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-8300 | Primes of form Power plus One | Let $q, n \in \Z_{>0}$ such that $q > 1$.
Then $q^n + 1$ is prime only if:
:$(1): \quad q$ is even
and
:$(2): \quad n$ is of the form $2^m$ for some positive integer $m$. | Note that if $q = 1$ then $q^n + 1 = 2$ which ''is'' prime.
Hence the condition on $q$ in the statement of the theorem.
So by hypothesis $q > 1$.
Let $q$ be odd.
Then by Two divides Power Plus One iff Odd, $q^n + 1$ is not prime.
Let $q$ be even.
Let $n$ be expressed in the form $r 2^m$ where $r$ is odd.
Then $q^n + 1$... | Let $q, n \in \Z_{>0}$ such that $q > 1$.
Then $q^n + 1$ is [[Definition:Prime Number|prime]] only if:
:$(1): \quad q$ is [[Definition:Even Integer|even]]
and
:$(2): \quad n$ is of the form $2^m$ for some [[Definition:Positive Integer|positive integer]] $m$. | Note that if $q = 1$ then $q^n + 1 = 2$ which ''is'' [[Definition:Prime Number|prime]].
Hence the condition on $q$ in the statement of the theorem.
So [[Definition:By Hypothesis|by hypothesis]] $q > 1$.
Let $q$ be [[Definition:Odd Integer|odd]].
Then by [[Two divides Power Plus One iff Odd]], $q^n + 1$ is not [[De... | Primes of form Power plus One | https://proofwiki.org/wiki/Primes_of_form_Power_plus_One | https://proofwiki.org/wiki/Primes_of_form_Power_plus_One | [
"Number Theory"
] | [
"Definition:Prime Number",
"Definition:Even Integer",
"Definition:Positive/Integer"
] | [
"Definition:Prime Number",
"Definition:By Hypothesis",
"Definition:Odd Integer",
"Two divides Power Plus One iff Odd",
"Definition:Prime Number",
"Definition:Even Integer",
"Definition:Odd Integer",
"Number Plus One divides Power Plus One iff Odd",
"Definition:Divisor (Algebra)/Integer",
"Definiti... |
proofwiki-8301 | Fermat Number whose Index is Sum of Integers | Let $F_n = 2^{\left({2^n}\right)} + 1$ be the $n$th Fermat number.
Let $k \in \Z_{>0}$.
Then:
:$F_{n + k} - 1 = \left({F_n - 1}\right)^{2^k}$ | By the definition of Fermat number
{{begin-eqn}}
{{eqn | l = F_{n + k} - 1
| r = 2^{2^{n + k} }
| c = {{Defof|Fermat Number}}
}}
{{eqn | r = 2^{2^n 2^k}
| c =
}}
{{eqn | r = \left({2^{2^n} }\right)^{2^k}
| c =
}}
{{eqn | r = \left({F_n - 1}\right)^{2^k}
| c = {{Defof|Fermat Number}}
}}
{... | Let $F_n = 2^{\left({2^n}\right)} + 1$ be the $n$th [[Definition:Fermat Number|Fermat number]].
Let $k \in \Z_{>0}$.
Then:
:$F_{n + k} - 1 = \left({F_n - 1}\right)^{2^k}$ | By the definition of [[Definition:Fermat Number|Fermat number]]
{{begin-eqn}}
{{eqn | l = F_{n + k} - 1
| r = 2^{2^{n + k} }
| c = {{Defof|Fermat Number}}
}}
{{eqn | r = 2^{2^n 2^k}
| c =
}}
{{eqn | r = \left({2^{2^n} }\right)^{2^k}
| c =
}}
{{eqn | r = \left({F_n - 1}\right)^{2^k}
| c ... | Fermat Number whose Index is Sum of Integers | https://proofwiki.org/wiki/Fermat_Number_whose_Index_is_Sum_of_Integers | https://proofwiki.org/wiki/Fermat_Number_whose_Index_is_Sum_of_Integers | [
"Fermat Numbers"
] | [
"Definition:Fermat Number"
] | [
"Definition:Fermat Number"
] |
proofwiki-8302 | Number of Boolean Interpretations for Finite Set of Variables | Let $\PP_0$ be the vocabulary of language of propositional logic.
Let $S \subseteq \PP_0$ be a finite set of $n$ letters from $\PP_0$.
Then there are $2^n$ different partial boolean interpretations for $S$. | A partial boolean interpretation for $S$ is a mapping from $S$ to the Boolean domain $\set {T, F}$.
By Cardinality of Set of All Mappings, the total number of mappings from $S$ to $T$ is:
:$\card {T^S} = \card T^{\card S}$
The result follows directly.
{{qed}} | Let $\PP_0$ be the [[Definition:Vocabulary of Propositional Logic|vocabulary]] of [[Definition:Language of Propositional Logic|language of propositional logic]].
Let $S \subseteq \PP_0$ be a [[Definition:Finite Set|finite set]] of $n$ [[Definition:Letter of Formal Language|letters]] from $\PP_0$.
Then there are $2^n... | A [[Definition:Partial Boolean Interpretation|partial boolean interpretation]] for $S$ is a [[Definition:Mapping|mapping]] from $S$ to the [[Definition:Boolean Domain|Boolean domain]] $\set {T, F}$.
By [[Cardinality of Set of All Mappings]], the total number of [[Definition:Mapping|mappings]] from $S$ to $T$ is:
:$\c... | Number of Boolean Interpretations for Finite Set of Variables | https://proofwiki.org/wiki/Number_of_Boolean_Interpretations_for_Finite_Set_of_Variables | https://proofwiki.org/wiki/Number_of_Boolean_Interpretations_for_Finite_Set_of_Variables | [
"Boolean Interpretations"
] | [
"Definition:Language of Propositional Logic/Alphabet/Letter",
"Definition:Language of Propositional Logic",
"Definition:Finite Set",
"Definition:Formal Language/Alphabet/Letter",
"Definition:Boolean Interpretation"
] | [
"Definition:Boolean Interpretation",
"Definition:Mapping",
"Definition:Boolean Domain",
"Cardinality of Set of All Mappings",
"Definition:Mapping"
] |
proofwiki-8303 | Integers under Addition form Monoid | The set of integers under addition $\struct {\Z, +}$ forms a monoid. | Follows directly from Integers under Addition form Abelian Group.
By definition, an abelian group is a group.
Also by definition, a group is a monoid.
Hence the result.
{{qed}} | The [[Definition:Set|set]] of [[Definition:Integer|integers]] under [[Definition:Integer Addition|addition]] $\struct {\Z, +}$ forms a [[Definition:Monoid|monoid]]. | Follows directly from [[Integers under Addition form Abelian Group]].
By definition, an [[Definition:Abelian Group|abelian group]] is a [[Definition:Group|group]].
Also by definition, a [[Definition:Group|group]] is a [[Definition:Monoid|monoid]].
Hence the result.
{{qed}} | Integers under Addition form Monoid | https://proofwiki.org/wiki/Integers_under_Addition_form_Monoid | https://proofwiki.org/wiki/Integers_under_Addition_form_Monoid | [
"Integer Addition",
"Examples of Monoids"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Monoid"
] | [
"Integers under Addition form Abelian Group",
"Definition:Abelian Group",
"Definition:Group",
"Definition:Group",
"Definition:Monoid"
] |
proofwiki-8304 | Rational Numbers under Addition form Monoid | The set of rational numbers under addition $\struct {\Q, +}$ forms a monoid. | Follows directly from Rational Numbers under Addition form Infinite Abelian Group.
By definition, an abelian group is a group.
Also by definition, a group is a monoid.
Hence the result.
{{qed}} | The [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] under [[Definition:Rational Addition|addition]] $\struct {\Q, +}$ forms a [[Definition:Monoid|monoid]]. | Follows directly from [[Rational Numbers under Addition form Infinite Abelian Group]].
By definition, an [[Definition:Abelian Group|abelian group]] is a [[Definition:Group|group]].
Also by definition, a [[Definition:Group|group]] is a [[Definition:Monoid|monoid]].
Hence the result.
{{qed}} | Rational Numbers under Addition form Monoid | https://proofwiki.org/wiki/Rational_Numbers_under_Addition_form_Monoid | https://proofwiki.org/wiki/Rational_Numbers_under_Addition_form_Monoid | [
"Rational Addition",
"Examples of Monoids"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Addition/Rational Numbers",
"Definition:Monoid"
] | [
"Rational Numbers under Addition form Infinite Abelian Group",
"Definition:Abelian Group",
"Definition:Group",
"Definition:Group",
"Definition:Monoid"
] |
proofwiki-8305 | Real Numbers form Field | The set of real numbers $\R$ forms a field under addition and multiplication: $\struct {\R, +, \times}$. | From Real Numbers under Addition form Infinite Abelian Group, we have that $\struct {\R, +}$ forms an abelian group.
From Non-Zero Real Numbers under Multiplication form Abelian Group, we have that $\struct {\R_{\ne 0}, \times}$ forms an abelian group.
Next we have that Real Multiplication Distributes over Addition.
Th... | The [[Definition:Real Number|set of real numbers]] $\R$ forms a [[Definition:Field (Abstract Algebra)|field]] under [[Definition:Real Addition|addition]] and [[Definition:Real Multiplication|multiplication]]: $\struct {\R, +, \times}$. | From [[Real Numbers under Addition form Infinite Abelian Group]], we have that $\struct {\R, +}$ forms an [[Definition:Abelian Group|abelian group]].
From [[Non-Zero Real Numbers under Multiplication form Abelian Group]], we have that $\struct {\R_{\ne 0}, \times}$ forms an [[Definition:Abelian Group|abelian group]].
... | Real Numbers form Field | https://proofwiki.org/wiki/Real_Numbers_form_Field | https://proofwiki.org/wiki/Real_Numbers_form_Field | [
"Examples of Fields",
"Real Numbers"
] | [
"Definition:Real Number",
"Definition:Field (Abstract Algebra)",
"Definition:Addition/Real Numbers",
"Definition:Multiplication/Real Numbers"
] | [
"Real Numbers under Addition form Infinite Abelian Group",
"Definition:Abelian Group",
"Non-Zero Real Numbers under Multiplication form Abelian Group",
"Definition:Abelian Group",
"Real Multiplication Distributes over Addition",
"Definition:Field (Abstract Algebra)"
] |
proofwiki-8306 | Rational Numbers form Field | Consider the algebraic structure $\struct {\Q, +, \times}$, where:
:$\Q$ is the set of all rational numbers
:$+$ is the operation of rational addition
:$\times$ is the operation of rational multiplication.
Then $\struct {\Q, +, \times}$ forms a field. | This is demonstrated in the formal definition of rational numbers.
{{Qed}} | Consider the [[Definition:Algebraic Structure with Two Operations|algebraic structure]] $\struct {\Q, +, \times}$, where:
:$\Q$ is the set of all [[Definition:Rational Number|rational numbers]]
:$+$ is the operation of [[Definition:Rational Addition|rational addition]]
:$\times$ is the operation of [[Definition:Ration... | This is demonstrated in the [[Definition:Rational Number/Formal Definition|formal definition of rational numbers]].
{{Qed}} | Rational Numbers form Field | https://proofwiki.org/wiki/Rational_Numbers_form_Field | https://proofwiki.org/wiki/Rational_Numbers_form_Field | [
"Examples of Fields",
"Rational Numbers"
] | [
"Definition:Algebraic Structure/Two Operations",
"Definition:Rational Number",
"Definition:Addition/Rational Numbers",
"Definition:Multiplication/Rational Numbers",
"Definition:Field (Abstract Algebra)"
] | [
"Definition:Rational Number/Formal Definition"
] |
proofwiki-8307 | Real Numbers under Addition form Monoid | The set of real numbers under addition $\struct {\R, +}$ forms a monoid. | Taking the monoid axioms in turn: | The [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] under [[Definition:Real Addition|addition]] $\struct {\R, +}$ forms a [[Definition:Monoid|monoid]]. | Taking the [[Axiom:Monoid Axioms|monoid axioms]] in turn: | Real Numbers under Addition form Monoid | https://proofwiki.org/wiki/Real_Numbers_under_Addition_form_Monoid | https://proofwiki.org/wiki/Real_Numbers_under_Addition_form_Monoid | [
"Real Addition",
"Examples of Monoids"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Addition/Real Numbers",
"Definition:Monoid"
] | [
"Axiom:Monoid Axioms"
] |
proofwiki-8308 | Integers Modulo m under Addition form Abelian Group | Let $\Z_m$ is the set of integers modulo $m$
Let $+_m$ be the operation of addition modulo $m$.
Then the structure $\struct {\Z_m, +_m}$ is an abelian group. | From Integers Modulo m under Addition form Cyclic Group, $\struct {\Z_m, +_m}$ is a cyclic group.
The result follows from Cyclic Group is Abelian.
{{Qed}} | Let $\Z_m$ is the set of [[Definition:Integers Modulo m|integers modulo $m$]]
Let $+_m$ be the [[Definition:Binary Operation|operation]] of [[Definition:Modulo Addition|addition modulo $m$]].
Then the [[Definition:Algebraic Structure with One Operation|structure]] $\struct {\Z_m, +_m}$ is an [[Definition:Abelian Grou... | From [[Integers Modulo m under Addition form Cyclic Group]], $\struct {\Z_m, +_m}$ is a [[Definition:Cyclic Group|cyclic group]].
The result follows from [[Cyclic Group is Abelian]].
{{Qed}} | Integers Modulo m under Addition form Abelian Group | https://proofwiki.org/wiki/Integers_Modulo_m_under_Addition_form_Abelian_Group | https://proofwiki.org/wiki/Integers_Modulo_m_under_Addition_form_Abelian_Group | [
"Additive Groups of Integers Modulo m",
"Modulo Addition",
"Examples of Abelian Groups"
] | [
"Definition:Integers Modulo m",
"Definition:Operation/Binary Operation",
"Definition:Modulo Addition",
"Definition:Algebraic Structure/One Operation",
"Definition:Abelian Group"
] | [
"Integers Modulo m under Addition form Cyclic Group",
"Definition:Cyclic Group",
"Cyclic Group is Abelian"
] |
proofwiki-8309 | Complex Numbers under Addition form Monoid | The set of complex numbers under addition $\left({\C, +}\right)$ forms a monoid. | Taking the monoid axioms in turn: | The [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] under [[Definition:Complex Addition|addition]] $\left({\C, +}\right)$ forms a [[Definition:Monoid|monoid]]. | Taking the [[Axiom:Monoid Axioms|monoid axioms]] in turn: | Complex Numbers under Addition form Monoid | https://proofwiki.org/wiki/Complex_Numbers_under_Addition_form_Monoid | https://proofwiki.org/wiki/Complex_Numbers_under_Addition_form_Monoid | [
"Complex Addition",
"Examples of Monoids"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Addition/Complex Numbers",
"Definition:Monoid"
] | [
"Axiom:Monoid Axioms"
] |
proofwiki-8310 | Rational Numbers under Multiplication form Monoid | The set of rational numbers under multiplication $\struct {\Q, \times}$ forms a monoid. | Taking the monoid axioms in turn: | The [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] under [[Definition:Rational Multiplication|multiplication]] $\struct {\Q, \times}$ forms a [[Definition:Monoid|monoid]]. | Taking the [[Axiom:Monoid Axioms|monoid axioms]] in turn: | Rational Numbers under Multiplication form Monoid | https://proofwiki.org/wiki/Rational_Numbers_under_Multiplication_form_Monoid | https://proofwiki.org/wiki/Rational_Numbers_under_Multiplication_form_Monoid | [
"Rational Multiplication",
"Examples of Monoids"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Multiplication/Rational Numbers",
"Definition:Monoid"
] | [
"Axiom:Monoid Axioms"
] |
proofwiki-8311 | Real Numbers under Multiplication form Monoid | The set of real numbers under multiplication $\struct {\R, \times}$ forms a monoid. | Taking the monoid axioms in turn: | The [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] under [[Definition:Real Multiplication|multiplication]] $\struct {\R, \times}$ forms a [[Definition:Monoid|monoid]]. | Taking the [[Axiom:Monoid Axioms|monoid axioms]] in turn: | Real Numbers under Multiplication form Monoid | https://proofwiki.org/wiki/Real_Numbers_under_Multiplication_form_Monoid | https://proofwiki.org/wiki/Real_Numbers_under_Multiplication_form_Monoid | [
"Real Multiplication",
"Examples of Monoids"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Multiplication/Real Numbers",
"Definition:Monoid"
] | [
"Axiom:Monoid Axioms"
] |
proofwiki-8312 | Complex Numbers under Multiplication form Monoid | The set of complex numbers under multiplication $\struct {\C, \times}$ forms a monoid. | Taking the monoid axioms in turn: | The [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] under [[Definition:Complex Multiplication|multiplication]] $\struct {\C, \times}$ forms a [[Definition:Monoid|monoid]]. | Taking the [[Axiom:Monoid Axioms|monoid axioms]] in turn: | Complex Numbers under Multiplication form Monoid | https://proofwiki.org/wiki/Complex_Numbers_under_Multiplication_form_Monoid | https://proofwiki.org/wiki/Complex_Numbers_under_Multiplication_form_Monoid | [
"Complex Multiplication",
"Examples of Monoids"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Multiplication/Complex Numbers",
"Definition:Monoid"
] | [
"Axiom:Monoid Axioms"
] |
proofwiki-8313 | Non-Zero Natural Numbers under Multiplication form Commutative Monoid | Let $\N_{>0}$ be the set of natural numbers without zero, i.e. $\N_{>0} = \N \setminus \set 0$.
The structure $\struct{\N_{>0}, \times}$ forms a commutative monoid. | From Non-Zero Natural Numbers under Multiplication form Commutative Semigroup, $\struct {\N_{>0}, \times}$ forms a commutative semigroup.
From Identity Element of Natural Number Multiplication is One, $\struct {\N_{>0}, \times}$ has an identity element which is $1$.
Hence the result, by definition of commutative monoid... | Let $\N_{>0}$ be the set of [[Definition:Natural Numbers|natural numbers]] without [[Definition:Zero (Number)|zero]], i.e. $\N_{>0} = \N \setminus \set 0$.
The [[Definition:Algebraic Structure with One Operation|structure]] $\struct{\N_{>0}, \times}$ forms a [[Definition:Commutative Monoid|commutative monoid]]. | From [[Non-Zero Natural Numbers under Multiplication form Commutative Semigroup]], $\struct {\N_{>0}, \times}$ forms a [[Definition:Commutative Semigroup|commutative semigroup]].
From [[Identity Element of Natural Number Multiplication is One]], $\struct {\N_{>0}, \times}$ has an [[Definition:Identity Element|identity... | Non-Zero Natural Numbers under Multiplication form Commutative Monoid | https://proofwiki.org/wiki/Non-Zero_Natural_Numbers_under_Multiplication_form_Commutative_Monoid | https://proofwiki.org/wiki/Non-Zero_Natural_Numbers_under_Multiplication_form_Commutative_Monoid | [
"Natural Number Multiplication",
"Examples of Commutative Monoids"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Algebraic Structure/One Operation",
"Definition:Commutative Monoid"
] | [
"Non-Zero Natural Numbers under Multiplication form Commutative Semigroup",
"Definition:Commutative Semigroup",
"Identity Element of Natural Number Multiplication is One",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutative Monoid"
] |
proofwiki-8314 | Non-Zero Natural Numbers under Addition do not form Monoid | Let $\N_{>0}$ be the set of natural numbers without zero, i.e. $\N_{>0} = \N \setminus \set 0$.
The structure $\struct {\N_{>0}, +}$ does ''not'' form a monoid. | From Natural Numbers under Addition form Commutative Monoid, $\struct {\N, +}$ forms a commutative monoid.
From Natural Numbers Bounded Below under Addition form Commutative Semigroup, $\struct {\N_{>0}, +}$ forms a commutative semigroup.
From Identity Element of Natural Number Addition is Zero, $0$ is the identity of ... | Let $\N_{>0}$ be the set of [[Definition:Natural Numbers|natural numbers]] without [[Definition:Zero (Number)|zero]], i.e. $\N_{>0} = \N \setminus \set 0$.
The [[Definition:Algebraic Structure with One Operation|structure]] $\struct {\N_{>0}, +}$ does ''not'' form a [[Definition:Monoid|monoid]]. | From [[Natural Numbers under Addition form Commutative Monoid]], $\struct {\N, +}$ forms a [[Definition:Commutative Monoid|commutative monoid]].
From [[Natural Numbers Bounded Below under Addition form Commutative Semigroup]], $\struct {\N_{>0}, +}$ forms a [[Definition:Commutative Semigroup|commutative semigroup]].
... | Non-Zero Natural Numbers under Addition do not form Monoid | https://proofwiki.org/wiki/Non-Zero_Natural_Numbers_under_Addition_do_not_form_Monoid | https://proofwiki.org/wiki/Non-Zero_Natural_Numbers_under_Addition_do_not_form_Monoid | [
"Natural Number Addition",
"Examples of Monoids"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Algebraic Structure/One Operation",
"Definition:Monoid"
] | [
"Natural Numbers under Addition form Commutative Monoid",
"Definition:Commutative Monoid",
"Natural Numbers Bounded Below under Addition form Commutative Semigroup",
"Definition:Commutative Semigroup",
"Identity Element of Natural Number Addition is Zero",
"Definition:Identity (Abstract Algebra)/Two-Sided... |
proofwiki-8315 | Natural Numbers Bounded Below under Addition form Commutative Semigroup | Let $m \in \N$ where $\N$ is the set of natural numbers.
Let $M \subseteq \N$ be defined as:
:$M := \set {x \in \N: x \ge m}$
That is, $M$ is the set of all natural numbers greater than or equal to $m$.
Then the algebraic structure $\struct {M, +}$ is a commutative semigroup. | We have that:
:Natural Number Addition is Associative
:Natural Number Addition is Commutative
From Restriction of Associative Operation is Associative, $+$ is associative on $\struct {M, +}$.
From Restriction of Commutative Operation is Commutative, $+$ is commutative on $\struct {M, +}$.
It remains to be shown that $+... | Let $m \in \N$ where $\N$ is the [[Definition:Natural Numbers|set of natural numbers]].
Let $M \subseteq \N$ be defined as:
:$M := \set {x \in \N: x \ge m}$
That is, $M$ is the set of all [[Definition:Natural Numbers|natural numbers]] greater than or equal to $m$.
Then the [[Definition:Algebraic Structure with One Op... | We have that:
:[[Natural Number Addition is Associative]]
:[[Natural Number Addition is Commutative]]
From [[Restriction of Associative Operation is Associative]], $+$ is [[Definition:Associative Operation|associative]] on $\struct {M, +}$.
From [[Restriction of Commutative Operation is Commutative]], $+$ is [[Defini... | Natural Numbers Bounded Below under Addition form Commutative Semigroup | https://proofwiki.org/wiki/Natural_Numbers_Bounded_Below_under_Addition_form_Commutative_Semigroup | https://proofwiki.org/wiki/Natural_Numbers_Bounded_Below_under_Addition_form_Commutative_Semigroup | [
"Examples of Commutative Semigroups",
"Natural Number Addition"
] | [
"Definition:Natural Numbers",
"Definition:Natural Numbers",
"Definition:Algebraic Structure/One Operation",
"Definition:Commutative Semigroup"
] | [
"Natural Number Addition is Associative",
"Natural Number Addition is Commutative",
"Restriction of Associative Operation is Associative",
"Definition:Associative Operation",
"Restriction of Commutative Operation is Commutative",
"Definition:Commutative/Operation",
"Definition:Closure (Abstract Algebra)... |
proofwiki-8316 | Positive Real Numbers under Max Operation form Monoid | Let $\R_{\ge 0}$ be the set of positive (that is, non-negative) real numbers.
Let $\max: \R_{\ge 0}^2 \to \R_{\ge 0}$ be the max operation on $\R_{\ge 0}$.
Then $\struct {\R_{\ge 0}, \max}$ is a monoid whose identity is $0$. | From Real Numbers are Totally Ordered, $\R$ is a totally ordered set.
From Max Operation on Toset forms Semigroup, $\struct {\R_{\ge 0}, \max}$ is a semigroup.
By definition of $\R_{\ge 0}$:
:$\forall x \in \R_{\ge 0}: 0 \le x$
Thus by definition of the max operation:
:$\forall x \in \R_{\ge 0}: \map \max {0, x} = x = ... | Let $\R_{\ge 0}$ be the set of [[Definition:Positive Real Number|positive (that is, non-negative) real numbers]].
Let $\max: \R_{\ge 0}^2 \to \R_{\ge 0}$ be the [[Definition:Max Operation|max operation]] on $\R_{\ge 0}$.
Then $\struct {\R_{\ge 0}, \max}$ is a [[Definition:Monoid|monoid]] whose [[Definition:Identity ... | From [[Real Numbers are Totally Ordered]], $\R$ is a [[Definition:Totally Ordered Set|totally ordered set]].
From [[Max Operation on Toset forms Semigroup]], $\struct {\R_{\ge 0}, \max}$ is a [[Definition:Semigroup|semigroup]].
By definition of $\R_{\ge 0}$:
:$\forall x \in \R_{\ge 0}: 0 \le x$
Thus by definition of... | Positive Real Numbers under Max Operation form Monoid | https://proofwiki.org/wiki/Positive_Real_Numbers_under_Max_Operation_form_Monoid | https://proofwiki.org/wiki/Positive_Real_Numbers_under_Max_Operation_form_Monoid | [
"Real Numbers",
"Max Operation",
"Examples of Monoids"
] | [
"Definition:Positive/Real Number",
"Definition:Max Operation",
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Real Numbers are Totally Ordered",
"Definition:Totally Ordered Set",
"Max Operation on Toset forms Semigroup",
"Definition:Semigroup",
"Definition:Max Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Monoid"
] |
proofwiki-8317 | Free Commutative Monoid on One Element is Isomorphic to Natural Numbers under Addition | Let $X = \set x$ be a singleton.
Let $M$ be the free commutative monoid on $X$.
Then $M$ is isomorphic to the additive monoid of natural numbers. | By definition, the free commutative monoid $M$ on $\set x$ is:
:$M = \set {e, x, x^2, x^3, \ldots}$
where $e$ denotes the null sequence of elements of $X$.
Let $\phi$ denote the mapping from $M$ to $\N$ as:
:$\forall a \in M: \map \phi a = \begin{cases}
0 & : a = e \\
n & : a = x^n \end{cases}$
By definition of $\phi$:... | Let $X = \set x$ be a [[Definition:Singleton|singleton]].
Let $M$ be the [[Definition:Free Commutative Monoid|free commutative monoid]] on $X$.
Then $M$ is [[Definition:Monoid Isomorphism|isomorphic]] to the [[Definition:Additive Monoid of Natural Numbers|additive monoid of natural numbers]]. | By definition, the [[Definition:Free Commutative Monoid|free commutative monoid]] $M$ on $\set x$ is:
:$M = \set {e, x, x^2, x^3, \ldots}$
where $e$ denotes the [[Definition:Null Sequence|null sequence]] of elements of $X$.
Let $\phi$ denote the [[Definition:Mapping|mapping]] from $M$ to $\N$ as:
:$\forall a \in M: \m... | Free Commutative Monoid on One Element is Isomorphic to Natural Numbers under Addition | https://proofwiki.org/wiki/Free_Commutative_Monoid_on_One_Element_is_Isomorphic_to_Natural_Numbers_under_Addition | https://proofwiki.org/wiki/Free_Commutative_Monoid_on_One_Element_is_Isomorphic_to_Natural_Numbers_under_Addition | [
"Natural Number Addition",
"Free Monoids"
] | [
"Definition:Singleton",
"Definition:Free Commutative Monoid",
"Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism",
"Definition:Additive Monoid of Natural Numbers"
] | [
"Definition:Free Commutative Monoid",
"Definition:Null Sequence",
"Definition:Mapping",
"Definition:Injection",
"Definition:Surjection",
"Definition:Monoid Homomorphism",
"Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism"
] |
proofwiki-8318 | Finite Monoid with Right Cancellable Operation is Group | Let $\struct {S, \circ}$ be a finite monoid.
Let $\circ$ be a right cancellable operation.
Then $\struct {S, \circ}$ is a group. | {{Group-axiom|0}}, {{Group-axiom|1}} and {{Group-axiom|2}} are satisfied by dint of $\struct {S, \circ}$ being a monoid.
Recall the definition of right cancellable operation:
:$\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$
Let $\rho_c: S \to S$ be the right regular representation of $\struct {S, \circ}$ ... | Let $\struct {S, \circ}$ be a [[Definition:Finite Monoid|finite monoid]].
Let $\circ$ be a [[Definition:Right Cancellable Operation|right cancellable operation]].
Then $\struct {S, \circ}$ is a [[Definition:Group|group]]. | {{Group-axiom|0}}, {{Group-axiom|1}} and {{Group-axiom|2}} are satisfied by dint of $\struct {S, \circ}$ being a [[Definition:Monoid|monoid]].
Recall the definition of [[Definition:Right Cancellable Operation|right cancellable operation]]:
:$\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$
Let $\rho_c: S ... | Finite Monoid with Right Cancellable Operation is Group | https://proofwiki.org/wiki/Finite_Monoid_with_Right_Cancellable_Operation_is_Group | https://proofwiki.org/wiki/Finite_Monoid_with_Right_Cancellable_Operation_is_Group | [
"Monoids",
"Finite Groups"
] | [
"Definition:Finite Monoid",
"Definition:Right Cancellable Operation",
"Definition:Group"
] | [
"Definition:Monoid",
"Definition:Right Cancellable Operation",
"Definition:Regular Representations/Right Regular Representation",
"Right Cancellable iff Right Regular Representation Injective",
"Definition:Injection",
"Right Regular Representation wrt Right Cancellable Element on Finite Semigroup is Bijec... |
proofwiki-8319 | Finite Monoid with Left Cancellable Operation is Group | Let $\struct {S, \circ}$ be a finite monoid.
Let $\circ$ be a left cancellable operation.
Then $\struct {S, \circ}$ is a group. | {{Group-axiom|0}}, {{Group-axiom|1}} and {{Group-axiom|2}} are satisfied by dint of $\paren {S, \circ}$ being a monoid.
Recall the definition of left cancellable operation:
:$\forall a, b, c \in S: c \circ a = c \circ b \implies a = b$
Let $\lambda_c: S \to S$ be the left regular representation of $\struct {S, \circ}$ ... | Let $\struct {S, \circ}$ be a [[Definition:Finite Monoid|finite monoid]].
Let $\circ$ be a [[Definition:Left Cancellable Operation|left cancellable operation]].
Then $\struct {S, \circ}$ is a [[Definition:Group|group]]. | {{Group-axiom|0}}, {{Group-axiom|1}} and {{Group-axiom|2}} are satisfied by dint of $\paren {S, \circ}$ being a [[Definition:Monoid|monoid]].
Recall the definition of [[Definition:Left Cancellable Operation|left cancellable operation]]:
:$\forall a, b, c \in S: c \circ a = c \circ b \implies a = b$
Let $\lambda_c: S ... | Finite Monoid with Left Cancellable Operation is Group | https://proofwiki.org/wiki/Finite_Monoid_with_Left_Cancellable_Operation_is_Group | https://proofwiki.org/wiki/Finite_Monoid_with_Left_Cancellable_Operation_is_Group | [
"Monoids",
"Finite Groups"
] | [
"Definition:Finite Monoid",
"Definition:Left Cancellable Operation",
"Definition:Group"
] | [
"Definition:Monoid",
"Definition:Left Cancellable Operation",
"Definition:Regular Representations/Left Regular Representation",
"Left Cancellable iff Left Regular Representation Injective",
"Definition:Injection",
"Left Regular Representation wrt Left Cancellable Element on Finite Semigroup is Bijection",... |
proofwiki-8320 | Multiplicative Group of Field is Abelian Group | Let $\struct {F, +, \times}$ be a field.
Let $F^* := F \setminus \set 0$ be the set $F$ less its zero.
The algebraic structure $\struct {F^*, \times}$ is an abelian group. | From the field axioms:
{{begin-axiom}}
{{axiom | n = \text M 0
| lc= Closure under product
| q = \forall x, y \in F
| m = x \circ y \in F
}}
{{axiom | n = \text M 1
| lc= Associativity of product
| q = \forall x, y, z \in F
| m = \paren {x \circ y} \circ z = x \circ \pare... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $F^* := F \setminus \set 0$ be the set $F$ less its [[Definition:Field Zero|zero]].
The [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\struct {F^*, \times}$ is an [[Definition:Abelian Group|abelian group... | From the [[Axiom:Field Axioms|field axioms]]:
{{begin-axiom}}
{{axiom | n = \text M 0
| lc= [[Definition:Closed Algebraic Structure|Closure]] under [[Definition:Field Product|product]]
| q = \forall x, y \in F
| m = x \circ y \in F
}}
{{axiom | n = \text M 1
| lc= [[Definition:Associati... | Multiplicative Group of Field is Abelian Group/Proof 1 | https://proofwiki.org/wiki/Multiplicative_Group_of_Field_is_Abelian_Group | https://proofwiki.org/wiki/Multiplicative_Group_of_Field_is_Abelian_Group/Proof_1 | [
"Field Theory",
"Abelian Groups",
"Multiplicative Group of Field is Abelian Group"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Algebraic Structure/One Operation",
"Definition:Abelian Group"
] | [
"Axiom:Field Axioms",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Field (Abstract Algebra)/Product",
"Definition:Associative Operation",
"Definition:Field (Abstract Algebra)/Product",
"Definition:Commutative/Operation",
"Definition:Field (Abstract Algebra)/Product",
"Defin... |
proofwiki-8321 | Multiplicative Group of Field is Abelian Group | Let $\struct {F, +, \times}$ be a field.
Let $F^* := F \setminus \set 0$ be the set $F$ less its zero.
The algebraic structure $\struct {F^*, \times}$ is an abelian group. | Recall that a field is a non-trivial commutative division ring.
The result follows from Non-Zero Elements of Division Ring form Group.
{{qed}} | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $F^* := F \setminus \set 0$ be the set $F$ less its [[Definition:Field Zero|zero]].
The [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\struct {F^*, \times}$ is an [[Definition:Abelian Group|abelian group... | Recall that a [[Definition:Field (Abstract Algebra)|field]] is a [[Definition:Non-Trivial Ring|non-trivial]] [[Definition:Commutative Division Ring|commutative division ring]].
The result follows from [[Non-Zero Elements of Division Ring form Group]].
{{qed}} | Multiplicative Group of Field is Abelian Group/Proof 2 | https://proofwiki.org/wiki/Multiplicative_Group_of_Field_is_Abelian_Group | https://proofwiki.org/wiki/Multiplicative_Group_of_Field_is_Abelian_Group/Proof_2 | [
"Field Theory",
"Abelian Groups",
"Multiplicative Group of Field is Abelian Group"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Algebraic Structure/One Operation",
"Definition:Abelian Group"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Non-Trivial Ring",
"Definition:Commutative Division Ring",
"Non-Zero Elements of Division Ring form Group"
] |
proofwiki-8322 | One and Minus One form Subgroup of Multiplicative Group of Rational Numbers | Let $\struct {\Q_{\ne 0}, \times}$ be the multiplicative group of rational numbers.
Let $S \subseteq \Q$ where $S = \set {1, -1}$.
Then $\struct {S, \times}$ is a subgroup of $\struct {\Q_{\ne 0}, \times}$. | By hypothesis, $S$ is not empty.
As $0 \notin S$, it follows that $S \subseteq \Q_{\ne 0}$.
Recall that $-1 \times -1 = 1$ and also $1 \times 1 = 1$.
Thus:
:$\forall x \in S: x \times y^{-1} \in S$
The result follows from the One-Step Subgroup Test.
{{qed}} | Let $\struct {\Q_{\ne 0}, \times}$ be the [[Definition:Multiplicative Group of Rational Numbers|multiplicative group of rational numbers]].
Let $S \subseteq \Q$ where $S = \set {1, -1}$.
Then $\struct {S, \times}$ is a [[Definition:Subgroup|subgroup]] of $\struct {\Q_{\ne 0}, \times}$. | [[Definition:By Hypothesis|By hypothesis]], $S$ is not [[Definition:Empty Set|empty]].
As $0 \notin S$, it follows that $S \subseteq \Q_{\ne 0}$.
Recall that $-1 \times -1 = 1$ and also $1 \times 1 = 1$.
Thus:
:$\forall x \in S: x \times y^{-1} \in S$
The result follows from the [[One-Step Subgroup Test]].
{{qed}} | One and Minus One form Subgroup of Multiplicative Group of Rational Numbers | https://proofwiki.org/wiki/One_and_Minus_One_form_Subgroup_of_Multiplicative_Group_of_Rational_Numbers | https://proofwiki.org/wiki/One_and_Minus_One_form_Subgroup_of_Multiplicative_Group_of_Rational_Numbers | [
"Rational Numbers",
"Examples of Subgroups"
] | [
"Definition:Multiplicative Group of Rational Numbers",
"Definition:Subgroup"
] | [
"Definition:By Hypothesis",
"Definition:Empty Set",
"One-Step Subgroup Test"
] |
proofwiki-8323 | Symmetric Group is not Abelian | Let $S_n$ be the symmetric group of order $n$ where $n \ge 3$.
Then $S_n$ is not abelian. | Let $\alpha \in S_n$ such that $\alpha$ is not the identity mapping.
From Center of Symmetric Group is Trivial, $\alpha$ is not in the center $\map Z {S_n}$ of $S_n$.
Thus $S_n \ne \map Z {S_n}$.
The result follows by definition of abelian group.
{{qed}} | Let $S_n$ be the [[Definition:Symmetric Group|symmetric group]] of order $n$ where $n \ge 3$.
Then $S_n$ is not [[Definition:Abelian Group|abelian]]. | Let $\alpha \in S_n$ such that $\alpha$ is not the [[Definition:Identity Mapping|identity mapping]].
From [[Center of Symmetric Group is Trivial]], $\alpha$ is not in the [[Definition:Center of Group|center]] $\map Z {S_n}$ of $S_n$.
Thus $S_n \ne \map Z {S_n}$.
The result follows by definition of [[Definition:Abeli... | Symmetric Group is not Abelian/Proof 1 | https://proofwiki.org/wiki/Symmetric_Group_is_not_Abelian | https://proofwiki.org/wiki/Symmetric_Group_is_not_Abelian/Proof_1 | [
"Symmetric Groups",
"Symmetric Group is not Abelian"
] | [
"Definition:Symmetric Group",
"Definition:Abelian Group"
] | [
"Definition:Identity Mapping",
"Center of Symmetric Group is Trivial",
"Definition:Center (Abstract Algebra)/Group",
"Definition:Abelian Group"
] |
proofwiki-8324 | Symmetric Group is not Abelian | Let $S_n$ be the symmetric group of order $n$ where $n \ge 3$.
Then $S_n$ is not abelian. | Let $a, b, c \in S$.
Let $\alpha$ be the transposition on $S$ which exchanges $a$ and $b$.
Let $\beta$ be the transposition on $S$ which exchanges $b$ and $c$.
Then:
:$\alpha \circ \beta$ maps $\tuple {a, b, c}$ to $\tuple {c, a, b}$
while:
:$\beta \circ \alpha$ maps $\tuple {a, b, c}$ to $\tuple {b, c, a}$
Thus $\alph... | Let $S_n$ be the [[Definition:Symmetric Group|symmetric group]] of order $n$ where $n \ge 3$.
Then $S_n$ is not [[Definition:Abelian Group|abelian]]. | Let $a, b, c \in S$.
Let $\alpha$ be the [[Definition:Transposition|transposition]] on $S$ which exchanges $a$ and $b$.
Let $\beta$ be the [[Definition:Transposition|transposition]] on $S$ which exchanges $b$ and $c$.
Then:
:$\alpha \circ \beta$ maps $\tuple {a, b, c}$ to $\tuple {c, a, b}$
while:
:$\beta \circ \al... | Symmetric Group is not Abelian/Proof 2 | https://proofwiki.org/wiki/Symmetric_Group_is_not_Abelian | https://proofwiki.org/wiki/Symmetric_Group_is_not_Abelian/Proof_2 | [
"Symmetric Groups",
"Symmetric Group is not Abelian"
] | [
"Definition:Symmetric Group",
"Definition:Abelian Group"
] | [
"Definition:Transposition",
"Definition:Transposition",
"Definition:Commutative/Elements",
"Definition:Abelian Group"
] |
proofwiki-8325 | Möbius Function is Multiplicative/Corollary | Let $\gcd \set {m, n} > 1$.
Then:
:$\map \mu {m n} = 0$ | Let $\gcd \set {m, n} = k$ where $k > 1$.
Then $m = k r$ and $n = k s$ for some $r, s \in \Z$.
Thus $m n = k^2 r s$.
From Integer is Expressible as Product of Primes there exists $p \in \Z$ such that $p$ is prime and $p \divides k$.
That is:
:$\exists t \in \Z: k = p t$
and so:
:$m n = p^2 t^2 r s$
That is:
:$p^2 \divi... | Let $\gcd \set {m, n} > 1$.
Then:
:$\map \mu {m n} = 0$ | Let $\gcd \set {m, n} = k$ where $k > 1$.
Then $m = k r$ and $n = k s$ for some $r, s \in \Z$.
Thus $m n = k^2 r s$.
From [[Integer is Expressible as Product of Primes]] there exists $p \in \Z$ such that $p$ is [[Definition:Prime Number|prime]] and $p \divides k$.
That is:
:$\exists t \in \Z: k = p t$
and so:
:$m ... | Möbius Function is Multiplicative/Corollary | https://proofwiki.org/wiki/Möbius_Function_is_Multiplicative/Corollary | https://proofwiki.org/wiki/Möbius_Function_is_Multiplicative/Corollary | [
"Multiplicative Functions",
"Möbius Function"
] | [] | [
"Integer is Expressible as Product of Primes",
"Definition:Prime Number",
"Definition:Möbius Function",
"Category:Multiplicative Functions",
"Category:Möbius Function"
] |
proofwiki-8326 | Affine Group of One Dimension is Group | Let $\map {\operatorname {Af}_1} \R$ be the $1$-dimensional affine group on $\R$.
Then $\map {\operatorname {Af}_1} \R$ is a group. | Taking the group axioms in turn:
=== {{Group-axiom|0|nolink}} ===
Let :
:$a, c \in \R_{\ne 0} \land b, d \in \R$
Let:
:$f_{ab}, f_{cd} \in \map {\operatorname {Af}_1} \R$
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {f_{ab} \circ f_{cd} } } x
| r = \map {f_{ab} } {\map {f_{cd} } x}
| c = {{Defof|Composition... | Let $\map {\operatorname {Af}_1} \R$ be the [[Definition:Affine Group of One Dimension|$1$-dimensional affine group on $\R$]].
Then $\map {\operatorname {Af}_1} \R$ is a [[Definition:Group|group]]. | Taking the [[Axiom:Group Axioms|group axioms]] in turn:
=== {{Group-axiom|0|nolink}} ===
Let :
:$a, c \in \R_{\ne 0} \land b, d \in \R$
Let:
:$f_{ab}, f_{cd} \in \map {\operatorname {Af}_1} \R$
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {f_{ab} \circ f_{cd} } } x
| r = \map {f_{ab} } {\map {f_{cd} } x}
... | Affine Group of One Dimension is Group/Proof 1 | https://proofwiki.org/wiki/Affine_Group_of_One_Dimension_is_Group | https://proofwiki.org/wiki/Affine_Group_of_One_Dimension_is_Group/Proof_1 | [
"Affine Group of One Dimension is Group",
"Affine Groups"
] | [
"Definition:Affine Group of One Dimension",
"Definition:Group"
] | [
"Axiom:Group Axioms",
"Axiom:Field Axioms",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Composition of Mappings is Associative",
"Definition:Associative Operation",
"Identity of Affine Group of One Dimension",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Inverse in A... |
proofwiki-8327 | Affine Group of One Dimension is Group | Let $\map {\operatorname {Af}_1} \R$ be the $1$-dimensional affine group on $\R$.
Then $\map {\operatorname {Af}_1} \R$ is a group. | The result follows from Affine Group of One Dimension as Semidirect Product and Semidirect Product of Groups is Group.
{{qed}} | Let $\map {\operatorname {Af}_1} \R$ be the [[Definition:Affine Group of One Dimension|$1$-dimensional affine group on $\R$]].
Then $\map {\operatorname {Af}_1} \R$ is a [[Definition:Group|group]]. | The result follows from [[Affine Group of One Dimension as Semidirect Product]] and [[Semidirect Product of Groups is Group]].
{{qed}} | Affine Group of One Dimension is Group/Proof 2 | https://proofwiki.org/wiki/Affine_Group_of_One_Dimension_is_Group | https://proofwiki.org/wiki/Affine_Group_of_One_Dimension_is_Group/Proof_2 | [
"Affine Group of One Dimension is Group",
"Affine Groups"
] | [
"Definition:Affine Group of One Dimension",
"Definition:Group"
] | [
"Affine Group of One Dimension as Semidirect Product",
"Semidirect Product of Groups is Group"
] |
proofwiki-8328 | Identity of Affine Group of One Dimension | Let $\map {\mathrm {Af}_1} \R$ denote the $1$-dimensional affine group on $\R$.
Then $\map {\mathrm {Af}_1} \R$ has $f_{1, 0}$ as an identity element. | Let $f_{a b} \in \map {\mathrm {Af}_1} \R$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {f_{a b} \circ f_{1, 0} } } x
| r = a \paren {1 x + 0} + b
| c =
}}
{{eqn | r = a x + b
| c =
}}
{{eqn | r = f_{a b}
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\paren {f_{1, 0} \circ f_{a b} } }... | Let $\map {\mathrm {Af}_1} \R$ denote the [[Definition:Affine Group of One Dimension|$1$-dimensional affine group on $\R$]].
Then $\map {\mathrm {Af}_1} \R$ has $f_{1, 0}$ as an [[Definition:Identity Element|identity element]]. | Let $f_{a b} \in \map {\mathrm {Af}_1} \R$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {f_{a b} \circ f_{1, 0} } } x
| r = a \paren {1 x + 0} + b
| c =
}}
{{eqn | r = a x + b
| c =
}}
{{eqn | r = f_{a b}
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map {\paren {f_{1, 0} \circ f_{a b} ... | Identity of Affine Group of One Dimension | https://proofwiki.org/wiki/Identity_of_Affine_Group_of_One_Dimension | https://proofwiki.org/wiki/Identity_of_Affine_Group_of_One_Dimension | [
"Affine Groups"
] | [
"Definition:Affine Group of One Dimension",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Category:Affine Groups"
] |
proofwiki-8329 | Inverse in Affine Group of One Dimension | Let $\map {\operatorname {Af}_1} \R$ denote the $1$-dimensional affine group on $\R$.
Let $f_{a b} \in \map {\operatorname {Af}_1} \R$.
Let $c = \dfrac 1 a$ and $d = \dfrac {-b} a$.
Then $f_{c d} \in \map {\operatorname {Af}_1} \R$ is the inverse of $f_{a b}$. | {{begin-eqn}}
{{eqn | l = y
| r = a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = y - b
| r = a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {y - b} a
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac y a + \frac {- b} a
| r = x
| c =
}}
{{end-eqn}}
As $a... | Let $\map {\operatorname {Af}_1} \R$ denote the [[Definition:Affine Group of One Dimension|$1$-dimensional affine group on $\R$]].
Let $f_{a b} \in \map {\operatorname {Af}_1} \R$.
Let $c = \dfrac 1 a$ and $d = \dfrac {-b} a$.
Then $f_{c d} \in \map {\operatorname {Af}_1} \R$ is the [[Definition:Inverse Element|inv... | {{begin-eqn}}
{{eqn | l = y
| r = a x + b
| c =
}}
{{eqn | ll= \leadsto
| l = y - b
| r = a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {y - b} a
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac y a + \frac {- b} a
| r = x
| c =
}}
{{end-eqn}}
As $... | Inverse in Affine Group of One Dimension | https://proofwiki.org/wiki/Inverse_in_Affine_Group_of_One_Dimension | https://proofwiki.org/wiki/Inverse_in_Affine_Group_of_One_Dimension | [
"Affine Groups"
] | [
"Definition:Affine Group of One Dimension",
"Definition:Inverse (Abstract Algebra)/Inverse"
] | [] |
proofwiki-8330 | Commutation with Group Elements implies Commuation with Product with Inverse | Let $G$ be a group.
Let $a, b, c \in G$ such that $a$ commutes with both $b$ and $c$.
Then $a$ commutes with $b c^{-1}$. | {{begin-eqn}}
{{eqn | l = a b c^{-1}
| r = b a c^{-1}
| c = as $a$ commutes with $b$
}}
{{eqn | r = b c^{-1} a
| c = Commutation with Inverse in Monoid
}}
{{end-eqn}}
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $a, b, c \in G$ such that $a$ [[Definition:Commute|commutes]] with both $b$ and $c$.
Then $a$ [[Definition:Commute|commutes]] with $b c^{-1}$. | {{begin-eqn}}
{{eqn | l = a b c^{-1}
| r = b a c^{-1}
| c = as $a$ [[Definition:Commute|commutes]] with $b$
}}
{{eqn | r = b c^{-1} a
| c = [[Commutation with Inverse in Monoid]]
}}
{{end-eqn}}
{{qed}} | Commutation with Group Elements implies Commuation with Product with Inverse | https://proofwiki.org/wiki/Commutation_with_Group_Elements_implies_Commuation_with_Product_with_Inverse | https://proofwiki.org/wiki/Commutation_with_Group_Elements_implies_Commuation_with_Product_with_Inverse | [
"Group Theory",
"Commutativity"
] | [
"Definition:Group",
"Definition:Commutative/Elements",
"Definition:Commutative/Elements"
] | [
"Definition:Commutative/Elements",
"Commutation with Inverse in Monoid"
] |
proofwiki-8331 | Möbius Strip has Euler Characteristic Zero | Let $M$ be a Möbius Strip.
Then:
:$\map \chi M = 0$
where $\map \chi M$ denotes the Euler characteristic of the graph $M$. | Let the number of vertices, edges and faces of $M$ be $V$, $E$ and $F$ respectively.
From Möbius Strip has no Vertices:
:$V = 0$
From Möbius Strip has 1 Edge:
:$E = 1$
From Möbius Strip has 1 Face:
:$F = 1$
By definition of the Euler characteristic:
{{begin-eqn}}
{{eqn | l = \map \chi M
| r = V - E + F
| c ... | Let $M$ be a [[Definition:Möbius Strip|Möbius Strip]].
Then:
:$\map \chi M = 0$
where $\map \chi M$ denotes the [[Definition:Euler Characteristic|Euler characteristic]] of the [[Definition:Graph (Graph Theory)|graph]] $M$. | Let the number of [[Definition:Vertex of Graph|vertices]], [[Definition:Edge of Graph|edges]] and [[Definition:Face of Graph|faces]] of $M$ be $V$, $E$ and $F$ respectively.
From [[Möbius Strip has no Vertices]]:
:$V = 0$
From [[Möbius Strip has 1 Edge]]:
:$E = 1$
From [[Möbius Strip has 1 Face]]:
:$F = 1$
By defin... | Möbius Strip has Euler Characteristic Zero | https://proofwiki.org/wiki/Möbius_Strip_has_Euler_Characteristic_Zero | https://proofwiki.org/wiki/Möbius_Strip_has_Euler_Characteristic_Zero | [
"Möbius Strip"
] | [
"Definition:Möbius Strip",
"Definition:Euler Characteristic",
"Definition:Graph (Graph Theory)"
] | [
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Planar Graph/Face",
"Möbius Strip has no Vertices",
"Möbius Strip has 1 Edge",
"Möbius Strip has 1 Face",
"Definition:Euler Characteristic"
] |
proofwiki-8332 | Centralizer of Subset is Intersection of Centralizers of Elements | Let $\struct {G, \circ}$ be a group.
Let $S \subseteq G$.
Let $\map {C_G} S$ be the centralizer of $S$ in $G$.
Then:
:$\ds \map {C_G} S = \bigcap_{x \mathop \in S} \map {C_G} x$
where $\map {C_G} z$ is the centralizer of $x$ in $G$. | {{begin-eqn}}
{{eqn | l = \map {C_G} S
| r = \set {g \in G: \forall x \in S: g \circ x = x \circ g}
| c = {{Defof|Centralizer of Group Subset}}
}}
{{eqn | r = \set {g \in G: \forall x \in S: g \in \map {C_G} x}
| c = {{Defof|Centralizer of Group Element}}
}}
{{eqn | r = \bigcap_{x \mathop \in S} \map ... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $S \subseteq G$.
Let $\map {C_G} S$ be the [[Definition:Centralizer of Group Subset|centralizer of $S$ in $G$]].
Then:
:$\ds \map {C_G} S = \bigcap_{x \mathop \in S} \map {C_G} x$
where $\map {C_G} z$ is the [[Definition:Centralizer of Group Element|cent... | {{begin-eqn}}
{{eqn | l = \map {C_G} S
| r = \set {g \in G: \forall x \in S: g \circ x = x \circ g}
| c = {{Defof|Centralizer of Group Subset}}
}}
{{eqn | r = \set {g \in G: \forall x \in S: g \in \map {C_G} x}
| c = {{Defof|Centralizer of Group Element}}
}}
{{eqn | r = \bigcap_{x \mathop \in S} \map ... | Centralizer of Subset is Intersection of Centralizers of Elements | https://proofwiki.org/wiki/Centralizer_of_Subset_is_Intersection_of_Centralizers_of_Elements | https://proofwiki.org/wiki/Centralizer_of_Subset_is_Intersection_of_Centralizers_of_Elements | [
"Centralizers",
"Subsets"
] | [
"Definition:Group",
"Definition:Centralizer/Group Subset",
"Definition:Centralizer/Group Element"
] | [] |
proofwiki-8333 | Permutation Representation defines Group Action | Let $G$ be a group whose identity is $e$.
Let $X$ be a set.
Let $\map \Gamma X$ be the symmetric group of $X$.
Let $\rho: G \to \map \Gamma X$ be a permutation representation, that is, a homomorphism.
The mapping $\phi: G \times X \to X$ associated to $\rho$, defined by:
:$\map \phi {g, x} = \map {\map \rho g} x$
is a ... | Let $g, h \in G$ and $x \in X$.
We verify that $g * \paren {h * x} = \paren {g h} * x$:
{{begin-eqn}}
{{eqn | l = g * \paren {h * x}
| r = \map {\map \rho g} {\map {\map \rho h} x}
| c = Definition of $\phi$
}}
{{eqn | r = \map {\paren {\map \rho g \circ \map \rho h} } x
| c = {{Defof|Composition of M... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $X$ be a [[Definition:Set|set]].
Let $\map \Gamma X$ be the [[Definition:Symmetric Group|symmetric group]] of $X$.
Let $\rho: G \to \map \Gamma X$ be a [[Definition:Permutation Representation|permutation representation... | Let $g, h \in G$ and $x \in X$.
We verify that $g * \paren {h * x} = \paren {g h} * x$:
{{begin-eqn}}
{{eqn | l = g * \paren {h * x}
| r = \map {\map \rho g} {\map {\map \rho h} x}
| c = Definition of $\phi$
}}
{{eqn | r = \map {\paren {\map \rho g \circ \map \rho h} } x
| c = {{Defof|Composition of... | Permutation Representation defines Group Action | https://proofwiki.org/wiki/Permutation_Representation_defines_Group_Action | https://proofwiki.org/wiki/Permutation_Representation_defines_Group_Action | [
"Group Actions"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Set",
"Definition:Symmetric Group",
"Definition:Group Representation/Permutation",
"Definition:Group Homomorphism",
"Definition:Mapping",
"Definition:Group Action/Permutation Representation",
"Definition:Gro... | [
"Definition:Group Homomorphism",
"Definition:Identity Mapping",
"Group Homomorphism Preserves Identity",
"Set of all Self-Maps under Composition forms Monoid"
] |
proofwiki-8334 | Group Action of Symmetric Group | Let $\N_n$ denote the set $\set {1, 2, \ldots, n}$.
Let $\struct {S_n, \circ}$ denote the symmetric group on $\N_n$.
The mapping $*: S_n \times \N_n \to \N_n$ defined as:
:$\forall \pi \in S_n, \forall n \in \N_n: \pi * n = \map \pi n$
is a group action. | The group action axioms are investigated in turn.
Let $\pi, \rho \in S_n$ and $n \in \N_n$.
Thus:
{{begin-eqn}}
{{eqn | l = \pi * \paren {\rho * n}
| r = \pi * \map \rho n
| c = Definition of $*$
}}
{{eqn | r = \map \pi {\map \rho n}
| c = Definition of $*$
}}
{{eqn | r = \map {\paren {\pi \circ \rho}... | Let $\N_n$ denote the [[Definition:Set|set]] $\set {1, 2, \ldots, n}$.
Let $\struct {S_n, \circ}$ denote the [[Definition:Symmetric Group on n Letters|symmetric group]] on $\N_n$.
The [[Definition:Mapping|mapping]] $*: S_n \times \N_n \to \N_n$ defined as:
:$\forall \pi \in S_n, \forall n \in \N_n: \pi * n = \map \p... | The [[Axiom:Group Action Axioms|group action axioms]] are investigated in turn.
Let $\pi, \rho \in S_n$ and $n \in \N_n$.
Thus:
{{begin-eqn}}
{{eqn | l = \pi * \paren {\rho * n}
| r = \pi * \map \rho n
| c = Definition of $*$
}}
{{eqn | r = \map \pi {\map \rho n}
| c = Definition of $*$
}}
{{eqn |... | Group Action of Symmetric Group | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group | [
"Symmetric Groups",
"Group Action of Symmetric Group"
] | [
"Definition:Set",
"Definition:Symmetric Group/n Letters",
"Definition:Mapping",
"Definition:Group Action"
] | [
"Axiom:Group Action Axioms",
"Definition:Identity Mapping"
] |
proofwiki-8335 | Group Action of Symmetric Group Acts Transitively | Let $S$ be a set.
Let $\struct {\map \Gamma S, \circ}$ be the symmetric group on $S$.
Let $*: \map \Gamma S \times S \to S$ be the group action defined as:
:$\forall \pi \in \map \Gamma S, \forall s \in S: \pi * s = \map \pi s$
Then $*$ is a transitive group action.
In other words, $\struct {\map \Gamma S, \circ}$ acts... | By Group Action of Symmetric Group, $*: \map \Gamma S \times S \to S$ is indeed a group action
Let $s, t \in S$.
As $\map \Gamma S$ is the symmetric group on $S$, there exists a permutation $\pi \in \map \Gamma S$ such that:
:$\map \pi s = t$
This holds for any $s, t \in S$.
Thus:
:$\forall t \in S: t \in \Orb s$
and s... | Let $S$ be a [[Definition:Set|set]].
Let $\struct {\map \Gamma S, \circ}$ be the [[Definition:Symmetric Group|symmetric group]] on $S$.
Let $*: \map \Gamma S \times S \to S$ be the [[Definition:Group Action|group action]] defined as:
:$\forall \pi \in \map \Gamma S, \forall s \in S: \pi * s = \map \pi s$
Then $*$ i... | By [[Group Action of Symmetric Group]], $*: \map \Gamma S \times S \to S$ is indeed a [[Definition:Group Action|group action]]
Let $s, t \in S$.
As $\map \Gamma S$ is the [[Definition:Symmetric Group|symmetric group]] on $S$, there exists a [[Definition:Permutation|permutation]] $\pi \in \map \Gamma S$ such that:
:$\... | Group Action of Symmetric Group Acts Transitively | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_Acts_Transitively | https://proofwiki.org/wiki/Group_Action_of_Symmetric_Group_Acts_Transitively | [
"Group Action of Symmetric Group",
"Transitive Group Actions"
] | [
"Definition:Set",
"Definition:Symmetric Group",
"Definition:Group Action",
"Definition:Transitive Group Action",
"Definition:Transitive Group Action"
] | [
"Group Action of Symmetric Group",
"Definition:Group Action",
"Definition:Symmetric Group",
"Definition:Permutation",
"Definition:Orbit (Group Theory)",
"Definition:Transitive Group Action"
] |
proofwiki-8336 | Trivial Group Action is Group Action | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $S$ be a non-empty set.
{{explain|Why does $S$ have to be non-empty?}}
Let $*: G \times S \to S$ be the trivial group action:
:$\forall \tuple {g, s} \in G \times S: g * s = s$
Then $*$ is indeed a group action. | The group action axioms are investigated in turn.
Let $g_1, g_2 \in G$ and $s \in S$.
Thus:
{{begin-eqn}}
{{eqn | l = g_1 * \paren {g_2 * s}
| r = g_1 * s
| c = {{Defof|Trivial Group Action}}
}}
{{eqn | r = s
| c = {{Defof|Trivial Group Action}}
}}
{{eqn | r = \paren {g_1 \circ g_2} * s
| c = {{... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $S$ be a [[Definition:Non-Empty Set|non-empty set]].
{{explain|Why does $S$ have to be non-empty?}}
Let $*: G \times S \to S$ be the [[Definition:Trivial Group Action|trivial group action]]:
:$\forall ... | The [[Axiom:Group Action Axioms|group action axioms]] are investigated in turn.
Let $g_1, g_2 \in G$ and $s \in S$.
Thus:
{{begin-eqn}}
{{eqn | l = g_1 * \paren {g_2 * s}
| r = g_1 * s
| c = {{Defof|Trivial Group Action}}
}}
{{eqn | r = s
| c = {{Defof|Trivial Group Action}}
}}
{{eqn | r = \paren ... | Trivial Group Action is Group Action | https://proofwiki.org/wiki/Trivial_Group_Action_is_Group_Action | https://proofwiki.org/wiki/Trivial_Group_Action_is_Group_Action | [
"Group Actions",
"Trivial Group Action"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Non-Empty Set",
"Definition:Trivial Group Action",
"Definition:Group Action"
] | [
"Axiom:Group Action Axioms",
"Category:Group Actions",
"Category:Trivial Group Action"
] |
proofwiki-8337 | Orbit of Trivial Group Action is Singleton | Let $\left({G, \circ}\right)$ be a group whose identity is $e$.
Let $S$ be a set.
Let $*: G \times S \to S$ be the trivial group action:
:$\forall \left({g, s}\right) \in G \times S: g * s = s$
Let $s \in S$.
Then the orbit of $s$ under $*$ is $\left\{{s}\right\}$. | By definition:
:$\operatorname{Orb} \left({s}\right) = \left\{{t \in S: \exists g \in G: g * s = t}\right\}$
By definition of the trivial group action:
:$\forall g \in G: g * s = s$
Hence the result.
{{qed}} | Let $\left({G, \circ}\right)$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $S$ be a [[Definition:Set|set]].
Let $*: G \times S \to S$ be the [[Definition:Trivial Group Action|trivial group action]]:
:$\forall \left({g, s}\right) \in G \times S: g * s = s$
Let $s \in S$.
... | By definition:
:$\operatorname{Orb} \left({s}\right) = \left\{{t \in S: \exists g \in G: g * s = t}\right\}$
By definition of the [[Definition:Trivial Group Action|trivial group action]]:
:$\forall g \in G: g * s = s$
Hence the result.
{{qed}} | Orbit of Trivial Group Action is Singleton | https://proofwiki.org/wiki/Orbit_of_Trivial_Group_Action_is_Singleton | https://proofwiki.org/wiki/Orbit_of_Trivial_Group_Action_is_Singleton | [
"Trivial Group Action"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Set",
"Definition:Trivial Group Action",
"Definition:Orbit (Group Theory)"
] | [
"Definition:Trivial Group Action"
] |
proofwiki-8338 | Right Regular Representation by Inverse is Group Action | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $*: G \times G \to G$ be the operation:
:$\forall g, h \in G: g * h = \map {\rho_{g^{-1} } } h$
where $\rho_g$ is the right regular representation of $G$ with respect to $g$.
Then $*$ is a group action. | The group action axioms are investigated in turn.
Let $g, h, a \in G$.
Thus:
{{begin-eqn}}
{{eqn | l = g * \paren {h * a}
| r = g * \map {\rho_{h^{-1} } } a
| c = Definition of $*$
}}
{{eqn | r = g * \paren {a \circ h^{-1} }
| c = {{Defof|Right Regular Representation}}
}}
{{eqn | r = \map {\rho_{g^{-1... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $*: G \times G \to G$ be the [[Definition:Binary Operation|operation]]:
:$\forall g, h \in G: g * h = \map {\rho_{g^{-1} } } h$
where $\rho_g$ is the [[Definition:Right Regular Representation|right regu... | The [[Axiom:Group Action Axioms|group action axioms]] are investigated in turn.
Let $g, h, a \in G$.
Thus:
{{begin-eqn}}
{{eqn | l = g * \paren {h * a}
| r = g * \map {\rho_{h^{-1} } } a
| c = Definition of $*$
}}
{{eqn | r = g * \paren {a \circ h^{-1} }
| c = {{Defof|Right Regular Representation}... | Right Regular Representation by Inverse is Group Action | https://proofwiki.org/wiki/Right_Regular_Representation_by_Inverse_is_Group_Action | https://proofwiki.org/wiki/Right_Regular_Representation_by_Inverse_is_Group_Action | [
"Group Actions"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Operation/Binary Operation",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Group Action"
] | [
"Axiom:Group Action Axioms",
"Inverse of Group Product",
"Inverse of Identity Element is Itself"
] |
proofwiki-8339 | Left Regular Representation is Group Action | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $*: G \times G \to G$ be the operation:
:$\forall g, h \in G: g * h = \map {\lambda_g} h$
where $\lambda_g$ is the left regular representation of $G$ with respect to $g$.
Then $*$ is a group action. | The group action axioms are investigated in turn.
Let $g, h, a \in G$.
Thus:
{{begin-eqn}}
{{eqn | l = g * \paren {h * a}
| r = g * \map {\lambda_h} a
| c = Definition of $*$
}}
{{eqn | r = g * \paren {h \circ a}
| c = {{Defof|Left Regular Representation}}
}}
{{eqn | r = \map {\lambda_g} {h \circ a}
... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $*: G \times G \to G$ be the [[Definition:Binary Operation|operation]]:
:$\forall g, h \in G: g * h = \map {\lambda_g} h$
where $\lambda_g$ is the [[Definition:Left Regular Representation|left regular r... | The [[Axiom:Group Action Axioms|group action axioms]] are investigated in turn.
Let $g, h, a \in G$.
Thus:
{{begin-eqn}}
{{eqn | l = g * \paren {h * a}
| r = g * \map {\lambda_h} a
| c = Definition of $*$
}}
{{eqn | r = g * \paren {h \circ a}
| c = {{Defof|Left Regular Representation}}
}}
{{eqn | ... | Left Regular Representation is Group Action | https://proofwiki.org/wiki/Left_Regular_Representation_is_Group_Action | https://proofwiki.org/wiki/Left_Regular_Representation_is_Group_Action | [
"Group Actions"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Operation/Binary Operation",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Group Action"
] | [
"Axiom:Group Action Axioms"
] |
proofwiki-8340 | Right Regular Representation by Inverse is Transitive Group Action | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $*: G \times G \to G$ be the group action:
:$\forall g, h \in G: g * h = \map {\rho_{g^{-1} } } h$
where $\rho_g$ is the right regular representation of $G$ with respect to $g$.
Then $*$ is a transitive group action. | Let $g, h \in G$.
Then:
{{begin-eqn}}
{{eqn | q = \exists a \in G
| l = h
| r = a \circ g^{-1}
| c = Group has Latin Square Property
}}
{{eqn | ll= \leadsto
| l = h
| r = \map {\rho_{g^{-1} } } a
| c = {{Defof|Right Regular Representation}}
}}
{{eqn | ll= \leadsto
| l = h
... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $*: G \times G \to G$ be the [[Definition:Group Action|group action]]:
:$\forall g, h \in G: g * h = \map {\rho_{g^{-1} } } h$
where $\rho_g$ is the [[Definition:Right Regular Representation|right regul... | Let $g, h \in G$.
Then:
{{begin-eqn}}
{{eqn | q = \exists a \in G
| l = h
| r = a \circ g^{-1}
| c = [[Group has Latin Square Property]]
}}
{{eqn | ll= \leadsto
| l = h
| r = \map {\rho_{g^{-1} } } a
| c = {{Defof|Right Regular Representation}}
}}
{{eqn | ll= \leadsto
| l = h
... | Right Regular Representation by Inverse is Transitive Group Action | https://proofwiki.org/wiki/Right_Regular_Representation_by_Inverse_is_Transitive_Group_Action | https://proofwiki.org/wiki/Right_Regular_Representation_by_Inverse_is_Transitive_Group_Action | [
"Regular Representations",
"Transitive Group Actions"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group Action",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Transitive Group Action"
] | [
"Group has Latin Square Property",
"Definition:Orbit (Group Theory)",
"Definition:Transitive Group Action"
] |
proofwiki-8341 | Left Regular Representation is Transitive Group Action | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $*: G \times G \to G$ be the group action:
:$\forall g, h \in G: g * h = \map {\lambda_g} h$
where $\lambda_g$ is the left regular representation of $G$ with respect to $g$.
Then $*$ is a transitive group action. | Let $g, h \in G$.
Then:
{{begin-eqn}}
{{eqn | q = \exists a \in G
| l = h
| r = g \circ a
| c = Group has Latin Square Property
}}
{{eqn | ll= \leadsto
| l = h
| r = \map {\lambda_g} a
| c = {{Defof|Left Regular Representation}}
}}
{{eqn | ll= \leadsto
| l = h
| r = g * a... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $*: G \times G \to G$ be the [[Definition:Group Action|group action]]:
:$\forall g, h \in G: g * h = \map {\lambda_g} h$
where $\lambda_g$ is the [[Definition:Left Regular Representation|left regular re... | Let $g, h \in G$.
Then:
{{begin-eqn}}
{{eqn | q = \exists a \in G
| l = h
| r = g \circ a
| c = [[Group has Latin Square Property]]
}}
{{eqn | ll= \leadsto
| l = h
| r = \map {\lambda_g} a
| c = {{Defof|Left Regular Representation}}
}}
{{eqn | ll= \leadsto
| l = h
| r = ... | Left Regular Representation is Transitive Group Action | https://proofwiki.org/wiki/Left_Regular_Representation_is_Transitive_Group_Action | https://proofwiki.org/wiki/Left_Regular_Representation_is_Transitive_Group_Action | [
"Transitive Group Actions",
"Regular Representations"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group Action",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Transitive Group Action"
] | [
"Group has Latin Square Property",
"Definition:Orbit (Group Theory)",
"Definition:Transitive Group Action"
] |
proofwiki-8342 | Conjugacy Action is not Transitive | Let $\struct {G, \circ}$ be a non-trivial group whose identity is $e$.
Let $*: G \times G \to G$ be the conjugacy group action:
:$\forall g, h \in G: g * h = g \circ h \circ g^{-1}$
Then $*$ is ''not'' a transitive group action. | Proof by Counterexample:
For $G$ to be a transitive group action, the orbit of any element of $G$ needs to be the whole of $G$.
Take $h = e$.
Then:
{{begin-eqn}}
{{eqn | q = \forall g \in G
| l = g * e
| r = g \circ e \circ g^{-1}
| c = {{Defof|Conjugacy Action}}
}}
{{eqn | r = g \circ g^{-1}
| ... | Let $\struct {G, \circ}$ be a [[Definition:Non-Trivial Group|non-trivial]] [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $*: G \times G \to G$ be the [[Definition:Conjugacy Action|conjugacy group action]]:
:$\forall g, h \in G: g * h = g \circ h \circ g^{-1}$
Then $*$ is ''not... | [[Proof by Counterexample]]:
For $G$ to be a [[Definition:Transitive Group Action|transitive group action]], the [[Definition:Orbit (Group Theory)|orbit]] of any [[Definition:Element|element]] of $G$ needs to be the whole of $G$.
Take $h = e$.
Then:
{{begin-eqn}}
{{eqn | q = \forall g \in G
| l = g * e
|... | Conjugacy Action is not Transitive | https://proofwiki.org/wiki/Conjugacy_Action_is_not_Transitive | https://proofwiki.org/wiki/Conjugacy_Action_is_not_Transitive | [
"Conjugacy Action",
"Transitive Group Actions"
] | [
"Definition:Non-Trivial Group",
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Conjugacy Action",
"Definition:Transitive Group Action"
] | [
"Proof by Counterexample",
"Definition:Transitive Group Action",
"Definition:Orbit (Group Theory)",
"Definition:Element",
"Definition:Orbit (Group Theory)",
"Definition:Trivial Group",
"Definition:Transitive Group Action"
] |
proofwiki-8343 | Conjugacy Action on Abelian Group is Trivial | Let $\struct {G, \circ}$ be an abelian group whose identity is $e$.
Let $*: G \times G \to G$ be the conjugacy group action:
:$\forall g, h \in G: g * h = g \circ h \circ g^{-1}$
Then $*$ is a trivial group action. | For $G$ to be a trivial group action, the orbit of any element of $G$ is a singleton containing only that element.
Take $h \in G$.
Then:
{{begin-eqn}}
{{eqn | q = \forall g \in G
| l = g * h
| r = g \circ h \circ g^{-1}
| c =
}}
{{eqn | r = h \circ g \circ g^{-1}
| c = {{Defof|Abelian Group}}: ... | Let $\struct {G, \circ}$ be an [[Definition:Abelian Group|abelian group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $*: G \times G \to G$ be the [[Definition:Conjugacy Action|conjugacy group action]]:
:$\forall g, h \in G: g * h = g \circ h \circ g^{-1}$
Then $*$ is a [[Definition:Trivial Group Acti... | For $G$ to be a [[Definition:Trivial Group Action|trivial group action]], the [[Definition:Orbit (Group Theory)|orbit]] of any [[Definition:Element|element]] of $G$ is a [[Definition:Singleton|singleton]] containing only that [[Definition:Element|element]].
Take $h \in G$.
Then:
{{begin-eqn}}
{{eqn | q = \forall g \i... | Conjugacy Action on Abelian Group is Trivial | https://proofwiki.org/wiki/Conjugacy_Action_on_Abelian_Group_is_Trivial | https://proofwiki.org/wiki/Conjugacy_Action_on_Abelian_Group_is_Trivial | [
"Conjugacy Action"
] | [
"Definition:Abelian Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Conjugacy Action",
"Definition:Trivial Group Action"
] | [
"Definition:Trivial Group Action",
"Definition:Orbit (Group Theory)",
"Definition:Element",
"Definition:Singleton",
"Definition:Element",
"Definition:Commutative/Elements",
"Definition:Orbit (Group Theory)",
"Definition:Trivial Group Action"
] |
proofwiki-8344 | Group Action on Subgroup by Left Regular Representation | Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Let $*: H \times G \to G$ be the operation defined as:
:$\forall h \in H: \forall g \in G: h * g = \map {\lambda_h} g$
where $\map {\lambda_h} g$ is the left regular representation of $g$ by $h$.
Then $*$ is a group action. | The group action axioms are investigated in turn.
Let $h_1, h_2 \in H$ and $g \in G$.
Thus:
{{begin-eqn}}
{{eqn | l = h_1 * \paren {h_2 * g}
| r = h_1 * \paren {\map {\lambda_{h_2} } g}
| c = Definition of $*$
}}
{{eqn | r = h_1 * \paren {h_2 \circ g}
| c = {{Defof|Left Regular Representation}}
}}
{{e... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $*: H \times G \to G$ be the [[Definition:Binary Operation|operation]] defined as:
:$\forall h \in H: \forall g \in G: h * g = \map {\lambda_h} g$
where $\map {\lambda_h} g$ is the [[Definition:Left Regular Representat... | The [[Axiom:Group Action Axioms|group action axioms]] are investigated in turn.
Let $h_1, h_2 \in H$ and $g \in G$.
Thus:
{{begin-eqn}}
{{eqn | l = h_1 * \paren {h_2 * g}
| r = h_1 * \paren {\map {\lambda_{h_2} } g}
| c = Definition of $*$
}}
{{eqn | r = h_1 * \paren {h_2 \circ g}
| c = {{Defof|Le... | Group Action on Subgroup by Left Regular Representation | https://proofwiki.org/wiki/Group_Action_on_Subgroup_by_Left_Regular_Representation | https://proofwiki.org/wiki/Group_Action_on_Subgroup_by_Left_Regular_Representation | [
"Group Actions"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Operation/Binary Operation",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Group Action"
] | [
"Axiom:Group Action Axioms",
"Axiom:Group Action Axioms",
"Definition:Group Action"
] |
proofwiki-8345 | Group Action on Subgroup by Right Regular Representation | Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Let $*: H \times G \to G$ be the operation defined as:
:$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$
where $\map {\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$.
Then $*$ is a group action. | The group action axioms are investigated in turn.
Let $h_1, h_2 \in H$ and $g \in G$.
Thus:
{{begin-eqn}}
{{eqn | l = h_1 * \paren {h_2 * g}
| r = h_1 * \paren {\map {\rho_{h_2^{-1} } } g}
| c = Definition of $*$
}}
{{eqn | r = h_1 * \paren {g \circ h_2^{-1} }
| c = {{Defof|Right Regular Representatio... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $*: H \times G \to G$ be the [[Definition:Binary Operation|operation]] defined as:
:$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$
where $\map {\rho_{h^{-1} } } g$ is the [[Definition:Right Re... | The [[Axiom:Group Action Axioms|group action axioms]] are investigated in turn.
Let $h_1, h_2 \in H$ and $g \in G$.
Thus:
{{begin-eqn}}
{{eqn | l = h_1 * \paren {h_2 * g}
| r = h_1 * \paren {\map {\rho_{h_2^{-1} } } g}
| c = Definition of $*$
}}
{{eqn | r = h_1 * \paren {g \circ h_2^{-1} }
| c = {... | Group Action on Subgroup by Right Regular Representation | https://proofwiki.org/wiki/Group_Action_on_Subgroup_by_Right_Regular_Representation | https://proofwiki.org/wiki/Group_Action_on_Subgroup_by_Right_Regular_Representation | [
"Group Actions"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Operation/Binary Operation",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Group Action"
] | [
"Axiom:Group Action Axioms",
"Inverse of Group Product",
"Inverse of Identity Element is Itself",
"Axiom:Group Action Axioms",
"Definition:Group Action"
] |
proofwiki-8346 | Group Action on Subgroup by Right Regular Representation is not Transitive | Let $G$ be a group.
Let $H$ be a proper subgroup of $G$.
Let $*: H \times G \to G$ be the group action defined as:
:$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$
where $\map {\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$.
Then $*$ is not transitive. | From Group Action on Subgroup by Right Regular Representation it is established that $*$ is a group action.
From Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset:
:$\forall x \in G: \Orb x = H x$
where $H x$ is the right coset of $H$ by $x$.
From Right Coset Space forms Partition it is a... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Proper Subgroup|proper subgroup]] of $G$.
Let $*: H \times G \to G$ be the [[Definition:Group Action|group action]] defined as:
:$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$
where $\map {\rho_{h^{-1} } } g$ is the [[Defini... | From [[Group Action on Subgroup by Right Regular Representation]] it is established that $*$ is a [[Definition:Group Action|group action]].
From [[Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset]]:
:$\forall x \in G: \Orb x = H x$
where $H x$ is the [[Definition:Right Coset|right coset... | Group Action on Subgroup by Right Regular Representation is not Transitive | https://proofwiki.org/wiki/Group_Action_on_Subgroup_by_Right_Regular_Representation_is_not_Transitive | https://proofwiki.org/wiki/Group_Action_on_Subgroup_by_Right_Regular_Representation_is_not_Transitive | [
"Transitive Group Actions"
] | [
"Definition:Group",
"Definition:Proper Subgroup",
"Definition:Group Action",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Transitive Group Action"
] | [
"Group Action on Subgroup by Right Regular Representation",
"Definition:Group Action",
"Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset",
"Definition:Coset/Right Coset",
"Right Coset Space forms Partition",
"Definition:Transitive Group Action"
] |
proofwiki-8347 | Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset | Let $G$ be a group.
Let $H$ be a proper subgroup of $G$.
Let $*: H \times G \to G$ be the group action defined as:
:$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$
where $\map {\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$.
Let $x \in G$.
Then the orbit of $x$ under... | We are given $G$ is a group and $H$ is a proper subgroup of $G$, so:
:$H < G$
We are given $*: H \times G \to G$ is the group action defined as:
:$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$
where $\map {\rho_{h^{-1} } } g$ is the right regular representation of $g$ by $h^{-1}$.
We are also ... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Proper Subgroup|proper subgroup]] of $G$.
Let $*: H \times G \to G$ be the [[Definition:Group Action|group action]] defined as:
:$\forall \tuple {h, g} \in H \times G: h * g = \map {\rho_{h^{-1} } } g$
where $\map {\rho_{h^{-1} } } g$ is the [[Definit... | We are [[Definition:Given|given]] $G$ is a [[Definition:Group|group]] and $H$ is a [[Definition:Proper Subgroup|proper subgroup]] of $G$, so:
:$H < G$
We are [[Definition:Given|given]] $*: H \times G \to G$ is the [[Definition:Group Action|group action]] defined as:
:$\forall \tuple {h, g} \in H \times G: h * g = \map... | Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset | https://proofwiki.org/wiki/Orbit_of_Group_Action_on_Subgroup_by_Right_Regular_Representation_is_Right_Coset | https://proofwiki.org/wiki/Orbit_of_Group_Action_on_Subgroup_by_Right_Regular_Representation_is_Right_Coset | [
"Group Actions"
] | [
"Definition:Group",
"Definition:Proper Subgroup",
"Definition:Group Action",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Orbit (Group Theory)",
"Definition:Coset/Right Coset"
] | [
"Definition:Given",
"Definition:Group",
"Definition:Proper Subgroup",
"Definition:Given",
"Definition:Group Action",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Given"
] |
proofwiki-8348 | Left Coset by Identity | Then:
: $e H = H$
where $e H$ is the left coset of $H$ by $e$. | We have:
{{begin-eqn}}
{{eqn | l = e H
| r = \set {y \in G: \exists h \in H: y = e h}
| c = {{Defof|Left Coset}} of $H$ by $e$
}}
{{eqn | r = \set {y \in G: \exists h \in H: y = h}
| c = {{Defof|Identity Element}}
}}
{{eqn | r = \set {y \in G: y \in H}
| c =
}}
{{eqn | r = H
| c =
}}
{{e... | Then:
: $e H = H$
where $e H$ is the [[Definition:Left Coset|left coset]] of $H$ by $e$. | We have:
{{begin-eqn}}
{{eqn | l = e H
| r = \set {y \in G: \exists h \in H: y = e h}
| c = {{Defof|Left Coset}} of $H$ by $e$
}}
{{eqn | r = \set {y \in G: \exists h \in H: y = h}
| c = {{Defof|Identity Element}}
}}
{{eqn | r = \set {y \in G: y \in H}
| c =
}}
{{eqn | r = H
| c =
}}
{{... | Left Coset by Identity | https://proofwiki.org/wiki/Left_Coset_by_Identity | https://proofwiki.org/wiki/Left_Coset_by_Identity | [
"Coset by Identity"
] | [
"Definition:Coset/Left Coset"
] | [] |
proofwiki-8349 | Right Coset by Identity | Then:
: $H = H e$
where $H e$ is the right coset of $H$ by $e$. | We have:
{{begin-eqn}}
{{eqn | l = H e
| r = \set {x \in G: \exists h \in H: x = h e}
| c = {{Defof|Right Coset}} of $H$ by $e$
}}
{{eqn | r = \set {x \in G: \exists h \in H: x = h}
| c = {{Defof|Identity Element}}
}}
{{eqn | r = \set {x \in G: x \in H}
| c =
}}
{{eqn | r = H
| c =
}}
{{... | Then:
: $H = H e$
where $H e$ is the [[Definition:Right Coset|right coset]] of $H$ by $e$. | We have:
{{begin-eqn}}
{{eqn | l = H e
| r = \set {x \in G: \exists h \in H: x = h e}
| c = {{Defof|Right Coset}} of $H$ by $e$
}}
{{eqn | r = \set {x \in G: \exists h \in H: x = h}
| c = {{Defof|Identity Element}}
}}
{{eqn | r = \set {x \in G: x \in H}
| c =
}}
{{eqn | r = H
| c =
}}
{... | Right Coset by Identity | https://proofwiki.org/wiki/Right_Coset_by_Identity | https://proofwiki.org/wiki/Right_Coset_by_Identity | [
"Coset by Identity"
] | [
"Definition:Coset/Right Coset"
] | [] |
proofwiki-8350 | Inversion Mapping is Permutation | Let $\struct {G, \circ}$ be a group.
Let $\iota: G \to G$ be the inversion mapping on $G$.
Then $\iota$ is a permutation on $G$. | The inversion mapping on $G$ is the mapping $\iota: G \to G$ defined by:
:$\forall g \in G: \map \iota g = g^{-1}$
where $g^{-1}$ is the inverse or $g$.
By Inversion Mapping is Involution, $\iota$ is an involution:
:$\forall g \in G: \map \iota {\map \iota g} = g$
The result follows from Involution is Permutation.
{{qe... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $\iota: G \to G$ be the [[Definition:Inversion Mapping|inversion mapping]] on $G$.
Then $\iota$ is a [[Definition:Permutation|permutation]] on $G$. | The [[Definition:Inversion Mapping|inversion mapping]] on $G$ is the [[Definition:Mapping|mapping]] $\iota: G \to G$ defined by:
:$\forall g \in G: \map \iota g = g^{-1}$
where $g^{-1}$ is the [[Definition:Inverse Element|inverse]] or $g$.
By [[Inversion Mapping is Involution]], $\iota$ is an [[Definition:Involution... | Inversion Mapping is Permutation/Proof 1 | https://proofwiki.org/wiki/Inversion_Mapping_is_Permutation | https://proofwiki.org/wiki/Inversion_Mapping_is_Permutation/Proof_1 | [
"Inversion Mapping is Permutation",
"Inversion Mappings",
"Permutations"
] | [
"Definition:Group",
"Definition:Inversion Mapping",
"Definition:Permutation"
] | [
"Definition:Inversion Mapping",
"Definition:Mapping",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Inversion Mapping is Involution",
"Definition:Involution (Mapping)",
"Involution is Permutation"
] |
proofwiki-8351 | Inversion Mapping is Permutation | Let $\struct {G, \circ}$ be a group.
Let $\iota: G \to G$ be the inversion mapping on $G$.
Then $\iota$ is a permutation on $G$. | === Proof of Surjection ===
Let $a \in G$.
By definition of $\iota$:
:$\iota(a^{-1}) = \left({a^{-1}}\right)^{-1}$
By Inverse of Inverse:
:$\left({a^{-1}}\right)^{-1} = a$
Hence $a$ has a preimage.
Since $a$ was arbitrary, $\iota$ is a surjection.
=== Proof of Injection ===
Suppose for some $a, b \in G$ that:
:$\iota \... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $\iota: G \to G$ be the [[Definition:Inversion Mapping|inversion mapping]] on $G$.
Then $\iota$ is a [[Definition:Permutation|permutation]] on $G$. | === Proof of Surjection ===
Let $a \in G$.
By definition of $\iota$:
:$\iota(a^{-1}) = \left({a^{-1}}\right)^{-1}$
By [[Inverse of Inverse]]:
:$\left({a^{-1}}\right)^{-1} = a$
Hence $a$ has a [[Definition:Preimage of Element under Mapping|preimage]].
Since $a$ was arbitrary, $\iota$ is a [[Definition:Surjection|... | Inversion Mapping is Permutation/Proof 2 | https://proofwiki.org/wiki/Inversion_Mapping_is_Permutation | https://proofwiki.org/wiki/Inversion_Mapping_is_Permutation/Proof_2 | [
"Inversion Mapping is Permutation",
"Inversion Mappings",
"Permutations"
] | [
"Definition:Group",
"Definition:Inversion Mapping",
"Definition:Permutation"
] | [
"Inverse of Inverse",
"Definition:Preimage/Mapping/Element",
"Definition:Surjection",
"Inverse in Group is Unique",
"Definition:Injection",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Set",
"Definition:Permutation"
] |
proofwiki-8352 | Inversion Mapping is Mapping | Let $\struct {G, \circ}$ be a group.
Let $\iota: G \to G$ be the inversion mapping on $G$.
Then $\iota$ is indeed a mapping. | To show that $\iota$ is a mapping, it is sufficient to show that:
:$\map \iota a \ne \map \iota b \implies a \ne b$:
{{begin-eqn}}
{{eqn | l = \map \iota a
| o = \ne
| r = \map \iota b
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = a^{-1}
| o = \ne
| r = b^{-1}
| c = Defini... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $\iota: G \to G$ be the [[Definition:Inversion Mapping|inversion mapping]] on $G$.
Then $\iota$ is indeed a [[Definition:Mapping|mapping]]. | To show that $\iota$ is a [[Definition:Mapping|mapping]], it is sufficient to show that:
:$\map \iota a \ne \map \iota b \implies a \ne b$:
{{begin-eqn}}
{{eqn | l = \map \iota a
| o = \ne
| r = \map \iota b
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = a^{-1}
| o = \ne
| r = ... | Inversion Mapping is Mapping | https://proofwiki.org/wiki/Inversion_Mapping_is_Mapping | https://proofwiki.org/wiki/Inversion_Mapping_is_Mapping | [
"Inversion Mappings"
] | [
"Definition:Group",
"Definition:Inversion Mapping",
"Definition:Mapping"
] | [
"Definition:Mapping",
"Cancellation Laws",
"Category:Inversion Mappings"
] |
proofwiki-8353 | Cartesian Product of Group Actions | Let $\struct {G, \circ}$ be a group.
Let $S$ and $T$ be sets.
Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be group actions.
Then the operation $*: G \times \paren {S \times T} \to S \times T$ defined as:
:$\forall \tuple {g, \tuple {s, t} } \in G \times \paren {S \times T}: g * \tuple {s, t} = \tuple {g *_S... | The group action axioms are investigated in turn.
Let $g, h \in G$ and $s, t \in S$.
Thus:
{{begin-eqn}}
{{eqn | l = g * \tuple {h * \tuple {s, t} }
| r = g * \tuple {h *_S s, h *_T t}
| c = Definition of $*$
}}
{{eqn | r = \tuple {g *_S \tuple {h *_S s}, g *_T \tuple {h *_T t} }
| c = Definition of $... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $S$ and $T$ be [[Definition:Set|sets]].
Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be [[Definition:Group Action|group actions]].
Then the [[Definition:Operation|operation]] $*: G \times \paren {S \times T} \to S \times T$ defined as:
:$\fora... | The [[Axiom:Group Action Axioms|group action axioms]] are investigated in turn.
Let $g, h \in G$ and $s, t \in S$.
Thus:
{{begin-eqn}}
{{eqn | l = g * \tuple {h * \tuple {s, t} }
| r = g * \tuple {h *_S s, h *_T t}
| c = Definition of $*$
}}
{{eqn | r = \tuple {g *_S \tuple {h *_S s}, g *_T \tuple {h *_... | Cartesian Product of Group Actions | https://proofwiki.org/wiki/Cartesian_Product_of_Group_Actions | https://proofwiki.org/wiki/Cartesian_Product_of_Group_Actions | [
"Group Actions",
"Cartesian Product"
] | [
"Definition:Group",
"Definition:Set",
"Definition:Group Action",
"Definition:Operation",
"Definition:Group Action"
] | [
"Axiom:Group Action Axioms",
"Axiom:Group Action Axioms",
"Definition:Group Action"
] |
proofwiki-8354 | Stabilizer of Cartesian Product of Group Actions | Let $\struct {G, \circ}$ be a group.
Let $S$ and $T$ be sets.
Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be group actions.
Let the group action $*: G \times \paren {S \times T} \to S \times T$ be defined as:
:$\forall \tuple {g, \tuple {s, t} } \in G \times \paren {S \times T}: g * \tuple {s, t} = \tuple {... | By definition, the stabilizer of an element $x$ of $S$ is defined as:
:$\Stab x := \set {g \in G: g * x = x}$
where $*$ denotes the group action.
So:
{{begin-eqn}}
{{eqn | l = \Stab {s, t}
| r = \set {g \in G: g * \tuple {s, t} = \tuple {s, t} }
| c = {{Defof|Stabilizer}}
}}
{{eqn | r = \set {g \in G: \tupl... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $S$ and $T$ be [[Definition:Set|sets]].
Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be [[Definition:Group Action|group actions]].
Let the [[Definition:Group Action|group action]] $*: G \times \paren {S \times T} \to S \times T$ be defined as:... | By definition, the [[Definition:Stabilizer|stabilizer]] of an [[Definition:Element|element]] $x$ of $S$ is defined as:
:$\Stab x := \set {g \in G: g * x = x}$
where $*$ denotes the [[Definition:Group Action|group action]].
So:
{{begin-eqn}}
{{eqn | l = \Stab {s, t}
| r = \set {g \in G: g * \tuple {s, t} = \tupl... | Stabilizer of Cartesian Product of Group Actions | https://proofwiki.org/wiki/Stabilizer_of_Cartesian_Product_of_Group_Actions | https://proofwiki.org/wiki/Stabilizer_of_Cartesian_Product_of_Group_Actions | [
"Group Actions",
"Cartesian Product",
"Stabilizers"
] | [
"Definition:Group",
"Definition:Set",
"Definition:Group Action",
"Definition:Group Action",
"Definition:Stabilizer",
"Definition:Stabilizer"
] | [
"Definition:Stabilizer",
"Definition:Element",
"Definition:Group Action"
] |
proofwiki-8355 | Index in Subgroup | Let $G$ be a group.
Let $H, K$ be subgroups of finite index of $G$.
Then:
:$\index H {H \cap K} \le \index G K$
where $\index G K$ denotes the index of $K$ in $G$.
Equality happens {{iff}} $G = H K$. | We list out all the left cosets of $H \cap K$ in $H$:
:$H / \paren {H \cap K} = \set {h_n \paren {H \cap K}: h_n \in H, n \in I}$
where $I$ is some finite indexing set.
For each pair $h_i, h_j \in H \subseteq G$, where $i \ne j$:
:$h_i^{-1} h_j \notin H \cap K \quad$ Cosets are Equal iff Product with Inverse in Subgrou... | Let $G$ be a [[Definition:Group|group]].
Let $H, K$ be [[Definition:Subgroup|subgroups]] of [[Definition:Finite Index|finite index]] of $G$.
Then:
:$\index H {H \cap K} \le \index G K$
where $\index G K$ denotes the [[Definition:Index of Subgroup|index]] of $K$ in $G$.
Equality happens {{iff}} $G = H K$. | We list out all the [[Definition:Left Coset|left cosets]] of $H \cap K$ in $H$:
:$H / \paren {H \cap K} = \set {h_n \paren {H \cap K}: h_n \in H, n \in I}$
where $I$ is some [[Definition:Finite Set|finite]] indexing set.
For each pair $h_i, h_j \in H \subseteq G$, where $i \ne j$:
:$h_i^{-1} h_j \notin H \cap K \qua... | Index in Subgroup | https://proofwiki.org/wiki/Index_in_Subgroup | https://proofwiki.org/wiki/Index_in_Subgroup | [
"Subgroups",
"Index of Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Index of Subgroup/Finite",
"Definition:Index of Subgroup"
] | [
"Definition:Coset/Left Coset",
"Definition:Finite Set",
"Cosets are Equal iff Product with Inverse in Subgroup",
"Definition:Subgroup",
"Cosets are Equal iff Product with Inverse in Subgroup",
"Definition:Coset/Left Coset",
"Definition:Coset/Left Coset",
"Definition:Coset/Left Coset",
"Definition:Co... |
proofwiki-8356 | Finite Cyclic Group has Euler Phi Generators | Let $C_n$ be a (finite) cyclic group of order $n$.
Then $C_n$ has $\map \phi n$ generators, where $\map \phi n$ denotes the Euler $\phi$ function. | From List of Elements in Finite Cyclic Group, the elements of $G$ are:
:$\set {g^k: g \in G, 0 \le k < n}$
From Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order, $g^k$ generates $G$ {{iff}} $k \perp n$.
The result follows by definition of the Euler $\phi$ function.
{{qed}} | Let $C_n$ be a [[Definition:Finite Cyclic Group|(finite) cyclic group]] of [[Definition:Order of Structure|order $n$]].
Then $C_n$ has $\map \phi n$ [[Definition:Generator of Cyclic Group|generators]], where $\map \phi n$ denotes the [[Definition:Euler Phi Function|Euler $\phi$ function]]. | From [[List of Elements in Finite Cyclic Group]], the [[Definition:Element|elements]] of $G$ are:
:$\set {g^k: g \in G, 0 \le k < n}$
From [[Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order]], $g^k$ [[Definition:Generator of Cyclic Group|generates]] $G$ {{iff}} $k \perp n$.
The result f... | Finite Cyclic Group has Euler Phi Generators | https://proofwiki.org/wiki/Finite_Cyclic_Group_has_Euler_Phi_Generators | https://proofwiki.org/wiki/Finite_Cyclic_Group_has_Euler_Phi_Generators | [
"Finite Cyclic Groups"
] | [
"Definition:Finite Cyclic Group",
"Definition:Order of Structure",
"Definition:Cyclic Group/Generator",
"Definition:Euler Phi Function"
] | [
"List of Elements in Finite Cyclic Group",
"Definition:Element",
"Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order",
"Definition:Cyclic Group/Generator",
"Definition:Euler Phi Function"
] |
proofwiki-8357 | Composition of Left Regular Representations | :$\lambda_x \circ \lambda_y = \lambda_{x * y}$ | Let $z \in S$.
{{begin-eqn}}
{{eqn | l = \map {\lambda_x \circ \lambda_y} z
| r = \map {\lambda_x} {\map {\lambda_y} z}
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {\lambda_x} {y * z}
| c = {{Defof|Left Regular Representation}}
}}
{{eqn | r = x * \paren {y * z}
| c = {{Defof|Left... | :$\lambda_x \circ \lambda_y = \lambda_{x * y}$ | Let $z \in S$.
{{begin-eqn}}
{{eqn | l = \map {\lambda_x \circ \lambda_y} z
| r = \map {\lambda_x} {\map {\lambda_y} z}
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {\lambda_x} {y * z}
| c = {{Defof|Left Regular Representation}}
}}
{{eqn | r = x * \paren {y * z}
| c = {{Defof|Lef... | Composition of Left Regular Representations | https://proofwiki.org/wiki/Composition_of_Left_Regular_Representations | https://proofwiki.org/wiki/Composition_of_Left_Regular_Representations | [
"Semigroups",
"Regular Representations"
] | [] | [] |
proofwiki-8358 | Composition of Right Regular Representations | :$\rho_x \circ \rho_y = \rho_{y * x}$ | Let $z \in S$.
{{begin-eqn}}
{{eqn | l = \map {\paren {\rho_x \circ \rho_y} } z
| r = \map {\rho_x} {\map {\rho_y} z}
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {\rho_x} {z * y}
| c = {{Defof|Right Regular Representation}}
}}
{{eqn | r = \paren {z * y} * x
| c = {{Defof|Right Re... | :$\rho_x \circ \rho_y = \rho_{y * x}$ | Let $z \in S$.
{{begin-eqn}}
{{eqn | l = \map {\paren {\rho_x \circ \rho_y} } z
| r = \map {\rho_x} {\map {\rho_y} z}
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {\rho_x} {z * y}
| c = {{Defof|Right Regular Representation}}
}}
{{eqn | r = \paren {z * y} * x
| c = {{Defof|Right R... | Composition of Right Regular Representations | https://proofwiki.org/wiki/Composition_of_Right_Regular_Representations | https://proofwiki.org/wiki/Composition_of_Right_Regular_Representations | [
"Semigroups",
"Regular Representations"
] | [] | [] |
proofwiki-8359 | Composition of Left Regular Representation with Right | :$\lambda_x \circ \rho_y = \rho_y \circ \lambda_x$ | Let $z \in S$.
{{begin-eqn}}
{{eqn | l = \map {\paren {\lambda_x \circ \rho_y} } z
| r = \map {\lambda_x} {\map {\rho_y} z}
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {\lambda_x} {z * y}
| c = {{Defof|Right Regular Representation}}
}}
{{eqn | r = x * \paren {z * y}
| c = {{Defof... | :$\lambda_x \circ \rho_y = \rho_y \circ \lambda_x$ | Let $z \in S$.
{{begin-eqn}}
{{eqn | l = \map {\paren {\lambda_x \circ \rho_y} } z
| r = \map {\lambda_x} {\map {\rho_y} z}
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {\lambda_x} {z * y}
| c = {{Defof|Right Regular Representation}}
}}
{{eqn | r = x * \paren {z * y}
| c = {{Defo... | Composition of Left Regular Representation with Right | https://proofwiki.org/wiki/Composition_of_Left_Regular_Representation_with_Right | https://proofwiki.org/wiki/Composition_of_Left_Regular_Representation_with_Right | [
"Semigroups",
"Regular Representations"
] | [] | [] |
proofwiki-8360 | Extendability Theorem for Derivatives Continuous on Open Intervals | Let $f$ be a continuous real function defined on an interval $\closedint a b$ where $a < b$.
Then $f$ is continuously differentiable on $\closedint a b$ {{iff}}:
:$f$ is continuously differentiable on $\openint a b$
and:
:$\ds \lim_{x \mathop \to a^+} \map {f'} x$ and $\ds \lim_{x \mathop \to b^-} \map {f'} x$ exist. | === Necessary Condition ===
Suppose that $f$ is continuously differentiable on $\closedint a b$.
We need to show that:
:$f$ is continuously differentiable on $\openint a b$
and:
:$\ds \lim_{x \mathop \to a^+} \map {f'} x$ and $\ds \lim_{x \mathop \to b^-} \map {f'} x$ exist.
$f$ is continuously differentiable on $\open... | Let $f$ be a [[Definition:Continuous Real Function|continuous real function]] defined on an [[Definition:Real Interval|interval]] $\closedint a b$ where $a < b$.
Then $f$ is [[Definition:Continuously Differentiable Real Function|continuously differentiable]] on $\closedint a b$ {{iff}}:
:$f$ is [[Definition:Continuou... | === Necessary Condition ===
Suppose that $f$ is [[Definition:Continuously Differentiable Real Function|continuously differentiable]] on $\closedint a b$.
We need to show that:
:$f$ is [[Definition:Continuously Differentiable Real Function|continuously differentiable]] on $\openint a b$
and:
:$\ds \lim_{x \mathop \... | Extendability Theorem for Derivatives Continuous on Open Intervals | https://proofwiki.org/wiki/Extendability_Theorem_for_Derivatives_Continuous_on_Open_Intervals | https://proofwiki.org/wiki/Extendability_Theorem_for_Derivatives_Continuous_on_Open_Intervals | [
"Differential Calculus"
] | [
"Definition:Continuous Real Function",
"Definition:Real Interval",
"Definition:Continuously Differentiable/Real Function",
"Definition:Continuously Differentiable/Real Function"
] | [
"Definition:Continuously Differentiable/Real Function",
"Definition:Continuously Differentiable/Real Function",
"Definition:Continuously Differentiable/Real Function",
"Definition:Continuously Differentiable/Real Function",
"Definition:Subset",
"Definition:Differentiable Mapping/Real Function",
"Definit... |
proofwiki-8361 | Universal Affirmative and Universal Negative are Contrary iff First Predicate is not Vacuous | Consider the categorical statements:
{{begin-axiom}}
{{axiom | ll = \mathbf A:
| lc= The universal affirmative:
| q = \forall x
| m = \map S x \implies \map P x
}}
{{axiom | ll= \mathbf E:
| lc= The universal negative:
| q = \forall x
| m = \map S x \implies \neg \map P x... | === Sufficient Condition ===
Let $\exists x: \map S x$.
Suppose $\mathbf A$ and $\mathbf E$ are both true.
As $\mathbf A$ is true, then by Modus Ponendo Ponens:
:$\map P x$
As $\mathbf E$ is true, then by Modus Ponendo Ponens:
:$\neg \map P x$
It follows by Proof by Contradiction that $\mathbf A$ and $\mathbf E$ are no... | Consider the [[Definition:Categorical Statement|categorical statements]]:
{{begin-axiom}}
{{axiom | ll = \mathbf A:
| lc= The [[Definition:Universal Affirmative|universal affirmative]]:
| q = \forall x
| m = \map S x \implies \map P x
}}
{{axiom | ll= \mathbf E:
| lc= The [[Definition:U... | === Sufficient Condition ===
Let $\exists x: \map S x$.
Suppose $\mathbf A$ and $\mathbf E$ are both [[Definition:True|true]].
As $\mathbf A$ is [[Definition:True|true]], then by [[Modus Ponendo Ponens]]:
:$\map P x$
As $\mathbf E$ is [[Definition:True|true]], then by [[Modus Ponendo Ponens]]:
:$\neg \map P x$
It ... | Universal Affirmative and Universal Negative are Contrary iff First Predicate is not Vacuous | https://proofwiki.org/wiki/Universal_Affirmative_and_Universal_Negative_are_Contrary_iff_First_Predicate_is_not_Vacuous | https://proofwiki.org/wiki/Universal_Affirmative_and_Universal_Negative_are_Contrary_iff_First_Predicate_is_not_Vacuous | [
"Categorical Statements"
] | [
"Definition:Categorical Statement",
"Definition:Universal Affirmative",
"Definition:Universal Negative",
"Definition:Contrary Statements",
"Definition:Symbolic Logic",
"Definition:Predicate Logic"
] | [
"Definition:True",
"Definition:True",
"Modus Ponendo Ponens",
"Definition:True",
"Modus Ponendo Ponens",
"Proof by Contradiction",
"Definition:True",
"Definition:Contrary Statements",
"Definition:Contrary Statements",
"Definition:True",
"Definition:True",
"Definition:Contrary Statements",
"P... |
proofwiki-8362 | Particular Affirmative and Particular Negative are Subcontrary iff First Predicate is not Vacuous | Consider the categorical statements:
{{begin-axiom}}
{{axiom | ll = \mathbf I:
| lc= The particular affirmative:
| q = \exists x
| m = \map S x \land \map P x
}}
{{axiom | ll= \mathbf O:
| lc= The particular negative:
| q = \exists x
| m = \map S x \land \neg \map P x
}}
... | === Sufficient Condition ===
Let $\exists x: \map S x$.
Suppose $\mathbf I$ and $\mathbf O$ are both false.
As $\mathbf I$ is false, then by the Rule of Conjunction:
:$\neg \map P x$
As $\mathbf O$ is false, then by the Rule of Conjunction:
:$\neg \neg \map P x$
and so by Double Negation:
:$\map P x$
It follows by Proo... | Consider the [[Definition:Categorical Statement|categorical statements]]:
{{begin-axiom}}
{{axiom | ll = \mathbf I:
| lc= The [[Definition:Particular Affirmative|particular affirmative]]:
| q = \exists x
| m = \map S x \land \map P x
}}
{{axiom | ll= \mathbf O:
| lc= The [[Definition:Pa... | === Sufficient Condition ===
Let $\exists x: \map S x$.
Suppose $\mathbf I$ and $\mathbf O$ are both [[Definition:False|false]].
As $\mathbf I$ is [[Definition:False|false]], then by the [[Rule of Conjunction/Proof Rule|Rule of Conjunction]]:
:$\neg \map P x$
As $\mathbf O$ is [[Definition:False|false]], then by th... | Particular Affirmative and Particular Negative are Subcontrary iff First Predicate is not Vacuous | https://proofwiki.org/wiki/Particular_Affirmative_and_Particular_Negative_are_Subcontrary_iff_First_Predicate_is_not_Vacuous | https://proofwiki.org/wiki/Particular_Affirmative_and_Particular_Negative_are_Subcontrary_iff_First_Predicate_is_not_Vacuous | [
"Categorical Statements"
] | [
"Definition:Categorical Statement",
"Definition:Particular Affirmative",
"Definition:Particular Negative",
"Definition:Subcontrary Statements",
"Definition:Symbolic Logic",
"Definition:Predicate Logic"
] | [
"Definition:False",
"Definition:False",
"Rule of Conjunction/Proof Rule",
"Definition:False",
"Rule of Conjunction/Proof Rule",
"Double Negation",
"Proof by Contradiction",
"Definition:False",
"Definition:Subcontrary Statements",
"Definition:Subcontrary Statements",
"Definition:False",
"Defini... |
proofwiki-8363 | Universal Affirmative and Particular Negative are Contradictory | Consider the categorical statements:
{{begin-axiom}}
{{axiom | ll = \mathbf A:
| lc= The universal affirmative:
| q = \forall x
| m = \map S x \implies \map P x
}}
{{axiom | ll= \mathbf O:
| lc= The particular negative:
| q = \exists x
| m = \map S x \land \neg \map P x
}... | {{begin-eqn}}
{{eqn | o =
| r = \mathbf A
| c =
}}
{{eqn | ll= \leadsto
| q = \forall x
| o =
| r = \map S x \implies \map P x
| c = Definition of $\mathbf A$
}}
{{eqn | ll= \leadsto
| q = \forall x
| o =
| r = \neg \paren {\map S x \land \neg \map P x}
| ... | Consider the [[Definition:Categorical Statement|categorical statements]]:
{{begin-axiom}}
{{axiom | ll = \mathbf A:
| lc= The [[Definition:Universal Affirmative|universal affirmative]]:
| q = \forall x
| m = \map S x \implies \map P x
}}
{{axiom | ll= \mathbf O:
| lc= The [[Definition:P... | {{begin-eqn}}
{{eqn | o =
| r = \mathbf A
| c =
}}
{{eqn | ll= \leadsto
| q = \forall x
| o =
| r = \map S x \implies \map P x
| c = Definition of $\mathbf A$
}}
{{eqn | ll= \leadsto
| q = \forall x
| o =
| r = \neg \paren {\map S x \land \neg \map P x}
| ... | Universal Affirmative and Particular Negative are Contradictory | https://proofwiki.org/wiki/Universal_Affirmative_and_Particular_Negative_are_Contradictory | https://proofwiki.org/wiki/Universal_Affirmative_and_Particular_Negative_are_Contradictory | [
"Categorical Statements"
] | [
"Definition:Categorical Statement",
"Definition:Universal Affirmative",
"Definition:Particular Negative",
"Definition:Contradictory/Statements",
"Definition:Symbolic Logic",
"Definition:Predicate Logic"
] | [
"Conditional is Equivalent to Negation of Conjunction with Negative",
"De Morgan's Laws (Predicate Logic)/Denial of Existence",
"Conjunction with Negative is Equivalent to Negation of Conditional",
"De Morgan's Laws (Predicate Logic)/Denial of Universality",
"Definition:Contradictory/Statements"
] |
proofwiki-8364 | Particular Affirmative and Universal Negative are Contradictory | Consider the categorical statements:
{{begin-axiom}}
{{axiom | ll = \mathbf I:
| lc= The particular affirmative:
| q = \exists x
| m = \map S x \land \map P x
}}
{{axiom | ll= \mathbf E:
| lc= The universal negative:
| q = \forall x
| m = \map S x \implies \neg \map P x
}... | {{begin-eqn}}
{{eqn | o =
| r = \mathbf E
| c =
}}
{{eqn | ll= \therefore
| l = \forall x:
| o =
| r = \map S x \implies \neg \map P x
| c = Definition of $\mathbf E$
}}
{{eqn | ll= \therefore
| l = \forall x:
| o =
| r = \neg \paren {\map S x \land \map P x}
... | Consider the [[Definition:Categorical Statement|categorical statements]]:
{{begin-axiom}}
{{axiom | ll = \mathbf I:
| lc= The [[Definition:Particular Affirmative|particular affirmative]]:
| q = \exists x
| m = \map S x \land \map P x
}}
{{axiom | ll= \mathbf E:
| lc= The [[Definition:Un... | {{begin-eqn}}
{{eqn | o =
| r = \mathbf E
| c =
}}
{{eqn | ll= \therefore
| l = \forall x:
| o =
| r = \map S x \implies \neg \map P x
| c = Definition of $\mathbf E$
}}
{{eqn | ll= \therefore
| l = \forall x:
| o =
| r = \neg \paren {\map S x \land \map P x}
... | Particular Affirmative and Universal Negative are Contradictory | https://proofwiki.org/wiki/Particular_Affirmative_and_Universal_Negative_are_Contradictory | https://proofwiki.org/wiki/Particular_Affirmative_and_Universal_Negative_are_Contradictory | [
"Categorical Statements"
] | [
"Definition:Categorical Statement",
"Definition:Particular Affirmative",
"Definition:Universal Negative",
"Definition:Contradictory/Statements",
"Definition:Symbolic Logic",
"Definition:Predicate Logic"
] | [
"Modus Ponendo Tollens/Variant",
"De Morgan's Laws (Predicate Logic)/Denial of Existence",
"Conjunction is Equivalent to Negation of Conditional of Negative",
"De Morgan's Laws (Predicate Logic)/Denial of Universality",
"Definition:Contradictory/Statements"
] |
proofwiki-8365 | Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous | Consider the categorical statements:
{{begin-axiom}}
{{axiom | ll = \map {\mathbf A} {S, P}:
| lc= The universal affirmative:
| q = \forall x
| m = \map S x \implies \map P x
}}
{{axiom | ll= \map {\mathbf I} {S, P}:
| lc= The particular affirmative:
| q = \exists x
| m =... | === Sufficient Condition ===
Let $\exists x: \map S x$.
Let $\map {\mathbf A} {S, P}$ be true.
As $\map {\mathbf A} {S, P}$ is true, then by Modus Ponendo Ponens:
:$\map P x$
From the Rule of Conjunction:
:$\map S x \land \map P x$
Thus $\map {\mathbf I} {S, P}$ holds.
So by the Rule of Implication:
:$\map {\mathbf A} ... | Consider the [[Definition:Categorical Statement|categorical statements]]:
{{begin-axiom}}
{{axiom | ll = \map {\mathbf A} {S, P}:
| lc= The [[Definition:Universal Affirmative|universal affirmative]]:
| q = \forall x
| m = \map S x \implies \map P x
}}
{{axiom | ll= \map {\mathbf I} {S, P}:
... | === Sufficient Condition ===
Let $\exists x: \map S x$.
Let $\map {\mathbf A} {S, P}$ be [[Definition:True|true]].
As $\map {\mathbf A} {S, P}$ is [[Definition:True|true]], then by [[Modus Ponendo Ponens]]:
:$\map P x$
From the [[Rule of Conjunction/Proof Rule|Rule of Conjunction]]:
:$\map S x \land \map P x$
Thus... | Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous | https://proofwiki.org/wiki/Universal_Affirmative_implies_Particular_Affirmative_iff_First_Predicate_is_not_Vacuous | https://proofwiki.org/wiki/Universal_Affirmative_implies_Particular_Affirmative_iff_First_Predicate_is_not_Vacuous | [
"Categorical Statements"
] | [
"Definition:Categorical Statement",
"Definition:Universal Affirmative",
"Definition:Particular Affirmative",
"Definition:Symbolic Logic",
"Definition:Predicate Logic"
] | [
"Definition:True",
"Definition:True",
"Modus Ponendo Ponens",
"Rule of Conjunction/Proof Rule",
"Rule of Implication",
"Definition:True",
"Rule of Conjunction/Proof Rule",
"Definition:True"
] |
proofwiki-8366 | Universal Negative implies Particular Negative iff First Predicate is not Vacuous | Consider the categorical statements:
{{begin-axiom}}
{{axiom | ll = \map {\mathbf E} {S, P}:
| lc= The universal negative:
| q = \forall x
| m = \map S x \implies \neg \map P x
}}
{{axiom | ll= \map {\mathbf O} {S, P}:
| lc= The particular negative:
| q = \exists x
| m = ... | === Sufficient Condition ===
Let $\exists x: \map S x$.
Let $\map {\mathbf E} {S, P}$ be true.
As $\map {\mathbf E} {S, P}$ is true, then by Modus Ponendo Ponens:
:$\neg \map P x$
From the Rule of Conjunction:
:$\map S x \land \neg \map P x$
Thus $\map {\mathbf O} {S, P}$ holds.
So by the Rule of Implication:
:$\map {\... | Consider the [[Definition:Categorical Statement|categorical statements]]:
{{begin-axiom}}
{{axiom | ll = \map {\mathbf E} {S, P}:
| lc= The [[Definition:Universal Negative|universal negative]]:
| q = \forall x
| m = \map S x \implies \neg \map P x
}}
{{axiom | ll= \map {\mathbf O} {S, P}:
... | === Sufficient Condition ===
Let $\exists x: \map S x$.
Let $\map {\mathbf E} {S, P}$ be [[Definition:True|true]].
As $\map {\mathbf E} {S, P}$ is [[Definition:True|true]], then by [[Modus Ponendo Ponens]]:
:$\neg \map P x$
From the [[Rule of Conjunction/Proof Rule|Rule of Conjunction]]:
:$\map S x \land \neg \map ... | Universal Negative implies Particular Negative iff First Predicate is not Vacuous | https://proofwiki.org/wiki/Universal_Negative_implies_Particular_Negative_iff_First_Predicate_is_not_Vacuous | https://proofwiki.org/wiki/Universal_Negative_implies_Particular_Negative_iff_First_Predicate_is_not_Vacuous | [
"Categorical Statements"
] | [
"Definition:Categorical Statement",
"Definition:Universal Negative",
"Definition:Particular Negative",
"Definition:Symbolic Logic",
"Definition:Predicate Logic"
] | [
"Definition:True",
"Definition:True",
"Modus Ponendo Ponens",
"Rule of Conjunction/Proof Rule",
"Rule of Implication",
"Definition:True",
"Rule of Conjunction/Proof Rule",
"Definition:True"
] |
proofwiki-8367 | Socrates is Mortal | :$(1): \quad$ ''All humans are mortal.''
:$(2): \quad$ ''{{AuthorRef|Socrates}} is human.''
:$(3): \quad$ ''Therefore {{AuthorRef|Socrates}} is mortal.'' | Let $x$ be an object variable from the universe of '''rational beings'''.
Let $\map H x$ denote the propositional function ''$x$ is '''human'''''.
Let $\map M x$ denote the propositional function ''$x$ is '''mortal'''''.
Let $S$ be a proper name that denotes {{AuthorRef|Socrates}}.
The argument can then be expressed as... | :$(1): \quad$ ''All humans are mortal.''
:$(2): \quad$ ''{{AuthorRef|Socrates}} is human.''
:$(3): \quad$ ''Therefore {{AuthorRef|Socrates}} is mortal.'' | Let $x$ be an [[Definition:Object Variable|object variable]] from the [[Definition:Universe of Discourse|universe]] of '''rational beings'''.
Let $\map H x$ denote the [[Definition:Propositional Function|propositional function]] ''$x$ is '''human'''''.
Let $\map M x$ denote the [[Definition:Propositional Function|pro... | Socrates is Mortal | https://proofwiki.org/wiki/Socrates_is_Mortal | https://proofwiki.org/wiki/Socrates_is_Mortal | [
"Socrates is Mortal",
"Logic",
"Classic Problems"
] | [] | [
"Definition:Variable",
"Definition:Universe of Discourse",
"Definition:Propositional Function",
"Definition:Propositional Function",
"Definition:Proper Name",
"Universal Instantiation",
"Modus Ponendo Ponens"
] |
proofwiki-8368 | Right-Hand Differentiable Function is Right-Continuous | Let $f$ be a real function defined on an interval $I$.
Let $a$ be a point in $I$ where $f$ is right-hand differentiable.
Then $f$ is right-continuous at $a$. | By hypothesis, $\map {f'_+} a$ exists.
First we note that $a$ cannot be the right hand end point of $I$ because values in $I$ greater than $a$ need to exist for $\map {f'_+} a$ to exist.
We form the following expression:
:$\ds \lim_{x \mathop \to a^+} \paren {\map f x - \map f a}$
We need to show that it is defined and... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Real Interval|interval]] $I$.
Let $a$ be a point in $I$ where $f$ is [[Definition:Right-Hand Derivative|right-hand differentiable]].
Then $f$ is [[Definition:Right-Continuous at Point|right-continuous]] at $a$. | By hypothesis, $\map {f'_+} a$ exists.
First we note that $a$ cannot be the right hand [[Definition:Endpoint of Real Interval|end point]] of $I$ because values in $I$ greater than $a$ need to exist for $\map {f'_+} a$ to exist.
We form the following expression:
:$\ds \lim_{x \mathop \to a^+} \paren {\map f x - \map... | Right-Hand Differentiable Function is Right-Continuous | https://proofwiki.org/wiki/Right-Hand_Differentiable_Function_is_Right-Continuous | https://proofwiki.org/wiki/Right-Hand_Differentiable_Function_is_Right-Continuous | [
"Continuous Real Functions",
"Differentiable Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Right-Hand Derivative",
"Definition:Continuous Real Function/Right-Continuous"
] | [
"Definition:Real Interval/Endpoints",
"Definition:Fraction/Denominator",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Combination Theorem for Limits of Functions/Real/Sum Rule",
"Definition:Continuous Real Function/Right-Continuous",
"Category:Continuous Real Functions",
"Category:D... |
proofwiki-8369 | Left-Hand Differentiable Function is Left-Continuous | Let $f$ be a real function defined on an interval $I$.
Let $a$ be a point in $I$ where $f$ is left-hand differentiable.
Then $f$ is left-continuous at $a$. | By hypothesis, $\map {f'_-} a$ exists.
First we note that $a$ cannot be the left-hand end point of $I$ because values in $I$ less than $a$ need to exist for $\map {f'_-} a$ to exist.
We form the following expression:
:$\ds \lim_{x \mathop \to a^-} \paren {\map f x - \map f a}$
We need to show that it is defined and to ... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Real Interval|interval]] $I$.
Let $a$ be a point in $I$ where $f$ is [[Definition:Left-Hand Derivative|left-hand differentiable]].
Then $f$ is [[Definition:Left-Continuous at Point|left-continuous]] at $a$. | By hypothesis, $\map {f'_-} a$ exists.
First we note that $a$ cannot be the left-hand [[Definition:Endpoint of Real Interval|end point]] of $I$ because values in $I$ less than $a$ need to exist for $\map {f'_-} a$ to exist.
We form the following expression:
:$\ds \lim_{x \mathop \to a^-} \paren {\map f x - \map f a... | Left-Hand Differentiable Function is Left-Continuous | https://proofwiki.org/wiki/Left-Hand_Differentiable_Function_is_Left-Continuous | https://proofwiki.org/wiki/Left-Hand_Differentiable_Function_is_Left-Continuous | [
"Continuous Real Functions",
"Differentiable Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Left-Hand Derivative",
"Definition:Continuous Real Function/Left-Continuous"
] | [
"Definition:Real Interval/Endpoints",
"Definition:Fraction/Denominator",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Combination Theorem for Limits of Functions/Real/Sum Rule",
"Definition:Continuous Real Function/Left-Continuous",
"Category:Continuous Real Functions",
"Category:Di... |
proofwiki-8370 | Left-Hand and Right-Hand Differentiable Function is Continuous | Let $f$ be a real function defined on an interval $I$.
Let $a$ be a point in $I$ where $f$ is left- and right-hand differentiable.
Then $f$ is continuous at $a$. | By Left-Hand Differentiable Function is Left-Continuous, $f$ is left-continuous at $a$.
By Right-Hand Differentiable Function is Right-Continuous, $f$ is right-continuous at $a$.
By Continuous at Point iff Left-Continuous and Right-Continuous, $f$ is continuous at $a$.
{{qed}}
Category:Continuous Real Functions
Categor... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Real Interval|interval]] $I$.
Let $a$ be a point in $I$ where $f$ is [[Definition:Left-Hand Derivative|left-]] and [[Definition:Right-Hand Derivative|right-hand differentiable]].
Then $f$ is [[Definition:Continuous Real Function at Po... | By [[Left-Hand Differentiable Function is Left-Continuous]], $f$ is [[Definition:Left-Continuous at Point|left-continuous]] at $a$.
By [[Right-Hand Differentiable Function is Right-Continuous]], $f$ is [[Definition:Right-Continuous at Point|right-continuous]] at $a$.
By [[Continuous at Point iff Left-Continuous and R... | Left-Hand and Right-Hand Differentiable Function is Continuous | https://proofwiki.org/wiki/Left-Hand_and_Right-Hand_Differentiable_Function_is_Continuous | https://proofwiki.org/wiki/Left-Hand_and_Right-Hand_Differentiable_Function_is_Continuous | [
"Continuous Real Functions",
"Differentiable Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Left-Hand Derivative",
"Definition:Right-Hand Derivative",
"Definition:Continuous Real Function/Point"
] | [
"Left-Hand Differentiable Function is Left-Continuous",
"Definition:Continuous Real Function/Left-Continuous",
"Right-Hand Differentiable Function is Right-Continuous",
"Definition:Continuous Real Function/Right-Continuous",
"Continuous at Point iff Left-Continuous and Right-Continuous",
"Definition:Conti... |
proofwiki-8371 | Universal Affirmative and Negative are both False iff Particular Affirmative and Negative are both True | Consider the categorical statements:
{{begin-axiom}}
{{axiom | q = \map {\mathbf A} {S, P}
| lc= The universal affirmative:
| ml= \forall x: \map S x
| mo= \implies
| mr= \map P x
}}
{{axiom | q = \map {\mathbf E} {S, P}
| lc= The universal negative:
| ml= \forall x: \map... | === Necessary Condition ===
Let $\map {\mathbf A} {S, P}$ and $\map {\mathbf E} {S, P}$ both be false.
{{begin-eqn}}
{{eqn | n = 1
| l = \neg \map {\mathbf A} {S, P}
| o = \land
| r = \neg \map {\mathbf E} {S, P}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \neg \map {\mathbf A} {S, P}
| o ... | Consider the [[Definition:Categorical Statement|categorical statements]]:
{{begin-axiom}}
{{axiom | q = \map {\mathbf A} {S, P}
| lc= The [[Definition:Universal Affirmative|universal affirmative]]:
| ml= \forall x: \map S x
| mo= \implies
| mr= \map P x
}}
{{axiom | q = \map {\mathbf E}... | === Necessary Condition ===
Let $\map {\mathbf A} {S, P}$ and $\map {\mathbf E} {S, P}$ both be [[Definition:False|false]].
{{begin-eqn}}
{{eqn | n = 1
| l = \neg \map {\mathbf A} {S, P}
| o = \land
| r = \neg \map {\mathbf E} {S, P}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \neg \map {\math... | Universal Affirmative and Negative are both False iff Particular Affirmative and Negative are both True | https://proofwiki.org/wiki/Universal_Affirmative_and_Negative_are_both_False_iff_Particular_Affirmative_and_Negative_are_both_True | https://proofwiki.org/wiki/Universal_Affirmative_and_Negative_are_both_False_iff_Particular_Affirmative_and_Negative_are_both_True | [
"Categorical Statements"
] | [
"Definition:Categorical Statement",
"Definition:Universal Affirmative",
"Definition:Universal Negative",
"Definition:Particular Affirmative",
"Definition:Particular Negative",
"Definition:False",
"Definition:True"
] | [
"Definition:False",
"Rule of Simplification",
"Universal Affirmative and Particular Negative are Contradictory",
"Rule of Simplification",
"Particular Affirmative and Universal Negative are Contradictory",
"Rule of Conjunction/Proof Rule",
"Definition:True",
"Definition:True",
"Rule of Simplificatio... |
proofwiki-8372 | Law of Simple Conversion of I | Consider the particular affirmative categorical statement ''Some $S$ is $P$'':
:$\map {\mathbf I} {S, P}: \exists x: \map S x \land \map P x$
Then ''Some $P$ is $S$'':
:$\map {\mathbf I} {P, S}$ | {{begin-eqn}}
{{eqn | o =
| r = \map {\mathbf I} {S, P}
| c =
}}
{{eqn | ll= \leadsto
| o =
| q = \exists x
| r = \map S x \land \map P x
| c = {{Defof|Particular Affirmative}}
}}
{{eqn | ll= \leadsto
| o =
| q = \exists x
| r = \map P x \land \map S x
| c... | Consider the [[Definition:Particular Affirmative|particular affirmative]] [[Definition:Categorical Statement|categorical statement]] ''Some $S$ is $P$'':
:$\map {\mathbf I} {S, P}: \exists x: \map S x \land \map P x$
Then ''Some $P$ is $S$'':
:$\map {\mathbf I} {P, S}$ | {{begin-eqn}}
{{eqn | o =
| r = \map {\mathbf I} {S, P}
| c =
}}
{{eqn | ll= \leadsto
| o =
| q = \exists x
| r = \map S x \land \map P x
| c = {{Defof|Particular Affirmative}}
}}
{{eqn | ll= \leadsto
| o =
| q = \exists x
| r = \map P x \land \map S x
| c... | Law of Simple Conversion of I | https://proofwiki.org/wiki/Law_of_Simple_Conversion_of_I | https://proofwiki.org/wiki/Law_of_Simple_Conversion_of_I | [
"Laws of Conversion",
"Particular Affirmative",
"Categorical Statements"
] | [
"Definition:Particular Affirmative",
"Definition:Categorical Statement"
] | [
"Rule of Commutation/Conjunction"
] |
proofwiki-8373 | Law of Simple Conversion of E | Consider the universal negative categorical statement ''No $S$ is $P$'':
:$\map {\mathbf E} {S, P}: \forall x: \map S x \implies \neg \map P x$
Then ''No $P$ is $S$'':
:$\map {\mathbf E} {P, S}$ | {{begin-eqn}}
{{eqn | o =
| r = \map {\mathbf E} {S, P}
| c =
}}
{{eqn | ll= \leadsto
| q = \forall x
| o =
| r = \map S x \implies \neg \map P x
| c = {{Defof|Universal Negative}}
}}
{{eqn | ll= \leadsto
| q = \forall x
| o =
| r = \neg \paren {\map S x \land \... | Consider the [[Definition:Universal Negative|universal negative]] [[Definition:Categorical Statement|categorical statement]] ''No $S$ is $P$'':
:$\map {\mathbf E} {S, P}: \forall x: \map S x \implies \neg \map P x$
Then ''No $P$ is $S$'':
:$\map {\mathbf E} {P, S}$ | {{begin-eqn}}
{{eqn | o =
| r = \map {\mathbf E} {S, P}
| c =
}}
{{eqn | ll= \leadsto
| q = \forall x
| o =
| r = \map S x \implies \neg \map P x
| c = {{Defof|Universal Negative}}
}}
{{eqn | ll= \leadsto
| q = \forall x
| o =
| r = \neg \paren {\map S x \land \... | Law of Simple Conversion of E | https://proofwiki.org/wiki/Law_of_Simple_Conversion_of_E | https://proofwiki.org/wiki/Law_of_Simple_Conversion_of_E | [
"Laws of Conversion",
"Universal Negative",
"Categorical Statements"
] | [
"Definition:Universal Negative",
"Definition:Categorical Statement"
] | [
"Modus Ponendo Tollens/Variant",
"Rule of Commutation/Conjunction",
"Modus Ponendo Tollens/Variant"
] |
proofwiki-8374 | Conversion per Accidens | Consider the categorical statements:
{{begin-axiom}}
{{axiom | ll = \map {\mathbf A} {S, P}:
| lc= The universal affirmative:
| q = \forall x
| m = \map S x \implies \map P x
}}
{{axiom | ll= \map {\mathbf I} {P, S}:
| lc= The particular affirmative:
| q = \exists x
| m =... | From Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous:
:$\exists x: \map S x \iff \paren {\paren {\forall x: \map S x \implies \map P x} \implies \paren {\exists x: \map S x \land \map P x} }$
From Law of Simple Conversion of I:
:$\paren {\exists x: \map S x \land \map P x} \impli... | Consider the [[Definition:Categorical Statement|categorical statements]]:
{{begin-axiom}}
{{axiom | ll = \map {\mathbf A} {S, P}:
| lc= The [[Definition:Universal Affirmative|universal affirmative]]:
| q = \forall x
| m = \map S x \implies \map P x
}}
{{axiom | ll= \map {\mathbf I} {P, S}:
... | From [[Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous]]:
:$\exists x: \map S x \iff \paren {\paren {\forall x: \map S x \implies \map P x} \implies \paren {\exists x: \map S x \land \map P x} }$
From [[Law of Simple Conversion of I]]:
:$\paren {\exists x: \map S x \land \map P ... | Conversion per Accidens | https://proofwiki.org/wiki/Conversion_per_Accidens | https://proofwiki.org/wiki/Conversion_per_Accidens | [
"Categorical Statements"
] | [
"Definition:Categorical Statement",
"Definition:Universal Affirmative",
"Definition:Particular Affirmative",
"Definition:Symbolic Logic",
"Definition:Predicate Logic"
] | [
"Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous",
"Law of Simple Conversion of I"
] |
proofwiki-8375 | Number of Standard Instances of Categorical Syllogism | There are $256$ distinct standard instances of the categorical syllogism. | Recall the four figures of the categorical syllogism:
:<nowiki>$\begin{array}{r|rl}
\text I & & \\
\hline \\
\text{Major Premise}: & \mathbf \Phi_1 & \tuple {M, P} \\
\text{Minor Premise}: & \mathbf \Phi_2 & \tuple {S, M} \\
\hline \\
\text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\
\end{array}
\qquad
\begin{arra... | There are $256$ [[Definition:Distinct Elements|distinct]] [[Definition:Standard Instance of Categorical Syllogism|standard instances]] of the [[Definition:Categorical Syllogism|categorical syllogism]]. | Recall the four [[Definition:Figure of Categorical Syllogism|figures]] of the [[Definition:Categorical Syllogism|categorical syllogism]]:
:<nowiki>$\begin{array}{r|rl}
\text I & & \\
\hline \\
\text{Major Premise}: & \mathbf \Phi_1 & \tuple {M, P} \\
\text{Minor Premise}: & \mathbf \Phi_2 & \tuple {S, M} \\
\hline \\
... | Number of Standard Instances of Categorical Syllogism | https://proofwiki.org/wiki/Number_of_Standard_Instances_of_Categorical_Syllogism | https://proofwiki.org/wiki/Number_of_Standard_Instances_of_Categorical_Syllogism | [
"Categorical Syllogisms",
"256"
] | [
"Definition:Distinct/Plural",
"Definition:Standard Instance of Categorical Syllogism",
"Definition:Categorical Syllogism"
] | [
"Definition:Figure of Categorical Syllogism",
"Definition:Categorical Syllogism",
"Definition:Categorical Syllogism",
"Definition:Categorical Statement",
"Definition:Categorical Syllogism/Premises/Major Premise",
"Definition:Categorical Syllogism/Premises/Minor Premise",
"Definition:Categorical Syllogis... |
proofwiki-8376 | Valid Patterns of Categorical Syllogism | The following categorical syllogisms are valid:
:<nowiki>$\begin{array}{rl}
\text{I} & AAA \\
\text{I} & AII \\
\text{I} & EAE \\
\text{I} & EIO \\
* \text{I} & AAI \\
* \text{I} & EAO \\
\end{array}
\qquad
\begin{array}{rl}
\text{II} & EAE \\
\text{II} & AEE \\
\text{II} & AOO \\
\text{II} & EIO \\
* \text{II} & EAO \... | From Elimination of all but 24 Categorical Syllogisms as Invalid, all but these $24$ patterns have been shown to be invalid.
It remains to be shown that these remaining syllogisms are in fact valid.
{{ProofWanted|Considerable work to be done yet.}} | The following [[Definition:Categorical Syllogism|categorical syllogisms]] are valid:
:<nowiki>$\begin{array}{rl}
\text{I} & AAA \\
\text{I} & AII \\
\text{I} & EAE \\
\text{I} & EIO \\
* \text{I} & AAI \\
* \text{I} & EAO \\
\end{array}
\qquad
\begin{array}{rl}
\text{II} & EAE \\
\text{II} & AEE \\
\text{II} & AOO \\
... | From [[Elimination of all but 24 Categorical Syllogisms as Invalid]], all but these $24$ patterns have been shown to be [[Definition:Invalid Argument|invalid]].
It remains to be shown that these remaining syllogisms are in fact [[Definition:Valid Argument|valid]].
{{ProofWanted|Considerable work to be done yet.}} | Valid Patterns of Categorical Syllogism | https://proofwiki.org/wiki/Valid_Patterns_of_Categorical_Syllogism | https://proofwiki.org/wiki/Valid_Patterns_of_Categorical_Syllogism | [
"Categorical Syllogisms"
] | [
"Definition:Categorical Syllogism",
"Definition:Figure of Categorical Syllogism",
"Definition:Categorical Syllogism",
"Definition:Universal Affirmative",
"Definition:Universal Negative",
"Definition:Particular Affirmative",
"Definition:Particular Negative",
"Definition:Categorical Syllogism/Shorthand"... | [
"Elimination of all but 24 Categorical Syllogisms as Invalid",
"Definition:Invalid Argument",
"Definition:Valid Argument"
] |
proofwiki-8377 | No Valid Categorical Syllogism contains two Particular Premises | Let $Q$ be a valid categorical syllogism.
Then at least one of the premises of $Q$ is universal. | Suppose both premises of $Q$ are particular.
Then the pattern of $Q$ is one of $\text{II}x$, $\text{IO}x$, $\text{OI}x$ or $\text{OO}x$, where $x$ is the conclusion.
$\text{I}$ is neither universal nor negative.
Thus the $\text{II}x$ pattern does not distribute the middle term of $Q$.
So $\text{II}x$ violates the rule ... | Let $Q$ be a [[Definition:Valid Argument|valid]] [[Definition:Categorical Syllogism|categorical syllogism]].
Then at least one of the [[Definition:Premise of Syllogism|premises]] of $Q$ is [[Definition:Universal Categorical Statement|universal]]. | Suppose both [[Definition:Premise of Syllogism|premises]] of $Q$ are [[Definition:Particular Categorical Statement|particular]].
Then the pattern of $Q$ is one of $\text{II}x$, $\text{IO}x$, $\text{OI}x$ or $\text{OO}x$, where $x$ is the [[Definition:Conclusion of Syllogism|conclusion]].
$\text{I}$ is neither [[Defi... | No Valid Categorical Syllogism contains two Particular Premises | https://proofwiki.org/wiki/No_Valid_Categorical_Syllogism_contains_two_Particular_Premises | https://proofwiki.org/wiki/No_Valid_Categorical_Syllogism_contains_two_Particular_Premises | [
"Categorical Syllogisms"
] | [
"Definition:Valid Argument",
"Definition:Categorical Syllogism",
"Definition:Categorical Syllogism/Premises",
"Definition:Universal Categorical Statement"
] | [
"Definition:Categorical Syllogism/Premises",
"Definition:Particular Categorical Statement",
"Definition:Categorical Syllogism/Conclusion",
"Definition:Universal Categorical Statement",
"Definition:Negative Categorical Statement",
"Definition:Distributed Term of Categorical Syllogism",
"Definition:Catego... |
proofwiki-8378 | No Valid Categorical Syllogism with Particular Premise has Universal Conclusion | Let $Q$ be a valid categorical syllogism.
Let one of the premises of $Q$ be particular.
Then the conclusion of $Q$ is also particular. | Let the major premise of $Q$ be denoted $\text{Maj}$.
Let the minor premise of $Q$ be denoted $\text{Min}$.
Let the conclusion of $Q$ be denoted $\text{C}$.
From No Valid Categorical Syllogism contains two Particular Premises, either $\text{Maj}$ or $\text{Min}$ has to be universal.
Let the other premise of $Q$ be part... | Let $Q$ be a [[Definition:Valid Argument|valid]] [[Definition:Categorical Syllogism|categorical syllogism]].
Let one of the [[Definition:Premise of Syllogism|premises]] of $Q$ be [[Definition:Particular Categorical Statement|particular]].
Then the [[Definition:Conclusion of Syllogism|conclusion]] of $Q$ is also [[De... | Let the [[Definition:Major Premise of Syllogism|major premise]] of $Q$ be denoted $\text{Maj}$.
Let the [[Definition:Minor Premise of Syllogism|minor premise]] of $Q$ be denoted $\text{Min}$.
Let the [[Definition:Conclusion of Syllogism|conclusion]] of $Q$ be denoted $\text{C}$.
From [[No Valid Categorical Syllogis... | No Valid Categorical Syllogism with Particular Premise has Universal Conclusion | https://proofwiki.org/wiki/No_Valid_Categorical_Syllogism_with_Particular_Premise_has_Universal_Conclusion | https://proofwiki.org/wiki/No_Valid_Categorical_Syllogism_with_Particular_Premise_has_Universal_Conclusion | [
"Categorical Syllogisms"
] | [
"Definition:Valid Argument",
"Definition:Categorical Syllogism",
"Definition:Categorical Syllogism/Premises",
"Definition:Particular Categorical Statement",
"Definition:Categorical Syllogism/Conclusion",
"Definition:Particular Categorical Statement"
] | [
"Definition:Categorical Syllogism/Premises/Major Premise",
"Definition:Categorical Syllogism/Premises/Minor Premise",
"Definition:Categorical Syllogism/Conclusion",
"No Valid Categorical Syllogism contains two Particular Premises",
"Definition:Universal Categorical Statement",
"Definition:Categorical Syll... |
proofwiki-8379 | Elimination of all but 48 Categorical Syllogisms as Invalid | Of the $256$ different types of categorical syllogism, all but $48$ can immediately be identified as invalid by consideration of the Rules of Quantity and the Rules of Quality. | There are $64$ patterns of categorical syllogism per figure:
:<nowiki>$\begin{array}{cccc}
AAA & AAE & AAI & AAO \\
AEA & AEE & AEI & AEO \\
AIA & AIE & AII & AIO \\
AOA & AOE & AOI & AOO \\
\end{array} \qquad
\begin{array}{cccc}
EAA & EAE & EAI & EAO \\
EEA & EEE & EEI & EEO \\
EIA & EIE & EII & EIO \\
EOA & EOE & EOI... | Of the $256$ different types of [[Definition:Categorical Syllogism|categorical syllogism]], all but $48$ can immediately be identified as [[Definition:Invalid Argument|invalid]] by consideration of the [[Rules of Quantity]] and the [[Rules of Quality]]. | There are $64$ patterns of [[Definition:Categorical Syllogism|categorical syllogism]] per [[Definition:Figure of Categorical Syllogism|figure]]:
:<nowiki>$\begin{array}{cccc}
AAA & AAE & AAI & AAO \\
AEA & AEE & AEI & AEO \\
AIA & AIE & AII & AIO \\
AOA & AOE & AOI & AOO \\
\end{array} \qquad
\begin{array}{cccc}
EAA &... | Elimination of all but 48 Categorical Syllogisms as Invalid | https://proofwiki.org/wiki/Elimination_of_all_but_48_Categorical_Syllogisms_as_Invalid | https://proofwiki.org/wiki/Elimination_of_all_but_48_Categorical_Syllogisms_as_Invalid | [
"Categorical Syllogisms"
] | [
"Definition:Categorical Syllogism",
"Definition:Invalid Argument",
"Rules of Quantity",
"Rules of Quality"
] | [
"Definition:Categorical Syllogism",
"Definition:Figure of Categorical Syllogism",
"No Valid Categorical Syllogism contains two Negative Premises",
"No Valid Categorical Syllogism contains two Particular Premises",
"Conclusion of Valid Categorical Syllogism is Negative iff one Premise is Negative",
"No Val... |
proofwiki-8380 | Conjunction implies Disjunction | :$\vdash \paren {p \land q} \implies \paren {p \lor q}$ | {{BeginTableau|\paren {p \land q} \implies \paren {p \lor q} }}
{{Assumption|1|p \land q}}
{{Simplification|2|1|p|1|1}}
{{Addition|3|1|p \lor q|2|1}}
{{Implication|4||\paren {p \land q} \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{qed}} | :$\vdash \paren {p \land q} \implies \paren {p \lor q}$ | {{BeginTableau|\paren {p \land q} \implies \paren {p \lor q} }}
{{Assumption|1|p \land q}}
{{Simplification|2|1|p|1|1}}
{{Addition|3|1|p \lor q|2|1}}
{{Implication|4||\paren {p \land q} \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{qed}} | Conjunction implies Disjunction/Proof 2 | https://proofwiki.org/wiki/Conjunction_implies_Disjunction | https://proofwiki.org/wiki/Conjunction_implies_Disjunction/Proof_2 | [
"Conjunction",
"Disjunction"
] | [] | [] |
proofwiki-8381 | Conjunction implies Disjunction | :$\vdash \paren {p \land q} \implies \paren {p \lor q}$ | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connective is true for all boolean interpretations.
:<nowiki>$\begin{array}{|ccc|c|ccc|} \hline
(p & \land & q) & \implies & (p & \lor & q) \\
\hline
\F & \F & \F & \T & \F & \F & \F \\
\F & \F & \T & \T & \F & \T & \T \\... | :$\vdash \paren {p \land q} \implies \paren {p \lor q}$ | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|... | Conjunction implies Disjunction/Proof by Truth Table | https://proofwiki.org/wiki/Conjunction_implies_Disjunction | https://proofwiki.org/wiki/Conjunction_implies_Disjunction/Proof_by_Truth_Table | [
"Conjunction",
"Disjunction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-8382 | Angle Bisector Vector | Let $\mathbf u$ and $\mathbf v$ be vectors of non-zero length.
Let $\norm {\mathbf u}$ and $\norm {\mathbf v}$ be their respective lengths.
Then $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the angle bisector of $\mathbf u$ and $\mathbf v$. | Let $\mathbf a = \norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$.
Then:
{{begin-eqn}}
{{eqn | l = \cos \angle \mathbf u, \mathbf a
| r = \frac {\mathbf u \cdot \mathbf a} {\norm {\mathbf u} \norm {\mathbf a} }
| c = Cosine Formula for Dot Product
}}
{{eqn | r = \frac {\mathbf u \cdot \paren {\nor... | Let $\mathbf u$ and $\mathbf v$ be [[Definition:Vector (Real Euclidean Space)|vectors]] of non-zero [[Definition:Vector Length|length]].
Let $\norm {\mathbf u}$ and $\norm {\mathbf v}$ be their respective [[Definition:Vector Length|lengths]].
Then $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the [[... | Let $\mathbf a = \norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$.
Then:
{{begin-eqn}}
{{eqn | l = \cos \angle \mathbf u, \mathbf a
| r = \frac {\mathbf u \cdot \mathbf a} {\norm {\mathbf u} \norm {\mathbf a} }
| c = [[Cosine Formula for Dot Product]]
}}
{{eqn | r = \frac {\mathbf u \cdot \paren... | Angle Bisector Vector/Algebraic Proof | https://proofwiki.org/wiki/Angle_Bisector_Vector | https://proofwiki.org/wiki/Angle_Bisector_Vector/Algebraic_Proof | [
"Angle Bisector Vector",
"Vector Algebra",
"Angle Bisectors",
"Euclidean Geometry"
] | [
"Definition:Vector/Real Euclidean Space",
"Definition:Vector Length",
"Definition:Vector Length",
"Definition:Angle Bisector"
] | [
"Cosine Formula for Dot Product",
"Dot Product Associates with Scalar Multiplication",
"Dot Product of Vector with Itself",
"Cosine Formula for Dot Product",
"Dot Product Associates with Scalar Multiplication",
"Dot Product of Vector with Itself",
"Definition:Cosine",
"Definition:Injection",
"Shape ... |
proofwiki-8383 | Angle Bisector Vector | Let $\mathbf u$ and $\mathbf v$ be vectors of non-zero length.
Let $\norm {\mathbf u}$ and $\norm {\mathbf v}$ be their respective lengths.
Then $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the angle bisector of $\mathbf u$ and $\mathbf v$. | :400px
As shown above:
:Let $\gamma$ be the angle between $\mathbf u$ and $\mathbf v$.
:Let $\alpha$ be the angle between $\norm {\mathbf u} \mathbf v$ and $\norm {\mathbf v} \mathbf u$.
:Let $\beta$ be the angle between $\mathbf u$ and $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$.
Note that $\norm {\mat... | Let $\mathbf u$ and $\mathbf v$ be [[Definition:Vector (Real Euclidean Space)|vectors]] of non-zero [[Definition:Vector Length|length]].
Let $\norm {\mathbf u}$ and $\norm {\mathbf v}$ be their respective [[Definition:Vector Length|lengths]].
Then $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the [[... | :[[File:Angular Bisector Vector Diagram.png|400px]]
As shown above:
:Let $\gamma$ be the [[Definition:Angle|angle]] between $\mathbf u$ and $\mathbf v$.
:Let $\alpha$ be the angle between $\norm {\mathbf u} \mathbf v$ and $\norm {\mathbf v} \mathbf u$.
:Let $\beta$ be the angle between $\mathbf u$ and $\norm {\math... | Angle Bisector Vector/Geometric Proof 1 | https://proofwiki.org/wiki/Angle_Bisector_Vector | https://proofwiki.org/wiki/Angle_Bisector_Vector/Geometric_Proof_1 | [
"Angle Bisector Vector",
"Vector Algebra",
"Angle Bisectors",
"Euclidean Geometry"
] | [
"Definition:Vector/Real Euclidean Space",
"Definition:Vector Length",
"Definition:Vector Length",
"Definition:Angle Bisector"
] | [
"File:Angular Bisector Vector Diagram.png",
"Definition:Angle",
"Vector Times Magnitude Same Length As Magnitude Times Vector",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Parallel Vectors",
"Parallelism implies Equal Corresponding Angles"
] |
proofwiki-8384 | Angle Bisector Vector | Let $\mathbf u$ and $\mathbf v$ be vectors of non-zero length.
Let $\norm {\mathbf u}$ and $\norm {\mathbf v}$ be their respective lengths.
Then $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the angle bisector of $\mathbf u$ and $\mathbf v$. | The vectors $\norm {\mathbf u} \mathbf v$ and $\norm {\mathbf v} \mathbf u$ have equal length from Vector Times Magnitude Same Length As Magnitude Times Vector.
Thus $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the diagonal of a rhombus.
The result follows from Diagonals of Rhombus Bisect Angles.
{{qe... | Let $\mathbf u$ and $\mathbf v$ be [[Definition:Vector (Real Euclidean Space)|vectors]] of non-zero [[Definition:Vector Length|length]].
Let $\norm {\mathbf u}$ and $\norm {\mathbf v}$ be their respective [[Definition:Vector Length|lengths]].
Then $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the [[... | The vectors $\norm {\mathbf u} \mathbf v$ and $\norm {\mathbf v} \mathbf u$ have equal length from [[Vector Times Magnitude Same Length As Magnitude Times Vector]].
Thus $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the diagonal of a [[Definition:Rhombus|rhombus]].
The result follows from [[Diagona... | Angle Bisector Vector/Geometric Proof 2 | https://proofwiki.org/wiki/Angle_Bisector_Vector | https://proofwiki.org/wiki/Angle_Bisector_Vector/Geometric_Proof_2 | [
"Angle Bisector Vector",
"Vector Algebra",
"Angle Bisectors",
"Euclidean Geometry"
] | [
"Definition:Vector/Real Euclidean Space",
"Definition:Vector Length",
"Definition:Vector Length",
"Definition:Angle Bisector"
] | [
"Vector Times Magnitude Same Length As Magnitude Times Vector",
"Definition:Quadrilateral/Rhombus",
"Diagonals of Rhombus Bisect Angles"
] |
proofwiki-8385 | Subgroup of Order 1 is Trivial | Let $\struct {G, \circ}$ be a group.
Then $\struct {G, \circ}$ has exactly $1$ subgroup of order $1$: the trivial subgroup. | From Trivial Subgroup is Subgroup, $\struct {\set e, \circ}$ is a subgroup of $\struct {G, \circ}$.
Suppose $\struct {\set g, \circ}$ is a subgroup of $\struct {G, \circ}$.
From Group is not Empty, $e \in \set g$.
Thus it follows trivially that $\struct {\set g, \circ} = \struct {\set e, \circ}$.
That is, $\struct {\se... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Then $\struct {G, \circ}$ has exactly $1$ [[Definition:Subgroup|subgroup]] of [[Definition:Order of Structure|order]] $1$: the [[Definition:Trivial Subgroup|trivial subgroup]]. | From [[Trivial Subgroup is Subgroup]], $\struct {\set e, \circ}$ is a [[Definition:Subgroup|subgroup]] of $\struct {G, \circ}$.
Suppose $\struct {\set g, \circ}$ is a [[Definition:Subgroup|subgroup]] of $\struct {G, \circ}$.
From [[Group is not Empty]], $e \in \set g$.
Thus it follows trivially that $\struct {\set ... | Subgroup of Order 1 is Trivial | https://proofwiki.org/wiki/Subgroup_of_Order_1_is_Trivial | https://proofwiki.org/wiki/Subgroup_of_Order_1_is_Trivial | [
"Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Order of Structure",
"Definition:Trivial Subgroup"
] | [
"Trivial Subgroup is Subgroup",
"Definition:Subgroup",
"Definition:Subgroup",
"Group is not Empty",
"Definition:Subgroup",
"Definition:Order of Structure"
] |
proofwiki-8386 | Injection from Finite Set to Itself is Surjection/Corollary | Let $S$ be a finite set.
Let $f: S \to S$ be an injection.
Then $f$ is a permutation. | From Injection from Finite Set to Itself is Surjection, $f$ is a surjection.
As $f$ is thus both an injection and a surjection, $f$ is a bijection by definition.
Thus as $f$ is a bijection to itself, it is by definition a permutation.
{{qed}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to S$ be an [[Definition:Injection|injection]].
Then $f$ is a [[Definition:Permutation|permutation]]. | From [[Injection from Finite Set to Itself is Surjection]], $f$ is a [[Definition:Surjection|surjection]].
As $f$ is thus both an [[Definition:Injection|injection]] and a [[Definition:Surjection|surjection]], $f$ is a [[Definition:Bijection|bijection]] by definition.
Thus as $f$ is a [[Definition:Bijection|bijection]... | Injection from Finite Set to Itself is Surjection/Corollary | https://proofwiki.org/wiki/Injection_from_Finite_Set_to_Itself_is_Surjection/Corollary | https://proofwiki.org/wiki/Injection_from_Finite_Set_to_Itself_is_Surjection/Corollary | [
"Injections",
"Permutations"
] | [
"Definition:Finite Set",
"Definition:Injection",
"Definition:Permutation"
] | [
"Injection from Finite Set to Itself is Surjection",
"Definition:Surjection",
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Permutation"
] |
proofwiki-8387 | Surjection from Finite Set to Itself is Permutation | Let $S$ be a finite set.
Let $f: S \to S$ be an surjection.
Then $f$ is a permutation. | From Surjection iff Right Inverse, $f$ has a right inverse $g: S \to S$.
From Right Inverse Mapping is Injection, $g$ is an injection.
From Injection from Finite Set to Itself is Permutation, $g$ is a permutation and so a bijection.
From Inverse of Bijection is Bijection, $f$ is also a bijection.
Thus as $f$ is a bijec... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $f: S \to S$ be an [[Definition:Surjection|surjection]].
Then $f$ is a [[Definition:Permutation|permutation]]. | From [[Surjection iff Right Inverse]], $f$ has a [[Definition:Right Inverse Mapping|right inverse]] $g: S \to S$.
From [[Right Inverse Mapping is Injection]], $g$ is an [[Definition:Injection|injection]].
From [[Injection from Finite Set to Itself is Permutation]], $g$ is a [[Definition:Permutation|permutation]] and ... | Surjection from Finite Set to Itself is Permutation | https://proofwiki.org/wiki/Surjection_from_Finite_Set_to_Itself_is_Permutation | https://proofwiki.org/wiki/Surjection_from_Finite_Set_to_Itself_is_Permutation | [
"Surjections",
"Permutations"
] | [
"Definition:Finite Set",
"Definition:Surjection",
"Definition:Permutation"
] | [
"Surjection iff Right Inverse",
"Definition:Right Inverse Mapping",
"Right Inverse Mapping is Injection",
"Definition:Injection",
"Injection from Finite Set to Itself is Surjection/Corollary",
"Definition:Permutation",
"Definition:Bijection",
"Inverse of Bijection is Bijection",
"Definition:Bijectio... |
proofwiki-8388 | Order of Power of Group Element | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $g \in G$ be an element of $G$ such that:
:$\order g = n$
where $\order g$ denotes the order of $g$.
Then:
:$\forall m \in \Z: \order {g^m} = \dfrac n {\gcd \set {m, n} }$
where $\gcd \set {m, n}$ denotes the greatest common divisor of $m$ and $n$. | Let $\gcd \set {m, n} = d$.
From Integers Divided by GCD are Coprime: there exists $m', n' \in \Z$ such that $m = d m'$, $n = d n'$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {g^m}^{n'}
| r = \paren {g^{d m'} }^{n'}
| c = Definition of $m'$
}}
{{eqn | r = \paren {g^{d n'} }^{m'}
| c =
}}
{{eqn | r = \pa... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $g \in G$ be an [[Definition:Element|element]] of $G$ such that:
:$\order g = n$
where $\order g$ denotes the [[Definition:Order of Group Element|order]] of $g$.
Then:
:$\forall m \in \Z: \order {g^m} =... | Let $\gcd \set {m, n} = d$.
From [[Integers Divided by GCD are Coprime]]: there exists $m', n' \in \Z$ such that $m = d m'$, $n = d n'$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {g^m}^{n'}
| r = \paren {g^{d m'} }^{n'}
| c = Definition of $m'$
}}
{{eqn | r = \paren {g^{d n'} }^{m'}
| c =
}}
{{eqn | ... | Order of Power of Group Element | https://proofwiki.org/wiki/Order_of_Power_of_Group_Element | https://proofwiki.org/wiki/Order_of_Power_of_Group_Element | [
"Order of Group Elements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Greatest Common Divisor/Integers"
] | [
"Integers Divided by GCD are Coprime",
"Definition:Order of Group Element",
"Element to Power of Multiple of Order is Identity",
"Bézout's Identity",
"Definition:Order of Group Element",
"Definition:Order of Group Element",
"Definition:Contradiction",
"Definition:Order of Group Element"
] |
proofwiki-8389 | Existence of Group of Finite Order | Let $n \in \Z_{>0}$.
Then there exists at least one group whose order is $n$. | From Existence of Cyclic Group of Order n, there exists a cyclic group whose order is $n$.
In particular, a concrete example of such a group is demonstrated in Roots of Unity under Multiplication form Cyclic Group.
{{qed}} | Let $n \in \Z_{>0}$.
Then there exists at least one [[Definition:Group|group]] whose [[Definition:Order of Structure|order]] is $n$. | From [[Existence of Cyclic Group of Order n]], there exists a [[Definition:Cyclic Group|cyclic group]] whose [[Definition:Order of Structure|order]] is $n$.
In particular, a concrete example of such a [[Definition:Group|group]] is demonstrated in [[Roots of Unity under Multiplication form Cyclic Group]].
{{qed}} | Existence of Group of Finite Order | https://proofwiki.org/wiki/Existence_of_Group_of_Finite_Order | https://proofwiki.org/wiki/Existence_of_Group_of_Finite_Order | [
"Order of Groups"
] | [
"Definition:Group",
"Definition:Order of Structure"
] | [
"Existence of Cyclic Group of Order n",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Group",
"Roots of Unity under Multiplication form Cyclic Group"
] |
proofwiki-8390 | Vitali Set Existence Theorem | There exists a set of real numbers which is not Lebesgue measurable. | === Lemma ===
{{:Vitali Set Existence Theorem/Lemma}}{{qed|lemma}}
Let $\map \mu X$ denote the Lebesgue measure of a set $X$ of real numbers.
We have that:
{{begin-itemize}}
{{item|(1):|$\map \mu X$ is a countably additive function}}
{{item|(2):|$\map \mu X$ is translation invariant}}
{{item|(3):|From Measure of Interv... | There exists a [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] which is not [[Definition:Lebesgue Measure|Lebesgue measurable]]. | === [[Vitali Set Existence Theorem/Lemma|Lemma]] ===
{{:Vitali Set Existence Theorem/Lemma}}{{qed|lemma}}
Let $\map \mu X$ denote the [[Definition:Lebesgue Measure|Lebesgue measure]] of a [[Definition:Set|set]] $X$ of [[Definition:Real Number|real numbers]].
We have that:
{{begin-itemize}}
{{item|(1):|[[Lebesgue Mea... | Vitali Set Existence Theorem/Proof 1 | https://proofwiki.org/wiki/Vitali_Set_Existence_Theorem | https://proofwiki.org/wiki/Vitali_Set_Existence_Theorem/Proof_1 | [
"Vitali Set Existence Theorem",
"Vitali Sets",
"Non-Measurable Sets",
"Measure Theory"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Lebesgue Measure"
] | [
"Vitali Set Existence Theorem/Lemma",
"Definition:Lebesgue Measure",
"Definition:Set",
"Definition:Real Number",
"Lebesgue Measure is Countably Additive",
"Lebesgue Measure is Invariant under Translations",
"Measure of Interval is Length",
"Definition:Real Interval/Closed",
"Definition:Real Number",... |
proofwiki-8391 | Vitali Set Existence Theorem | There exists a set of real numbers which is not Lebesgue measurable. | We construct such a set.
For $x, y \in \hointr 0 1$, define the sum modulo 1:
:$x +_1 y = \begin {cases} x + y & : x + y < 1 \\ x + y - 1 & : x + y \ge 1 \end {cases}$
Let $E \subset \hointr 0 1$ be a measurable set.
Let $E_1 = E \cap \hointr 0 {1 - x}$ and $E_2 = E \cap \hointr {1 - x} 1$.
By Measure of Interval is Le... | There exists a [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] which is not [[Definition:Lebesgue Measure|Lebesgue measurable]]. | We construct such a [[Definition:Set|set]].
For $x, y \in \hointr 0 1$, define the [[Definition:Modulo Addition|sum modulo 1]]:
:$x +_1 y = \begin {cases} x + y & : x + y < 1 \\ x + y - 1 & : x + y \ge 1 \end {cases}$
Let $E \subset \hointr 0 1$ be a [[Definition:Measurable Set|measurable set]].
Let $E_1 = E \cap \... | Vitali Set Existence Theorem/Proof 2 | https://proofwiki.org/wiki/Vitali_Set_Existence_Theorem | https://proofwiki.org/wiki/Vitali_Set_Existence_Theorem/Proof_2 | [
"Vitali Set Existence Theorem",
"Vitali Sets",
"Non-Measurable Sets",
"Measure Theory"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Lebesgue Measure"
] | [
"Definition:Set",
"Definition:Modulo Addition",
"Definition:Measurable Set",
"Measure of Interval is Length",
"Definition:Disjoint Sets",
"Definition:Real Interval",
"Definition:Measurable Set",
"Measurable Sets form Algebra of Sets",
"Definition:Set Intersection",
"Lebesgue Measure is Invariant u... |
proofwiki-8392 | Finite Number of Groups of Given Finite Order | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then there exists a finite number of types of group of order $n$. | For any group $\struct {G, \circ}$ of order $n$ and any set of $n$ elements, $X$ can be the underlying set of a group which is isomorphic to $\struct {G, \circ}$, as follows:
Choose a bijection $\phi: G \to X$.
Define the group operation $*$ on $X$ by the rule:
:$\map \phi {g_1} * \map \phi {g_2} = \map \phi {g_1 \circ... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then there exists a [[Definition:Finite|finite number]] of [[Definition:Group Type|types of group]] of [[Definition:Order of Structure|order $n$]]. | For any [[Definition:Group|group]] $\struct {G, \circ}$ of [[Definition:Order of Structure|order $n$]] and any [[Definition:Set|set]] of $n$ [[Definition:Element|elements]], $X$ can be the [[Definition:Underlying Set of Structure|underlying set]] of a [[Definition:Group|group]] which is [[Definition:Group Isomorphism|i... | Finite Number of Groups of Given Finite Order | https://proofwiki.org/wiki/Finite_Number_of_Groups_of_Given_Finite_Order | https://proofwiki.org/wiki/Finite_Number_of_Groups_of_Given_Finite_Order | [
"Order of Groups"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Finite",
"Definition:Group Type",
"Definition:Order of Structure"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Definition:Set",
"Definition:Element",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Bijection",
"Definition:Group Product/Group Law",
"Definition:Is... |
proofwiki-8393 | Nu of Prime Number is 1 | Let $p$ be a prime number.
Then:
:$\map \nu p = 1$
where $\nu$ denotes the $\nu$ function: the number of types of group of a given order. | Let $G_1$ and $G_2$ be groups of order $p$.
From Prime Group is Cyclic, $G_1$ and $G_2$ are both cyclic groups.
From Cyclic Groups of Same Order are Isomorphic, $G_1$ and $G_2$ are isomorphic.
Thus by definition, $G_1$ and $G_2$ are of the same type.
Hence the result.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Then:
:$\map \nu p = 1$
where $\nu$ denotes the [[Definition:Nu Function|$\nu$ function]]: the number of [[Definition:Group Type|types of group]] of a given [[Definition:Order of Group|order]]. | Let $G_1$ and $G_2$ be [[Definition:Group|groups]] of [[Definition:Order of Group|order]] $p$.
From [[Prime Group is Cyclic]], $G_1$ and $G_2$ are both [[Definition:Cyclic Group|cyclic groups]].
From [[Cyclic Groups of Same Order are Isomorphic]], $G_1$ and $G_2$ are [[Definition:Group Isomorphism|isomorphic]].
Thus... | Nu of Prime Number is 1 | https://proofwiki.org/wiki/Nu_of_Prime_Number_is_1 | https://proofwiki.org/wiki/Nu_of_Prime_Number_is_1 | [
"Prime Groups",
"Nu Function"
] | [
"Definition:Prime Number",
"Definition:Nu Function",
"Definition:Group Type",
"Definition:Order of Structure"
] | [
"Definition:Group",
"Definition:Order of Structure",
"Prime Group is Cyclic",
"Definition:Cyclic Group",
"Cyclic Groups of Same Order are Isomorphic",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Group Type"
] |
proofwiki-8394 | Image of Subset under Relation equals Union of Images of Elements | Let $S$ and $T$ be sets.
Let $\RR \subseteq S \times T$ be a relation on $S \times T$.
Let $X \subseteq S$ be a subset of $S$.
Then:
:$\ds \RR \sqbrk X = \bigcup_{x \mathop \in X} \map \RR x$
where:
:$\RR \sqbrk X$ is the image of the subset $X$ under $\RR$
:$\map \RR x$ is the image of the element $x$ under $\RR$. | By definition:
:$\RR \sqbrk X = \set {y \in T: \exists x \in X: \tuple {x, y} \in \RR}$
:$\map \RR x = \set {y \in T:\tuple {x, y} \in \RR}$
First:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \RR \sqbrk X
}}
{{eqn | ll= \leadsto
| q = \exists x \in X
| l = \tuple {x, y}
| o = \in
| ... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]] on $S \times T$.
Let $X \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Then:
:$\ds \RR \sqbrk X = \bigcup_{x \mathop \in X} \map \RR x$
where:
:$\RR \sqbrk X$ is the [[Definition:Image of Subset ... | By definition:
:$\RR \sqbrk X = \set {y \in T: \exists x \in X: \tuple {x, y} \in \RR}$
:$\map \RR x = \set {y \in T:\tuple {x, y} \in \RR}$
First:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \RR \sqbrk X
}}
{{eqn | ll= \leadsto
| q = \exists x \in X
| l = \tuple {x, y}
| o = \in
... | Image of Subset under Relation equals Union of Images of Elements | https://proofwiki.org/wiki/Image_of_Subset_under_Relation_equals_Union_of_Images_of_Elements | https://proofwiki.org/wiki/Image_of_Subset_under_Relation_equals_Union_of_Images_of_Elements | [
"Relation Theory",
"Subsets",
"Set Union"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Subset",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Image (Set Theory)/Relation/Element"
] | [
"Definition:Set Equality/Definition 2",
"Category:Relation Theory",
"Category:Subsets",
"Category:Set Union"
] |
proofwiki-8395 | Preimage of Subset under Relation equals Union of Preimages of Elements | Let $S$ and $T$ be sets.
Let $\RR \subseteq S \times T$ be a relation on $S \times T$.
Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$
Let $Y \subseteq T$ be a subset of $T$.
Then:
:$\RR^{-1} \sqbrk Y = \ds \bigcup_{y \mathop \in Y} \map {\RR^{-1} } y$
where:
:$\RR^{-1} \sqbrk Y$ is the preimage of... | By definition, $\RR^{-1} \subseteq T \times S$ is a relation on $T \times S$.
Thus Image of Subset under Relation equals Union of Images of Elements can be applied directly.
{{qed}}
Category:Relation Theory
h2kcezt8nswgbhvmdszqs4q0igjmyeo | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]] on $S \times T$.
Let $\RR^{-1} \subseteq T \times S$ be the [[Definition:Inverse Relation|inverse relation]] to $\RR$
Let $Y \subseteq T$ be a [[Definition:Subset|subset]] of $T$.
Then:
:$\RR^{-1} \sqb... | By definition, $\RR^{-1} \subseteq T \times S$ is a [[Definition:Relation|relation]] on $T \times S$.
Thus [[Image of Subset under Relation equals Union of Images of Elements]] can be applied directly.
{{qed}}
[[Category:Relation Theory]]
h2kcezt8nswgbhvmdszqs4q0igjmyeo | Preimage of Subset under Relation equals Union of Preimages of Elements | https://proofwiki.org/wiki/Preimage_of_Subset_under_Relation_equals_Union_of_Preimages_of_Elements | https://proofwiki.org/wiki/Preimage_of_Subset_under_Relation_equals_Union_of_Preimages_of_Elements | [
"Relation Theory"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Inverse Relation",
"Definition:Subset",
"Definition:Preimage/Relation/Subset",
"Definition:Subset",
"Definition:Preimage/Relation/Element",
"Definition:Element"
] | [
"Definition:Relation",
"Image of Subset under Relation equals Union of Images of Elements",
"Category:Relation Theory"
] |
proofwiki-8396 | Preimage of Subset under Mapping equals Union of Preimages of Elements | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping from $S$ to $T$.
Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, defined as:
:$f^{-1} = \set {\tuple {t, s}: \map f s = t}$
Let $Y \subseteq T$ be a subset of $T$.
Then:
:$\ds f^{-1} \sqbrk Y = \bigcup_{y \mathop \in Y} \map {f^{-1} } y$
where:
:$f^{-1} \s... | By definition, $f^{-1} \subseteq T \times S$ is a relation on $T \times S$.
Thus Image of Subset under Relation equals Union of Images of Elements can be applied directly.
{{qed}}
Category:Preimages under Mappings
Category:Subsets
Category:Set Union
3dwfblsxgav2kxtnvq5h5v003e9y11i | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] from $S$ to $T$.
Let $f^{-1} \subseteq T \times S$ be the [[Definition:Inverse of Mapping|inverse]] of $f$, defined as:
:$f^{-1} = \set {\tuple {t, s}: \map f s = t}$
Let $Y \subseteq T$ be a [[Definition:Subset|subset... | By definition, $f^{-1} \subseteq T \times S$ is a [[Definition:Relation|relation]] on $T \times S$.
Thus [[Image of Subset under Relation equals Union of Images of Elements]] can be applied directly.
{{qed}}
[[Category:Preimages under Mappings]]
[[Category:Subsets]]
[[Category:Set Union]]
3dwfblsxgav2kxtnvq5h5v003e9y... | Preimage of Subset under Mapping equals Union of Preimages of Elements | https://proofwiki.org/wiki/Preimage_of_Subset_under_Mapping_equals_Union_of_Preimages_of_Elements | https://proofwiki.org/wiki/Preimage_of_Subset_under_Mapping_equals_Union_of_Preimages_of_Elements | [
"Preimages under Mappings",
"Subsets",
"Set Union"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Subset",
"Definition:Preimage/Mapping/Subset",
"Definition:Preimage/Mapping/Element"
] | [
"Definition:Relation",
"Image of Subset under Relation equals Union of Images of Elements",
"Category:Preimages under Mappings",
"Category:Subsets",
"Category:Set Union"
] |
proofwiki-8397 | Image of Domain of Mapping is Image Set | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
The image of $S$ is the image set of $f$:
:$f \sqbrk S = \Img f$ | By definition, a mapping is a relation.
Thus Image of Domain of Relation is Image Set applies.
{{qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
The [[Definition:Image of Subset under Mapping|image of $S$]] is the [[Definition:Image of Mapping|image set of $f$]]:
:$f \sqbrk S = \Img f$ | By definition, a [[Definition:Mapping|mapping]] is a [[Definition:Relation|relation]].
Thus [[Image of Domain of Relation is Image Set]] applies.
{{qed}} | Image of Domain of Mapping is Image Set | https://proofwiki.org/wiki/Image_of_Domain_of_Mapping_is_Image_Set | https://proofwiki.org/wiki/Image_of_Domain_of_Mapping_is_Image_Set | [
"Images"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Mapping",
"Definition:Relation",
"Image of Domain of Relation is Image Set"
] |
proofwiki-8398 | Image of Singleton under Mapping | Let $f: S \to T$ be a mapping.
Then the image of an element of $S$ is equal to the image of a singleton containing that element, the singleton being a subset of $S$:
:$\forall s \in S: \set {\map f s} = f \sqbrk {\set s}$ | By definition, a mapping is a relation.
Thus Image of Singleton under Relation applies.
{{Qed}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then the [[Definition:Image of Element under Mapping|image]] of an [[Definition:Element|element]] of $S$ is equal to the [[Definition:Image of Subset under Mapping|image]] of a [[Definition:Singleton|singleton]] containing that element, the singleton being a [[Defi... | By definition, a [[Definition:Mapping|mapping]] is a [[Definition:Relation|relation]].
Thus [[Image of Singleton under Relation]] applies.
{{Qed}} | Image of Singleton under Mapping | https://proofwiki.org/wiki/Image_of_Singleton_under_Mapping | https://proofwiki.org/wiki/Image_of_Singleton_under_Mapping | [
"Images",
"Singletons"
] | [
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Element",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Singleton",
"Definition:Subset"
] | [
"Definition:Mapping",
"Definition:Relation",
"Image of Singleton under Relation"
] |
proofwiki-8399 | Image of Subset under Mapping equals Union of Images of Elements | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping from $S$ to $T$.
Let $X \subseteq S$ be a subset of $S$.
Then:
:$\ds f \sqbrk X = \bigcup_{x \mathop \in X} \map f x$
where:
:$f \sqbrk X$ is the image of the subset $X$ under $f$
:$\map f x$ is the image of the element $x$ under $f$. | By definition, a mapping is a relation.
Thus Image of Subset under Relation equals Union of Images of Elements applies.
{{Qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] from $S$ to $T$.
Let $X \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Then:
:$\ds f \sqbrk X = \bigcup_{x \mathop \in X} \map f x$
where:
:$f \sqbrk X$ is the [[Definition:Image of Subset under Mapping|image of... | By definition, a [[Definition:Mapping|mapping]] is a [[Definition:Relation|relation]].
Thus [[Image of Subset under Relation equals Union of Images of Elements]] applies.
{{Qed}} | Image of Subset under Mapping equals Union of Images of Elements | https://proofwiki.org/wiki/Image_of_Subset_under_Mapping_equals_Union_of_Images_of_Elements | https://proofwiki.org/wiki/Image_of_Subset_under_Mapping_equals_Union_of_Images_of_Elements | [
"Images",
"Set Union"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Image (Set Theory)/Mapping/Element"
] | [
"Definition:Mapping",
"Definition:Relation",
"Image of Subset under Relation equals Union of Images of Elements"
] |
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