | The set of Gaussian integers $\Z \sqbrk i$ is closed under negation:
:$\forall x \in \Z \sqbrk i: -x \in \Z \sqbrk i$ | Let $x$ be a Gaussian integer.
Then:
{{begin-eqn}}
{{eqn | q = \exists a, b \in \Z
| l = x
| r = a + b i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | ll= \leadsto
| l = -x
| r = -a - b i
| c = {{Defof|Complex Negation Function}}
}}
{{eqn | ll= \leadsto
| l = -x
| o = \in
... | The [[Definition:Set|set]] of [[Definition:Gaussian Integer|Gaussian integers]] $\Z \sqbrk i$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Negation Function|negation]]:
:$\forall x \in \Z \sqbrk i: -x \in \Z \sqbrk i$ | Let $x$ be a [[Definition:Gaussian Integer|Gaussian integer]].
Then:
{{begin-eqn}}
{{eqn | q = \exists a, b \in \Z
| l = x
| r = a + b i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | ll= \leadsto
| l = -x
| r = -a - b i
| c = {{Defof|Complex Negation Function}}
}}
{{eqn | ll= \leadst... | Gaussian Integers are Closed under Negation | https://proofwiki.org/wiki/Gaussian_Integers_are_Closed_under_Negation | https://proofwiki.org/wiki/Gaussian_Integers_are_Closed_under_Negation | [
"Gaussian Integers",
"Negation Functions",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Gaussian Integer",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Negation Function/Complex"
] | [
"Definition:Gaussian Integer",
"Integer Subtraction is Closed",
"Category:Gaussian Integers",
"Category:Negation Functions",
"Category:Algebraic Closure"
] |
proofwiki-21751 | Gaussian Integers are Closed under Subtraction | The set of Gaussian integers $\Z \sqbrk i$ is closed under subtraction:
:$\forall x, y \in \Z \sqbrk i: x - y \in \Z \sqbrk i$ | Let $x$ and $y$ be Gaussian integers.
Then:
{{begin-eqn}}
{{eqn | l = x - y
| r = x + \paren {-y}
| c = {{Defof|Complex Subtraction}}
}}
{{eqn | ll= \leadsto
| l = x + \paren {-y}
| o = \in
| r = \Z \sqbrk i
| c = Gaussian Integers are Closed under Addition and Gaussian Integers are ... | The [[Definition:Set|set]] of [[Definition:Gaussian Integer|Gaussian integers]] $\Z \sqbrk i$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Subtraction|subtraction]]:
:$\forall x, y \in \Z \sqbrk i: x - y \in \Z \sqbrk i$ | Let $x$ and $y$ be [[Definition:Gaussian Integer|Gaussian integers]].
Then:
{{begin-eqn}}
{{eqn | l = x - y
| r = x + \paren {-y}
| c = {{Defof|Complex Subtraction}}
}}
{{eqn | ll= \leadsto
| l = x + \paren {-y}
| o = \in
| r = \Z \sqbrk i
| c = [[Gaussian Integers are Closed under... | Gaussian Integers are Closed under Subtraction | https://proofwiki.org/wiki/Gaussian_Integers_are_Closed_under_Subtraction | https://proofwiki.org/wiki/Gaussian_Integers_are_Closed_under_Subtraction | [
"Gaussian Integers",
"Complex Subtraction",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Gaussian Integer",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Subtraction/Complex Numbers"
] | [
"Definition:Gaussian Integer",
"Gaussian Integers are Closed under Addition",
"Gaussian Integers are Closed under Negation"
] |
proofwiki-21752 | Gaussian Integers are Closed under Multiplication | The set of Gaussian integers $\Z \sqbrk i$ is closed under multiplication:
:$\forall x, y \in \Z \sqbrk i: x \times y \in \Z \sqbrk i$ | Let $x$ and $y$ be Gaussian integers.
Then:
{{begin-eqn}}
{{eqn | q = \exists a, b \in \Z
| l = x
| r = a + b i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | q = \exists c, d \in \Z
| l = y
| r = c + d i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | ll= \leadsto
| l = x \times y
... | The [[Definition:Set|set]] of [[Definition:Gaussian Integer|Gaussian integers]] $\Z \sqbrk i$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Multiplication|multiplication]]:
:$\forall x, y \in \Z \sqbrk i: x \times y \in \Z \sqbrk i$ | Let $x$ and $y$ be [[Definition:Gaussian Integer|Gaussian integers]].
Then:
{{begin-eqn}}
{{eqn | q = \exists a, b \in \Z
| l = x
| r = a + b i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | q = \exists c, d \in \Z
| l = y
| r = c + d i
| c = {{Defof|Gaussian Integer}}
}}
{{eqn | ll= \... | Gaussian Integers are Closed under Multiplication | https://proofwiki.org/wiki/Gaussian_Integers_are_Closed_under_Multiplication | https://proofwiki.org/wiki/Gaussian_Integers_are_Closed_under_Multiplication | [
"Gaussian Integers",
"Complex Multiplication",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Gaussian Integer",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Multiplication/Complex Numbers"
] | [
"Definition:Gaussian Integer",
"Integer Addition is Closed",
"Integer Subtraction is Closed",
"Integer Multiplication is Closed"
] |
proofwiki-21753 | Gaussian Integers are not Closed under Division | The set of Gaussian integers $\Z \sqbrk i$ is not closed under division. | Proof by Counterexample:
Let:
{{begin-eqn}}
{{eqn | l = x
| r = 1 + 2 i
}}
{{eqn | l = y
| r = 3 + 4 i
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | ll= \leadsto
| l = x \div y
| r = \dfrac {1 + 2 i} {3 + 4 i}
| c = {{Defof|Complex Division}}
}}
{{eqn | r = \dfrac {\paren {1 + 2 i} \paren {3 ... | The [[Definition:Set|set]] of [[Definition:Gaussian Integer|Gaussian integers]] $\Z \sqbrk i$ is not [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Division|division]]. | [[Proof by Counterexample]]:
Let:
{{begin-eqn}}
{{eqn | l = x
| r = 1 + 2 i
}}
{{eqn | l = y
| r = 3 + 4 i
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | ll= \leadsto
| l = x \div y
| r = \dfrac {1 + 2 i} {3 + 4 i}
| c = {{Defof|Complex Division}}
}}
{{eqn | r = \dfrac {\paren {1 + 2 i} \par... | Gaussian Integers are not Closed under Division | https://proofwiki.org/wiki/Gaussian_Integers_are_not_Closed_under_Division | https://proofwiki.org/wiki/Gaussian_Integers_are_not_Closed_under_Division | [
"Gaussian Integers",
"Complex Multiplication",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Gaussian Integer",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Division/Field/Complex Numbers"
] | [
"Proof by Counterexample",
"Definition:Gaussian Integer"
] |
proofwiki-21754 | Gauss-Markov Theorem | The least-squares estimator gives the unbiased (linear) estimator of a parameter having minimum variance. | {{ProofWanted}}
{{Namedfor|Carl Friedrich Gauss|name2 = Andrey Andreyevich Markov|cat = Gauss|cat2 = Markov A A}} | The [[Definition:Least-Squares Estimator|least-squares estimator]] gives the [[Definition:Unbiased Linear Estimator|unbiased (linear) estimator]] of a [[Definition:Population Parameter|parameter]] having [[Definition:Minimum|minimum]] [[Definition:Variance|variance]]. | {{ProofWanted}}
{{Namedfor|Carl Friedrich Gauss|name2 = Andrey Andreyevich Markov|cat = Gauss|cat2 = Markov A A}} | Gauss-Markov Theorem | https://proofwiki.org/wiki/Gauss-Markov_Theorem | https://proofwiki.org/wiki/Gauss-Markov_Theorem | [
"Statistics"
] | [
"Definition:Least-Squares Estimator",
"Definition:Unbiased Linear Estimator",
"Definition:Population Parameter",
"Definition:Minimum Value of Real Function",
"Definition:Variance"
] | [] |
proofwiki-21755 | Generalized Eigenvalues as Roots of Equation | Let $\mathbf A$ be a square matrix of order $n$.
Let $\lambda$ be a generalized eigenvalue of $\mathbf A$.
Then:
:$\map \det {\mathbf A - \lambda \mathbf B} = 0$
where:
:$\mathbf B$ is another square matrix of order $n$
:$\det$ denotes the determinant. | By definition of generalized eigenvalue:
{{begin-eqn}}
{{eqn | l = \mathbf A \mathbf x
| r = \lambda \mathbf B \mathbf x
| c = for some non-zero vector $\mathbf x$
}}
{{eqn | ll= \leadsto
| l = \mathbf A \mathbf x - \lambda \mathbf B \mathbf x
| r = 0
| c =
}}
{{eqn | ll= \leadsto
|... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\lambda$ be a [[Definition:Generalized Eigenvalue|generalized eigenvalue]] of $\mathbf A$.
Then:
:$\map \det {\mathbf A - \lambda \mathbf B} = 0$
where:
:$\mathbf B$ is another [[Definition:Square... | By definition of [[Definition:Generalized Eigenvalue|generalized eigenvalue]]:
{{begin-eqn}}
{{eqn | l = \mathbf A \mathbf x
| r = \lambda \mathbf B \mathbf x
| c = for some non-[[Definition:Zero Vector|zero]] [[Definition:Vector|vector]] $\mathbf x$
}}
{{eqn | ll= \leadsto
| l = \mathbf A \mathbf x ... | Generalized Eigenvalues as Roots of Equation | https://proofwiki.org/wiki/Generalized_Eigenvalues_as_Roots_of_Equation | https://proofwiki.org/wiki/Generalized_Eigenvalues_as_Roots_of_Equation | [
"Generalized Eigenvalues"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Generalized Eigenvalue",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix"
] | [
"Definition:Generalized Eigenvalue",
"Definition:Zero Vector",
"Definition:Vector",
"Definition:Zero Vector",
"Matrix is Singular iff Product with Non-Zero Vector is Zero",
"Definition:Matrix/Square Matrix",
"Definition:Singular Matrix",
"Definition:Singular Matrix"
] |
proofwiki-21756 | Eigenvalue is Instance of Generalized Eigenvalue | Let $\mathbf A$ be a square matrix of order $n$.
Let $\lambda$ be an eigenvalue of $\mathbf A$.
Then:
:$\lambda$ is a generalized eigenvalue of $\mathbf A$
:the corresponding eigenvector $\mathbf x$ corresponding to $\lambda$ is the generalized eigenvector of $\mathbf A$ corresponding to $\lambda$. | By the definition of eigenvalue of $\mathbf A$:
:$\map \det {\mathbf I_n \mathbf x - \mathbf A} = 0$
for some non-zero vector $\mathbf x$.
Recall the definition of generalized eigenvalue of $\mathbf A$:
:$\mathbf A \mathbf x = \lambda \mathbf B \mathbf x$
for:
:some non-zero vector $\mathbf x$
:some square matrix $\mat... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix of order $n$]].
Let $\lambda$ be an [[Definition:Eigenvalue of Square Matrix|eigenvalue]] of $\mathbf A$.
Then:
:$\lambda$ is a [[Definition:Generalized Eigenvalue|generalized eigenvalue]] of $\mathbf A$
:the corresponding [[Definition:Eigenvector of Squ... | By the definition of [[Definition:Eigenvalue of Square Matrix|eigenvalue]] of $\mathbf A$:
:$\map \det {\mathbf I_n \mathbf x - \mathbf A} = 0$
for some non-[[Definition:Zero Vector|zero]] [[Definition:Vector|vector]] $\mathbf x$.
Recall the definition of [[Definition:Generalized Eigenvalue|generalized eigenvalue]] o... | Eigenvalue is Instance of Generalized Eigenvalue | https://proofwiki.org/wiki/Eigenvalue_is_Instance_of_Generalized_Eigenvalue | https://proofwiki.org/wiki/Eigenvalue_is_Instance_of_Generalized_Eigenvalue | [
"Generalized Eigenvalues",
"Eigenvalues of Square Matrices",
"Generalized Eigenvectors",
"Eigenvectors of Square Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Eigenvalue/Square Matrix",
"Definition:Generalized Eigenvalue",
"Definition:Eigenvector/Square Matrix",
"Definition:Generalized Eigenvector"
] | [
"Definition:Eigenvalue/Square Matrix",
"Definition:Zero Vector",
"Definition:Vector",
"Definition:Generalized Eigenvalue",
"Definition:Zero Vector",
"Definition:Vector",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Generalized Eigenvalues as Roots of Equation",
"Def... |
proofwiki-21757 | Special Linear Group is Normal Subgroup of General Linear Group | Let $K$ be a field whose zero is $0_K$ and unity is $1_K$.
Let $\SL {n, K}$ be the special linear group of order $n$ over $K$.
Then $\SL {n, K}$ is a normal subgroup of the general linear group $\GL {n, K}$. | From Special Linear Group is Subgroup of General Linear Group, we have that $\SL {n, K}$ is a subgroup of $\GL {n, K}$.
{{Proofread}}
Let $\mathbf A \in \SL {n, K}$.
Let $\mathbf B \in \GL {n, K}$.
Then:
{{begin-eqn}}
{{eqn | l = \map \det {\mathbf B \mathbf A \mathbf B^{-1} }
| r = \map \det {\mathbf B} \map \d... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_K$ and [[Definition:Unity of Field|unity]] is $1_K$.
Let $\SL {n, K}$ be the [[Definition:Special Linear Group|special linear group of order $n$ over $K$]].
Then $\SL {n, K}$ is a [[Definition:Normal Subgroup|normal... | From [[Special Linear Group is Subgroup of General Linear Group]], we have that $\SL {n, K}$ is a [[Definition:Subgroup|subgroup]] of $\GL {n, K}$.
{{Proofread}}
Let $\mathbf A \in \SL {n, K}$.
Let $\mathbf B \in \GL {n, K}$.
Then:
{{begin-eqn}}
{{eqn | l = \map \det {\mathbf B \mathbf A \mathbf B^{-1} }
| ... | Special Linear Group is Normal Subgroup of General Linear Group | https://proofwiki.org/wiki/Special_Linear_Group_is_Normal_Subgroup_of_General_Linear_Group | https://proofwiki.org/wiki/Special_Linear_Group_is_Normal_Subgroup_of_General_Linear_Group | [
"Special Linear Group",
"General Linear Group",
"Examples of Normal Subgroups"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Special Linear Group",
"Definition:Normal Subgroup",
"Definition:General Linear Group"
] | [
"Special Linear Group is Subgroup of General Linear Group",
"Definition:Subgroup",
"Determinant of Matrix Product",
"Determinant of Inverse Matrix",
"Definition:Special Linear Group",
"Definition:Normal Subgroup"
] |
proofwiki-21758 | List of Elements in Infinite Cyclic Group | Let $\gen a$ be the infinite cyclic subgroup generated by $a$.
Then:
:$\set {\ldots, a^{-2}, a^{-1}, a^0, a^1, a^2, \ldots}$
is a complete repetition-free list of the elements of $\gen a$.
That is:
:$G = \set {a^n: n \in \Z}$ | From Infinite Cyclic Group is Isomorphic to Integers:
:$\gen a \cong \struct {\Z, +}$
That is, elements of $\gen a$ correspond one-to-one with $\struct {\Z, +}$ by the bijection $\phi: \Z \to \gen a$ defined as:
:$\forall x \in \Z: \map \phi x = a^x$
Hence the result.
{{qed}} | Let $\gen a$ be the [[Definition:Infinite Cyclic Group|infinite cyclic subgroup]] [[Definition:Generated Subgroup|generated]] by $a$.
Then:
:$\set {\ldots, a^{-2}, a^{-1}, a^0, a^1, a^2, \ldots}$
is a complete repetition-free list of the [[Definition:Element|elements]] of $\gen a$.
That is:
:$G = \set {a^n: n \in \Z... | From [[Infinite Cyclic Group is Isomorphic to Integers]]:
:$\gen a \cong \struct {\Z, +}$
That is, [[Definition:Element|elements]] of $\gen a$ [[Definition:One-to-One Correspondence|correspond one-to-one]] with $\struct {\Z, +}$ by the [[Definition:Bijection|bijection]] $\phi: \Z \to \gen a$ defined as:
:$\forall x \i... | List of Elements in Infinite Cyclic Group | https://proofwiki.org/wiki/List_of_Elements_in_Infinite_Cyclic_Group | https://proofwiki.org/wiki/List_of_Elements_in_Infinite_Cyclic_Group | [
"Infinite Cyclic Group"
] | [
"Definition:Infinite Cyclic Group",
"Definition:Generated Subgroup",
"Definition:Element"
] | [
"Infinite Cyclic Group is Isomorphic to Integers",
"Definition:Element",
"Definition:Bijection",
"Definition:Bijection"
] |
proofwiki-21759 | Definitions of Genera are Equivalent | In appropriate circumstances, the following definitions of genus are equivalent:
=== Genus of Manifold ===
{{:Definition:Genus of Manifold}}
=== Genus of Riemann Surface ===
{{:Definition:Genus of Riemann Surface}}
=== Genus of Plane Algebraic Curve ===
{{:Definition:Genus of Plane Algebraic Curve}} | {{tidy|''please'' make an attempt to comply with house style and rules. It's good manners.}}
{{MissingLinks}}
{{ProofWanted}}
'''Theorem:''' Let $C$ be a non-singular projective algebraic curve of degree $n$.
Then the genus of $C$ as a Riemann surface is given by
:$g = \dfrac {\paren {n - 1} \paren {n - 2} } 2$
'''Proo... | In appropriate circumstances, the following definitions of [[Definition:Genus|genus]] are equivalent:
=== [[Definition:Genus of Manifold|Genus of Manifold]] ===
{{:Definition:Genus of Manifold}}
=== [[Definition:Genus of Riemann Surface|Genus of Riemann Surface]] ===
{{:Definition:Genus of Riemann Surface}}
=== [[De... | {{tidy|''please'' make an attempt to comply with house style and rules. It's good manners.}}
{{MissingLinks}}
{{ProofWanted}}
'''Theorem:''' Let $C$ be a [[non-singular]] projective algebraic curve of degree $n$.
Then the genus of $C$ as a Riemann surface is given by
:$g = \dfrac {\paren {n - 1} \paren {n - 2} } 2$
... | Definitions of Genera are Equivalent | https://proofwiki.org/wiki/Definitions_of_Genera_are_Equivalent | https://proofwiki.org/wiki/Definitions_of_Genera_are_Equivalent | [
"Genera"
] | [
"Definition:Genus",
"Definition:Genus of Manifold",
"Definition:Genus of Riemann Surface",
"Definition:Genus of Plane Algebraic Curve"
] | [
"non-singular",
"Ramification points of y/x",
"Definition:Ramification",
"multiplicity",
"Definition:Point of Inflection",
"Definition:Central Projection",
"Riemann-Hurwitz Formula",
"Bézout's Theorem",
"Hessian",
"Definition:Point of Inflection ",
"symmetric bilinear form",
"Euler's Homogeneo... |
proofwiki-21760 | Galois Group Acts Faithfully on Generating Set | Let $G$ be the Galois group of a finite field extension $E/ F$.
Let $\set {\alpha_1, \ldots , \alpha_n}$ be a generating set for field $E$ as an $F$-vector space.
Let $f_1, \ldots, f_n$ be their minimal polynomials.
Let $f = f_1 \dots f_n$ be the product of them.
Then the action of $G$ on roots of $f$ is faithful. | {{tidy|reduce the use of compound sentences}}
Suppose that $\sigma \in G$ fixes each $\alpha_i$.
Then since $\sigma$ is an $F$-linear automorphism of $E$ and since $\set {\alpha_1, \ldots, \alpha_n}$ is a generating set for $E$, we see that $\sigma$ fixes every other element of $E$.
Then $\sigma$ must be the identity m... | Let $G$ be the [[Definition:Galois Group of Field Extension|Galois group]] of a [[Definition:Finite Field Extension|finite field extension]] $E/ F$.
Let $\set {\alpha_1, \ldots , \alpha_n}$ be a [[Definition:Generating Set of Vector Space|generating set]] for field $E$ as an $F$-vector space.
Let $f_1, \ldots, f_n$ b... | {{tidy|reduce the use of compound sentences}}
Suppose that $\sigma \in G$ [[Definition:Fixed Element|fixes]] each $\alpha_i$.
Then since $\sigma$ is an $F$-linear [[Definition:Field Automorphism|automorphism]] of $E$ and since $\set {\alpha_1, \ldots, \alpha_n}$ is a generating set for $E$, we see that $\sigma$ fixes... | Galois Group Acts Faithfully on Generating Set | https://proofwiki.org/wiki/Galois_Group_Acts_Faithfully_on_Generating_Set | https://proofwiki.org/wiki/Galois_Group_Acts_Faithfully_on_Generating_Set | [
"Galois Groups of Field Extensions"
] | [
"Definition:Galois Group of Field Extension",
"Definition:Field Extension/Degree/Finite",
"Definition:Generator of Vector Space",
"Definition:Minimal Polynomial",
"Definition:Faithful Group Action"
] | [
"Definition:Fixed Element",
"Definition:Field Automorphism",
"Definition:Faithful Group Action",
"Category:Galois Groups of Field Extensions"
] |
proofwiki-21761 | Euler's Homogeneous Function Theorem | Let $\map f{x,y}$ be a homogeneous function of order $n$ so that:
:$\map f {t x, t y} = t^n \map f {x, y}$
Then:
:$x \dfrac {\partial f} {\partial x} + y \dfrac {\partial f} {\partial y} = n \map f {x, y}$ | Define $x' = x t$ and $y' = y t$.
By Chain Rule for Derivatives:
{{begin-eqn}}
{{eqn | l = n t^{n - 1} \map f {x, y}
| r = \dfrac {\partial f} {\partial x'} \dfrac {\partial x'} {\partial t} + \dfrac {\partial f} {\partial y'} \dfrac {\partial y'} {\partial t}
}}
{{eqn | r = x \dfrac {\partial f} {\partial x'} +... | Let $\map f{x,y}$ be a [[Definition:Homogeneous Function|homogeneous function]] of order $n$ so that:
:$\map f {t x, t y} = t^n \map f {x, y}$
Then:
:$x \dfrac {\partial f} {\partial x} + y \dfrac {\partial f} {\partial y} = n \map f {x, y}$ | Define $x' = x t$ and $y' = y t$.
By [[Chain Rule for Derivatives]]:
{{begin-eqn}}
{{eqn | l = n t^{n - 1} \map f {x, y}
| r = \dfrac {\partial f} {\partial x'} \dfrac {\partial x'} {\partial t} + \dfrac {\partial f} {\partial y'} \dfrac {\partial y'} {\partial t}
}}
{{eqn | r = x \dfrac {\partial f} {\partial... | Euler's Homogeneous Function Theorem | https://proofwiki.org/wiki/Euler's_Homogeneous_Function_Theorem | https://proofwiki.org/wiki/Euler's_Homogeneous_Function_Theorem | [
"Calculus"
] | [
"Definition:Homogeneous Function"
] | [
"Derivative of Composite Function"
] |
proofwiki-21762 | Euler Phi Function by Argument is Injective | Let $f: \N \to \N$ be the arithmetic function defined as:
:$\map f n = n \map \phi n$
where $\map \phi n$ is the Euler Phi function (which is not injective).
Then $f$ is injective. | By Product of Multiplicative Functions is Multiplicative and Euler Phi Function is Multiplicative, $f$ is a multiplicative function.
Suppose that $\map f {n_1} = \map f {n_2}$.
The case $n_1 = 1$ is trivial.
Let $n_1 > 1$.
Let $p$ be the largest prime factor of $n_1$.
By Largest Prime Factor of Euler Phi Function, the ... | Let $f: \N \to \N$ be the [[Definition:Arithmetic Function|arithmetic function]] defined as:
:$\map f n = n \map \phi n$
where $\map \phi n$ is the [[Definition:Euler Phi Function|Euler Phi function]] (which is not [[Definition:Injective|injective]]).
Then $f$ is [[Definition:Injective|injective]]. | By [[Product of Multiplicative Functions is Multiplicative]] and [[Euler Phi Function is Multiplicative]], $f$ is a [[Definition:Multiplicative Arithmetic Function|multiplicative function]].
Suppose that $\map f {n_1} = \map f {n_2}$.
The case $n_1 = 1$ is trivial.
Let $n_1 > 1$.
Let $p$ be the largest prime fact... | Euler Phi Function by Argument is Injective | https://proofwiki.org/wiki/Euler_Phi_Function_by_Argument_is_Injective | https://proofwiki.org/wiki/Euler_Phi_Function_by_Argument_is_Injective | [
"Euler Phi Function"
] | [
"Definition:Arithmetic Function",
"Definition:Euler Phi Function",
"Definition:Injective",
"Definition:Injective"
] | [
"Product of Multiplicative Functions is Multiplicative",
"Euler Phi Function is Multiplicative",
"Definition:Multiplicative Arithmetic Function",
"Largest Prime Factor of Euler Phi Function",
"Definition:Multiplicative Arithmetic Function"
] |
proofwiki-21763 | Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n | Let $n \in \Z_{\ge 0}$ be an integer.
Let $\struct {\Z / n \Z, +, \cdot}$ be the ring of integers modulo $n$.
Let $U = \struct {\paren {\Z / n \Z}^\times, \cdot}$ denote the group of units of $\struct {\Z / n \Z, +, \cdot}$.
Let $C_8$ denote the cyclic group of order $8$
Then:
:$U$ and $C_8$ are not isomorphic. | === Lemma ===
{{:Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Lemma}}{{qed|lemma}}
{{AimForCont}} $U$ and $C_8$ are isomorphic.
:$\order U = \order {C_8} = 8$
From Order of Group of Units of Integers Modulo n we have that
:$8 = \order U = \map \phi n$
where $\phi$ denotes the Euler $... | Let $n \in \Z_{\ge 0}$ be an [[Definition:Integer|integer]].
Let $\struct {\Z / n \Z, +, \cdot}$ be the [[Definition:Ring of Integers Modulo m|ring of integers modulo $n$]].
Let $U = \struct {\paren {\Z / n \Z}^\times, \cdot}$ denote the [[Definition:Group of Units of Ring|group of units]] of $\struct {\Z / n \Z, +, ... | === [[Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Lemma|Lemma]] ===
{{:Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Lemma}}{{qed|lemma}}
{{AimForCont}} $U$ and $C_8$ are isomorphic.
:$\order U = \order {C_8} = 8$
From [[Order of Group of Units... | Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Proof 1 | https://proofwiki.org/wiki/Cyclic_Group_of_Order_8_is_not_isomorphic_to_Group_of_Units_of_Integers_Modulo_n | https://proofwiki.org/wiki/Cyclic_Group_of_Order_8_is_not_isomorphic_to_Group_of_Units_of_Integers_Modulo_n/Proof_1 | [
"Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n",
"Ring of Integers Modulo m",
"Cyclic Group of Order 8"
] | [
"Definition:Integer",
"Definition:Ring of Integers Modulo m",
"Definition:Group of Units/Ring",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Lemma",
"Order of Group of Units of Integers Modulo n",
"Definition:Euler Phi Function",
"Cyclicity Condition for Units of Ring of Integers Modulo n",
"Isomorphism between Group of Units Ring of Integers Modulo 2^n and C2 x C2... |
proofwiki-21764 | Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n | Let $n \in \Z_{\ge 0}$ be an integer.
Let $\struct {\Z / n \Z, +, \cdot}$ be the ring of integers modulo $n$.
Let $U = \struct {\paren {\Z / n \Z}^\times, \cdot}$ denote the group of units of $\struct {\Z / n \Z, +, \cdot}$.
Let $C_8$ denote the cyclic group of order $8$
Then:
:$U$ and $C_8$ are not isomorphic. | {{AimForCont}} $U$ and $C_8$ are isomorphic.
:$\order U = \order {C_8} = 8$
From Order of Group of Units of Integers Modulo n we have that
:$8 = \order U = \map \phi n$
where $\phi$ denotes the Euler $\phi$-function.
$\map \phi 1 = \map \phi 2 = 1$, so $n > 2$.
By Cyclicity Condition for Units of Ring of Integers Modul... | Let $n \in \Z_{\ge 0}$ be an [[Definition:Integer|integer]].
Let $\struct {\Z / n \Z, +, \cdot}$ be the [[Definition:Ring of Integers Modulo m|ring of integers modulo $n$]].
Let $U = \struct {\paren {\Z / n \Z}^\times, \cdot}$ denote the [[Definition:Group of Units of Ring|group of units]] of $\struct {\Z / n \Z, +, ... | {{AimForCont}} $U$ and $C_8$ are isomorphic.
:$\order U = \order {C_8} = 8$
From [[Order of Group of Units of Integers Modulo n]] we have that
:$8 = \order U = \map \phi n$
where $\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$-function]].
$\map \phi 1 = \map \phi 2 = 1$, so $n > 2$.
By [[Cyclicity... | Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Proof 2 | https://proofwiki.org/wiki/Cyclic_Group_of_Order_8_is_not_isomorphic_to_Group_of_Units_of_Integers_Modulo_n | https://proofwiki.org/wiki/Cyclic_Group_of_Order_8_is_not_isomorphic_to_Group_of_Units_of_Integers_Modulo_n/Proof_2 | [
"Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n",
"Ring of Integers Modulo m",
"Cyclic Group of Order 8"
] | [
"Definition:Integer",
"Definition:Ring of Integers Modulo m",
"Definition:Group of Units/Ring",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Order of Group of Units of Integers Modulo n",
"Definition:Euler Phi Function",
"Cyclicity Condition for Units of Ring of Integers Modulo n",
"Definition:Prime Number"
] |
proofwiki-21765 | Units of Ring Direct Product are Ring Direct Product of Units | Let $R_1, R_2, \ldots, R_m$ be rings with identity.
{{MissingLinks|group isomorphism? This is rings we're talking about here.}}
Then the following group isomorphism holds:
:$\map U {R_1 \oplus R_2 \oplus \cdots \oplus R_n} \simeq \map U {R_1} \oplus \map U {R_2} \oplus \cdots \oplus \map U {R_n}$.
{{explain|what's $U$?... | It suffices to prove the case $m = 2$.
{{explain|no it doesn't, we need an inductive argument}}
By {{Defof|Ring Direct Sum}} the finite direct sum
{{explain|Is ring direct sum the same thing as finite direct sum? At least be consistent.}}
is isomorphic to the finite direct product.
{{mistake|I believe the wrong link h... | Let $R_1, R_2, \ldots, R_m$ be [[Definition:Ring (Abstract Algebra)|rings with identity]].
{{MissingLinks|group isomorphism? This is rings we're talking about here.}}
Then the following group isomorphism holds:
:$\map U {R_1 \oplus R_2 \oplus \cdots \oplus R_n} \simeq \map U {R_1} \oplus \map U {R_2} \oplus \cdots \o... | It suffices to prove the case $m = 2$.
{{explain|no it doesn't, we need an inductive argument}}
By {{Defof|Ring Direct Sum}} the finite direct sum
{{explain|Is ring direct sum the same thing as finite direct sum? At least be consistent.}}
is isomorphic to the finite [[Definition:Product (Abstract Algebra)/Ring|dir... | Units of Ring Direct Product are Ring Direct Product of Units | https://proofwiki.org/wiki/Units_of_Ring_Direct_Product_are_Ring_Direct_Product_of_Units | https://proofwiki.org/wiki/Units_of_Ring_Direct_Product_are_Ring_Direct_Product_of_Units | [
"Ring Direct Products"
] | [
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Ring (Abstract Algebra)/Product",
"Category:Ring Direct Products"
] |
proofwiki-21766 | Phi is 8 has only 5 solutions | $\phi$ denotes the Euler $\phi$-function.
There are $5$ numbers $n$ with the property that $\map \phi n = 8$, and they are $15$, $16$, $20$, $24$ and $30$. | Suppose $\phi(n)=8$.
By Fundamental Theorem of Arithmetic, $n$ has a prime factorization: $$n=\prod_{i=1}^k p_i^{\alpha_i}$$
where $p_i$ are all distinct primes. Since $\phi(1)=1\neq8$, we assume $k>0$. Say that $n$ has $k$ distinct prime factors, so that each $\alpha_i > 0$.
By Euler Phi Function is Multiplicative, $$... | $\phi$ denotes the [[Definition:Euler Phi Function|Euler $\phi$-function]].
There are $5$ numbers $n$ with the property that $\map \phi n = 8$, and they are $15$, $16$, $20$, $24$ and $30$. | Suppose $\phi(n)=8$.
By [[Fundamental Theorem of Arithmetic]], $n$ has a prime factorization: $$n=\prod_{i=1}^k p_i^{\alpha_i}$$
where $p_i$ are all distinct primes. Since $\phi(1)=1\neq8$, we assume $k>0$. Say that $n$ has $k$ distinct prime factors, so that each $\alpha_i > 0$.
By [[Euler Phi Function is Multiplic... | Phi is 8 has only 5 solutions | https://proofwiki.org/wiki/Phi_is_8_has_only_5_solutions | https://proofwiki.org/wiki/Phi_is_8_has_only_5_solutions | [] | [
"Definition:Euler Phi Function"
] | [
"Fundamental Theorem of Arithmetic",
"Euler Phi Function is Multiplicative",
"Euler Phi Function of Prime Power",
"Euler Phi Function is Multiplicative",
"Euler Phi Function of Prime Power"
] |
proofwiki-21767 | Ramaré's Theorem | Every positive integer can be expressed as the sum of no more than $6$ prime numbers. | {{ProofWanted}}
{{Namedfor|Olivier Ramaré|cat = Ramaré}} | Every [[Definition:Positive Integer|positive integer]] can be expressed as the [[Definition:Integer Addition|sum]] of no more than $6$ [[Definition:Prime Number|prime numbers]]. | {{ProofWanted}}
{{Namedfor|Olivier Ramaré|cat = Ramaré}} | Ramaré's Theorem | https://proofwiki.org/wiki/Ramaré's_Theorem | https://proofwiki.org/wiki/Ramaré's_Theorem | [
"Prime Numbers",
"Number Theory"
] | [
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Prime Number"
] | [] |
proofwiki-21768 | Dirichlet's Integral Form of Digamma Function | :$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac 1 {t \paren {1 + t}^z } } \rd t$ | We have:
{{begin-eqn}}
{{eqn | l = \map \psi z
| r = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t
| c = Gauss's Integral Form of Digamma Function
}}
{{eqn | r = \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {\paren {e^{-t} }^z } {1 - e^{-t} } \rd t
... | :$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac 1 {t \paren {1 + t}^z } } \rd t$ | We have:
{{begin-eqn}}
{{eqn | l = \map \psi z
| r = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t
| c = [[Gauss's Integral Form of Digamma Function]]
}}
{{eqn | r = \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {\paren {e^{-t} }^z } {1 - e^{-t} } \rd t
... | Dirichlet's Integral Form of Digamma Function | https://proofwiki.org/wiki/Dirichlet's_Integral_Form_of_Digamma_Function | https://proofwiki.org/wiki/Dirichlet's_Integral_Form_of_Digamma_Function | [
"Digamma Function",
"Complex Analysis",
"Definite Integrals"
] | [] | [
"Gauss's Integral Form of Digamma Function",
"Linear Combination of Integrals",
"Integration by Substitution",
"Natural Logarithm of 1 is 0",
"Logarithm Tends to Infinity",
"Linear Combination of Integrals"
] |
proofwiki-21769 | Lüroth's Theorem | {{MissingLinks|throughout}}
Let $K$ be a field.
Let $\map K X$ be the rational function field, for some indeterminate $X$.
Let $M$ be an intermediate field between $K$ and $\map K X$.
Then there exists a rational function $\map f X \in \map K X$ such that:
{{explain|Domain and codomain of $\map f X$}}
:$M = \map K {\ma... | By Gauss's Lemma (Polynomial Theory),
{{ProofWanted}}
{{Namedfor|Jacob Lüroth|cat = Lüroth}} | {{MissingLinks|throughout}}
Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\map K X$ be the rational function field, for some [[Definition:Indeterminate (Polynomial Theory)|indeterminate]] $X$.
Let $M$ be an [[Definition:Intermediate Field|intermediate field]] between $K$ and $\map K X$.
Then ther... | By [[Gauss's Lemma (Polynomial Theory)]],
{{ProofWanted}}
{{Namedfor|Jacob Lüroth|cat = Lüroth}} | Lüroth's Theorem | https://proofwiki.org/wiki/Lüroth's_Theorem | https://proofwiki.org/wiki/Lüroth's_Theorem | [
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Polynomial Ring/Indeterminate",
"Definition:Intermediate Field",
"Definition:Rational Function",
"Definition:Intermediate Field",
"Definition:Simple Field Extension"
] | [
"Gauss's Lemma (Polynomial Theory)"
] |
proofwiki-21770 | Abel-Plana Formula | :$\ds \sum_{n \mathop = 0}^\infty \map f n = \int_0^\infty \map f x \rd x + \dfrac 1 2 \map f 0 + i \int_0^\infty \dfrac {\map f {i t} - \map f {-i t} } {e^{2 \pi t} - 1} \rd t$ | {{ProofWanted}}
{{Namedfor|Niels Henrik Abel|name2 = Giovanni Antonio Amedeo Plana|cat = Abel|cat2 = Plana}} | :$\ds \sum_{n \mathop = 0}^\infty \map f n = \int_0^\infty \map f x \rd x + \dfrac 1 2 \map f 0 + i \int_0^\infty \dfrac {\map f {i t} - \map f {-i t} } {e^{2 \pi t} - 1} \rd t$ | {{ProofWanted}}
{{Namedfor|Niels Henrik Abel|name2 = Giovanni Antonio Amedeo Plana|cat = Abel|cat2 = Plana}} | Abel-Plana Formula | https://proofwiki.org/wiki/Abel-Plana_Formula | https://proofwiki.org/wiki/Abel-Plana_Formula | [
"Analytic Number Theory"
] | [] | [] |
proofwiki-21771 | Isomorphism between Group of Units Ring of Integers Modulo 2^n and C2 x C2^(n-2) | Let $n \in \Z_{\ge 0}$ be a positive integer and $n \ge 2$.
Let $R = \struct {\Z / 2^n \Z, +, \times}$ be the ring of integers modulo $2^n$.
Let $U = \struct {\paren {\Z / 2^n \Z}^\times, \times}$ denote the group of units of $R$.
Let $C_2$ be cyclic group of order $2$.
Let $C_{2^{n - 2} }$ be the cyclic group of order... | {{tidy|coherence needs to be improved}} | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]] and $n \ge 2$.
Let $R = \struct {\Z / 2^n \Z, +, \times}$ be the [[Definition:Ring (Abstract Algebra)|ring]] of [[Definition:Integers Modulo m|integers modulo $2^n$]].
Let $U = \struct {\paren {\Z / 2^n \Z}^\times, \times}$ denote the [[Defi... | {{tidy|coherence needs to be improved}} | Isomorphism between Group of Units Ring of Integers Modulo 2^n and C2 x C2^(n-2) | https://proofwiki.org/wiki/Isomorphism_between_Group_of_Units_Ring_of_Integers_Modulo_2^n_and_C2_x_C2^(n-2) | https://proofwiki.org/wiki/Isomorphism_between_Group_of_Units_Ring_of_Integers_Modulo_2^n_and_C2_x_C2^(n-2) | [
"Ring of Integers Modulo m"
] | [
"Definition:Positive/Integer",
"Definition:Ring (Abstract Algebra)",
"Definition:Integers Modulo m",
"Definition:Group of Units/Ring",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)",
... | [] |
proofwiki-21772 | Isomorphism between Group of Units of Ring of Integers Modulo p^n and C((p-1)p^(n-1)) | Let $n \in \Z_{\ge 0}$ be a positive integer.
Let $p$ be an odd prime.
Let $R = \struct {\Z / p^n \Z, +, \times}$ be the ring of integers modulo $p^n$.
Let $U = \struct {\paren {\Z / p^n \Z}^\times, \times}$ denote the group of units of $R$.
Let $C_{(p - 1)p^{n - 1}}$ be be cyclic group of order $(p - 1)p^{n - 1}$.
The... | The case $n = 1$ is proved in Ring of Integers Modulo Prime is Field.
Suppose $n \ge 2$.
{{ProofWanted}} | Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let $R = \struct {\Z / p^n \Z, +, \times}$ be the [[Definition:Ring (Abstract Algebra)|ring]] of [[Definition:Integers Modulo m|integers modulo $p^n$]].
Let $U = \struct {\paren {\Z / p^n \... | The case $n = 1$ is proved in [[Ring of Integers Modulo Prime is Field]].
Suppose $n \ge 2$.
{{ProofWanted}} | Isomorphism between Group of Units of Ring of Integers Modulo p^n and C((p-1)p^(n-1)) | https://proofwiki.org/wiki/Isomorphism_between_Group_of_Units_of_Ring_of_Integers_Modulo_p^n_and_C((p-1)p^(n-1)) | https://proofwiki.org/wiki/Isomorphism_between_Group_of_Units_of_Ring_of_Integers_Modulo_p^n_and_C((p-1)p^(n-1)) | [
"Examples of Group Isomorphisms"
] | [
"Definition:Positive/Integer",
"Definition:Odd Prime",
"Definition:Ring (Abstract Algebra)",
"Definition:Integers Modulo m",
"Definition:Group of Units/Ring",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Isomorphism (Abstract Algebra)"
] | [
"Ring of Integers Modulo Prime is Field"
] |
proofwiki-21773 | Kernel of Linear Transformation is Linear Subspace | Let $V, W$ be normed vector spaces on a field $F$.
{{explain|Why do they need to be normed?}}
Let $T: V \to W$ be a linear transformation.
Then the kernel of $T$ is a linear subspace of $V$.
{{explain|We need to decide whether we want to use the term "linear subspace" in the first place. The term has a number of differ... | By Kernel of Linear Transformation contains Zero Vector, $0_V \in \ker T$. | Let $V, W$ be [[Definition:Normed Vector Space|normed vector spaces]] on a [[Definition:Field (Abstract Algebra)|field]] $F$.
{{explain|Why do they need to be normed?}}
Let $T: V \to W$ be a [[Definition:Linear Transformation|linear transformation]].
Then the [[Definition:Kernel of Linear Transformation on Vector S... | By [[Kernel of Linear Transformation contains Zero Vector]], $0_V \in \ker T$. | Kernel of Linear Transformation is Linear Subspace | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_is_Linear_Subspace | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_is_Linear_Subspace | [
"Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Linear Transformation",
"Definition:Kernel of Linear Transformation/Vector Space",
"Definition:Linear Subspace"
] | [
"Kernel of Linear Transformation contains Zero Vector"
] |
proofwiki-21774 | Complete Graph K5 is not Planar | The complete graph $K_5$ is not planar. | Recall the definition of planar graph:
{{:Definition:Planar Graph}}
First we note that the complete graph $K_4$ is planar by demonstrating its planarity:
:200px
Now we attempt to create $K_5$ by building it from its subgraph $K_4$.
Each of the faces of $K_4$ is in the same form as each of the others.
That is, each face... | The [[Definition:Complete Graph|complete graph]] $K_5$ is not [[Definition:Planar Graph|planar]]. | Recall the definition of [[Definition:Planar Graph|planar graph]]:
{{:Definition:Planar Graph}}
First we note that the [[Definition:Complete Graph|complete graph]] $K_4$ is [[Definition:Planar Graph|planar]] by demonstrating its planarity:
:[[File:K4.png|200px]]
Now we attempt to create $K_5$ by building it from it... | Complete Graph K5 is not Planar | https://proofwiki.org/wiki/Complete_Graph_K5_is_not_Planar | https://proofwiki.org/wiki/Complete_Graph_K5_is_not_Planar | [
"Planar Graphs",
"Complete Graphs"
] | [
"Definition:Complete Graph",
"Definition:Planar Graph"
] | [
"Definition:Planar Graph",
"Definition:Complete Graph",
"Definition:Planar Graph",
"File:K4.png",
"Definition:Subgraph",
"Definition:Planar Graph/Face",
"Definition:Planar Graph/Face",
"Definition:Triangle (Graph Theory)",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Incident (Graph Theor... |
proofwiki-21775 | Thomsen Graph is not Planar | The Thomsen graph is not planar. | {{Recall|Thomsen Graph|Thomsen graph}}
{{:Definition:Thomsen Graph}}
{{Recall|Planar Graph|planar graph}}
{{:Definition:Planar Graph}}
Recall that the complete bipartite graph $K_{2, 3}$ is planar:
:280px
Let $A = \set {A_1, A_2, A_3}$ and $B = \set {B_1, B_2, B_3}$ be the two subsets which partition the vertex set of ... | The [[Definition:Thomsen Graph|Thomsen graph]] is not [[Definition:Planar Graph|planar]]. | {{Recall|Thomsen Graph|Thomsen graph}}
{{:Definition:Thomsen Graph}}
{{Recall|Planar Graph|planar graph}}
{{:Definition:Planar Graph}}
Recall that the [[Planar Graph/Examples/Complete Bipartite Graph K2,3|complete bipartite graph $K_{2, 3}$ is planar]]:
:[[File:Complete-bipartite-graph-2-3.png|280px]]
Let $A = \set... | Thomsen Graph is not Planar | https://proofwiki.org/wiki/Thomsen_Graph_is_not_Planar | https://proofwiki.org/wiki/Thomsen_Graph_is_not_Planar | [
"Thomsen Graph",
"Planar Graphs",
"Complete Bipartite Graphs"
] | [
"Definition:Thomsen Graph",
"Definition:Planar Graph"
] | [
"Planar Graph/Examples/Complete Bipartite Graph K2,3",
"File:Complete-bipartite-graph-2-3.png",
"Definition:Subset",
"Definition:Set Partition",
"Definition:Vertex Set",
"Definition:Planar Graph",
"Definition:Subgraph",
"Definition:Set Partition",
"Definition:Vertex Set",
"Definition:Planar Graph/... |
proofwiki-21776 | Sobolev Space is Banach Space | Let $k \in \Z_{\ge 0}$.
Let $p \in \R_{\ge 1}$ or $p = \infty$.
Let $U \subset \R^n$ be an open set.
Then the Sobolev space $\map {W^{k, p} } U$ equipped with the Sobolev norm is a Banach space. | {{tidy}}
{{MissingLinks|still some missing}}
By Sobolev Norm is Norm, it remains to show that $\map {W^{k, p} } U$ is complete.
Assume $\sequence {u_m}$ is a Cauchy sequence in $\map {W^{k, p} } U$.
Then for each multiindex $\size \alpha \le k$, $\sequence {D^\alpha u_m}$ is a Cauchy sequence in $\map {L^p} U$.
From th... | Let $k \in \Z_{\ge 0}$.
Let $p \in \R_{\ge 1}$ or $p = \infty$.
Let $U \subset \R^n$ be an [[Definition:Open Subset of Real Euclidean Space|open set]].
Then the [[Definition:Sobolev Space|Sobolev space]] $\map {W^{k, p} } U$ equipped with the [[Definition:Sobolev Norm|Sobolev norm]] is a [[Definition:Banach Space|Ba... | {{tidy}}
{{MissingLinks|still some missing}}
By [[Sobolev Norm is Norm]], it remains to show that $\map {W^{k, p} } U$ is [[Definition:Complete Metric Space|complete]].
Assume $\sequence {u_m}$ is a [[Definition:Cauchy Sequence|Cauchy sequence]] in $\map {W^{k, p} } U$.
Then for each [[Definition:Multiindex|multiind... | Sobolev Space is Banach Space | https://proofwiki.org/wiki/Sobolev_Space_is_Banach_Space | https://proofwiki.org/wiki/Sobolev_Space_is_Banach_Space | [
"Sobolev Spaces",
"Banach Spaces"
] | [
"Definition:Open Set/Real Analysis/Real Euclidean Space",
"Definition:Sobolev Space",
"Definition:Sobolev Norm",
"Definition:Banach Space"
] | [
"Sobolev Norm is Norm",
"Definition:Complete Metric Space",
"Definition:Cauchy Sequence",
"Definition:Multiindex",
"Definition:Cauchy Sequence",
"Riesz-Fischer Theorem",
"Definition:Complete Metric Space",
"Integration by Parts",
"Definition:Distributional Derivative"
] |
proofwiki-21777 | Sobolev Norm is Norm | Let $k \in \Z_+$ and $1 \le p \le \infty$.
Let $U \subset \R^n$ be an open set.
Then the Sobolev norm is a norm on the Sobolev space $\map {W^{k, p} } U$. | === {{NormAxiomVector|1|nolink}} ===
Let $u, v \in \map {W^{k, p} } U$.
From P-Seminorm of Function Zero iff A.E. Zero:
:$\norm u_{\map {W^{k, p} } U} = 0$
for $u \in \map {W^{k, p} } U$ {{iff}} $u = 0$ almost everywhere.
{{MissingLinks|The above is stated as a fact without reference to the question in hand. }}
So {{N... | Let $k \in \Z_+$ and $1 \le p \le \infty$.
Let $U \subset \R^n$ be an [[Definition:Open Subset of Real Euclidean Space|open set]].
Then the [[Definition:Sobolev Norm|Sobolev norm]] is a [[Definition:Norm on Vector Space|norm]] on the [[Definition:Sobolev Space|Sobolev space]] $\map {W^{k, p} } U$. | === {{NormAxiomVector|1|nolink}} ===
Let $u, v \in \map {W^{k, p} } U$.
From [[P-Seminorm of Function Zero iff A.E. Zero]]:
:$\norm u_{\map {W^{k, p} } U} = 0$
for $u \in \map {W^{k, p} } U$ {{iff}} $u = 0$ [[Definition:Almost Everywhere|almost everywhere]].
{{MissingLinks|The above is stated as a fact without refe... | Sobolev Norm is Norm | https://proofwiki.org/wiki/Sobolev_Norm_is_Norm | https://proofwiki.org/wiki/Sobolev_Norm_is_Norm | [
"Sobolev Spaces"
] | [
"Definition:Open Set/Real Analysis/Real Euclidean Space",
"Definition:Sobolev Norm",
"Definition:Norm/Vector Space",
"Definition:Sobolev Space"
] | [
"P-Seminorm of Function Zero iff A.E. Zero",
"Definition:Almost Everywhere"
] |
proofwiki-21778 | Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods/Necessary Condition | Let $\struct{S, \tau}$ be a $\struct{S, \tau}$ is a sober space.
For each $x \in S$, let:
:$\map \UU x$ denote the system of open neighborhoods of $x$
Then:
:for each completely prime filter $\FF$ in the complete lattice $\struct{\tau, \subseteq}$:
::$\exists ! x \in S : \FF = \map \UU x$ | Let $\FF$ be a completely prime filter of $\struct{\tau, \subseteq}$.
Let $W = \bigcup \set{U \in \tau : U \notin \FF}$.
By the contrapositive statement of the definition of completely prime filter:
:$W \notin \FF$
From Completely Prime Filter Induces Meet-Irreducible Open Set:
:$W$ is a meet-irreducible open set
By de... | Let $\struct{S, \tau}$ be a $\struct{S, \tau}$ is a [[Definition:Sober Space|sober space]].
For each $x \in S$, let:
:$\map \UU x$ denote the [[Definition:System of Open Neighborhoods|system of open neighborhoods]] of $x$
Then:
:for each [[Definition:Completely Prime Filter|completely prime filter]] $\FF$ in the [[... | Let $\FF$ be a [[Definition:Completely Prime Filter|completely prime filter]] of $\struct{\tau, \subseteq}$.
Let $W = \bigcup \set{U \in \tau : U \notin \FF}$.
By the [[Definition:Contrapositive Statement|contrapositive statement]] of the definition of [[Definition:Completely Prime Filter|completely prime filter]]:... | Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods/Necessary Condition | https://proofwiki.org/wiki/Sober_Space_iff_Completely_Prime_Filter_is_Unique_System_of_Open_Neighborhoods/Necessary_Condition | https://proofwiki.org/wiki/Sober_Space_iff_Completely_Prime_Filter_is_Unique_System_of_Open_Neighborhoods/Necessary_Condition | [
"Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods"
] | [
"Definition:Sober Space",
"Definition:System of Open Neighborhoods",
"Definition:Completely Prime Filter",
"Definition:Complete Lattice"
] | [
"Definition:Completely Prime Filter",
"Definition:Contrapositive Statement",
"Definition:Completely Prime Filter",
"Completely Prime Filter Induces Meet-Irreducible Open Set",
"Definition:Meet-Irreducible Open Set",
"Definition:Sober Space",
"Set is Subset of Union",
"Open Set Not in System of Open Ne... |
proofwiki-21779 | Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods/Sufficient Condition | Let $\struct{S, \tau}$ be a topological space.
For each $x \in S$, let:
:$\map \UU x$ denote the system of open neighborhoods of $x$
For each completely prime filter $\FF$ in the complete lattice $\struct{\tau, \subseteq}$, let:
:$\exists ! x \in S : \FF = \map \UU x$
Then $\struct{S, \tau}$ is a sober space. | For each completely prime filter $\FF$ in the complete lattice $\struct{\tau, \subseteq}$, let:
:$\exists ! x \in S : \FF = \map \UU x$
Let $W \in \tau$ be a meet-irreducible open set.
Let $\FF = \set{U \in \tau : U \nsubseteq W}$.
From Meet-Irreducible Open Set Induces Completely Prime Filter:
:$\FF$ is a completely ... | Let $\struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
For each $x \in S$, let:
:$\map \UU x$ denote the [[Definition:System of Open Neighborhoods|system of open neighborhoods]] of $x$
For each [[Definition:Completely Prime Filter|completely prime filter]] $\FF$ in the [[Definition:Complete ... | For each [[Definition:Completely Prime Filter|completely prime filter]] $\FF$ in the [[Definition:Complete Lattice|complete lattice]] $\struct{\tau, \subseteq}$, let:
:$\exists ! x \in S : \FF = \map \UU x$
Let $W \in \tau$ be a [[Definition:Meet-Irreducible Open Set|meet-irreducible open set]].
Let $\FF = \set{U ... | Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods/Sufficient Condition | https://proofwiki.org/wiki/Sober_Space_iff_Completely_Prime_Filter_is_Unique_System_of_Open_Neighborhoods/Sufficient_Condition | https://proofwiki.org/wiki/Sober_Space_iff_Completely_Prime_Filter_is_Unique_System_of_Open_Neighborhoods/Sufficient_Condition | [
"Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:System of Open Neighborhoods",
"Definition:Completely Prime Filter",
"Definition:Complete Lattice",
"Definition:Sober Space"
] | [
"Definition:Completely Prime Filter",
"Definition:Complete Lattice",
"Definition:Meet-Irreducible Open Set",
"Meet-Irreducible Open Set Induces Completely Prime Filter",
"Definition:Completely Prime Filter",
"Open Set Not in System of Open Neighborhoods Iff Subset of Complement of Singleton Closure",
"O... |
proofwiki-21780 | Prime whose Divisor Sum is Square is 3 | There is exactly $1$ prime number whose divisor sum is a square number, and that is $3$:
:$\map {\sigma_1} 3 = 4$ | That $\map {\sigma_1} 3 = 4$ is shown at {{DSFLink|3}}.
It remains to be shown there are no more.
Let $n \in \N$ such that $\map {\sigma_1} n$ is square.
:$\map {\sigma_1} n = m^2$
Suppose $n$ is prime.
From Divisor Sum of Prime Number:
:$\map {\sigma_1} n = n + 1$
So we have:
{{begin-eqn}}
{{eqn | l = n + 1
| r ... | There is exactly $1$ [[Definition:Prime Number|prime number]] whose [[Definition:Divisor Sum Function|divisor sum]] is a [[Definition:Square Number|square number]], and that is $3$:
:$\map {\sigma_1} 3 = 4$ | That $\map {\sigma_1} 3 = 4$ is shown at {{DSFLink|3}}.
It remains to be shown there are no more.
Let $n \in \N$ such that $\map {\sigma_1} n$ is [[Definition:Square Number|square]].
:$\map {\sigma_1} n = m^2$
Suppose $n$ is [[Definition:Prime Number|prime]].
From [[Divisor Sum of Prime Number]]:
:$\map {\sigma_1}... | Prime whose Divisor Sum is Square is 3 | https://proofwiki.org/wiki/Prime_whose_Divisor_Sum_is_Square_is_3 | https://proofwiki.org/wiki/Prime_whose_Divisor_Sum_is_Square_is_3 | [
"Divisor Sum Function",
"3"
] | [
"Definition:Prime Number",
"Definition:Divisor Sum Function",
"Definition:Square Number"
] | [
"Definition:Square Number",
"Definition:Prime Number",
"Divisor Sum of Prime Number",
"Difference of Two Squares",
"Definition:Prime Number",
"Category:Divisor Sum Function",
"Category:3"
] |
proofwiki-21781 | Linear Combination of Riemann-Stieltjes Integrals/Integrand | Let $f, g, \alpha$ be real functions that are bounded on $\closedint a b$.
Let $c_1, c_2 \in \R$.
Suppose that $f$ and $g$ are Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$ and:
:$\ds \int_a^b f \rd \alpha = A$
:$\ds \int_a^b g \rd \alpha = B$
Then, the real function $h : \closedint a b \to ... | Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann-Stieltjes integral, let $P'_\epsilon, P' '_\epsilon$ be subdivisions of $\closedint a b$ such that:
:For every $P$ finer than $P'_\epsilon$, $\size {\map S {P, f, \alpha} - A} < \dfrac \epsilon {\size {c_1} + \size {c_2} + 1}$
:For every $P$ finer than $P' '... | Let $f, g, \alpha$ be [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$.
Let $c_1, c_2 \in \R$.
Suppose that $f$ and $g$ are [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integrable]] with respect to $\alpha$ on $\closedint a b$ an... | Let $\epsilon > 0$ be arbitrary.
By definition of the [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integral]], let $P'_\epsilon, P' '_\epsilon$ be [[Definition:Finite Subdivision|subdivisions]] of $\closedint a b$ such that:
:For every $P$ [[Definition:Finer Subdivision|finer]] than $P'_\epsilon$, $\size ... | Linear Combination of Riemann-Stieltjes Integrals/Integrand | https://proofwiki.org/wiki/Linear_Combination_of_Riemann-Stieltjes_Integrals/Integrand | https://proofwiki.org/wiki/Linear_Combination_of_Riemann-Stieltjes_Integrals/Integrand | [
"Riemann-Stieltjes Integral",
"Integral Calculus"
] | [
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral",
"Definition:Real Function",
"Definition:Riemann-Stieltjes Integral"
] | [
"Definition:Riemann-Stieltjes Integral",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Definition:Refinement of Finite Subdivision",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Definition:Subdivision of Inter... |
proofwiki-21782 | Linear Combination of Riemann-Stieltjes Integrals/Integrator | Let $f, \alpha, \beta$ be real functions that are bounded on $\closedint a b$.
Let $c_1, c_2 \in \R$.
Suppose that $f$ is Riemann-Stieltjes integrable with respect both $\alpha$ and $\beta$ on $\closedint a b$ and:
:$\ds \int_a^b f \rd \alpha = A$
:$\ds \int_a^b f \rd \beta = B$
Then, $f$ is Riemann-Stieltjes integrabl... | Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann-Stieltjes integral, let $P'_\epsilon, P' '_\epsilon$ be subdivisions of $\closedint a b$ such that:
:For every $P$ finer than $P'_\epsilon$, $\size {\map S {P, f, \alpha} - A} < \dfrac \epsilon {\size {c_1} + \size {c_2} + 1}$
:For every $P$ finer than $P' '... | Let $f, \alpha, \beta$ be [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$.
Let $c_1, c_2 \in \R$.
Suppose that $f$ is [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integrable]] with respect both $\alpha$ and $\beta$ on $\closedin... | Let $\epsilon > 0$ be arbitrary.
By definition of the [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integral]], let $P'_\epsilon, P' '_\epsilon$ be [[Definition:Finite Subdivision|subdivisions]] of $\closedint a b$ such that:
:For every $P$ [[Definition:Finer Subdivision|finer]] than $P'_\epsilon$, $\size ... | Linear Combination of Riemann-Stieltjes Integrals/Integrator | https://proofwiki.org/wiki/Linear_Combination_of_Riemann-Stieltjes_Integrals/Integrator | https://proofwiki.org/wiki/Linear_Combination_of_Riemann-Stieltjes_Integrals/Integrator | [
"Riemann-Stieltjes Integral",
"Integral Calculus"
] | [
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral",
"Definition:Riemann-Stieltjes Integral"
] | [
"Definition:Riemann-Stieltjes Integral",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Definition:Refinement of Finite Subdivision",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Definition:Subdivision of Inter... |
proofwiki-21783 | Integration by Parts/Riemann-Stieltjes Integral | Let $f, \alpha$ be a real functions that are bounded on $\closedint a b$.
Suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$.
Then, $\alpha$ is Riemann-Stieltjes integrable with respect to $f$ on $\closedint a b$ and:
:$\ds \int_a^b f \rd \alpha + \int_a^b \alpha \rd f = \map ... | Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann-Stieltjes integral, let $P_\epsilon$ be a subdivision of $\closedint a b$ such that:
:For every $P$ finer than $P_\epsilon$, $\size {\map S {P, f, \alpha} - \int_a^b f \rd \alpha} < \epsilon$
Let $P = \set {x_0, \dotsc, x_n}$ be an arbitrary subdivision fine... | Let $f, \alpha$ be a [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$.
Suppose that $f$ is [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integrable]] with respect to $\alpha$ on $\closedint a b$.
Then, $\alpha$ is [[Definition:Ri... | Let $\epsilon > 0$ be arbitrary.
By definition of the [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integral]], let $P_\epsilon$ be a [[Definition:Finite Subdivision|subdivision]] of $\closedint a b$ such that:
:For every $P$ [[Definition:Finer Subdivision|finer]] than $P_\epsilon$, $\size {\map S {P, f, \... | Integration by Parts/Riemann-Stieltjes Integral | https://proofwiki.org/wiki/Integration_by_Parts/Riemann-Stieltjes_Integral | https://proofwiki.org/wiki/Integration_by_Parts/Riemann-Stieltjes_Integral | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral",
"Definition:Riemann-Stieltjes Integral"
] | [
"Definition:Riemann-Stieltjes Integral",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Linear Combination of Indexed Summations",
"Telescoping Series/Example 2",
"... |
proofwiki-21784 | Integration by Substitution/Riemann-Stieltjes Integral/Increasing | Let $g$ be a real function that is continuous and strictly increasing on $\closedint a b$.
Let $f, \alpha$ be real functions that are bounded on $\closedint {\map g a} {\map g b}$.
Further suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint {\map g a} {\map g b}$.
Let $h, \beta : \c... | For any $x \in \closedint a b$, $\map g x < \map g a$ implies $x < a$ by definition of strictly increasing, so:
:$\forall x \in \closedint a b: \map g x \ge \map g a$
Likewise, $\map g x > \map g b$ implies $x > b$, so:
:$\forall x \in \closedint a b: \map g x \le \map g b$
Together:
:$\Img g \subseteq \closedint {\map... | Let $g$ be a [[Definition:Real Function|real function]] that is [[Definition:Continuous Real Function on Subset|continuous]] and [[Definition:Strictly Increasing Real Function|strictly increasing]] on $\closedint a b$.
Let $f, \alpha$ be [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Va... | For any $x \in \closedint a b$, $\map g x < \map g a$ implies $x < a$ by definition of [[Definition:Strictly Increasing Real Function|strictly increasing]], so:
:$\forall x \in \closedint a b: \map g x \ge \map g a$
Likewise, $\map g x > \map g b$ implies $x > b$, so:
:$\forall x \in \closedint a b: \map g x \le \map ... | Integration by Substitution/Riemann-Stieltjes Integral/Increasing | https://proofwiki.org/wiki/Integration_by_Substitution/Riemann-Stieltjes_Integral/Increasing | https://proofwiki.org/wiki/Integration_by_Substitution/Riemann-Stieltjes_Integral/Increasing | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Subset",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral",
"Definition:Riemann-Stieltjes Integral"
] | [
"Definition:Strictly Increasing/Real Function",
"Intermediate Value Theorem",
"Inverse of Strictly Monotone Function",
"Definition:Inverse Mapping",
"Definition:Strictly Increasing/Real Function",
"Definition:Riemann-Stieltjes Integral",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinem... |
proofwiki-21785 | Riemann-Stieltjes Integral of Constant Integrand | Let $\alpha$ be a real function that is bounded on $\closedint a b$.
Let $f : \closedint a b \to \R$ be defined as:
:$\map f x = c$
for some $c \in \R$.
Then, $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$, and:
:$\ds \int_a^b f \rd \alpha = c \map \alpha b - c \map \alpha a$ | Let $\epsilon > 0$
Define $P_\epsilon = \set {a, b}$.
Let $P = \set {x_0, \dotsc, x_n}$ be a subdivision of $\closedint a b$ that is finer than $P_\epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \map S {P, f, \alpha}
| r = \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } }
... | Let $\alpha$ be a [[Definition:Real Function|real function]] that is [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$.
Let $f : \closedint a b \to \R$ be defined as:
:$\map f x = c$
for some $c \in \R$.
Then, $f$ is [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integrable]] with re... | Let $\epsilon > 0$
Define $P_\epsilon = \set {a, b}$.
Let $P = \set {x_0, \dotsc, x_n}$ be a [[Definition:Finite Subdivision|subdivision]] of $\closedint a b$ that is [[Definition:Finer Subdivision|finer]] than $P_\epsilon$.
Then:
{{begin-eqn}}
{{eqn | l = \map S {P, f, \alpha}
| r = \sum_{k \mathop = 1}^n \ma... | Riemann-Stieltjes Integral of Constant Integrand | https://proofwiki.org/wiki/Riemann-Stieltjes_Integral_of_Constant_Integrand | https://proofwiki.org/wiki/Riemann-Stieltjes_Integral_of_Constant_Integrand | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral"
] | [
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Telescoping Series/Example 2",
"Definition:Riemann-Stieltjes Integral"
] |
proofwiki-21786 | Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Part to Whole | Let $a < c < b$ be real numbers.
Let $f, \alpha$ be real functions that are bounded on $\closedint a b$
Suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a c$, and also on $\closedint c b$.
Then, $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$, and... | Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann-Stieltjes integral, let $P'_\epsilon$ be a subdivision of $\closedint a c$ and $P' '_\epsilon$ be a subdivision of $\closedint c b$ such that:
:For every $P'$ finer than $P'_\epsilon$, $\ds \size {\map S {P', f, \alpha} - \int_a^c f \rd \alpha} < \frac \epsi... | Let $a < c < b$ be [[Definition:Real Number|real numbers]].
Let $f, \alpha$ be [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$
Suppose that $f$ is [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integrable]] with respect to $\alpha... | Let $\epsilon > 0$ be arbitrary.
By definition of the [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integral]], let $P'_\epsilon$ be a [[Definition:Finite Subdivision|subdivision]] of $\closedint a c$ and $P' '_\epsilon$ be a [[Definition:Finite Subdivision|subdivision]] of $\closedint c b$ such that:
:For... | Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Part to Whole | https://proofwiki.org/wiki/Sum_of_Riemann-Stieltjes_Integrals_on_Adjacent_Intervals/Part_to_Whole | https://proofwiki.org/wiki/Sum_of_Riemann-Stieltjes_Integrals_on_Adjacent_Intervals/Part_to_Whole | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral",
"Definition:Riemann-Stieltjes Integral"
] | [
"Definition:Riemann-Stieltjes Integral",
"Definition:Subdivision of Interval/Finite",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Definition:Refinement of Finite Subdivision",
"Definition:Subdivision of Interval/Finite",
"Definition:Subdivision of Interva... |
proofwiki-21787 | Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Whole to Part | Let $a < c < b$ be real numbers.
Let $f, \alpha$ be real functions that are bounded on $\closedint a b$
Suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$, and also on one of the two intervals $\closedint a c$ and $\closedint c b$.
Then, $f$ is Riemann-Stieltjes integrable wit... | Let $\closedint p q$ be the interval from among $\closedint a c$ and $\closedint c b$ on which we know $f$ is Riemann-Stieltjes integrable with respect to $\alpha$.
Let $\closedint u v$ be the other interval.
Then, we want to prove that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint u v$, a... | Let $a < c < b$ be [[Definition:Real Number|real numbers]].
Let $f, \alpha$ be [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$
Suppose that $f$ is [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integrable]] with respect to $\alpha... | Let $\closedint p q$ be the [[Definition:Closed Real Interval|interval]] from among $\closedint a c$ and $\closedint c b$ on which we know $f$ is [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integrable]] with respect to $\alpha$.
Let $\closedint u v$ be the other interval.
Then, we want to prove that $f$... | Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Whole to Part | https://proofwiki.org/wiki/Sum_of_Riemann-Stieltjes_Integrals_on_Adjacent_Intervals/Whole_to_Part | https://proofwiki.org/wiki/Sum_of_Riemann-Stieltjes_Integrals_on_Adjacent_Intervals/Whole_to_Part | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral",
"Definition:Real Interval/Closed",
"Definition:Riemann-Stieltjes Integral",
"Definition:Real Interval/Closed"
] | [
"Definition:Real Interval/Closed",
"Definition:Riemann-Stieltjes Integral",
"Definition:Riemann-Stieltjes Integral",
"Definition:Riemann-Stieltjes Integral",
"Definition:Subdivision of Interval/Finite",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Defini... |
proofwiki-21788 | Sum of Riemann-Stieltjes Integrals on Adjacent Intervals | Let $a, b, c \in \R$.
Let $f, \alpha$ be real functions that are bounded on some closed interval containing $a, b, c$.
Suppose that two of the following three Riemann-Stieltjes integrals exist, with the limits of integration interpreted in the general sense:
:$\ds \int_a^b f \rd \alpha$
:$\ds \int_b^c f \rd \alpha$
:$\... | We have the following special cases: | Let $a, b, c \in \R$.
Let $f, \alpha$ be [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Valued Function|bounded]] on some [[Definition:Closed Real Interval|closed interval]] containing $a, b, c$.
Suppose that two of the following three [[Definition:Riemann-Stieltjes Integral|Riemann-St... | We have the following special cases: | Sum of Riemann-Stieltjes Integrals on Adjacent Intervals | https://proofwiki.org/wiki/Sum_of_Riemann-Stieltjes_Integrals_on_Adjacent_Intervals | https://proofwiki.org/wiki/Sum_of_Riemann-Stieltjes_Integrals_on_Adjacent_Intervals | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Interval/Closed",
"Definition:Riemann-Stieltjes Integral",
"Definition:Riemann-Stieltjes Integral/General Limits of Integration"
] | [] |
proofwiki-21789 | Riemann-Stieltjes Integral by Norm of Subdivision | Let $f, \alpha$ be real functions that are bounded on $\closedint a b$.
Suppose there exists some $A \in \R$ where, for every $\epsilon > 0$, there exists some $\delta_\epsilon > 0$ such that:
:For every finite subdivision $P$ of $\closedint a b$, if the norm $\norm P < \delta_\epsilon$, then:
::For every Riemann-Stiel... | Let $\epsilon > 0$ be arbitrary.
By hypothesis, let $\delta_\epsilon > 0$ be such that, for every subdivision $P$ with $\norm P < \delta_\epsilon$:
:$\size {\map S {P, f, \alpha} - A} < \epsilon$
By Existence of Subdivision with Small Norm, let $P_\epsilon$ be a subdivision of $\closedint a b$ such that:
:$\norm {P_\ep... | Let $f, \alpha$ be [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$.
Suppose there exists some $A \in \R$ where, for every $\epsilon > 0$, there exists some $\delta_\epsilon > 0$ such that:
:For every [[Definition:Finite Subdivision|finite sub... | Let $\epsilon > 0$ be arbitrary.
By hypothesis, let $\delta_\epsilon > 0$ be such that, for every [[Definition:Finite Subdivision|subdivision]] $P$ with $\norm P < \delta_\epsilon$:
:$\size {\map S {P, f, \alpha} - A} < \epsilon$
By [[Existence of Subdivision with Small Norm]], let $P_\epsilon$ be a [[Definition:Fini... | Riemann-Stieltjes Integral by Norm of Subdivision | https://proofwiki.org/wiki/Riemann-Stieltjes_Integral_by_Norm_of_Subdivision | https://proofwiki.org/wiki/Riemann-Stieltjes_Integral_by_Norm_of_Subdivision | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Subdivision of Interval/Finite",
"Definition:Norm of Subdivision",
"Definition:Riemann-Stieltjes Sum",
"Definition:Riemann-Stieltjes Integral"
] | [
"Definition:Subdivision of Interval/Finite",
"Existence of Subdivision with Small Norm",
"Definition:Subdivision of Interval/Finite",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Norm of Refinement is no Greater than Norm of Subdivision",
"Definition:Refin... |
proofwiki-21790 | Riemann-Stieltjes Integral by Norm of Subdivision/Riemann Integral | Let $f$ be a real function that is bounded on $\closedint a b$.
Suppose $f$ is Riemann integrable on $\closedint a b$.
Let $\iota$ be the identity mapping on $\closedint a b$.
Then, $f$ is Riemann-Stieltjes integrable with respect to $\iota$ on $\closedint a b$ and:
:$\ds \int_a^b f \rd \iota = \int_a^b \map f x \rd x$... | Let $\epsilon > 0$ be arbitrary.
By definition of the Riemann integral, there exists some $\delta_\epsilon$ such that:
:For every finite subdivision $\Delta = \set {x_0, \dotsc, x_n}$ of $\closedint a b$, and every $C = \paren {c_i}_{1 \mathop \le i \mathop \le n}$ such that $c_i \in \closedint {x_{i - 1}} {x_i}$:
::If... | Let $f$ be a [[Definition:Real Function|real function]] that is [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$.
Suppose $f$ is [[Definition:Riemann Integral|Riemann integrable]] on $\closedint a b$.
Let $\iota$ be the [[Definition:Identity Mapping|identity mapping]] on $\closedint a b$.
The... | Let $\epsilon > 0$ be arbitrary.
By definition of the [[Definition:Riemann Integral|Riemann integral]], there exists some $\delta_\epsilon$ such that:
:For every [[Definition:Finite Subdivision|finite subdivision]] $\Delta = \set {x_0, \dotsc, x_n}$ of $\closedint a b$, and every $C = \paren {c_i}_{1 \mathop \le i \ma... | Riemann-Stieltjes Integral by Norm of Subdivision/Riemann Integral | https://proofwiki.org/wiki/Riemann-Stieltjes_Integral_by_Norm_of_Subdivision/Riemann_Integral | https://proofwiki.org/wiki/Riemann-Stieltjes_Integral_by_Norm_of_Subdivision/Riemann_Integral | [
"Definite Integrals",
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Definite Integral/Riemann",
"Definition:Identity Mapping",
"Definition:Riemann-Stieltjes Integral",
"Definition:Definite Integral",
"Definition:Definite Integral/Riemann"
] | [
"Definition:Definite Integral/Riemann",
"Definition:Subdivision of Interval/Finite",
"Definition:Norm of Subdivision",
"Definition:Riemann Sum",
"Definition:Subdivision of Interval/Finite",
"Definition:Subdivision of Interval/Finite",
"Riemann-Stieltjes Integral by Norm of Subdivision",
"Category:Defi... |
proofwiki-21791 | Norm of Refinement is no Greater than Norm of Subdivision | Let $P$ be a finite subdivision of $\closedint a b$.
Let $P'$ be a refinement of $P$.
Then:
:$\norm {P'} \le \norm P$
where $\norm P$ denotes the norm of $P$. | Let $P = \set {x_0, x_1, \dotsc, x_{n - 1}, x_n}$.
Let $P' = \set {y_0, y_1, \dotsc, y_{m - 1}, y_m}$.
By definition of refinement:
:$P \subseteq P'$
Therefore, every $x_i \in P'$.
Thus, every $i \in \set {0, 1, \dotsc, n - 1, n}$, define $k_i \in \set {0, 1, \dotsc, m - 1, m}$ such that:
:$y_{k_i} = x_i$
Now, for each... | Let $P$ be a [[Definition:Finite Subdivision|finite subdivision]] of $\closedint a b$.
Let $P'$ be a [[Definition:Refinement of Finite Subdivision|refinement]] of $P$.
Then:
:$\norm {P'} \le \norm P$
where $\norm P$ denotes the [[Definition:Norm of Subdivision|norm]] of $P$. | Let $P = \set {x_0, x_1, \dotsc, x_{n - 1}, x_n}$.
Let $P' = \set {y_0, y_1, \dotsc, y_{m - 1}, y_m}$.
By definition of [[Definition:Refinement of Finite Subdivision|refinement]]:
:$P \subseteq P'$
Therefore, every $x_i \in P'$.
Thus, every $i \in \set {0, 1, \dotsc, n - 1, n}$, define $k_i \in \set {0, 1, \dotsc, ... | Norm of Refinement is no Greater than Norm of Subdivision | https://proofwiki.org/wiki/Norm_of_Refinement_is_no_Greater_than_Norm_of_Subdivision | https://proofwiki.org/wiki/Norm_of_Refinement_is_no_Greater_than_Norm_of_Subdivision | [
"Subdivisions (Real Analysis)"
] | [
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Definition:Norm of Subdivision"
] | [
"Definition:Refinement of Finite Subdivision",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Proof by Contradiction",
"Definition:Norm of Subdivision",
"Definition:Norm of Subdivision",
"Category:Subdivisions (Real Analysis)"
] |
proofwiki-21792 | Existence of Subdivision with Small Norm | Let $\closedint a b$ be a closed real interval.
Let $\epsilon > 0$ be a positive real number.
Then, there exists a finite subdivision $P$ of $\closedint a b$ such that:
:$\norm P < \epsilon$
where $\norm P$ denotes the norm of $P$. | By the Axiom of Archimedes, choose $N \in \N$ such that:
:$N > \dfrac {b - a} \epsilon > 0$
For each $k \in \set {0, 1, \dotsc, N - 1, N}$, define:
:$x_k = a + \dfrac k N \paren {b - a}$
We have:
{{begin-eqn}}
{{eqn | l = x_0
| r = a + \frac 0 N \paren {b - a}
| c = Definition of $x_k$
}}
{{eqn | r = a
}}
{... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\epsilon > 0$ be a [[Definition:Strictly Positive Real Number|positive real number]].
Then, there exists a [[Definition:Finite Subdivision|finite subdivision]] $P$ of $\closedint a b$ such that:
:$\norm P < \epsilon$
where $\norm... | By the [[Axiom of Archimedes]], choose $N \in \N$ such that:
:$N > \dfrac {b - a} \epsilon > 0$
For each $k \in \set {0, 1, \dotsc, N - 1, N}$, define:
:$x_k = a + \dfrac k N \paren {b - a}$
We have:
{{begin-eqn}}
{{eqn | l = x_0
| r = a + \frac 0 N \paren {b - a}
| c = Definition of $x_k$
}}
{{eqn | r = ... | Existence of Subdivision with Small Norm | https://proofwiki.org/wiki/Existence_of_Subdivision_with_Small_Norm | https://proofwiki.org/wiki/Existence_of_Subdivision_with_Small_Norm | [
"Subdivisions (Real Analysis)"
] | [
"Definition:Real Interval/Closed",
"Definition:Strictly Positive/Real Number",
"Definition:Subdivision of Interval/Finite",
"Definition:Norm of Subdivision"
] | [
"Axiom of Archimedes",
"Definition:Subdivision of Interval/Finite",
"Category:Subdivisions (Real Analysis)"
] |
proofwiki-21793 | Integers under Addition form Group | The set of integers under addition $\struct {\Z, +}$ forms a group. | From Integers under Addition form Abelian Group, $\struct {\Z, +}$ forms an abelian group.
An abelian group is {{afortiori}} a group.
Hence the result.
{{qed}} | The [[Definition:Set|set]] of [[Definition:Integer|integers]] under [[Definition:Integer Addition|addition]] $\struct {\Z, +}$ forms a [[Definition:Group|group]]. | From [[Integers under Addition form Abelian Group]], $\struct {\Z, +}$ forms an [[Definition:Abelian Group|abelian group]].
An [[Definition:Abelian Group|abelian group]] is {{afortiori}} a [[Definition:Group|group]].
Hence the result.
{{qed}} | Integers under Addition form Group | https://proofwiki.org/wiki/Integers_under_Addition_form_Group | https://proofwiki.org/wiki/Integers_under_Addition_form_Group | [
"Additive Group of Integers"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Group"
] | [
"Integers under Addition form Abelian Group",
"Definition:Abelian Group",
"Definition:Abelian Group",
"Definition:Group"
] |
proofwiki-21794 | Reduction of Riemann-Stieltjes Integral to Identity Integrator | Let $f, \alpha$ be real functions that are bounded on $\closedint a b$.
Suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$.
Also suppose that $\alpha$ is continuously differentiable on $\closedint a b$.
Then, let $\alpha'$ be the derivative of $\alpha$ on $\closedint a b$.
Let... | As $f$ is bounded on $\closedint a b$, let $M \in \R$ such that:
:$\forall x \in \closedint a b: \size {\map f x} \le M$
{{WLOG}}, let $M > 0$.
Let $\epsilon > 0$ be arbitrary.
By Continuous Function on Closed Real Interval is Uniformly Continuous:
:$\alpha'$ is uniformly continuous on $\closedint a b$
Therefore, there... | Let $f, \alpha$ be [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$.
Suppose that $f$ is [[Definition:Riemann-Stieltjes Integral|Riemann-Stieltjes integrable]] with respect to $\alpha$ on $\closedint a b$.
Also suppose that $\alpha$ is [[Defi... | As $f$ is [[Definition:Bounded Real-Valued Function|bounded]] on $\closedint a b$, let $M \in \R$ such that:
:$\forall x \in \closedint a b: \size {\map f x} \le M$
{{WLOG}}, let $M > 0$.
Let $\epsilon > 0$ be arbitrary.
By [[Continuous Function on Closed Real Interval is Uniformly Continuous]]:
:$\alpha'$ is [[Def... | Reduction of Riemann-Stieltjes Integral to Identity Integrator | https://proofwiki.org/wiki/Reduction_of_Riemann-Stieltjes_Integral_to_Identity_Integrator | https://proofwiki.org/wiki/Reduction_of_Riemann-Stieltjes_Integral_to_Identity_Integrator | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral",
"Definition:Continuously Differentiable Real Function on Closed Interval",
"Definition:Derivative of Real Function on Closed Interval",
"Definition:Identity Mapping",
"Definition:Riemann-Stielt... | [
"Definition:Bounded Mapping/Real-Valued",
"Continuous Function on Closed Real Interval is Uniformly Continuous",
"Definition:Uniform Continuity/Real Function",
"Existence of Subdivision with Small Norm",
"Definition:Subdivision of Interval/Finite",
"Definition:Norm of Subdivision",
"Definition:Riemann-S... |
proofwiki-21795 | Integration by Substitution/Riemann-Stieltjes Integral | Let $g$ be a real function that is continuous and strictly monotone on $\closedint a b$.
Let $\Bbb I = g \closedint a b$ be the image of $g$ under $\closedint a b$.
Let $f, \alpha$ be real functions that are bounded on $\Bbb I$.
Further suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\Bbb ... | We have the following special cases: | Let $g$ be a [[Definition:Real Function|real function]] that is [[Definition:Continuous Real Function on Subset|continuous]] and [[Definition:Strictly Monotone Real Function|strictly monotone]] on $\closedint a b$.
Let $\Bbb I = g \closedint a b$ be the [[Definition:Image of Set under Mapping|image]] of $g$ under $\cl... | We have the following special cases: | Integration by Substitution/Riemann-Stieltjes Integral | https://proofwiki.org/wiki/Integration_by_Substitution/Riemann-Stieltjes_Integral | https://proofwiki.org/wiki/Integration_by_Substitution/Riemann-Stieltjes_Integral | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Subset",
"Definition:Strictly Monotone/Real Function",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral",
"Definition:Riemann-Sti... | [] |
proofwiki-21796 | Integration by Substitution/Riemann-Stieltjes Integral/Decreasing | Let $g$ be a real function that is continuous and strictly decreasing on $\closedint a b$.
Let $f, \alpha$ be real functions that are bounded on $\closedint {\map g b} {\map g a}$.
Further suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint {\map g b} {\map g a}$.
Let $h, \beta : \c... | For any $x \in \closedint a b$, $\map g x > \map g a$ implies $x < a$ by definition of strictly decreasing, so:
:$\forall x \in \closedint a b: \map g x \le \map g a$
Likewise, $\map g x > \map g b$ implies $x < b$, so:
:$\forall x \in \closedint a b: \map g x \ge \map g b$
Together:
:$\Img g \subseteq \closedint {\map... | Let $g$ be a [[Definition:Real Function|real function]] that is [[Definition:Continuous Real Function on Subset|continuous]] and [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on $\closedint a b$.
Let $f, \alpha$ be [[Definition:Real Function|real functions]] that are [[Definition:Bounded Real-Va... | For any $x \in \closedint a b$, $\map g x > \map g a$ implies $x < a$ by definition of [[Definition:Strictly Decreasing Real Function|strictly decreasing]], so:
:$\forall x \in \closedint a b: \map g x \le \map g a$
Likewise, $\map g x > \map g b$ implies $x < b$, so:
:$\forall x \in \closedint a b: \map g x \ge \map ... | Integration by Substitution/Riemann-Stieltjes Integral/Decreasing | https://proofwiki.org/wiki/Integration_by_Substitution/Riemann-Stieltjes_Integral/Decreasing | https://proofwiki.org/wiki/Integration_by_Substitution/Riemann-Stieltjes_Integral/Decreasing | [
"Riemann-Stieltjes Integral"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Subset",
"Definition:Strictly Decreasing/Real Function",
"Definition:Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Riemann-Stieltjes Integral",
"Definition:Riemann-Stieltjes Integral"
] | [
"Definition:Strictly Decreasing/Real Function",
"Intermediate Value Theorem",
"Inverse of Strictly Monotone Function",
"Definition:Inverse Mapping",
"Definition:Strictly Decreasing/Real Function",
"Definition:Riemann-Stieltjes Integral",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinem... |
proofwiki-21797 | Continuous at Point iff Left-Continuous and Right-Continuous | Let $A \subseteq \R$ be an open set of real numbers.
Let $f : A \to \R$ be a real function.
Let $x_0 \in A$.
Then:
:$f$ is continuous at $x_0$
{{iff}}:
:$f$ is both left-continuous and right-continuous at $x_0$ | === Necessary Condition ===
Suppose $f$ is continuous at $x_0$.
Then, by definition:
:$\ds \lim_{x \mathop \to x_0} \map f x = \map f {x_0}$
By Limit iff Limits from Left and Right:
:$\ds \lim_{x \mathop \to x_0^-} \map f x = \map f {x_0}$
:$\ds \lim_{x \mathop \to x_0^+} \map f x = \map f {x_0}$
Therefore, by definiti... | Let $A \subseteq \R$ be an [[Definition:Open Set of Real Numbers|open set of real numbers]].
Let $f : A \to \R$ be a [[Definition:Real Function|real function]].
Let $x_0 \in A$.
Then:
:$f$ is [[Definition:Continuous Real Function at Point|continuous]] at $x_0$
{{iff}}:
:$f$ is both [[Definition:Left-Continuous at P... | === Necessary Condition ===
Suppose $f$ is [[Definition:Continuous Real Function at Point|continuous]] at $x_0$.
Then, by definition:
:$\ds \lim_{x \mathop \to x_0} \map f x = \map f {x_0}$
By [[Limit iff Limits from Left and Right]]:
:$\ds \lim_{x \mathop \to x_0^-} \map f x = \map f {x_0}$
:$\ds \lim_{x \mathop \t... | Continuous at Point iff Left-Continuous and Right-Continuous | https://proofwiki.org/wiki/Continuous_at_Point_iff_Left-Continuous_and_Right-Continuous | https://proofwiki.org/wiki/Continuous_at_Point_iff_Left-Continuous_and_Right-Continuous | [
"Continuous Real Functions"
] | [
"Definition:Open Set/Real Analysis/Real Numbers",
"Definition:Real Function",
"Definition:Continuous Real Function/Point",
"Definition:Continuous Real Function/Left-Continuous",
"Definition:Continuous Real Function/Right-Continuous"
] | [
"Definition:Continuous Real Function/Point",
"Limit iff Limits from Left and Right",
"Definition:Continuous Real Function/Left-Continuous",
"Definition:Continuous Real Function/Right-Continuous",
"Definition:Continuous Real Function/Left-Continuous",
"Definition:Continuous Real Function/Right-Continuous",... |
proofwiki-21798 | Integers under Subtraction form Magma | The set of integers $\Z$ under the operation of subtraction forms a magma. | Recall the definition of magma:
:$\struct {S, \circ}$ is a magma {{iff}} $\forall a, b \in S: a \circ b \in S$:
That is, a magma is closed under its operation.
Recall that the operation of subtraction is defined as:
:$\forall a, b \in \Z: a - b := a + \paren {-b}$
Recall that the Integers under Addition form Group.
Hen... | The [[Definition:Set|set]] of [[Definition:Integer|integers]] $\Z$ under the [[Definition:Binary Operation|operation]] of [[Definition:Integer Subtraction|subtraction]] forms a [[Definition:Magma|magma]]. | Recall the definition of [[Definition:Magma|magma]]:
:$\struct {S, \circ}$ is a [[Definition:Magma|magma]] {{iff}} $\forall a, b \in S: a \circ b \in S$:
That is, a [[Definition:Magma|magma]] is [[Definition:Closed Algebraic Structure|closed]] under its [[Definition:Binary Operation|operation]].
Recall that the [[Def... | Integers under Subtraction form Magma | https://proofwiki.org/wiki/Integers_under_Subtraction_form_Magma | https://proofwiki.org/wiki/Integers_under_Subtraction_form_Magma | [
"Integers",
"Subtraction",
"Examples of Magmas"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Operation/Binary Operation",
"Definition:Subtraction/Integers",
"Definition:Magma"
] | [
"Definition:Magma",
"Definition:Magma",
"Definition:Magma",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Operation/Binary Operation",
"Definition:Subtraction/Integers",
"Integers under Addition form Group",
"Definition:Magma"
] |
proofwiki-21799 | Natural Numbers under Subtraction do not form Magma | The set of natural numbers $\N$ under the operation of subtraction does not form a magma. | Proof by Counterexample:
Recall the definition of magma:
:$\struct {S, \circ}$ is a magma {{iff}} $\forall a, b \in S: a \circ b \in S$:
That is, a magma is closed under its operation.
Recall that the operation of natural number subtraction of $n - m$ is defined as:
:$n - m = p$
where $p \in \N$ such that $n = m + p$.
... | The [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ under the [[Definition:Binary Operation|operation]] of [[Definition:Natural Number Subtraction|subtraction]] does not form a [[Definition:Magma|magma]]. | [[Proof by Counterexample]]:
Recall the definition of [[Definition:Magma|magma]]:
:$\struct {S, \circ}$ is a [[Definition:Magma|magma]] {{iff}} $\forall a, b \in S: a \circ b \in S$:
That is, a [[Definition:Magma|magma]] is [[Definition:Closed Algebraic Structure|closed]] under its [[Definition:Binary Operation|operat... | Natural Numbers under Subtraction do not form Magma | https://proofwiki.org/wiki/Natural_Numbers_under_Subtraction_do_not_form_Magma | https://proofwiki.org/wiki/Natural_Numbers_under_Subtraction_do_not_form_Magma | [
"Natural Numbers",
"Subtraction",
"Examples of Magmas"
] | [
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Operation/Binary Operation",
"Definition:Subtraction/Natural Numbers",
"Definition:Magma"
] | [
"Proof by Counterexample",
"Definition:Magma",
"Definition:Magma",
"Definition:Magma",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Operation/Binary Operation",
"Definition:Subtraction/Natural Numbers",
"Definition:Algebraic Struct... |
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