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-18700 | Canonical P-adic Expansion of Rational is Eventually Periodic | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $x \in \Q_p$.
Then:
:$x$ is a rational number {{iff}} the canonical expansion of $x$ is eventually periodic. | === Necessary Condition ===
Let $x$ be a rational number.
{{:Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition}}{{qed|lemma}} | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $x \in \Q_p$.
Then:
:$x$ is a [[Definition:Rational Number|rational number]] {{iff}} the [[Definition:Canonical P-adic Expansion|canonical expansion]] of... | === [[Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition|Necessary Condition]] ===
Let $x$ be a [[Definition:Rational Number|rational number]].
{{:Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition}}{{qed|lemma}} | Canonical P-adic Expansion of Rational is Eventually Periodic | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic | [
"P-adic Number Theory",
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:Rational Number",
"Definition:Canonical P-adic Expansion",
"Definition:Eventually Periodic P-adic Expansion"
] | [
"Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition",
"Definition:Rational Number"
] |
proofwiki-18701 | Sum of Hyperbolic Sine and Cosine equals Exponential | Let $z \in \C$ be a complex number.
Then:
:$e^z = \cosh z + \sinh z$ | {{begin-eqn}}
{{eqn | l = \cosh z + \sinh z
| r = \dfrac {e^z + e^{-z} } 2 + \dfrac {e^z - e^{-z} } 2
| c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \dfrac {e^z + e^z + e^{-z} - e^{-z} } 2
| c =
}}
{{eqn | r = e^z
| c =
}}
{{end-eqn}}
{{Qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$e^z = \cosh z + \sinh z$ | {{begin-eqn}}
{{eqn | l = \cosh z + \sinh z
| r = \dfrac {e^z + e^{-z} } 2 + \dfrac {e^z - e^{-z} } 2
| c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \dfrac {e^z + e^z + e^{-z} - e^{-z} } 2
| c =
}}
{{eqn | r = e^z
| c =
}}
{{end-eqn}}
{{Qed}} | Sum of Hyperbolic Sine and Cosine equals Exponential | https://proofwiki.org/wiki/Sum_of_Hyperbolic_Sine_and_Cosine_equals_Exponential | https://proofwiki.org/wiki/Sum_of_Hyperbolic_Sine_and_Cosine_equals_Exponential | [
"Exponential Function",
"Hyperbolic Sine Function",
"Hyperbolic Cosine Function"
] | [
"Definition:Complex Number"
] | [] |
proofwiki-18702 | Naturally Ordered Semigroup Axioms are Independent | Consider the naturally ordered semigroup axioms:
{{:Axiom:Naturally Ordered Semigroup Axioms}}
Each of the naturally ordered semigroup axioms is independent of all the others.
That is, you cannot drop any one of them and still uniquely define a naturally ordered semigroup. | This will be proved by demonstrating that for each of the naturally ordered semigroup axioms, it is possible to create an algebraic structure which fulfils all the others, but is not a naturally ordered semigroup. | Consider the [[Axiom:Naturally Ordered Semigroup Axioms|naturally ordered semigroup axioms]]:
{{:Axiom:Naturally Ordered Semigroup Axioms}}
Each of the [[Axiom:Naturally Ordered Semigroup Axioms|naturally ordered semigroup axioms]] is independent of all the others.
That is, you cannot drop any one of them and still u... | This will be proved by demonstrating that for each of the [[Axiom:Naturally Ordered Semigroup Axioms|naturally ordered semigroup axioms]], it is possible to create an [[Definition:Algebraic Structure|algebraic structure]] which fulfils all the others, but is not a [[Definition:Naturally Ordered Semigroup|naturally orde... | Naturally Ordered Semigroup Axioms are Independent | https://proofwiki.org/wiki/Naturally_Ordered_Semigroup_Axioms_are_Independent | https://proofwiki.org/wiki/Naturally_Ordered_Semigroup_Axioms_are_Independent | [
"Naturally Ordered Semigroup"
] | [
"Axiom:Naturally Ordered Semigroup Axioms",
"Axiom:Naturally Ordered Semigroup Axioms",
"Definition:Naturally Ordered Semigroup"
] | [
"Axiom:Naturally Ordered Semigroup Axioms",
"Definition:Algebraic Structure",
"Definition:Naturally Ordered Semigroup"
] |
proofwiki-18703 | Naturally Ordered Semigroup Axioms imply Commutativity | Consider the naturally ordered semigroup axioms:
{{:Axiom:Naturally Ordered Semigroup Axioms}}
Axioms $\text {NO} 1$, $\text {NO} 2$ and $\text {NO} 3$ together imply the commutativity of the naturally ordered semigroup $\struct {S, \circ, \preceq}$. | From {{NOSAxiom|1}}, $\struct {S, \circ, \preceq}$ has a smallest element.
This is identified as '''zero: $0$'''.
From Zero is Identity in Naturally Ordered Semigroup, $0$ is the identity element of $\struct {S, \circ, \preceq}$,
It may be the case that $S$ is a singleton such that $S = \set 0$ .
Then $\struct {S, \cir... | Consider the [[Axiom:Naturally Ordered Semigroup Axioms|naturally ordered semigroup axioms]]:
{{:Axiom:Naturally Ordered Semigroup Axioms}}
[[Axiom:Naturally Ordered Semigroup Axioms|Axioms $\text {NO} 1$, $\text {NO} 2$ and $\text {NO} 3$]] together imply the [[Definition:Commutative Operation|commutativity]] of the ... | From {{NOSAxiom|1}}, $\struct {S, \circ, \preceq}$ has a [[Definition:Smallest Element|smallest element]].
This is identified as '''[[Definition:Zero of Naturally Ordered Semigroup|zero: $0$]]'''.
From [[Zero is Identity in Naturally Ordered Semigroup]], $0$ is the [[Definition:Identity Element|identity element]] of ... | Naturally Ordered Semigroup Axioms imply Commutativity | https://proofwiki.org/wiki/Naturally_Ordered_Semigroup_Axioms_imply_Commutativity | https://proofwiki.org/wiki/Naturally_Ordered_Semigroup_Axioms_imply_Commutativity | [
"Naturally Ordered Semigroup"
] | [
"Axiom:Naturally Ordered Semigroup Axioms",
"Axiom:Naturally Ordered Semigroup Axioms",
"Definition:Commutative/Operation",
"Definition:Naturally Ordered Semigroup"
] | [
"Definition:Smallest Element",
"Definition:Zero (Number)/Naturally Ordered Semigroup",
"Zero is Identity in Naturally Ordered Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Singleton",
"Definition:Degenerate",
"Definition:Trivial Group",
"Trivial Group is Abelian"... |
proofwiki-18704 | Positive Rational Numbers under Addition fulfil Naturally Ordered Semigroup Axioms 2 to 4 | The algebraic structure:
:$\struct {\Q_{\ge 0}, +, \le}$
is an ordered semigroup which fulfils the axioms:
:{{NOSAxiom|2}}
:{{NOSAxiom|3}}
:{{NOSAxiom|4}}
but:
:does not fulfil {{NOSAxiom|1}}
:$\struct {\Q_{\ge 0}, +}$is not isomorphic to $\struct {\N, +}$. | First we note that from Positive Rational Numbers under Addition form Ordered Semigroup:
:$\struct {\Q_{\ge 0}, +, \le}$ is an ordered semigroup. | The [[Definition:Algebraic Structure|algebraic structure]]:
:$\struct {\Q_{\ge 0}, +, \le}$
is an [[Definition:Ordered Semigroup|ordered semigroup]] which fulfils the [[Axiom:Naturally Ordered Semigroup Axioms|axioms]]:
:{{NOSAxiom|2}}
:{{NOSAxiom|3}}
:{{NOSAxiom|4}}
but:
:does not fulfil {{NOSAxiom|1}}
:$\struct {\Q... | First we note that from [[Positive Rational Numbers under Addition form Ordered Semigroup]]:
:$\struct {\Q_{\ge 0}, +, \le}$ is an [[Definition:Ordered Semigroup|ordered semigroup]]. | Positive Rational Numbers under Addition fulfil Naturally Ordered Semigroup Axioms 2 to 4 | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_fulfil_Naturally_Ordered_Semigroup_Axioms_2_to_4 | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_fulfil_Naturally_Ordered_Semigroup_Axioms_2_to_4 | [
"Naturally Ordered Semigroup"
] | [
"Definition:Algebraic Structure",
"Definition:Ordered Semigroup",
"Axiom:Naturally Ordered Semigroup Axioms",
"Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism"
] | [
"Positive Rational Numbers under Addition form Ordered Semigroup",
"Definition:Ordered Semigroup"
] |
proofwiki-18705 | Subsemigroup of Ordered Semigroup is Ordered | Let $\struct {S, \circ, \preceq}$ be an ordered semigroup.
Let $\struct {T, \circ_T}$ be a subsemigroup of $\struct {S, \circ}$.
Then the ordered structure $\struct {T, \circ_T, \preceq_T}$ is also an ordered semigroup.
In the above:
:$\circ_T$ denotes the operation induced on $T$ by $\circ$
:$\preceq_T$ denotes the re... | It is necessary to ascertain that $\struct {T, \circ {\restriction_T} }$ fulfils the ordered semigroup axioms:
{{:Axiom:Ordered Semigroup Axioms}}
In this context, we see that $\text {OS} 0$ and $\text {OS} 1$ are fulfilled a fortiori by dint of $\struct {T, \circ {\restriction_T} }$ being a subsemigroup of $\struct {S... | Let $\struct {S, \circ, \preceq}$ be an [[Definition:Ordered Semigroup|ordered semigroup]].
Let $\struct {T, \circ_T}$ be a [[Definition:Subsemigroup|subsemigroup]] of $\struct {S, \circ}$.
Then the [[Definition:Ordered Structure|ordered structure]] $\struct {T, \circ_T, \preceq_T}$ is also an [[Definition:Ordered S... | It is necessary to ascertain that $\struct {T, \circ {\restriction_T} }$ fulfils the [[Axiom:Ordered Semigroup Axioms|ordered semigroup axioms]]:
{{:Axiom:Ordered Semigroup Axioms}}
In this context, we see that $\text {OS} 0$ and $\text {OS} 1$ are fulfilled [[Definition:A Fortiori|a fortiori]] by dint of $\struct {T,... | Subsemigroup of Ordered Semigroup is Ordered | https://proofwiki.org/wiki/Subsemigroup_of_Ordered_Semigroup_is_Ordered | https://proofwiki.org/wiki/Subsemigroup_of_Ordered_Semigroup_is_Ordered | [
"Ordered Semigroups"
] | [
"Definition:Ordered Semigroup",
"Definition:Subsemigroup",
"Definition:Ordered Structure",
"Definition:Ordered Semigroup",
"Definition:Operation Induced by Restriction",
"Definition:Restriction/Relation"
] | [
"Axiom:Ordered Semigroup Axioms",
"Definition:A Fortiori",
"Definition:Subsemigroup",
"Definition:Ordered Semigroup",
"Restriction of Ordering is Ordering",
"Definition:Ordering",
"Axiom:Ordered Semigroup Axioms",
"Axiom:Ordered Semigroup Axioms",
"Axiom:Ordered Semigroup Axioms"
] |
proofwiki-18706 | Positive Rational Numbers are Closed under Addition | Let $\Q_{\ge 0}$ denote the set of positive rational numbers:
:$\Q_{\ge 0} := \set {x \in \Q: x \ge 0}$
where $\Q$ denotes the set of rational numbers.
Then the algebraic structure $\struct {\Q_{\ge 0}, +}$ is closed in the sense that:
:$\forall a, b \in \Q_{\ge 0}: a + b \in \Q_{\ge 0}$
where $+$ denotes rational addi... | Let $a$ and $b$ be expressed in canonical form:
:$a = \dfrac {p_1} {q_1}, b = \dfrac {p_2} {q_2}$
where $p_1, p_2 \in \Z$ and $q_1, q_2 \in \Z_{>0}$.
As $\forall a, b \in \Q_{\ge 0}$ it follows that:
:$p_1, p_2 \in \Z_{\ge 0}$
By definition of rational addition:
:$\dfrac {p_1} {q_1} + \dfrac {p_2} {q_2} = \dfrac {p_1 q... | Let $\Q_{\ge 0}$ denote the [[Definition:Set|set]] of [[Definition:Positive Rational Number|positive rational numbers]]:
:$\Q_{\ge 0} := \set {x \in \Q: x \ge 0}$
where $\Q$ denotes the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]].
Then the [[Definition:Algebraic Structure|algebraic struc... | Let $a$ and $b$ be expressed in [[Definition:Canonical Form of Rational Number|canonical form]]:
:$a = \dfrac {p_1} {q_1}, b = \dfrac {p_2} {q_2}$
where $p_1, p_2 \in \Z$ and $q_1, q_2 \in \Z_{>0}$.
As $\forall a, b \in \Q_{\ge 0}$ it follows that:
:$p_1, p_2 \in \Z_{\ge 0}$
By definition of [[Definition:Rational ... | Positive Rational Numbers are Closed under Addition | https://proofwiki.org/wiki/Positive_Rational_Numbers_are_Closed_under_Addition | https://proofwiki.org/wiki/Positive_Rational_Numbers_are_Closed_under_Addition | [
"Rational Addition",
"Rational Numbers"
] | [
"Definition:Set",
"Definition:Positive/Rational Number",
"Definition:Set",
"Definition:Rational Number",
"Definition:Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Addition/Rational Numbers"
] | [
"Definition:Rational Number/Canonical Form",
"Definition:Addition/Rational Numbers",
"Integers form Ordered Integral Domain",
"Category:Rational Addition",
"Category:Rational Numbers"
] |
proofwiki-18707 | Positive Rational Numbers under Addition form Ordered Semigroup | Let $\Q_{\ge 0}$ denote the set of positive rational numbers.
The algebraic structure:
:$\struct {\Q_{\ge 0}, +, \le}$
forms an ordered semigroup. | It is necessary to ascertain that $\struct {\Q_{\ge 0}, +, \le}$ fulfils the ordered semigroup axioms:
{{:Axiom:Ordered Semigroup Axioms}}
From Rational Numbers form Totally Ordered Field, $\struct {\Q, +, \times, \le}$ is a totally ordered field.
Hence $\struct {\Q, +, \le}$ is an ordered group, and so an ordered semi... | Let $\Q_{\ge 0}$ denote the [[Definition:Set|set]] of [[Definition:Positive Rational Number|positive rational numbers]].
The [[Definition:Algebraic Structure|algebraic structure]]:
:$\struct {\Q_{\ge 0}, +, \le}$
forms an [[Definition:Ordered Semigroup|ordered semigroup]]. | It is necessary to ascertain that $\struct {\Q_{\ge 0}, +, \le}$ fulfils the [[Axiom:Ordered Semigroup Axioms|ordered semigroup axioms]]:
{{:Axiom:Ordered Semigroup Axioms}}
From [[Rational Numbers form Totally Ordered Field]], $\struct {\Q, +, \times, \le}$ is a [[Definition:Totally Ordered Field|totally ordered fie... | Positive Rational Numbers under Addition form Ordered Semigroup/Proof 1 | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_form_Ordered_Semigroup | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_form_Ordered_Semigroup/Proof_1 | [
"Examples of Ordered Semigroups",
"Rational Numbers",
"Positive Rational Numbers under Addition form Ordered Semigroup"
] | [
"Definition:Set",
"Definition:Positive/Rational Number",
"Definition:Algebraic Structure",
"Definition:Ordered Semigroup"
] | [
"Axiom:Ordered Semigroup Axioms",
"Rational Numbers form Totally Ordered Field",
"Definition:Totally Ordered Field",
"Definition:Ordered Group",
"Definition:Ordered Semigroup",
"Positive Rational Numbers are Closed under Addition",
"Restriction of Associative Operation is Associative",
"Restriction of... |
proofwiki-18708 | Positive Rational Numbers under Addition form Ordered Semigroup | Let $\Q_{\ge 0}$ denote the set of positive rational numbers.
The algebraic structure:
:$\struct {\Q_{\ge 0}, +, \le}$
forms an ordered semigroup. | From Rational Numbers form Totally Ordered Field, $\struct {\Q, +, \times, \le}$ is a totally ordered field.
Hence $\struct {\Q, +, \le}$ is an ordered group, and so an ordered semigroup.
From Positive Rational Numbers are Closed under Addition we have that $\struct {\Q_{\ge 0}, +}$ is closed.
Hence from Subsemigroup C... | Let $\Q_{\ge 0}$ denote the [[Definition:Set|set]] of [[Definition:Positive Rational Number|positive rational numbers]].
The [[Definition:Algebraic Structure|algebraic structure]]:
:$\struct {\Q_{\ge 0}, +, \le}$
forms an [[Definition:Ordered Semigroup|ordered semigroup]]. | From [[Rational Numbers form Totally Ordered Field]], $\struct {\Q, +, \times, \le}$ is a [[Definition:Totally Ordered Field|totally ordered field]].
Hence $\struct {\Q, +, \le}$ is an [[Definition:Ordered Group|ordered group]], and so an [[Definition:Ordered Semigroup|ordered semigroup]].
From [[Positive Rational N... | Positive Rational Numbers under Addition form Ordered Semigroup/Proof 2 | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_form_Ordered_Semigroup | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_form_Ordered_Semigroup/Proof_2 | [
"Examples of Ordered Semigroups",
"Rational Numbers",
"Positive Rational Numbers under Addition form Ordered Semigroup"
] | [
"Definition:Set",
"Definition:Positive/Rational Number",
"Definition:Algebraic Structure",
"Definition:Ordered Semigroup"
] | [
"Rational Numbers form Totally Ordered Field",
"Definition:Totally Ordered Field",
"Definition:Ordered Group",
"Definition:Ordered Semigroup",
"Positive Rational Numbers are Closed under Addition",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Subsemigroup Closure Test",
"Definition:Su... |
proofwiki-18709 | Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $x$ be a rational number.
Then:
:the canonical expansion of $x$ is eventually periodic. | Let $\ldots d_n \ldots d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_{-m}$ be the canonical expansion of $x$.
It is sufficient to show that the canonical expansion $\ldots d_n \ldots d_2 d_1 d_0$ is eventually periodic.
Let $y$ be the $p$-adic number with canonical expansion:
:$\ldots d_n \ldots d_2 d_1 d_0$
We have:
:$y = x - ... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $x$ be a [[Definition:Rational Number|rational number]].
Then:
:the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x$ is [[Definition:... | Let $\ldots d_n \ldots d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_{-m}$ be the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x$.
It is sufficient to show that the [[Definition:Canonical P-adic Expansion|canonical expansion]] $\ldots d_n \ldots d_2 d_1 d_0$ is [[Definition:Eventually Periodic P-adic Expan... | Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Necessary_Condition | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Necessary_Condition | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:Rational Number",
"Definition:Canonical P-adic Expansion",
"Definition:Eventually Periodic P-adic Expansion"
] | [
"Definition:Canonical P-adic Expansion",
"Definition:Canonical P-adic Expansion",
"Definition:Eventually Periodic P-adic Expansion",
"Definition:P-adic Number",
"Definition:Canonical P-adic Expansion",
"Definition:Rational Number",
"Definition:P-adic Integer",
"Definition:P-adic Integer",
"Definitio... |
proofwiki-18710 | Canonical P-adic Expansion of Rational is Eventually Periodic/Sufficient Condition | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $x \in \Q_p$.
Let the canonical expansion of $x$ be eventually periodic.
Then:
:$x$ be a rational number | Let the canonical expansion of $x$ be eventually periodic.
=== Lemma 6 ===
{{:Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6}}{{qed|lemma}}
To show that $x$ is a rational number it is sufficient to show that $y$ is a rational number.
Let:
:$\dots d_{k - 1} \ldots d_1 d_0 d_{k - 1} \ldots d_1 d_0 ... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $x \in \Q_p$.
Let the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x$ be [[Definition:Eventually Periodic P-adic Expansion|eventually... | Let the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x$ be [[Definition:Eventually Periodic P-adic Expansion|eventually periodic]].
=== [[Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6|Lemma 6]] ===
{{:Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6}}{... | Canonical P-adic Expansion of Rational is Eventually Periodic/Sufficient Condition | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Sufficient_Condition | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Sufficient_Condition | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:Canonical P-adic Expansion",
"Definition:Eventually Periodic P-adic Expansion",
"Definition:Rational Number"
] | [
"Definition:Canonical P-adic Expansion",
"Definition:Eventually Periodic P-adic Expansion",
"Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6",
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Periodic P-adic Expansion",
"Definition:Canonical P-adic Expansion"... |
proofwiki-18711 | Positive Rational Numbers under Addition form Commutative Monoid | Let $\Q_{\ge 0}$ denote the set of positive rational numbers.
The algebraic structure:
:$\struct {\Q_{\ge 0}, +}$
forms a commutative monoid. | From Rational Numbers form Field, $\struct {\Q, +, \times}$ is a field.
Hence $\struct {\Q, +}$ is an abelian group.
From Positive Rational Numbers are Closed under Addition we have that $\struct {\Q_{\ge 0}, +}$ is closed.
Hence from Subsemigroup Closure Test, $\struct {\Q_{\ge 0}, +}$ is a subsemigroup of $\struct {\... | Let $\Q_{\ge 0}$ denote the [[Definition:Set|set]] of [[Definition:Positive Rational Number|positive rational numbers]].
The [[Definition:Algebraic Structure|algebraic structure]]:
:$\struct {\Q_{\ge 0}, +}$
forms a [[Definition:Commutative Monoid|commutative monoid]]. | From [[Rational Numbers form Field]], $\struct {\Q, +, \times}$ is a [[Definition:Field (Abstract Algebra)|field]].
Hence $\struct {\Q, +}$ is an [[Definition:Abelian Group|abelian group]].
From [[Positive Rational Numbers are Closed under Addition]] we have that $\struct {\Q_{\ge 0}, +}$ is [[Definition:Closed Alge... | Positive Rational Numbers under Addition form Commutative Monoid | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_form_Commutative_Monoid | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_form_Commutative_Monoid | [
"Examples of Commutative Monoids",
"Rational Numbers"
] | [
"Definition:Set",
"Definition:Positive/Rational Number",
"Definition:Algebraic Structure",
"Definition:Commutative Monoid"
] | [
"Rational Numbers form Field",
"Definition:Field (Abstract Algebra)",
"Definition:Abelian Group",
"Positive Rational Numbers are Closed under Addition",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Subsemigroup Closure Test",
"Definition:Subsemigroup",
"Restriction of Commutative Oper... |
proofwiki-18712 | Positive Rational Numbers under Addition not Isomorphic to Natural Numbers | The positive rational numbers $\Q_{\ge 0}$ under addition:
:$\struct {\Q_{\ge 0}, +}$
is not isomorphic to the natural numbers under addition:
:$\struct {\N, +}$ | From:
:Positive Rational Numbers under Addition form Commutative Monoid
:Natural Numbers under Addition form Commutative Monoid
both $\struct {\Q_{\ge 0}, +}$ and $\struct {\N, +}$ form commutative monoids.
{{AimForCont}} there exists an semigroup isomorphism $\phi$ from $\struct {\Q_{\ge 0}, +}$ to $\struct {\N, +}$.
... | The [[Definition:Positive Rational Number|positive rational numbers]] $\Q_{\ge 0}$ under [[Definition:Rational Addition|addition]]:
:$\struct {\Q_{\ge 0}, +}$
is not [[Definition:Semigroup Isomorphism|isomorphic]] to the [[Definition:Natural Numbers|natural numbers]] under [[Definition:Natural Number Addition|addition]... | From:
:[[Positive Rational Numbers under Addition form Commutative Monoid]]
:[[Natural Numbers under Addition form Commutative Monoid]]
both $\struct {\Q_{\ge 0}, +}$ and $\struct {\N, +}$ form [[Definition:Commutative Monoid|commutative monoids]].
{{AimForCont}} there exists an [[Definition:Semigroup Isomorphism|se... | Positive Rational Numbers under Addition not Isomorphic to Natural Numbers | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_not_Isomorphic_to_Natural_Numbers | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Addition_not_Isomorphic_to_Natural_Numbers | [
"Natural Number Addition",
"Rational Addition"
] | [
"Definition:Positive/Rational Number",
"Definition:Addition/Rational Numbers",
"Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism",
"Definition:Natural Numbers",
"Definition:Addition/Natural Numbers"
] | [
"Positive Rational Numbers under Addition form Commutative Monoid",
"Natural Numbers under Addition form Commutative Monoid",
"Definition:Commutative Monoid",
"Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism",
"Definition:S... |
proofwiki-18713 | Natural Numbers without 1 fulfil Naturally Ordered Semigroup Axioms 1, 2 and 4 | Let $S \subseteq \N$ be the subset of the natural numbers defined as:
:$S = \N \setminus \set 1 = \set {0, 2, 3, 4, \ldots}$
Then the algebraic structure:
:$\struct {S, +, \le}$
is an ordered semigroup which fulfils the axioms:
:{{NOSAxiom|1}}
:{{NOSAxiom|2}}
:{{NOSAxiom|4}}
but:
:does not fulfil {{NOSAxiom|3}}
:$\stru... | Recall the axioms:
{{:Axiom:Naturally Ordered Semigroup Axioms}} | Let $S \subseteq \N$ be the [[Definition:Subset|subset]] of the [[Definition:Natural Numbers|natural numbers]] defined as:
:$S = \N \setminus \set 1 = \set {0, 2, 3, 4, \ldots}$
Then the [[Definition:Algebraic Structure|algebraic structure]]:
:$\struct {S, +, \le}$
is an [[Definition:Ordered Semigroup|ordered semigro... | Recall the [[Axiom:Naturally Ordered Semigroup Axioms|axioms]]:
{{:Axiom:Naturally Ordered Semigroup Axioms}} | Natural Numbers without 1 fulfil Naturally Ordered Semigroup Axioms 1, 2 and 4 | https://proofwiki.org/wiki/Natural_Numbers_without_1_fulfil_Naturally_Ordered_Semigroup_Axioms_1,_2_and_4 | https://proofwiki.org/wiki/Natural_Numbers_without_1_fulfil_Naturally_Ordered_Semigroup_Axioms_1,_2_and_4 | [
"Naturally Ordered Semigroup"
] | [
"Definition:Subset",
"Definition:Natural Numbers",
"Definition:Algebraic Structure",
"Definition:Ordered Semigroup",
"Axiom:Naturally Ordered Semigroup Axioms",
"Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism"
] | [
"Axiom:Naturally Ordered Semigroup Axioms"
] |
proofwiki-18714 | Singleton fulfils Naturally Ordered Semigroup Axioms 1 to 3 | Let $S$ be a singleton:
:$S = \set s$
for an arbitrary object $s$.
Let $+$ be the operation on $S$ defined as:
:$\forall s \in S: s + s = s$
Let $\le$ be the relation defined on $S$ as:
:$s \le s$
Then the algebraic structure:
:$\struct {S, +, \le}$
is an ordered semigroup which fulfils the axioms:
:{{NOSAxiom|1}}
:{{... | Recall the axioms:
{{:Axiom:Naturally Ordered Semigroup Axioms}} | Let $S$ be a [[Definition:Singleton|singleton]]:
:$S = \set s$
for an arbitrary [[Definition:Object|object]] $s$.
Let $+$ be the [[Definition:Binary Operation|operation]] on $S$ defined as:
:$\forall s \in S: s + s = s$
Let $\le$ be the [[Definition:Relation|relation]] defined on $S$ as:
:$s \le s$
Then the [[Def... | Recall the [[Axiom:Naturally Ordered Semigroup Axioms|axioms]]:
{{:Axiom:Naturally Ordered Semigroup Axioms}} | Singleton fulfils Naturally Ordered Semigroup Axioms 1 to 3 | https://proofwiki.org/wiki/Singleton_fulfils_Naturally_Ordered_Semigroup_Axioms_1_to_3 | https://proofwiki.org/wiki/Singleton_fulfils_Naturally_Ordered_Semigroup_Axioms_1_to_3 | [
"Naturally Ordered Semigroup"
] | [
"Definition:Singleton",
"Definition:Object",
"Definition:Operation/Binary Operation",
"Definition:Relation",
"Definition:Algebraic Structure",
"Definition:Ordered Semigroup",
"Axiom:Naturally Ordered Semigroup Axioms",
"Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism"
] | [
"Axiom:Naturally Ordered Semigroup Axioms"
] |
proofwiki-18715 | Reflexive Relation on Singleton is Well-Ordering | Let $S = \set s$ be a singleton.
Let $\RR$ be a reflexive relation on $S$.
Then $\RR$ is a well-ordering on $S$. | Let $S = \set s$.
By definition of reflexive relation:
:$s \mathrel \RR s$
It trivially holds that:
:$\forall a, b \in S: a \mathrel \RR b \land b \mathrel \RR a \implies a = b$
and so $\RR$ is antisymmetric.
It also trivially holds that:
:$\forall a, b, c \in S: a \mathrel \RR b \land b \mathrel \RR c \implies a \math... | Let $S = \set s$ be a [[Definition:Singleton|singleton]].
Let $\RR$ be a [[Definition:Reflexive Relation|reflexive relation]] on $S$.
Then $\RR$ is a [[Definition:Well-Ordering|well-ordering]] on $S$. | Let $S = \set s$.
By definition of [[Definition:Reflexive Relation|reflexive relation]]:
:$s \mathrel \RR s$
It trivially holds that:
:$\forall a, b \in S: a \mathrel \RR b \land b \mathrel \RR a \implies a = b$
and so $\RR$ is [[Definition:Antisymmetric Relation|antisymmetric]].
It also trivially holds that:
:$\f... | Reflexive Relation on Singleton is Well-Ordering | https://proofwiki.org/wiki/Reflexive_Relation_on_Singleton_is_Well-Ordering | https://proofwiki.org/wiki/Reflexive_Relation_on_Singleton_is_Well-Ordering | [
"Singletons",
"Reflexive Relations",
"Well-Orderings"
] | [
"Definition:Singleton",
"Definition:Reflexive Relation",
"Definition:Well-Ordering"
] | [
"Definition:Reflexive Relation",
"Definition:Antisymmetric Relation",
"Definition:Ordering",
"Definition:Total Ordering",
"Finite Totally Ordered Set is Well-Ordered",
"Definition:Well-Ordered Set",
"Category:Singletons",
"Category:Reflexive Relations",
"Category:Well-Orderings"
] |
proofwiki-18716 | Power Function with Cancellable Element Preserves Strict Ordering in Ordered Semigroup | Let $\struct {S, \circ, \preceq}$ be an ordered semigroup.
Let $x, y \in S$ be such that:
:$(1): \quad x \prec y$
:$(2): \quad$ either $X$ or $y$ (or both) is cancellable for $\circ$.
Let $n \in \N_{>0}$ be a strictly positive integer.
Then:
:$x^n \prec y^n$
where $x^n$ is the $n$th power of $x$. | {{WLOG}}, suppose $x$ is cancellable for $\circ$.
The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$x \prec y \implies x^n \prec y^n$
$\map P 1$ is the case:
:$x \prec y \implies x \prec y$
which is trivially true.
Thus $\map P 1$ is seen to hold. | Let $\struct {S, \circ, \preceq}$ be an [[Definition:Ordered Semigroup|ordered semigroup]].
Let $x, y \in S$ be such that:
:$(1): \quad x \prec y$
:$(2): \quad$ either $X$ or $y$ (or both) is [[Definition:Cancellable Element|cancellable]] for $\circ$.
Let $n \in \N_{>0}$ be a [[Definition:Strictly Positive Integer|st... | {{WLOG}}, suppose $x$ is [[Definition:Cancellable Element|cancellable]] for $\circ$.
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$x \prec y \implies x^n \prec y^n$
$\map P 1$ is the case:
:$x \prec ... | Power Function with Cancellable Element Preserves Strict Ordering in Ordered Semigroup | https://proofwiki.org/wiki/Power_Function_with_Cancellable_Element_Preserves_Strict_Ordering_in_Ordered_Semigroup | https://proofwiki.org/wiki/Power_Function_with_Cancellable_Element_Preserves_Strict_Ordering_in_Ordered_Semigroup | [
"Ordered Semigroups"
] | [
"Definition:Ordered Semigroup",
"Definition:Cancellable Element",
"Definition:Strictly Positive/Integer",
"Definition:Power of Element/Semigroup"
] | [
"Definition:Cancellable Element",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-18717 | Trivial Group is Abelian | The trivial group is an abelian group. | From Trivial Group is Cyclic Group, it is shown that the algebraic structure $\struct {\set e, \circ}$ such that $e \circ e = e$ is a cyclic group.
The result follows from Cyclic Group is Abelian.
{{qed}}
Category:Trivial Group
Category:Abelian Groups
jeq7bjs861u6z3t1pxsjr5053col511 | The [[Definition:Trivial Group|trivial group]] is an [[Definition:Abelian Group|abelian group]]. | From [[Trivial Group is Cyclic Group]], it is shown that the [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\struct {\set e, \circ}$ such that $e \circ e = e$ is a [[Definition:Cyclic Group|cyclic group]].
The result follows from [[Cyclic Group is Abelian]].
{{qed}}
[[Category:Trivial Grou... | Trivial Group is Abelian | https://proofwiki.org/wiki/Trivial_Group_is_Abelian | https://proofwiki.org/wiki/Trivial_Group_is_Abelian | [
"Trivial Group",
"Abelian Groups"
] | [
"Definition:Trivial Group",
"Definition:Abelian Group"
] | [
"Trivial Group is Cyclic Group",
"Definition:Algebraic Structure/One Operation",
"Definition:Cyclic Group",
"Cyclic Group is Abelian",
"Category:Trivial Group",
"Category:Abelian Groups"
] |
proofwiki-18718 | Equivalence of Definitions of Dipper Semigroup | Let $m \in \N$ be a natural number.
Let $n \in \N_{>0}$ be a non-zero natural number.
{{TFAE|def = Dipper Semigroup}}
=== Definition 1 ===
{{:Definition:Dipper Semigroup/Definition 1}}
=== Definition 2 ===
{{:Definition:Dipper Semigroup/Definition 2}} | We have established in Dipper Semigroup is Commutative Semigroup that $\struct {\N_{< \paren {m + n} }, \oplus_{m, n} }$ is a (commutative) semigroup.
Let $\phi_{m, n}$ be the canonical surjection from $\N$ onto $\map D {m, n}$.
It will be established that the restriction of $\phi_{m, n}$ to $\N_{< \paren {m + n} }$ is... | Let $m \in \N$ be a [[Definition:Natural Number|natural number]].
Let $n \in \N_{>0}$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural number]].
{{TFAE|def = Dipper Semigroup}}
=== [[Definition:Dipper Semigroup/Definition 1|Definition 1]] ===
{{:Definition:Dipper Semigroup/Definition 1}... | We have established in [[Dipper Semigroup is Commutative Semigroup]] that $\struct {\N_{< \paren {m + n} }, \oplus_{m, n} }$ is a [[Definition:Commutative Semigroup|(commutative) semigroup]].
Let $\phi_{m, n}$ be the [[Definition:Canonical Surjection|canonical surjection]] from $\N$ onto $\map D {m, n}$.
It will be ... | Equivalence of Definitions of Dipper Semigroup | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Dipper_Semigroup | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Dipper_Semigroup | [
"Equivalence of Definitions of Dipper Semigroup",
"Dipper Semigroups",
"Semigroup Isomorphisms"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Dipper Semigroup/Definition 1",
"Definition:Dipper Semigroup/Definition 2"
] | [
"Dipper Semigroup is Commutative Semigroup",
"Definition:Commutative Semigroup",
"Definition:Quotient Mapping",
"Definition:Restriction/Operation",
"Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism",
"Definition:Semigroup",
"Dipper Relation is Equivalence Relation",
"Definition:Equivale... |
proofwiki-18719 | Conditional is not Associative | :$p \implies \paren {q \implies r} \not \vdash \paren {p \implies q} \implies r$ | <onlyinclude>
We apply the Method of Truth Tables:
:<nowiki>$\begin{array}{|ccccc||ccccc|} \hline
p & \implies & (q & \implies & r) & (p & \implies & q) & \implies & r \\
\hline
\F & \T & \F & \T & \F & \F & \T & \F & \F & \F \\
\F & \T & \F & \T & \T & \F & \T & \F & \T & \T \\
\F & \T & \T & \F & \F & \F & \T & \T & ... | :$p \implies \paren {q \implies r} \not \vdash \paren {p \implies q} \implies r$ | <onlyinclude>
We apply the [[Method of Truth Tables]]:
:<nowiki>$\begin{array}{|ccccc||ccccc|} \hline
p & \implies & (q & \implies & r) & (p & \implies & q) & \implies & r \\
\hline
\F & \T & \F & \T & \F & \F & \T & \F & \F & \F \\
\F & \T & \F & \T & \T & \F & \T & \F & \T & \T \\
\F & \T & \T & \F & \F & \F & \T & ... | Conditional is not Associative | https://proofwiki.org/wiki/Conditional_is_not_Associative | https://proofwiki.org/wiki/Conditional_is_not_Associative | [
"Conditional"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-18720 | Dipper Operation is Commutative | Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
The '''dipper operation''' on $\N_{< \paren {m \mathop + n} }$ is commutative. | Recall the definition of the '''dipper operation''' on $\N_{< \paren {m \mathop + n} }$ defined as:
:$\forall a, b \in \Z_{>0}: a +_{m, n} b = \begin{cases} a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$
where $k$ is the largest integer satisfying:
:$m + k n \le a + b$
Let $a + b < m$.
Then:
{{begin-eq... | Let $m, n \in \Z$ be [[Definition:Integer|integers]] such that $m \ge 0, n > 0$.
Let $\N_{< \paren {m \mathop + n} }$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment]] of the [[Definition:Natural Numbers|natural numbers]]:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m ... | Recall the definition of the '''[[Definition:Dipper Operation|dipper operation]]''' on $\N_{< \paren {m \mathop + n} }$ defined as:
:$\forall a, b \in \Z_{>0}: a +_{m, n} b = \begin{cases} a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$
where $k$ is the largest [[Definition:Integer|integer]] satisfying... | Dipper Operation is Commutative | https://proofwiki.org/wiki/Dipper_Operation_is_Commutative | https://proofwiki.org/wiki/Dipper_Operation_is_Commutative | [
"Dipper Operations",
"Examples of Commutative Operations"
] | [
"Definition:Integer",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Natural Numbers",
"Definition:Dipper Operation",
"Definition:Commutative/Operation"
] | [
"Definition:Dipper Operation",
"Definition:Integer",
"Natural Number Addition is Commutative",
"Natural Number Addition is Commutative",
"Definition:Commutative/Operation"
] |
proofwiki-18721 | Dipper Operation is Associative | Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
The '''dipper operation''' on $\N_{< \paren {m \mathop + n} }$ is associative. | Recall the definition of the '''dipper operation''' on $\N_{< \paren {m \mathop + n} }$ defined as:
:$\forall a, b \in \Z_{>0}: a +_{m, n} b = \begin{cases} a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$
where $k$ is the largest integer satisfying:
:$m + k n \le a + b$
Let $a, b, c \in \N_{< \paren {m ... | Let $m, n \in \Z$ be [[Definition:Integer|integers]] such that $m \ge 0, n > 0$.
Let $\N_{< \paren {m \mathop + n} }$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment]] of the [[Definition:Natural Numbers|natural numbers]]:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m ... | Recall the definition of the '''[[Definition:Dipper Operation|dipper operation]]''' on $\N_{< \paren {m \mathop + n} }$ defined as:
:$\forall a, b \in \Z_{>0}: a +_{m, n} b = \begin{cases} a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$
where $k$ is the largest [[Definition:Integer|integer]] satisfying... | Dipper Operation is Associative | https://proofwiki.org/wiki/Dipper_Operation_is_Associative | https://proofwiki.org/wiki/Dipper_Operation_is_Associative | [
"Dipper Operations",
"Examples of Associative Operations"
] | [
"Definition:Integer",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Natural Numbers",
"Definition:Dipper Operation",
"Definition:Associative Operation"
] | [
"Definition:Dipper Operation",
"Definition:Integer",
"Dipper Operation is Commutative",
"Natural Number Addition is Associative",
"Natural Number Addition is Associative",
"Natural Number Addition is Associative",
"Natural Number Addition is Associative",
"Definition:Associative Operation"
] |
proofwiki-18722 | Characteristic of Cayley Table of Left Operation | Let $S$ be a finite set.
Let $\leftarrow$ denote the left operation on $S$.
The Cayley table of the algebraic structure $\struct {S, \leftarrow}$ is characterised by the fact that each row contains just one distinct element. | A row of a Cayley table headed by $x$ contains all those elements of the form $x \leftarrow y$.
By definition of the left operation:
:$x \leftarrow y = x$
Hence the result.
{{qed}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\leftarrow$ denote the [[Definition:Left Operation|left operation]] on $S$.
The [[Definition:Cayley Table|Cayley table]] of the [[Definition:Algebraic Structure|algebraic structure]] $\struct {S, \leftarrow}$ is characterised by the fact that each [[Definition:... | A [[Definition:Row of Array|row]] of a [[Definition:Cayley Table|Cayley table]] headed by $x$ contains all those [[Definition:Element|elements]] of the form $x \leftarrow y$.
By definition of the [[Definition:Left Operation|left operation]]:
:$x \leftarrow y = x$
Hence the result.
{{qed}} | Characteristic of Cayley Table of Left Operation | https://proofwiki.org/wiki/Characteristic_of_Cayley_Table_of_Left_Operation | https://proofwiki.org/wiki/Characteristic_of_Cayley_Table_of_Left_Operation | [
"Cayley Tables",
"Left Operation"
] | [
"Definition:Finite Set",
"Definition:Left Operation",
"Definition:Cayley Table",
"Definition:Algebraic Structure",
"Definition:Array/Row",
"Definition:Distinct",
"Definition:Element"
] | [
"Definition:Array/Row",
"Definition:Cayley Table",
"Definition:Element",
"Definition:Left Operation"
] |
proofwiki-18723 | Characteristic of Cayley Table of Right Operation | Let $S$ be a finite set.
Let $\rightarrow$ denote the right operation on $S$.
The Cayley table of the algebraic structure $\struct {S, \rightarrow}$ is characterised by the fact that each column contains just one distinct element. | A column of a Cayley table headed by $y$ contains all those elements of the form $x \rightarrow y$.
By definition of the right operation:
:$x \rightarrow y = y$
Hence the result.
{{qed}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rightarrow$ denote the [[Definition:Right Operation|right operation]] on $S$.
The [[Definition:Cayley Table|Cayley table]] of the [[Definition:Algebraic Structure|algebraic structure]] $\struct {S, \rightarrow}$ is characterised by the fact that each [[Definit... | A [[Definition:Column of Array|column]] of a [[Definition:Cayley Table|Cayley table]] headed by $y$ contains all those [[Definition:Element|elements]] of the form $x \rightarrow y$.
By definition of the [[Definition:Right Operation|right operation]]:
:$x \rightarrow y = y$
Hence the result.
{{qed}} | Characteristic of Cayley Table of Right Operation | https://proofwiki.org/wiki/Characteristic_of_Cayley_Table_of_Right_Operation | https://proofwiki.org/wiki/Characteristic_of_Cayley_Table_of_Right_Operation | [
"Cayley Tables",
"Right Operation"
] | [
"Definition:Finite Set",
"Definition:Right Operation",
"Definition:Cayley Table",
"Definition:Algebraic Structure",
"Definition:Array/Column",
"Definition:Distinct",
"Definition:Element"
] | [
"Definition:Array/Column",
"Definition:Cayley Table",
"Definition:Element",
"Definition:Right Operation"
] |
proofwiki-18724 | Left Operation is not Commutative | Let $S$ be a finite set.
Let $\leftarrow$ denote the left operation on $S$.
Then $\leftarrow$ is not commutative on $S$ unless $S$ is a singleton. | Let $S$ be a singleton, $S = \set s$, say.
Then:
:$s \leftarrow s = s$
and so $\leftarrow$ is trivially commutative on $S$
Otherwise, $\exists s, t \in S$ such that $s \ne t$.
Then:
:$s \leftarrow t = s$
but:
:$t \leftarrow s = t$
and the result follows by definition of commutative operation.
{{qed}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\leftarrow$ denote the [[Definition:Left Operation|left operation]] on $S$.
Then $\leftarrow$ is not [[Definition:Commutative Operation|commutative]] on $S$ unless $S$ is a [[Definition:Singleton|singleton]]. | Let $S$ be a [[Definition:Singleton|singleton]], $S = \set s$, say.
Then:
:$s \leftarrow s = s$
and so $\leftarrow$ is trivially [[Definition:Commutative Operation|commutative]] on $S$
Otherwise, $\exists s, t \in S$ such that $s \ne t$.
Then:
:$s \leftarrow t = s$
but:
:$t \leftarrow s = t$
and the result follows... | Left Operation is not Commutative | https://proofwiki.org/wiki/Left_Operation_is_not_Commutative | https://proofwiki.org/wiki/Left_Operation_is_not_Commutative | [
"Left Operation",
"Examples of Commutative Operations"
] | [
"Definition:Finite Set",
"Definition:Left Operation",
"Definition:Commutative/Operation",
"Definition:Singleton"
] | [
"Definition:Singleton",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation"
] |
proofwiki-18725 | Right Operation is not Commutative | Let $S$ be a finite set.
Let $\rightarrow$ denote the right operation on $S$.
Then $\rightarrow$ is not commutative on $S$ unless $S$ is a singleton. | Let $S$ be a singleton, $S = \set s$, say.
Then:
:$s \rightarrow s = s$
and so $\rightarrow$ is trivially commutative on $S$
Otherwise, $\exists s, t \in S$ such that $s \ne t$.
Then:
:$s \rightarrow t = t$
but:
:$t \rightarrow s = s$
and the result follows by definition of commutative operation.
{{qed}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rightarrow$ denote the [[Definition:Right Operation|right operation]] on $S$.
Then $\rightarrow$ is not [[Definition:Commutative Operation|commutative]] on $S$ unless $S$ is a [[Definition:Singleton|singleton]]. | Let $S$ be a [[Definition:Singleton|singleton]], $S = \set s$, say.
Then:
:$s \rightarrow s = s$
and so $\rightarrow$ is trivially [[Definition:Commutative Operation|commutative]] on $S$
Otherwise, $\exists s, t \in S$ such that $s \ne t$.
Then:
:$s \rightarrow t = t$
but:
:$t \rightarrow s = s$
and the result fol... | Right Operation is not Commutative | https://proofwiki.org/wiki/Right_Operation_is_not_Commutative | https://proofwiki.org/wiki/Right_Operation_is_not_Commutative | [
"Right Operation",
"Examples of Commutative Operations"
] | [
"Definition:Finite Set",
"Definition:Right Operation",
"Definition:Commutative/Operation",
"Definition:Singleton"
] | [
"Definition:Singleton",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation"
] |
proofwiki-18726 | Induced Metric on Surface of Revolution | Let $\struct {\R^3, d}$ be the Euclidean space.
Let $S_C \subseteq \R^3$ be the surface of revolution.
Let the smooth local parametrization of $C$ be:
:$\map \gamma t = \tuple {\map x t, \map y t}$
Then the induced metric on $S_C$ is:
:$g = \paren {\map {x'} t^2 + \map {y'} t^2} d t^2 + \map y t^2 d \theta^2$ | By Smooth Local Parametrization of Surface of Revolution, the smooth local parametrization of $S_C$ can be written as:
:$\map X {t, \theta} = \tuple {\map y t \cos \theta, \map y t \sin \theta, \map x t}$
By definition, the induced metric on $S_C$ is:
{{begin-eqn}}
{{eqn | l = g
| r = X^* \tilde g
}}
{{eqn | r = ... | Let $\struct {\R^3, d}$ be the [[Definition:Euclidean Space|Euclidean space]].
Let $S_C \subseteq \R^3$ be the [[Definition:Surface of Revolution|surface of revolution]].
Let the [[Definition:Smooth Local Parametrization|smooth local parametrization]] of $C$ be:
:$\map \gamma t = \tuple {\map x t, \map y t}$
Then ... | By [[Smooth Local Parametrization of Surface of Revolution]], the [[Definition:Smooth Local Parametrization|smooth local parametrization]] of $S_C$ can be written as:
:$\map X {t, \theta} = \tuple {\map y t \cos \theta, \map y t \sin \theta, \map x t}$
By definition, the [[Definition:Induced Metric on Submanifold|ind... | Induced Metric on Surface of Revolution | https://proofwiki.org/wiki/Induced_Metric_on_Surface_of_Revolution | https://proofwiki.org/wiki/Induced_Metric_on_Surface_of_Revolution | [
"Surfaces of Revolution",
"Solid Geometry",
"Induced Metrics"
] | [
"Definition:Euclidean Space",
"Definition:Surface of Revolution",
"Definition:Smooth Local Parametrization",
"Definition:Induced Metric on Submanifold"
] | [
"Smooth Local Parametrization of Surface of Revolution",
"Definition:Smooth Local Parametrization",
"Definition:Induced Metric on Submanifold"
] |
proofwiki-18727 | Primitive of Reciprocal of 1 plus Cosine of x | :$\ds \int \frac {\d x} {1 + \cos x} = \tan \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \tan \frac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 + \cos x}
| r = \int \frac {\dfrac {2 \rd u} {1 + u^2} } {1 + \dfrac {1 - u^2} {1 + u^2} }
| c = Weierstrass Substitution
}}
{{eqn | r = \int \frac {2 \rd u} {1 + u^2 + \paren {1 - u^... | :$\ds \int \frac {\d x} {1 + \cos x} = \tan \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \tan \frac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 + \cos x}
| r = \int \frac {\dfrac {2 \rd u} {1 + u^2} } {1 + \dfrac {1 - u^2} {1 + u^2} }
| c = [[Weierstrass Substitution]]
}}
{{eqn | r = \int \frac {2 \rd u} {1 + u^2 + \paren {1 ... | Primitive of Reciprocal of 1 plus Cosine of x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_plus_Cosine_of_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_plus_Cosine_of_x | [
"Primitive of Reciprocal of 1 plus Cosine of x",
"Primitives involving Cosine Function"
] | [] | [
"Weierstrass Substitution",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Constant"
] |
proofwiki-18728 | Primitive of Reciprocal of 1 minus Cosine of x | :$\ds \int \frac {\d x} {1 - \cos x} = -\cot \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \tan \frac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 - \cos x}
| r = \int \frac {\dfrac {2 \rd u} {1 + u^2} } {1 - \dfrac {1 - u^2} {1 + u^2} }
| c = Weierstrass Substitution
}}
{{eqn | r = \int \frac {2 \rd u} {1 + u^2 - \paren {1 - u^... | :$\ds \int \frac {\d x} {1 - \cos x} = -\cot \frac x 2 + C$ | {{begin-eqn}}
{{eqn | l = u
| r = \tan \frac x 2
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 - \cos x}
| r = \int \frac {\dfrac {2 \rd u} {1 + u^2} } {1 - \dfrac {1 - u^2} {1 + u^2} }
| c = [[Weierstrass Substitution]]
}}
{{eqn | r = \int \frac {2 \rd u} {1 + u^2 - \paren {1 ... | Primitive of Reciprocal of 1 minus Cosine of x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Cosine_of_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Cosine_of_x | [
"Primitive of Reciprocal of 1 minus Cosine of x",
"Primitives involving Cosine Function"
] | [] | [
"Weierstrass Substitution",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Power",
"Cotangent is Reciprocal of Tangent"
] |
proofwiki-18729 | Primitive of Reciprocal of Sine of x by Cosine of x | :$\ds \int \frac {\d x} {\sin x \cos x} = \ln \size {\tan x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x \cos x}
| r = \int \frac {\sec x \rd x} {\sin x}
| c = Secant is Reciprocal of Cosine
}}
{{eqn | r = \int \frac {\sec^2 x \rd x} {\sin x \sec x}
| c = multiplying top and bottom by $\sec x$
}}
{{eqn | r = \int \frac {\sec^2 x \rd x} {\frac {\sin x} {... | :$\ds \int \frac {\d x} {\sin x \cos x} = \ln \size {\tan x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x \cos x}
| r = \int \frac {\sec x \rd x} {\sin x}
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | r = \int \frac {\sec^2 x \rd x} {\sin x \sec x}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\sec x$
}}
{... | Primitive of Reciprocal of Sine of x by Cosine of x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_x_by_Cosine_of_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_x_by_Cosine_of_x | [
"Primitive of Reciprocal of Sine of x by Cosine of x",
"Primitives involving Sine Function and Cosine Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Secant is Reciprocal of Cosine",
"Tangent is Sine divided by Cosine",
"Primitive of Square of Secant of a x over Tangent of a x"
] |
proofwiki-18730 | Primitive of Arcsine Function | :$\ds \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin x
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \sin u
| r = x
| c = {{Defof|Real Arcsine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - x^2}
| c = Sum of Squares of Sine and Cosine
}}
{{end-eqn}... | :$\ds \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsin x
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \sin u
| r = x
| c = {{Defof|Real Arcsine}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \cos u
| r = \sqrt {1 - x^2}
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{end-... | Primitive of Arcsine Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Arcsine_Function | https://proofwiki.org/wiki/Primitive_of_Arcsine_Function/Proof_1 | [
"Primitive of Arcsine Function",
"Primitives involving Inverse Sine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Primitive of Function of Arcsine",
"Primitive of x by Cosine of a x"
] |
proofwiki-18731 | Primitive of Arctangent Function | :$\ds \int \arctan x \rd x = x \arctan x - \frac {\map \ln {x^2 + 1} } 2 + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan x
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \tan u
| r = x
| c = {{Defof|Real Arctangent}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sec u
| r = \sqrt {1 + x^2}
| c = Difference of Squares of Secant and Tangent
... | :$\ds \int \arctan x \rd x = x \arctan x - \frac {\map \ln {x^2 + 1} } 2 + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan x
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \tan u
| r = x
| c = {{Defof|Real Arctangent}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \sec u
| r = \sqrt {1 + x^2}
| c = [[Difference of Squares of Secant and Tangen... | Primitive of Arctangent Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Arctangent_Function | https://proofwiki.org/wiki/Primitive_of_Arctangent_Function/Proof_1 | [
"Primitive of Arctangent Function",
"Primitives involving Inverse Tangent Function"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Function of Arctangent",
"Primitive of x by Square of Secant of a x",
"Logarithm of Reciprocal",
"Secant is Reciprocal of Cosine",
"Logarithm of Power",
"Definition:Positive/Real Number"
] |
proofwiki-18732 | Primitive of Arccotangent Function | :$\ds \int \arccot x \rd x = x \arccot x + \frac {\map \ln {x^2 + 1} } 2 + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot x
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cot u
| r = x
| c = {{Defof|Arccotangent}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \csc u
| r = \sqrt {1 + x^2}
| c = Difference of Squares of Cosecant and Cotangent... | :$\ds \int \arccot x \rd x = x \arccot x + \frac {\map \ln {x^2 + 1} } 2 + C$ | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot x
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = \cot u
| r = x
| c = {{Defof|Arccotangent}}
}}
{{eqn | n = 2
| ll= \leadsto
| l = \csc u
| r = \sqrt {1 + x^2}
| c = [[Difference of Squares of Cosecant and Cotange... | Primitive of Arccotangent Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Arccotangent_Function | https://proofwiki.org/wiki/Primitive_of_Arccotangent_Function/Proof_1 | [
"Primitive of Arccotangent Function",
"Primitives involving Inverse Cotangent Function"
] | [] | [
"Sum of Squares of Sine and Cosine/Corollary 2",
"Primitive of Function of Arccotangent",
"Primitive of x by Square of Cosecant of a x",
"Logarithm of Reciprocal",
"Cosecant is Reciprocal of Sine",
"Logarithm of Power",
"Definition:Positive/Real Number"
] |
proofwiki-18733 | Primitive of Reciprocal of x by Root of x squared minus a squared/Arcsine Form | :$\ds \int \frac {\d x} {x \sqrt {x^2 - a^2} } = -\frac 1 a \arcsin \size {\frac a x} + C$
for $0 < a < \size x$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
:$x > a$
or:
:$x < -a$
where it is assumed that $a > 0$.
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 - a^2} }
| r = \frac 1 a \arccos \size {\frac a x} + C
| c = Primitive of $\dfrac 1 {x \sqrt {x^2 - a^... | :$\ds \int \frac {\d x} {x \sqrt {x^2 - a^2} } = -\frac 1 a \arcsin \size {\frac a x} + C$
for $0 < a < \size x$. | We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
:$x > a$
or:
:$x < -a$
where it is assumed that $a > 0$.
Hence:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 - a^2} }
| r = \frac 1 a \arccos \size {\frac a x} + C
| c = [[Primitive of Reciprocal of x by Root... | Primitive of Reciprocal of x by Root of x squared minus a squared/Arcsine Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_minus_a_squared/Arcsine_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_minus_a_squared/Arcsine_Form | [
"Arcsine Function",
"Primitive of Reciprocal of x by Root of x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of x by Root of x squared minus a squared/Arccosine Form",
"Sum of Arcsine and Arccosine",
"Definition:Arbitrary Constant"
] |
proofwiki-18734 | Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Sine Form | :$\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \sinh^{-1} {\frac a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 + a^2} }
| r = -\frac 1 a \csch^{-1} {\frac x a} + C
| c = Primitive of $\dfrac 1 {x \sqrt {x^2 + a^2} }$: Inverse Hyperbolic Cosecant Form
}}
{{eqn | r = -\frac 1 a \sinh^{-1} {\frac a x} + C
| c = Real Area Hyperbolic Sine of Reciprocal equal... | :$\ds \int \frac {\d x} {x \sqrt {x^2 + a^2} } = -\frac 1 a \sinh^{-1} {\frac a x} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \sqrt {x^2 + a^2} }
| r = -\frac 1 a \csch^{-1} {\frac x a} + C
| c = [[Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form|Primitive of $\dfrac 1 {x \sqrt {x^2 + a^2} }$: Inverse Hyperbolic Cosecant Form]]
}}
{{eqn... | Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Sine Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Sine_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_x_squared_plus_a_squared/Inverse_Hyperbolic_Sine_Form | [
"Primitive of Reciprocal of x by Root of x squared plus a squared",
"Expressions whose Primitives are Inverse Hyperbolic Functions",
"Inverse Hyperbolic Sine"
] | [] | [
"Primitive of Reciprocal of x by Root of x squared plus a squared/Inverse Hyperbolic Cosecant Form",
"Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant"
] |
proofwiki-18735 | Primitive of Root of a squared minus x squared/Arccosine Form | :$\ds \int \sqrt {a^2 - x^2} \rd x = \frac {x \sqrt {a^2 - x^2} } 2 - \frac {a^2} 2 \arccos \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \sqrt {a^2 - x^2} \rd x
| r = \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C
}}
{{eqn | r = \frac {x \sqrt {a^2 - x^2} } 2 + \paren {\dfrac \pi 2 - \frac {a^2} 2 \arcsin \frac x a} + C
| c = Sum of Arcsine and Arccosine
}}
{{eqn | r = \frac {x \sqrt ... | :$\ds \int \sqrt {a^2 - x^2} \rd x = \frac {x \sqrt {a^2 - x^2} } 2 - \frac {a^2} 2 \arccos \frac x a + C$ | Let:
{{begin-eqn}}
{{eqn | l = \int \sqrt {a^2 - x^2} \rd x
| r = \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C
}}
{{eqn | r = \frac {x \sqrt {a^2 - x^2} } 2 + \paren {\dfrac \pi 2 - \frac {a^2} 2 \arcsin \frac x a} + C
| c = [[Sum of Arcsine and Arccosine]]
}}
{{eqn | r = \frac {x \s... | Primitive of Root of a squared minus x squared/Arccosine Form | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared/Arccosine_Form | https://proofwiki.org/wiki/Primitive_of_Root_of_a_squared_minus_x_squared/Arccosine_Form | [
"Primitive of Root of a squared minus x squared"
] | [] | [
"Sum of Arcsine and Arccosine",
"Definition:Arbitrary Constant"
] |
proofwiki-18736 | Isomorphisms between Additive Group of Integers Modulo 4 and Reduced Residue System Modulo 5 under Multiplication | Let $\struct {\Z_4, +_4}$ denote the additive group of integers modulo $4$.
Let $\struct {\Z'_5, \times_5}$ denote the multiplicative group of reduced residues modulo $5$.
There are $2$ (group) isomorphisms from $\struct {\Z_4, +_4}$ onto $\struct {\Z'_5, \times_5}$. | Let us recall the Cayley table of $\struct {\Z_4, +_4}$:
{{:Modulo Addition/Cayley Table/Modulo 4}}
and the Cayley Table of $\struct {\Z'_5, \times_5}$:
{{:Multiplicative Group of Reduced Residues Modulo 5/Cayley Table}}
Each of these is the cyclic group of order $4$.
Each has $2$ generators, each of $1$ element.
Hence... | Let $\struct {\Z_4, +_4}$ denote the [[Definition:Additive Group of Integers Modulo m|additive group of integers modulo $4$]].
Let $\struct {\Z'_5, \times_5}$ denote the [[Multiplicative Group of Reduced Residues Modulo 5|multiplicative group of reduced residues modulo $5$]].
There are $2$ [[Definition:Group Isomorp... | Let us recall the [[Modulo Addition/Cayley Table/Modulo 4|Cayley table of $\struct {\Z_4, +_4}$]]:
{{:Modulo Addition/Cayley Table/Modulo 4}}
and the [[Multiplicative Group of Reduced Residues Modulo 5/Cayley Table|Cayley Table of $\struct {\Z'_5, \times_5}$]]:
{{:Multiplicative Group of Reduced Residues Modulo 5/Cayl... | Isomorphisms between Additive Group of Integers Modulo 4 and Reduced Residue System Modulo 5 under Multiplication | https://proofwiki.org/wiki/Isomorphisms_between_Additive_Group_of_Integers_Modulo_4_and_Reduced_Residue_System_Modulo_5_under_Multiplication | https://proofwiki.org/wiki/Isomorphisms_between_Additive_Group_of_Integers_Modulo_4_and_Reduced_Residue_System_Modulo_5_under_Multiplication | [
"Examples of Group Isomorphisms",
"Multiplicative Group of Reduced Residues Modulo 5",
"Cyclic Group of Order 4"
] | [
"Definition:Additive Group of Integers Modulo m",
"Multiplicative Group of Reduced Residues/Examples/Modulo 5",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Modulo Addition/Cayley Table/Modulo 4",
"Multiplicative Group of Reduced Residues Modulo 5/Cayley Table",
"Definition:Cyclic Group",
"Definition:Generator of Group",
"Definition:Element",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Mapping"
] |
proofwiki-18737 | Max and Min Operations on Real Numbers are Isomorphic | Let $\R$ denote the set of real numbers.
Let $\vee$ and $\wedge$ denote the max operation and min operation respectively.
Let $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ denote the algebraic structures formed from the above.
Then $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ are isomorphic. | First we note that from:
:Min Operation on Toset forms Semigroup
and:
:Max Operation on Toset forms Semigroup
both $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ are semigroups.
Let $\phi: \R \to \R$ defined as:
:$\forall x \in \R: \map \phi x = -x$
We have that:
:$-x = -y \iff x = y$
which demonstrates that $\phi$ is... | Let $\R$ denote the [[Definition:Real Number|set of real numbers]].
Let $\vee$ and $\wedge$ denote the [[Definition:Max Operation|max operation]] and [[Definition:Min Operation|min operation]] respectively.
Let $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ denote the [[Definition:Algebraic Structure|algebraic stru... | First we note that from:
:[[Min Operation on Toset forms Semigroup]]
and:
:[[Max Operation on Toset forms Semigroup]]
both $\struct {\R, \vee}$ and $\struct {\R, \wedge}$ are [[Definition:Semigroup|semigroups]].
Let $\phi: \R \to \R$ defined as:
:$\forall x \in \R: \map \phi x = -x$
We have that:
:$-x = -y \iff x ... | Max and Min Operations on Real Numbers are Isomorphic | https://proofwiki.org/wiki/Max_and_Min_Operations_on_Real_Numbers_are_Isomorphic | https://proofwiki.org/wiki/Max_and_Min_Operations_on_Real_Numbers_are_Isomorphic | [
"Max and Min Operations",
"Examples of Semigroup Isomorphisms"
] | [
"Definition:Real Number",
"Definition:Max Operation",
"Definition:Min Operation",
"Definition:Algebraic Structure",
"Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism"
] | [
"Min Operation on Toset forms Semigroup",
"Max Operation on Toset forms Semigroup",
"Definition:Semigroup",
"Definition:Bijection",
"Definition:Semigroup Homomorphism",
"Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism"
] |
proofwiki-18738 | Mapping on Quadratic Integers over 3 to Conjugate is Automorphism | Let $\Z \sqbrk {\sqrt 3}$ denote the set of quadratic integers over $3$:
:$\Z \sqbrk {\sqrt 3} := \set {a + b \sqrt 3: a, b \in \Z}$
that is, all numbers of the form $a + b \sqrt 3$ where $a$ and $b$ are integers.
Then the mapping $\phi: \Z \sqbrk {\sqrt 3} \to \Z \sqbrk {\sqrt 3}$ defined as:
:$\forall x = a + b \sqrt... | We have Quadratic Integers over 2 form Subdomain of Reals.
First we note that:
:$\forall x \in \Z \sqbrk {\sqrt 3}: \map \phi x \in \Z \sqbrk {\sqrt 3}$ | Let $\Z \sqbrk {\sqrt 3}$ denote the [[Definition:Set|set]] of [[Definition:Quadratic Integer|quadratic integers]] over $3$:
:$\Z \sqbrk {\sqrt 3} := \set {a + b \sqrt 3: a, b \in \Z}$
that is, all numbers of the form $a + b \sqrt 3$ where $a$ and $b$ are [[Definition:Integer|integers]].
Then the [[Definition:Mapping... | We have [[Quadratic Integers over 2 form Subdomain of Reals]].
First we note that:
:$\forall x \in \Z \sqbrk {\sqrt 3}: \map \phi x \in \Z \sqbrk {\sqrt 3}$ | Mapping on Quadratic Integers over 3 to Conjugate is Automorphism | https://proofwiki.org/wiki/Mapping_on_Quadratic_Integers_over_3_to_Conjugate_is_Automorphism | https://proofwiki.org/wiki/Mapping_on_Quadratic_Integers_over_3_to_Conjugate_is_Automorphism | [
"Examples of Ring Automorphisms",
"Examples of Integral Domains",
"Quadratic Integers"
] | [
"Definition:Set",
"Definition:Algebraic Integer/Quadratic",
"Definition:Integer",
"Definition:Mapping",
"Definition:Ring Automorphism"
] | [
"Quadratic Integers over 2 form Subdomain of Reals"
] |
proofwiki-18739 | Multiplicative Group of Complex Numbers is not Isomorphic to Multiplicative Group of Real Numbers | Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.
Let $\struct {\R_{\ne 0}, \times}$ be the multiplicative group of real numbers.
Then $\struct {\C_{\ne 0}, \times}$ is not isomorphic to $\struct {\R_{\ne 0}, \times}$. | {{AimForCont}} $\struct {\C_{\ne 0}, \times}$ is isomorphic to $\struct {\R_{\ne 0}, \times}$.
Let $\phi: \C_{\ne 0} \to \R_{\ne 0}$ be an isomorphism.
Note that $\order i = 4$ in $\struct {\C_{\ne 0}, \times}$.
By Group Isomorphism Preserves Order of Group Element:
:$\order {\map \phi i} = 4$ in $\struct {\R_{\ne 0}, ... | Let $\struct {\C_{\ne 0}, \times}$ be the [[Definition:Multiplicative Group of Complex Numbers|multiplicative group of complex numbers]].
Let $\struct {\R_{\ne 0}, \times}$ be the [[Definition:Multiplicative Group of Real Numbers|multiplicative group of real numbers]].
Then $\struct {\C_{\ne 0}, \times}$ is not [[De... | {{AimForCont}} $\struct {\C_{\ne 0}, \times}$ is [[Definition:Group Isomorphism|isomorphic]] to $\struct {\R_{\ne 0}, \times}$.
Let $\phi: \C_{\ne 0} \to \R_{\ne 0}$ be an [[Definition:Group Isomorphism|isomorphism]].
Note that $\order i = 4$ in $\struct {\C_{\ne 0}, \times}$.
By [[Group Isomorphism Preserves Order ... | Multiplicative Group of Complex Numbers is not Isomorphic to Multiplicative Group of Real Numbers | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Numbers_is_not_Isomorphic_to_Multiplicative_Group_of_Real_Numbers | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Numbers_is_not_Isomorphic_to_Multiplicative_Group_of_Real_Numbers | [
"Examples of Group Isomorphisms",
"Multiplicative Group of Complex Numbers",
"Multiplicative Group of Real Numbers"
] | [
"Definition:Multiplicative Group of Complex Numbers",
"Definition:Multiplicative Group of Real Numbers",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Group Isomorphism Preserves Order of Group Element",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Proof by Contradiction"
] |
proofwiki-18740 | Multiplicative Group of Complex Numbers is not Isomorphic to Additive Group of Complex Numbers | Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.
Let $\struct {\C, +}$ be the additive group of complex numbers.
Then $\struct {\C_{\ne 0}, \times}$ is not isomorphic to $\struct {\C, +}$. | A direct application of Additive Group and Multiplicative Group of Field are not Isomorphic.
{{qed}} | Let $\struct {\C_{\ne 0}, \times}$ be the [[Definition:Multiplicative Group of Complex Numbers|multiplicative group of complex numbers]].
Let $\struct {\C, +}$ be the [[Definition:Additive Group of Complex Numbers|additive group of complex numbers]].
Then $\struct {\C_{\ne 0}, \times}$ is not [[Definition:Group Isom... | A direct application of [[Additive Group and Multiplicative Group of Field are not Isomorphic]].
{{qed}} | Multiplicative Group of Complex Numbers is not Isomorphic to Additive Group of Complex Numbers | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Numbers_is_not_Isomorphic_to_Additive_Group_of_Complex_Numbers | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Numbers_is_not_Isomorphic_to_Additive_Group_of_Complex_Numbers | [
"Examples of Group Isomorphisms",
"Multiplicative Group of Complex Numbers",
"Additive Group of Complex Numbers"
] | [
"Definition:Multiplicative Group of Complex Numbers",
"Definition:Additive Group of Complex Numbers",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Additive Group and Multiplicative Group of Field are not Isomorphic"
] |
proofwiki-18741 | Multiplicative Group of Complex Numbers is not Isomorphic to Additive Group of Real Numbers | Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.
Let $\struct {\R, +}$ be the additive group of real numbers.
Then $\struct {\C_{\ne 0}, \times}$ is not isomorphic to $\struct {\R, +}$. | {{AimForCont}} $\struct {\C_{\ne 0}, \times}$ is isomorphic to $\struct {\R, +}$.
Let $\phi: \C_{\ne 0} \to \R$ be an isomorphism.
Note that $\order i = 4$ in $\struct {\C_{\ne 0}, \times}$.
By Group Isomorphism Preserves Order of Group Element:
:$\order {\map \phi i} = 4$ in $\struct {\R, +}$
Then $4 \map \phi i = 0$.... | Let $\struct {\C_{\ne 0}, \times}$ be the [[Definition:Multiplicative Group of Complex Numbers|multiplicative group of complex numbers]].
Let $\struct {\R, +}$ be the [[Definition:Additive Group of Real Numbers|additive group of real numbers]].
Then $\struct {\C_{\ne 0}, \times}$ is not [[Definition:Group Isomorphis... | {{AimForCont}} $\struct {\C_{\ne 0}, \times}$ is [[Definition:Group Isomorphism|isomorphic]] to $\struct {\R, +}$.
Let $\phi: \C_{\ne 0} \to \R$ be an [[Definition:Group Isomorphism|isomorphism]].
Note that $\order i = 4$ in $\struct {\C_{\ne 0}, \times}$.
By [[Group Isomorphism Preserves Order of Group Element]]:
:... | Multiplicative Group of Complex Numbers is not Isomorphic to Additive Group of Real Numbers | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Numbers_is_not_Isomorphic_to_Additive_Group_of_Real_Numbers | https://proofwiki.org/wiki/Multiplicative_Group_of_Complex_Numbers_is_not_Isomorphic_to_Additive_Group_of_Real_Numbers | [
"Examples of Group Isomorphisms",
"Multiplicative Group of Complex Numbers",
"Additive Group of Real Numbers"
] | [
"Definition:Multiplicative Group of Complex Numbers",
"Definition:Additive Group of Real Numbers",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Group Isomorphism Preserves Order of Group Element",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Proof by Contradiction"
] |
proofwiki-18742 | Additive Group of Rational Numbers is not Isomorphic to Multiplicative Group of Rational Numbers | Let $\struct {\Q, +}$ be the additive group of rational numbers.
Let $\struct {\Q_{\ne 0}, \times}$ be the multiplicative group of rational numbers.
Then $\struct {\Q_{\ne 0}, \times}$ is not isomorphic to $\struct {\Q, +}$. | A direct application of Additive Group and Multiplicative Group of Field are not Isomorphic.
{{qed}} | Let $\struct {\Q, +}$ be the [[Definition:Additive Group of Rational Numbers|additive group of rational numbers]].
Let $\struct {\Q_{\ne 0}, \times}$ be the [[Definition:Multiplicative Group of Rational Numbers|multiplicative group of rational numbers]].
Then $\struct {\Q_{\ne 0}, \times}$ is not [[Definition:Group ... | A direct application of [[Additive Group and Multiplicative Group of Field are not Isomorphic]].
{{qed}} | Additive Group of Rational Numbers is not Isomorphic to Multiplicative Group of Rational Numbers | https://proofwiki.org/wiki/Additive_Group_of_Rational_Numbers_is_not_Isomorphic_to_Multiplicative_Group_of_Rational_Numbers | https://proofwiki.org/wiki/Additive_Group_of_Rational_Numbers_is_not_Isomorphic_to_Multiplicative_Group_of_Rational_Numbers | [
"Examples of Group Isomorphisms",
"Multiplicative Group of Rational Numbers",
"Additive Group of Rational Numbers"
] | [
"Definition:Additive Group of Rational Numbers",
"Definition:Multiplicative Group of Rational Numbers",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Additive Group and Multiplicative Group of Field are not Isomorphic"
] |
proofwiki-18743 | Permutation of Set is Automorphism of Set under Left Operation | Let $S$ be a set.
Let $\struct {S, \gets}$ be the algebraic structure formed from $S$ under the left operation.
Let $f$ be a permutation on $S$.
Then $f$ is an automorphism of $f$. | We have {{hypothesis}} that $f$ is a permutation and so {{afortiori}} a bijection.
It remains to show that $f$ is a homomorphism.
So, let $a, b \in S$ be arbitrary.
We have:
{{begin-eqn}}
{{eqn | l = \map f {a \gets b}
| r = \map f a
| c = {{Defof|Left Operation}}
}}
{{eqn | r = \map f a \gets \map f b
... | Let $S$ be a [[Definition:Set|set]].
Let $\struct {S, \gets}$ be the [[Definition:Algebraic Structure|algebraic structure]] formed from $S$ under the [[Definition:Left Operation|left operation]].
Let $f$ be a [[Definition:Permutation|permutation]] on $S$.
Then $f$ is an [[Definition:Automorphism (Abstract Algebra)|... | We have {{hypothesis}} that $f$ is a [[Definition:Permutation|permutation]] and so {{afortiori}} a [[Definition:Bijection|bijection]].
It remains to show that $f$ is a [[Definition:Homomorphism (Abstract Algebra)|homomorphism]].
So, let $a, b \in S$ be [[Definition:Arbitrary|arbitrary]].
We have:
{{begin-eqn}}
{{e... | Permutation of Set is Automorphism of Set under Left Operation | https://proofwiki.org/wiki/Permutation_of_Set_is_Automorphism_of_Set_under_Left_Operation | https://proofwiki.org/wiki/Permutation_of_Set_is_Automorphism_of_Set_under_Left_Operation | [
"Examples of Automorphisms",
"Left Operation"
] | [
"Definition:Set",
"Definition:Algebraic Structure",
"Definition:Left Operation",
"Definition:Permutation",
"Definition:Automorphism (Abstract Algebra)"
] | [
"Definition:Permutation",
"Definition:Bijection",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Arbitrary"
] |
proofwiki-18744 | Permutation of Set is Automorphism of Set under Right Operation | Let $S$ be a set.
Let $\struct {S, \to}$ be the algebraic structure formed from $S$ under the right operation.
Let $f$ be a permutation on $S$.
Then $f$ is an automorphism of $f$. | We have {{hypothesis}} that $f$ is a permutation and so a fortiori a bijection.
It remains to show that $f$ is a homomorphism.
So, let $a, b \in S$ be arbitrary.
We have:
{{begin-eqn}}
{{eqn | l = \map f {a \to b}
| r = \map f b
| c = {{Defof|Right Operation}}
}}
{{eqn | r = \map f a \to \map f b
| c ... | Let $S$ be a [[Definition:Set|set]].
Let $\struct {S, \to}$ be the [[Definition:Algebraic Structure|algebraic structure]] formed from $S$ under the [[Definition:Right Operation|right operation]].
Let $f$ be a [[Definition:Permutation|permutation]] on $S$.
Then $f$ is an [[Definition:Automorphism (Abstract Algebra)|... | We have {{hypothesis}} that $f$ is a [[Definition:Permutation|permutation]] and so [[Definition:A Fortiori|a fortiori]] a [[Definition:Bijection|bijection]].
It remains to show that $f$ is a [[Definition:Homomorphism (Abstract Algebra)|homomorphism]].
So, let $a, b \in S$ be arbitrary.
We have:
{{begin-eqn}}
{{eqn... | Permutation of Set is Automorphism of Set under Right Operation | https://proofwiki.org/wiki/Permutation_of_Set_is_Automorphism_of_Set_under_Right_Operation | https://proofwiki.org/wiki/Permutation_of_Set_is_Automorphism_of_Set_under_Right_Operation | [
"Examples of Automorphisms",
"Right Operation"
] | [
"Definition:Set",
"Definition:Algebraic Structure",
"Definition:Right Operation",
"Definition:Permutation",
"Definition:Automorphism (Abstract Algebra)"
] | [
"Definition:Permutation",
"Definition:A Fortiori",
"Definition:Bijection",
"Definition:Homomorphism (Abstract Algebra)"
] |
proofwiki-18745 | Algebraic Structures formed by Left and Right Operations are not Isomorphic for Cardinality Greater than 1 | Let $S$ be a set.
Let $\gets$ and $\to$ denote the left operation and right operation respectively.
Let $\card S > 1$.
The algebraic structures $\struct {S, \gets}$ and $\struct {S, \to}$ are not isomorphic. | {{AimForCont}} there exists an isomorphism $\phi$ from $\struct {S, \gets}$ to $\struct {S, \to}$.
Because $\card S > 1$ we have that there exist at least $2$ distinct elements of $S$.
Let these be $x$ and $y$.
Hence:
{{begin-eqn}}
{{eqn | l = \map \phi {x \gets y}
| r = \map \phi x
| c = {{Defof|Left Opera... | Let $S$ be a [[Definition:Set|set]].
Let $\gets$ and $\to$ denote the [[Definition:Left Operation|left operation]] and [[Definition:Right Operation|right operation]] respectively.
Let $\card S > 1$.
The [[Definition:Algebraic Structure|algebraic structures]] $\struct {S, \gets}$ and $\struct {S, \to}$ are not [[Def... | {{AimForCont}} there exists an [[Definition:Isomorphism (Abstract Algebra)|isomorphism]] $\phi$ from $\struct {S, \gets}$ to $\struct {S, \to}$.
Because $\card S > 1$ we have that there exist at least $2$ [[Definition:Distinct Elements|distinct elements]] of $S$.
Let these be $x$ and $y$.
Hence:
{{begin-eqn}}
{{eq... | Algebraic Structures formed by Left and Right Operations are not Isomorphic for Cardinality Greater than 1 | https://proofwiki.org/wiki/Algebraic_Structures_formed_by_Left_and_Right_Operations_are_not_Isomorphic_for_Cardinality_Greater_than_1 | https://proofwiki.org/wiki/Algebraic_Structures_formed_by_Left_and_Right_Operations_are_not_Isomorphic_for_Cardinality_Greater_than_1 | [
"Left and Right Operations",
"Examples of Isomorphisms (Abstract Algebra)"
] | [
"Definition:Set",
"Definition:Left Operation",
"Definition:Right Operation",
"Definition:Algebraic Structure",
"Definition:Isomorphism (Abstract Algebra)"
] | [
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Distinct/Plural",
"Definition:Injection",
"Definition:Bijection",
"Definition:Isomorphism (Abstract Algebra)"
] |
proofwiki-18746 | Induced Metric on Surface of Revolution/Corollary | Let $\struct {\R^3, d}$ be the Euclidean space.
Let $S_C \subseteq \R^3$ be the surface of revolution.
Let the smooth local parametrization of $C$ be:
:$\map \gamma t = \tuple {\map x t, \map y t}$
Let $\gamma$ be a unit-speed curve.
Then the induced metric on $S_C$ is:
:$g = d t^2 + \map y t^2 d \theta^2$ | By definition of the unit-speed curve:
:$\size {\map {\gamma'} t}_g = 1$
In our case we are working with the Euclidean space.
Hence:
:$\sqrt {\map {x'^2} t + \map {y'^2} t} = 1$
or:
:$ \map {x'^2} t + \map {y'^2} t = 1$
Substitution of this into the induced metric of $S_C$ yields the desired result.
{{qed}} | Let $\struct {\R^3, d}$ be the [[Definition:Euclidean Space|Euclidean space]].
Let $S_C \subseteq \R^3$ be the [[Definition:Surface of Revolution|surface of revolution]].
Let the [[Definition:Smooth Local Parametrization|smooth local parametrization]] of $C$ be:
:$\map \gamma t = \tuple {\map x t, \map y t}$
Let $\... | By definition of the [[Definition:Unit-Speed Curve|unit-speed curve]]:
:$\size {\map {\gamma'} t}_g = 1$
In our case we are working with the [[Definition:Euclidean Space|Euclidean space]].
Hence:
:$\sqrt {\map {x'^2} t + \map {y'^2} t} = 1$
or:
:$ \map {x'^2} t + \map {y'^2} t = 1$
Substitution of this into the ... | Induced Metric on Surface of Revolution/Corollary | https://proofwiki.org/wiki/Induced_Metric_on_Surface_of_Revolution/Corollary | https://proofwiki.org/wiki/Induced_Metric_on_Surface_of_Revolution/Corollary | [
"Surfaces of Revolution",
"Solid Geometry",
"Induced Metrics"
] | [
"Definition:Euclidean Space",
"Definition:Surface of Revolution",
"Definition:Smooth Local Parametrization",
"Definition:Unit-Speed Curve",
"Definition:Induced Metric on Submanifold"
] | [
"Definition:Unit-Speed Curve",
"Definition:Euclidean Space",
"Induced Metric on Surface of Revolution"
] |
proofwiki-18747 | Structure of Cardinality 3+ where Every Permutation is Automorphism is Idempotent | Let $S$ be a set whose cardinality is at least $3$.
Let $\struct {S, \circ}$ be an algebraic structure on $S$ such that every permutation on $S$ is an automorphism on $\struct {S, \circ}$.
Then $\circ$ is an idempotent operation. | {{AimForCont}} $\circ$ is not idempotent.
Then there exists $x \in S$ such that:
:$\exists y \in S: x \circ x = y$
where $x \ne y$.
Because there are at least $3$ distinct elements of $S$ {{hypothesis}}:
:$\exists z \in S: z \ne x, z \ne y$
Let $f: S \to S$ be a permutation on $S$ such that:
:$\map f x = x$
:$\map f y ... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is at least $3$.
Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]] on $S$ such that every [[Definition:Permutation|permutation]] on $S$ is an [[Definition:Automorphism|automorphism]] on $\struct {S, \ci... | {{AimForCont}} $\circ$ is not [[Definition:Idempotent Operation|idempotent]].
Then there exists $x \in S$ such that:
:$\exists y \in S: x \circ x = y$
where $x \ne y$.
Because there are at least $3$ [[Definition:Distinct Elements|distinct elements]] of $S$ {{hypothesis}}:
:$\exists z \in S: z \ne x, z \ne y$
Let $... | Structure of Cardinality 3+ where Every Permutation is Automorphism is Idempotent | https://proofwiki.org/wiki/Structure_of_Cardinality_3+_where_Every_Permutation_is_Automorphism_is_Idempotent | https://proofwiki.org/wiki/Structure_of_Cardinality_3+_where_Every_Permutation_is_Automorphism_is_Idempotent | [
"Idempotence"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Algebraic Structure",
"Definition:Permutation",
"Definition:Automorphism",
"Definition:Idempotence/Operation"
] | [
"Definition:Idempotence/Operation",
"Definition:Distinct/Plural",
"Definition:Permutation",
"Definition:Automorphism",
"Definition:Contradiction",
"Definition:Distinct/Plural",
"Proof by Contradiction",
"Definition:Idempotence/Operation",
"Definition:Idempotence/Operation"
] |
proofwiki-18748 | Operation on Structure of Cardinality 4+ where Every Permutation is Automorphism is Left or Right Operation | Let $S$ be a set whose cardinality is at least $4$.
Let $\struct {S, \circ}$ be an algebraic structure on $S$ such that every permutation on $S$ is an automorphism on $\struct {S, \circ}$.
Then $\circ$ is either the left operation or the right operation. | From Structure of Cardinality 3+ where Every Permutation is Automorphism is Idempotent, we have that $\circ$ is idempotent:
:$\forall a \in S: a \circ a = a$
{{AimForCont}} $\circ$ is such that:
:$x \circ y = z$
for some distinct $x, y, z \in S$.
As $S$ has cardinality of at least $4$, there exists $w \in S$ such that ... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is at least $4$.
Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]] on $S$ such that every [[Definition:Permutation|permutation]] on $S$ is an [[Definition:Automorphism|automorphism]] on $\struct {S, \ci... | From [[Structure of Cardinality 3+ where Every Permutation is Automorphism is Idempotent]], we have that $\circ$ is [[Definition:Idempotent Operation|idempotent]]:
:$\forall a \in S: a \circ a = a$
{{AimForCont}} $\circ$ is such that:
:$x \circ y = z$
for some [[Definition:Distinct Elements|distinct]] $x, y, z \in S$... | Operation on Structure of Cardinality 4+ where Every Permutation is Automorphism is Left or Right Operation | https://proofwiki.org/wiki/Operation_on_Structure_of_Cardinality_4+_where_Every_Permutation_is_Automorphism_is_Left_or_Right_Operation | https://proofwiki.org/wiki/Operation_on_Structure_of_Cardinality_4+_where_Every_Permutation_is_Automorphism_is_Left_or_Right_Operation | [
"Left and Right Operations"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Algebraic Structure",
"Definition:Permutation",
"Definition:Automorphism",
"Definition:Left Operation",
"Definition:Right Operation"
] | [
"Structure of Cardinality 3+ where Every Permutation is Automorphism is Idempotent",
"Definition:Idempotence/Operation",
"Definition:Distinct/Plural",
"Definition:Cardinality",
"Definition:Permutation",
"Definition:Automorphism",
"Definition:Contradiction",
"Definition:Distinct/Plural",
"Definition:... |
proofwiki-18749 | Unit Sphere as Surface of Revolution | {{WIP|Resolving consistency issues with previous results}}
Let $\struct {\R^3, d}$ be the Euclidean space.
Let $S_C \subseteq \R^3$ be the surface of revolution.
Let $C$ be a semi-circle defined by $x^2 + y^2 = 1$ in the open upper half-plane.
Let the smooth local parametrization of $C$ be:
:$\map \gamma \phi = \tuple ... | By Smooth Local Parametrization of Surface of Revolution, the smooth local parametrization of $S_C$ can be written as:
:$\map X {\phi, \theta} = \tuple {\sin \phi \cos \theta, \sin \phi \sin \theta, \cos \phi}$
By Induced Metric on Surface of Revolution:
{{begin-eqn}}
{{eqn | l = g
| r = \paren {\paren {\cos' \ph... | {{WIP|Resolving consistency issues with previous results}}
Let $\struct {\R^3, d}$ be the [[Definition:Euclidean Space|Euclidean space]].
Let $S_C \subseteq \R^3$ be the [[Definition:Surface of Revolution|surface of revolution]].
Let $C$ be a semi-circle defined by $x^2 + y^2 = 1$ in the [[Definition:Open Upper Half-... | By [[Smooth Local Parametrization of Surface of Revolution]], the [[Definition:Smooth Local Parametrization|smooth local parametrization]] of $S_C$ can be written as:
:$\map X {\phi, \theta} = \tuple {\sin \phi \cos \theta, \sin \phi \sin \theta, \cos \phi}$
By [[Induced Metric on Surface of Revolution]]:
{{begin-eq... | Unit Sphere as Surface of Revolution | https://proofwiki.org/wiki/Unit_Sphere_as_Surface_of_Revolution | https://proofwiki.org/wiki/Unit_Sphere_as_Surface_of_Revolution | [
"Examples of Surfaces of Revolution",
"Induced Metrics"
] | [
"Definition:Euclidean Space",
"Definition:Surface of Revolution",
"Definition:Half-Plane/Open/Upper",
"Definition:Smooth Local Parametrization",
"Definition:Induced Metric on Submanifold"
] | [
"Smooth Local Parametrization of Surface of Revolution",
"Definition:Smooth Local Parametrization",
"Induced Metric on Surface of Revolution",
"Definition:Riemannian Metric",
"Definition:Unit Sphere",
"Definition:Point",
"Definition:Axis/X-Axis"
] |
proofwiki-18750 | Conjugation of Bijection between Symmetric Groups is Isomorphism | Let $A$ and $B$ be sets
Let $f$ be a bijection from $E$ to $F$.
Let $S_A$ and $S_B$ denote the set of all permutations on $A$ and $B$ respectively.
Let $\Phi: S_A \to S_B$ be the mapping defined as:
:$\forall u \in S_A: \map \Phi u = f \circ u \circ f^{-1}$
where $\circ$ denotes composition of mappings.
Then $\Phi$ is ... | We have that $\struct {S_A, \circ}$ and $\struct {S_B, \circ}$ are the symmetric group on $S_A$ and $S_B$ respectively.
Hence we are about to prove that $\Phi$ is actually a group isomorphism.
Because $f$ is a bijection it follows from Inverse of Bijection is Bijection that $f^{-1}$ is also a bijection.
From Composite ... | Let $A$ and $B$ be [[Definition:Set|sets]]
Let $f$ be a [[Definition:Bijection|bijection]] from $E$ to $F$.
Let $S_A$ and $S_B$ denote the [[Definition:Set|set]] of all [[Definition:Permutation|permutations]] on $A$ and $B$ respectively.
Let $\Phi: S_A \to S_B$ be the [[Definition:Mapping|mapping]] defined as:
:$\f... | We have that $\struct {S_A, \circ}$ and $\struct {S_B, \circ}$ are the [[Definition:Symmetric Group|symmetric group]] on $S_A$ and $S_B$ respectively.
Hence we are about to prove that $\Phi$ is actually a [[Definition:Group Isomorphism|group isomorphism]].
Because $f$ is a [[Definition:Bijection|bijection]] it follo... | Conjugation of Bijection between Symmetric Groups is Isomorphism | https://proofwiki.org/wiki/Conjugation_of_Bijection_between_Symmetric_Groups_is_Isomorphism | https://proofwiki.org/wiki/Conjugation_of_Bijection_between_Symmetric_Groups_is_Isomorphism | [
"Symmetric Groups",
"Examples of Group Isomorphisms"
] | [
"Definition:Set",
"Definition:Bijection",
"Definition:Set",
"Definition:Permutation",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Isomorphism (Abstract Algebra)"
] | [
"Definition:Symmetric Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Bijection",
"Inverse of Bijection is Bijection",
"Definition:Bijection",
"Composite of Bijections is Bijection",
"Definition:Bijection",
"Definition:Permutation",
"Definition:Permutation",
"Def... |
proofwiki-18751 | Product Inverse Operation is Self-Inverse | :$\forall x \in G: x \oplus x = e$ | {{begin-eqn}}
{{eqn | q = \forall x \in G
| l = x \oplus x
| r = x \circ x^{-1}
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = e
| c = {{Group-axiom|3}}
}}
{{end-eqn}}
{{qed}} | :$\forall x \in G: x \oplus x = e$ | {{begin-eqn}}
{{eqn | q = \forall x \in G
| l = x \oplus x
| r = x \circ x^{-1}
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = e
| c = {{Group-axiom|3}}
}}
{{end-eqn}}
{{qed}} | Product Inverse Operation is Self-Inverse | https://proofwiki.org/wiki/Product_Inverse_Operation_is_Self-Inverse | https://proofwiki.org/wiki/Product_Inverse_Operation_is_Self-Inverse | [
"Product Inverse Operation"
] | [] | [] |
proofwiki-18752 | Group Identity is Right Identity for Product Inverse Operation | :$\forall x \in G: x \oplus e = x$ | {{begin-eqn}}
{{eqn | q = \forall x \in G
| l = x \oplus e
| r = x \circ e^{-1}
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = x \circ e
| c = Inverse of Identity Element is Itself
}}
{{eqn | r = x
| c = {{Group-axiom|2}}
}}
{{end-eqn}}
{{qed}} | :$\forall x \in G: x \oplus e = x$ | {{begin-eqn}}
{{eqn | q = \forall x \in G
| l = x \oplus e
| r = x \circ e^{-1}
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = x \circ e
| c = [[Inverse of Identity Element is Itself]]
}}
{{eqn | r = x
| c = {{Group-axiom|2}}
}}
{{end-eqn}}
{{qed}} | Group Identity is Right Identity for Product Inverse Operation | https://proofwiki.org/wiki/Group_Identity_is_Right_Identity_for_Product_Inverse_Operation | https://proofwiki.org/wiki/Group_Identity_is_Right_Identity_for_Product_Inverse_Operation | [
"Product Inverse Operation",
"Identity Elements"
] | [] | [
"Inverse of Identity Element is Itself"
] |
proofwiki-18753 | Group Product Inverse Operation with Identity | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $\oplus: G \times G \to G$ be the product inverse of $\circ$ on $G$.
Then:
:$\forall x, y \in G: e \oplus \paren {x \oplus y} = y \oplus x$ | {{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = e \oplus \paren {x \oplus y}
| r = e \oplus \paren {x \circ y^{-1} }
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = e \circ \paren {x \circ y^{-1} }^{-1}
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = e \circ \paren {y \circ ... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\oplus: G \times G \to G$ be the [[Definition:Product Inverse Operation|product inverse]] of $\circ$ on $G$.
Then:
:$\forall x, y \in G: e \oplus \paren {x \oplus y} = y \oplus x$ | {{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = e \oplus \paren {x \oplus y}
| r = e \oplus \paren {x \circ y^{-1} }
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = e \circ \paren {x \circ y^{-1} }^{-1}
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = e \circ \paren {y \circ ... | Group Product Inverse Operation with Identity | https://proofwiki.org/wiki/Group_Product_Inverse_Operation_with_Identity | https://proofwiki.org/wiki/Group_Product_Inverse_Operation_with_Identity | [
"Product Inverse Operation",
"Identity Elements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Product Inverse Operation"
] | [
"Inverse of Group Product"
] |
proofwiki-18754 | Cancellation Property of Product Inverse Operator | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $\oplus: G \times G \to G$ be the product inverse of $\circ$ on $G$.
Then:
:$\forall x, y, z \in G: \paren {x \oplus z} \oplus \paren {y \oplus z} = x \oplus y$ | {{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = \paren {x \oplus z} \oplus \paren {y \oplus z}
| r = \paren {x \circ z^{-1} } \oplus \paren {y \circ z^{-1} }
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = \paren {x \circ z^{-1} } \circ \paren {y \circ z^{-1} }^{-1}
| c = {{Defof|... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\oplus: G \times G \to G$ be the [[Definition:Product Inverse Operation|product inverse]] of $\circ$ on $G$.
Then:
:$\forall x, y, z \in G: \paren {x \oplus z} \oplus \paren {y \oplus z} = x \oplus y$ | {{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = \paren {x \oplus z} \oplus \paren {y \oplus z}
| r = \paren {x \circ z^{-1} } \oplus \paren {y \circ z^{-1} }
| c = {{Defof|Product Inverse Operation}}
}}
{{eqn | r = \paren {x \circ z^{-1} } \circ \paren {y \circ z^{-1} }^{-1}
| c = {{Defof|... | Cancellation Property of Product Inverse Operator | https://proofwiki.org/wiki/Cancellation_Property_of_Product_Inverse_Operator | https://proofwiki.org/wiki/Cancellation_Property_of_Product_Inverse_Operator | [
"Product Inverse Operation"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Product Inverse Operation"
] | [
"Inverse of Group Product"
] |
proofwiki-18755 | Product Inverse Operation Properties induce Group | Let $\struct {G, \oplus}$ be a closed algebraic structure on which the following properties hold:
{{:Axiom:Product Inverse Operation Axioms}}
Let $\circ$ be the operation on $G$ defined as:
:$\forall x, y \in G: x \circ y = x \oplus \paren {e \oplus y}$
Then $\struct {G, \circ}$ is a group. | Taking the group axioms in turn: | Let $\struct {G, \oplus}$ be a [[Definition:Closed Algebraic Structure|closed]] [[Definition:Algebraic Structure|algebraic structure]] on which the following properties hold:
{{:Axiom:Product Inverse Operation Axioms}}
Let $\circ$ be the [[Definition:Binary Operation|operation]] on $G$ defined as:
:$\forall x, y \in G... | Taking the [[Axiom:Group Axioms|group axioms]] in turn: | Product Inverse Operation Properties induce Group | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties_induce_Group | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties_induce_Group | [
"Product Inverse Operation",
"Examples of Groups"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Group"
] | [
"Axiom:Group Axioms",
"Axiom:Group Axioms"
] |
proofwiki-18756 | Torus as Surface of Revolution | Let $\struct {\R^3, d}$ be the Euclidean space.
Let $S_C \subseteq \R^3$ be the surface of revolution.
{{explain|Surface of revolution of what?}}
Let $C$ be a circle defined by $x^2 + \paren {y - 2}^2 = 1$ in the open upper half-plane.
Let the smooth local parametrization of $C$ be:
:$\map \gamma t = \tuple {\sin t, 2 ... | We have that:
:$\map {\gamma'} t = \tuple {\cos t, - \sin t}$
Furthermore:
:$\paren {\cos t}^2 + \paren {- \sin t}^2 = 1$
Hence, $\map \gamma t$ is a unit-speed curve.
By {{Corollary|Induced Metric on Surface of Revolution}}:
:$g = d t^2 + \paren {2 + \cos t}^2 \, d \theta^2$
{{qed}} | Let $\struct {\R^3, d}$ be the [[Definition:Euclidean Space|Euclidean space]].
Let $S_C \subseteq \R^3$ be the [[Definition:Surface of Revolution|surface of revolution]].
{{explain|Surface of revolution of what?}}
Let $C$ be a [[Definition:Circle|circle]] defined by $x^2 + \paren {y - 2}^2 = 1$ in the [[Definition:O... | We have that:
:$\map {\gamma'} t = \tuple {\cos t, - \sin t}$
Furthermore:
:$\paren {\cos t}^2 + \paren {- \sin t}^2 = 1$
Hence, $\map \gamma t$ is a [[Definition:Unit-Speed Curve|unit-speed curve]].
By {{Corollary|Induced Metric on Surface of Revolution}}:
:$g = d t^2 + \paren {2 + \cos t}^2 \, d \theta^2$
{{qed... | Torus as Surface of Revolution | https://proofwiki.org/wiki/Torus_as_Surface_of_Revolution | https://proofwiki.org/wiki/Torus_as_Surface_of_Revolution | [
"Examples of Surfaces of Revolution",
"Induced Metrics"
] | [
"Definition:Euclidean Space",
"Definition:Surface of Revolution",
"Definition:Circle",
"Definition:Half-Plane/Open/Upper",
"Definition:Smooth Local Parametrization",
"Definition:Induced Metric on Submanifold"
] | [
"Definition:Unit-Speed Curve"
] |
proofwiki-18757 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 1 | :$\forall n \in \N: \exists r_n \in \Z : \dfrac a b - \paren{p^{n + 1} \dfrac {r_n} b} \in \set{0, 1, \ldots, p^{n + 1} - 1}$ | Let $n \in \N$.
From Integer Coprime to all Factors is Coprime to Whole:
:$b, p^{n + 1}$ are coprime
From Integer Combination of Coprime Integers:
:$\exists c_n, d_n \in \Z : c_n b + d_n p^{n + 1} = 1$
Multiplying both sides by $a$:
:$a = a c_n b + a d_n p^{n + 1}$
Let $A_n$ be the least positive residue of $a c_n \pmo... | :$\forall n \in \N: \exists r_n \in \Z : \dfrac a b - \paren{p^{n + 1} \dfrac {r_n} b} \in \set{0, 1, \ldots, p^{n + 1} - 1}$ | Let $n \in \N$.
From [[Integer Coprime to all Factors is Coprime to Whole]]:
:$b, p^{n + 1}$ are [[Definition:Coprime Integers|coprime]]
From [[Integer Combination of Coprime Integers]]:
:$\exists c_n, d_n \in \Z : c_n b + d_n p^{n + 1} = 1$
Multiplying both sides by $a$:
:$a = a c_n b + a d_n p^{n + 1}$
Let $A_n$ ... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 1 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_1 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_1 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Integer Coprime to all Factors is Coprime to Whole",
"Definition:Coprime/Integers",
"Integer Combination of Coprime Integers",
"Definition:Set of Residue Classes/Least Positive",
"Definition:Set of Residue Classes/Least Positive",
"Definition:Divisor (Algebra)/Integer",
"Definition:Set of Residue Class... |
proofwiki-18758 | Product Inverse Operation Properties/Lemma 1 | :$\forall x, y, z \in G: \paren {x \circ z} \oplus \paren {y \circ z} = x \oplus y$ | {{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = \paren {x \circ z} \oplus \paren {y \circ z}
| r = \paren {x \oplus \paren {e \oplus z} } \oplus \paren {y \oplus \paren {e \oplus z} }
| c = Definition of $\circ$
}}
{{eqn | r = x \oplus y
| c = $\text {PI} 4$: Cancellation Property
}}
{{e... | :$\forall x, y, z \in G: \paren {x \circ z} \oplus \paren {y \circ z} = x \oplus y$ | {{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = \paren {x \circ z} \oplus \paren {y \circ z}
| r = \paren {x \oplus \paren {e \oplus z} } \oplus \paren {y \oplus \paren {e \oplus z} }
| c = Definition of $\circ$
}}
{{eqn | r = x \oplus y
| c = $\text {PI} 4$: Cancellation Property
}}
{{e... | Product Inverse Operation Properties/Lemma 1 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_1 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_1 | [
"Product Inverse Operation"
] | [] | [] |
proofwiki-18759 | Product Inverse Operation Properties/Lemma 2 | :$\forall x, y, z \in G: \paren {x \oplus z} \circ \paren {z \oplus y} = x \oplus y$ | {{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = \paren {x \oplus z} \circ \paren {z \oplus y}
| r = \paren {x \oplus z} \oplus \paren {e \oplus \paren {z \oplus y} }
| c = Definition of $\circ$
}}
{{eqn | r = \paren {x \oplus z} \oplus \paren {y \oplus z}
| c = $\text {PI} 3$: Product Inv... | :$\forall x, y, z \in G: \paren {x \oplus z} \circ \paren {z \oplus y} = x \oplus y$ | {{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = \paren {x \oplus z} \circ \paren {z \oplus y}
| r = \paren {x \oplus z} \oplus \paren {e \oplus \paren {z \oplus y} }
| c = Definition of $\circ$
}}
{{eqn | r = \paren {x \oplus z} \oplus \paren {y \oplus z}
| c = $\text {PI} 3$: [[Definitio... | Product Inverse Operation Properties/Lemma 2 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_2 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_2 | [
"Product Inverse Operation"
] | [] | [
"Definition:Product Inverse",
"Definition:Identity (Abstract Algebra)/Right Identity"
] |
proofwiki-18760 | Product Inverse Operation Properties/Lemma 3 | :$\forall x, y \in G: \quad x \oplus y = e \implies x = y$ | Let $x \oplus y = e$.
Then we have:
{{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = \paren {x \oplus y} \oplus \paren {e \oplus y}
| r = e \oplus \paren {e \oplus y}
| c = from $x \oplus y = e$
}}
{{eqn | r = y \oplus e
| c = $\text {PI} 3$: Product Inverse with Right Identity
}}
{{eqn | r = ... | :$\forall x, y \in G: \quad x \oplus y = e \implies x = y$ | Let $x \oplus y = e$.
Then we have:
{{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = \paren {x \oplus y} \oplus \paren {e \oplus y}
| r = e \oplus \paren {e \oplus y}
| c = from $x \oplus y = e$
}}
{{eqn | r = y \oplus e
| c = $\text {PI} 3$: [[Definition:Product Inverse|Product Inverse]] wi... | Product Inverse Operation Properties/Lemma 3 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_3 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_3 | [
"Product Inverse Operation"
] | [] | [
"Definition:Product Inverse",
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Identity (Abstract Algebra)/Right Identity"
] |
proofwiki-18761 | Product Inverse Operation Properties/Lemma 4 | :$\forall x, y, z \in G: x \oplus z = y \oplus z \implies x = y$ | Let $x \oplus z = y \oplus z$.
Then we have:
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = \paren {x \oplus z} \oplus \paren {y \oplus z}
| r = \paren {x \oplus z} \oplus \paren {x \oplus z}
| c = from $x \oplus z = y \oplus z$
}}
{{eqn | r = x \oplus x
| c = $\text {PI} 4$: Cancellation... | :$\forall x, y, z \in G: x \oplus z = y \oplus z \implies x = y$ | Let $x \oplus z = y \oplus z$.
Then we have:
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = \paren {x \oplus z} \oplus \paren {y \oplus z}
| r = \paren {x \oplus z} \oplus \paren {x \oplus z}
| c = from $x \oplus z = y \oplus z$
}}
{{eqn | r = x \oplus x
| c = $\text {PI} 4$: Cancellati... | Product Inverse Operation Properties/Lemma 4 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_4 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_4 | [
"Product Inverse Operation"
] | [] | [
"Definition:Self-Inverse Element",
"Product Inverse Operation Properties/Lemma 3"
] |
proofwiki-18762 | Product Inverse Operation Properties/Lemma 5 | :$\forall x, y \in G: \paren {x \circ y} \oplus y = x$ | {{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = \paren {x \circ y} \oplus y
| r = \paren {x \oplus \paren {e \oplus y} } \oplus y
| c = Definition of $\circ$
}}
{{eqn | r = \paren {x \oplus \paren {e \oplus y} } \oplus \paren {y \oplus e}
| c = $\text {PI} 2$: Right Identity
}}
{{eqn | r = \p... | :$\forall x, y \in G: \paren {x \circ y} \oplus y = x$ | {{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = \paren {x \circ y} \oplus y
| r = \paren {x \oplus \paren {e \oplus y} } \oplus y
| c = Definition of $\circ$
}}
{{eqn | r = \paren {x \oplus \paren {e \oplus y} } \oplus \paren {y \oplus e}
| c = $\text {PI} 2$: [[Definition:Right Identity|Righ... | Product Inverse Operation Properties/Lemma 5 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_5 | https://proofwiki.org/wiki/Product_Inverse_Operation_Properties/Lemma_5 | [
"Product Inverse Operation"
] | [] | [
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Product Inverse",
"Definition:Identity (Abstract Algebra)/Right Identity"
] |
proofwiki-18763 | Operation Induced by Permutation on Magma is Closed | Let $\struct {S, \circ}$ be a magma.
Let $\sigma: S \to S$ be a permutation on $S$.
Let $\circ_\sigma$ be the operation on $S$ induced by $\sigma$:
:$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$
Then $\circ_\sigma$ is closed on $S$ | {{Proofread|Check the validity of the proof for the case when S is the empty set}}
Suppose $S$ is the empty set.
Let $\sigma: S \to S$ be a permutation on $S$.
Since $S$ is empty, by definition, $\sigma$ is the empty map.
Since $\circ_\sigma$ is the operation on $S$ induced by $\sigma$, it follows that $\circ_\sigma$ ... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $\sigma: S \to S$ be a [[Definition:Permutation|permutation]] on $S$.
Let $\circ_\sigma$ be the [[Definition:Operation Induced by Permutation|operation on $S$ induced by $\sigma$]]:
:$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$
Then ... | {{Proofread|Check the validity of the proof for the case when S is the empty set}}
Suppose $S$ is the [[Definition:Empty Set|empty]] set.
Let $\sigma: S \to S$ be a [[Definition:Permutation|permutation]] on $S$.
Since $S$ is empty, by definition, $\sigma$ is the [[Definition:Empty Mapping|empty map]].
Since $\circ_... | Operation Induced by Permutation on Magma is Closed | https://proofwiki.org/wiki/Operation_Induced_by_Permutation_on_Magma_is_Closed | https://proofwiki.org/wiki/Operation_Induced_by_Permutation_on_Magma_is_Closed | [
"Operations Induced by Permutations",
"Magmas"
] | [
"Definition:Magma",
"Definition:Permutation",
"Definition:Operation Induced by Permutation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Empty Set",
"Definition:Permutation",
"Definition:Empty Mapping",
"Definition:Operation Induced by Permutation",
"Definition:Empty Mapping",
"Definition:Vacuous Truth",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Non-Empty Set",
"Definition:Magma",
"Definit... |
proofwiki-18764 | Structure Induced by Permutation on Semigroup is not necessarily Semigroup | Let $\struct {S, \circ}$ be a semigroup.
Let $\sigma: S \to S$ be a permutation on $S$.
Let $\struct {S, \circ_\sigma}$ be the structure induced by $\sigma$ on $\circ$:
:$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$
Then $\struct {S, \circ_\sigma}$ is not necessarily itself a semigroup. | From Operation Induced by Permutation on Magma is Closed we have that $\struct {S, \circ_\sigma}$ is a closed structure.
Hence {{Semigroup-axiom|0}} holds.
However, we have that Operation Induced by Permutation on Semigroup is not necessarily Associative.
Hence {{Semigroup-axiom|1}} does not necessarily hold for $\stru... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $\sigma: S \to S$ be a [[Definition:Permutation|permutation]] on $S$.
Let $\struct {S, \circ_\sigma}$ be the [[Definition:Structure Induced by Permutation|structure induced by $\sigma$ on $\circ$]]:
:$\forall x, y \in S: x \circ_\sigma y := \map \s... | From [[Operation Induced by Permutation on Magma is Closed]] we have that $\struct {S, \circ_\sigma}$ is a [[Definition:Closed Algebraic Structure|closed structure]].
Hence {{Semigroup-axiom|0}} holds.
However, we have that [[Operation Induced by Permutation on Semigroup is not necessarily Associative]].
Hence {{Sem... | Structure Induced by Permutation on Semigroup is not necessarily Semigroup | https://proofwiki.org/wiki/Structure_Induced_by_Permutation_on_Semigroup_is_not_necessarily_Semigroup | https://proofwiki.org/wiki/Structure_Induced_by_Permutation_on_Semigroup_is_not_necessarily_Semigroup | [
"Operations Induced by Permutations",
"Semigroups"
] | [
"Definition:Semigroup",
"Definition:Permutation",
"Definition:Structure Induced by Permutation",
"Definition:Semigroup"
] | [
"Operation Induced by Permutation on Magma is Closed",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Operation Induced by Permutation on Semigroup is not necessarily Associative",
"Definition:Semigroup"
] |
proofwiki-18765 | Operation Induced by Permutation on Semigroup is not necessarily Associative | Let $\struct {S, \circ}$ be a semigroup.
Let $\sigma: S \to S$ be a permutation on $S$.
Let $\circ_\sigma$ be the operation on $S$ induced by $\sigma$:
:$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$
Then $\circ_\sigma$ is not necessarily associative on $S$. | ;Proof by Counterexample
Let $S = \set {a, b, c}$.
Let $\circ$ denote the right operation on $S$:
:$\forall x, y \in S: x \to y = y$
From Structure under Right Operation is Semigroup, $\struct {S, \circ}$ is a semigroup.
Hence we have:
:$a \circ \paren {b \circ c} = c = \paren {a \circ b} \circ c$
Let $\sigma$ denote t... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $\sigma: S \to S$ be a [[Definition:Permutation|permutation]] on $S$.
Let $\circ_\sigma$ be the [[Definition:Operation Induced by Permutation|operation on $S$ induced by $\sigma$]]:
:$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$... | ;[[Proof by Counterexample]]
Let $S = \set {a, b, c}$.
Let $\circ$ denote the [[Definition:Right Operation|right operation]] on $S$:
:$\forall x, y \in S: x \to y = y$
From [[Structure under Right Operation is Semigroup]], $\struct {S, \circ}$ is a [[Definition:Semigroup|semigroup]].
Hence we have:
:$a \circ \pare... | Operation Induced by Permutation on Semigroup is not necessarily Associative | https://proofwiki.org/wiki/Operation_Induced_by_Permutation_on_Semigroup_is_not_necessarily_Associative | https://proofwiki.org/wiki/Operation_Induced_by_Permutation_on_Semigroup_is_not_necessarily_Associative | [
"Operations Induced by Permutations",
"Semigroups",
"Associativity"
] | [
"Definition:Semigroup",
"Definition:Permutation",
"Definition:Operation Induced by Permutation",
"Definition:Associative Operation"
] | [
"Proof by Counterexample",
"Definition:Right Operation",
"Structure under Right Operation is Semigroup",
"Definition:Semigroup",
"Definition:Permutation",
"Category:Operations Induced by Permutations",
"Category:Semigroups",
"Category:Associativity"
] |
proofwiki-18766 | Structure under Right Operation is Semigroup | Let $\struct {S, \to}$ be an algebraic structure in which the operation $\to$ is the right operation.
Then $\struct {S, \to}$ is a semigroup. | We need to verify the semigroup axioms:
{{:Axiom:Semigroup Axioms}}
By the nature of the right operation, $\struct {S, \to}$ is closed:
:$\forall x, y \in S: x \to y = y \in S$
whatever $S$ may be.
Hence {{Semigroup-axiom|0}} holds.
From Right Operation is Associative, $\to$ is associative.
Hence {{Semigroup-axiom|1}} ... | Let $\struct {S, \to}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]] in which the [[Definition:Binary Operation|operation]] $\to$ is the [[Definition:Right Operation|right operation]].
Then $\struct {S, \to}$ is a [[Definition:Semigroup|semigroup]]. | We need to verify the [[Axiom:Semigroup Axioms|semigroup axioms]]:
{{:Axiom:Semigroup Axioms}}
By the nature of the [[Definition:Right Operation|right operation]], $\struct {S, \to}$ is [[Definition:Closed Algebraic Structure|closed]]:
:$\forall x, y \in S: x \to y = y \in S$
whatever $S$ may be.
Hence {{Semigroup-ax... | Structure under Right Operation is Semigroup | https://proofwiki.org/wiki/Structure_under_Right_Operation_is_Semigroup | https://proofwiki.org/wiki/Structure_under_Right_Operation_is_Semigroup | [
"Right Operation",
"Examples of Semigroups"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Right Operation",
"Definition:Semigroup"
] | [
"Axiom:Semigroup Axioms",
"Definition:Right Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Right Operation is Associative",
"Definition:Associative Operation",
"Definition:Semigroup"
] |
proofwiki-18767 | Structure under Left Operation is Semigroup | Let $\struct {S, \gets}$ be an algebraic structure in which the operation $\gets$ is the left operation.
Then $\struct {S, \gets}$ is a semigroup. | We need to verify the semigroup axioms:
{{:Axiom:Semigroup Axioms}}
By the nature of the right operation, $\struct {S, \to}$ is closed:
:$\forall x, y \in S: x \gets y = y \in S$
whatever $S$ may be.
Hence {{Semigroup-axiom|0}} holds.
From Right Operation is Associative, $\gets$ is associative.
Hence {{Semigroup-axiom|... | Let $\struct {S, \gets}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]] in which the [[Definition:Binary Operation|operation]] $\gets$ is the [[Definition:Left Operation|left operation]].
Then $\struct {S, \gets}$ is a [[Definition:Semigroup|semigroup]]. | We need to verify the [[Axiom:Semigroup Axioms|semigroup axioms]]:
{{:Axiom:Semigroup Axioms}}
By the nature of the [[Definition:Right Operation|right operation]], $\struct {S, \to}$ is [[Definition:Closed Algebraic Structure|closed]]:
:$\forall x, y \in S: x \gets y = y \in S$
whatever $S$ may be.
Hence {{Semigroup-... | Structure under Left Operation is Semigroup | https://proofwiki.org/wiki/Structure_under_Left_Operation_is_Semigroup | https://proofwiki.org/wiki/Structure_under_Left_Operation_is_Semigroup | [
"Left Operation",
"Examples of Semigroups"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Left Operation",
"Definition:Semigroup"
] | [
"Axiom:Semigroup Axioms",
"Definition:Right Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Right Operation is Associative",
"Definition:Associative Operation",
"Definition:Semigroup"
] |
proofwiki-18768 | Right Identity in Semigroup may not be Unique | Let $\struct {S, \circ}$ be a semigroup.
Let $e_R$ be a right identity of $\struct {S, \circ}$.
Then it is not necessarily the case that $e_R$ is unique. | ;Proof by Counterexample
Let $\struct {S, \gets}$ be an algebraic structure in which the operation $\gets$ is the left operation.
From Structure under Left Operation is Semigroup, $\struct {S, \gets}$ is a semigroup.
From Element under Left Operation is Right Identity, every element of $\struct {S, \gets}$ is a right i... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $e_R$ be a [[Definition:Right Identity|right identity]] of $\struct {S, \circ}$.
Then it is not necessarily the case that $e_R$ is [[Definition:Unique|unique]]. | ;[[Proof by Counterexample]]
Let $\struct {S, \gets}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]] in which the [[Definition:Binary Operation|operation]] $\gets$ is the [[Definition:Left Operation|left operation]].
From [[Structure under Left Operation is Semigroup]], $\struct {S, \... | Right Identity in Semigroup may not be Unique | https://proofwiki.org/wiki/Right_Identity_in_Semigroup_may_not_be_Unique | https://proofwiki.org/wiki/Right_Identity_in_Semigroup_may_not_be_Unique | [
"Identity Elements",
"Semigroups"
] | [
"Definition:Semigroup",
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Unique"
] | [
"Proof by Counterexample",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Left Operation",
"Structure under Left Operation is Semigroup",
"Definition:Semigroup",
"Element under Left Operation is Right Identity",
"Definition:Element",
"Definition:... |
proofwiki-18769 | Left Operation has no Left Identities | Let $S$ be a set with more than $1$ element.
Let $\struct {S, \gets}$ be an algebraic structure in which the operation $\gets$ is the left operation.
Then $\struct {S, \gets}$ has no left identities. | From Element under Left Operation is Right Identity, every element of $\struct {S, \gets}$ is a right identity.
Because there are at least $2$ elements in $\struct {S, \gets}$, it follows that $\struct {S, \gets}$ has more than one right identity.
From More than one Right Identity then no Left Identity, it follows that... | Let $S$ be a [[Definition:Set|set]] with more than $1$ [[Definition:Element|element]].
Let $\struct {S, \gets}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]] in which the [[Definition:Binary Operation|operation]] $\gets$ is the [[Definition:Left Operation|left operation]].
Then $\st... | From [[Element under Left Operation is Right Identity]], every [[Definition:Element|element]] of $\struct {S, \gets}$ is a [[Definition:Right Identity|right identity]].
Because there are at least $2$ [[Definition:Element|elements]] in $\struct {S, \gets}$, it follows that $\struct {S, \gets}$ has more than one [[Defin... | Left Operation has no Left Identities | https://proofwiki.org/wiki/Left_Operation_has_no_Left_Identities | https://proofwiki.org/wiki/Left_Operation_has_no_Left_Identities | [
"Identity Elements",
"Left Operation"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Left Operation",
"Definition:Identity (Abstract Algebra)/Left Identity"
] | [
"Element under Left Operation is Right Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Right Identity",
"More than one Right Identity then no Left Identity",
"Definition:Identity (Abstract Algebra)/Le... |
proofwiki-18770 | Right Operation has no Right Identities | Let $S$ be a set with more than $1$ element.
Let $\struct {S, \to}$ be an algebraic structure in which the operation $\to$ is the right operation.
Then $\struct {S, \to}$ has no right identities. | From Element under Right Operation is Left Identity, every element of $\struct {S, \to}$ is a left identity.
Because there are at least $2$ elements in $\struct {S, \to}$, it follows that $\struct {S, \to}$ has more than one left identity.
From More than one Left Identity then no Right Identity, it follows that $\struc... | Let $S$ be a [[Definition:Set|set]] with more than $1$ [[Definition:Element|element]].
Let $\struct {S, \to}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]] in which the [[Definition:Binary Operation|operation]] $\to$ is the [[Definition:Right Operation|right operation]].
Then $\stru... | From [[Element under Right Operation is Left Identity]], every [[Definition:Element|element]] of $\struct {S, \to}$ is a [[Definition:Left Identity|left identity]].
Because there are at least $2$ [[Definition:Element|elements]] in $\struct {S, \to}$, it follows that $\struct {S, \to}$ has more than one [[Definition:Le... | Right Operation has no Right Identities | https://proofwiki.org/wiki/Right_Operation_has_no_Right_Identities | https://proofwiki.org/wiki/Right_Operation_has_no_Right_Identities | [
"Identity Elements",
"Right Operation"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Right Operation",
"Definition:Identity (Abstract Algebra)/Right Identity"
] | [
"Element under Right Operation is Left Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Left Identity",
"More than one Left Identity then no Right Identity",
"Definition:Identity (Abstract Algebra)/Righ... |
proofwiki-18771 | Left Identity in Semigroup may not be Unique | Let $\struct {S, \circ}$ be a semigroup.
Let $e_L$ be a left identity of $\struct {S, \circ}$.
Then it is not necessarily the case that $e_L$ is unique. | ;Proof by Counterexample
Let $\struct {S, \gets}$ be an algebraic structure in which the operation $\to$ is the right operation.
From Structure under Right Operation is Semigroup, $\struct {S, \to}$ is a semigroup.
From Element under Right Operation is Left Identity, every element of $\struct {S, \to}$ is a left identi... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $e_L$ be a [[Definition:Left Identity|left identity]] of $\struct {S, \circ}$.
Then it is not necessarily the case that $e_L$ is [[Definition:Unique|unique]]. | ;[[Proof by Counterexample]]
Let $\struct {S, \gets}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]] in which the [[Definition:Binary Operation|operation]] $\to$ is the [[Definition:Right Operation|right operation]].
From [[Structure under Right Operation is Semigroup]], $\struct {S, ... | Left Identity in Semigroup may not be Unique | https://proofwiki.org/wiki/Left_Identity_in_Semigroup_may_not_be_Unique | https://proofwiki.org/wiki/Left_Identity_in_Semigroup_may_not_be_Unique | [
"Identity Elements",
"Semigroups"
] | [
"Definition:Semigroup",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Definition:Unique"
] | [
"Proof by Counterexample",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Right Operation",
"Structure under Right Operation is Semigroup",
"Definition:Semigroup",
"Element under Right Operation is Left Identity",
"Definition:Element",
"Definitio... |
proofwiki-18772 | Structure Induced by Permutation on Quasigroup is Quasigroup | Let $\struct {S, \circ}$ be a quasigroup.
Let $\sigma: S \to S$ be a permutation on $S$.
Let $\struct {S, \circ_\sigma}$ be the structure induced by $\sigma$ on $\circ$:
:$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$
Then $\struct {S, \circ_\sigma}$ is also a quasigroup. | By definition of quasigroup:
:$\forall a, b \in S: \exists ! x \in S: x \circ a = b$
:$\forall a, b \in S: \exists ! y \in S: a \circ y = b$
Let $a, b \in S$.
As $\sigma$ is a permutation, it is by definition both surjective and injective.
We have that:
:$\exists ! x: x \circ a = b$
Thus:
{{begin-eqn}}
{{eqn | q = \exi... | Let $\struct {S, \circ}$ be a [[Definition:Quasigroup|quasigroup]].
Let $\sigma: S \to S$ be a [[Definition:Permutation|permutation]] on $S$.
Let $\struct {S, \circ_\sigma}$ be the [[Definition:Structure Induced by Permutation|structure induced by $\sigma$ on $\circ$]]:
:$\forall x, y \in S: x \circ_\sigma y := \map ... | By definition of [[Definition:Quasigroup|quasigroup]]:
:$\forall a, b \in S: \exists ! x \in S: x \circ a = b$
:$\forall a, b \in S: \exists ! y \in S: a \circ y = b$
Let $a, b \in S$.
As $\sigma$ is a [[Definition:Permutation|permutation]], it is by definition both [[Definition:Surjection|surjective]] and [[Defini... | Structure Induced by Permutation on Quasigroup is Quasigroup | https://proofwiki.org/wiki/Structure_Induced_by_Permutation_on_Quasigroup_is_Quasigroup | https://proofwiki.org/wiki/Structure_Induced_by_Permutation_on_Quasigroup_is_Quasigroup | [
"Operations Induced by Permutations",
"Quasigroups"
] | [
"Definition:Quasigroup",
"Definition:Permutation",
"Definition:Structure Induced by Permutation",
"Definition:Quasigroup"
] | [
"Definition:Quasigroup",
"Definition:Permutation",
"Definition:Surjection",
"Definition:Injection"
] |
proofwiki-18773 | Structure Induced by Permutation on Algebra Loop is not necessarily Algebra Loop | Let $\struct {S, \circ}$ be an algebra loop.
Let $\sigma: S \to S$ be a permutation on $S$.
Let $\struct {S, \circ_\sigma}$ be the structure induced by $\sigma$ on $\circ$:
:$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$
Then $\struct {S, \circ_\sigma}$ is not necessarily also an algebra loop. | Consider the Cayley table of the algebra loop on $S = \set {e, a, b}$:
:$\begin{array}{r|rrr}
\circ & e & a & b \\
\hline
e & e & a & b
\\
a & a & b & e
\\
b & b & e & a
\\
\end{array}$
Consider the permutation on $S$:
Let $\sigma$ denote the permutation on $S$ defined as:
{{begin-eqn}}
{{eqn | l = \map \sigma e
... | Let $\struct {S, \circ}$ be an [[Definition:Algebra Loop|algebra loop]].
Let $\sigma: S \to S$ be a [[Definition:Permutation|permutation]] on $S$.
Let $\struct {S, \circ_\sigma}$ be the [[Definition:Structure Induced by Permutation|structure induced by $\sigma$ on $\circ$]]:
:$\forall x, y \in S: x \circ_\sigma y := ... | Consider the [[Definition:Cayley Table|Cayley table]] of the [[Definition:Algebra Loop|algebra loop]] on $S = \set {e, a, b}$:
:$\begin{array}{r|rrr}
\circ & e & a & b \\
\hline
e & e & a & b
\\
a & a & b & e
\\
b & b & e & a
\\
\end{array}$
Consider the [[Definition:Permutation|permutation]] on $S$:
Let $\sigma$ de... | Structure Induced by Permutation on Algebra Loop is not necessarily Algebra Loop | https://proofwiki.org/wiki/Structure_Induced_by_Permutation_on_Algebra_Loop_is_not_necessarily_Algebra_Loop | https://proofwiki.org/wiki/Structure_Induced_by_Permutation_on_Algebra_Loop_is_not_necessarily_Algebra_Loop | [
"Operations Induced by Permutations",
"Algebra Loops"
] | [
"Definition:Algebra Loop",
"Definition:Permutation",
"Definition:Structure Induced by Permutation",
"Definition:Algebra Loop"
] | [
"Definition:Cayley Table",
"Definition:Algebra Loop",
"Definition:Permutation",
"Definition:Permutation",
"Definition:Cayley Table",
"Definition:Structure Induced by Permutation",
"Definition:Cayley Table",
"Definition:Quasigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Defi... |
proofwiki-18774 | Structure Induced by Permutation on Commutative Quasigroup is Commutative Quasigroup | Let $\struct {S, \circ}$ be a quasigroup such that $\circ$ is a commutative operation.
Let $\sigma: S \to S$ be a permutation on $S$.
Let $\struct {S, \circ_\sigma}$ be the structure induced by $\sigma$ on $\circ$:
:$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$
Then $\struct {S, \circ_\sigma}$ is al... | From definition of a quasigroup, $\struct {S, \circ}$ is closed under $\circ$.
From Structure Induced by Permutation on Quasigroup is Quasigroup, $\struct {S, \circ_\sigma}$ is a quasigroup.
Again, from definition of a quasigroup, $\struct {S, \circ_\sigma}$ is closed under $\circ_\sigma$.
Hence:
{{begin-eqn}}
{{eqn | ... | Let $\struct {S, \circ}$ be a [[Definition:Quasigroup|quasigroup]] such that $\circ$ is a [[Definition:Commutative Operation|commutative operation]].
Let $\sigma: S \to S$ be a [[Definition:Permutation|permutation]] on $S$.
Let $\struct {S, \circ_\sigma}$ be the [[Definition:Structure Induced by Permutation|structure... | From [[Definition:Quasigroup|definition of a quasigroup]], $\struct {S, \circ}$ is [[Definition:Closure (Abstract Algebra)|closed]] under $\circ$.
From [[Structure Induced by Permutation on Quasigroup is Quasigroup]], $\struct {S, \circ_\sigma}$ is a [[Definition:Quasigroup|quasigroup]].
Again, from [[Definition:Quas... | Structure Induced by Permutation on Commutative Quasigroup is Commutative Quasigroup | https://proofwiki.org/wiki/Structure_Induced_by_Permutation_on_Commutative_Quasigroup_is_Commutative_Quasigroup | https://proofwiki.org/wiki/Structure_Induced_by_Permutation_on_Commutative_Quasigroup_is_Commutative_Quasigroup | [
"Operations Induced by Permutations",
"Quasigroups",
"Commutativity"
] | [
"Definition:Quasigroup",
"Definition:Commutative/Operation",
"Definition:Permutation",
"Definition:Structure Induced by Permutation",
"Definition:Quasigroup",
"Definition:Commutative/Operation"
] | [
"Definition:Quasigroup",
"Definition:Closure (Abstract Algebra)",
"Structure Induced by Permutation on Quasigroup is Quasigroup",
"Definition:Quasigroup",
"Definition:Quasigroup",
"Definition:Closure (Abstract Algebra)"
] |
proofwiki-18775 | Condition for Isomorphism between Structures Induced by Permutations | Let $S$ be a set.
Let $\oplus$ and $\otimes$ be closed operations on $S$ such that both $\oplus$ and $\otimes$ have the same identity.
Let $\sigma$ and $\tau$ be permutations on $S$.
Let $\oplus_\sigma$ and $\otimes_\tau$ be the operations on $S$ induced on $\oplus$ by $\sigma$ and on $\otimes$ by $\tau$ respectively:
... | Recall that:
:an isomorphism is a bijection which is a homomorphism
:a permutation is a bijection from a set to itself.
Hence on both sides of the double implication:
:$f$ is a permutation on $S$
:both $f \circ \sigma$ and $\tau \circ f$ are permutations on $S$.
So bijectivity of all relevant mappings can be taken for ... | Let $S$ be a [[Definition:Set|set]].
Let $\oplus$ and $\otimes$ be [[Definition:Closed Operation|closed]] [[Definition:Binary Operation|operations]] on $S$ such that both $\oplus$ and $\otimes$ have the same [[Definition:Identity Element|identity]].
Let $\sigma$ and $\tau$ be [[Definition:Permutation|permutations]] o... | Recall that:
:an [[Definition:Isomorphism (Abstract Algebra)|isomorphism]] is a [[Definition:Bijection|bijection]] which is a [[Definition:Homomorphism|homomorphism]]
:a [[Definition:Permutation|permutation]] is a [[Definition:Bijection|bijection]] from a [[Definition:Set|set]] to itself.
Hence on both sides of the ... | Condition for Isomorphism between Structures Induced by Permutations | https://proofwiki.org/wiki/Condition_for_Isomorphism_between_Structures_Induced_by_Permutations | https://proofwiki.org/wiki/Condition_for_Isomorphism_between_Structures_Induced_by_Permutations | [
"Operations Induced by Permutations",
"Isomorphisms (Abstract Algebra)"
] | [
"Definition:Set",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Permutation",
"Definition:Operation Induced by Permutation",
"Definition:Mapping",
"Definition:Isomorphism (A... | [
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Bijection",
"Definition:Homomorphism",
"Definition:Permutation",
"Definition:Bijection",
"Definition:Set",
"Definition:Permutation",
"Definition:Permutation",
"Definition:Bijection",
"Definition:Mapping",
"Definition:Isomorphism (Abstract ... |
proofwiki-18776 | Count of Commutative Quasigroups on Set given Count of Commutative Algebra Loops | Let $S$ be a finite set of cardinality $n$.
Let $e \in S$.
Let there be $m$ commutative operations $\oplus$ on $S$ such that $\struct {S, \oplus}$ is an algebra loop whose identity is $e$.
Then there are $n! m$ commutative binary operations $\otimes$ on $S$ such that $\struct {S, \otimes}$ is a quasigroup. | Consider a commutative algebra loop $\struct {S, \oplus}$ whose identity is $e$.
Consider the row of the Cayley table of $S$ which is headed by $e$.
There are $n!$ permutations of the elements of this row.
Each of these corresponds to a different commutative operation on $S$, as the corresponding column headed by $e$ i... | Let $S$ be a [[Definition:Finite Set|finite set]] of [[Definition:Cardinality|cardinality]] $n$.
Let $e \in S$.
Let there be $m$ [[Definition:Commutative Operation|commutative]] [[Definition:Binary Operation|operations]] $\oplus$ on $S$ such that $\struct {S, \oplus}$ is an [[Definition:Algebra Loop|algebra loop]] wh... | Consider a [[Definition:Commutative Operation|commutative]] [[Definition:Algebra Loop|algebra loop]] $\struct {S, \oplus}$ whose [[Definition:Identity Element|identity]] is $e$.
Consider the [[Definition:Row of Array|row]] of the [[Definition:Cayley Table|Cayley table]] of $S$ which is headed by $e$.
There are $n!$ [... | Count of Commutative Quasigroups on Set given Count of Commutative Algebra Loops | https://proofwiki.org/wiki/Count_of_Commutative_Quasigroups_on_Set_given_Count_of_Commutative_Algebra_Loops | https://proofwiki.org/wiki/Count_of_Commutative_Quasigroups_on_Set_given_Count_of_Commutative_Algebra_Loops | [
"Algebra Loops",
"Quasigroups",
"Count of Commutative Quasigroups on Set given Count of Commutative Algebra Loops"
] | [
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Commutative/Operation",
"Definition:Operation/Binary Operation",
"Definition:Algebra Loop",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutative/Operation",
"Definition:Operation/Binary Operation",
"Defini... | [
"Definition:Commutative/Operation",
"Definition:Algebra Loop",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Array/Row",
"Definition:Cayley Table",
"Definition:Permutation",
"Definition:Element",
"Definition:Array/Row",
"Definition:Commutative/Operation",
"Definition:Arr... |
proofwiki-18777 | Regular Representations wrt Element are Permutations then Element is Invertible | Let $\struct {S, \circ}$ be a semigroup.
Let $\lambda_a: S \to S$ and $\rho_a: S \to S$ be the left regular representation and right regular representation with respect to $a$ respectively:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\lambda_a} x
| r = a \circ x
}}
{{eqn | q = \forall x \in S
... | We have that $\rho_a$ is a permutation on $S$.
Hence:
{{begin-eqn}}
{{eqn | q = \exists g \in S
| l = a
| r = \map {\rho_a} g
| c =
}}
{{eqn | r = g \circ a
| c = {{Defof|Right Regular Representation}}
}}
{{end-eqn}}
Then we have:
{{begin-eqn}}
{{eqn | q = \forall b \in S
| l = \paren {b ... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $\lambda_a: S \to S$ and $\rho_a: S \to S$ be the [[Definition:Left Regular Representation|left regular representation]] and [[Definition:Right Regular Representation|right regular representation]] with respect to $a$ respectively:
{{begin-eqn}}
{{... | We have that $\rho_a$ is a [[Definition:Permutation|permutation]] on $S$.
Hence:
{{begin-eqn}}
{{eqn | q = \exists g \in S
| l = a
| r = \map {\rho_a} g
| c =
}}
{{eqn | r = g \circ a
| c = {{Defof|Right Regular Representation}}
}}
{{end-eqn}}
Then we have:
{{begin-eqn}}
{{eqn | q = \forall... | Regular Representations wrt Element are Permutations then Element is Invertible | https://proofwiki.org/wiki/Regular_Representations_wrt_Element_are_Permutations_then_Element_is_Invertible | https://proofwiki.org/wiki/Regular_Representations_wrt_Element_are_Permutations_then_Element_is_Invertible | [
"Regular Representations",
"Semigroups"
] | [
"Definition:Semigroup",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Permutation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Invertible Element"
] | [
"Definition:Permutation",
"Right Cancellable iff Right Regular Representation Injective",
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Permutation",
"Left Cancellable iff Left Regular Representation Injective",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Definition:I... |
proofwiki-18778 | Regular Representations in Semigroup are Permutations then Structure is Group | Let $\struct {S, \circ}$ be a semigroup.
For $a \in S$, let $\lambda_a: S \to S$ and $\rho_a: S \to S$ denote the left regular representation and right regular representation with respect to $a$ respectively:
{{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\lambda_a} x
| r = a \circ x
}}
{{eqn | q = \... | We have that $\lambda_a$ be a permutation on $S$ for all $a \in S$.
In particular this applies to $b$.
So:
:$\lambda_b$ is a permutation on $S$
:$\rho_b$ is a permutation on $S$
and so from Regular Representations wrt Element are Permutations then Element is Invertible:
:$\struct {S, \circ}$ has an identity element
:$b... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
For $a \in S$, let $\lambda_a: S \to S$ and $\rho_a: S \to S$ denote the [[Definition:Left Regular Representation|left regular representation]] and [[Definition:Right Regular Representation|right regular representation]] with respect to $a$ respectively... | We have that $\lambda_a$ be a [[Definition:Permutation|permutation]] on $S$ for all $a \in S$.
In particular this applies to $b$.
So:
:$\lambda_b$ is a [[Definition:Permutation|permutation]] on $S$
:$\rho_b$ is a [[Definition:Permutation|permutation]] on $S$
and so from [[Regular Representations wrt Element are Per... | Regular Representations in Semigroup are Permutations then Structure is Group | https://proofwiki.org/wiki/Regular_Representations_in_Semigroup_are_Permutations_then_Structure_is_Group | https://proofwiki.org/wiki/Regular_Representations_in_Semigroup_are_Permutations_then_Structure_is_Group | [
"Regular Representations",
"Semigroups",
"Group Theory"
] | [
"Definition:Semigroup",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Permutation",
"Definition:Permutation",
"Definition:Group"
] | [
"Definition:Permutation",
"Definition:Permutation",
"Definition:Permutation",
"Regular Representations wrt Element are Permutations then Element is Invertible",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Id... |
proofwiki-18779 | Structure is Group iff Semigroup and Quasigroup | Let $\struct {S, \circ}$ be an algebraic structure.
Then:
:$\struct {S, \circ}$ is a group
{{iff}}
:$\struct {S, \circ}$ is both a semigroup and a quasigroup. | === Sufficient Condition ===
Let $\struct {S, \circ}$ be a group.
Then a fortiori $\struct {S, \circ}$ is a semigroup.
From Regular Representations in Group are Permutations:
:for all $a \in S$, the left regular representation and the rightt regular representation are permutations of $S$.
Hence by definition $\struct {... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Then:
:$\struct {S, \circ}$ is a [[Definition:Group|group]]
{{iff}}
:$\struct {S, \circ}$ is both a [[Definition:Semigroup|semigroup]] and a [[Definition:Quasigroup|quasigroup]]. | === Sufficient Condition ===
Let $\struct {S, \circ}$ be a [[Definition:Group|group]].
Then [[Definition:A Fortiori|a fortiori]] $\struct {S, \circ}$ is a [[Definition:Semigroup|semigroup]].
From [[Regular Representations in Group are Permutations]]:
:for all $a \in S$, the [[Definition:Left Regular Representation|l... | Structure is Group iff Semigroup and Quasigroup | https://proofwiki.org/wiki/Structure_is_Group_iff_Semigroup_and_Quasigroup | https://proofwiki.org/wiki/Structure_is_Group_iff_Semigroup_and_Quasigroup | [
"Quasigroups",
"Semigroups",
"Group Theory"
] | [
"Definition:Algebraic Structure",
"Definition:Group",
"Definition:Semigroup",
"Definition:Quasigroup"
] | [
"Definition:Group",
"Definition:A Fortiori",
"Definition:Semigroup",
"Regular Representations in Group are Permutations",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Permutation",
"Definition:Quasigrou... |
proofwiki-18780 | Condition for Group given Semigroup with Idempotent Element | Let $\struct {S, \circ}$ be a semigroup.
Let there exist an idempotent element $e$ of $S$ such that for all $a \in S$:
:there exists at least one element $x$ of $S$ satisfying $x \circ a = e$
:there exists at most one element $y$ of $S$ satisfying $a \circ y = e$.
Then $\struct {S, \circ}$ is a group. | Let $a$ be arbitrary.
We have:
{{begin-eqn}}
{{eqn | q = \exists x \in S
| l = x \circ a
| r = e
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {x \circ a} \circ \paren {x \circ a}
| r = e \circ e
| c =
}}
{{eqn | r = e
| c = {{hypothesis}}: $e$ is idempotent
}}
{{eqn | ll= \lead... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let there exist an [[Definition:Idempotent Element|idempotent element]] $e$ of $S$ such that for all $a \in S$:
:there exists at least one [[Definition:Element|element]] $x$ of $S$ satisfying $x \circ a = e$
:there exists at most one [[Definition:Elemen... | Let $a$ be arbitrary.
We have:
{{begin-eqn}}
{{eqn | q = \exists x \in S
| l = x \circ a
| r = e
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {x \circ a} \circ \paren {x \circ a}
| r = e \circ e
| c =
}}
{{eqn | r = e
| c = {{hypothesis}}: $e$ is [[Definition:Idempotent Eleme... | Condition for Group given Semigroup with Idempotent Element | https://proofwiki.org/wiki/Condition_for_Group_given_Semigroup_with_Idempotent_Element | https://proofwiki.org/wiki/Condition_for_Group_given_Semigroup_with_Idempotent_Element | [
"Semigroups",
"Group Theory",
"Idempotence"
] | [
"Definition:Semigroup",
"Definition:Idempotence/Element",
"Definition:Element",
"Definition:Element",
"Definition:Group"
] | [
"Definition:Idempotence/Element",
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Definition:Inverse (Abstract Algebra)/Left Inverse",
"Definition:Algebraic Str... |
proofwiki-18781 | Existence and Uniqueness of Distributional Primitive | Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Then there exists a Schwartz distribution $S \in \map {\DD'} \R$ such that in the distributional sense:
:$S' = T$
Furthermore, $S$ is unique up to an arbitrary constant. | Let $\mathbf 0 : \R \to 0$ be the zero mapping.
Let $\phi_0 \in \map \DD \R \setminus \set {\mathbf 0}$ be a test function.
Let $\psi \in \map \DD \R$ be a test function.
Let $\phi : \R \to \R$ be a real function such that:
:$\ds \phi := \psi - \frac {\int_{-\infty}^\infty \map \psi x \rd x} {\int_{-\infty}^\infty \map... | Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Then there exists a [[Definition:Schwartz Distribution|Schwartz distribution]] $S \in \map {\DD'} \R$ such that in the [[Definition:Distributional Derivative|distributional]] sense:
:$S' = T$
Furthermore, $S$ is unique up to... | Let $\mathbf 0 : \R \to 0$ be the [[Definition:Zero Mapping|zero mapping]].
Let $\phi_0 \in \map \DD \R \setminus \set {\mathbf 0}$ be a [[Definition:Test Function|test function]].
Let $\psi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
Let $\phi : \R \to \R$ be a [[Definition:Real Function|real ... | Existence and Uniqueness of Distributional Primitive | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Distributional_Primitive | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Distributional_Primitive | [
"Distributional Derivatives"
] | [
"Definition:Schwartz Distribution",
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative",
"Definition:Arbitrary Constant"
] | [
"Definition:Zero Mapping",
"Definition:Test Function",
"Definition:Test Function",
"Definition:Real Function",
"Definition:Test Function",
"Definition:Test Function"
] |
proofwiki-18782 | Conditions under which Commutative Semigroup is Group | Suppose the following:
{{:Conditions under which Commutative Semigroup is Group/Statement of Conditions}}
Then $\struct {S, \circ}$ is a group. | Some lemmata: | Suppose [[Conditions under which Commutative Semigroup is Group/Statement of Conditions|the following]]:
{{:Conditions under which Commutative Semigroup is Group/Statement of Conditions}}
Then $\struct {S, \circ}$ is a [[Definition:Group|group]]. | Some [[Definition:Lemma|lemmata]]: | Conditions under which Commutative Semigroup is Group | https://proofwiki.org/wiki/Conditions_under_which_Commutative_Semigroup_is_Group | https://proofwiki.org/wiki/Conditions_under_which_Commutative_Semigroup_is_Group | [
"Conditions under which Commutative Semigroup is Group",
"Group Theory",
"Commutative Semigroups"
] | [
"Conditions under which Commutative Semigroup is Group/Statement of Conditions",
"Definition:Group"
] | [
"Definition:Lemma"
] |
proofwiki-18783 | Operation is Right Operation iff Anticommutative with Left Cancellable Element | Let $\struct {S, \circ}$ be a semigroup.
Then:
:$\circ$ is the right operation
{{iff}}:
:$\circ$ is anticommutative and has a left cancellable element. | === Sufficient Condition ===
Let $\circ$ be the right operation.
Then from Right Operation is Anticommutative we have that $\circ$ is anticommutative.
Let $x \in S$ be arbitrary.
Let $y, z \in S$ such that:
:$x \circ z = x \circ y$
Then:
{{begin-eqn}}
{{eqn | l = x \circ z
| r = z
| c = {{Defof|Right Operat... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Then:
:$\circ$ is the [[Definition:Right Operation|right operation]]
{{iff}}:
:$\circ$ is [[Definition:Anticommutative Structure with One Operation|anticommutative]] and has a [[Definition:Left Cancellable Element|left cancellable element]]. | === Sufficient Condition ===
Let $\circ$ be the [[Definition:Right Operation|right operation]].
Then from [[Right Operation is Anticommutative]] we have that $\circ$ is [[Definition:Anticommutative Structure with One Operation|anticommutative]].
Let $x \in S$ be arbitrary.
Let $y, z \in S$ such that:
:$x \circ z =... | Operation is Right Operation iff Anticommutative with Left Cancellable Element | https://proofwiki.org/wiki/Operation_is_Right_Operation_iff_Anticommutative_with_Left_Cancellable_Element | https://proofwiki.org/wiki/Operation_is_Right_Operation_iff_Anticommutative_with_Left_Cancellable_Element | [
"Right Operation",
"Anticommutativity",
"Cancellability"
] | [
"Definition:Semigroup",
"Definition:Right Operation",
"Definition:Anticommutative/Structure with One Operation",
"Definition:Cancellable Element/Left Cancellable"
] | [
"Definition:Right Operation",
"Right Operation is Anticommutative",
"Definition:Anticommutative/Structure with One Operation",
"Definition:Cancellable Element/Left Cancellable",
"Definition:Anticommutative/Structure with One Operation",
"Definition:Cancellable Element/Left Cancellable",
"Definition:Anti... |
proofwiki-18784 | Operation is Left Operation iff Anticommutative with Right Cancellable Element | Let $\struct {S, \circ}$ be an semigroup.
Then:
:$\circ$ is the left operation
{{iff}}:
:$\circ$ is anticommutative and has a right cancellable element. | === Sufficient Condition ===
Let $\circ$ be the left operation.
Then from Left Operation is Anticommutative we have that $\circ$ is anticommutative.
Let $x \in S$ be arbitrary.
Let $y, z \in S$ such that:
:$z \circ x = y \circ x$
Then:
{{begin-eqn}}
{{eqn | l = z \circ x
| r = z
| c = {{Defof|Left Operation... | Let $\struct {S, \circ}$ be an [[Definition:Semigroup|semigroup]].
Then:
:$\circ$ is the [[Definition:Left Operation|left operation]]
{{iff}}:
:$\circ$ is [[Definition:Anticommutative Structure with One Operation|anticommutative]] and has a [[Definition:Right Cancellable Element|right cancellable element]]. | === Sufficient Condition ===
Let $\circ$ be the [[Definition:Left Operation|left operation]].
Then from [[Left Operation is Anticommutative]] we have that $\circ$ is [[Definition:Anticommutative Structure with One Operation|anticommutative]].
Let $x \in S$ be arbitrary.
Let $y, z \in S$ such that:
:$z \circ x = y ... | Operation is Left Operation iff Anticommutative with Right Cancellable Element | https://proofwiki.org/wiki/Operation_is_Left_Operation_iff_Anticommutative_with_Right_Cancellable_Element | https://proofwiki.org/wiki/Operation_is_Left_Operation_iff_Anticommutative_with_Right_Cancellable_Element | [
"Left Operation",
"Anticommutativity",
"Cancellability"
] | [
"Definition:Semigroup",
"Definition:Left Operation",
"Definition:Anticommutative/Structure with One Operation",
"Definition:Cancellable Element/Right Cancellable"
] | [
"Definition:Left Operation",
"Left Operation is Anticommutative",
"Definition:Anticommutative/Structure with One Operation",
"Definition:Cancellable Element/Right Cancellable",
"Definition:Anticommutative/Structure with One Operation",
"Definition:Cancellable Element/Right Cancellable",
"Definition:Anti... |
proofwiki-18785 | Count of Operations on Finite Set which are Closed on Every Subset | Let $S$ be a finite set with $n$ elements.
There are $2^{n \paren {n - 1} }$ binary operations on $S$ which are closed on all subsets of $S$. | Suppose $\circ$ is a binary operation on $S$ such that $a \circ b = c$ for $a \ne b \ne c \ne a$.
Then $\circ$ is not closed on $\set {a, b} \subseteq S$.
Similarly, suppose $\circ$ is a binary operation on $S$ such that $a \circ a = b$ for $a \ne b$.
Then $\circ$ is not closed on $\set a \subseteq S$.
Hence for $\circ... | Let $S$ be a [[Definition:Finite Set|finite set]] with $n$ [[Definition:Element|elements]].
There are $2^{n \paren {n - 1} }$ [[Definition:Binary Operation|binary operations]] on $S$ which are [[Definition:Closed Operation|closed]] on all [[Definition:Subset|subsets]] of $S$. | Suppose $\circ$ is a [[Definition:Binary Operation|binary operation]] on $S$ such that $a \circ b = c$ for $a \ne b \ne c \ne a$.
Then $\circ$ is not [[Definition:Closed Operation|closed]] on $\set {a, b} \subseteq S$.
Similarly, suppose $\circ$ is a [[Definition:Binary Operation|binary operation]] on $S$ such that $... | Count of Operations on Finite Set which are Closed on Every Subset | https://proofwiki.org/wiki/Count_of_Operations_on_Finite_Set_which_are_Closed_on_Every_Subset | https://proofwiki.org/wiki/Count_of_Operations_on_Finite_Set_which_are_Closed_on_Every_Subset | [
"Closed Algebraic Structures"
] | [
"Definition:Finite Set",
"Definition:Element",
"Definition:Operation/Binary Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Subset"
] | [
"Definition:Operation/Binary Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Subset",
"Definition:Ordere... |
proofwiki-18786 | Count of Commutative Operations on Finite Set which are Closed on Every Subset | Let $S$ be a finite set with $n$ elements.
There are $2^{n \paren {n - 1} / 2}$ commutative binary operations on $S$ which are closed on all subsets of $S$. | From Count of Operations on Finite Set which are Closed on Every Subset, there are $2^{n \paren {n - 1} }$ binary operations on $S$ which are closed on all subsets of $S$.
For each $a, b \in S$, we have that
:there are $2$ possible products for $a \circ b$
:there are $2$ possible products for $b \circ a$
and so on the ... | Let $S$ be a [[Definition:Finite Set|finite set]] with $n$ [[Definition:Element|elements]].
There are $2^{n \paren {n - 1} / 2}$ [[Definition:Commutative Operation|commutative]] [[Definition:Binary Operation|binary operations]] on $S$ which are [[Definition:Closed Operation|closed]] on all [[Definition:Subset|subsets]... | From [[Count of Operations on Finite Set which are Closed on Every Subset]], there are $2^{n \paren {n - 1} }$ [[Definition:Binary Operation|binary operations]] on $S$ which are [[Definition:Closed Operation|closed]] on all [[Definition:Subset|subsets]] of $S$.
For each $a, b \in S$, we have that
:there are $2$ possib... | Count of Commutative Operations on Finite Set which are Closed on Every Subset | https://proofwiki.org/wiki/Count_of_Commutative_Operations_on_Finite_Set_which_are_Closed_on_Every_Subset | https://proofwiki.org/wiki/Count_of_Commutative_Operations_on_Finite_Set_which_are_Closed_on_Every_Subset | [
"Closed Algebraic Structures",
"Commutativity"
] | [
"Definition:Finite Set",
"Definition:Element",
"Definition:Commutative/Operation",
"Definition:Operation/Binary Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Subset"
] | [
"Count of Operations on Finite Set which are Closed on Every Subset",
"Definition:Operation/Binary Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Subset",
"Definition:Product (Abstract Algebra)",
"Definition:Product (Abstract Algebra)",
"Definition:Set",
"Definiti... |
proofwiki-18787 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 4 | :$\forall n \in \N: r_n = d_{n + 1} b + p r_{n + 1}$ | We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = \dfrac a b
| r = A_n + p^{n + 1} \dfrac {r_n} b
| c = {{hypothesis}}
}}
{{eqn | r = A_{n + 1} + p^{n + 2} \dfrac {r_{n + 1} } b
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| q = \forall n \in \N
| l = A_{n + 1} + p^{n + 2} ... | :$\forall n \in \N: r_n = d_{n + 1} b + p r_{n + 1}$ | We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N
| l = \dfrac a b
| r = A_n + p^{n + 1} \dfrac {r_n} b
| c = {{hypothesis}}
}}
{{eqn | r = A_{n + 1} + p^{n + 2} \dfrac {r_{n + 1} } b
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| q = \forall n \in \N
| l = A_{n + 1} + p^{n + 2}... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 4 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_4 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_4 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Characterization of Rational P-adic Integer",
"Prime not Divisor implies Coprime",
"Definition:Coprime/Integers",
"Integer Coprime to all Factors is Coprime to Whole",
"Definition:Coprime/Integers",
"Euclid's Lemma",
"Definition:P-adically Coherent Sequence",
"Definition:Sequence",
"Definition:P-ad... |
proofwiki-18788 | Unit Cylinder as Surface of Revolution | Let $\struct {\R^3, d}$ be the Euclidean space.
Let $S_C \subseteq \R^3$ be the surface of revolution.
Let $C$ be a straight line in the open upper half-plane.
Let the smooth local parametrization of $C$ be:
:$\map \gamma t = \tuple {t, 1}$
Then the induced metric on $S_C$ is:
:$g = d t^2 + d \theta^2$ | We have that:
:$\map {\gamma'} t = \tuple {1, 0}$
Hence, $\map \gamma t$ is a unit-speed curve.
By the corollary of the induced metric on the surface of revolution:
:$g = d t^2 + d \theta^2$
{{qed}} | Let $\struct {\R^3, d}$ be the [[Definition:Euclidean Space|Euclidean space]].
Let $S_C \subseteq \R^3$ be the [[Definition:Surface of Revolution|surface of revolution]].
Let $C$ be a [[Definition:Straight Line|straight line]] in the [[Definition:Open Upper Half-Plane|open upper half-plane]].
Let the [[Definition:Sm... | We have that:
:$\map {\gamma'} t = \tuple {1, 0}$
Hence, $\map \gamma t$ is a [[Definition:Unit-Speed Curve|unit-speed curve]].
By [[Induced Metric on Surface of Revolution/Corollary|the corollary of the induced metric on the surface of revolution]]:
:$g = d t^2 + d \theta^2$
{{qed}} | Unit Cylinder as Surface of Revolution | https://proofwiki.org/wiki/Unit_Cylinder_as_Surface_of_Revolution | https://proofwiki.org/wiki/Unit_Cylinder_as_Surface_of_Revolution | [
"Examples of Surfaces of Revolution",
"Induced Metrics"
] | [
"Definition:Euclidean Space",
"Definition:Surface of Revolution",
"Definition:Line/Straight Line",
"Definition:Half-Plane/Open/Upper",
"Definition:Smooth Local Parametrization",
"Definition:Induced Metric on Submanifold"
] | [
"Definition:Unit-Speed Curve",
"Induced Metric on Surface of Revolution/Corollary"
] |
proofwiki-18789 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 2 | :$\exists n_0 \in \N : \forall n \ge n_0 : -b \le r_n \le 0$ | === Lemma 11 ===
{{:Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 11}}{{qed|lemma}} | :$\exists n_0 \in \N : \forall n \ge n_0 : -b \le r_n \le 0$ | === [[Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 11|Lemma 11]] ===
{{:Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 11}}{{qed|lemma}} | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 2 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_2 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_2 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 11"
] |
proofwiki-18790 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 3 | :$\ds \lim_{n \mathop \to \infty} A_n = \dfrac a b$ | Let $\epsilon \in \R_{> 0}$.
Let $M = \max \set {\norm{r_0}_p, \norm{r_1}_p, \ldots, \norm{r_{n_0}}_p, \norm{-1}_p, \norm{-2}_p, \ldots, \norm{-b}_p}$
From Power Function is Unbounded Above:
:$\exists N \in \N: p^{N+1} > \dfrac M {\epsilon \norm b_p}$
We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N: n \ge N
... | :$\ds \lim_{n \mathop \to \infty} A_n = \dfrac a b$ | Let $\epsilon \in \R_{> 0}$.
Let $M = \max \set {\norm{r_0}_p, \norm{r_1}_p, \ldots, \norm{r_{n_0}}_p, \norm{-1}_p, \norm{-2}_p, \ldots, \norm{-b}_p}$
From [[Power Function is Unbounded Above]]:
:$\exists N \in \N: p^{N+1} > \dfrac M {\epsilon \norm b_p}$
We have:
{{begin-eqn}}
{{eqn | q = \forall n \in \N: n \ge... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 3 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_3 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_3 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Limit at Infinity of x^n",
"Power Function on Base between Zero and One is Strictly Decreasing",
"Definition:Convergent Sequence/Normed Division Ring",
"Category:Canonical P-adic Expansion of Rational is Eventually Periodic"
] |
proofwiki-18791 | Powerset of Subset is Closed under Union | Let $S$ be a set.
Let $T \subseteq S$ be a subset of $S$.
Let $\powerset S$ denote the power set of $S$.
Then $\powerset T$ is a closed subset of $\powerset S$ under set union:
:$\forall A, B \in \powerset T: A \cup B \in \powerset T$ | A direct application of Power Set is Closed under Union.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $\powerset S$ denote the [[Definition:Power Set|power set]] of $S$.
Then $\powerset T$ is a [[Definition:Submagma|closed subset]] of $\powerset S$ under [[Definition:Set Union|set union]]:
:$\forall A, B \in \pow... | A direct application of [[Power Set is Closed under Union]].
{{qed}} | Powerset of Subset is Closed under Union | https://proofwiki.org/wiki/Powerset_of_Subset_is_Closed_under_Union | https://proofwiki.org/wiki/Powerset_of_Subset_is_Closed_under_Union | [
"Closed Algebraic Structures",
"Power Set",
"Set Union",
"Subsets"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Submagma",
"Definition:Set Union"
] | [
"Power Set is Closed under Union"
] |
proofwiki-18792 | Powerset of Subset is Closed under Intersection | Let $S$ be a set.
Let $T \subseteq S$ be a subset of $S$.
Let $\powerset S$ denote the power set of $S$.
Then $\powerset T$ is a closed subset of $\powerset S$ under set intersection:
:$\forall A, B \in \powerset T: A \cap B \in \powerset T$ | A direct application of Power Set is Closed under Intersection.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $\powerset S$ denote the [[Definition:Power Set|power set]] of $S$.
Then $\powerset T$ is a [[Definition:Submagma|closed subset]] of $\powerset S$ under [[Definition:Set Intersection|set intersection]]:
:$\forall... | A direct application of [[Power Set is Closed under Intersection]].
{{qed}} | Powerset of Subset is Closed under Intersection | https://proofwiki.org/wiki/Powerset_of_Subset_is_Closed_under_Intersection | https://proofwiki.org/wiki/Powerset_of_Subset_is_Closed_under_Intersection | [
"Closed Algebraic Structures",
"Power Set",
"Set Intersection",
"Subsets"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Submagma",
"Definition:Set Intersection"
] | [
"Power Set is Closed under Intersection"
] |
proofwiki-18793 | Powerset of Subset is Closed under Symmetric Difference | Let $S$ be a set.
Let $T \subseteq S$ be a subset of $S$.
Let $\powerset S$ denote the power set of $S$.
Then $\powerset T$ is a closed subset of $\powerset S$ under set intersection:
:$\forall A, B \in \powerset T: A \symdif B \in \powerset T$
where $\symdif$ denotes symmetric difference. | A direct application of Power Set is Closed under Symmetric Difference.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $\powerset S$ denote the [[Definition:Power Set|power set]] of $S$.
Then $\powerset T$ is a [[Definition:Submagma|closed subset]] of $\powerset S$ under [[Definition:Set Intersection|set intersection]]:
:$\forall... | A direct application of [[Power Set is Closed under Symmetric Difference]].
{{qed}} | Powerset of Subset is Closed under Symmetric Difference | https://proofwiki.org/wiki/Powerset_of_Subset_is_Closed_under_Symmetric_Difference | https://proofwiki.org/wiki/Powerset_of_Subset_is_Closed_under_Symmetric_Difference | [
"Closed Algebraic Structures",
"Power Set",
"Symmetric Difference",
"Subsets"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Submagma",
"Definition:Set Intersection",
"Definition:Symmetric Difference"
] | [
"Power Set is Closed under Symmetric Difference"
] |
proofwiki-18794 | Dipper Semigroup is Commutative Semigroup | The dipper semigroup is a commutative semigroup. | Recall the definition of the '''dipper semigroup''':
Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
Let $+_{m, n}$ be the dipper operation on $\N_{< \par... | The [[Definition:Dipper Semigroup|dipper semigroup]] is a [[Definition:Commutative Semigroup|commutative semigroup]]. | Recall the definition of the '''[[Definition:Dipper Semigroup|dipper semigroup]]''':
Let $m, n \in \Z$ be [[Definition:Integer|integers]] such that $m \ge 0, n > 0$.
Let $\N_{< \paren {m \mathop + n} }$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment]] of the [[Definition:Natural... | Dipper Semigroup is Commutative Semigroup | https://proofwiki.org/wiki/Dipper_Semigroup_is_Commutative_Semigroup | https://proofwiki.org/wiki/Dipper_Semigroup_is_Commutative_Semigroup | [
"Dipper Semigroups",
"Examples of Commutative Semigroups"
] | [
"Definition:Dipper Semigroup",
"Definition:Commutative Semigroup"
] | [
"Definition:Dipper Semigroup",
"Definition:Integer",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Natural Numbers",
"Definition:Dipper Operation",
"Definition:Integer",
"Axiom:Semigroup Axioms",
"Axiom:Semigroup Axioms"
] |
proofwiki-18795 | Weak Solution to Dx u = u | thumbright$\map u {x, t} = \map H t e^x$
Let $H$ be the Heaviside step function.
Let $\map u {x, t} = \map H t e^x$
Then $u$ is a weak solution of the partial differential equation $\ds \dfrac {\partial u} {\partial x} = u$.
That is, for the Schwartz distribution $T_u \in \map {\DD'} {\R^2}$ associated with $u$ in the ... | Let $\phi \in \map \DD {\R^2}$ be a test function.
Then in the distributional sense we have that:
{{begin-eqn}}
{{eqn | l = \paren {\dfrac \partial {\partial x} - 1} \map {T_u} \phi
| r = -\map {T_u} {\dfrac {\partial \phi} {\partial x} } - \map {T_u} \phi
| c = {{Defof|Distributional Partial Derivative}}
}... | [[File:U(x,t)=exp(x)H(t).png|thumb|right|$\map u {x, t} = \map H t e^x$]]
Let $H$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $\map u {x, t} = \map H t e^x$
Then $u$ is a [[Definition:Weak Solution|weak solution]] of the [[Definition:Partial Differential Equation|partial differential eq... | Let $\phi \in \map \DD {\R^2}$ be a [[Definition:Test Function|test function]].
Then in the [[Definition:Distributional Derivative|distributional sense]] we have that:
{{begin-eqn}}
{{eqn | l = \paren {\dfrac \partial {\partial x} - 1} \map {T_u} \phi
| r = -\map {T_u} {\dfrac {\partial \phi} {\partial x} } - ... | Weak Solution to Dx u = u | https://proofwiki.org/wiki/Weak_Solution_to_Dx_u_=_u | https://proofwiki.org/wiki/Weak_Solution_to_Dx_u_=_u | [
"Examples of Distributional Solutions",
"Examples of Weak Solutions"
] | [
"File:U(x,t)=exp(x)H(t).png",
"Definition:Heaviside Step Function",
"Definition:Differential Equation/Solution/Weak Solution",
"Definition:Differential Equation/Partial",
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative"
] | [
"Definition:Test Function",
"Definition:Distributional Derivative",
"Product Rule for Derivatives",
"Fundamental Theorem of Calculus/Second Part"
] |
proofwiki-18796 | Existence of Subgroup of Dipper Semigroup | Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$.
Let $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ denote the dipper semigroup.
Consider the subset $H \subseteq N_{< \paren {m \mathop + n} }$ defined as:
:$H = \set {k \in \N: m \le k < m + n} = \set {m, m + 1, \ldots, m + n - 1}$
Then the substructure $... | Recall the definition of the '''dipper semigroup''':
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
Let $+_{m, n}$ be the dipper operation on $\N_{< \paren {m \mathop + n} }$:
:$\forall a, b \in \N_{< \paren {m ... | Let $m, n \in \Z$ be [[Definition:Integer|integers]] such that $m \ge 0, n > 0$.
Let $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ denote the [[Definition:Dipper Semigroup|dipper semigroup]].
Consider the [[Definition:Subset|subset]] $H \subseteq N_{< \paren {m \mathop + n} }$ defined as:
:$H = \set {k \in \N:... | Recall the definition of the '''[[Definition:Dipper Semigroup|dipper semigroup]]''':
Let $\N_{< \paren {m \mathop + n} }$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment]] of the [[Definition:Natural Numbers|natural numbers]]:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots... | Existence of Subgroup of Dipper Semigroup | https://proofwiki.org/wiki/Existence_of_Subgroup_of_Dipper_Semigroup | https://proofwiki.org/wiki/Existence_of_Subgroup_of_Dipper_Semigroup | [
"Dipper Semigroups",
"Examples of Subgroups",
"Existence of Subgroup of Dipper Semigroup"
] | [
"Definition:Integer",
"Definition:Dipper Semigroup",
"Definition:Subset",
"Definition:Algebraic Substructure",
"Definition:Subgroup"
] | [
"Definition:Dipper Semigroup",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Natural Numbers",
"Definition:Dipper Operation",
"Definition:Integer",
"Axiom:Group Axioms",
"Axiom:Group Axioms"
] |
proofwiki-18797 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 5 | :$\exists \mathop m, l \in \N : \forall n \ge m: r_n = r_{n + l}$ and $d_n = d_{n + l}$ | We have {{hypothesis}} that the set $\set {r_n : n \in \N}$ of values of $\sequence {r_n}$ is a subset of:
:$\set {r_0, r_1, \ldots, r_{n_0} } \cup \set {-b, -b + 1, -b + 2, \ldots, 2, 1, 0}$
It follows that $\set {r_n : n \in \N}$ takes only finitely many values.
Hence:
:$\exists m_0, l \in \N : l > 0 : r_{m_0} = r_{ ... | :$\exists \mathop m, l \in \N : \forall n \ge m: r_n = r_{n + l}$ and $d_n = d_{n + l}$ | We have {{hypothesis}} that the [[Definition:Set|set]] $\set {r_n : n \in \N}$ of values of $\sequence {r_n}$ is a [[Definition:Subset|subset]] of:
:$\set {r_0, r_1, \ldots, r_{n_0} } \cup \set {-b, -b + 1, -b + 2, \ldots, 2, 1, 0}$
It follows that $\set {r_n : n \in \N}$ takes only [[Definition:Finite Set|finitely ma... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 5 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_5 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_5 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Definition:Set",
"Definition:Subset",
"Definition:Finite Set"
] |
proofwiki-18798 | If n is Triangular then so is (2m+1)^2 n + m(m+1)/2 | Let $n$ be a triangular number.
Let $m \in \Z_{\ge 0}$ be a positive integer.
Then $\paren {2 m + 1}^2 n + \dfrac {m \paren {m + 1} } 2$ is also a triangular number. | Let $n$ be a triangular number.
Then:
:$\exists k \in \Z: n = \dfrac {k \paren {k + 1} } 2$
So:
{{begin-eqn}}
{{eqn | l = \paren {2 m + 1}^2 n + \dfrac {m \paren {m + 1} } 2
| r = \paren {2 m + 1}^2 \dfrac {k \paren {k + 1} } 2 + \dfrac {m \paren {m + 1} } 2
| c =
}}
{{eqn | r = \dfrac {\paren {2 m + 1}^2 ... | Let $n$ be a [[Definition:Triangular Number|triangular number]].
Let $m \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then $\paren {2 m + 1}^2 n + \dfrac {m \paren {m + 1} } 2$ is also a [[Definition:Triangular Number|triangular number]]. | Let $n$ be a [[Definition:Triangular Number|triangular number]].
Then:
:$\exists k \in \Z: n = \dfrac {k \paren {k + 1} } 2$
So:
{{begin-eqn}}
{{eqn | l = \paren {2 m + 1}^2 n + \dfrac {m \paren {m + 1} } 2
| r = \paren {2 m + 1}^2 \dfrac {k \paren {k + 1} } 2 + \dfrac {m \paren {m + 1} } 2
| c =
}}
{{e... | If n is Triangular then so is (2m+1)^2 n + m(m+1)/2 | https://proofwiki.org/wiki/If_n_is_Triangular_then_so_is_(2m+1)^2_n_+_m(m+1)/2 | https://proofwiki.org/wiki/If_n_is_Triangular_then_so_is_(2m+1)^2_n_+_m(m+1)/2 | [
"Triangular Numbers"
] | [
"Definition:Triangular Number",
"Definition:Positive/Integer",
"Definition:Triangular Number"
] | [
"Definition:Triangular Number",
"Category:Triangular Numbers"
] |
proofwiki-18799 | Identity of Subgroup of Dipper Semigroup is not Identity of Dipper | Let $m, n \in \Z$ be integers such that $m, n > 0$.
Let $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ denote the dipper semigroup.
Let $\struct {H, +_{m, n} }$ be the subgroup of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ where $H = \set {k \in \N: m \le k < m + n}$
Then the identity of $\struct {H, +_... | This is demonstrated by Proof by Counterexample.
First we note that by Existence of Subgroup of Dipper Semigroup:
:$\struct {H, +_{m, n} }$ is indeed a subgroup of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$
:the identity of $\struct {H, +_{m, n} }$ is $n$.
But we note that by definition of $+_{m, n}$:
:$0 +_{... | Let $m, n \in \Z$ be [[Definition:Integer|integers]] such that $m, n > 0$.
Let $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ denote the [[Definition:Dipper Semigroup|dipper semigroup]].
Let $\struct {H, +_{m, n} }$ be the [[Definition:Subgroup|subgroup]] of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$... | This is demonstrated by [[Proof by Counterexample]].
First we note that by [[Existence of Subgroup of Dipper Semigroup]]:
:$\struct {H, +_{m, n} }$ is indeed a [[Definition:Subgroup|subgroup]] of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$
:the [[Definition:Identity Element|identity]] of $\struct {H, +_{m, n}... | Identity of Subgroup of Dipper Semigroup is not Identity of Dipper | https://proofwiki.org/wiki/Identity_of_Subgroup_of_Dipper_Semigroup_is_not_Identity_of_Dipper | https://proofwiki.org/wiki/Identity_of_Subgroup_of_Dipper_Semigroup_is_not_Identity_of_Dipper | [
"Dipper Semigroups",
"Examples of Identity Elements",
"Identity of Subgroup of Dipper Semigroup is not Identity of Dipper"
] | [
"Definition:Integer",
"Definition:Dipper Semigroup",
"Definition:Subgroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Proof by Counterexample",
"Existence of Subgroup of Dipper Semigroup",
"Definition:Subgroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
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