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-4400 | Definite Integral of Odd Function | Let $f$ be an odd function with a primitive on the open interval $\closedint {-a} a$, where $a > 0$.
Then:
:$\ds \int_{-a}^a \map f x \rd x = 0$ | Let $F$ be a primitive for $f$ on the interval $\closedint {-a} a$.
Then, by Sum of Integrals on Adjacent Intervals for Integrable Functions, we have:
{{begin-eqn}}
{{eqn | l = \int_{-a}^a \map f x \rd x
| r = \int_{-a}^0 \map f x \rd x + \int_0^a \map f x \rd x
| c =
}}
{{end-eqn}}
Therefore, it suffices t... | Let $f$ be an [[Definition:Odd Function|odd function]] with a [[Definition:Primitive (Calculus)|primitive]] on the [[Definition:Open Real Interval|open interval]] $\closedint {-a} a$, where $a > 0$.
Then:
:$\ds \int_{-a}^a \map f x \rd x = 0$ | Let $F$ be a [[Definition:Primitive (Calculus)|primitive]] for $f$ on the interval $\closedint {-a} a$.
Then, by [[Sum of Integrals on Adjacent Intervals for Integrable Functions]], we have:
{{begin-eqn}}
{{eqn | l = \int_{-a}^a \map f x \rd x
| r = \int_{-a}^0 \map f x \rd x + \int_0^a \map f x \rd x
| c... | Definite Integral of Odd Function | https://proofwiki.org/wiki/Definite_Integral_of_Odd_Function | https://proofwiki.org/wiki/Definite_Integral_of_Odd_Function | [
"Integral Calculus",
"Odd Functions",
"Definite Integrals"
] | [
"Definition:Odd Function",
"Definition:Primitive (Calculus)",
"Definition:Real Interval/Open"
] | [
"Definition:Primitive (Calculus)",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Derivative of Identity Function",
"Derivative of Constant Multiple",
"Integration by Substitution",
"Definition:Odd Function"
] |
proofwiki-4401 | Union of Equivalence Classes is Whole Set | Let $\RR \subseteq S \times S$ be an equivalence on a set $S$.
Then the set of $\RR$-classes constitutes the whole of $S$. | We have that:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = x
| o = \in
| r = \eqclass x \RR
| c = {{Defof|Equivalence Class}}
}}
{{eqn | n = 1
| ll= \leadsto
| l = x
| o = \in
| r = \set {y \in S: x \mathrel \RR y}
| c = {{Defof|Equivalence Class}}
}}
{{end-eqn... | Let $\RR \subseteq S \times S$ be an [[Definition:Equivalence Relation|equivalence]] on a [[Definition:Set|set]] $S$.
Then the set of [[Definition:Equivalence Class|$\RR$-classes]] constitutes the whole of $S$. | We have that:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = x
| o = \in
| r = \eqclass x \RR
| c = {{Defof|Equivalence Class}}
}}
{{eqn | n = 1
| ll= \leadsto
| l = x
| o = \in
| r = \set {y \in S: x \mathrel \RR y}
| c = {{Defof|Equivalence Class}}
}}
{{end-eq... | Union of Equivalence Classes is Whole Set | https://proofwiki.org/wiki/Union_of_Equivalence_Classes_is_Whole_Set | https://proofwiki.org/wiki/Union_of_Equivalence_Classes_is_Whole_Set | [
"Equivalence Relations",
"Quotient Sets",
"Fundamental Theorem on Equivalence Relations"
] | [
"Definition:Equivalence Relation",
"Definition:Set",
"Definition:Equivalence Class"
] | [
"Set Union Preserves Subsets",
"Set Union Preserves Subsets"
] |
proofwiki-4402 | Ratio of Consecutive Fibonacci Numbers | For $n \in \N$, let $f_n$ be the $n$th Fibonacci number.
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {f_{n + 1} } {f_n} = \phi$
where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean. | Let:
{{begin-eqn}}
{{eqn | l = \phi
| o = :=
| r = \dfrac {1 + \sqrt 5} 2
}}
{{eqn | l = \hat \phi
| o = :=
| r = \paren {1 - \phi}
| rr= = \dfrac {1 - \sqrt 5} 2
}}
{{eqn | l = \alpha
| o = :=
| r = \dfrac {\phi} {\hat \phi}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \alph... | For $n \in \N$, let $f_n$ be the $n$th [[Definition:Fibonacci Number|Fibonacci number]].
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {f_{n + 1} } {f_n} = \phi$
where $\phi = \dfrac {1 + \sqrt 5} 2$ is the [[Definition:Golden Mean|golden mean]]. | Let:
{{begin-eqn}}
{{eqn | l = \phi
| o = :=
| r = \dfrac {1 + \sqrt 5} 2
}}
{{eqn | l = \hat \phi
| o = :=
| r = \paren {1 - \phi}
| rr= = \dfrac {1 - \sqrt 5} 2
}}
{{eqn | l = \alpha
| o = :=
| r = \dfrac {\phi} {\hat \phi}
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \al... | Ratio of Consecutive Fibonacci Numbers/Proof 1 | https://proofwiki.org/wiki/Ratio_of_Consecutive_Fibonacci_Numbers | https://proofwiki.org/wiki/Ratio_of_Consecutive_Fibonacci_Numbers/Proof_1 | [
"Ratio of Consecutive Fibonacci Numbers",
"Fibonacci Numbers",
"Golden Mean"
] | [
"Definition:Fibonacci Number",
"Definition:Golden Mean"
] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Euler-Binet Formula",
"Definition:Fraction/Numerator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-4403 | Ratio of Consecutive Fibonacci Numbers | For $n \in \N$, let $f_n$ be the $n$th Fibonacci number.
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {f_{n + 1} } {f_n} = \phi$
where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean. | From Continued Fraction Expansion of Golden Mean: Successive Convergents, the $n$th convergent of the continued fraction expansion of $\phi$ is:
:$C_n = \dfrac {f_{n + 1} } {f_n}$
The result follows from Continued Fraction Expansion of Irrational Number Converges to Number Itself.
{{qed}} | For $n \in \N$, let $f_n$ be the $n$th [[Definition:Fibonacci Number|Fibonacci number]].
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {f_{n + 1} } {f_n} = \phi$
where $\phi = \dfrac {1 + \sqrt 5} 2$ is the [[Definition:Golden Mean|golden mean]]. | From [[Continued Fraction Expansion of Golden Mean/Successive Convergents|Continued Fraction Expansion of Golden Mean: Successive Convergents]], the $n$th [[Definition:Convergent of Continued Fraction|convergent]] of the [[Definition:Continued Fraction Expansion of Real Number|continued fraction expansion]] of $\phi$ i... | Ratio of Consecutive Fibonacci Numbers/Proof 2 | https://proofwiki.org/wiki/Ratio_of_Consecutive_Fibonacci_Numbers | https://proofwiki.org/wiki/Ratio_of_Consecutive_Fibonacci_Numbers/Proof_2 | [
"Ratio of Consecutive Fibonacci Numbers",
"Fibonacci Numbers",
"Golden Mean"
] | [
"Definition:Fibonacci Number",
"Definition:Golden Mean"
] | [
"Continued Fraction Expansion of Golden Mean/Successive Convergents",
"Definition:Convergent of Continued Fraction",
"Definition:Continued Fraction Expansion/Real Number",
"Continued Fraction Expansion of Irrational Number Converges to Number Itself"
] |
proofwiki-4404 | Ratio of Consecutive Fibonacci Numbers | For $n \in \N$, let $f_n$ be the $n$th Fibonacci number.
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {f_{n + 1} } {f_n} = \phi$
where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean. | Let:
{{begin-eqn}}
{{eqn | l = a_n
| o = :=
| r = \dfrac {f_{n + 1} } {f_n}
}}
{{eqn | l = \map g x
| o = :=
| r = 1 + \dfrac 1 x
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \map g {a_n}
| r = 1 + \dfrac {f_n} {f_{n + 1} }
| c =
}}
{{eqn | r = \dfrac {f_{n + 1} + f_n} {f_{n + 1}... | For $n \in \N$, let $f_n$ be the $n$th [[Definition:Fibonacci Number|Fibonacci number]].
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {f_{n + 1} } {f_n} = \phi$
where $\phi = \dfrac {1 + \sqrt 5} 2$ is the [[Definition:Golden Mean|golden mean]]. | Let:
{{begin-eqn}}
{{eqn | l = a_n
| o = :=
| r = \dfrac {f_{n + 1} } {f_n}
}}
{{eqn | l = \map g x
| o = :=
| r = 1 + \dfrac 1 x
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \map g {a_n}
| r = 1 + \dfrac {f_n} {f_{n + 1} }
| c =
}}
{{eqn | r = \dfrac {f_{n + 1} + f_n} {f_{n + ... | Ratio of Consecutive Fibonacci Numbers/Proof 3 | https://proofwiki.org/wiki/Ratio_of_Consecutive_Fibonacci_Numbers | https://proofwiki.org/wiki/Ratio_of_Consecutive_Fibonacci_Numbers/Proof_3 | [
"Ratio of Consecutive Fibonacci Numbers",
"Fibonacci Numbers",
"Golden Mean"
] | [
"Definition:Fibonacci Number",
"Definition:Golden Mean"
] | [
"Definition:Dynamical System",
"Banach Fixed-Point Theorem",
"Definition:Complete Metric Space",
"Definition:Contraction Mapping (Metric Space)",
"Definition:Fixed Point",
"Definition:Contraction Mapping (Metric Space)",
"Definition:Real Interval/Closed",
"Definition:Fixed Point",
"Definition:Comple... |
proofwiki-4405 | Cardinality of Image of Injection | Let $f: S \rightarrowtail T$ be an injection.
Let $A \subseteq S$ be a finite subset of $S$.
Then:
:$\card {f \paren A} = \card A$
where $\card A$ denotes the cardinality of $A$. | Proof by induction:
For all $n \in \N_{>0}$, let $P_n$ be the proposition:
:$\card {f \paren A} = \card A$ when $\card A = n$
Suppose $\card A = 0$.
{{begin-eqn}}
{{eqn | l = \card A
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = A
| r = \O
| c = Cardinality of Empty Set
}}
{{eqn | ll= \leads... | Let $f: S \rightarrowtail T$ be an [[Definition:Injection|injection]].
Let $A \subseteq S$ be a [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $S$.
Then:
:$\card {f \paren A} = \card A$
where $\card A$ denotes the [[Definition:Cardinality|cardinality]] of $A$. | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $P_n$ be the [[Definition:Proposition|proposition]]:
:$\card {f \paren A} = \card A$ when $\card A = n$
Suppose $\card A = 0$.
{{begin-eqn}}
{{eqn | l = \card A
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = A
... | Cardinality of Image of Injection | https://proofwiki.org/wiki/Cardinality_of_Image_of_Injection | https://proofwiki.org/wiki/Cardinality_of_Image_of_Injection | [
"Injections",
"Cardinality"
] | [
"Definition:Injection",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Cardinality"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Cardinality of Empty Set",
"Image of Empty Set is Empty Set",
"Cardinality of Empty Set",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical ... |
proofwiki-4406 | Composition of Mappings is not Commutative | The composition of mappings is '''not''' in general a commutative binary operation:
:$f_2 \circ f_1 \ne f_1 \circ f_2$ | ;Proof by Counterexample:
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings.
First note that unless $S_1 = S_3$ then $f_2 \circ f_1$ is not even defined.
So in that case $f_2 \circ f_1$ is definitely not the same thing as $f_1 \circ f_2$.
So, let us suppose $S_1 = S_3$ and so we define $f_1: S_1 \to S_2$ and $f... | The [[Definition:Composition of Mappings|composition of mappings]] is '''not''' in general a [[Definition:Commutative Operation|commutative]] [[Definition:Binary Operation|binary operation]]:
:$f_2 \circ f_1 \ne f_1 \circ f_2$ | ;[[Proof by Counterexample]]:
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be [[Definition:Mapping|mappings]].
First note that unless $S_1 = S_3$ then $f_2 \circ f_1$ is not even defined.
So in that case $f_2 \circ f_1$ is definitely not the same thing as $f_1 \circ f_2$.
So, let us suppose $S_1 = S_3$ and so we ... | Composition of Mappings is not Commutative | https://proofwiki.org/wiki/Composition_of_Mappings_is_not_Commutative | https://proofwiki.org/wiki/Composition_of_Mappings_is_not_Commutative | [
"Composition of Mappings is not Commutative",
"Composite Mappings",
"Commutativity"
] | [
"Definition:Composition of Mappings",
"Definition:Commutative/Operation",
"Definition:Operation/Binary Operation"
] | [
"Proof by Counterexample",
"Definition:Mapping",
"Equality of Mappings",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Mapping"
] |
proofwiki-4407 | Intersection of Injective Image with Relative Complement | Let $f: S \to T$ be a mapping.
Then $f$ is an injection {{iff}}:
:$\forall A \subseteq S: f \sqbrk A \cap f \sqbrk {\relcomp S A} = \O$ | From Intersection with Relative Complement is Empty:
:$A \cap \relcomp S A = \O$
From Image of Intersection under Injection:
:$\forall A, B \subseteq S: f \sqbrk {A \cap B} = f \sqbrk A \cap f \sqbrk B$
{{iff}} $f$ is an injection.
Hence the result.
{{qed}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then $f$ is an [[Definition:Injection|injection]] {{iff}}:
:$\forall A \subseteq S: f \sqbrk A \cap f \sqbrk {\relcomp S A} = \O$ | From [[Intersection with Relative Complement is Empty]]:
:$A \cap \relcomp S A = \O$
From [[Image of Intersection under Injection]]:
:$\forall A, B \subseteq S: f \sqbrk {A \cap B} = f \sqbrk A \cap f \sqbrk B$
{{iff}} $f$ is an [[Definition:Injection|injection]].
Hence the result.
{{qed}} | Intersection of Injective Image with Relative Complement | https://proofwiki.org/wiki/Intersection_of_Injective_Image_with_Relative_Complement | https://proofwiki.org/wiki/Intersection_of_Injective_Image_with_Relative_Complement | [
"Injections",
"Set Intersection",
"Relative Complement"
] | [
"Definition:Mapping",
"Definition:Injection"
] | [
"Intersection with Relative Complement is Empty",
"Image of Intersection under Injection",
"Definition:Injection"
] |
proofwiki-4408 | Injection to Image is Bijection | Let $f: S \rightarrowtail T$ be an injection.
Let $X \subseteq T$ be the image of $f$.
Then the restriction $f {\restriction_{S \times X}}: S \to X$ of $f$ to the image of $f$ is a bijection of $S$ onto $X$. | We have:
:Restriction of Injection is Injection
:Restriction of Mapping to Image is Surjection
Thus we have that:
:$f {\restriction_{S \times X}}: S \to X$ is an injection
and
:$f {\restriction_{S \times X}}: S \to X$ is a surjection
Hence the result by definition of bijection.
{{qed}} | Let $f: S \rightarrowtail T$ be an [[Definition:Injection|injection]].
Let $X \subseteq T$ be the [[Definition:Image of Mapping|image]] of $f$.
Then the [[Definition:Restriction of Mapping|restriction]] $f {\restriction_{S \times X}}: S \to X$ of $f$ to the [[Definition:Image of Mapping|image]] of $f$ is a [[Definit... | We have:
:[[Restriction of Injection is Injection]]
:[[Restriction of Mapping to Image is Surjection]]
Thus we have that:
:$f {\restriction_{S \times X}}: S \to X$ is an [[Definition:Injection|injection]]
and
:$f {\restriction_{S \times X}}: S \to X$ is a [[Definition:Surjection|surjection]]
Hence the result by defi... | Injection to Image is Bijection | https://proofwiki.org/wiki/Injection_to_Image_is_Bijection | https://proofwiki.org/wiki/Injection_to_Image_is_Bijection | [
"Injections",
"Bijections",
"Restrictions"
] | [
"Definition:Injection",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Restriction/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Bijection"
] | [
"Restriction of Injection is Injection",
"Restriction of Mapping to Image is Surjection",
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection"
] |
proofwiki-4409 | Cardinality of Set of Induced Equivalence Classes of Injection | Let $f: S \to T$ be a mapping.
Let $\RR_f \subseteq S \times S$ be the relation induced by $f$:
:$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$
Let $f$ be an injection.
Then there are $\card S$ different $\RR_f$-classes. | From Cardinality of Image of Injection we have that $\card {f \sqbrk S} = \card S$.
From the nature of an injection, for all $s \in S$, the $\RR_f$-class of $s$ is a singleton.
Hence the result.
{{qed}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $\RR_f \subseteq S \times S$ be the [[Definition:Equivalence Relation Induced by Mapping|relation induced by $f$]]:
:$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$
Let $f$ be an [[Definition:Injection|injection]].
Then there are $\card S$ diff... | From [[Cardinality of Image of Injection]] we have that $\card {f \sqbrk S} = \card S$.
From the nature of an [[Definition:Injection|injection]], for all $s \in S$, the [[Definition:Equivalence Class|$\RR_f$-class]] of $s$ is a [[Definition:Singleton|singleton]].
Hence the result.
{{qed}} | Cardinality of Set of Induced Equivalence Classes of Injection | https://proofwiki.org/wiki/Cardinality_of_Set_of_Induced_Equivalence_Classes_of_Injection | https://proofwiki.org/wiki/Cardinality_of_Set_of_Induced_Equivalence_Classes_of_Injection | [
"Injections",
"Equivalence Relations"
] | [
"Definition:Mapping",
"Definition:Equivalence Relation Induced by Mapping",
"Definition:Injection",
"Definition:Equivalence Class"
] | [
"Cardinality of Image of Injection",
"Definition:Injection",
"Definition:Equivalence Class",
"Definition:Singleton"
] |
proofwiki-4410 | Cardinality of Set of Induced Equivalence Classes of Surjection | Let $f: S \to T$ be a mapping.
Let $\RR_f \subseteq S \times S$ be the relation induced by $f$:
:$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$
Let $f$ be a surjection.
Then there are $\card T$ different $\RR_f$-classes. | From the definition of a surjection:
:$\forall t \in T: \exists s \in S: \map f s = t$
Thus there are as many $\RR_f$-classes of $f$ as there are elements of $T$.
Hence the result.
{{qed}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $\RR_f \subseteq S \times S$ be the [[Definition:Equivalence Relation Induced by Mapping|relation induced by $f$]]:
:$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$
Let $f$ be a [[Definition:Surjection|surjection]].
Then there are $\card T$ di... | From the definition of a [[Definition:Surjection|surjection]]:
:$\forall t \in T: \exists s \in S: \map f s = t$
Thus there are as many [[Definition:Equivalence Class|$\RR_f$-classes]] of $f$ as there are [[Definition:Element|elements]] of $T$.
Hence the result.
{{qed}} | Cardinality of Set of Induced Equivalence Classes of Surjection | https://proofwiki.org/wiki/Cardinality_of_Set_of_Induced_Equivalence_Classes_of_Surjection | https://proofwiki.org/wiki/Cardinality_of_Set_of_Induced_Equivalence_Classes_of_Surjection | [
"Surjections",
"Equivalence Relations"
] | [
"Definition:Mapping",
"Definition:Equivalence Relation Induced by Mapping",
"Definition:Surjection",
"Definition:Equivalence Class"
] | [
"Definition:Surjection",
"Definition:Equivalence Class",
"Definition:Element"
] |
proofwiki-4411 | Power of Elements is Subgroup | Let $G$ be an abelian group whose identity is $e$.
Then for any $n \in \Z$, the set $G^n = \set {x^n: x \in G}$ is a subgroup of $G$.
Moreover, if:
:$(1): \quad G$ is finite
and:
:$(2): \quad n \ne 1$ is a divisor of the order of $G$
then $G^n$ is a proper subgroup of $G$. | As $e^n = e \in G^n$, the set $G^n$ is not empty.
Let $x^n, y^n \in G^n$.
From Power of Product in Abelian Group, we have:
:$x^n \paren {y^n}^{-1} = x^n \paren {y^{-1} }^n = \paren {x y^{-1} }^n$
It follows that $x^n \paren {y^n}^{-1} \in G^n$.
From the One-Step Subgroup Test, we conclude that $G^n$ is a subgroup of $G... | Let $G$ be an [[Definition:Abelian Group|abelian group]] whose [[Definition:Identity Element|identity]] is $e$.
Then for any $n \in \Z$, the set $G^n = \set {x^n: x \in G}$ is a [[Definition:Subgroup|subgroup]] of $G$.
Moreover, if:
:$(1): \quad G$ is [[Definition:Finite Set|finite]]
and:
:$(2): \quad n \ne 1$ is a... | As $e^n = e \in G^n$, the set $G^n$ is [[Definition:Non-Empty Set|not empty]].
Let $x^n, y^n \in G^n$.
From [[Power of Product in Abelian Group]], we have:
:$x^n \paren {y^n}^{-1} = x^n \paren {y^{-1} }^n = \paren {x y^{-1} }^n$
It follows that $x^n \paren {y^n}^{-1} \in G^n$.
From the [[One-Step Subgroup Test]], w... | Power of Elements is Subgroup | https://proofwiki.org/wiki/Power_of_Elements_is_Subgroup | https://proofwiki.org/wiki/Power_of_Elements_is_Subgroup | [
"Abelian Groups"
] | [
"Definition:Abelian Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Finite Set",
"Definition:Divisor (Algebra)/Integer",
"Definition:Order of Structure",
"Definition:Proper Subgroup"
] | [
"Definition:Non-Empty Set",
"Power of Product in Abelian Group",
"One-Step Subgroup Test",
"Definition:Finite Set",
"Definition:Proper Subgroup",
"Definition:Injection",
"Fundamental Theorem of Arithmetic",
"Definition:Prime Number",
"Cauchy's Lemma (Group Theory)",
"Definition:Element",
"Defini... |
proofwiki-4412 | Index Laws/Sum of Indices/Semigroup | Let $\struct {S, \circ}$ be a semigroup.
For $a \in S$, let $\circ^n a = a^n$ be defined as the $n$th power of $a$:
:<nowiki>$a^n = \begin{cases}
a & : n = 1 \\
a^x \circ a & : n = x + 1
\end{cases}$</nowiki>
That is:
:$a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \circ^n \paren a$
Then... | Let $a \in S$.
Because $\struct {S, \circ}$ is a semigroup, $\circ$ is associative on $S$.
The proof proceeds by the Principle of Mathematical Induction.
Let $\map P m$ be the proposition:
:$\forall n \in \N_{>0}: a^{n + m} = a^n \circ a^m$
that is:
:$\forall n \in \N_{>0}: \circ^{n + m} a = \paren {\circ^n a} \circ \p... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
For $a \in S$, let $\circ^n a = a^n$ be defined as the [[Definition:Power of Element of Semigroup|$n$th power of $a$]]:
:<nowiki>$a^n = \begin{cases}
a & : n = 1 \\
a^x \circ a & : n = x + 1
\end{cases}$</nowiki>
That is:
:$a^n = \underbrace {a \circ ... | Let $a \in S$.
Because $\struct {S, \circ}$ is a [[Definition:Semigroup|semigroup]], $\circ$ is [[Definition:Associative Operation|associative]] on $S$.
The proof proceeds by the [[Principle of Mathematical Induction]].
Let $\map P m$ be the [[Definition:Proposition|proposition]]:
:$\forall n \in \N_{>0}: a^{n + m}... | Index Laws/Sum of Indices/Semigroup | https://proofwiki.org/wiki/Index_Laws/Sum_of_Indices/Semigroup | https://proofwiki.org/wiki/Index_Laws/Sum_of_Indices/Semigroup | [
"Index Laws",
"Semigroups"
] | [
"Definition:Semigroup",
"Definition:Power of Element/Semigroup"
] | [
"Definition:Semigroup",
"Definition:Associative Operation",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-4413 | Index Laws/Product of Indices/Semigroup | Let $\struct {S, \circ}$ be a semigroup.
For $a \in S$, let $\circ^n a = a^n$ be the $n$th power of $a$.
Then:
:$\forall m, n \in \N_{>0}: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$ | Let $b = \circ^m a$.
Let $h: \N_{>0} \to S$ be the mapping defined as:
:$\forall n \in \N_{>0}: \map h n = \circ^{n m} a$
Let the mapping $f_b: \N_{>0} \to S$ be recursively defined as:
:$\forall n \in \N_{>0}: \map {f_b} n = \circ^n b$
From the Principle of Recursive Definition:
:$f_b$ is the unique mapping which sati... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
For $a \in S$, let $\circ^n a = a^n$ be the [[Definition:Power of Element of Semigroup|$n$th power of $a$]].
Then:
:$\forall m, n \in \N_{>0}: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$ | Let $b = \circ^m a$.
Let $h: \N_{>0} \to S$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall n \in \N_{>0}: \map h n = \circ^{n m} a$
Let the [[Definition:Mapping|mapping]] $f_b: \N_{>0} \to S$ be [[Definition:Recursively Defined Mapping/Natural Numbers|recursively defined]] as:
:$\forall n \in \N_{>0}... | Index Laws/Product of Indices/Semigroup | https://proofwiki.org/wiki/Index_Laws/Product_of_Indices/Semigroup | https://proofwiki.org/wiki/Index_Laws/Product_of_Indices/Semigroup | [
"Index Laws",
"Semigroups"
] | [
"Definition:Semigroup",
"Definition:Power of Element/Semigroup"
] | [
"Definition:Mapping",
"Definition:Mapping",
"Definition:Recursively Defined Mapping/Natural Numbers",
"Principle of Recursive Definition",
"Definition:Unique",
"Definition:Mapping",
"Natural Number Multiplication Distributes over Addition",
"Index Laws/Sum of Indices/Semigroup",
"Natural Number Mult... |
proofwiki-4414 | Semigroup is Subsemigroup of Itself | Let $\struct {S, \circ}$ be a semigroup.
Then $\struct {S, \circ}$ is a subsemigroup of itself. | For all sets $S$, $S \subseteq S$, that is, $S$ is a subset of itself.
Thus $\struct {S, \circ}$ is a semigroup which is a subset of $\struct {S, \circ}$, and therefore a subsemigroup of $\struct {S, \circ}$.
{{Qed}} | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Then $\struct {S, \circ}$ is a [[Definition:Subsemigroup|subsemigroup]] of itself. | For all [[Definition:Set|sets]] $S$, $S \subseteq S$, that is, [[Set is Subset of Itself|$S$ is a subset of itself]].
Thus $\struct {S, \circ}$ is a [[Definition:Semigroup|semigroup]] which is a [[Definition:Subset|subset]] of $\struct {S, \circ}$, and therefore a [[Definition:Subsemigroup|subsemigroup]] of $\struct {... | Semigroup is Subsemigroup of Itself | https://proofwiki.org/wiki/Semigroup_is_Subsemigroup_of_Itself | https://proofwiki.org/wiki/Semigroup_is_Subsemigroup_of_Itself | [
"Subsemigroups",
"Semigroups"
] | [
"Definition:Semigroup",
"Definition:Subsemigroup"
] | [
"Definition:Set",
"Set is Subset of Itself",
"Definition:Semigroup",
"Definition:Subset",
"Definition:Subsemigroup"
] |
proofwiki-4415 | Identity of Monoid is Unique | Let $\struct {S, \circ}$ be a monoid.
Then $S$ has a unique identity. | As $\struct {S, \circ}$ is an algebraic structure, the result Identity is Unique can be applied directly.
{{qed}} | Let $\struct {S, \circ}$ be a [[Definition:Monoid|monoid]].
Then $S$ has a [[Definition:Unique|unique]] [[Definition:Identity Element|identity]]. | As $\struct {S, \circ}$ is an [[Definition:Algebraic Structure|algebraic structure]], the result [[Identity is Unique]] can be applied directly.
{{qed}} | Identity of Monoid is Unique | https://proofwiki.org/wiki/Identity_of_Monoid_is_Unique | https://proofwiki.org/wiki/Identity_of_Monoid_is_Unique | [
"Monoids"
] | [
"Definition:Monoid",
"Definition:Unique",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Algebraic Structure",
"Identity is Unique"
] |
proofwiki-4416 | Identity of Group is Unique | Let $\struct {G, \circ}$ be a group which has an identity element $e \in G$.
Then $e$ is unique. | By the definition of a group, $\struct {G, \circ}$ is also a monoid.
The result follows by applying the result Identity of Monoid is Unique.
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] which has an [[Definition:Identity Element|identity element]] $e \in G$.
Then $e$ is [[Definition:Unique|unique]]. | By the definition of a [[Definition:Group|group]], $\struct {G, \circ}$ is also a [[Definition:Monoid|monoid]].
The result follows by applying the result [[Identity of Monoid is Unique]].
{{qed}} | Identity of Group is Unique/Proof 1 | https://proofwiki.org/wiki/Identity_of_Group_is_Unique | https://proofwiki.org/wiki/Identity_of_Group_is_Unique/Proof_1 | [
"Group Theory",
"Identity Elements",
"Identity of Group is Unique"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unique"
] | [
"Definition:Group",
"Definition:Monoid",
"Identity of Monoid is Unique"
] |
proofwiki-4417 | Identity of Group is Unique | Let $\struct {G, \circ}$ be a group which has an identity element $e \in G$.
Then $e$ is unique. | Let $e$ and $f$ both be identity elements of a group $\struct {G, \circ}$.
Then:
{{begin-eqn}}
{{eqn | l = e
| r = e \circ f
| c = $f$ is an identity
}}
{{eqn | r = f
| c = $e$ is an identity
}}
{{end-eqn}}
So $e = f$ and there is only one identity after all.
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] which has an [[Definition:Identity Element|identity element]] $e \in G$.
Then $e$ is [[Definition:Unique|unique]]. | Let $e$ and $f$ both be [[Definition:Identity Element|identity elements]] of a [[Definition:Group|group]] $\struct {G, \circ}$.
Then:
{{begin-eqn}}
{{eqn | l = e
| r = e \circ f
| c = $f$ is an [[Definition:Identity Element|identity]]
}}
{{eqn | r = f
| c = $e$ is an [[Definition:Identity Element|id... | Identity of Group is Unique/Proof 2 | https://proofwiki.org/wiki/Identity_of_Group_is_Unique | https://proofwiki.org/wiki/Identity_of_Group_is_Unique/Proof_2 | [
"Group Theory",
"Identity Elements",
"Identity of Group is Unique"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unique"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-4418 | Identity of Group is Unique | Let $\struct {G, \circ}$ be a group which has an identity element $e \in G$.
Then $e$ is unique. | From Group has Latin Square Property, there exists a unique $x \in G$ such that:
:$a x = b$
and there exists a unique $y \in G$ such that:
:$y a = b$
Setting $b = a$, this becomes:
There exists a unique $x \in G$ such that:
:$a x = a$
and there exists a unique $y \in G$ such that:
:$y a = a$
These $x$ and $y$ are both ... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] which has an [[Definition:Identity Element|identity element]] $e \in G$.
Then $e$ is [[Definition:Unique|unique]]. | From [[Group has Latin Square Property]], there exists a [[Definition:Unique|unique]] $x \in G$ such that:
:$a x = b$
and there exists a [[Definition:Unique|unique]] $y \in G$ such that:
:$y a = b$
Setting $b = a$, this becomes:
There exists a [[Definition:Unique|unique]] $x \in G$ such that:
:$a x = a$
and there e... | Identity of Group is Unique/Proof 3 | https://proofwiki.org/wiki/Identity_of_Group_is_Unique | https://proofwiki.org/wiki/Identity_of_Group_is_Unique/Proof_3 | [
"Group Theory",
"Identity Elements",
"Identity of Group is Unique"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unique"
] | [
"Group has Latin Square Property",
"Definition:Unique",
"Definition:Unique",
"Definition:Unique",
"Definition:Unique",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-4419 | Unity of Integral Domain is Unique | Let $\struct {D, +, \times}$ be an integral domain.
Then the unity of $\struct {D, +, \times}$ is unique. | From the definition of an integral domain, $\struct {D, +, \times}$ is a commutative ring such that $\struct {D^*, \circ}$ is a monoid.
The result follows from Identity of Monoid is Unique.
{{qed}} | Let $\struct {D, +, \times}$ be an [[Definition:Integral Domain|integral domain]].
Then the [[Definition:Unity of Ring|unity]] of $\struct {D, +, \times}$ is [[Definition:Unique|unique]]. | From the definition of an [[Definition:Integral Domain|integral domain]], $\struct {D, +, \times}$ is a [[Definition:Commutative Ring|commutative ring]] such that $\struct {D^*, \circ}$ is a [[Definition:Monoid|monoid]].
The result follows from [[Identity of Monoid is Unique]].
{{qed}} | Unity of Integral Domain is Unique | https://proofwiki.org/wiki/Unity_of_Integral_Domain_is_Unique | https://proofwiki.org/wiki/Unity_of_Integral_Domain_is_Unique | [
"Integral Domains"
] | [
"Definition:Integral Domain",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Unique"
] | [
"Definition:Integral Domain",
"Definition:Commutative Ring",
"Definition:Monoid",
"Identity of Monoid is Unique"
] |
proofwiki-4420 | Zero of Integral Domain is Unique | Let $\struct {D, +, \times}$ be an integral domain.
Then the zero of $\struct {D, +, \times}$ is unique. | By definition, an integral domain is a ring.
The result the follows from Ring Zero is Unique.
{{qed}} | Let $\struct {D, +, \times}$ be an [[Definition:Integral Domain|integral domain]].
Then the [[Definition:Ring Zero|zero]] of $\struct {D, +, \times}$ is [[Definition:Unique|unique]]. | By definition, an [[Definition:Integral Domain|integral domain]] is a [[Definition:Ring (Abstract Algebra)|ring]].
The result the follows from [[Ring Zero is Unique]].
{{qed}} | Zero of Integral Domain is Unique | https://proofwiki.org/wiki/Zero_of_Integral_Domain_is_Unique | https://proofwiki.org/wiki/Zero_of_Integral_Domain_is_Unique | [
"Integral Domains"
] | [
"Definition:Integral Domain",
"Definition:Ring Zero",
"Definition:Unique"
] | [
"Definition:Integral Domain",
"Definition:Ring (Abstract Algebra)",
"Ring Zero is Unique"
] |
proofwiki-4421 | Negative in Integral Domain is Unique | Let $\struct {D, +, \times}$ be an integral domain.
Let $a \in R$.
Then the negative $-a$ of $a$ is unique. | From the definition of an integral domain, $\struct {D, +, \times}$ is a ring.
The result follows from Ring Negative is Unique.
{{qed}} | Let $\struct {D, +, \times}$ be an [[Definition:Integral Domain|integral domain]].
Let $a \in R$.
Then the [[Definition:Ring Negative|negative]] $-a$ of $a$ is [[Definition:Unique|unique]]. | From the definition of an [[Definition:Integral Domain|integral domain]], $\struct {D, +, \times}$ is a [[Definition:Ring (Abstract Algebra)|ring]].
The result follows from [[Ring Negative is Unique]].
{{qed}} | Negative in Integral Domain is Unique | https://proofwiki.org/wiki/Negative_in_Integral_Domain_is_Unique | https://proofwiki.org/wiki/Negative_in_Integral_Domain_is_Unique | [
"Integral Domains"
] | [
"Definition:Integral Domain",
"Definition:Ring Negative",
"Definition:Unique"
] | [
"Definition:Integral Domain",
"Definition:Ring (Abstract Algebra)",
"Ring Negative is Unique"
] |
proofwiki-4422 | Inverse in Group is Unique | Let $\struct {G, \circ}$ be a group.
Then every element $x \in G$ has exactly one inverse:
:$\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x^{-1} \circ x$
where $e$ is the identity element of $\struct {G, \circ}$. | By the definition of a group, $\struct {G, \circ}$ is a monoid each of whose elements has an inverse.
The result follows directly from Inverse in Monoid is Unique.
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Then every element $x \in G$ has [[Definition:Unique|exactly one]] [[Definition:Inverse Element|inverse]]:
:$\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x^{-1} \circ x$
where $e$ is the [[Definition:Identity Element|identity element]] of... | By the definition of a [[Definition:Group|group]], $\struct {G, \circ}$ is a [[Definition:Monoid|monoid]] each of whose elements has an [[Definition:Inverse Element|inverse]].
The result follows directly from [[Inverse in Monoid is Unique]].
{{qed}} | Inverse in Group is Unique/Proof 1 | https://proofwiki.org/wiki/Inverse_in_Group_is_Unique | https://proofwiki.org/wiki/Inverse_in_Group_is_Unique/Proof_1 | [
"Group Theory",
"Inverse Elements",
"Inverse in Group is Unique"
] | [
"Definition:Group",
"Definition:Unique",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Group",
"Definition:Monoid",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Inverse in Monoid is Unique"
] |
proofwiki-4423 | Inverse in Group is Unique | Let $\struct {G, \circ}$ be a group.
Then every element $x \in G$ has exactly one inverse:
:$\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x^{-1} \circ x$
where $e$ is the identity element of $\struct {G, \circ}$. | Let $\struct {G, \circ}$ be a group whose identity element is $e$.
By {{Group-axiom|3}}, every element of $G$ has at least one inverse.
Suppose that:
:$\exists b, c \in G: a \circ b = e, a \circ c = e$
that is, that $b$ and $c$ are both inverse elements of $a$.
Then:
{{begin-eqn}}
{{eqn | l = b
| r = b \circ e
... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Then every element $x \in G$ has [[Definition:Unique|exactly one]] [[Definition:Inverse Element|inverse]]:
:$\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x^{-1} \circ x$
where $e$ is the [[Definition:Identity Element|identity element]] of... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity element]] is $e$.
By {{Group-axiom|3}}, every [[Definition:Element|element]] of $G$ has at least one [[Definition:Inverse Element|inverse]].
Suppose that:
:$\exists b, c \in G: a \circ b = e, a \circ c = e$
that is,... | Inverse in Group is Unique/Proof 2 | https://proofwiki.org/wiki/Inverse_in_Group_is_Unique | https://proofwiki.org/wiki/Inverse_in_Group_is_Unique/Proof_2 | [
"Group Theory",
"Inverse Elements",
"Inverse in Group is Unique"
] | [
"Definition:Group",
"Definition:Unique",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inv... |
proofwiki-4424 | Inverse in Group is Unique | Let $\struct {G, \circ}$ be a group.
Then every element $x \in G$ has exactly one inverse:
:$\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x^{-1} \circ x$
where $e$ is the identity element of $\struct {G, \circ}$. | Let $x, y \in G$.
We already have, from the definition of inverse element, that:
:$\forall x \in G: \exists x^{-1} \in G: x \circ x^{-1} = e = x^{-1} \circ x$
By Group has Latin Square Property, there exists exactly one $a \in G$ such that $a \circ x = y$.
Similarly, there exists exactly one $b \in G$ such that $x \cir... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Then every element $x \in G$ has [[Definition:Unique|exactly one]] [[Definition:Inverse Element|inverse]]:
:$\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x^{-1} \circ x$
where $e$ is the [[Definition:Identity Element|identity element]] of... | Let $x, y \in G$.
We already have, from the definition of [[Definition:Inverse Element|inverse element]], that:
:$\forall x \in G: \exists x^{-1} \in G: x \circ x^{-1} = e = x^{-1} \circ x$
By [[Group has Latin Square Property]], there exists [[Definition:Unique|exactly one]] $a \in G$ such that $a \circ x = y$.
Sim... | Inverse in Group is Unique/Proof 3 | https://proofwiki.org/wiki/Inverse_in_Group_is_Unique | https://proofwiki.org/wiki/Inverse_in_Group_is_Unique/Proof_3 | [
"Group Theory",
"Inverse Elements",
"Inverse in Group is Unique"
] | [
"Definition:Group",
"Definition:Unique",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Inverse (Abstract Algebra)/Inverse",
"Group has Latin Square Property",
"Definition:Unique",
"Definition:Unique"
] |
proofwiki-4425 | Product Inverse in Ring is Unique | Let $\struct {R, +, \circ}$ be a ring with unity.
Let $x \in R$ be a unit of $R$.
Then the product inverse $x^{-1}$ of $x$ is unique. | By definition of ring with unity, the algebraic structure $\struct {R, \circ}$ is a monoid.
The result follows from Inverse in Monoid is Unique.
{{qed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $x \in R$ be a [[Definition:Unit of Ring|unit]] of $R$.
Then the [[Definition:Product Inverse|product inverse]] $x^{-1}$ of $x$ is [[Definition:Unique|unique]]. | By definition of [[Definition:Ring with Unity|ring with unity]], the [[Definition:Algebraic Structure|algebraic structure]] $\struct {R, \circ}$ is a [[Definition:Monoid|monoid]].
The result follows from [[Inverse in Monoid is Unique]].
{{qed}} | Product Inverse in Ring is Unique | https://proofwiki.org/wiki/Product_Inverse_in_Ring_is_Unique | https://proofwiki.org/wiki/Product_Inverse_in_Ring_is_Unique | [
"Rings with Unity"
] | [
"Definition:Ring with Unity",
"Definition:Unit of Ring",
"Definition:Product Inverse",
"Definition:Unique"
] | [
"Definition:Ring with Unity",
"Definition:Algebraic Structure",
"Definition:Monoid",
"Inverse in Monoid is Unique"
] |
proofwiki-4426 | Units of Gaussian Integers form Group | Let $U_\C$ be the set of units of the Gaussian integers:
:$U_\C = \set {1, i, -1, -i}$
where $i$ is the imaginary unit: $i = \sqrt {-1}$.
Let $\struct {U_\C, \times}$ be the algebraic structure formed by $U_\C$ under the operation of complex multiplication.
Then $\struct {U_\C, \times}$ forms a cyclic group under compl... | By definition of the imaginary unit $i$:
{{begin-eqn}}
{{eqn | l = i^2
| r = -1
}}
{{eqn | l = i^3
| r = -i
}}
{{eqn | l = i^4
| r = 1
}}
{{end-eqn}}
thus demonstrating that $U_\C$ is generated by $i$.
Thus $\struct {U_\C, \times}$ is by definition a cyclic group of order $4$.
{{qed}} | Let $U_\C$ be the [[Definition:Set|set]] of [[Definition:Unit of Ring|units]] of the [[Definition:Gaussian Integer|Gaussian integers]]:
:$U_\C = \set {1, i, -1, -i}$
where $i$ is the [[Definition:Imaginary Unit|imaginary unit]]: $i = \sqrt {-1}$.
Let $\struct {U_\C, \times}$ be the [[Definition:Algebraic Structure wi... | By definition of the [[Definition:Imaginary Unit|imaginary unit]] $i$:
{{begin-eqn}}
{{eqn | l = i^2
| r = -1
}}
{{eqn | l = i^3
| r = -i
}}
{{eqn | l = i^4
| r = 1
}}
{{end-eqn}}
thus demonstrating that $U_\C$ is [[Definition:Generator of Cyclic Group|generated]] by $i$.
Thus $\struct {U_\C, \time... | Units of Gaussian Integers form Group/Proof 1 | https://proofwiki.org/wiki/Units_of_Gaussian_Integers_form_Group | https://proofwiki.org/wiki/Units_of_Gaussian_Integers_form_Group/Proof_1 | [
"Units of Gaussian Integers form Group",
"Group of Gaussian Integer Units",
"Complex Roots of Unity"
] | [
"Definition:Set",
"Definition:Unit of Ring",
"Definition:Gaussian Integer",
"Definition:Complex Number/Imaginary Unit",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Multiplication/Complex Numbers",
"Definition:Cyclic Group",
"Definition:Multipl... | [
"Definition:Complex Number/Imaginary Unit",
"Definition:Cyclic Group/Generator",
"Definition:Cyclic Group",
"Definition:Order of Structure"
] |
proofwiki-4427 | Units of Gaussian Integers form Group | Let $U_\C$ be the set of units of the Gaussian integers:
:$U_\C = \set {1, i, -1, -i}$
where $i$ is the imaginary unit: $i = \sqrt {-1}$.
Let $\struct {U_\C, \times}$ be the algebraic structure formed by $U_\C$ under the operation of complex multiplication.
Then $\struct {U_\C, \times}$ forms a cyclic group under compl... | From Gaussian Integer Units are 4th Roots of Unity:
: $\left\{{1, i, -1, -i}\right\}$ constitutes the set of the $4$th roots of unity.
The result follows from Roots of Unity under Multiplication form Cyclic Group.
{{qed}} | Let $U_\C$ be the [[Definition:Set|set]] of [[Definition:Unit of Ring|units]] of the [[Definition:Gaussian Integer|Gaussian integers]]:
:$U_\C = \set {1, i, -1, -i}$
where $i$ is the [[Definition:Imaginary Unit|imaginary unit]]: $i = \sqrt {-1}$.
Let $\struct {U_\C, \times}$ be the [[Definition:Algebraic Structure wi... | From [[Gaussian Integer Units are 4th Roots of Unity]]:
: $\left\{{1, i, -1, -i}\right\}$ constitutes the [[Definition:Set|set]] of the [[Definition:Complex Roots of Unity|$4$th roots of unity]].
The result follows from [[Roots of Unity under Multiplication form Cyclic Group]].
{{qed}} | Units of Gaussian Integers form Group/Proof 2 | https://proofwiki.org/wiki/Units_of_Gaussian_Integers_form_Group | https://proofwiki.org/wiki/Units_of_Gaussian_Integers_form_Group/Proof_2 | [
"Units of Gaussian Integers form Group",
"Group of Gaussian Integer Units",
"Complex Roots of Unity"
] | [
"Definition:Set",
"Definition:Unit of Ring",
"Definition:Gaussian Integer",
"Definition:Complex Number/Imaginary Unit",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Multiplication/Complex Numbers",
"Definition:Cyclic Group",
"Definition:Multipl... | [
"Gaussian Integer Units are 4th Roots of Unity",
"Definition:Set",
"Definition:Root of Unity/Complex",
"Roots of Unity under Multiplication form Cyclic Group"
] |
proofwiki-4428 | Units of Gaussian Integers form Group | Let $U_\C$ be the set of units of the Gaussian integers:
:$U_\C = \set {1, i, -1, -i}$
where $i$ is the imaginary unit: $i = \sqrt {-1}$.
Let $\struct {U_\C, \times}$ be the algebraic structure formed by $U_\C$ under the operation of complex multiplication.
Then $\struct {U_\C, \times}$ forms a cyclic group under compl... | From Units of Gaussian Integers, $U_\C$ is the set of units of the ring of Gaussian integers.
From Group of Units is Group, $\left({U_\C, \times}\right)$ forms a group.
It remains to note that:
{{begin-eqn}}
{{eqn | l = i^2
| r = -1
}}
{{eqn | l = i^3
| r = -i
}}
{{eqn | l = i^4
| r = 1
}}
{{end-eqn}}... | Let $U_\C$ be the [[Definition:Set|set]] of [[Definition:Unit of Ring|units]] of the [[Definition:Gaussian Integer|Gaussian integers]]:
:$U_\C = \set {1, i, -1, -i}$
where $i$ is the [[Definition:Imaginary Unit|imaginary unit]]: $i = \sqrt {-1}$.
Let $\struct {U_\C, \times}$ be the [[Definition:Algebraic Structure wi... | From [[Units of Gaussian Integers]], $U_\C$ is the [[Definition:Set|set]] of [[Definition:Unit of Ring|units]] of the [[Definition:Ring of Gaussian Integers|ring of Gaussian integers]].
From [[Group of Units is Group]], $\left({U_\C, \times}\right)$ forms a [[Definition:Group|group]].
It remains to note that:
{{begi... | Units of Gaussian Integers form Group/Proof 3 | https://proofwiki.org/wiki/Units_of_Gaussian_Integers_form_Group | https://proofwiki.org/wiki/Units_of_Gaussian_Integers_form_Group/Proof_3 | [
"Units of Gaussian Integers form Group",
"Group of Gaussian Integer Units",
"Complex Roots of Unity"
] | [
"Definition:Set",
"Definition:Unit of Ring",
"Definition:Gaussian Integer",
"Definition:Complex Number/Imaginary Unit",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Multiplication/Complex Numbers",
"Definition:Cyclic Group",
"Definition:Multipl... | [
"Units of Gaussian Integers",
"Definition:Set",
"Definition:Unit of Ring",
"Definition:Ring of Gaussian Integers",
"Group of Units is Group",
"Definition:Group",
"Definition:Cyclic Group"
] |
proofwiki-4429 | Gaussian Integer Units form Multiplicative Subgroup of Complex Numbers | The group of Gaussian integer units under complex multiplication:
:$\struct {U_\C, \times} = \struct {\set {1, i, -1, -i}, \times}$
forms a subgroup of the multiplicative group of complex numbers. | By Units of Gaussian Integers form Group, $\struct {U_\C, \times}$ forms a group.
Each of the elements of $U_\C$ is a complex number, and non-zero, and therefore $U_\C \subseteq \C \setminus \set 0$.
The result follows by definition of subgroup.
{{qed}} | The [[Definition:Group of Gaussian Integer Units|group of Gaussian integer units]] under [[Definition:Complex Multiplication|complex multiplication]]:
:$\struct {U_\C, \times} = \struct {\set {1, i, -1, -i}, \times}$
forms a [[Definition:Subgroup|subgroup]] of the [[Definition:Multiplicative Group of Complex Numbers|m... | By [[Units of Gaussian Integers form Group]], $\struct {U_\C, \times}$ forms a [[Definition:Group|group]].
Each of the elements of $U_\C$ is a [[Definition:Complex Number|complex number]], and non-zero, and therefore $U_\C \subseteq \C \setminus \set 0$.
The result follows by definition of [[Definition:Subgroup|subgr... | Gaussian Integer Units form Multiplicative Subgroup of Complex Numbers | https://proofwiki.org/wiki/Gaussian_Integer_Units_form_Multiplicative_Subgroup_of_Complex_Numbers | https://proofwiki.org/wiki/Gaussian_Integer_Units_form_Multiplicative_Subgroup_of_Complex_Numbers | [
"Multiplicative Group of Complex Numbers",
"Gaussian Integers",
"Examples of Subgroups"
] | [
"Definition:Group of Gaussian Integer Units",
"Definition:Multiplication/Complex Numbers",
"Definition:Subgroup",
"Definition:Multiplicative Group of Complex Numbers"
] | [
"Units of Gaussian Integers form Group",
"Definition:Group",
"Definition:Complex Number",
"Definition:Subgroup"
] |
proofwiki-4430 | Permutation Group is Subgroup of Symmetric Group | Let $S$ be a set.
Let $\struct {\map \Gamma S, \circ}$ be the symmetric group on $S$, where $\circ$ denotes the composition operation.
Let $\struct {H, \circ}$ be a set of permutations of $S$ which forms a group under $\circ$.
Then $\struct {H, \circ}$ is a subgroup of $\struct {\map \Gamma S, \circ}$. | Follows directly from the definition of subgroup:
$H$ is a subset of $\map \Gamma S$, and $\struct {H, \circ}$ is a group.
Hence the result.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\struct {\map \Gamma S, \circ}$ be the [[Definition:Symmetric Group|symmetric group on $S$]], where $\circ$ denotes the [[Definition:Composition of Mappings|composition]] operation.
Let $\struct {H, \circ}$ be a [[Definition:Set|set]] of [[Definition:Permutation|permutations]... | Follows directly from the definition of [[Definition:Subgroup|subgroup]]:
$H$ is a [[Definition:Subset|subset]] of $\map \Gamma S$, and $\struct {H, \circ}$ is a [[Definition:Group|group]].
Hence the result.
{{qed}} | Permutation Group is Subgroup of Symmetric Group | https://proofwiki.org/wiki/Permutation_Group_is_Subgroup_of_Symmetric_Group | https://proofwiki.org/wiki/Permutation_Group_is_Subgroup_of_Symmetric_Group | [
"Symmetric Groups",
"Subgroups"
] | [
"Definition:Set",
"Definition:Symmetric Group",
"Definition:Composition of Mappings",
"Definition:Set",
"Definition:Permutation",
"Definition:Group",
"Definition:Subgroup"
] | [
"Definition:Subgroup",
"Definition:Subset",
"Definition:Group"
] |
proofwiki-4431 | Units of Field of Complex Numbers form Group | :$\C^\times = \C \setminus \set 0$
where $\C^\times$ denotes the group of units of $\C$. | By Complex Numbers form Field, $\C$ is a field.
From Group of Units of Field it follows that:
: $\C^\times = \C \setminus \set 0$
{{qed}}
Category:Examples of Groups
Category:Complex Numbers
tss7spdnxtbd4mireozplygblsf6ij5 | :$\C^\times = \C \setminus \set 0$
where $\C^\times$ denotes the [[Definition:Group of Units of Ring|group of units]] of $\C$. | By [[Complex Numbers form Field]], $\C$ is a [[Definition:Field (Abstract Algebra)|field]].
From [[Group of Units of Field]] it follows that:
: $\C^\times = \C \setminus \set 0$
{{qed}}
[[Category:Examples of Groups]]
[[Category:Complex Numbers]]
tss7spdnxtbd4mireozplygblsf6ij5 | Units of Field of Complex Numbers form Group | https://proofwiki.org/wiki/Units_of_Field_of_Complex_Numbers_form_Group | https://proofwiki.org/wiki/Units_of_Field_of_Complex_Numbers_form_Group | [
"Examples of Groups",
"Complex Numbers"
] | [
"Definition:Group of Units/Ring"
] | [
"Complex Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Group of Units of Field",
"Category:Examples of Groups",
"Category:Complex Numbers"
] |
proofwiki-4432 | Determinant of Block Diagonal Matrix | Let $\mathbf A$ be a block diagonal matrix of order $n$.
Let $\mathbf A_1, \ldots, \mathbf A_k$ be the square matrices on the diagonal:
:<nowiki>$\ds \mathbf A = \begin {bmatrix}
\mathbf A_1 & 0 & \cdots & 0 \\
0 & \mathbf A_2 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \mathbf A_k
\end {bmat... | {{tidy}}
{{MissingLinks}}
To prove this fact, we need to prove additional helper propositions.
{{ExtractTheorem}}
$\textbf{Claim 1}$
The determinant of the block-diagonal matrix of type $M = \begin{pmatrix}
A & 0 \\
0 & I \\
\end{pmatrix}$ or $M = \begin{pmatrix}
I & 0 \\
0 & A \\
\end{pmatrix}$ equals $\map \det A$.
$... | Let $\mathbf A$ be a [[Definition:Block Diagonal Matrix|block diagonal matrix]] of order $n$.
Let $\mathbf A_1, \ldots, \mathbf A_k$ be the [[Definition:Square Matrix|square matrices]] on the diagonal:
:<nowiki>$\ds \mathbf A = \begin {bmatrix}
\mathbf A_1 & 0 & \cdots & 0 \\
0 & \mathbf A_2 & \cdots & 0 \\
\vdots & \... | {{tidy}}
{{MissingLinks}}
To prove this fact, we need to prove additional helper propositions.
{{ExtractTheorem}}
$\textbf{Claim 1}$
The determinant of the block-diagonal matrix of type $M = \begin{pmatrix}
A & 0 \\
0 & I \\
\end{pmatrix}$ or $M = \begin{pmatrix}
I & 0 \\
0 & A \\
\end{pmatrix}$ equals $\map \det A$... | Determinant of Block Diagonal Matrix | https://proofwiki.org/wiki/Determinant_of_Block_Diagonal_Matrix | https://proofwiki.org/wiki/Determinant_of_Block_Diagonal_Matrix | [
"Determinants"
] | [
"Definition:Block Diagonal Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Determinant/Matrix"
] | [] |
proofwiki-4433 | Multiplicative Group of Positive Rationals is Non-Cyclic | Let $\struct {\Q_{>0}, \times}$ be the multiplicative group of positive rational numbers.
Then $\struct {\Q_{>0}, \times}$ is not a cyclic group. | {{AimForCont}} $\struct {\Q_{>0}, \times}$ is cyclic.
Then $\struct {\Q_{>0}, \times}$ has a generator $x$ such that $x > 1$.
It would follow that:
:$\Q = \set {\ldots, x^{-2}, x^{-1}, 1, x, x^2, \ldots}$
where the elements are arranged in ascending order.
But then consider $y \in \Q: y = \dfrac {1 + x} 2$.
So $1 < y <... | Let $\struct {\Q_{>0}, \times}$ be the [[Definition:Multiplicative Group of Positive Rational Numbers|multiplicative group of positive rational numbers]].
Then $\struct {\Q_{>0}, \times}$ is not a [[Definition:Cyclic Group|cyclic group]]. | {{AimForCont}} $\struct {\Q_{>0}, \times}$ is [[Definition:Cyclic Group|cyclic]].
Then $\struct {\Q_{>0}, \times}$ has a [[Definition:Generator of Cyclic Group|generator]] $x$ such that $x > 1$.
It would follow that:
:$\Q = \set {\ldots, x^{-2}, x^{-1}, 1, x, x^2, \ldots}$
where the [[Definition:Element|elements]] ar... | Multiplicative Group of Positive Rationals is Non-Cyclic | https://proofwiki.org/wiki/Multiplicative_Group_of_Positive_Rationals_is_Non-Cyclic | https://proofwiki.org/wiki/Multiplicative_Group_of_Positive_Rationals_is_Non-Cyclic | [
"Rational Numbers"
] | [
"Definition:Multiplicative Group of Positive Rational Numbers",
"Definition:Cyclic Group"
] | [
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator",
"Definition:Element",
"Definition:Contradiction",
"Definition:Cyclic Group/Generator",
"Definition:Cyclic Group"
] |
proofwiki-4434 | General Linear Group to Determinant is Homomorphism | Let $\GL {n, \R}$ be the general linear group over the field of real numbers.
Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers.
Let $\det: \GL {n, \R} \to \struct {\R_{\ne 0}, \times}$ be the mapping:
:$\mathbf A \mapsto \map \det {\mathbf A}$
where $\map \det {\mathbf A}$ is the deter... | From Determinant of Matrix Product:
: $\map \det {\mathbf A \mathbf B} = \map \det {\mathbf A} \, \map \det {\mathbf B}$
which is seen to be a group homomorphism by definition.
{{qed}} | Let $\GL {n, \R}$ be the [[Definition:General Linear Group|general linear group]] over the [[Definition:Field of Real Numbers|field of real numbers]].
Let $\struct {\R_{\ne 0}, \times}$ denote the [[Definition:Multiplicative Group of Real Numbers|multiplicative group of real numbers]].
Let $\det: \GL {n, \R} \to \st... | From [[Determinant of Matrix Product]]:
: $\map \det {\mathbf A \mathbf B} = \map \det {\mathbf A} \, \map \det {\mathbf B}$
which is seen to be a [[Definition:Group Homomorphism|group homomorphism]] by definition.
{{qed}} | General Linear Group to Determinant is Homomorphism | https://proofwiki.org/wiki/General_Linear_Group_to_Determinant_is_Homomorphism | https://proofwiki.org/wiki/General_Linear_Group_to_Determinant_is_Homomorphism | [
"Examples of Group Homomorphisms",
"General Linear Group",
"Determinants"
] | [
"Definition:General Linear Group",
"Definition:Field of Real Numbers",
"Definition:Multiplicative Group of Real Numbers",
"Definition:Mapping",
"Definition:Determinant/Matrix",
"Definition:Group Homomorphism"
] | [
"Determinant of Matrix Product",
"Definition:Group Homomorphism"
] |
proofwiki-4435 | Logarithm on Positive Real Numbers is Group Isomorphism | Let $\struct {\R_{>0}, \times}$ be the multiplicative group of positive real numbers.
Let $\struct {\R, +}$ be the additive group of real numbers.
Let $b$ be any real number such that $b > 1$.
Let $\log_b: \struct {\R_{>0}, \times} \to \struct {\R, +}$ be the mapping:
:$x \mapsto \map {\log_b} x$
where $\log_b$ is the ... | From Sum of Logarithms we have:
:$\forall x, y \in \R_{>0}: \map {\log_b} {x y} = \map {\log_b} x + \map {\log_b} y$
That is $\log_b$ is a group homomorphism.
From Change of Base of Logarithm, $\log_b$ is a constant multiplied by the natural logarithm function.
Then we have that Logarithm is Strictly Increasing.
From S... | Let $\struct {\R_{>0}, \times}$ be the [[Definition:Multiplicative Group of Positive Real Numbers|multiplicative group of positive real numbers]].
Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $b$ be any [[Definition:Real Number|real number]] such that ... | From [[Sum of Logarithms]] we have:
:$\forall x, y \in \R_{>0}: \map {\log_b} {x y} = \map {\log_b} x + \map {\log_b} y$
That is $\log_b$ is a [[Definition:Group Homomorphism|group homomorphism]].
From [[Change of Base of Logarithm]], $\log_b$ is a constant multiplied by the [[Definition:Natural Logarithm|natural log... | Logarithm on Positive Real Numbers is Group Isomorphism | https://proofwiki.org/wiki/Logarithm_on_Positive_Real_Numbers_is_Group_Isomorphism | https://proofwiki.org/wiki/Logarithm_on_Positive_Real_Numbers_is_Group_Isomorphism | [
"Examples of Group Isomorphisms",
"Logarithms"
] | [
"Definition:Multiplicative Group of Positive Real Numbers",
"Definition:Additive Group of Real Numbers",
"Definition:Real Number",
"Definition:Mapping",
"Definition:General Logarithm",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Sum of Logarithms",
"Definition:Group Homomorphism",
"Change of Base of Logarithm",
"Definition:Natural Logarithm",
"Logarithm is Strictly Increasing",
"Strictly Monotone Real Function is Bijective",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Group Homomorphism",
"Definition:Isom... |
proofwiki-4436 | Kernel is Trivial iff Monomorphism/Group | Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a group homomorphism.
Let $\map \ker \phi$ be the kernel of $\phi$.
Then $\phi$ is a group monomorphism {{iff}} $\map \ker \phi$ is trivial. | === Necessary Condition ===
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a group monomorphism.
By Homomorphism to Group Preserves Identity, $e_S \in \map \ker \phi$.
If $\map \ker \phi$ contained another element $s \ne e_S$, then $\map \phi s = \map \phi {e_S} = e_T$ and $\phi$ would not be injective, thus not ... | Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a [[Definition:Group Homomorphism|group homomorphism]].
Let $\map \ker \phi$ be the [[Definition:Kernel of Group Homomorphism|kernel]] of $\phi$.
Then $\phi$ is a [[Definition:Group Monomorphism|group monomorphism]] {{iff}} $\map \ker \phi$ is [[Definition:Trivial... | === Necessary Condition ===
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a [[Definition:Group Monomorphism|group monomorphism]].
By [[Homomorphism to Group Preserves Identity]], $e_S \in \map \ker \phi$.
If $\map \ker \phi$ contained another element $s \ne e_S$, then $\map \phi s = \map \phi {e_S} = e_T$ and... | Kernel is Trivial iff Monomorphism/Group | https://proofwiki.org/wiki/Kernel_is_Trivial_iff_Monomorphism/Group | https://proofwiki.org/wiki/Kernel_is_Trivial_iff_Monomorphism/Group | [
"Kernel is Trivial iff Monomorphism",
"Group Monomorphisms"
] | [
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Group Monomorphism",
"Definition:Trivial Group"
] | [
"Definition:Group Monomorphism",
"Homomorphism to Group Preserves Identity",
"Definition:Injection",
"Definition:Group Monomorphism",
"Definition:Trivial Subgroup",
"Definition:Injection",
"Definition:Group Monomorphism"
] |
proofwiki-4437 | Kernel is Trivial iff Monomorphism/Ring | Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism.
Let $\map \ker \phi$ be the kernel of $\phi$.
Then $\phi$ is a ring monomorphism {{iff}} $\map \ker \phi = 0_{R_1}$. | The proof for the ring monomorphism follows directly from:
:Ring Homomorphism of Addition is Group Homomorphism
and:
:Kernel is Trivial iff Group Monomorphism for the group monomorphism.
{{qed}} | Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
Let $\map \ker \phi$ be the [[Definition:Kernel of Ring Homomorphism|kernel]] of $\phi$.
Then $\phi$ is a [[Definition:Ring Monomorphism|ring monomorphism]] {{iff}} $\map \ker \phi = 0_{R_... | The proof for the [[Definition:Ring Monomorphism|ring monomorphism]] follows directly from:
:[[Ring Homomorphism of Addition is Group Homomorphism]]
and:
:[[Kernel is Trivial iff Group Monomorphism]] for the [[Definition:Group Monomorphism|group monomorphism]].
{{qed}} | Kernel is Trivial iff Monomorphism/Ring | https://proofwiki.org/wiki/Kernel_is_Trivial_iff_Monomorphism/Ring | https://proofwiki.org/wiki/Kernel_is_Trivial_iff_Monomorphism/Ring | [
"Kernel is Trivial iff Monomorphism",
"Kernels of Ring Homomorphisms",
"Ring Monomorphisms"
] | [
"Definition:Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Definition:Ring Monomorphism"
] | [
"Definition:Ring Monomorphism",
"Ring Homomorphism of Addition is Group Homomorphism",
"Kernel is Trivial iff Monomorphism/Group",
"Definition:Group Monomorphism"
] |
proofwiki-4438 | Exponential on Real Numbers is Group Isomorphism | Let $\struct {\R, +}$ be the additive group of real numbers.
Let $\struct {\R_{> 0}, \times}$ be the multiplicative group of positive real numbers.
Let $\exp: \struct {\R, +} \to \struct {\R_{> 0}, \times}$ be the mapping:
:$x \mapsto \map \exp x$
where $\exp$ is the exponential function.
Then $\exp$ is a group isomorp... | From Exponential of Sum we have:
:$\forall x, y \in \R: \map \exp {x + y} = \exp x \cdot \exp y$
That is, $\exp$ is a group homomorphism.
Then we have that Exponential is Strictly Increasing.
From Strictly Monotone Real Function is Bijective, it follows that $\exp$ is a bijection.
So $\exp$ is a bijective group homomor... | Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $\struct {\R_{> 0}, \times}$ be the [[Definition:Multiplicative Group of Positive Real Numbers|multiplicative group of positive real numbers]].
Let $\exp: \struct {\R, +} \to \struct {\R_{> 0}, \times}$ be ... | From [[Exponential of Sum]] we have:
:$\forall x, y \in \R: \map \exp {x + y} = \exp x \cdot \exp y$
That is, $\exp$ is a [[Definition:Group Homomorphism|group homomorphism]].
Then we have that [[Exponential is Strictly Increasing]].
From [[Strictly Monotone Real Function is Bijective]], it follows that $\exp$ is a ... | Exponential on Real Numbers is Group Isomorphism/Proof 1 | https://proofwiki.org/wiki/Exponential_on_Real_Numbers_is_Group_Isomorphism | https://proofwiki.org/wiki/Exponential_on_Real_Numbers_is_Group_Isomorphism/Proof_1 | [
"Examples of Group Isomorphisms",
"Exponential Function",
"Exponential on Real Numbers is Group Isomorphism"
] | [
"Definition:Additive Group of Real Numbers",
"Definition:Multiplicative Group of Positive Real Numbers",
"Definition:Mapping",
"Definition:Exponential Function/Real",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Exponential of Sum",
"Definition:Group Homomorphism",
"Exponential is Strictly Increasing",
"Strictly Monotone Real Function is Bijective",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Group Homomorphism",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] |
proofwiki-4439 | Exponential on Real Numbers is Group Isomorphism | Let $\struct {\R, +}$ be the additive group of real numbers.
Let $\struct {\R_{> 0}, \times}$ be the multiplicative group of positive real numbers.
Let $\exp: \struct {\R, +} \to \struct {\R_{> 0}, \times}$ be the mapping:
:$x \mapsto \map \exp x$
where $\exp$ is the exponential function.
Then $\exp$ is a group isomorp... | From Real Numbers under Addition form Group, $\struct {\R, +}$ is a group.
From Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group, $\struct {\R_{> 0}, \times}$ is a group.
We have that for all $y \in R_{> 0}$ there exists $x = \map \ln y \in R$ which satisfies $\map \exp x = y$.
{{expla... | Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Let $\struct {\R_{> 0}, \times}$ be the [[Definition:Multiplicative Group of Positive Real Numbers|multiplicative group of positive real numbers]].
Let $\exp: \struct {\R, +} \to \struct {\R_{> 0}, \times}$ be ... | From [[Real Numbers under Addition form Group]], $\struct {\R, +}$ is a [[Definition:Group|group]].
From [[Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group]], $\struct {\R_{> 0}, \times}$ is a [[Definition:Group|group]].
We have that for all $y \in R_{> 0}$ there exists $x = \map \ln... | Exponential on Real Numbers is Group Isomorphism/Proof 2 | https://proofwiki.org/wiki/Exponential_on_Real_Numbers_is_Group_Isomorphism | https://proofwiki.org/wiki/Exponential_on_Real_Numbers_is_Group_Isomorphism/Proof_2 | [
"Examples of Group Isomorphisms",
"Exponential Function",
"Exponential on Real Numbers is Group Isomorphism"
] | [
"Definition:Additive Group of Real Numbers",
"Definition:Multiplicative Group of Positive Real Numbers",
"Definition:Mapping",
"Definition:Exponential Function/Real",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Real Numbers under Addition form Group",
"Definition:Group",
"Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group",
"Definition:Group",
"Definition:Surjection",
"Exponential on Real Numbers is Injection",
"Definition:Injection",
"Definition:Bijection",
"Exponential of... |
proofwiki-4440 | First Isomorphism Theorem/Groups | Let $\phi: G_1 \to G_2$ be a group homomorphism.
Let $\map \ker \phi$ be the kernel of $\phi$.
Then:
:$\Img \phi \cong G_1 / \map \ker \phi$
where $\cong$ denotes group isomorphism. | Let $K = \map \ker \phi$.
By Kernel is Normal Subgroup of Domain, $G_1 / K$ exists.
We need to establish that the mapping $\theta: G_1 / K \to G_2$ defined as:
:$\forall x \in G_1: \map \theta {x K} = \map \phi x$
is well-defined.
That is, we need to ensure that:
:$\forall x, y \in G_1: x K = y K \implies \map \theta {... | Let $\phi: G_1 \to G_2$ be a [[Definition:Group Homomorphism|group homomorphism]].
Let $\map \ker \phi$ be the [[Definition:Kernel of Group Homomorphism|kernel]] of $\phi$.
Then:
:$\Img \phi \cong G_1 / \map \ker \phi$
where $\cong$ denotes [[Definition:Group Isomorphism|group isomorphism]]. | Let $K = \map \ker \phi$.
By [[Kernel is Normal Subgroup of Domain]], $G_1 / K$ exists.
We need to establish that the [[Definition:Mapping|mapping]] $\theta: G_1 / K \to G_2$ defined as:
:$\forall x \in G_1: \map \theta {x K} = \map \phi x$
is [[Definition:Well-Defined Mapping|well-defined]].
That is, we need to ens... | First Isomorphism Theorem/Groups | https://proofwiki.org/wiki/First_Isomorphism_Theorem/Groups | https://proofwiki.org/wiki/First_Isomorphism_Theorem/Groups | [
"First Isomorphism Theorem",
"Group Isomorphisms",
"Group Homomorphisms"
] | [
"Definition:Group Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Kernel is Normal Subgroup of Domain",
"Definition:Mapping",
"Definition:Well-Defined/Mapping",
"Left Cosets are Equal iff Product with Inverse in Subgroup",
"Definition:Injection",
"Definition:Group Homomorphism",
"Definition:Group Monomorphism",
"Definition:Image (Set Theory)/Mapping/Mapping"
] |
proofwiki-4441 | First Isomorphism Theorem/Rings | Let $\phi: R \to S$ be a ring homomorphism.
Let $\map \ker \phi$ be the kernel of $\phi$.
Then:
:$\Img \phi \cong R / \map \ker \phi$
where $\cong$ denotes ring isomorphism. | From Ring Homomorphism whose Kernel contains Ideal, let $J = \map \ker \phi$.
This gives the ring homomorphism $\mu: R / \map \ker \phi \to S$ as follows:
::<nowiki>$\begin {xy} \xymatrix@L + 2mu@ + 1em {
R \ar[rr]^*{\nu}
\ar[rdrd]_*{\phi}
& & R / \map \ker \phi \ar[dd]_*{\mu} \\
\\
& & S
} \end {xy}$</nowiki>
Tha... | Let $\phi: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
Let $\map \ker \phi$ be the [[Definition:Kernel of Ring Homomorphism|kernel]] of $\phi$.
Then:
:$\Img \phi \cong R / \map \ker \phi$
where $\cong$ denotes [[Definition:Ring Isomorphism|ring isomorphism]]. | From [[Ring Homomorphism whose Kernel contains Ideal]], let $J = \map \ker \phi$.
This gives the [[Definition:Ring Homomorphism|ring homomorphism]] $\mu: R / \map \ker \phi \to S$ as follows:
::<nowiki>$\begin {xy} \xymatrix@L + 2mu@ + 1em {
R \ar[rr]^*{\nu}
\ar[rdrd]_*{\phi}
& & R / \map \ker \phi \ar[dd]_*{\mu... | First Isomorphism Theorem/Rings | https://proofwiki.org/wiki/First_Isomorphism_Theorem/Rings | https://proofwiki.org/wiki/First_Isomorphism_Theorem/Rings | [
"First Ring Isomorphism Theorem",
"Ring Homomorphisms",
"Ring Isomorphisms",
"First Isomorphism Theorem"
] | [
"Definition:Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism"
] | [
"Ring Homomorphism whose Kernel contains Ideal",
"Definition:Ring Homomorphism",
"Definition:Null Ring",
"Quotient Ring Defined by Ring Itself is Null Ring",
"Kernel is Trivial iff Monomorphism",
"Definition:Ring Monomorphism",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism"
] |
proofwiki-4442 | Second Isomorphism Theorem/Groups | Let $G$ be a group, and let:
:$(1): \quad H$ be a subgroup of $G$
:$(2): \quad N$ be a normal subgroup of $G$.
Then:
:$\dfrac H {H \cap N} \cong \dfrac {H N} N$
where $\cong$ denotes group isomorphism. | The fact that $N$ is normal, together with Intersection with Normal Subgroup is Normal, gives us that $N \cap H \lhd H$.
Also, $N \lhd N H = \gen {H, N}$ follows from Subset Product with Normal Subgroup as Generator.
Now we define a mapping $\phi: H \to H N / N$ by the rule:
:$\map \phi h = h N$
Note that $N$ need not ... | Let $G$ be a [[Definition:Group|group]], and let:
:$(1): \quad H$ be a [[Definition:Subgroup|subgroup]] of $G$
:$(2): \quad N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Then:
:$\dfrac H {H \cap N} \cong \dfrac {H N} N$
where $\cong$ denotes [[Definition:Group Isomorphism|group isomorphism]]. | The fact that $N$ is [[Definition:Normal Subgroup|normal]], together with [[Intersection with Normal Subgroup is Normal]], gives us that $N \cap H \lhd H$.
Also, $N \lhd N H = \gen {H, N}$ follows from [[Subset Product with Normal Subgroup as Generator]].
Now we define a [[Definition:Mapping|mapping]] $\phi: H \to H... | Second Isomorphism Theorem/Groups | https://proofwiki.org/wiki/Second_Isomorphism_Theorem/Groups | https://proofwiki.org/wiki/Second_Isomorphism_Theorem/Groups | [
"Group Isomorphisms",
"Normal Subgroups",
"Isomorphism Theorems"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Normal Subgroup",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Normal Subgroup",
"Intersection with Normal Subgroup is Normal",
"Subset Product with Normal Subgroup as Generator",
"Definition:Mapping",
"Definition:Subset",
"Definition:Coset",
"Definition:Group Homomorphism",
"Definition:Surjection",
"First Isomorphism Theorem/Groups"
] |
proofwiki-4443 | Second Isomorphism Theorem/Rings | Let $R$ be a ring, and let:
:$S$ be a subring of $R$
:$J$ be an ideal of $R$.
Then:
:$(1): \quad S + J$ is a subring of $R$
:$(2): \quad J$ is an ideal of $S + J$
:$(3): \quad S \cap J$ is an ideal of $S$
:$(4): \quad \dfrac S {S \cap J} \cong \dfrac {S + J} J$
where $\cong$ denotes group isomorphism. | The relations being defined can be illustrated by this commutative diagram:
:620px
$(1): \quad S + J$ is a subring of $R$
From Sum of All Ring Products is Additive Subgroup, $S + J$ is an additive subgroup of $R$.
Suppose $s, s' \in S, j, j' \in J$.
Then:
:$\paren {s + j} \paren {s' + j'}$
{{begin-eqn}}
{{eqn | l = \pa... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]], and let:
:$S$ be a [[Definition:Subring|subring]] of $R$
:$J$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
Then:
:$(1): \quad S + J$ is a [[Definition:Subring|subring]] of $R$
:$(2): \quad J$ is an [[Definition:Ideal of Ring|ideal]] of $S + J$
:$(3): \qu... | The relations being defined can be illustrated by this [[Definition:Commutative Diagram|commutative diagram]]:
:[[File:CommDiagSecondIsomTheorem.png|620px]]
$(1): \quad S + J$ is a [[Definition:Subring|subring]] of $R$
From [[Sum of All Ring Products is Additive Subgroup]], $S + J$ is an [[Definition:Additive Subgr... | Second Isomorphism Theorem/Rings | https://proofwiki.org/wiki/Second_Isomorphism_Theorem/Rings | https://proofwiki.org/wiki/Second_Isomorphism_Theorem/Rings | [
"Second Ring Isomorphism Theorem",
"Ring Isomorphisms",
"Ideal Theory",
"Isomorphism Theorems"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Subring",
"Definition:Ideal of Ring",
"Definition:Subring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Commutative Diagram",
"File:CommDiagSecondIsomTheorem.png",
"Definition:Subring",
"Sum of All Ring Products is Additive Subgroup",
"Definition:Additive Subgroup",
"Definition:Ideal of Ring",
"Subring Test",
"Definition:Subring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
... |
proofwiki-4444 | Third Isomorphism Theorem/Groups | Let $G$ be a group, and let:
:$H, N$ be normal subgroups of $G$
:$N$ be a subset of $H$.
Then:
:$(1): \quad N$ is a normal subgroup of $H$
:$(2): \quad H / N$ is a normal subgroup of $G / N$
:::where $H / N$ denotes the quotient group of $H$ by $N$
:$(3): \quad \dfrac {G / N} {H / N} \cong \dfrac G H$
:::where $\cong$ ... | From Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup, $N$ is a normal subgroup of $H$.
We define a mapping:
:$\phi: G / N \to G / H$ by $\map \phi {g N} = g H$
Since $\phi$ is defined on cosets, we need to check that $\phi$ is well-defined.
Suppose $x N = y N$.
Then:
:$y^{-1} x \in N$
Then:
:$N... | Let $G$ be a [[Definition:Group|group]], and let:
:$H, N$ be [[Definition:Normal Subgroup|normal subgroups]] of $G$
:$N$ be a [[Definition:Subset|subset]] of $H$.
Then:
:$(1): \quad N$ is a [[Definition:Normal Subgroup|normal subgroup]] of $H$
:$(2): \quad H / N$ is a [[Definition:Normal Subgroup|normal subgroup]] ... | From [[Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup]], $N$ is a [[Definition:Normal Subgroup|normal subgroup]] of $H$.
We define a [[Definition:Mapping|mapping]]:
:$\phi: G / N \to G / H$ by $\map \phi {g N} = g H$
Since $\phi$ is defined on [[Definition:Coset|cosets]], we need to check th... | Third Isomorphism Theorem/Groups/Proof 1 | https://proofwiki.org/wiki/Third_Isomorphism_Theorem/Groups | https://proofwiki.org/wiki/Third_Isomorphism_Theorem/Groups/Proof_1 | [
"Group Isomorphisms",
"Quotient Groups",
"Normal Subgroups",
"Isomorphism Theorems",
"Third Isomorphism Theorem"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Subset",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup",
"Definition:Normal Subgroup",
"Definition:Mapping",
"Definition:Coset",
"Definition:Well-Defined/Mapping",
"Definition:Group Homomorphism",
"Definition:Surjection",
"First Isomorphism Theorem/Groups"
] |
proofwiki-4445 | Third Isomorphism Theorem/Groups | Let $G$ be a group, and let:
:$H, N$ be normal subgroups of $G$
:$N$ be a subset of $H$.
Then:
:$(1): \quad N$ is a normal subgroup of $H$
:$(2): \quad H / N$ is a normal subgroup of $G / N$
:::where $H / N$ denotes the quotient group of $H$ by $N$
:$(3): \quad \dfrac {G / N} {H / N} \cong \dfrac G H$
:::where $\cong$ ... | From Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup, $N$ is a normal subgroup of $H$.
Let $q_H$ and $q_N$ be the quotient mappings from $G$ to $\dfrac G H$ and $G$ to $\dfrac G N$ respectively.
Hence:
:$N \subseteq \map \ker {q_H}$
{{explain|This is confusing me: having difficulty identifying ... | Let $G$ be a [[Definition:Group|group]], and let:
:$H, N$ be [[Definition:Normal Subgroup|normal subgroups]] of $G$
:$N$ be a [[Definition:Subset|subset]] of $H$.
Then:
:$(1): \quad N$ is a [[Definition:Normal Subgroup|normal subgroup]] of $H$
:$(2): \quad H / N$ is a [[Definition:Normal Subgroup|normal subgroup]] ... | From [[Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup]], $N$ is a [[Definition:Normal Subgroup|normal subgroup]] of $H$.
Let $q_H$ and $q_N$ be the [[Definition:Quotient Mapping|quotient mappings]] from $G$ to $\dfrac G H$ and $G$ to $\dfrac G N$ respectively.
Hence:
:$N \subseteq \map \ker ... | Third Isomorphism Theorem/Groups/Proof 2 | https://proofwiki.org/wiki/Third_Isomorphism_Theorem/Groups | https://proofwiki.org/wiki/Third_Isomorphism_Theorem/Groups/Proof_2 | [
"Group Isomorphisms",
"Quotient Groups",
"Normal Subgroups",
"Isomorphism Theorems",
"Third Isomorphism Theorem"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Subset",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup",
"Definition:Normal Subgroup",
"Definition:Quotient Mapping",
"Quotient Theorem for Group Homomorphisms/Corollary 2",
"Definition:Group Epimorphism",
"Definition:Group Epimorphism",
"Definition:Composition of Mappings",
"Defini... |
proofwiki-4446 | Third Isomorphism Theorem/Groups | Let $G$ be a group, and let:
:$H, N$ be normal subgroups of $G$
:$N$ be a subset of $H$.
Then:
:$(1): \quad N$ is a normal subgroup of $H$
:$(2): \quad H / N$ is a normal subgroup of $G / N$
:::where $H / N$ denotes the quotient group of $H$ by $N$
:$(3): \quad \dfrac {G / N} {H / N} \cong \dfrac G H$
:::where $\cong$ ... | From Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup, $N$ is a normal subgroup of $H$.
Let $q_H$ denote the quotient mapping from $G$ to $\dfrac G H$.
Let $q_N$ denote the quotient mapping from $G$ to $\dfrac G N$.
Let $\RR$ be the congruence relation defined by $N$ in $G$.
Let $\TT$ be the con... | Let $G$ be a [[Definition:Group|group]], and let:
:$H, N$ be [[Definition:Normal Subgroup|normal subgroups]] of $G$
:$N$ be a [[Definition:Subset|subset]] of $H$.
Then:
:$(1): \quad N$ is a [[Definition:Normal Subgroup|normal subgroup]] of $H$
:$(2): \quad H / N$ is a [[Definition:Normal Subgroup|normal subgroup]] ... | From [[Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup]], $N$ is a [[Definition:Normal Subgroup|normal subgroup]] of $H$.
Let $q_H$ denote the [[Definition:Quotient Mapping|quotient mapping]] from $G$ to $\dfrac G H$.
Let $q_N$ denote the [[Definition:Quotient Mapping|quotient mapping]] from ... | Third Isomorphism Theorem/Groups/Proof 3 | https://proofwiki.org/wiki/Third_Isomorphism_Theorem/Groups | https://proofwiki.org/wiki/Third_Isomorphism_Theorem/Groups/Proof_3 | [
"Group Isomorphisms",
"Quotient Groups",
"Normal Subgroups",
"Isomorphism Theorems",
"Third Isomorphism Theorem"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Subset",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup",
"Definition:Normal Subgroup",
"Definition:Quotient Mapping",
"Definition:Quotient Mapping",
"Definition:Congruence Modulo Subgroup",
"Definition:Congruence Modulo Subgroup",
"Congruence Relation induces Normal Subgroup",
"Defi... |
proofwiki-4447 | Third Isomorphism Theorem/Rings | Let $R$ be a ring.
Let:
:$J, K$ be ideals of $R$
:$J$ be a subset of $K$.
Then:
:$(1): \quad K / J$ is an ideal of $R / J$
::where $K / J$ denotes the quotient ring of $K$ by $J$
:$(2): \quad \dfrac {R / J} {K / J} \cong \dfrac R K$
::where $\cong$ denotes ring isomorphism. | By Ideal Contained in Larger Ideal is Ideal of Larger Ideal, $J$ is indeed an ideal of $K$.
Hence $K / J$ is adequately defined.
In Ring Homomorphism whose Kernel contains Ideal, take $\phi: R \to R / K$ to be the quotient epimorphism.
Then (from the same source) its kernel is $K$.
Thus we have that:
:$\phi = \psi \cir... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let:
:$J, K$ be [[Definition:Ideal of Ring|ideals]] of $R$
:$J$ be a [[Definition:Subset|subset]] of $K$.
Then:
:$(1): \quad K / J$ is an [[Definition:Ideal of Ring|ideal]] of $R / J$
::where $K / J$ denotes the [[Definition:Quotient Ring|quotient ring]] of $... | By [[Ideal Contained in Larger Ideal is Ideal of Larger Ideal]], $J$ is indeed an [[Definition:Ideal of Ring|ideal]] of $K$.
Hence $K / J$ is adequately defined.
In [[Ring Homomorphism whose Kernel contains Ideal]], take $\phi: R \to R / K$ to be the [[Definition:Quotient Ring Epimorphism|quotient epimorphism]].
Th... | Third Isomorphism Theorem/Rings | https://proofwiki.org/wiki/Third_Isomorphism_Theorem/Rings | https://proofwiki.org/wiki/Third_Isomorphism_Theorem/Rings | [
"Third Ring Isomorphism Theorem",
"Ring Isomorphisms",
"Ideal Theory",
"Isomorphism Theorems"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring",
"Definition:Subset",
"Definition:Ideal of Ring",
"Definition:Quotient Ring",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism"
] | [
"Ideal Contained in Larger Ideal is Ideal of Larger Ideal",
"Definition:Ideal of Ring",
"Ring Homomorphism whose Kernel contains Ideal",
"Definition:Quotient Epimorphism/Ring",
"Definition:Kernel of Ring Homomorphism",
"Definition:Ring Homomorphism",
"Definition:Commutative Diagram",
"Definition:Ring ... |
proofwiki-4448 | Inertia Principle | Let $\sequence {a_n}$ be a sequence in $\R$.
Let $a_n \to l$ as $n \to \infty$.
Let $c \in \R$: such that $c < l$.
Then $\exists N \in \N$ such that:
: $\forall n \in \N: n \ge N \implies c < a_n$ | Pick $\epsilon = l - c > 0$ (as $c < l$).
As $a_n \to l$ as $n \to \infty$, then $\exists N \in \N$ such that:
:$\forall n \in \N: n \ge N \implies \size {a_n - l} < \epsilon$
Equivalently:
:$\forall n \in \N: n \ge N \implies \size {l - a_n} < l - c$
For each $a_n$, either $a_n \ge l$ or $a_n < l$.
If $a_n < l$, then ... | Let $\sequence {a_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Let $a_n \to l$ as $n \to \infty$.
Let $c \in \R$: such that $c < l$.
Then $\exists N \in \N$ such that:
: $\forall n \in \N: n \ge N \implies c < a_n$ | Pick $\epsilon = l - c > 0$ (as $c < l$).
As $a_n \to l$ as $n \to \infty$, then $\exists N \in \N$ such that:
:$\forall n \in \N: n \ge N \implies \size {a_n - l} < \epsilon$
Equivalently:
:$\forall n \in \N: n \ge N \implies \size {l - a_n} < l - c$
For each $a_n$, either $a_n \ge l$ or $a_n < l$.
If $a_n < l$, ... | Inertia Principle | https://proofwiki.org/wiki/Inertia_Principle | https://proofwiki.org/wiki/Inertia_Principle | [
"Convergence",
"Sequences",
"Named Theorems"
] | [
"Definition:Real Sequence"
] | [] |
proofwiki-4449 | Composite of Homomorphisms is Homomorphism/Algebraic Structure | Let:
:$\struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n}$
:$\struct {S_2, *_1, *_2, \ldots, *_n}$
:$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be algebraic structures.
Let:
:$\phi: \struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
:$\psi: \struct {S_2, *_1, ... | Let $\psi \circ \phi$ denote the composite of $\phi$ and $\psi$.
Then what we are trying to prove is denoted:
:$\paren {\psi \circ \phi}: \struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ is a homomorphism.
To prove the above is the case, we need to demonstr... | Let:
:$\struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n}$
:$\struct {S_2, *_1, *_2, \ldots, *_n}$
:$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be [[Definition:Algebraic Structure|algebraic structures]].
Let:
:$\phi: \struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n} \to \struct {S_2, *_1, *_2, \ldot... | Let $\psi \circ \phi$ denote the [[Definition:Composition of Mappings|composite]] of $\phi$ and $\psi$.
Then what we are trying to prove is denoted:
:$\paren {\psi \circ \phi}: \struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ is a [[Definition:Homomorphis... | Composite of Homomorphisms is Homomorphism/Algebraic Structure | https://proofwiki.org/wiki/Composite_of_Homomorphisms_is_Homomorphism/Algebraic_Structure | https://proofwiki.org/wiki/Composite_of_Homomorphisms_is_Homomorphism/Algebraic_Structure | [
"Composite of Homomorphisms is Homomorphism",
"Homomorphisms (Abstract Algebra)",
"Composite Mappings"
] | [
"Definition:Algebraic Structure",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Composition of Mappings",
"Definition:Homomorphism (Abstract Algebra)"
] | [
"Definition:Composition of Mappings",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Morphism Property",
"Definition:Morphism Property",
"Definition:Operation/Binary Operation",
"Definition:Homomorphism (Abstract Algebra)"
] |
proofwiki-4450 | Composite of Homomorphisms is Homomorphism/R-Algebraic Structure | Let $\struct {R, +_R, \times_R}$ be a ring.
Let:
:$\struct {S_1, \odot_1, \ldots, \odot_n}$
:$\struct {S_2, {\odot'}_1, \ldots, {\odot'}_n}$
:$\struct {S_3, {\odot' '}_1, \ldots, {\odot' '}_n}$
be algebraic structures each with $n$ operations.
Let:
:$\struct {S_1, *_1}_R$
:$\struct {S_2, *_2}_R$
:$\struct {S_3, *_3}_R$... | What we are trying to prove is that $\paren {\psi \circ \phi}: \struct {S_1, *_1}_R \to \struct {S_3, *_3}_R$ is a homomorphism.
That is:
:$(1): \quad \forall k \in \closedint 1 n: \forall x, y \in S_1: \map {\paren {\psi \circ \phi} } {x \odot_k y} = \map {\paren {\psi \circ \phi} } x {\odot' '}_k \map {\paren {\psi \... | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let:
:$\struct {S_1, \odot_1, \ldots, \odot_n}$
:$\struct {S_2, {\odot'}_1, \ldots, {\odot'}_n}$
:$\struct {S_3, {\odot' '}_1, \ldots, {\odot' '}_n}$
be [[Definition:Algebraic Structure|algebraic structures]] each with $n$ [[Definition:... | What we are trying to prove is that $\paren {\psi \circ \phi}: \struct {S_1, *_1}_R \to \struct {S_3, *_3}_R$ is a [[Definition:R-Algebraic Structure Homomorphism|homomorphism]].
That is:
:$(1): \quad \forall k \in \closedint 1 n: \forall x, y \in S_1: \map {\paren {\psi \circ \phi} } {x \odot_k y} = \map {\paren {\ps... | Composite of Homomorphisms is Homomorphism/R-Algebraic Structure | https://proofwiki.org/wiki/Composite_of_Homomorphisms_is_Homomorphism/R-Algebraic_Structure | https://proofwiki.org/wiki/Composite_of_Homomorphisms_is_Homomorphism/R-Algebraic_Structure | [
"Composite of Homomorphisms is Homomorphism",
"Homomorphisms (Abstract Algebra)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:R-Algebraic Structure",
"Definition:R-Algebraic Structure Homomorphism",
"Definition:Composition of Mappings",
"Definition:R-Algebraic Structure Homomorphism"
] | [
"Definition:R-Algebraic Structure Homomorphism",
"Definition:Closed Interval/Integer Interval",
"Definition:Morphism Property",
"Definition:Morphism Property",
"Definition:Operation/Binary Operation",
"Definition:Morphism Property",
"Definition:Morphism Property",
"Definition:Operation/Binary Operatio... |
proofwiki-4451 | Subgroup is Subgroup of Normalizer | Let $G$ be a group.
A subgroup $H \le G$ is a subgroup of its normalizer:
:$H \le G \implies H \le \map {N_G} H$ | === Subset of Normalizer ===
First we show that $H$ is a subset of $\map {N_G} H$.
This follows directly from Left Coset Equals Subgroup iff Element in Subgroup:
:$x \in H \implies x H = H$
As $x \in H \implies x^{-1} \in H$ it also follows that $x \in H \implies H x^{-1} = H$.
Thus:
:$x \in H \implies x H x^{-1} = H^x... | Let $G$ be a [[Definition:Group|group]].
A [[Definition:Subgroup|subgroup]] $H \le G$ is a [[Definition:Subgroup|subgroup]] of its [[Definition:Normalizer|normalizer]]:
:$H \le G \implies H \le \map {N_G} H$ | === Subset of Normalizer ===
First we show that $H$ is a [[Definition:Subset|subset]] of $\map {N_G} H$.
This follows directly from [[Left Coset Equals Subgroup iff Element in Subgroup]]:
:$x \in H \implies x H = H$
As $x \in H \implies x^{-1} \in H$ it also follows that $x \in H \implies H x^{-1} = H$.
Thus:
:$x \... | Subgroup is Subgroup of Normalizer | https://proofwiki.org/wiki/Subgroup_is_Subgroup_of_Normalizer | https://proofwiki.org/wiki/Subgroup_is_Subgroup_of_Normalizer | [
"Subgroups",
"Normalizers"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Normalizer"
] | [
"Definition:Subset",
"Left Coset Equals Subgroup iff Element in Subgroup"
] |
proofwiki-4452 | Inclusion Mappings to Topological Sum from Components | Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces.
Let $\struct {Z, \tau_3}$ be the topological sum of $X$ and $Y$ where $\tau_3$ is the topology generated by $\tau_1$ and $\tau_2$.
Then $\tau_3$ is the finest topology on $Z$ in which the inclusion mappings from $\struct {X, \tau_1}$ and $\struc... | {{Recall|Topological Sum|topological sum}}
{{:Definition:Topological Sum}}
Let $\phi_1$ and $\phi_2$ be continuous inclusion mappings from $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ to $\struct {Z, \tau_3}$ respectively:
{{begin-eqn}}
{{eqn | l = U \in \tau_3
| o = \implies
| r = {\phi_1}^{-1} \sqbrk U... | Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $\struct {Z, \tau_3}$ be the [[Definition:Topological Sum|topological sum]] of $X$ and $Y$ where $\tau_3$ is the [[Definition:Topology Generated by Synthetic Sub-Basis|topology generated]] by $\tau_1$ and $\... | {{Recall|Topological Sum|topological sum}}
{{:Definition:Topological Sum}}
Let $\phi_1$ and $\phi_2$ be [[Definition:Continuous Mapping (Topology)|continuous]] [[Definition:Inclusion Mapping|inclusion mappings]] from $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ to $\struct {Z, \tau_3}$ respectively:
{{begin-eqn}}
... | Inclusion Mappings to Topological Sum from Components | https://proofwiki.org/wiki/Inclusion_Mappings_to_Topological_Sum_from_Components | https://proofwiki.org/wiki/Inclusion_Mappings_to_Topological_Sum_from_Components | [
"Topological Sum",
"Inclusion Mappings",
"Finer Topology"
] | [
"Definition:Topological Space",
"Definition:Topological Sum",
"Definition:Topology Generated by Synthetic Sub-Basis",
"Definition:Finer Topology",
"Definition:Inclusion Mapping",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Continuous Mapping (Topology)",
"Definition:Inclusion Mapping",
"Definition:Coarser Topology",
"Definition:Element",
"Definition:Open Set/Topology",
"Definition:Finer Topology",
"Definition:Topology",
"Definition:Inclusion Mapping",
"Definition:Continuous Mapping (Topology)",
"Definiti... |
proofwiki-4453 | Principal Ultrafilter is All Sets Containing Cluster Point | Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
Let $\FF \subset \powerset S$ be a principal ultrafilter on $S$.
Let its cluster point be $x$.
Then $\FF$ is the set of all subsets $T$ of $S$ such that $x \in T$. | We have {{hypothesis}} that the cluster point of $\FF$ is $x$.
Let $A \subseteq S$ such that $x \in A$.
Then, by definition of relative complement:
:$x \notin \relcomp S A$
By definition of cluster point, $x$ is in every element of $\FF$.
But as $x \notin \relcomp S A$, it follows that:
:$\relcomp S A \notin \FF$
So by... | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ denote the [[Definition:Power Set|power set]] of $S$.
Let $\FF \subset \powerset S$ be a [[Definition:Principal Ultrafilter|principal ultrafilter]] on $S$.
Let its [[Definition:Cluster Point of Filter|cluster point]] be $x$.
Then $\FF$ is the [[Definition:Set ... | We have {{hypothesis}} that the [[Definition:Cluster Point of Filter|cluster point]] of $\FF$ is $x$.
Let $A \subseteq S$ such that $x \in A$.
Then, by definition of [[Definition:Relative Complement|relative complement]]:
:$x \notin \relcomp S A$
By definition of [[Definition:Cluster Point of Filter|cluster point]]... | Principal Ultrafilter is All Sets Containing Cluster Point | https://proofwiki.org/wiki/Principal_Ultrafilter_is_All_Sets_Containing_Cluster_Point | https://proofwiki.org/wiki/Principal_Ultrafilter_is_All_Sets_Containing_Cluster_Point | [
"Filter Theory"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Principal Ultrafilter",
"Definition:Cluster Point of Filter",
"Definition:Set of Sets"
] | [
"Definition:Cluster Point of Filter",
"Definition:Relative Complement",
"Definition:Cluster Point of Filter",
"Definition:Element",
"Ultrafilter Contains Set or Complement",
"Filter Contains no Disjoint Sets"
] |
proofwiki-4454 | Equivalence of Definitions of T4 Space | {{TFAE|def = T4 Space|view = $T_4$ space}}
Let $T = \struct {S, \tau}$ be a topological space. | === Definition by Open Sets implies Definition by Closed Neighborhoods ===
Let $T$ satisfy the definition by open sets of a $T_4$ space.
Let $A$ be a closed set in $T$, and let $U_A$ be an open neighborhood of $A$.
By definition of open neighborhood:
:$A \subseteq U_A$
Let $B := \relcomp S {U_A}$.
By Intersection with ... | {{TFAE|def = T4 Space|view = $T_4$ space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]]. | === Definition by Open Sets implies Definition by Closed Neighborhoods ===
Let $T$ satisfy the [[Definition:T4 Space/Definition 1|definition by open sets of a $T_4$ space]].
Let $A$ be a [[Definition:Closed Set (Topology)|closed set]] in $T$, and let $U_A$ be an [[Definition:Open Neighborhood|open neighborhood]] of ... | Equivalence of Definitions of T4 Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T4_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T4_Space | [
"T4 Spaces"
] | [
"Definition:Topological Space"
] | [
"Definition:T4 Space/Definition 1",
"Definition:Closed Set/Topology",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Intersection with Complement is Empty iff Subset",
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Disjoint Sets",
"Definition:Closed... |
proofwiki-4455 | Product Space is T2 iff Factor Spaces are T2 | :$T$ is a $T_2$ (Hausdorff) space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_2$ (Hausdorff) space. | === Sufficient Condition ===
{{:Product Space is T2 iff Factor Spaces are T2/Sufficient Condition/Proof 1}}{{qed|lemma}} | :$T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | === [[Product Space is T2 iff Factor Spaces are T2/Sufficient Condition|Sufficient Condition]] ===
{{:Product Space is T2 iff Factor Spaces are T2/Sufficient Condition/Proof 1}}{{qed|lemma}} | Product Space is T2 iff Factor Spaces are T2 | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2 | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2 | [
"Product Space is T2 iff Factor Spaces are T2",
"Hausdorff Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:T2 Space",
"Definition:T2 Space"
] | [
"Product Space is T2 iff Factor Spaces are T2/Sufficient Condition"
] |
proofwiki-4456 | Product Space is T2 iff Factor Spaces are T2 | :$T$ is a $T_2$ (Hausdorff) space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_2$ (Hausdorff) space. | Let $T$ be a $T_2$ space.
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the axiom of choice.
From Subspace of Product Space is Homeomorphic to Factor Space, $\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to a subspace $T_\alpha$ of $T$.
From $T_2$ property is hereditary, $T_\alpha$ is a $T_2$ space.
From $T_2$ Pro... | :$T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | Let $T$ be a [[Definition:T2 Space|$T_2$ space]].
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the [[Axiom:Axiom of Choice|axiom of choice]].
From [[Subspace of Product Space is Homeomorphic to Factor Space]], $\struct {S_\alpha, \tau_\alpha}$ is [[Definition:Homeomorphism (Topological Spaces)|homeomorphic]] to a ... | Product Space is T2 iff Factor Spaces are T2/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2 | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Sufficient_Condition/Proof_1 | [
"Product Space is T2 iff Factor Spaces are T2",
"Hausdorff Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:T2 Space",
"Definition:T2 Space"
] | [
"Definition:T2 Space",
"Axiom:Axiom of Choice",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subspace",
"T2 Property is Hereditary",
"Definition:T2 Space",
"T2 Property is Preserved under Homeomorphism",
"Definition:T2 Space"... |
proofwiki-4457 | Product Space is T2 iff Factor Spaces are T2 | :$T$ is a $T_2$ (Hausdorff) space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_2$ (Hausdorff) space. | Let $\alpha \in I$.
Let $x, y \in S_\alpha$.
By the Axiom of Choice, there exists $z \in S$.
Define $x', y' \in S$ by:
:$x'_\beta = \begin{cases} z_\beta & \beta \ne \alpha \\ x & \beta = \alpha \end{cases}$
and
:$y'_\beta = \begin{cases} z_\beta & \beta \ne \alpha \\ y & \beta = \alpha \end{cases}$
By definition of a ... | :$T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | Let $\alpha \in I$.
Let $x, y \in S_\alpha$.
By the [[Axiom:Axiom of Choice|Axiom of Choice]], there exists $z \in S$.
Define $x', y' \in S$ by:
:$x'_\beta = \begin{cases} z_\beta & \beta \ne \alpha \\ x & \beta = \alpha \end{cases}$
and
:$y'_\beta = \begin{cases} z_\beta & \beta \ne \alpha \\ y & \beta = \alpha \en... | Product Space is T2 iff Factor Spaces are T2/Sufficient Condition/Proof 2 | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2 | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Sufficient_Condition/Proof_2 | [
"Product Space is T2 iff Factor Spaces are T2",
"Hausdorff Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:T2 Space",
"Definition:T2 Space"
] | [
"Axiom:Axiom of Choice",
"Definition:T2 Space",
"Definition:Topology",
"Definition:Product Space (Topology)",
"Definition:Product Topology",
"Definition:Product Topology",
"Definition:Product Topology/Natural Basis",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Product Topology",
"De... |
proofwiki-4458 | Product Space is T2.5 iff Factor Spaces are T2.5 | :$T$ is a $T_{2 \frac 1 2}$ space {{iff}} each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_{2 \frac 1 2}$ space. | === Sufficient Condition ===
{{:Product Space is T2.5 iff Factor Spaces are T2.5/Sufficient Condition}}{{qed|lemma}} | :$T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]] {{iff}} each of $\struct{S_\alpha, \tau_\alpha}$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | === [[Product Space is T2.5 iff Factor Spaces are T2.5/Sufficient Condition|Sufficient Condition]] ===
{{:Product Space is T2.5 iff Factor Spaces are T2.5/Sufficient Condition}}{{qed|lemma}} | Product Space is T2.5 iff Factor Spaces are T2.5 | https://proofwiki.org/wiki/Product_Space_is_T2.5_iff_Factor_Spaces_are_T2.5 | https://proofwiki.org/wiki/Product_Space_is_T2.5_iff_Factor_Spaces_are_T2.5 | [
"Product Space is T2.5 iff Factor Spaces are T2.5",
"T2.5 Spaces",
"Separation Properties Preserved under Topological Product",
"Product Topology"
] | [
"Definition:T2.5 Space",
"Definition:T2.5 Space"
] | [
"Product Space is T2.5 iff Factor Spaces are T2.5/Sufficient Condition"
] |
proofwiki-4459 | Product Space is T3 iff Factor Spaces are T3 | :$T$ is a $T_3$ space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_3$ space. | === Sufficient Condition ===
{{:Product Space is T3 iff Factor Spaces are T3/Sufficient Condition}}{{qed|lemma}} | :$T$ is a [[Definition:T3 Space|$T_3$ space]] {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a [[Definition:T3 Space|$T_3$ space]]. | === [[Product Space is T3 iff Factor Spaces are T3/Sufficient Condition|Sufficient Condition]] ===
{{:Product Space is T3 iff Factor Spaces are T3/Sufficient Condition}}{{qed|lemma}} | Product Space is T3 iff Factor Spaces are T3 | https://proofwiki.org/wiki/Product_Space_is_T3_iff_Factor_Spaces_are_T3 | https://proofwiki.org/wiki/Product_Space_is_T3_iff_Factor_Spaces_are_T3 | [
"Product Space is T3 iff Factor Spaces are T3",
"T3 Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:T3 Space",
"Definition:T3 Space"
] | [
"Product Space is T3 iff Factor Spaces are T3/Sufficient Condition"
] |
proofwiki-4460 | Product Space is T3.5 iff Factor Spaces are T3.5 | :$T$ is a $T_{3 \frac 1 2}$ space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_{3 \frac 1 2}$ space. | === Sufficient Condition ===
{{:Product Space is T3.5 iff Factor Spaces are T3.5/Sufficient Condition}}{{qed|lemma}} | :$T$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]] {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | === [[Product Space is T3.5 iff Factor Spaces are T3.5/Sufficient Condition|Sufficient Condition]] ===
{{:Product Space is T3.5 iff Factor Spaces are T3.5/Sufficient Condition}}{{qed|lemma}} | Product Space is T3.5 iff Factor Spaces are T3.5 | https://proofwiki.org/wiki/Product_Space_is_T3.5_iff_Factor_Spaces_are_T3.5 | https://proofwiki.org/wiki/Product_Space_is_T3.5_iff_Factor_Spaces_are_T3.5 | [
"Product Space is T3.5 iff Factor Spaces are T3.5",
"T3.5 Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:T3.5 Space",
"Definition:T3.5 Space"
] | [
"Product Space is T3.5 iff Factor Spaces are T3.5/Sufficient Condition"
] |
proofwiki-4461 | Factor Spaces are T4 if Product Space is T4 | Let $\SS = \family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$.
Let $\ds T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a $T_4$ space.... | Let $T$ be a $T_4$ space.
Since $S_\alpha \ne \O$ we also have $S \ne \O$ by the axiom of choice.
Let $z \in S$.
From Subspace of Product Space is Homeomorphic to Factor Space, every $\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to the subspace $T_\alpha$ of $T$ defined by:
:$T_\alpha = \set {x \in S: \forall \beta... | Let $\SS = \family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty|non-empty]] [[Definition:Topological Space|topological spaces]] for $\alpha$ in some [[Definition:Indexing Set|indexing set]] $I$.
Let $\ds T = \struct {S, \tau} = \... | Let $T$ be a [[Definition:T4 Space|$T_4$ space]].
Since $S_\alpha \ne \O$ we also have $S \ne \O$ by the [[Axiom:Axiom of Choice|axiom of choice]].
Let $z \in S$.
From [[Subspace of Product Space is Homeomorphic to Factor Space]], every $\struct {S_\alpha, \tau_\alpha}$ is [[Definition:Homeomorphism|homeomorphic]] t... | Factor Spaces are T4 if Product Space is T4 | https://proofwiki.org/wiki/Factor_Spaces_are_T4_if_Product_Space_is_T4 | https://proofwiki.org/wiki/Factor_Spaces_are_T4_if_Product_Space_is_T4 | [
"T4 Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T4 Space",
"Definition:T4 Space"
] | [
"Definition:T4 Space",
"Axiom:Axiom of Choice",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism",
"Definition:Homeomorphism",
"Inclusion Mapping is Continuous",
"Definition:Inclusion Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Composite of... |
proofwiki-4462 | Factor Spaces are T5 if Product Space is T5 | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$.
Let $\ds T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a $T_5$ space.
Then eac... | Let $T$ be a $T_5$ space.
Let $\struct {S_\alpha, \tau_\alpha}$ be arbitrary.
By Subspace of Product Space is Homeomorphic to Factor Space:
:$\struct {S_\alpha, \tau_\alpha}$
is homeomorphic to a subspace of $T$.
By $T_5$ Property is Hereditary, this subspace is also $T_5$.
Finally, by $T_5$ Property is Preserved under... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] for $\alpha$ in some [[Definition:Indexing Set|indexing set]] $I$.
Let $\ds T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \stru... | Let $T$ be a [[Definition:T5 Space|$T_5$ space]].
Let $\struct {S_\alpha, \tau_\alpha}$ be arbitrary.
By [[Subspace of Product Space is Homeomorphic to Factor Space]]:
:$\struct {S_\alpha, \tau_\alpha}$
is [[Definition:Homeomorphism|homeomorphic]] to a [[Definition:Topological Subspace|subspace]] of $T$.
By [[T5 Pro... | Factor Spaces are T5 if Product Space is T5 | https://proofwiki.org/wiki/Factor_Spaces_are_T5_if_Product_Space_is_T5 | https://proofwiki.org/wiki/Factor_Spaces_are_T5_if_Product_Space_is_T5 | [
"T5 Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T5 Space",
"Definition:T5 Space"
] | [
"Definition:T5 Space",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism",
"Definition:Topological Subspace",
"T5 Property is Hereditary",
"Definition:Topological Subspace",
"T5 Property is Preserved under Homeomorphism",
"Definition:T5 Space",
"Category:T5 Space... |
proofwiki-4463 | T4 Property is not Hereditary | Let $T = \struct {S, \tau}$ be a topological space which is a $T_4$ space.
Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$.
Then it does not necessarily follow that $T_H$ is a $T_4$ space. | Let $T$ be the Tychonoff plank.
Let $T'$ be the deleted Tychonoff plank.
By definition, $T'$ is a subspace of $T$.
From Tychonoff Plank is Normal, $T$ is a normal space.
From Deleted Tychonoff Plank is Not Normal, $T'$ is not a normal space.
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Defi... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is a [[Definition:T4 Space|$T_4$ space]].
Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a [[Definition:Topological Subspace|subspace]] of $T$.
Then it does not necessarily follow that $T_H$ is a [[Defini... | Let $T$ be the [[Definition:Tychonoff Plank|Tychonoff plank]].
Let $T'$ be the [[Definition:Deleted Tychonoff Plank|deleted Tychonoff plank]].
By definition, $T'$ is a [[Definition:Topological Subspace|subspace]] of $T$.
From [[Tychonoff Plank is Normal]], $T$ is a [[Definition:Normal Space|normal space]].
From [[... | T4 Property is not Hereditary | https://proofwiki.org/wiki/T4_Property_is_not_Hereditary | https://proofwiki.org/wiki/T4_Property_is_not_Hereditary | [
"T4 Spaces",
"Examples of Hereditary Properties",
"Separation Properties Preserved in Subspace"
] | [
"Definition:Topological Space",
"Definition:T4 Space",
"Definition:Topological Subspace",
"Definition:T4 Space"
] | [
"Definition:Tychonoff Plank",
"Definition:Tychonoff Plank/Deleted",
"Definition:Topological Subspace",
"Tychonoff Plank is Normal",
"Definition:Normal Space",
"Deleted Tychonoff Plank is Not Normal",
"Definition:Normal Space",
"T1 Property is Hereditary",
"Definition:T1 Space",
"Definition:Normal ... |
proofwiki-4464 | Derivative of Arcsecant Function | :<nowiki>$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {+1} {x \sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\
\dfrac {-1} {x \sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x < \pi \ (\text {that is: $x < -1$}) \\
\end{cases}$</nowi... | {{:Graph of Arcsecant Function}}
Let $y = \arcsec x$ where $\size x > 1$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = \arcsec x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \sec y
| c = where $y \in \closedint 0 \pi \land y \ne \dfrac \pi 2$
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d... | :<nowiki>$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {+1} {x \sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\
\dfrac {-1} {x \sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x < \pi \ (\text {that is: $x < -1$}) \\
\end{cases}$</nowi... | {{:Graph of Arcsecant Function}}
Let $y = \arcsec x$ where $\size x > 1$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = \arcsec x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \sec y
| c = where $y \in \closedint 0 \pi \land y \ne \dfrac \pi 2$
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {... | Derivative of Arcsecant Function | https://proofwiki.org/wiki/Derivative_of_Arcsecant_Function | https://proofwiki.org/wiki/Derivative_of_Arcsecant_Function | [
"Derivatives of Inverse Trigonometric Functions",
"Arcsecant Function",
"Derivative of Arcsecant Function"
] | [] | [
"Derivative of Secant Function",
"Derivative of Inverse Function",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Sine and Cosine are Periodic on Reals",
"Definition:Negative Real Function",
"Definition:Inverse Secant/Real/Arcsecant"
] |
proofwiki-4465 | Paracompactness is Preserved under Projections | Let $I$ be an indexing set.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \ma... | We are given that $\struct {S, \tau}$ is paracompact.
So every open cover of $S$ has an open refinement which is locally finite.
We are given that for each $\alpha \in I$, $\pr_{\alpha}: S \to S_{\alpha}$ is the $\alpha$-th projection.
For each $\alpha \in I$, let $\pr_{\alpha}^\gets: \powerset {S_{\alpha}} \to \powers... | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexed]] by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mat... | We are [[Definition:Given|given]] that $\struct {S, \tau}$ is [[Definition:Paracompact Space|paracompact]].
So every [[Definition:Open Cover|open cover]] of $S$ has an [[Definition:Open Refinement|open refinement]] which is [[Definition:Locally Finite Cover|locally finite]].
We are [[Definition:Given|given]] that fo... | Paracompactness is Preserved under Projections | https://proofwiki.org/wiki/Paracompactness_is_Preserved_under_Projections | https://proofwiki.org/wiki/Paracompactness_is_Preserved_under_Projections | [
"Paracompact Spaces",
"Continuous Mappings",
"Product Spaces",
"Projections"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Projection (Mapping Theory)",
"Definition:Paracompact Space",
"Definition:Paracompact Space"
] | [
"Definition:Given",
"Definition:Paracompact Space",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Locally Finite Cover",
"Definition:Given",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Indexing Set/Family of Se... |
proofwiki-4466 | Connected Space is Connected Between Two Points | Let $T$ be a topological space which is connected.
Then $T$ is connected between two points. | By definition of connected space, $T$ admits no separation.
Therefore, vacuously, every partition has one open containing $t_1, t_2 \in T$, for all $t_1, t_2 \in T$.
That is, for all $t_1, t_2 \in T$, $T$ is connected between $t_1$ and $t_2$.
{{qed}} | Let $T$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Connected Topological Space|connected]].
Then $T$ is [[Definition:Connected Between Two Points|connected between two points]]. | By definition of [[Definition:Connected Topological Space|connected space]], $T$ admits no [[Definition:Separation (Topology)|separation]].
Therefore, [[Definition:Vacuous Truth|vacuously]], every [[Definition:Partition (Topology)|partition]] has one [[Definition:Open Set (Topology)|open]] containing $t_1, t_2 \in T$,... | Connected Space is Connected Between Two Points | https://proofwiki.org/wiki/Connected_Space_is_Connected_Between_Two_Points | https://proofwiki.org/wiki/Connected_Space_is_Connected_Between_Two_Points | [
"Connectedness Between Two Points",
"Connected Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Connected Topological Space",
"Definition:Connected Between Two Points"
] | [
"Definition:Connected Topological Space",
"Definition:Separation (Topology)",
"Definition:Vacuous Truth",
"Definition:Separation (Topology)",
"Definition:Open Set/Topology",
"Definition:Connected Between Two Points"
] |
proofwiki-4467 | Equality is Reflexive | :$\forall a: a = a$ | This proof depends on Leibniz's law:
:$x = y \dashv \vdash \map P x \iff \map P y$
We are trying to prove $a = a$.
Our assertion, then, is:
:$a = a \dashv \vdash \map P a \iff \map P a$
From Law of Identity, $\map P a \iff \map P a$ is a tautology.
Thus $a = a$ is also tautologous, and the theorem holds.
{{qed}} | :$\forall a: a = a$ | This proof depends on [[Axiom:Leibniz's Law|Leibniz's law]]:
:$x = y \dashv \vdash \map P x \iff \map P y$
We are trying to prove $a = a$.
Our assertion, then, is:
:$a = a \dashv \vdash \map P a \iff \map P a$
From [[Law of Identity]], $\map P a \iff \map P a$ is a [[Definition:Tautology|tautology]].
Thus $a = a... | Equality is Reflexive | https://proofwiki.org/wiki/Equality_is_Reflexive | https://proofwiki.org/wiki/Equality_is_Reflexive | [
"Logic",
"Equality"
] | [] | [
"Axiom:Leibniz's Law",
"Law of Identity",
"Definition:Tautology"
] |
proofwiki-4468 | Closed Set in Metric Space is G-Delta | Let $M = \struct {A, d}$ be a metric space.
Let $F \subset A$ be a closed set of $M$.
Then $F$ is a $G_\delta$ set of $A$. | {{Recall|G-Delta Set|$G_\delta$ set}}
{{:Definition:G-Delta Set}}
Let $n \in \N$.
Let $\ds F_{\frac 1 n} = \bigcup \limits_{x \mathop \in F} \map B {x, \dfrac 1 n}$, where $\map B {x, \dfrac 1 n}$ is the open ball around $x$ with radius $\dfrac 1 n$.
$F_{\frac 1 n}$ is an open set by definition of open ball.
Also, $F_{... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $F \subset A$ be a [[Definition:Closed Set (Metric Space)|closed set]] of $M$.
Then $F$ is a [[Definition:G-Delta Set|$G_\delta$ set]] of $A$. | {{Recall|G-Delta Set|$G_\delta$ set}}
{{:Definition:G-Delta Set}}
Let $n \in \N$.
Let $\ds F_{\frac 1 n} = \bigcup \limits_{x \mathop \in F} \map B {x, \dfrac 1 n}$, where $\map B {x, \dfrac 1 n}$ is the [[Definition:Open Ball|open ball]] around $x$ with radius $\dfrac 1 n$.
$F_{\frac 1 n}$ is an [[Definition:Open S... | Closed Set in Metric Space is G-Delta | https://proofwiki.org/wiki/Closed_Set_in_Metric_Space_is_G-Delta | https://proofwiki.org/wiki/Closed_Set_in_Metric_Space_is_G-Delta | [
"G-Delta Sets",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:G-Delta Set"
] | [
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Definition:Open Ball",
"Definition:Set Union",
"Definition:Open Ball",
"Definition:Limit Point/Topology/Set",
"Definition:Closed Set/Metric Space",
"Definition:Set Intersection/Countable Intersection",
"Definition:Open Set/Metric Space",
... |
proofwiki-4469 | Totally Disconnected Space is Totally Pathwise Disconnected | Let $T = \struct {S, \tau}$ be a topological space which is totally disconnected.
Then $T$ is a totally pathwise disconnected space. | Let $T = \struct {S, \tau}$ be a topological space which is totally disconnected.
Then by definition $T$ contains no non-degenerate connected sets.
{{AimForCont}} $T$ is not a totally pathwise disconnected space.
That is, there exist two points $x, y \in S$ such that there exists a path between $x$ and $y$.
That is, $x... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Totally Disconnected Space|totally disconnected]].
Then $T$ is a [[Definition:Totally Pathwise Disconnected Space|totally pathwise disconnected space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Totally Disconnected Space|totally disconnected]].
Then by definition $T$ contains no [[Definition:Non-Degenerate Connected Set|non-degenerate connected sets]].
{{AimForCont}} $T$ is not a [[Definition:Totally Pa... | Totally Disconnected Space is Totally Pathwise Disconnected | https://proofwiki.org/wiki/Totally_Disconnected_Space_is_Totally_Pathwise_Disconnected | https://proofwiki.org/wiki/Totally_Disconnected_Space_is_Totally_Pathwise_Disconnected | [
"Totally Pathwise Disconnected Spaces",
"Totally Disconnected Spaces",
"Sequence of Implications of Disconnectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Totally Disconnected Space",
"Definition:Totally Pathwise Disconnected Space"
] | [
"Definition:Topological Space",
"Definition:Totally Disconnected Space",
"Definition:Degenerate Connected Set/Non-Degenerate",
"Definition:Totally Pathwise Disconnected Space",
"Definition:Path (Topology)",
"Definition:Path-Connected/Points",
"Definition:Path Component",
"Path-Connected Space is Conne... |
proofwiki-4470 | Second-Countable T3 Space is T5 | Let $T = \struct {S, \tau}$ be a $T_3$ space which is also second-countable.
Then $T$ is a $T_5$ space. | Let $A, B \subseteq S$ with $A^- \cap B = A \cap B^- = \O$.
For each $x \in A$, since $T$ is $T_3$:
:$\exists P, Q \in \tau: x \in P, B^- \subseteq Q, P \cap Q = \O$
Let $\BB$ be a basis for $T$.
Then:
:$\exists U \in \BB: x \in U \subseteq P$
Notice that:
{{begin-eqn}}
{{eqn | o =
| r = U^- \cap B
}}
{{eqn | o =... | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ space]] which is also [[Definition:Second-Countable Space|second-countable]].
Then $T$ is a [[Definition:T5 Space|$T_5$ space]]. | Let $A, B \subseteq S$ with $A^- \cap B = A \cap B^- = \O$.
For each $x \in A$, since $T$ is [[Definition:T3 Space|$T_3$]]:
:$\exists P, Q \in \tau: x \in P, B^- \subseteq Q, P \cap Q = \O$
Let $\BB$ be a [[Definition:Basis (Topology)|basis]] for $T$.
Then:
:$\exists U \in \BB: x \in U \subseteq P$
Notice that:
{{b... | Second-Countable T3 Space is T5 | https://proofwiki.org/wiki/Second-Countable_T3_Space_is_T5 | https://proofwiki.org/wiki/Second-Countable_T3_Space_is_T5 | [
"Second-Countable Spaces",
"T3 Spaces",
"T5 Spaces",
"Sequence of Implications of Compactness Properties in T3 Space"
] | [
"Definition:T3 Space",
"Definition:Second-Countable Space",
"Definition:T5 Space"
] | [
"Definition:T3 Space",
"Definition:Basis (Topology)",
"Set Intersection Preserves Subsets",
"Set is Subset of its Topological Closure",
"Set Intersection Preserves Subsets",
"Topological Closure of Subset is Subset of Topological Closure",
"Open Set Disjoint from Set is Disjoint from Closure",
"Subset... |
proofwiki-4471 | Mean Value Theorem for Integrals | Let $f$ be a continuous real function on the closed interval $\closedint a b$.
Then there exists a real number $k \in \closedint a b$ such that:
:$\ds \int_a^b \map f x \rd x = \map f k \paren {b - a}$ | From Continuous Real Function is Darboux Integrable, $f$ is Darboux integrable on $\closedint a b$.
By the Extreme Value Theorem, there exist $m, M \in \closedint a b$ such that:
:$\ds \map f m = \min_{x \mathop \in \closedint a b} \map f x$
:$\ds \map f M = \max_{x \mathop \in \closedint a b} \map f x$
Then, from Darb... | Let $f$ be a [[Definition:Continuous Real Function on Closed Interval|continuous real function]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then there exists a [[Definition:Real Number|real number]] $k \in \closedint a b$ such that:
:$\ds \int_a^b \map f x \rd x = \map f k \paren {b ... | From [[Continuous Real Function is Darboux Integrable]], $f$ is [[Definition:Darboux Integrable Function|Darboux integrable]] on $\closedint a b$.
By the [[Extreme Value Theorem]], there exist $m, M \in \closedint a b$ such that:
:$\ds \map f m = \min_{x \mathop \in \closedint a b} \map f x$
:$\ds \map f M = \max_{x ... | Mean Value Theorem for Integrals/Proof 1 | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Integrals | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Integrals/Proof_1 | [
"Mean Value Theorem for Integrals",
"Definite Integrals",
"Named Theorems"
] | [
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Real Number"
] | [
"Continuous Real Function is Darboux Integrable",
"Definition:Darboux Integrable Function",
"Extreme Value Theorem",
"Darboux's Theorem",
"Intermediate Value Theorem"
] |
proofwiki-4472 | Mean Value Theorem for Integrals | Let $f$ be a continuous real function on the closed interval $\closedint a b$.
Then there exists a real number $k \in \closedint a b$ such that:
:$\ds \int_a^b \map f x \rd x = \map f k \paren {b - a}$ | From Continuous Real Function is Darboux Integrable, $f$ is Darboux integrable on $\closedint a b$.
Let $F : \closedint a b \to \R$ be a real function defined by:
:$\ds \map F x = \int_a^x \map f x \rd x$
We are assured that this function is well-defined, since $f$ is integrable on $\closedint a b$.
From Fundamental T... | Let $f$ be a [[Definition:Continuous Real Function on Closed Interval|continuous real function]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Then there exists a [[Definition:Real Number|real number]] $k \in \closedint a b$ such that:
:$\ds \int_a^b \map f x \rd x = \map f k \paren {b ... | From [[Continuous Real Function is Darboux Integrable]], $f$ is [[Definition:Darboux Integrable Function|Darboux integrable]] on $\closedint a b$.
Let $F : \closedint a b \to \R$ be a [[Definition:Real Function|real function]] defined by:
:$\ds \map F x = \int_a^x \map f x \rd x$
We are assured that this function i... | Mean Value Theorem for Integrals/Proof 2 | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Integrals | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Integrals/Proof_2 | [
"Mean Value Theorem for Integrals",
"Definite Integrals",
"Named Theorems"
] | [
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Real Number"
] | [
"Continuous Real Function is Darboux Integrable",
"Definition:Darboux Integrable Function",
"Definition:Real Function",
"Definition:Well-Defined",
"Definition:Darboux Integrable Function",
"Fundamental Theorem of Calculus/First Part",
"Definition:Continuous Real Function",
"Definition:Differentiable M... |
proofwiki-4473 | Closure in Infinite Particular Point Space is not Compact | Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Let $A \in \tau_p$ be open in $T$.
Let $A^-$ be the closure of $A$.
Then $A^-$ is not compact. | From Closure of Open Set of Particular Point Space, we have that $A^- = S$.
The result follows from Infinite Particular Point Space is not Compact.
{{qed}} | Let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
Let $A \in \tau_p$ be [[Definition:Open Set (Topology)|open]] in $T$.
Let $A^-$ be the [[Definition:Closure (Topology)|closure]] of $A$.
Then $A^-$ is not [[Definition:Compact Topological Subspace|... | From [[Closure of Open Set of Particular Point Space]], we have that $A^- = S$.
The result follows from [[Infinite Particular Point Space is not Compact]].
{{qed}} | Closure in Infinite Particular Point Space is not Compact | https://proofwiki.org/wiki/Closure_in_Infinite_Particular_Point_Space_is_not_Compact | https://proofwiki.org/wiki/Closure_in_Infinite_Particular_Point_Space_is_not_Compact | [
"Infinite Particular Point Topologies",
"Examples of Compact Topological Spaces",
"Set Closures"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Open Set/Topology",
"Definition:Closure (Topology)",
"Definition:Compact Topological Space/Subspace"
] | [
"Closure of Open Set of Particular Point Space",
"Infinite Particular Point Space is not Compact"
] |
proofwiki-4474 | Cover of Doubletons of Infinite Particular Point Space has no Locally Finite Refinement | Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Let $\CC$ be the open cover of $T$ defined as:
:$\CC = \set {\set {x, p}: x \in S, x \ne p}$
Then $\CC$ has no open refinement which is locally finite. | Suppose $T$ is an infinite particular point space.
As $S$ is infinite, $\CC$ is also infinite.
Let $x \in S, x \ne p$.
Then any neighborhood of $x$ must contain $p$, by the nature of the particular point topology.
But $p$ is contained in all elements of $\CC$.
That is:
:$\forall C \in \CC: p \in C$
So any neighborhood ... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
Let $\CC$ be the [[Definition:Open Cover|open cover]] of $T$ defined as:
:$\CC = \set {\set {x, p}: x \in S, x \ne p}$
Then $\CC$ has no [[Definition:Open Refinement|open refinement]] which is [[Def... | Suppose $T$ is an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
As $S$ is [[Definition:Infinite Set|infinite]], $\CC$ is also [[Definition:Infinite Set|infinite]].
Let $x \in S, x \ne p$.
Then any [[Definition:Neighborhood of Point|neighborhood]] of $x$ must contain $p$, by the ... | Cover of Doubletons of Infinite Particular Point Space has no Locally Finite Refinement | https://proofwiki.org/wiki/Cover_of_Doubletons_of_Infinite_Particular_Point_Space_has_no_Locally_Finite_Refinement | https://proofwiki.org/wiki/Cover_of_Doubletons_of_Infinite_Particular_Point_Space_has_no_Locally_Finite_Refinement | [
"Infinite Particular Point Topologies",
"Examples of Covers of Sets",
"Examples of Open Refinements"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Locally Finite Cover"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Infinite Set",
"Definition:Infinite Set",
"Definition:Neighborhood (Topology)/Point",
"Definition:Particular Point Topology",
"Definition:Element",
"Definition:Neighborhood (Topology)/Point",
"Definition:Set Intersection",
"Definition:Elem... |
proofwiki-4475 | Path-Connected iff Path-Connected to Point | Let $T = \struct {S, \tau}$ be a topological space.
Let $U \subseteq S$ be a non-empty subset of $T$.
Then $U$ is path-connected {{iff}}:
:$\exists p \in U: \forall q \in U: \exists f: \closedint 0 1 \to U: f$ continuous, $\map f 0 = p$ and $\map f 1 = q$ | === Necessary Condition ===
When $U$ is path-connected, any $p \in U$ satisfies the assertion.
Because $U$ is non-empty, such a $p$ exists.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U \subseteq S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $T$.
Then $U$ is [[Definition:Path-Connected Set|path-connected]] {{iff}}:
:$\exists p \in U: \forall q \in U: \exists f: \closedint 0 ... | === Necessary Condition ===
When $U$ is [[Definition:Path-Connected Set|path-connected]], any $p \in U$ satisfies the assertion.
Because $U$ is [[Definition:Non-Empty Set|non-empty]], such a $p$ exists.
{{qed|lemma}} | Path-Connected iff Path-Connected to Point | https://proofwiki.org/wiki/Path-Connected_iff_Path-Connected_to_Point | https://proofwiki.org/wiki/Path-Connected_iff_Path-Connected_to_Point | [
"Path-Connected Sets"
] | [
"Definition:Topological Space",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Path-Connected/Set",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Definition:Path-Connected/Set",
"Definition:Non-Empty Set",
"Definition:Path-Connected/Set"
] |
proofwiki-4476 | Uncountable Fort Space is not Separable | Let $T = \struct {S, \tau_p}$ be a Fort space on an uncountable set $S$.
Then $T$ is not a separable space. | Let $C \subseteq S$ be a countable set.
Since $S$ is uncountable, by Uncountable Set less Countable Set is Uncountable, so is $\relcomp S C$.
Thus there exists some point $x \in \relcomp S C$ such that $x \ne p$.
By Clopen Points in Fort Space, $\set x \in \tau_p$.
By Empty Intersection iff Subset of Complement, we ha... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]] on an [[Definition:Uncountable Set|uncountable set]] $S$.
Then $T$ is not a [[Definition:Separable Space|separable space]]. | Let $C \subseteq S$ be a [[Definition:Countable Set|countable set]].
Since $S$ is [[Definition:Uncountable Set|uncountable]], by [[Uncountable Set less Countable Set is Uncountable]], so is $\relcomp S C$.
Thus there exists some [[Definition:Point|point]] $x \in \relcomp S C$ such that $x \ne p$.
By [[Clopen Points... | Uncountable Fort Space is not Separable | https://proofwiki.org/wiki/Uncountable_Fort_Space_is_not_Separable | https://proofwiki.org/wiki/Uncountable_Fort_Space_is_not_Separable | [
"Uncountable Fort Spaces",
"Examples of Separable Spaces"
] | [
"Definition:Fort Space",
"Definition:Uncountable/Set",
"Definition:Separable Space"
] | [
"Definition:Countable Set",
"Definition:Uncountable/Set",
"Uncountable Set less Countable Set is Uncountable",
"Definition:Point",
"Clopen Points in Fort Space",
"Empty Intersection iff Subset of Complement",
"Definition:Everywhere Dense",
"Definition:Arbitrary",
"Definition:Separable Space"
] |
proofwiki-4477 | Countable Fort Space is Separable | Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$.
Then $T$ is a separable space. | This result follows trivially from Countable Space is Separable.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]] on a [[Definition:Countably Infinite Set|countably infinite set]] $S$.
Then $T$ is a [[Definition:Separable Space|separable space]]. | This result follows trivially from [[Countable Space is Separable]].
{{qed}} | Countable Fort Space is Separable | https://proofwiki.org/wiki/Countable_Fort_Space_is_Separable | https://proofwiki.org/wiki/Countable_Fort_Space_is_Separable | [
"Countable Fort Spaces",
"Examples of Separable Spaces"
] | [
"Definition:Fort Space",
"Definition:Countably Infinite/Set",
"Definition:Separable Space"
] | [
"Countable Space is Separable"
] |
proofwiki-4478 | Countable Fort Space is Second-Countable | Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$.
Then $T$ is a second-countable space. | From Subset of Countably Infinite Set is Countable, $S \setminus \set p$ is countable.
Let $f: \N \to S$ be an enumeration of $S \setminus \set p$.
For brevity, let us write $s_n$ for $\map f n$.
Now define, for $n \in \N$, $S_n = S \setminus \set {s_0, \ldots, s_{n - 1} }$.
Note that $s_n \ne p$ for all $n \in \N$, so... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]] on a [[Definition:Countably Infinite Set|countably infinite set]] $S$.
Then $T$ is a [[Definition:Second-Countable Space|second-countable space]]. | From [[Subset of Countably Infinite Set is Countable]], $S \setminus \set p$ is [[Definition:Countable Set|countable]].
Let $f: \N \to S$ be an [[Definition:Enumeration|enumeration]] of $S \setminus \set p$.
For brevity, let us write $s_n$ for $\map f n$.
Now define, for $n \in \N$, $S_n = S \setminus \set {s_0, \l... | Countable Fort Space is Second-Countable | https://proofwiki.org/wiki/Countable_Fort_Space_is_Second-Countable | https://proofwiki.org/wiki/Countable_Fort_Space_is_Second-Countable | [
"Countable Fort Spaces",
"Examples of Second-Countable Spaces"
] | [
"Definition:Fort Space",
"Definition:Countably Infinite/Set",
"Definition:Second-Countable Space"
] | [
"Subset of Countably Infinite Set is Countable",
"Definition:Countable Set",
"Definition:Enumeration",
"Relative Complement of Relative Complement",
"Definition:Open Set/Topology",
"Finite Union of Countable Sets is Countable",
"Definition:Countable Set",
"Definition:Basis (Topology)",
"Definition:O... |
proofwiki-4479 | Fort Space is Regular | Let $T = \struct {S, \tau_p}$ be a Fort space.
Then $T$ is a regular space. | We have that the Fort Space is Completely Normal.
The result follows from tracing the relevant implications on Sequence of Implications of Separation Axioms.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]].
Then $T$ is a [[Definition:Regular Space|regular space]]. | We have that the [[Fort Space is Completely Normal]].
The result follows from tracing the relevant implications on [[Sequence of Implications of Separation Axioms]].
{{qed}} | Fort Space is Regular | https://proofwiki.org/wiki/Fort_Space_is_Regular | https://proofwiki.org/wiki/Fort_Space_is_Regular | [
"Fort Spaces",
"Examples of Regular Spaces"
] | [
"Definition:Fort Space",
"Definition:Regular Space"
] | [
"Fort Space is Completely Normal",
"Sequence of Implications of Separation Axioms"
] |
proofwiki-4480 | Countable Fort Space is Metrizable | Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$.
Then $T$ is a metrizable space. | We have:
:Fort Space is Regular
:Countable Fort Space is Second-Countable.
The result follows from Urysohn's Metrization Theorem.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]] on a [[Definition:Countably Infinite Set|countably infinite set]] $S$.
Then $T$ is a [[Definition:Metrizable Space|metrizable space]]. | We have:
:[[Fort Space is Regular]]
:[[Countable Fort Space is Second-Countable]].
The result follows from [[Urysohn's Metrization Theorem]].
{{qed}} | Countable Fort Space is Metrizable | https://proofwiki.org/wiki/Countable_Fort_Space_is_Metrizable | https://proofwiki.org/wiki/Countable_Fort_Space_is_Metrizable | [
"Countable Fort Spaces",
"Examples of Metrizable Spaces"
] | [
"Definition:Fort Space",
"Definition:Countably Infinite/Set",
"Definition:Metrizable Space"
] | [
"Fort Space is Regular",
"Countable Fort Space is Second-Countable",
"Urysohn's Metrization Theorem"
] |
proofwiki-4481 | Fortissimo Space is not Compact | Let $T = \struct {S, \tau_p}$ be a Fortissimo space.
Then $T$ is not a compact space. | From the definition, $T$ is a compact space if every open cover of $T$ has a finite subcover.
Let $N \subset S$ be a countably infinite set such that $p \notin N$.
Then $\relcomp S N$ is open in $T$ by the definition of the Fortissimo space.
Let $q \in S$ such that $q \ne p$.
As $p \in \relcomp S {\set q}$ it follows t... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Then $T$ is not a [[Definition:Compact Topological Space|compact space]]. | From the definition, $T$ is a [[Definition:Compact Topological Space|compact space]] if every [[Definition:Open Cover|open cover]] of $T$ has a [[Definition:Finite Subcover|finite subcover]].
Let $N \subset S$ be a [[Definition:Countably Infinite Set|countably infinite]] set such that $p \notin N$.
Then $\relcomp S ... | Fortissimo Space is not Compact | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Compact | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Compact | [
"Fortissimo Spaces",
"Examples of Compact Topological Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Compact Topological Space"
] | [
"Definition:Compact Topological Space",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Countably Infinite/Set",
"Definition:Open Set/Topology",
"Definition:Fortissimo Space",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Definition:Pairwise Disjoint",
"Definition:Su... |
proofwiki-4482 | Fortissimo Space is not Sequentially Compact | Let $T = \struct {S, \tau_p}$ be a Fortissimo space.
Then $T$ is not sequentially compact. | Let $T' = \struct {S \setminus \set p, \tau_p}$ be the subspace of $T$ induced by $S \setminus \set p$.
From Discrete Subspace of Fortissimo Space, $T'$ is a discrete topological space.
Let $H = \sequence {x_n}_{n \mathop \in \N}$ be a sequence of distinct terms of $S \setminus \set p$.
That is:
:$\forall n \in \N: x_n... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Then $T$ is not [[Definition:Sequentially Compact Space|sequentially compact]]. | Let $T' = \struct {S \setminus \set p, \tau_p}$ be the [[Definition:Topological Subspace|subspace]] of $T$ induced by $S \setminus \set p$.
From [[Discrete Subspace of Fortissimo Space]], $T'$ is a [[Definition:Discrete Topology|discrete topological space]].
Let $H = \sequence {x_n}_{n \mathop \in \N}$ be a [[Defini... | Fortissimo Space is not Sequentially Compact | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Sequentially_Compact | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Sequentially_Compact | [
"Fortissimo Spaces",
"Examples of Sequentially Compact Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Topological Subspace",
"Discrete Subspace of Fortissimo Space",
"Definition:Discrete Topology",
"Definition:Sequence of Distinct Terms",
"Definition:Sequence of Distinct Terms",
"Definition:Convergent Sequence/Topology",
"Definition:Countable Set",
"Closed Sets of Fortissimo Space",
"Def... |
proofwiki-4483 | Fortissimo Space is not Weakly Countably Compact | Let $T = \struct {S, \tau_p}$ be a Fortissimo space.
Then $T$ is not weakly countably compact. | It suffices to show that $T$ has an infinite subset without limit points.
Consider the set $S \setminus \set p$.
Let $x \in S$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {S \setminus \paren {S \setminus \set p} } \cup \set x
| r = \set p \cup \set x
| c =
}}
{{eqn | r = \set {p, x}
| c =
}}
{{end-eq... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Then $T$ is not [[Definition:Weakly Countably Compact Space|weakly countably compact]]. | It suffices to show that $T$ has an [[Definition:Infinite Set|infinite]] [[Definition:Subset|subset]] without [[Definition:Limit Point of Set|limit points]].
Consider the set $S \setminus \set p$.
Let $x \in S$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {S \setminus \paren {S \setminus \set p} } \cup \set x
|... | Fortissimo Space is not Weakly Countably Compact | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Weakly_Countably_Compact | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Weakly_Countably_Compact | [
"Fortissimo Spaces",
"Examples of Weakly Countably Compact Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Weakly Countably Compact Space"
] | [
"Definition:Infinite Set",
"Definition:Subset",
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Set",
"Definition:Neighborhood (Topology)/Point",
"Definition:Fortissimo Space",
"Definition:Open Set",
"Definition:Open Neighborhood/Point",
"Definition:Limit Point/Topology/Set",... |
proofwiki-4484 | Fortissimo Space is not Pseudocompact | Let $T = \struct {S, \tau_p}$ be a Fortissimo space.
Then $T$ is not a pseudocompact space. | Let $N_p$ be a neighborhood of $p$ such that $\relcomp S {N_p}$ is countable.
Let $\psi: \relcomp S {N_p} \to \Z_{\ne 0}$ be a bijection between $\relcomp S {N_p}$ and the non-zero integers $\Z_{\ne 0}$.
Let $\phi: S \to \set {0, 1}$ be the mapping defined as:
:$\forall x \in S: \map \phi x = \begin {cases} 0 & : x \in... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Then $T$ is not a [[Definition:Pseudocompact Space|pseudocompact space]]. | Let $N_p$ be a [[Definition:Neighborhood of Point|neighborhood]] of $p$ such that $\relcomp S {N_p}$ is [[Definition:Countable Set|countable]].
Let $\psi: \relcomp S {N_p} \to \Z_{\ne 0}$ be a [[Definition:Bijection|bijection]] between $\relcomp S {N_p}$ and the non-[[Definition:Zero (Number)|zero]] [[Definition:Integ... | Fortissimo Space is not Pseudocompact | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Pseudocompact | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Pseudocompact | [
"Fortissimo Spaces",
"Examples of Pseudocompact Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Pseudocompact Space"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Countable Set",
"Definition:Bijection",
"Definition:Zero (Number)",
"Definition:Integer",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Bounded Mapping",
"Definition:Pseudocompact Space"
] |
proofwiki-4485 | Double Pointed Fortissimo Space is Weakly Countably Compact | Let $T = \struct {S, \tau_p}$ be a Fortissimo space.
Let $T \times D$ be the double pointed topology on $T$.
Then $T \times D$ is weakly countably compact. | Let $D = \set {0, 1}$.
Let $\tuple {p, 0}$ belong to some infinite $A \subseteq S$.
Then its twin $\tuple {p, 1}$ is a limit point of $A$.
Hence the result by definition of weakly countably compact.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Let $T \times D$ be the [[Definition:Double Pointed Topology|double pointed topology]] on $T$.
Then $T \times D$ is [[Definition:Weakly Countably Compact Space|weakly countably compact]]. | Let $D = \set {0, 1}$.
Let $\tuple {p, 0}$ belong to some infinite $A \subseteq S$.
Then its twin $\tuple {p, 1}$ is a [[Definition:Limit Point of Set|limit point]] of $A$.
Hence the result by definition of [[Definition:Weakly Countably Compact Space|weakly countably compact]].
{{qed}} | Double Pointed Fortissimo Space is Weakly Countably Compact | https://proofwiki.org/wiki/Double_Pointed_Fortissimo_Space_is_Weakly_Countably_Compact | https://proofwiki.org/wiki/Double_Pointed_Fortissimo_Space_is_Weakly_Countably_Compact | [
"Double Pointed Topologies",
"Fortissimo Spaces",
"Examples of Weakly Countably Compact Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Double Pointed Topology",
"Definition:Weakly Countably Compact Space"
] | [
"Definition:Limit Point/Topology/Set",
"Definition:Weakly Countably Compact Space"
] |
proofwiki-4486 | Double Pointed Fortissimo Space is Lindelöf | Let $T = \struct {S, \tau}$ be a Fortissimo space.
Let $T \times D$ be the double pointed topology on $T$.
Then $T \times D$ is a Lindelöf space. | Let $D = \set {0, 1}$.
Let $\CC$ be an open cover of $T \times D$.
Then $\exists A \times B \in \CC$ such that $\tuple {p, 0} \in A \times B$.
We must have $\relcomp S A$ is countable and $B = D$.
Hence $\relcomp S A \times D$, a product of countable sets, must be countable.
So $A \times D$, together with an open neigh... | Let $T = \struct {S, \tau}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Let $T \times D$ be the [[Definition:Double Pointed Topology|double pointed topology]] on $T$.
Then $T \times D$ is a [[Definition:Lindelöf Space|Lindelöf space]]. | Let $D = \set {0, 1}$.
Let $\CC$ be an [[Definition:Open Cover|open cover]] of $T \times D$.
Then $\exists A \times B \in \CC$ such that $\tuple {p, 0} \in A \times B$.
We must have $\relcomp S A$ is [[Definition:Countable Set|countable]] and $B = D$.
Hence $\relcomp S A \times D$, a product of [[Definition:Countab... | Double Pointed Fortissimo Space is Lindelöf | https://proofwiki.org/wiki/Double_Pointed_Fortissimo_Space_is_Lindelöf | https://proofwiki.org/wiki/Double_Pointed_Fortissimo_Space_is_Lindelöf | [
"Double Pointed Topologies",
"Fortissimo Spaces",
"Examples of Lindelöf Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Double Pointed Topology",
"Definition:Lindelöf Space"
] | [
"Definition:Open Cover",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Open Neighborhood/Point",
"Definition:Element",
"Definition:Subcover/Countable",
"Definition:Lindelöf Space"
] |
proofwiki-4487 | Double Pointed Fortissimo Space is not Sigma-Compact | Let $T = \struct {S, \tau}$ be a Fortissimo space.
Let $T \times D$ be the double pointed topology on $T$.
Then $T \times D$ is not $\sigma$-compact. | {{proof wanted|Use the same argument as Fortissimo Space is not Sigma-Compact}} | Let $T = \struct {S, \tau}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Let $T \times D$ be the [[Definition:Double Pointed Topology|double pointed topology]] on $T$.
Then $T \times D$ is not [[Definition:Sigma-Compact Space|$\sigma$-compact]]. | {{proof wanted|Use the same argument as [[Fortissimo Space is not Sigma-Compact]]}} | Double Pointed Fortissimo Space is not Sigma-Compact | https://proofwiki.org/wiki/Double_Pointed_Fortissimo_Space_is_not_Sigma-Compact | https://proofwiki.org/wiki/Double_Pointed_Fortissimo_Space_is_not_Sigma-Compact | [
"Fortissimo Spaces",
"Double Pointed Topologies",
"Examples of Sigma-Compact Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Double Pointed Topology",
"Definition:Sigma-Compact Space"
] | [
"Fortissimo Space is not Sigma-Compact"
] |
proofwiki-4488 | Double Pointed Fortissimo Space is not Pseudocompact | Let $T = \struct {S, \tau}$ be a Fortissimo space.
Let $T \times D$ be the double pointed topology on $T$.
Then $T \times D$ is not pseudocompact. | {{proof wanted|Use the same argument as Fortissimo Space is not Pseudocompact}} | Let $T = \struct {S, \tau}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Let $T \times D$ be the [[Definition:Double Pointed Topology|double pointed topology]] on $T$.
Then $T \times D$ is not [[Definition:Pseudocompact Space|pseudocompact]]. | {{proof wanted|Use the same argument as [[Fortissimo Space is not Pseudocompact]]}} | Double Pointed Fortissimo Space is not Pseudocompact | https://proofwiki.org/wiki/Double_Pointed_Fortissimo_Space_is_not_Pseudocompact | https://proofwiki.org/wiki/Double_Pointed_Fortissimo_Space_is_not_Pseudocompact | [
"Double Pointed Topologies",
"Fortissimo Spaces",
"Examples of Pseudocompact Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Double Pointed Topology",
"Definition:Pseudocompact Space"
] | [
"Fortissimo Space is not Pseudocompact"
] |
proofwiki-4489 | Arens-Fort Space is Non-Meager | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a non-meager space. | From Meager Sets in Arens-Fort Space, we have that $A \subseteq S$ is meager in $T$ {{iff}} $A = \set {\tuple {0, 0} }$.
So as $\set {\tuple{0, 0} } \ne S \subseteq S$, it follows that $T$ is non-meager.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Non-Meager Space|non-meager space]]. | From [[Meager Sets in Arens-Fort Space]], we have that $A \subseteq S$ is [[Definition:Meager Space|meager]] in $T$ {{iff}} $A = \set {\tuple {0, 0} }$.
So as $\set {\tuple{0, 0} } \ne S \subseteq S$, it follows that $T$ is [[Definition:Non-Meager Space|non-meager]].
{{qed}} | Arens-Fort Space is Non-Meager | https://proofwiki.org/wiki/Arens-Fort_Space_is_Non-Meager | https://proofwiki.org/wiki/Arens-Fort_Space_is_Non-Meager | [
"Arens-Fort Space",
"Examples of Non-Meager Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Meager Space/Non-Meager"
] | [
"Meager Sets in Arens-Fort Space",
"Definition:Meager Space",
"Definition:Meager Space/Non-Meager"
] |
proofwiki-4490 | Arens-Fort Space is not Extremally Disconnected | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is not extremally disconnected. | Let $\map {S_m} V$ denote the set:
:$\map {S_m} V := \set {n: \tuple {m, n} \notin V}$
where $V \subseteq \Z_{\ge 0} \times \Z_{\ge 0}$ (the same set $S_m$ used in the definition of the Arens-Fort space).
Let $U = \set {\tuple {n, m}: \exists k: m = 2 k} \setminus \set {\tuple {0, 0} }$.
From the definition of the Aren... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is not [[Definition:Extremally Disconnected Space|extremally disconnected]]. | Let $\map {S_m} V$ denote the [[Definition:Set|set]]:
:$\map {S_m} V := \set {n: \tuple {m, n} \notin V}$
where $V \subseteq \Z_{\ge 0} \times \Z_{\ge 0}$ (the same [[Definition:Set|set]] $S_m$ used in the definition of the [[Definition:Arens-Fort Space|Arens-Fort space]]).
Let $U = \set {\tuple {n, m}: \exists k: m ... | Arens-Fort Space is not Extremally Disconnected | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Extremally_Disconnected | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Extremally_Disconnected | [
"Arens-Fort Space",
"Examples of Extremally Disconnected Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Extremally Disconnected Space"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Arens-Fort Space",
"Definition:Arens-Fort Space",
"Definition:Open Set/Topology",
"Definition:Infinite Set",
"Definition:Arens-Fort Space",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closure (Topology)",
"Definiti... |
proofwiki-4491 | Modified Fort Space is Totally Disconnected | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is totally disconnected. | We have:
:Modified Fort Space is Scattered
:Modified Fort Space is $T_1$
We also have:
:Scattered $T_1$ Space is Totally Disconnected
Hence the result.
{{qed}} | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is [[Definition:Totally Disconnected Space|totally disconnected]]. | We have:
:[[Modified Fort Space is Scattered]]
:[[Modified Fort Space is T1|Modified Fort Space is $T_1$]]
We also have:
:[[Scattered T1 Space is Totally Disconnected|Scattered $T_1$ Space is Totally Disconnected]]
Hence the result.
{{qed}} | Modified Fort Space is Totally Disconnected | https://proofwiki.org/wiki/Modified_Fort_Space_is_Totally_Disconnected | https://proofwiki.org/wiki/Modified_Fort_Space_is_Totally_Disconnected | [
"Modified Fort Spaces",
"Examples of Totally Disconnected Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:Totally Disconnected Space"
] | [
"Modified Fort Space is Scattered",
"Modified Fort Space is T1",
"Scattered T1 Space is Totally Disconnected"
] |
proofwiki-4492 | Modified Fort Space is not Locally Connected | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not locally connected. | We have:
:Modified Fort Space is Totally Disconnected
:Totally Disconnected and Locally Connected Space is Discrete
We also have:
:Modified Fort Space is not $T_2$
:Discrete Space satisfies all Separation Properties
Hence modified Fort space is not the discrete space, and the result follows.
{{qed}} | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not [[Definition:Locally Connected Space|locally connected]]. | We have:
:[[Modified Fort Space is Totally Disconnected]]
:[[Totally Disconnected and Locally Connected Space is Discrete]]
We also have:
:[[Modified Fort Space is not T2|Modified Fort Space is not $T_2$]]
:[[Discrete Space satisfies all Separation Properties]]
Hence [[Definition:Modified Fort Space|modified Fort spa... | Modified Fort Space is not Locally Connected | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Locally_Connected | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Locally_Connected | [
"Modified Fort Spaces",
"Examples of Locally Connected Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:Locally Connected Space"
] | [
"Modified Fort Space is Totally Disconnected",
"Totally Disconnected and Locally Connected Space is Discrete",
"Modified Fort Space is not T2",
"Discrete Space satisfies all Separation Properties",
"Definition:Modified Fort Space",
"Definition:Discrete Topology"
] |
proofwiki-4493 | Sets in Modified Fort Space are Disconnected | Let $T = \struct {S, \tau_{a, b}}$ be a modified Fort space.
Let $H$ be a subset of $S$ with more than one point.
Then $H$ is disconnected. | By Isolated Points in Subsets of Modified Fort Space:
:$\exists x \in H: x$ is isolated
By Point in Topological Space is Open iff Isolated, $\set x$ is open in $T$.
By Modified Fort Space is $T_1$ and definition of $T_1$ space, $\set x$ is closed in $T$.
Therefore $\relcomp S {\set x}$ is open in $T$.
Then we have:
{{b... | Let $T = \struct {S, \tau_{a, b}}$ be a [[Definition:Modified Fort Space|modified Fort space]].
Let $H$ be a [[Definition:Subset|subset]] of $S$ with more than one [[Definition:Point of Set|point]].
Then $H$ is [[Definition:Disconnected Set|disconnected]]. | By [[Isolated Points in Subsets of Modified Fort Space]]:
:$\exists x \in H: x$ is [[Definition:Isolated Point of Subset|isolated]]
By [[Point in Topological Space is Open iff Isolated]], $\set x$ is [[Definition:Open Set (Topology)|open]] in $T$.
By [[Modified Fort Space is T1|Modified Fort Space is $T_1$]] and def... | Sets in Modified Fort Space are Disconnected | https://proofwiki.org/wiki/Sets_in_Modified_Fort_Space_are_Disconnected | https://proofwiki.org/wiki/Sets_in_Modified_Fort_Space_are_Disconnected | [
"Modified Fort Spaces",
"Examples of Disconnected Sets"
] | [
"Definition:Modified Fort Space",
"Definition:Subset",
"Definition:Element",
"Definition:Disconnected (Topology)/Set"
] | [
"Isolated Points in Subsets of Modified Fort Space",
"Definition:Isolated Point (Topology)/Subset",
"Point in Topological Space is Open iff Isolated",
"Definition:Open Set/Topology",
"Modified Fort Space is T1",
"Definition:T1 Space",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
... |
proofwiki-4494 | Clopen Sets in Modified Fort Space | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Let $H \subseteq S$ be clopen in $T$.
If $a \in H$, then $b \in H$ as well.
That is, any clopen set of $T$ must contain '''both''' or '''neither''' of $a$ and $b$. | Let $A$ be both closed and open in $T$.
Let $S = A \mid B$ be a separation of $S$ into the open sets $A$ and $B$.
Then by definition $B$ is also both closed and open in $T$.
Let $a \in A$.
Suppose $b \in B$.
As $A$ is open it must contain all but a finite number of points of $S$.
Thus $A$ must itself be infinite.
But t... | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Let $H \subseteq S$ be [[Definition:Clopen Set|clopen]] in $T$.
If $a \in H$, then $b \in H$ as well.
That is, any [[Definition:Clopen Set|clopen set]] of $T$ must contain '''both''' or '''neither''' of $a$ and $b$. | Let $A$ be [[Definition:Clopen Set|both closed and open]] in $T$.
Let $S = A \mid B$ be a [[Definition:Separation (Topology)|separation]] of $S$ into the [[Definition:Open Set (Topology)|open sets]] $A$ and $B$.
Then by definition $B$ is also [[Definition:Clopen Set|both closed and open]] in $T$.
Let $a \in A$.
Su... | Clopen Sets in Modified Fort Space | https://proofwiki.org/wiki/Clopen_Sets_in_Modified_Fort_Space | https://proofwiki.org/wiki/Clopen_Sets_in_Modified_Fort_Space | [
"Modified Fort Spaces",
"Examples of Clopen Sets"
] | [
"Definition:Modified Fort Space",
"Definition:Clopen Set",
"Definition:Clopen Set"
] | [
"Definition:Clopen Set",
"Definition:Separation (Topology)",
"Definition:Open Set/Topology",
"Definition:Clopen Set",
"Definition:Open Set/Topology",
"Definition:Finite Set",
"Definition:Infinite Set",
"Definition:Open Set/Topology",
"Definition:Finite Set"
] |
proofwiki-4495 | Modified Fort Space is not Totally Separated | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not totally separated. | We have:
:Totally Separated Space is $T_{2 \frac 1 2}$
But we have:
:Modified Fort Space is not $T_{2 \frac 1 2}$
The result follows from Modus Tollendo Tollens.
{{qed}} | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not [[Definition:Totally Separated Space|totally separated]]. | We have:
:[[Totally Separated Space is T2.5|Totally Separated Space is $T_{2 \frac 1 2}$]]
But we have:
:[[Modified Fort Space is not T2.5|Modified Fort Space is not $T_{2 \frac 1 2}$]]
The result follows from [[Modus Tollendo Tollens]].
{{qed}} | Modified Fort Space is not Totally Separated | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Totally_Separated | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Totally_Separated | [
"Modified Fort Spaces",
"Examples of Totally Separated Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:Totally Separated Space"
] | [
"Totally Separated Space is T2.5",
"Modified Fort Space is not T2.5",
"Modus Tollendo Tollens"
] |
proofwiki-4496 | Modified Fort Space is not Zero Dimensional | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not zero dimensional. | We have Modified Fort Space is not $T_3$.
We also have Zero Dimensional Space is $T_3$.
The result follows by Modus Tollendo Tollens.
{{qed}} | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not [[Definition:Zero Dimensional Space|zero dimensional]]. | We have [[Modified Fort Space is not T3|Modified Fort Space is not $T_3$]].
We also have [[Zero Dimensional Space is T3|Zero Dimensional Space is $T_3$]].
The result follows by [[Modus Tollendo Tollens]].
{{qed}} | Modified Fort Space is not Zero Dimensional | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Zero_Dimensional | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_Zero_Dimensional | [
"Modified Fort Spaces",
"Examples of Zero Dimensional Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:Zero Dimensional Space"
] | [
"Modified Fort Space is not T3",
"Zero Dimensional Space is T3",
"Modus Tollendo Tollens"
] |
proofwiki-4497 | Modified Fort Space is Scattered | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is scattered. | We have that a modified Fort space is $T_1$.
We also have that a dense-in-itself subset of a $T_1$ space is infinite.
But from Isolated Points in Subsets of Modified Fort Space, we have that any subset of $T$ with more than two points has at least one isolated point.
So any dense-in-itself subset of $T$ must have an is... | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is [[Definition:Scattered Space|scattered]]. | We have that a [[Modified Fort Space is T1|modified Fort space is $T_1$]].
We also have that a [[Dense-in-itself Subset of T1 Space is Infinite|dense-in-itself subset of a $T_1$ space is infinite]].
But from [[Isolated Points in Subsets of Modified Fort Space]], we have that any [[Definition:Subset|subset]] of $T$ wi... | Modified Fort Space is Scattered | https://proofwiki.org/wiki/Modified_Fort_Space_is_Scattered | https://proofwiki.org/wiki/Modified_Fort_Space_is_Scattered | [
"Modified Fort Spaces",
"Examples of Scattered Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:Scattered Space"
] | [
"Modified Fort Space is T1",
"Dense-in-itself Subset of T1 Space is Infinite",
"Isolated Points in Subsets of Modified Fort Space",
"Definition:Subset",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Dense-in-itself",
"Definition:Subset",
"Definition:Isolated Point (Topology)/Subset",
"D... |
proofwiki-4498 | Isolated Points in Subsets of Modified Fort Space | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Let $H \subseteq S$ contain more than two points.
Then $H$ contains an isolated point. | Let $H \subseteq S$ contain more than two points.
From {{Defof|Modified Fort Space}}, we can write:
:$S = N \cup \set a \cup \set b$
where $N$ is an infinite set.
Suppose $H \cap N \ne \O$.
Let $x \in H \cap N$.
Then:
:$\set x \subset N$
Therefore:
:$\set x \in \tau_{a, b}$
This shows that $x$ is isolated.
Suppose $H \... | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Let $H \subseteq S$ contain more than two [[Definition:Point|points]].
Then $H$ contains an [[Definition:Isolated Point of Subset|isolated point]]. | Let $H \subseteq S$ contain more than two [[Definition:Point|points]].
From {{Defof|Modified Fort Space}}, we can write:
:$S = N \cup \set a \cup \set b$
where $N$ is an [[Definition:Infinite Set|infinite set]].
Suppose $H \cap N \ne \O$.
Let $x \in H \cap N$.
Then:
:$\set x \subset N$
Therefore:
:$\set x \in \ta... | Isolated Points in Subsets of Modified Fort Space | https://proofwiki.org/wiki/Isolated_Points_in_Subsets_of_Modified_Fort_Space | https://proofwiki.org/wiki/Isolated_Points_in_Subsets_of_Modified_Fort_Space | [
"Modified Fort Spaces",
"Examples of Isolated Points"
] | [
"Definition:Modified Fort Space",
"Definition:Point",
"Definition:Isolated Point (Topology)/Subset"
] | [
"Definition:Point",
"Definition:Infinite Set",
"Definition:Isolated Point (Topology)/Subset",
"Empty Intersection iff Subset of Complement",
"Definition:Element",
"Definition:Finite Set",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Isolated Point (Topology)/Subset"
] |
proofwiki-4499 | Modified Fort Space is Sequentially Compact | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is sequentially compact. | Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $T$.
Suppose $\sequence {x_n}$ takes an infinite number of distinct values in $S$.
Then there is an infinite subsequence $\sequence {x_{n_r} }_{r \mathop \in \N}$ with distinct terms.
Let $U$ be a neighborhood of $a$.
Then $S \setminus U$ is a finite s... | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is [[Definition:Sequentially Compact Space|sequentially compact]]. | Let $\sequence {x_n}_{n \mathop \in \N}$ be an [[Definition:Infinite Sequence|infinite sequence]] in $T$.
Suppose $\sequence {x_n}$ takes an [[Definition:Infinite Set|infinite number]] of [[Definition:Distinct|distinct]] values in $S$.
Then there is an [[Definition:Infinite Sequence|infinite]] [[Definition:Subsequenc... | Modified Fort Space is Sequentially Compact | https://proofwiki.org/wiki/Modified_Fort_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Modified_Fort_Space_is_Sequentially_Compact | [
"Modified Fort Spaces",
"Examples of Sequentially Compact Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Infinite Set",
"Definition:Distinct",
"Definition:Sequence/Infinite Sequence",
"Definition:Subsequence",
"Definition:Sequence of Distinct Terms",
"Definition:Neighborhood (Topology)/Point",
"Definition:Finite Set",
"Definition:Convergent Sequence/... |
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