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-19100 | Subspace of Complete Metric Space is Relatively Compact iff Every Sequence has Cauchy Subsequence | Let $\struct {X, d}$ be a complete metric space.
Let $\struct {H, d_H}$ be a metric subspace of $\struct {X, d}$.
Then $\struct {H, d_H}$ is relatively compact {{iff}}:
:every sequence in $H$ has a Cauchy subsequence. | === Necessary Condition ===
Suppose that:
:$\struct {H, d_H}$ is relatively compact
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $H$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is a sequence in $\map \cl H$, where $\map \cl H$ is the closure of $H$ in $\struct {X, d}$.
Since $\struct {H, d_H}$ is relativel... | Let $\struct {X, d}$ be a [[Definition:Complete Metric Space|complete metric space]].
Let $\struct {H, d_H}$ be a [[Definition:Metric Subspace|metric subspace]] of $\struct {X, d}$.
Then $\struct {H, d_H}$ is [[Definition:Relatively Compact Subspace|relatively compact]] {{iff}}:
:every [[Definition:Sequence|sequen... | === Necessary Condition ===
Suppose that:
:$\struct {H, d_H}$ is [[Definition:Relatively Compact Subspace|relatively compact]]
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $H$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is a [[Definition:Sequence|sequence]] in $\map \cl H$, wher... | Subspace of Complete Metric Space is Relatively Compact iff Every Sequence has Cauchy Subsequence | https://proofwiki.org/wiki/Subspace_of_Complete_Metric_Space_is_Relatively_Compact_iff_Every_Sequence_has_Cauchy_Subsequence | https://proofwiki.org/wiki/Subspace_of_Complete_Metric_Space_is_Relatively_Compact_iff_Every_Sequence_has_Cauchy_Subsequence | [
"Relatively Compact Subspaces",
"Complete Metric Spaces",
"Cauchy Sequences"
] | [
"Definition:Complete Metric Space",
"Definition:Metric Subspace",
"Definition:Relatively Compact Subspace",
"Definition:Sequence",
"Definition:Cauchy Sequence"
] | [
"Definition:Relatively Compact Subspace",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Closure (Topology)/Metric Space",
"Definition:Relatively Compact Subspace",
"Definition:Compact Space/Metric Space",
"Compact Subspace of Metric Space is Sequentially Compact in Itself",
"Definition:Seq... |
proofwiki-19101 | Power Law Delta Sequence | thumb600pxThe graph of the $\dfrac {\size x^{\frac 1 n - 1} } {2 n}$ delta sequence. As $n$ grows, the graph becomes steeper and thinner. The area under each graph is infinite unless the range of integration is a finite interval say, $\closedint {-a} a$. However, in the limit $n \to \infty$, this area approaches $1$ re... | Let $a \in \R_{> 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int_{-a}^a \frac {\size x^{\frac 1 n - 1} } {2 n} \rd x
| r = \int_{-a}^0 \frac {\size x^{\frac 1 n - 1} } {2 n} \rd x + \int_0^a \frac {\size x^{\frac 1 n - 1} } {2 n} \rd x
}}
{{eqn | r = \int_{-a}^0 \frac {\paren {-x}^{\frac 1 n - 1} } {2 n} \rd x + \int_0... | [[File:PowerLawDeltaSequence.png|thumb|600px|The graph of the $\dfrac {\size x^{\frac 1 n - 1} } {2 n}$ delta sequence. As $n$ grows, the graph becomes steeper and thinner. The area under each graph is infinite unless the range of integration is a finite interval say, $\closedint {-a} a$. However, in the limit $n \to \... | Let $a \in \R_{> 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int_{-a}^a \frac {\size x^{\frac 1 n - 1} } {2 n} \rd x
| r = \int_{-a}^0 \frac {\size x^{\frac 1 n - 1} } {2 n} \rd x + \int_0^a \frac {\size x^{\frac 1 n - 1} } {2 n} \rd x
}}
{{eqn | r = \int_{-a}^0 \frac {\paren {-x}^{\frac 1 n - 1} } {2 n} \rd x + \int... | Power Law Delta Sequence | https://proofwiki.org/wiki/Power_Law_Delta_Sequence | https://proofwiki.org/wiki/Power_Law_Delta_Sequence | [
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:PowerLawDeltaSequence.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Integration by Substitution/Definite Integral",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Mean Value Theorem for Integrals/Generalization",
"Definition:Arbitrary Constant",
"Definition:Positive/Number",
"Defini... |
proofwiki-19102 | Five Cube Theorem | Every integer can be expressed as a sum of five cube numbers. | Let $r$ be an integer.
Then $r$ can be expressed in exactly one the following forms:
:$\paren {6 m + 0}$
:$\paren {6 m + 1}$
:$\paren {6 m + 2}$
:$\paren {6 m + 3}$
:$\paren {6 m + 4}$
:$\paren {6 m + 5}$
for some $m \in \Z$.
:$\paren {6 m + 0} = \paren {m + 1}^3 + \paren {m - 1}^3 + \paren {- m}^3 + \paren {- m}^3 + 0... | Every [[Definition:Integer|integer]] can be expressed as a [[Definition:Integer Addition|sum]] of five [[Definition:Cube Number|cube numbers]]. | Let $r$ be an [[Definition:Integer|integer]].
Then $r$ can be expressed in exactly one the following forms:
:$\paren {6 m + 0}$
:$\paren {6 m + 1}$
:$\paren {6 m + 2}$
:$\paren {6 m + 3}$
:$\paren {6 m + 4}$
:$\paren {6 m + 5}$
for some $m \in \Z$.
:$\paren {6 m + 0} = \paren {m + 1}^3 + \paren {m - 1}^3 + \paren {-... | Five Cube Theorem | https://proofwiki.org/wiki/Five_Cube_Theorem | https://proofwiki.org/wiki/Five_Cube_Theorem | [
"Cube Numbers",
"Sums of Cubes"
] | [
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Cube Number"
] | [
"Definition:Integer"
] |
proofwiki-19103 | Pell's Equation/Examples/8 | :$x^2 - 8 y^2 = 1$
has the positive integral solutions:
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| r = \tuple {3, 1}
}}
{{eqn | l = \tuple {x, y}
| r = \tuple {17, 6}
}}
{{eqn | l = \tuple {x, y}
| r = \tuple {99, 35}
}}
{{eqn | l = \tuple {x, y}
| r = \tuple {577, 204}
}}
{{eqn | l = \tuple {x, ... | From Continued Fraction Expansion of $\sqrt 8$:
:$\sqrt 8 = \sqbrk {2, \sequence {1, 4} }$
The cycle is of length $2$.
By Solution of Pell's Equation, the only solutions of $x^2 - 8 y^2 = 1$ are:
:${p_{2 r} }^2 - 8 {q_{2 r} }^2 = \paren {-1}^{2 r}$
for $r = 1, 2, 3, \ldots$
When $r = 1$ this gives:
:${p_2}^2 - 8 {q_2}^... | :$x^2 - 8 y^2 = 1$
has the [[Definition:Positive Integer|positive integral]] solutions:
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| r = \tuple {3, 1}
}}
{{eqn | l = \tuple {x, y}
| r = \tuple {17, 6}
}}
{{eqn | l = \tuple {x, y}
| r = \tuple {99, 35}
}}
{{eqn | l = \tuple {x, y}
| r = \tuple {577... | From [[Continued Fraction Expansion of Irrational Square Root/Examples/8|Continued Fraction Expansion of $\sqrt 8$]]:
:$\sqrt 8 = \sqbrk {2, \sequence {1, 4} }$
The [[Definition:Cycle of Periodic Continued Fraction|cycle]] is of [[Definition:Cycle Length of Periodic Continued Fraction|length]] $2$.
By [[Solution of P... | Pell's Equation/Examples/8 | https://proofwiki.org/wiki/Pell's_Equation/Examples/8 | https://proofwiki.org/wiki/Pell's_Equation/Examples/8 | [
"Pell's Equation",
"8"
] | [
"Definition:Positive/Integer"
] | [
"Continued Fraction Expansion of Irrational Square Root/Examples/8",
"Definition:Periodic Continued Fraction/Cycle",
"Definition:Periodic Continued Fraction/Cycle/Length",
"Solution to Pell's Equation",
"Continued Fraction Expansion of Irrational Square Root/Examples/8/Convergents",
"Category:Pell's Equat... |
proofwiki-19104 | Digital Root of Square | Let $n^2$ be a square number.
Then the digital root of $n^2$ is $1$, $4$, $7$ or $9$. | Let $\map d n$ denote the digital root base $10$ of $n$.
From Digital Root is Congruent to Number Modulo Base minus 1, $\map d n \equiv n \pmod 9$.
So, let $n = 9 k + m$ where $1 \le m \le 9$.
Thus:
{{begin-eqn}}
{{eqn | l = n^2
| r = \paren {9 k + m}^2
| c =
}}
{{eqn | r = 81 k^2 + 18 k m + m^2
| c ... | Let $n^2$ be a [[Definition:Square Number|square number]].
Then the [[Definition:Digital Root|digital root]] of $n^2$ is $1$, $4$, $7$ or $9$. | Let $\map d n$ denote the [[Definition:Digital Root|digital root base $10$]] of $n$.
From [[Digital Root is Congruent to Number Modulo Base minus 1]], $\map d n \equiv n \pmod 9$.
So, let $n = 9 k + m$ where $1 \le m \le 9$.
Thus:
{{begin-eqn}}
{{eqn | l = n^2
| r = \paren {9 k + m}^2
| c =
}}
{{eqn | ... | Digital Root of Square | https://proofwiki.org/wiki/Digital_Root_of_Square | https://proofwiki.org/wiki/Digital_Root_of_Square | [
"Digital Roots",
"Square Numbers"
] | [
"Definition:Square Number",
"Definition:Digital Root"
] | [
"Definition:Digital Root",
"Digital Root is Congruent to Number Modulo Base minus 1",
"Definition:Square/Function",
"Definition:Digit",
"Category:Digital Roots",
"Category:Square Numbers"
] |
proofwiki-19105 | Tuning Fork Delta Sequence | thumb300pxThe graph of the tuning fork delta sequence. As $n$ grows, the rectangles becomes thinner and longer. The area of each shape is equal to $1$, where the area under the axis contributes negatively. Note that at $x = 0$ the graph is always negative, and its value here approaches $-\infty$. Only taking the entire... | Let $\phi \in \map \DD \R$ be a test function.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x
| r = \lim_{n \mathop \to \infty} \paren {\int_{- \infty}^{-\frac 1 n } \map \phi x \map {\delta_n} x \rd x + \int_{- \frac 1 n }^{- \frac 1 {2n} } \... | [[File:TuningForkDeltaSequence.png|thumb|300px|The graph of the tuning fork delta sequence. As $n$ grows, the rectangles becomes thinner and longer. The area of each shape is equal to $1$, where the area under the axis contributes negatively. Note that at $x = 0$ the graph is always negative, and its value here approac... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x
| r = \lim_{n \mathop \to \infty} \paren {\int_{- \infty}^{-\frac 1 n } \map \phi x \map {\delta_n} x \rd x + \int_{- ... | Tuning Fork Delta Sequence/Proof 1 | https://proofwiki.org/wiki/Tuning_Fork_Delta_Sequence | https://proofwiki.org/wiki/Tuning_Fork_Delta_Sequence/Proof_1 | [
"Tuning Fork Delta Sequence",
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:TuningForkDeltaSequence.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Definition:Test Function",
"Mean Value Theorem for Integrals",
"Limit of Image of Sequence/Real Number Line",
"Squeeze Theorem/Sequences/Real Numbers"
] |
proofwiki-19106 | Tuning Fork Delta Sequence | thumb300pxThe graph of the tuning fork delta sequence. As $n$ grows, the rectangles becomes thinner and longer. The area of each shape is equal to $1$, where the area under the axis contributes negatively. Note that at $x = 0$ the graph is always negative, and its value here approaches $-\infty$. Only taking the entire... | We have that:
{{begin-eqn}}
{{eqn | l = \int_{-\infty}^\infty \map {\delta_n} x \rd x
| r = 2n \int_{- \frac 1 n }^{- \frac 1 {2n} } \rd x - n \int_{- \frac 1 {2n} }^{\frac 1 {2n} } \rd x + 2n \int_{\frac 1 {2n} }^{\frac 1 n} \rd x
}}
{{eqn | r = n \paren {2 \paren {- \frac 1 {2n} - \paren {- \frac 1 n} } - \par... | [[File:TuningForkDeltaSequence.png|thumb|300px|The graph of the tuning fork delta sequence. As $n$ grows, the rectangles becomes thinner and longer. The area of each shape is equal to $1$, where the area under the axis contributes negatively. Note that at $x = 0$ the graph is always negative, and its value here approac... | We have that:
{{begin-eqn}}
{{eqn | l = \int_{-\infty}^\infty \map {\delta_n} x \rd x
| r = 2n \int_{- \frac 1 n }^{- \frac 1 {2n} } \rd x - n \int_{- \frac 1 {2n} }^{\frac 1 {2n} } \rd x + 2n \int_{\frac 1 {2n} }^{\frac 1 n} \rd x
}}
{{eqn | r = n \paren {2 \paren {- \frac 1 {2n} - \paren {- \frac 1 n} } - \pa... | Tuning Fork Delta Sequence/Proof 2 | https://proofwiki.org/wiki/Tuning_Fork_Delta_Sequence | https://proofwiki.org/wiki/Tuning_Fork_Delta_Sequence/Proof_2 | [
"Tuning Fork Delta Sequence",
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:TuningForkDeltaSequence.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Definition:Test Function",
"Definition:Smooth Real Function",
"Differentiable Function is Continuous",
"Definition:Continuous Real Function at Point/Definition 1",
"Definition:Limit of Sequence/Real Numbers",
"Combination Theorem for Limits of Functions/Real/Sum Rule"
] |
proofwiki-19107 | Count of All Permutations on n Objects | Let $S$ be a set of $n$ objects.
Let $N$ be the number of permutations of $k$ objects from $S$, where $1 \le k \le N$.
Then:
:$\ds N = n! \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {k!}$ | The number of permutations on $k$ objects, from $n$ is denoted ${}^n P_k$.
From Number of Permutations:
:${}^n P_k = \dfrac {n!} {\paren {n - k}!}$
Hence:
{{begin-eqn}}
{{eqn | q =
| l = N
| r = \sum_{k \mathop = 1}^n {}^n P_k
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^n \dfrac {n!} {\paren {n - k}!}... | Let $S$ be a [[Definition:Set|set]] of $n$ [[Definition:Object|objects]].
Let $N$ be the number of [[Definition:Permutation (Ordered Selection)|permutations]] of $k$ [[Definition:Object|objects]] from $S$, where $1 \le k \le N$.
Then:
:$\ds N = n! \sum_{k \mathop = 0}^{n - 1} \dfrac 1 {k!}$ | The number of [[Definition:Permutation (Ordered Selection)|permutations]] on $k$ objects, from $n$ is denoted ${}^n P_k$.
From [[Number of Permutations]]:
:${}^n P_k = \dfrac {n!} {\paren {n - k}!}$
Hence:
{{begin-eqn}}
{{eqn | q =
| l = N
| r = \sum_{k \mathop = 1}^n {}^n P_k
| c =
}}
{{eqn | r... | Count of All Permutations on n Objects | https://proofwiki.org/wiki/Count_of_All_Permutations_on_n_Objects | https://proofwiki.org/wiki/Count_of_All_Permutations_on_n_Objects | [
"Count of All Permutations on n Objects",
"Permutations (Ordered Selections)"
] | [
"Definition:Set",
"Definition:Object",
"Definition:Permutation/Ordered Selection",
"Definition:Object"
] | [
"Definition:Permutation/Ordered Selection",
"Number of Permutations"
] |
proofwiki-19108 | Time when Hour Hand and Minute Hand at Right Angle | Let the time of day be such that the hour hand and minute hand are at a right angle to each other.
Then the time happens $22$ times in every $12$ hour period:
:when the minute hand is $15$ minutes ahead of the hour hand
:when the minute hand is $15$ minutes behind the hour hand.
In the first case, this happens at $09:0... | Obviously the hands are at right angles at $3$ and $9$ o'clock.
Thus we only need to show that the angle between the hands will be the same after every $1$ hour, $5$ minutes and $27 . \dot 2 \dot 7$ seconds.
Note that:
{{begin-eqn}}
{{eqn | l = 1 h \ 5 m \ 27. \dot 2 \dot 7 s
| r = 65 m \ 27 \tfrac {27}{99} s
}}... | Let the [[Definition:Time of Day|time of day]] be such that the [[Definition:Hour Hand|hour hand]] and [[Definition:Minute Hand|minute hand]] are at a [[Definition:Right Angle|right angle]] to each other.
Then the [[Definition:Time of Day|time]] happens $22$ times in every $12$ [[Definition:Hour|hour period]]:
:when ... | Obviously the hands are at [[Definition:Right Angle|right angles]] at $3$ and $9$ o'clock.
Thus we only need to show that the angle between the hands will be the same after every $1$ [[Definition:Hour|hour]], $5$ [[Definition:Minute of Time|minutes]] and $27 . \dot 2 \dot 7$ [[Definition:Second of Time|seconds]].
Not... | Time when Hour Hand and Minute Hand at Right Angle | https://proofwiki.org/wiki/Time_when_Hour_Hand_and_Minute_Hand_at_Right_Angle | https://proofwiki.org/wiki/Time_when_Hour_Hand_and_Minute_Hand_at_Right_Angle | [
"Clocks"
] | [
"Definition:Time of Day",
"Definition:Clock/Hour Hand",
"Definition:Clock/Minute Hand",
"Definition:Right Angle",
"Definition:Time of Day",
"Definition:Time/Unit/Hour",
"Definition:Clock/Minute Hand",
"Definition:Time/Unit/Minute",
"Definition:Clock/Hour Hand",
"Definition:Clock/Minute Hand",
"D... | [
"Definition:Right Angle",
"Definition:Time/Unit/Hour",
"Definition:Time/Unit/Minute",
"Definition:Time/Unit/Second",
"Definition:Time/Unit/Hour",
"Definition:Clock/Minute Hand",
"Definition:Clock/Hour Hand",
"Definition:Time/Unit/Hour",
"Category:Clocks"
] |
proofwiki-19109 | Convergence in Normed Dual Space implies Weak-* Convergence | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a convergent sequence in $X^\ast$.
Then $\sequence {f_n}_{n \mathop \in \N}$ converges we... | From Convergent Sequence in Normed Vector Space is Weakly Convergent, $\sequence {f_n}_{n \mathop \in \N}$ converges weakly.
From Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent, $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$.
{{qed}} | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition:Conv... | From [[Convergent Sequence in Normed Vector Space is Weakly Convergent]], $\sequence {f_n}_{n \mathop \in \N}$ [[Definition:Weak Convergence (Normed Vector Space)|converges weakly]].
From [[Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent]], $\sequence {f_n}_{n \mathop \in \N}$ [[Definition:Weak-... | Convergence in Normed Dual Space implies Weak-* Convergence/Proof 1 | https://proofwiki.org/wiki/Convergence_in_Normed_Dual_Space_implies_Weak-*_Convergence | https://proofwiki.org/wiki/Convergence_in_Normed_Dual_Space_implies_Weak-*_Convergence/Proof_1 | [
"Convergence in Normed Dual Space implies Weak-* Convergence",
"Weak-* Convergence (Normed Vector Spaces)",
"Convergence in Normed Dual Space implies Weak-* Convergence"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Weak-* Convergence (Normed Vector Space)"
] | [
"Convergent Sequence in Normed Vector Space is Weakly Convergent",
"Definition:Weak Convergence (Normed Vector Space)",
"Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent",
"Definition:Weak-* Convergence (Normed Vector Space)"
] |
proofwiki-19110 | Convergence in Normed Dual Space implies Weak-* Convergence | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a convergent sequence in $X^\ast$.
Then $\sequence {f_n}_{n \mathop \in \N}$ converges we... | Let $f \in X^\ast$ be the limit of $\sequence {f_n}_{n \mathop \in \N}$.
That is:
:$\norm {f_n - f}_{X^\ast} \stackrel {n \mathop \to \infty} \longrightarrow 0$
Thus, for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \size {\map {f_n} x - \map f x}
| r = \size {\map {\paren {f_n - f} } x}
| c = {{Defof|Vector S... | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition:Conv... | Let $f \in X^\ast$ be the [[Definition:Limit of Sequence in Normed Vector Space|limit]] of $\sequence {f_n}_{n \mathop \in \N}$.
That is:
:$\norm {f_n - f}_{X^\ast} \stackrel {n \mathop \to \infty} \longrightarrow 0$
Thus, for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \size {\map {f_n} x - \map f x}
| r = \siz... | Convergence in Normed Dual Space implies Weak-* Convergence/Proof 2 | https://proofwiki.org/wiki/Convergence_in_Normed_Dual_Space_implies_Weak-*_Convergence | https://proofwiki.org/wiki/Convergence_in_Normed_Dual_Space_implies_Weak-*_Convergence/Proof_2 | [
"Convergence in Normed Dual Space implies Weak-* Convergence",
"Weak-* Convergence (Normed Vector Spaces)",
"Convergence in Normed Dual Space implies Weak-* Convergence"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Weak-* Convergence (Normed Vector Space)"
] | [
"Definition:Limit of Sequence/Normed Vector Space",
"Fundamental Property of Norm on Bounded Linear Functional"
] |
proofwiki-19111 | Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent | Let $\mathbb F$ be a subfield of $\C$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\mathbb F$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $f \in X^\ast$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence ... | Let $x \in X$.
We aim to show that:
:$\map {f_n} x \to \map f x$
Then, since $x \in X$ was arbitrary, we will obtain that $\sequence {f_n}_{n \mathop \in \N}$ converges weakly-$\ast$ to $f$.
Since $\sequence {f_n}_{n \mathop \in \N}$ converges weakly to $f$, we have:
:$\map F {f_n} \to \map F f$ for each $F \in \pare... | Let $\mathbb F$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\mathbb F$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {... | Let $x \in X$.
We aim to show that:
:$\map {f_n} x \to \map f x$
Then, since $x \in X$ was arbitrary, we will obtain that $\sequence {f_n}_{n \mathop \in \N}$ [[Definition:Weak-* Convergence (Normed Vector Space)|converges weakly-$\ast$]] to $f$.
Since $\sequence {f_n}_{n \mathop \in \N}$ [[Definition:Weak Converg... | Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent/Proof 1 | https://proofwiki.org/wiki/Weakly_Convergent_Sequence_in_Normed_Dual_Space_is_Weakly-*_Convergent | https://proofwiki.org/wiki/Weakly_Convergent_Sequence_in_Normed_Dual_Space_is_Weakly-*_Convergent/Proof_1 | [
"Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent",
"Weak Convergence (Normed Vector Spaces)",
"Weak-* Convergence (Normed Vector Spaces)",
"Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent"
] | [
"Definition:Subfield",
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)",
"Definition:Weak-* Convergence (Normed Vector Space)"
] | [
"Definition:Weak-* Convergence (Normed Vector Space)",
"Definition:Weak Convergence (Normed Vector Space)",
"Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual"
] |
proofwiki-19112 | Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent | Let $\mathbb F$ be a subfield of $\C$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\mathbb F$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $f \in X^\ast$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence ... | Let $J$ be the evaluation linear transformation on $X$.
By Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual:
:$J : X \to X^{\ast \ast}$
Thus, for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \map {f_n} x
| r = \map {\map J x} {f_n}
}}
{{eqn | o =\stack... | Let $\mathbb F$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\mathbb F$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {... | Let $J$ be the [[Definition:Evaluation Linear Transformation on Normed Vector Space|evaluation linear transformation]] on $X$.
By [[Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual]]:
:$J : X \to X^{\ast \ast}$
Thus, for each $x \in X$:
{{begin-eqn}}
{... | Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent/Proof 2 | https://proofwiki.org/wiki/Weakly_Convergent_Sequence_in_Normed_Dual_Space_is_Weakly-*_Convergent | https://proofwiki.org/wiki/Weakly_Convergent_Sequence_in_Normed_Dual_Space_is_Weakly-*_Convergent/Proof_2 | [
"Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent",
"Weak Convergence (Normed Vector Spaces)",
"Weak-* Convergence (Normed Vector Spaces)",
"Weakly Convergent Sequence in Normed Dual Space is Weakly-* Convergent"
] | [
"Definition:Subfield",
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)",
"Definition:Weak-* Convergence (Normed Vector Space)"
] | [
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual"
] |
proofwiki-19113 | Weak Convergence in Normed Dual Space of Reflexive Normed Vector Space is Equivalent to Weak-* Convergence | Let $\mathbb F$ be a subfield of $\C$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a reflexive normed vector space over $\mathbb F$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $f \in X^\ast$.
Then:
:$\sequence {f_n}_{n \mathop \in \N}$ c... | Let $J : X \to X^{**}$ be the evaluation linear transformation.
In view of {{Defof|Reflexive Space}}, $J$ is especially a surjection.
Therefore:
{{begin-eqn}}
{{eqn | l = f_n
| o = \weakconv
| r = f
}}
{{eqn | ll = \leadstoandfrom
| q = \forall x' ' \in X^{**}
| l = \map {x' '} {f_n}
| o =... | Let $\mathbb F$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Reflexive Space|reflexive]] [[Definition:Normed Vector Space|normed vector space]] over $\mathbb F$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|no... | Let $J : X \to X^{**}$ be the [[Definition:Evaluation Linear Transformation on Normed Vector Space|evaluation linear transformation]].
In view of {{Defof|Reflexive Space}}, $J$ is especially a [[Definition:Surjection|surjection]].
Therefore:
{{begin-eqn}}
{{eqn | l = f_n
| o = \weakconv
| r = f
}}
{{eqn |... | Weak Convergence in Normed Dual Space of Reflexive Normed Vector Space is Equivalent to Weak-* Convergence/Proof 2 | https://proofwiki.org/wiki/Weak_Convergence_in_Normed_Dual_Space_of_Reflexive_Normed_Vector_Space_is_Equivalent_to_Weak-*_Convergence | https://proofwiki.org/wiki/Weak_Convergence_in_Normed_Dual_Space_of_Reflexive_Normed_Vector_Space_is_Equivalent_to_Weak-*_Convergence/Proof_2 | [
"Reflexive Spaces",
"Weak Convergence (Normed Vector Spaces)",
"Weak-* Convergence (Normed Vector Spaces)"
] | [
"Definition:Subfield",
"Definition:Reflexive Space",
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak Convergence (Normed Vector Space)",
"Definition:Weak-* Convergence (Normed Vector Space)"
] | [
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Surjection",
"Definition:Surjection"
] |
proofwiki-19114 | Power Structure Operation on Set of Singleton Subsets is Closed | Let $\struct {S, \circ}$ be a magma.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $S$.
Let $S'$ denote the set of singleton elements of $\powerset S$.
Then the algebraic structure $\struct {S', \circ_\PP}$ is closed. | {{Recall|Magma}}
{{:Definition:Magma}}
{{Recall|Singleton}}
{{:Definition:Singleton}}
Let $A, B \in S'$.
Then:
:$\exists a, b \in S: A = \set a, B = \set b$
Hence:
{{begin-eqn}}
{{eqn | l = A \circ_\PP B
| r = \set {x \circ y: x \in A, y \in B}
| c = {{Defof|Subset Product}}
}}
{{eqn | r = \set {a \circ b}
... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $\struct {\powerset S, \circ_\PP}$ be the [[Definition:Power Structure|power structure]] of $S$.
Let $S'$ denote the [[Definition:Set|set]] of [[Definition:Singleton|singleton]] [[Definition:Element|elements]] of $\powerset S$.
Then the [[Definition:Alge... | {{Recall|Magma}}
{{:Definition:Magma}}
{{Recall|Singleton}}
{{:Definition:Singleton}}
Let $A, B \in S'$.
Then:
:$\exists a, b \in S: A = \set a, B = \set b$
Hence:
{{begin-eqn}}
{{eqn | l = A \circ_\PP B
| r = \set {x \circ y: x \in A, y \in B}
| c = {{Defof|Subset Product}}
}}
{{eqn | r = \set {a \cir... | Power Structure Operation on Set of Singleton Subsets is Closed | https://proofwiki.org/wiki/Power_Structure_Operation_on_Set_of_Singleton_Subsets_is_Closed | https://proofwiki.org/wiki/Power_Structure_Operation_on_Set_of_Singleton_Subsets_is_Closed | [
"Power Structures",
"Singletons"
] | [
"Definition:Magma",
"Definition:Power Structure",
"Definition:Set",
"Definition:Singleton",
"Definition:Element",
"Definition:Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [] |
proofwiki-19115 | Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets | Let $\struct {S, \circ}$ be a magma.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Let $S'$ denote the set of singleton elements of $\powerset S$.
Then $\struct {S, \circ}$ is isomorphic to $\struct {S', \circ_\PP}$. | Let $\phi: S \to S'$ be the mapping defined as:
:$\forall x \in S: \map \phi x = \set x$
We have that:
{{begin-eqn}}
{{eqn | q = \forall a, b \in S
| l = \map \phi a
| r = \map \phi b
| c =
}}
{{eqn | ll= \leadsto
| l = \set a
| r = \set b
| c = Definition of $\phi$
}}
{{eqn | ll= \... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $\struct {\powerset S, \circ_\PP}$ be the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$.
Let $S'$ denote the [[Definition:Set|set]] of [[Definition:Singleton|singleton]] [[Definition:Element|elements]] of $\powerset S$.
Then $\st... | Let $\phi: S \to S'$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in S: \map \phi x = \set x$
We have that:
{{begin-eqn}}
{{eqn | q = \forall a, b \in S
| l = \map \phi a
| r = \map \phi b
| c =
}}
{{eqn | ll= \leadsto
| l = \set a
| r = \set b
| c = Definition o... | Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets | https://proofwiki.org/wiki/Operation_is_Isomorphic_to_Power_Structure_Operation_on_Set_of_Singleton_Subsets | https://proofwiki.org/wiki/Operation_is_Isomorphic_to_Power_Structure_Operation_on_Set_of_Singleton_Subsets | [
"Power Structures",
"Singletons",
"Examples of Isomorphisms (Abstract Algebra)"
] | [
"Definition:Magma",
"Definition:Power Structure",
"Definition:Set",
"Definition:Singleton",
"Definition:Element",
"Definition:Isomorphism (Abstract Algebra)"
] | [
"Definition:Mapping",
"Singleton Equality",
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Magma",
"Power Structure of Magma is Magma",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Bijection",
"Definition:Homomorphism (Abstract Algebra)",
"Defin... |
proofwiki-19116 | Power Structure Operation on Set of Singleton Subsets preserves Associativity | Let $\struct {S, \circ}$ be a magma.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Let $S'$ denote the set of singleton elements of $\powerset S$.
Then $\circ_\PP$ is associative {{iff}} $\circ$ is associative. | From Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets:
:$\struct {S, \circ}$ is isomorphic to $\struct {S', \circ_\PP}$
The result follows from Isomorphism Preserves Associativity.
{{qed}} | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $\struct {\powerset S, \circ_\PP}$ be the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$.
Let $S'$ denote the [[Definition:Set|set]] of [[Definition:Singleton|singleton]] [[Definition:Element|elements]] of $\powerset S$.
Then $\ci... | From [[Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets]]:
:$\struct {S, \circ}$ is [[Definition:Isomorphism (Abstract Algebra)|isomorphic]] to $\struct {S', \circ_\PP}$
The result follows from [[Isomorphism Preserves Associativity]].
{{qed}} | Power Structure Operation on Set of Singleton Subsets preserves Associativity | https://proofwiki.org/wiki/Power_Structure_Operation_on_Set_of_Singleton_Subsets_preserves_Associativity | https://proofwiki.org/wiki/Power_Structure_Operation_on_Set_of_Singleton_Subsets_preserves_Associativity | [
"Power Structures",
"Associativity"
] | [
"Definition:Magma",
"Definition:Power Structure",
"Definition:Set",
"Definition:Singleton",
"Definition:Element",
"Definition:Associative Operation",
"Definition:Associative Operation"
] | [
"Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets",
"Definition:Isomorphism (Abstract Algebra)",
"Isomorphism Preserves Associativity"
] |
proofwiki-19117 | Power Structure Operation on Set of Singleton Subsets preserves Commutativity | Let $\struct {S, \circ}$ be a magma.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Let $S'$ denote the set of singleton elements of $\powerset S$.
Then $\circ_\PP$ is commutative {{iff}} $\circ$ is commutative. | From Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets:
:$\struct {S, \circ}$ is isomorphic to $\struct {S', \circ_\PP}$
The result follows from Isomorphism Preserves Commutativity.
{{qed}} | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $\struct {\powerset S, \circ_\PP}$ be the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$.
Let $S'$ denote the [[Definition:Set|set]] of [[Definition:Singleton|singleton]] [[Definition:Element|elements]] of $\powerset S$.
Then $\ci... | From [[Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets]]:
:$\struct {S, \circ}$ is [[Definition:Isomorphism (Abstract Algebra)|isomorphic]] to $\struct {S', \circ_\PP}$
The result follows from [[Isomorphism Preserves Commutativity]].
{{qed}} | Power Structure Operation on Set of Singleton Subsets preserves Commutativity | https://proofwiki.org/wiki/Power_Structure_Operation_on_Set_of_Singleton_Subsets_preserves_Commutativity | https://proofwiki.org/wiki/Power_Structure_Operation_on_Set_of_Singleton_Subsets_preserves_Commutativity | [
"Power Structures",
"Commutativity",
"Singletons"
] | [
"Definition:Magma",
"Definition:Power Structure",
"Definition:Set",
"Definition:Singleton",
"Definition:Element",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation"
] | [
"Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets",
"Definition:Isomorphism (Abstract Algebra)",
"Isomorphism Preserves Commutativity"
] |
proofwiki-19118 | Identity Element for Power Structure | Let $\struct {S, \circ}$ be a magma whose underlying set $S$ is non-empty.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Then:
:a subset $J$ of $S$ is an identity element of $\struct {\powerset S, \circ_\PP}$
{{iff}}:
:there exists an identity element $e$ of $\struct {S, \circ}$... | === Sufficient Condition ===
Let $J \subseteq S$ such that $J$ is an identity element of $\struct {\powerset S, \circ_\PP}$.
We have:
{{begin-eqn}}
{{eqn | q = \forall A \in \powerset S
| l = J \circ_\PP A
| r = A
| c = {{Defof|Identity Element}}
}}
{{eqn | ll= \leadsto
| l = \set {j \circ a: j ... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]] whose [[Definition:Underlying Set of Structure|underlying set]] $S$ is [[Definition:Non-Empty Set|non-empty]].
Let $\struct {\powerset S, \circ_\PP}$ be the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$.
Then:
:a [[Definition:Subset|su... | === Sufficient Condition ===
Let $J \subseteq S$ such that $J$ is an [[Definition:Identity Element|identity element]] of $\struct {\powerset S, \circ_\PP}$.
We have:
{{begin-eqn}}
{{eqn | q = \forall A \in \powerset S
| l = J \circ_\PP A
| r = A
| c = {{Defof|Identity Element}}
}}
{{eqn | ll= \lead... | Identity Element for Power Structure | https://proofwiki.org/wiki/Identity_Element_for_Power_Structure | https://proofwiki.org/wiki/Identity_Element_for_Power_Structure | [
"Power Structures",
"Identity Elements"
] | [
"Definition:Magma",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Non-Empty Set",
"Definition:Power Structure",
"Definition:Subset",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Defin... |
proofwiki-19119 | Invertible Element containing Identity in Power Structure | Let $\struct {S, \circ}$ be a magma.
Let identity element $e \in S$ be an identity element of $\struct {S, \circ}$.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Let $X \subseteq S$ such that:
:$e \in X$
:$X$ is invertible for $\circ_PP$.
Then $X = \set e$. | Let $X \subseteq S$ be invertible for $\circ_PP$ and such that $e \in X$.
From Identity Element for Power Structure, $\struct {\powerset S, \circ_\PP}$ has an identity element $J = \set e$.
We have:
{{begin-eqn}}
{{eqn | n = 1
| q = \exists Y \in \powerset S
| l = X \circ_\PP Y
| r = J
| c = {{D... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let [[Definition:Identity Element|identity element]] $e \in S$ be an [[Definition:Identity Element|identity element]] of $\struct {S, \circ}$.
Let $\struct {\powerset S, \circ_\PP}$ be the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$.... | Let $X \subseteq S$ be [[Definition:Invertible Element|invertible]] for $\circ_PP$ and such that $e \in X$.
From [[Identity Element for Power Structure]], $\struct {\powerset S, \circ_\PP}$ has an [[Definition:Identity Element|identity element]] $J = \set e$.
We have:
{{begin-eqn}}
{{eqn | n = 1
| q = \exists ... | Invertible Element containing Identity in Power Structure | https://proofwiki.org/wiki/Invertible_Element_containing_Identity_in_Power_Structure | https://proofwiki.org/wiki/Invertible_Element_containing_Identity_in_Power_Structure | [
"Power Structures",
"Inverse Elements"
] | [
"Definition:Magma",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Power Structure",
"Definition:Invertible Element"
] | [
"Definition:Invertible Element",
"Identity Element for Power Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Element"
] |
proofwiki-19120 | Condition for Invertibility in Power Structure on Associative or Cancellable Operation | Let $\struct {S, \circ}$ be a unital magma.
Let $\struct {\powerset S, \circ_\PP}$ denote the power structure of $\struct {S, \circ}$.
Let identity element $e \in S$ be an identity element of $\struct {S, \circ}$.
Let $\circ$ be either:
:an associative operation
:a cancellable operation.
Let $X \subseteq S$ be a subset... | First we note that from Identity Element for Power Structure, the algebraic structure $\struct {\powerset S, \circ_\PP}$ has an identity element $J = \set e$. | Let $\struct {S, \circ}$ be a [[Definition:Unital Magma|unital magma]].
Let $\struct {\powerset S, \circ_\PP}$ denote the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$.
Let [[Definition:Identity Element|identity element]] $e \in S$ be an [[Definition:Identity Element|identity element]] of $\s... | First we note that from [[Identity Element for Power Structure]], the [[Definition:Algebraic Structure|algebraic structure]] $\struct {\powerset S, \circ_\PP}$ has an [[Definition:Identity Element|identity element]] $J = \set e$. | Condition for Invertibility in Power Structure on Associative or Cancellable Operation | https://proofwiki.org/wiki/Condition_for_Invertibility_in_Power_Structure_on_Associative_or_Cancellable_Operation | https://proofwiki.org/wiki/Condition_for_Invertibility_in_Power_Structure_on_Associative_or_Cancellable_Operation | [
"Power Structures",
"Associativity",
"Cancellability",
"Inverse Elements",
"Magmas"
] | [
"Definition:Unital Magma",
"Definition:Power Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Associative Operation",
"Definition:Cancellable Operation",
"Definition:Subset",
"Definition:Invertible Element... | [
"Identity Element for Power Structure",
"Definition:Algebraic Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-19121 | Set of Finite Subsets under Induced Operation is Closed | Let $\struct {S, \circ}$ be a magma.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$
Let $T \subseteq \powerset S$ be the set of all finite subsets of $S$.
Then the algebraic structure $\struct {T, \circ_\PP}$ is closed. | Let $X, Y \in T$.
Then:
{{begin-eqn}}
{{eqn | q =
| l = X \circ_\PP Y
| r = \set {x \circ y: x \in X, y \in Y}
| c = {{Defof|Operation Induced on Power Set}}
}}
{{eqn | ll= \leadsto
| l = \card {X \circ_\PP Y}
| o = \le
| r = \card X \times \card Y
| c =
}}
{{eqn | ll= \leads... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $\struct {\powerset S, \circ_\PP}$ be the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$
Let $T \subseteq \powerset S$ be the [[Definition:Set|set]] of all [[Definition:Finite Set|finite]] [[Definition:Subset|subsets]] of $S$.
The... | Let $X, Y \in T$.
Then:
{{begin-eqn}}
{{eqn | q =
| l = X \circ_\PP Y
| r = \set {x \circ y: x \in X, y \in Y}
| c = {{Defof|Operation Induced on Power Set}}
}}
{{eqn | ll= \leadsto
| l = \card {X \circ_\PP Y}
| o = \le
| r = \card X \times \card Y
| c =
}}
{{eqn | ll= \lea... | Set of Finite Subsets under Induced Operation is Closed | https://proofwiki.org/wiki/Set_of_Finite_Subsets_under_Induced_Operation_is_Closed | https://proofwiki.org/wiki/Set_of_Finite_Subsets_under_Induced_Operation_is_Closed | [
"Power Structures"
] | [
"Definition:Magma",
"Definition:Power Structure",
"Definition:Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Algebraic Structure/One Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Order of Subset Product is not Greater than Cardinality of Cartesian Product"
] |
proofwiki-19122 | Power Structure of Subset is Closed iff Subset is Closed | Let $\struct {S, \circ}$ be a magma.
Let $\circ_\PP$ be the operation induced on $\powerset S$, the power set of $S$.
Let $T \subseteq S$.
Then:
:the algebraic structure $\struct {\powerset T, \circ_\PP}$ is closed
{{iff}}:
:the algebraic structure $\struct {T, \circ}$ is closed. | === Sufficient Condition ===
Let $\struct {\powerset T, \circ_\PP}$ be closed.
Then:
{{begin-eqn}}
{{eqn | q = \forall X, Y \in \powerset T
| l = X \circ_\PP Y
| o = \in
| r = \powerset T
| c = {{Defof|Closed Algebraic Structure}}
}}
{{eqn | l = \set {x \circ y: x \in X, y \in Y}
| o = \su... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $\circ_\PP$ be the [[Definition:Operation Induced on Power Set|operation induced]] on $\powerset S$, the [[Definition:Power Set|power set]] of $S$.
Let $T \subseteq S$.
Then:
:the [[Definition:Algebraic Structure|algebraic structure]] $\struct {\powerset... | === Sufficient Condition ===
Let $\struct {\powerset T, \circ_\PP}$ be [[Definition:Closed Algebraic Structure|closed]].
Then:
{{begin-eqn}}
{{eqn | q = \forall X, Y \in \powerset T
| l = X \circ_\PP Y
| o = \in
| r = \powerset T
| c = {{Defof|Closed Algebraic Structure}}
}}
{{eqn | l = \set ... | Power Structure of Subset is Closed iff Subset is Closed | https://proofwiki.org/wiki/Power_Structure_of_Subset_is_Closed_iff_Subset_is_Closed | https://proofwiki.org/wiki/Power_Structure_of_Subset_is_Closed_iff_Subset_is_Closed | [
"Power Set",
"Subset Products",
"Power Structures"
] | [
"Definition:Magma",
"Definition:Subset Product",
"Definition:Power Set",
"Definition:Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-19123 | Pell Number as Sum of Squares | Let $P_n$ be a Pell Number:
:$P_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ 2 P_{n - 1} + P_{n - 2} & : \text {otherwise}\end{cases}$
Then:
:$P_{2 n + 1} = P_{n + 1}^2 + P_n^2$ | This proof proceeds by induction. | Let $P_n$ be a [[Definition:Pell Numbers|Pell Number]]:
:$P_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ 2 P_{n - 1} + P_{n - 2} & : \text {otherwise}\end{cases}$
Then:
:$P_{2 n + 1} = P_{n + 1}^2 + P_n^2$ | This proof proceeds by [[Principle of Mathematical Induction|induction]]. | Pell Number as Sum of Squares | https://proofwiki.org/wiki/Pell_Number_as_Sum_of_Squares | https://proofwiki.org/wiki/Pell_Number_as_Sum_of_Squares | [
"Pell Numbers"
] | [
"Definition:Pell Numbers"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-19124 | Multiple of Repdigit Base minus 1/Generalization | Let $b \in \Z_{>1}$ be an integer greater than $1$.
Let $n$ be a repdigit number of $k$ instances of the digit $b - 1$ for some integer $k$ such that $k \ge 1$.
Let $m \in \N$ be an integer such that $1 \le m \le b^k$.
Then $m \times n$, when expressed in base $b$, is the concatenation of $m - 1$ with $b^k - m$, that i... | {{begin-eqn}}
{{eqn | l = n
| r = \sum_{j \mathop = 0}^{k - 1} \paren {b - 1} b^j
| c = Basis Representation Theorem
}}
{{eqn | r = b^k - 1
| c = Sum of Geometric Sequence
}}
{{eqn | ll= \leadsto
| l = m n
| r = m \paren {b^k - 1}
| c =
}}
{{eqn | r = \paren {m - 1} b^k + b^k - m
... | Let $b \in \Z_{>1}$ be an [[Definition:Integer|integer]] greater than $1$.
Let $n$ be a [[Definition:Repdigit Number|repdigit number]] of $k$ instances of the [[Definition:Digit|digit]] $b - 1$ for some [[Definition:Integer|integer]] $k$ such that $k \ge 1$.
Let $m \in \N$ be an [[Definition:Integer|integer]] such th... | {{begin-eqn}}
{{eqn | l = n
| r = \sum_{j \mathop = 0}^{k - 1} \paren {b - 1} b^j
| c = [[Basis Representation Theorem]]
}}
{{eqn | r = b^k - 1
| c = [[Sum of Geometric Sequence]]
}}
{{eqn | ll= \leadsto
| l = m n
| r = m \paren {b^k - 1}
| c =
}}
{{eqn | r = \paren {m - 1} b^k + b^... | Multiple of Repdigit Base minus 1/Generalization | https://proofwiki.org/wiki/Multiple_of_Repdigit_Base_minus_1/Generalization | https://proofwiki.org/wiki/Multiple_of_Repdigit_Base_minus_1/Generalization | [
"Repdigit Numbers",
"Multiple of Repdigit Base minus 1"
] | [
"Definition:Integer",
"Definition:Repdigit Number",
"Definition:Digit",
"Definition:Integer",
"Definition:Integer",
"Definition:Number Base",
"Definition:Digit",
"Definition:Digit"
] | [
"Basis Representation Theorem",
"Sum of Geometric Sequence"
] |
proofwiki-19125 | Set of Subsemigroups of Commutative Semigroup form Subsemigroup of Power Structure | Let $\struct {S, \circ}$ be a commutative semigroup.
Let $\struct {\powerset S, \circ_\PP}$ denote the power structure of $\struct {S, \circ}$.
Let $\TT$ be the set of all subsemigroups of $S$.
Then $\struct {\TT, \circ_\PP}$ is a subsemigroup of $\struct {\powerset S, \circ_\PP}$. | First we establish that from Power Structure of Semigroup is Semigroup:
:$\struct {\powerset S, \circ_\PP}$ is a semigroup.
From Subset Product within Commutative Structure is Commutative:
:$\struct {\powerset S, \circ_\PP}$ is a commutative semigroup.
Let $A$ and $B$ be arbitrary subsemigroups of $S$.
As $A$ and $B$ a... | Let $\struct {S, \circ}$ be a [[Definition:Commutative Semigroup|commutative semigroup]].
Let $\struct {\powerset S, \circ_\PP}$ denote the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$.
Let $\TT$ be the [[Definition:Set|set]] of all [[Definition:Subsemigroup|subsemigroups]] of $S$.
Then $\... | First we establish that from [[Power Structure of Semigroup is Semigroup]]:
:$\struct {\powerset S, \circ_\PP}$ is a [[Definition:Semigroup|semigroup]].
From [[Subset Product within Commutative Structure is Commutative]]:
:$\struct {\powerset S, \circ_\PP}$ is a [[Definition:Commutative Semigroup|commutative semigroup... | Set of Subsemigroups of Commutative Semigroup form Subsemigroup of Power Structure | https://proofwiki.org/wiki/Set_of_Subsemigroups_of_Commutative_Semigroup_form_Subsemigroup_of_Power_Structure | https://proofwiki.org/wiki/Set_of_Subsemigroups_of_Commutative_Semigroup_form_Subsemigroup_of_Power_Structure | [
"Commutative Semigroups",
"Power Structures",
"Subsemigroups"
] | [
"Definition:Commutative Semigroup",
"Definition:Power Structure",
"Definition:Set",
"Definition:Subsemigroup",
"Definition:Subsemigroup"
] | [
"Power Structure of Semigroup is Semigroup",
"Definition:Semigroup",
"Subset Product within Commutative Structure is Commutative",
"Definition:Commutative Semigroup",
"Definition:Subsemigroup",
"Definition:Subsemigroup",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Subset P... |
proofwiki-19126 | Smith Numbers are Infinite in Number/Lemma | Let $\map S m$ denote the sum of the digits of a positive integer $m$.
Let $\map {S_p} m$ denote the sum of the digits of the prime decomposition of $m$.
Let $\map N m$ denote the number of digits in $m$.
Suppose $m = p_1 p_2 \dots p_r$, where $p_i$ are prime numbers, not necessarily distinct.
Then $\map {S_p} m < 9 \m... | Let $b_i = \map N {p_i} - 1, i = 1, 2, \dots, r$.
Let $b = b_1 + b_2 + \dots + b_r$.
Since a prime cannot be a multiple of $9$, $\map S {p_i}$ cannot be a multiple of $9$ either.
Hence:
:$\map S {p_i} \le 9 \map N {p_i} - 1 = 9 b_i + 8$
We partition the $p_i$ into $9$ disjoint classes by means of the following:
Define ... | Let $\map S m$ denote the [[Definition:Integer Addition|sum]] of the [[Definition:Digit|digits]] of a [[Definition:Positive Integer|positive integer]] $m$.
Let $\map {S_p} m$ denote the [[Definition:Integer Addition|sum]] of the [[Definition:Digit|digits]] of the [[Definition:Prime Decomposition|prime decomposition]] ... | Let $b_i = \map N {p_i} - 1, i = 1, 2, \dots, r$.
Let $b = b_1 + b_2 + \dots + b_r$.
Since a [[Definition:Prime Number|prime]] cannot be a [[Definition:Integer Multiple|multiple]] of $9$, $\map S {p_i}$ cannot be a [[Definition:Integer Multiple|multiple]] of $9$ either.
Hence:
:$\map S {p_i} \le 9 \map N {p_i} - 1 =... | Smith Numbers are Infinite in Number/Lemma | https://proofwiki.org/wiki/Smith_Numbers_are_Infinite_in_Number/Lemma | https://proofwiki.org/wiki/Smith_Numbers_are_Infinite_in_Number/Lemma | [
"Smith Numbers are Infinite in Number"
] | [
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Positive/Integer",
"Definition:Addition/Integers",
"Definition:Digit",
"Definition:Prime Decomposition",
"Definition:Digit",
"Definition:Prime Number",
"Definition:Distinct"
] | [
"Definition:Prime Number",
"Definition:Integral Multiple/Real Numbers",
"Definition:Integral Multiple/Real Numbers",
"Definition:Strictly Negative/Integer",
"Definition:Strictly Negative/Integer"
] |
proofwiki-19127 | Set of Subgroups of Abelian Group form Subsemigroup of Power Structure | Let $\struct {G, \circ}$ be an abelian group.
Let $\struct {\powerset G, \circ_\PP}$ denote the power structure of $\struct {G, \circ}$.
Let $\SS$ be the set of all subgroups of $G$.
Then $\struct {\SS, \circ_\PP}$ is a subsemigroup of $\struct {\powerset G, \circ_\PP}$. | From Power Structure of Semigroup is Semigroup:
:$\struct {\powerset S, \circ_\PP}$ is a semigroup.
Let $A$ and $B$ be arbitrary subgroups of $G$.
We need to show that $A \circ_\PP B$ is also a subgroup of $G$.
Let $x$ and $y$ be arbitrary elements of $A \circ_\PP B$.
Then:
{{begin-eqn}}
{{eqn | q = \exists a_x \in A, ... | Let $\struct {G, \circ}$ be an [[Definition:Abelian Group|abelian group]].
Let $\struct {\powerset G, \circ_\PP}$ denote the [[Definition:Power Structure|power structure]] of $\struct {G, \circ}$.
Let $\SS$ be the [[Definition:Set|set]] of all [[Definition:Subgroup|subgroups]] of $G$.
Then $\struct {\SS, \circ_\PP}... | From [[Power Structure of Semigroup is Semigroup]]:
:$\struct {\powerset S, \circ_\PP}$ is a [[Definition:Semigroup|semigroup]].
Let $A$ and $B$ be arbitrary [[Definition:Subgroup|subgroups]] of $G$.
We need to show that $A \circ_\PP B$ is also a [[Definition:Subgroup|subgroup]] of $G$.
Let $x$ and $y$ be arbitrary ... | Set of Subgroups of Abelian Group form Subsemigroup of Power Structure | https://proofwiki.org/wiki/Set_of_Subgroups_of_Abelian_Group_form_Subsemigroup_of_Power_Structure | https://proofwiki.org/wiki/Set_of_Subgroups_of_Abelian_Group_form_Subsemigroup_of_Power_Structure | [
"Abelian Groups",
"Power Structures",
"Subsemigroups"
] | [
"Definition:Abelian Group",
"Definition:Power Structure",
"Definition:Set",
"Definition:Subgroup",
"Definition:Subsemigroup"
] | [
"Power Structure of Semigroup is Semigroup",
"Definition:Semigroup",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Element",
"Inverse of Group Product",
"Definition:Commutative/Operation",
"One-Step Subgroup Test"
] |
proofwiki-19128 | Lamé's Theorem/Lemma | Suppose it takes $n$ cycles around the Euclidean Algorithm to find $\gcd \set {a, b}$.
Then $\min \set {a, b} \ge F_{n + 2}$, where $F_n$ denotes the $n$th Fibonacci number. | {{WLOG}}, suppose $a \ge b$.
Let $q_i, r_i$ be the quotients and remainders of each step of the Euclidean Algorithm, that is:
{{begin-eqn}}
{{eqn | l = a
| r = q_1 b + r_1
| c =
}}
{{eqn | l = b
| r = q_2 r_1 + r_2
| c =
}}
{{eqn | l = r_1
| r = q_3 r_2 + r_3
| c =
}}
{{eqn | l = ... | Suppose it takes $n$ cycles around the [[Euclidean Algorithm]] to find $\gcd \set {a, b}$.
Then $\min \set {a, b} \ge F_{n + 2}$, where $F_n$ denotes the $n$th [[Definition:Fibonacci Number|Fibonacci number]]. | {{WLOG}}, suppose $a \ge b$.
Let $q_i, r_i$ be the [[Definition:Quotient|quotients]] and [[Definition:Remainder|remainders]] of each step of the [[Euclidean Algorithm]], that is:
{{begin-eqn}}
{{eqn | l = a
| r = q_1 b + r_1
| c =
}}
{{eqn | l = b
| r = q_2 r_1 + r_2
| c =
}}
{{eqn | l = r_1... | Lamé's Theorem/Lemma | https://proofwiki.org/wiki/Lamé's_Theorem/Lemma | https://proofwiki.org/wiki/Lamé's_Theorem/Lemma | [
"Lamé's Theorem"
] | [
"Euclidean Algorithm",
"Definition:Fibonacci Number"
] | [
"Definition:Quotient",
"Definition:Remainder",
"Euclidean Algorithm",
"Principle of Mathematical Induction",
"Euclidean Algorithm",
"Euclidean Algorithm",
"Euclidean Algorithm",
"Euclidean Algorithm"
] |
proofwiki-19129 | Product of Subgroup with Inverse | Let $\struct {G, \circ}$ be a group.
Then:
:$\forall H \le \struct {G, \circ}:$
::$H^{-1} \circ H = H$
::$H \circ H^{-1} = H$
where $H \le G$ denotes that $H$ is a subgroup of $G$. | From Inverse of Subgroup:
:$H = H^{-1}$
From Product of Subgroup with Itself:
:$H \circ H = H$
The result follows.
{{qed}} | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Then:
:$\forall H \le \struct {G, \circ}:$
::$H^{-1} \circ H = H$
::$H \circ H^{-1} = H$
where $H \le G$ denotes that $H$ is a [[Definition:Subgroup|subgroup]] of $G$. | From [[Inverse of Subgroup]]:
:$H = H^{-1}$
From [[Product of Subgroup with Itself]]:
:$H \circ H = H$
The result follows.
{{qed}} | Product of Subgroup with Inverse | https://proofwiki.org/wiki/Product_of_Subgroup_with_Inverse | https://proofwiki.org/wiki/Product_of_Subgroup_with_Inverse | [
"Subgroups",
"Subset Products"
] | [
"Definition:Group",
"Definition:Subgroup"
] | [
"Inverse of Subgroup",
"Product of Subgroup with Itself"
] |
proofwiki-19130 | Equivalence Relation inducing Closed Quotient Set of Magma is Congruence Relation | Let $\struct {S, \circ}$ be a magma.
Let $\circ_\PP$ be the operation induced by $\circ$ on $\powerset S$, the power set of $S$.
Let $\RR$ be an equivalence relation on $S$.
Let $S / \RR$ denote the quotient set of $S$ induced by $\RR$.
Let the algebraic structure $\struct {S / \RR, \circ_\PP}$ be closed.
Then:
:$\RR$ ... | Let $x_1, y_1, x_2, y_2 \in S$ be arbitrary, such that:
{{begin-eqn}}
{{eqn | l = x_1
| o = \RR
| r = x_2
}}
{{eqn | l = y_1
| o = \RR
| r = y_2
}}
{{end-eqn}}
To demonstrate that $\RR$ is a congruence relation for $\circ$, we need to show that:
:$\paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 ... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $\circ_\PP$ be the [[Definition:Operation Induced on Power Set|operation induced]] by $\circ$ on $\powerset S$, the [[Definition:Power Set|power set]] of $S$.
Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $S / \RR$ d... | Let $x_1, y_1, x_2, y_2 \in S$ be arbitrary, such that:
{{begin-eqn}}
{{eqn | l = x_1
| o = \RR
| r = x_2
}}
{{eqn | l = y_1
| o = \RR
| r = y_2
}}
{{end-eqn}}
To demonstrate that $\RR$ is a [[Definition:Congruence Relation|congruence relation]] for $\circ$, we need to show that:
:$\paren {x_1... | Equivalence Relation inducing Closed Quotient Set of Magma is Congruence Relation | https://proofwiki.org/wiki/Equivalence_Relation_inducing_Closed_Quotient_Set_of_Magma_is_Congruence_Relation | https://proofwiki.org/wiki/Equivalence_Relation_inducing_Closed_Quotient_Set_of_Magma_is_Congruence_Relation | [
"Quotient Structures",
"Congruence Relations"
] | [
"Definition:Magma",
"Definition:Subset Product",
"Definition:Power Set",
"Definition:Equivalence Relation",
"Definition:Quotient Set",
"Definition:Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Congruence Relation",
"Definition:Operation Induced on Quo... | [
"Definition:Congruence Relation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Quotient Set",
"Definition:Subset Product",
"Definition:Equivalence Class",
"Definition:Congruence Relation",
"Definition:Logical Equivalence",
"Equivalence Class Equivalent Statements",
"Defin... |
proofwiki-19131 | Set of Normal Subgroups of Group is Subsemigroup of Power Set Semigroup | Let $\struct {G, \circ}$ be a group.
Let $\circ_\PP$ be the operation induced by $\circ$ on $\powerset G$, the power set of $G$.
Let $\HH$ be the set of all normal subgroups of $\struct {G, \circ}$.
Then the algebraic structure $\struct {\HH, \circ_\PP}$ is a subsemigroup of the algebraic structure $\struct {\powerset ... | From Power Structure of Group is Semigroup, we have that $\struct {\powerset G, \circ_\PP}$ is a semigroup.
Note that $\HH \subseteq \powerset G$.
From Subset Product of Normal Subgroups is Normal, $\struct {\HH, \circ_\PP}$ is closed.
By Subsemigroup Closure Test, $\struct {\HH, \circ_\PP}$ is a subsemigroup of $\stru... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $\circ_\PP$ be the [[Definition:Operation Induced on Power Set|operation induced]] by $\circ$ on $\powerset G$, the [[Definition:Power Set|power set]] of $G$.
Let $\HH$ be the [[Definition:Set|set]] of all [[Definition:Normal Subgroup|normal subgroups]] of... | From [[Power Structure of Group is Semigroup]], we have that $\struct {\powerset G, \circ_\PP}$ is a [[Definition:Semigroup|semigroup]].
Note that $\HH \subseteq \powerset G$.
From [[Subset Product of Normal Subgroups is Normal]], $\struct {\HH, \circ_\PP}$ is [[Definition:Closed Algebraic Structure|closed]].
By [[S... | Set of Normal Subgroups of Group is Subsemigroup of Power Set Semigroup | https://proofwiki.org/wiki/Set_of_Normal_Subgroups_of_Group_is_Subsemigroup_of_Power_Set_Semigroup | https://proofwiki.org/wiki/Set_of_Normal_Subgroups_of_Group_is_Subsemigroup_of_Power_Set_Semigroup | [
"Normal Subgroups",
"Subset Products",
"Semigroups"
] | [
"Definition:Group",
"Definition:Subset Product",
"Definition:Power Set",
"Definition:Set",
"Definition:Normal Subgroup",
"Definition:Algebraic Structure",
"Definition:Subsemigroup",
"Definition:Algebraic Structure"
] | [
"Power Structure of Group is Semigroup",
"Definition:Semigroup",
"Subset Product of Normal Subgroups is Normal",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Subsemigroup Closure Test",
"Definition:Subsemigroup"
] |
proofwiki-19132 | Set of Normal Subgroups of Group is Subsemigroup of Power Set under Intersection | Let $\struct {G, \circ}$ be a group.
Let $\HH$ be the set of all normal subgroups of $\struct {G, \circ}$.
Then the algebraic structure $\struct {\HH, \cap}$ is a subsemigroup of the algebraic structure $\struct {\powerset G, \cap}$. | From Power Set with Intersection is Commutative Monoid, we have that $\struct {\powerset G, \cap}$ is ''a fortiori'' a semigroup.
Note that $\HH \subseteq \powerset G$.
Let $H_1$ and $H_2$ be normal subgroups of $\struct {G, \circ}$.
From Intersection of Normal Subgroups is Normal, $H_1 \cap H_2$ is also a normal subgr... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $\HH$ be the [[Definition:Set|set]] of all [[Definition:Normal Subgroup|normal subgroups]] of $\struct {G, \circ}$.
Then the [[Definition:Algebraic Structure|algebraic structure]] $\struct {\HH, \cap}$ is a [[Definition:Subsemigroup|subsemigroup]] of the ... | From [[Power Set with Intersection is Commutative Monoid]], we have that $\struct {\powerset G, \cap}$ is ''[[Definition:A Fortiori|a fortiori]]'' a [[Definition:Semigroup|semigroup]].
Note that $\HH \subseteq \powerset G$.
Let $H_1$ and $H_2$ be [[Definition:Normal Subgroup|normal subgroups]] of $\struct {G, \circ}$... | Set of Normal Subgroups of Group is Subsemigroup of Power Set under Intersection | https://proofwiki.org/wiki/Set_of_Normal_Subgroups_of_Group_is_Subsemigroup_of_Power_Set_under_Intersection | https://proofwiki.org/wiki/Set_of_Normal_Subgroups_of_Group_is_Subsemigroup_of_Power_Set_under_Intersection | [
"Normal Subgroups",
"Set Intersection",
"Power Set",
"Semigroups"
] | [
"Definition:Group",
"Definition:Set",
"Definition:Normal Subgroup",
"Definition:Algebraic Structure",
"Definition:Subsemigroup",
"Definition:Algebraic Structure"
] | [
"Power Set with Intersection is Commutative Monoid",
"Definition:A Fortiori",
"Definition:Semigroup",
"Definition:Normal Subgroup",
"Intersection of Normal Subgroups is Normal",
"Definition:Normal Subgroup",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Subsemigroup Closure Test",
"D... |
proofwiki-19133 | Condition for Subgroup of Power Set of Group to be Quotient Group | Let $\struct {G, \circ}$ be a group.
Let $\circ_\PP$ be the operation induced by $\circ$ on $\powerset G$, the power set of $G$.
Let $\struct {\LL, \circ_\PP}$ be a subgroup of the algebraic structure $\struct {\powerset G, \circ_\PP}$.
Then:
:there exists a subgroup $H$ of $G$
:and a normal subgroup $K$ of $H$
:such t... | From Power Structure of Group is Semigroup, we have that $\struct {\powerset G, \circ_\PP}$ is a semigroup. | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $\circ_\PP$ be the [[Definition:Operation Induced on Power Set|operation induced]] by $\circ$ on $\powerset G$, the [[Definition:Power Set|power set]] of $G$.
Let $\struct {\LL, \circ_\PP}$ be a [[Definition:Subgroup|subgroup]] of the [[Definition:Algebrai... | From [[Power Structure of Group is Semigroup]], we have that $\struct {\powerset G, \circ_\PP}$ is a [[Definition:Semigroup|semigroup]]. | Condition for Subgroup of Power Set of Group to be Quotient Group | https://proofwiki.org/wiki/Condition_for_Subgroup_of_Power_Set_of_Group_to_be_Quotient_Group | https://proofwiki.org/wiki/Condition_for_Subgroup_of_Power_Set_of_Group_to_be_Quotient_Group | [
"Quotient Groups",
"Subset Products",
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Subset Product",
"Definition:Power Set",
"Definition:Subgroup",
"Definition:Algebraic Structure",
"Definition:Subgroup",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup"
] | [
"Power Structure of Group is Semigroup",
"Definition:Semigroup"
] |
proofwiki-19134 | Operation on Set for which Every Equivalence Relation is Congruence | Let $S$ be a set with at least $3$ elements.
Let $\circ$ be an operation on $S$ such that every equivalence relation on $S$ is a congruence relation for $\circ$.
Then $\circ$ is one of the following:
:the right operation $\to$
:the left operation $\gets$
:the constant operation $\sqbrk c$ for some $c \in S$. | First we note that from:
:Equivalence Relation is Congruence for Constant Operation
:Equivalence Relation is Congruence for Left Operation
:Equivalence Relation is Congruence for Right Operation
every equivalence relation on $S$ is a congruence relation for the right operation, the left operation and the constant opera... | Let $S$ be a [[Definition:Set|set]] with at least $3$ [[Definition:Element|elements]].
Let $\circ$ be an [[Definition:Binary Operation|operation]] on $S$ such that every [[Definition:Equivalence Relation|equivalence relation]] on $S$ is a [[Definition:Congruence Relation|congruence relation]] for $\circ$.
Then $\cir... | First we note that from:
:[[Equivalence Relation is Congruence for Constant Operation]]
:[[Equivalence Relation is Congruence for Left Operation]]
:[[Equivalence Relation is Congruence for Right Operation]]
every [[Definition:Equivalence Relation|equivalence relation]] on $S$ is a [[Definition:Congruence Relation|cong... | Operation on Set for which Every Equivalence Relation is Congruence | https://proofwiki.org/wiki/Operation_on_Set_for_which_Every_Equivalence_Relation_is_Congruence | https://proofwiki.org/wiki/Operation_on_Set_for_which_Every_Equivalence_Relation_is_Congruence | [
"Congruence Relations"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Operation/Binary Operation",
"Definition:Equivalence Relation",
"Definition:Congruence Relation",
"Definition:Right Operation",
"Definition:Left Operation",
"Definition:Constant Operation"
] | [
"Equivalence Relation is Congruence for Constant Operation",
"Equivalence Relation is Congruence for Left Operation",
"Equivalence Relation is Congruence for Right Operation",
"Definition:Equivalence Relation",
"Definition:Congruence Relation",
"Definition:Right Operation",
"Definition:Left Operation",
... |
proofwiki-19135 | Condition for Cosets of Subgroup of Monoid to be Partition | Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.
Let $\struct {H, \circ}$ be a subgroup of $\struct {S, \circ}$.
Let the identity element of $\struct {H, \circ}$ be $e$.
Then:
:the set of left cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$
and:
:the set of right cosets... | Because of the fact that it is not necessarily the case that $x \in S$ has an inverse, we cannot invoke the result Left Congruence Modulo Subgroup is Equivalence Relation.
Instead we prove partitionhood directly.
First we recall the definition of left coset and right coset:
:The '''left coset of $H$ by $x$''', is:
::$x... | Let $\struct {S, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity element]] is $e$.
Let $\struct {H, \circ}$ be a [[Definition:Subgroup|subgroup]] of $\struct {S, \circ}$.
Let the [[Definition:Identity Element|identity element]] of $\struct {H, \circ}$ be $e$.
Then:
:the [[Defi... | Because of the fact that it is not necessarily the case that $x \in S$ has an [[Definition:Inverse Element|inverse]], we cannot invoke the result [[Left Congruence Modulo Subgroup is Equivalence Relation]].
Instead we prove [[Definition:Set Partition|partitionhood]] directly.
First we recall the definition of [[Defi... | Condition for Cosets of Subgroup of Monoid to be Partition | https://proofwiki.org/wiki/Condition_for_Cosets_of_Subgroup_of_Monoid_to_be_Partition | https://proofwiki.org/wiki/Condition_for_Cosets_of_Subgroup_of_Monoid_to_be_Partition | [
"Cosets",
"Monoids"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Set",
"Definition:Coset/Left Coset",
"Definition:Set Partition",
"Definition:Set",
"Definition:Coset/Right Coset",
"Defi... | [
"Definition:Inverse (Abstract Algebra)/Inverse",
"Left Congruence Modulo Subgroup is Equivalence Relation",
"Definition:Set Partition",
"Definition:Coset/Left Coset",
"Definition:Coset/Right Coset",
"Definition:Coset/Left Coset",
"Definition:Coset/Right Coset",
"Definition:Set Partition",
"Definitio... |
proofwiki-19136 | Condition for Subgroup of Monoid to be Normal | Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.
Let $\struct {H, \circ}$ be a subgroup of $\struct {S, \circ}$.
Then:
::the set of left cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of $S$
:and:
::the set of right cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form ... | === Necessary Condition ===
Let $\struct {H, \circ}$ be a normal subgroup of $\struct {S, \circ}$.
Then by definition:
:$e \in H$
Hence from Condition for Cosets of Subgroup of Monoid to be Partition, the set of left cosets and the set of right cosets of $\struct {H, \circ}$ in $\struct {S, \circ}$ form a partition of ... | Let $\struct {S, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity element]] is $e$.
Let $\struct {H, \circ}$ be a [[Definition:Subgroup|subgroup]] of $\struct {S, \circ}$.
Then:
::the [[Definition:Set|set]] of [[Definition:Left Coset|left cosets]] of $\struct {H, \circ}$ in $\st... | === Necessary Condition ===
Let $\struct {H, \circ}$ be a [[Definition:Normal Subgroup of Monoid|normal subgroup]] of $\struct {S, \circ}$.
Then by definition:
:$e \in H$
Hence from [[Condition for Cosets of Subgroup of Monoid to be Partition]], the [[Definition:Set|set]] of [[Definition:Left Coset|left cosets]] and... | Condition for Subgroup of Monoid to be Normal | https://proofwiki.org/wiki/Condition_for_Subgroup_of_Monoid_to_be_Normal | https://proofwiki.org/wiki/Condition_for_Subgroup_of_Monoid_to_be_Normal | [
"Normal Subgroups",
"Monoids"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Set",
"Definition:Coset/Left Coset",
"Definition:Set Partition",
"Definition:Set",
"Definition:Coset/Right Coset",
"Definition:Set Partition",
"Definition:Equivalence Relation Induc... | [
"Definition:Normal Subgroup of Monoid",
"Condition for Cosets of Subgroup of Monoid to be Partition",
"Definition:Set",
"Definition:Coset/Left Coset",
"Definition:Set",
"Definition:Coset/Right Coset",
"Definition:Set Partition",
"Definition:Set",
"Definition:Coset/Left Coset",
"Definition:Set Part... |
proofwiki-19137 | Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid | Let $\struct {S, \circ}$ be a monoid whose identity element is $e$ such that $\struct {S, \circ}$ is specifically ''not'' a group.
Let $\struct {H, \circ}$ be the subgroup of $\struct {S, \circ}$ consisting of its invertible elements.
Let $N$ be the set of non-invertible elements of $\struct {S, \circ}$.
Let $\RR$ be t... | First we confirm from Invertible Elements of Monoid form Subgroup that $\struct {H, \circ}$ is in fact a subgroup of $\struct {S, \circ}$.
Let $h_1, h_2 \in H$ and $n_1, n_2 \in N$.
Immediately we have by {{Group-axiom|0}}:
:$h_1 \circ h_2 \in H$
{{AimForCont}}:
:$h_1 \circ n_1 \in H$
By {{Group-axiom|3}}:
:${h_1}^{-1}... | Let $\struct {S, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity element]] is $e$ such that $\struct {S, \circ}$ is specifically ''not'' a [[Definition:Group|group]].
Let $\struct {H, \circ}$ be the [[Definition:Subgroup|subgroup]] of $\struct {S, \circ}$ consisting of its [[Defi... | First we confirm from [[Invertible Elements of Monoid form Subgroup]] that $\struct {H, \circ}$ is in fact a [[Definition:Subgroup|subgroup]] of $\struct {S, \circ}$.
Let $h_1, h_2 \in H$ and $n_1, n_2 \in N$.
Immediately we have by {{Group-axiom|0}}:
:$h_1 \circ h_2 \in H$
{{AimForCont}}:
:$h_1 \circ n_1 \in H$
B... | Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid | https://proofwiki.org/wiki/Condition_for_Partition_between_Invertible_and_Non-Invertible_Elements_to_induce_Congruence_Relation_on_Monoid | https://proofwiki.org/wiki/Condition_for_Partition_between_Invertible_and_Non-Invertible_Elements_to_induce_Congruence_Relation_on_Monoid | [
"Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid",
"Monoids",
"Subgroups",
"Inverse Elements"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group",
"Definition:Subgroup",
"Definition:Invertible Element",
"Definition:Set",
"Definition:Invertible Element",
"Definition:Equivalence Relation Induced by Partition",
"Definition:Set Partition",
"Defi... | [
"Invertible Elements of Monoid form Subgroup",
"Definition:Subgroup",
"Definition:Contradiction",
"Definition:Right Inverse",
"Definition:Cancellable Operation",
"Definition:Commutative/Operation",
"Definition:Left Inverse",
"Definition:Invertible Element",
"Definition:Contradiction",
"Definition:... |
proofwiki-19138 | Existence of One-Sided Inverses on Natural Numbers whose Composition is Identity Mapping | Consider the set of natural numbers $\N$.
There exist mappings $f: \N \to \N$ and $g: \N \to \N$ such that:
:$g \circ f = I_\N$
where:
:$\circ$ denotes composition of mappings
:$I_\N$ denotes the identity mapping on $\N$
such that neither $f$ nor $g$ are permutations on $\N$. | Let $f: \N \to \N$ be the mapping defined as:
:$\forall x \in \N: \map f x = x + 1$
Let $g: \N \to \N$ be the mapping defined as:
:$\forall x \in \N: \map g x = \begin {cases} x - 1 & : x > 0 \\ 0 & : x = 0 \end {cases}$
It is apparent by inspection that:
:$f$ is injective
but
:$f$ is not surjective, as there exists no... | Consider the [[Definition:Natural Numbers|set of natural numbers]] $\N$.
There exist [[Definition:Mapping|mappings]] $f: \N \to \N$ and $g: \N \to \N$ such that:
:$g \circ f = I_\N$
where:
:$\circ$ denotes [[Definition:Composition of Mappings|composition of mappings]]
:$I_\N$ denotes the [[Definition:Identity Mappin... | Let $f: \N \to \N$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in \N: \map f x = x + 1$
Let $g: \N \to \N$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in \N: \map g x = \begin {cases} x - 1 & : x > 0 \\ 0 & : x = 0 \end {cases}$
It is apparent by inspection that:
:$f$ is [[Defini... | Existence of One-Sided Inverses on Natural Numbers whose Composition is Identity Mapping | https://proofwiki.org/wiki/Existence_of_One-Sided_Inverses_on_Natural_Numbers_whose_Composition_is_Identity_Mapping | https://proofwiki.org/wiki/Existence_of_One-Sided_Inverses_on_Natural_Numbers_whose_Composition_is_Identity_Mapping | [
"Natural Numbers",
"Inverse Mappings"
] | [
"Definition:Natural Numbers",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Identity Mapping",
"Definition:Permutation"
] | [
"Definition:Mapping",
"Definition:Mapping",
"Definition:Injection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Injection",
"Definition:Bijection",
"Definition:Permutation"
] |
proofwiki-19139 | Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid/Counterexample | Let $\circ$ be such that it is neither a cancellable operation nor a commutative operation.
Then it is not necessarily the case that either:
:$\RR$ is a congruence relation on $\circ$
or:
:the quotient structure $\struct {S / \RR, \circ_\RR}$ is isomorphic to $\struct {\Z_2, \times_2}$, the multiplicative monoid of int... | Consider the set of natural numbers $\N$.
Let $\N^\N$ denote the set of all mappings from $\N$ to $\N$.
Now consider the algebraic structure $\struct {\N^\N, \circ}$, where $\circ$ in this context denotes composition of mappings.
From Set of all Self-Maps under Composition forms Monoid, $\struct {\N^\N, \circ}$ is a mo... | Let $\circ$ be such that it is neither a [[Definition:Cancellable Operation|cancellable operation]] nor a [[Definition:Commutative Operation|commutative operation]].
Then it is not necessarily the case that either:
:$\RR$ is a [[Definition:Congruence Relation|congruence relation]] on $\circ$
or:
:the [[Definition:Quo... | Consider the [[Definition:Natural Numbers|set of natural numbers]] $\N$.
Let $\N^\N$ denote the [[Definition:Set of Mappings|set of all mappings]] from $\N$ to $\N$.
Now consider the [[Definition:Algebraic Structure|algebraic structure]] $\struct {\N^\N, \circ}$, where $\circ$ in this context denotes [[Definition:Com... | Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid/Counterexample | https://proofwiki.org/wiki/Condition_for_Partition_between_Invertible_and_Non-Invertible_Elements_to_induce_Congruence_Relation_on_Monoid/Counterexample | https://proofwiki.org/wiki/Condition_for_Partition_between_Invertible_and_Non-Invertible_Elements_to_induce_Congruence_Relation_on_Monoid/Counterexample | [
"Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid"
] | [
"Definition:Cancellable Operation",
"Definition:Commutative/Operation",
"Definition:Congruence Relation",
"Definition:Quotient Structure",
"Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism",
"Definition:Multiplicative Monoid of Integers Modulo m"
] | [
"Definition:Natural Numbers",
"Definition:Set of All Mappings",
"Definition:Algebraic Structure",
"Definition:Composition of Mappings",
"Set of all Self-Maps under Composition forms Monoid",
"Definition:Monoid",
"Composition of Mappings is not Commutative",
"Definition:Commutative/Operation",
"Defin... |
proofwiki-19140 | Condition on Congruence Relations for Cancellable Monoid to be Group | Let $\struct {S, \circ}$ be a cancellable monoid whose identity element is $e$.
Then:
:$\struct {S, \circ}$ is a group
{{iff}}:
:every non-trivial congruence relation on $\struct {S, \circ}$ is induced by a normal subgroup of $\struct {S, \circ}$. | === Necessary Condition ===
Let $\struct {S, \circ}$ be such that every non-trivial congruence relation on $\struct {S, \circ}$ is induced by a normal subgroup of $\struct {S, \circ}$.
Hence, let $\RR$ be an arbitrary non-trivial congruence relation.
From Condition for Subgroup of Monoid to be Normal, there exists a no... | Let $\struct {S, \circ}$ be a [[Definition:Cancellable Monoid|cancellable monoid]] whose [[Definition:Identity Element|identity element]] is $e$.
Then:
:$\struct {S, \circ}$ is a [[Definition:Group|group]]
{{iff}}:
:every non-[[Definition:Trivial Equivalence Relation|trivial]] [[Definition:Congruence Relation|congrue... | === Necessary Condition ===
Let $\struct {S, \circ}$ be such that every non-[[Definition:Trivial Equivalence Relation|trivial]] [[Definition:Congruence Relation|congruence relation]] on $\struct {S, \circ}$ is induced by a [[Definition:Normal Subgroup of Monoid|normal subgroup]] of $\struct {S, \circ}$.
Hence, let $\... | Condition on Congruence Relations for Cancellable Monoid to be Group | https://proofwiki.org/wiki/Condition_on_Congruence_Relations_for_Cancellable_Monoid_to_be_Group | https://proofwiki.org/wiki/Condition_on_Congruence_Relations_for_Cancellable_Monoid_to_be_Group | [
"Condition on Congruence Relations for Cancellable Monoid to be Group",
"Normal Subgroups",
"Cancellable Monoids"
] | [
"Definition:Cancellable Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group",
"Definition:Trivial Equivalence Relation",
"Definition:Congruence Relation",
"Definition:Normal Subgroup of Monoid"
] | [
"Definition:Trivial Equivalence Relation",
"Definition:Congruence Relation",
"Definition:Normal Subgroup of Monoid",
"Definition:Trivial Equivalence Relation",
"Definition:Congruence Relation",
"Condition for Subgroup of Monoid to be Normal",
"Definition:Normal Subgroup of Monoid",
"Definition:Set",
... |
proofwiki-19141 | Condition on Congruence Relations for Cancellable Monoid to be Group/Counterexample | Let $\struct {S, \circ}$ be a monoid which is not cancellable.
Let every non-trivial congruence relation on $\struct {S, \circ}$ be induced by a normal subgroup of $\struct {S, \circ}$.
Then it is not necessarily the case that $\struct {S, \circ}$ is a group. | {{Improve|Is all this necessary, considering we have Group is Cancellable Monoid?}}
Consider the Multiplicative Monoid of Integers Modulo $3$ $\struct {\Z_3, \times_3}$, defined by its Cayley table:
{{:Modulo Multiplication/Cayley Table/Modulo 3}}
It is noted that $\struct {\Z_3, \times_3}$ is not a cancellable monoid.... | Let $\struct {S, \circ}$ be a [[Definition:Cancellable Monoid|monoid]] which is not [[Definition:Cancellable Monoid|cancellable]].
Let every non-[[Definition:Trivial Equivalence Relation|trivial]] [[Definition:Congruence Relation|congruence relation]] on $\struct {S, \circ}$ be induced by a [[Definition:Normal Subgrou... | {{Improve|Is all this necessary, considering we have [[Group is Cancellable Monoid]]?}}
Consider the [[Definition:Multiplicative Monoid of Integers Modulo m|Multiplicative Monoid of Integers Modulo $3$]] $\struct {\Z_3, \times_3}$, defined by its [[Modulo Multiplication/Cayley Table/Modulo 3|Cayley table]]:
{{:Modulo ... | Condition on Congruence Relations for Cancellable Monoid to be Group/Counterexample | https://proofwiki.org/wiki/Condition_on_Congruence_Relations_for_Cancellable_Monoid_to_be_Group/Counterexample | https://proofwiki.org/wiki/Condition_on_Congruence_Relations_for_Cancellable_Monoid_to_be_Group/Counterexample | [
"Condition on Congruence Relations for Cancellable Monoid to be Group"
] | [
"Definition:Cancellable Monoid",
"Definition:Cancellable Monoid",
"Definition:Trivial Equivalence Relation",
"Definition:Congruence Relation",
"Definition:Normal Subgroup of Monoid",
"Definition:Group"
] | [
"Group is Cancellable Monoid",
"Definition:Multiplicative Monoid of Integers Modulo m",
"Modulo Multiplication/Cayley Table/Modulo 3",
"Definition:Cancellable Monoid",
"Definition:Cancellable Operation",
"Group is Cancellable Monoid",
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sid... |
proofwiki-19142 | Riesz-Markov-Kakutani Representation Theorem | Let $\struct {X, \tau}$ be a locally compact Hausdorff space.
Let $\map {C_c} X$ be the space of continuous complex functions with compact support on $X$.
Let $\Lambda$ be a positive linear functional on $\map {C_c} X$.
There exists a $\sigma$-algebra $\MM$ over $X$ which contains the Borel $\sigma$-algebra of $\struct... | === Lemma $1$ ===
{{:Riesz-Markov-Kakutani Representation Theorem/Lemma 1}}{{qed|lemma}} | Let $\struct {X, \tau}$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $\map {C_c} X$ be the [[Definition:Vector Space|space]] of [[Definition:Continuous Complex Function|continuous complex functions]] with [[Definition:Compact Topological Subspace|compact]] [[Definition:Suppo... | === [[Riesz-Markov-Kakutani Representation Theorem/Lemma 1|Lemma $1$]] ===
{{:Riesz-Markov-Kakutani Representation Theorem/Lemma 1}}{{qed|lemma}} | Riesz-Markov-Kakutani Representation Theorem | https://proofwiki.org/wiki/Riesz-Markov-Kakutani_Representation_Theorem | https://proofwiki.org/wiki/Riesz-Markov-Kakutani_Representation_Theorem | [
"Measure Theory",
"Riesz-Markov-Kakutani Representation Theorem"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Vector Space",
"Definition:Continuous Complex Function",
"Definition:Compact Topological Space/Subspace",
"Definition:Support of Continuous Mapping",
"Definition:Positive Linear Functional",
"Definition:Sigma-Algebra",
"Definition:Borel Sigma-A... | [
"Riesz-Markov-Kakutani Representation Theorem/Lemma 1"
] |
proofwiki-19143 | Vitali-Carathéodory Theorem | Let $\struct {X, \tau}$ be a locally compact Hausdorff space.
Let $\MM$ be a $\sigma$-algebra over $X$ which contains the Borel $\sigma$-algebra generated by $\tau$.
Let $\mu$ be a Radon measure on $\MM$.
Let $f \in \map {\LL^1} \mu$, where $\map {\LL^1} \mu$ denotes the (real) Lebesgue 1-space of $\mu$.
For all $\epsi... | Let:
:$\forall x \in X: \map f x \ge 0$
and:
:$\exists x \in X: \map f x \ne 0$
By Measurable Function is Pointwise Limit of Simple Functions, there exists a sequence:
:$\sequence {s_n} \in \paren {\map \EE \MM}^\N$
where $\map \EE \MM$ denotes the space of simple functions on $\struct {X, \MM}$.
By Pointwise Differen... | Let $\struct {X, \tau}$ be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]].
Let $\MM$ be a [[Definition:Sigma-Algebra|$\sigma$-algebra]] over $X$ which contains the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra generated by $\tau$]].
Let $\mu$ be a [[Definition:Radon Measure... | Let:
:$\forall x \in X: \map f x \ge 0$
and:
:$\exists x \in X: \map f x \ne 0$
By [[Measurable Function is Pointwise Limit of Simple Functions]], there exists a [[Definition:Sequence|sequence]]:
:$\sequence {s_n} \in \paren {\map \EE \MM}^\N$
where $\map \EE \MM$ denotes the [[Definition:Space of Simple Functions|sp... | Vitali-Carathéodory Theorem | https://proofwiki.org/wiki/Vitali-Carathéodory_Theorem | https://proofwiki.org/wiki/Vitali-Carathéodory_Theorem | [
"Lower Semicontinuity",
"Upper Semicontinuity",
"Real Analysis",
"Hausdorff Spaces",
"Lower Semicontinuity",
"Upper Semicontinuity"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Sigma-Algebra",
"Definition:Borel Sigma-Algebra",
"Definition:Radon Measure",
"Definition:Lebesgue Space",
"Definition:Upper Semicontinuous",
"Definition:Bounded Above Mapping/Real-Valued",
"Definition:Lower Semicontinuous",
"Definition:Bound... | [
"Measurable Function is Pointwise Limit of Simple Functions",
"Definition:Sequence",
"Definition:Space of Simple Functions",
"Pointwise Difference of Simple Functions is Simple Function",
"Definition:Sequence",
"Definition:Simple Function",
"Definition:Simple Function",
"Limit of Sequence is Sum of Di... |
proofwiki-19144 | Even Integers not Sum of Two Abundant Numbers | The even integers which are not the sum of $2$ abundant numbers are:
:All even integers less than $24$;
:$26, 28, 34, 46$ | From Sequence of Abundant Numbers, the first few abundant numbers are:
:$12, 18, 20, 24, 30, 36, 40, 42, 48$
Immediately we see that any number less than $2 \times 12 = 24$ cannot be expressed as a sum of $2$ abundant numbers.
The sum of the first $2$ abundant numbers is $12 + 18 = 30$, so $26$ and $28$ are not sums of... | The [[Definition:Even Integer|even integers]] which are not the [[Definition:Integer Addition|sum]] of $2$ [[Definition:Abundant Number|abundant numbers]] are:
:All [[Definition:Even Integer|even integers]] less than $24$;
:$26, 28, 34, 46$ | From [[Sequence of Abundant Numbers]], the first few [[Definition:Abundant Number|abundant numbers]] are:
:$12, 18, 20, 24, 30, 36, 40, 42, 48$
Immediately we see that any number less than $2 \times 12 = 24$ cannot be expressed as a [[Definition:Integer Addition|sum]] of $2$ [[Definition:Abundant Number|abundant numbe... | Even Integers not Sum of Two Abundant Numbers | https://proofwiki.org/wiki/Even_Integers_not_Sum_of_Two_Abundant_Numbers | https://proofwiki.org/wiki/Even_Integers_not_Sum_of_Two_Abundant_Numbers | [
"Abundant Numbers"
] | [
"Definition:Even Integer",
"Definition:Addition/Integers",
"Definition:Abundant Number",
"Definition:Even Integer"
] | [
"Abundant Number/Sequence",
"Definition:Abundant Number",
"Definition:Addition/Integers",
"Definition:Abundant Number",
"Definition:Addition/Integers",
"Definition:Abundant Number",
"Definition:Addition/Integers",
"Definition:Abundant Number",
"Definition:Subtraction/Difference",
"Definition:Abund... |
proofwiki-19145 | Sum of Reciprocals of Sequence of Pairs of Odd Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} }
| r = \dfrac 1 {1 \times 2} + \dfrac 1 {2 \times 5} + \dfrac 1 {5 \times 13} + \dfrac 1 {13 \times 34} + \cdots
| c =
}}
{{eqn | r = \phi^{-1}
| c =
}}
{{end-eqn}}
where:
:$F_k$ denotes the $k$th Fibonacci number
:$... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} }
| r = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} } \paren {\dfrac {F_{2 k + 1} - F_{2 k - 1} } {F_{2 k + 1} - F_{2 k - 1} } }
| c = multiplying by $1$
}}
{{eqn | r = \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} }
| r = \dfrac 1 {1 \times 2} + \dfrac 1 {2 \times 5} + \dfrac 1 {5 \times 13} + \dfrac 1 {13 \times 34} + \cdots
| c =
}}
{{eqn | r = \phi^{-1}
| c =
}}
{{end-eqn}}
where:
:$F_k$ denotes the $k$th [[Definition:Fibon... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} }
| r = \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k - 1} F_{2 k + 1} } \paren {\dfrac {F_{2 k + 1} - F_{2 k - 1} } {F_{2 k + 1} - F_{2 k - 1} } }
| c = multiplying by $1$
}}
{{eqn | r = \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_... | Sum of Reciprocals of Sequence of Pairs of Odd Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Sequence_of_Pairs_of_Odd_Index_Consecutive_Fibonacci_Numbers_is_Reciprocal_of_Golden_Mean | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Sequence_of_Pairs_of_Odd_Index_Consecutive_Fibonacci_Numbers_is_Reciprocal_of_Golden_Mean | [
"Fibonacci Numbers",
"Golden Mean"
] | [
"Definition:Fibonacci Number",
"Definition:Golden Mean"
] | [
"Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared",
"Power of Golden Mean as Sum of Smaller Powers"
] |
proofwiki-19146 | Condition for Equivalence Relation for Max Operation on Natural Numbers to be Congruence | Let $\RR$ be an equivalence relation on the set of natural numbers $\N$.
Let $\vee$ denote the max operation on $\N$:
:$\forall a, b \in \N: a \vee b := \max \set {a, b}$
Then:
:$\RR$ is a congruence relation for $\vee$ on $\N$
{{iff}}:
:every equivalence class under $\RR$ is an order-convex subset of $\N$. | === Necessary Condition ===
Suppose every equivalence class under $\RR$ is a order-convex subset of $\N$.
Let $x_1, x_2, y_1, y_2 \in \N$ such that $x_1 \mathrel \RR x_2$ and $y_1 \mathrel \RR y_2$.
{{WLOG}}, suppose:
:$x_1 \le x_2$
:$y_1 \le y_2$
:$x_1 \le y_1$
Then we have two cases: | Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on the [[Definition:Natural Numbers|set of natural numbers]] $\N$.
Let $\vee$ denote the [[Definition:Max Operation|max operation]] on $\N$:
:$\forall a, b \in \N: a \vee b := \max \set {a, b}$
Then:
:$\RR$ is a [[Definition:Congruence Relation... | === Necessary Condition ===
Suppose every [[Definition:Equivalence Class|equivalence class]] under $\RR$ is a [[Definition:Order-Convex Subset of Natural Numbers|order-convex subset]] of $\N$.
Let $x_1, x_2, y_1, y_2 \in \N$ such that $x_1 \mathrel \RR x_2$ and $y_1 \mathrel \RR y_2$.
{{WLOG}}, suppose:
:$x_1 \le x_... | Condition for Equivalence Relation for Max Operation on Natural Numbers to be Congruence | https://proofwiki.org/wiki/Condition_for_Equivalence_Relation_for_Max_Operation_on_Natural_Numbers_to_be_Congruence | https://proofwiki.org/wiki/Condition_for_Equivalence_Relation_for_Max_Operation_on_Natural_Numbers_to_be_Congruence | [
"Congruence Relations",
"Max Operation",
"Natural Numbers"
] | [
"Definition:Equivalence Relation",
"Definition:Natural Numbers",
"Definition:Max Operation",
"Definition:Congruence Relation",
"Definition:Equivalence Class",
"Definition:Order-Convex Subset of Natural Numbers"
] | [
"Definition:Equivalence Class",
"Definition:Order-Convex Subset of Natural Numbers",
"Definition:Order-Convex Subset of Natural Numbers",
"Definition:Equivalence Class",
"Definition:Order-Convex Subset of Natural Numbers",
"Definition:Equivalence Class",
"Definition:Order-Convex Subset of Natural Number... |
proofwiki-19147 | Binary Operation on Natural Numbers on which Congruence Relations induce Order-Convex Equivalence Classes | Let $\N$ denote the set of natural numbers: $\set {0, 1, 2, \ldots}$
Let $\circ$ be a binary operation on $\N$ with the following properties:
:$\paren {\text O 1}: \quad$ $\circ$ has an identity element $e$
:$\paren {\text O 2}: \quad$ Every equivalence relation $\RR$ on $\N$ whose equivalence classes are order-convex ... | Let $a, b \in \N$ where $e \le a < b$.
Let $\RR_1$ be an equivalence relation on $\N$ with equivalence classes:
:$\closedint 0 {b - 1}, \set b, \hointr {b + 1} \infty$
By inspection, each of these equivalence classes are non-empty order-convex subsets of $\N$.
Hence $\RR_1$ is a congruence relation for $\circ$.
We have... | Let $\N$ denote the [[Definition:Natural Number|set of natural numbers]]: $\set {0, 1, 2, \ldots}$
Let $\circ$ be a [[Definition:Binary Operation|binary operation]] on $\N$ with the following properties:
:$\paren {\text O 1}: \quad$ $\circ$ has an [[Definition:Identity Element|identity element]] $e$
:$\paren {\text O... | Let $a, b \in \N$ where $e \le a < b$.
Let $\RR_1$ be an [[Definition:Equivalence Relation|equivalence relation]] on $\N$ with [[Definition:Equivalence Class|equivalence classes]]:
:$\closedint 0 {b - 1}, \set b, \hointr {b + 1} \infty$
By inspection, each of these [[Definition:Equivalence Class|equivalence classes]]... | Binary Operation on Natural Numbers on which Congruence Relations induce Order-Convex Equivalence Classes | https://proofwiki.org/wiki/Binary_Operation_on_Natural_Numbers_on_which_Congruence_Relations_induce_Order-Convex_Equivalence_Classes | https://proofwiki.org/wiki/Binary_Operation_on_Natural_Numbers_on_which_Congruence_Relations_induce_Order-Convex_Equivalence_Classes | [
"Congruence Relations",
"Natural Numbers",
"Order-Convex Sets"
] | [
"Definition:Natural Numbers",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Equivalence Relation",
"Definition:Equivalence Class",
"Definition:Order-Convex Subset of Natural Numbers",
"Definition:Congruence Relation",
"Definition:Max ... | [
"Definition:Equivalence Relation",
"Definition:Equivalence Class",
"Definition:Equivalence Class",
"Definition:Non-Empty Set",
"Definition:Order-Convex Subset of Natural Numbers",
"Definition:Congruence Relation",
"Definition:Congruence Relation",
"Definition:Congruence Relation",
"Definition:Equiva... |
proofwiki-19148 | Equivalent Characterizations of Nonsingular Matrix | Let $\mathbf A$ be a square matrix of order $n$ over a field $K$.
{{TFAE}}
:$(1):\quad$ $\mathbf A$ is nonsingular
:$(2):\quad$ The transpose of $\mathbf A$ is nonsingular
:$(3):\quad$ $\mathbf A$ row-reduces to the identity matrix $\mathbf I_n$
:$(4):\quad$ The null space of $\mathbf A$ contains only the zero vector $... | === $(3)$ iff $(5)$ ===
See Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 5.
{{qed|lemma}} | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$ over a [[Definition:Field (Abstract Algebra)|field]] $K$.
{{TFAE}}
:$(1):\quad$ $\mathbf A$ is [[Definition:Nonsingular Matrix|nonsingular]]
:$(2):\quad$ The [[Definition:Transpose of Matrix|transpose]... | === $(3)$ iff $(5)$ ===
See [[Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 5]].
{{qed|lemma}} | Equivalent Characterizations of Nonsingular Matrix | https://proofwiki.org/wiki/Equivalent_Characterizations_of_Nonsingular_Matrix | https://proofwiki.org/wiki/Equivalent_Characterizations_of_Nonsingular_Matrix | [
"Nonsingular Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Nonsingular Matrix",
"Definition:Transpose of Matrix",
"Definition:Nonsingular Matrix",
"Definition:Echelon Matrix/Reduced Echelon Form",
"Definition:Unit Matrix",
"Definiti... | [
"Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 5",
"Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations"
] |
proofwiki-19149 | Mapping on Integers is Endomorphism of Max or Min Operation iff Increasing | Let $\Z$ denote the set of integers.
Let $f: \Z \to \Z$ be a mapping on $\Z$.
Let $\vee$ and $\wedge$ be the max operation and min operation on $\Z$ defined as:
{{begin-eqn}}
{{eqn | q = \forall x, y \in \Z
| l = x \vee y
| r = \max \set {x, y}
}}
{{eqn | l = x \wedge y
| r = \min \set {x, y}
}}
{{end... | === Necessary Condition ===
Suppose $f$ is an increasing mapping.
Let $x, y \in \Z$ and suppose $x \le y$.
By definition of an increasing mapping, we have $\map f x \le \map f y$.
Therefore:
:$\map f x \vee \map f y = \map f y = \map f {x \vee y}$
:$\map f x \wedge \map f y = \map f x = \map f {x \wedge y}$
Hence $f$ i... | Let $\Z$ denote the [[Definition:Integer|set of integers]].
Let $f: \Z \to \Z$ be a [[Definition:Mapping|mapping]] on $\Z$.
Let $\vee$ and $\wedge$ be the [[Definition:Max Operation|max operation]] and [[Definition:Min Operation|min operation]] on $\Z$ defined as:
{{begin-eqn}}
{{eqn | q = \forall x, y \in \Z
... | === Necessary Condition ===
Suppose $f$ is an [[Definition:Increasing Mapping|increasing mapping]].
Let $x, y \in \Z$ and suppose $x \le y$.
By definition of an [[Definition:Increasing Mapping|increasing mapping]], we have $\map f x \le \map f y$.
Therefore:
:$\map f x \vee \map f y = \map f y = \map f {x \vee y}$
... | Mapping on Integers is Endomorphism of Max or Min Operation iff Increasing | https://proofwiki.org/wiki/Mapping_on_Integers_is_Endomorphism_of_Max_or_Min_Operation_iff_Increasing | https://proofwiki.org/wiki/Mapping_on_Integers_is_Endomorphism_of_Max_or_Min_Operation_iff_Increasing | [
"Endomorphisms",
"Min Operation",
"Max Operation",
"Increasing Mappings"
] | [
"Definition:Integer",
"Definition:Mapping",
"Definition:Max Operation",
"Definition:Min Operation",
"Definition:Endomorphism",
"Definition:Increasing/Mapping"
] | [
"Definition:Increasing/Mapping",
"Definition:Increasing/Mapping",
"Definition:Endomorphism",
"Conjunction implies Disjunction",
"Definition:Endomorphism",
"Definition:Endomorphism",
"Definition:Increasing/Mapping",
"Definition:Endomorphism",
"Definition:Increasing/Mapping",
"Definition:Endomorphis... |
proofwiki-19150 | Mapping on Integers is Homomorphism between Max or Min Operation iff Decreasing | Let $\Z$ denote the set of integers.
Let $f: \Z \to \Z$ be a mapping on $\Z$.
{{MissingLinks|min and max operations are defined on PW}}
Let $\vee$ and $\wedge$ be the operations on $\Z$ defined as:
{{begin-eqn}}
{{eqn | q = \forall x, y \in \Z
| l = x \vee y
| r = \max \set {x, y}
}}
{{eqn | l = x \wedge y
... | === Necessary Condition ===
Let $f$ be a decreasing mapping.
Let $x, y \in \Z$ such that $x \le y$.
By definition of a decreasing mapping, we have $\map f x \ge \map f y$.
Therefore:
:$\map f x \vee \map f y = \map f x = \map f {x \wedge y}$
:$\map f x \wedge \map f y = \map f y = \map f {x \vee y}$
Hence:
:$f$ is a ho... | Let $\Z$ denote the [[Definition:Integer|set of integers]].
Let $f: \Z \to \Z$ be a [[Definition:Mapping|mapping]] on $\Z$.
{{MissingLinks|min and max operations are defined on PW}}
Let $\vee$ and $\wedge$ be the [[Definition:Binary Operation|operations]] on $\Z$ defined as:
{{begin-eqn}}
{{eqn | q = \forall x, y \i... | === Necessary Condition ===
Let $f$ be a [[Definition:Decreasing Mapping|decreasing mapping]].
Let $x, y \in \Z$ such that $x \le y$.
By definition of a [[Definition:Decreasing Mapping|decreasing mapping]], we have $\map f x \ge \map f y$.
Therefore:
:$\map f x \vee \map f y = \map f x = \map f {x \wedge y}$
:$\map... | Mapping on Integers is Homomorphism between Max or Min Operation iff Decreasing | https://proofwiki.org/wiki/Mapping_on_Integers_is_Homomorphism_between_Max_or_Min_Operation_iff_Decreasing | https://proofwiki.org/wiki/Mapping_on_Integers_is_Homomorphism_between_Max_or_Min_Operation_iff_Decreasing | [
"Homomorphisms (Abstract Algebra)",
"Min Operation",
"Max Operation",
"Decreasing Mappings"
] | [
"Definition:Integer",
"Definition:Mapping",
"Definition:Operation/Binary Operation",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Decreasing/Mapping"
] | [
"Definition:Decreasing/Mapping",
"Definition:Decreasing/Mapping",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Homomorphism (Abstract Algebra)",
"Conjunction implies Disjunction",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Decre... |
proofwiki-19151 | Inner Automorphism Maps Subgroup to Itself iff Normal/Sufficient Condition | Let $G$ be a group.
For $x \in G$, let $\kappa_x$ denote the inner automorphism of $x$ in $G$.
Let $H$ be a normal subgroup of $G$.
Then:
:$\forall x \in G: \kappa_x \sqbrk H = H$ | Let $H$ be a normal subgroup of $G$.
Let $x \in G$ be arbitrary.
By definition, $\kappa_x: G \to G$ is a mapping defined as:
:$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$
Let $n \in N$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\kappa_x} n
| r = x n x^{-1}
| c =
}}
{{eqn | o = \in
| r = N
| c ... | Let $G$ be a [[Definition:Group|group]].
For $x \in G$, let $\kappa_x$ denote the [[Definition:Inner Automorphism|inner automorphism]] of $x$ in $G$.
Let $H$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Then:
:$\forall x \in G: \kappa_x \sqbrk H = H$ | Let $H$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $x \in G$ be arbitrary.
By definition, $\kappa_x: G \to G$ is a [[Definition:Mapping|mapping]] defined as:
:$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$
Let $n \in N$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\kappa_x} n
| r = x n x^{-... | Inner Automorphism Maps Subgroup to Itself iff Normal/Sufficient Condition | https://proofwiki.org/wiki/Inner_Automorphism_Maps_Subgroup_to_Itself_iff_Normal/Sufficient_Condition | https://proofwiki.org/wiki/Inner_Automorphism_Maps_Subgroup_to_Itself_iff_Normal/Sufficient_Condition | [
"Inner Automorphism Maps Subgroup to Itself iff Normal"
] | [
"Definition:Group",
"Definition:Inner Automorphism",
"Definition:Normal Subgroup"
] | [
"Definition:Normal Subgroup",
"Definition:Mapping"
] |
proofwiki-19152 | Inner Automorphism Maps Subgroup to Itself iff Normal/Necessary Condition | Let $G$ be a group.
For $x \in G$, let $\kappa_x$ denote the inner automorphism of $x$ in $G$.
Suppose that:
:$\forall x \in G: \kappa_x \sqbrk H = H$
Then $H$ is a normal subgroup of $G$. | Suppose that:
:$\forall x \in G: \kappa_x \sqbrk H = H$
Let $x \in G$ be arbitrary.
By definition of inner automorphism of $x$ in $G$:
:$\forall h \in H: x h x^{-1} \in H$
So, by definition, $H$ is a normal subgroup of $G$ | Let $G$ be a [[Definition:Group|group]].
For $x \in G$, let $\kappa_x$ denote the [[Definition:Inner Automorphism|inner automorphism]] of $x$ in $G$.
Suppose that:
:$\forall x \in G: \kappa_x \sqbrk H = H$
Then $H$ is a [[Definition:Normal Subgroup|normal subgroup]] of $G$. | Suppose that:
:$\forall x \in G: \kappa_x \sqbrk H = H$
Let $x \in G$ be arbitrary.
By definition of [[Definition:Inner Automorphism|inner automorphism]] of $x$ in $G$:
:$\forall h \in H: x h x^{-1} \in H$
So, by definition, $H$ is a [[Definition:Normal Subgroup|normal subgroup]] of $G$ | Inner Automorphism Maps Subgroup to Itself iff Normal/Necessary Condition | https://proofwiki.org/wiki/Inner_Automorphism_Maps_Subgroup_to_Itself_iff_Normal/Necessary_Condition | https://proofwiki.org/wiki/Inner_Automorphism_Maps_Subgroup_to_Itself_iff_Normal/Necessary_Condition | [
"Inner Automorphism Maps Subgroup to Itself iff Normal"
] | [
"Definition:Group",
"Definition:Inner Automorphism",
"Definition:Normal Subgroup"
] | [
"Definition:Inner Automorphism",
"Definition:Normal Subgroup"
] |
proofwiki-19153 | Subset of Domain is Subset of Preimage of Image/Equality does Not Necessarily Hold | It is not necessarily the case that:
:$A \subseteq S \implies A = \paren {f^{-1} \circ f} \sqbrk A$ | Proof by Counterexample:
Let:
:$S = \set {0, 1}$
:$T = \set 2$
Let $f: S \to T$ be defined as:
:$\map f 0 = 2$
:$\map f 1 = 2$
Let $A \subseteq S$ be defined as:
:$A = \set 0$
Then we have:
:$f \sqbrk A = \set 2$
but:
:$f^{-1} \circ f \sqbrk A = f^{-1} \sqbrk 2 = \set {0, 1}$
That is:
:$A \subseteq \paren {f^{-1} \circ... | It is not necessarily the case that:
:$A \subseteq S \implies A = \paren {f^{-1} \circ f} \sqbrk A$ | [[Proof by Counterexample]]:
Let:
:$S = \set {0, 1}$
:$T = \set 2$
Let $f: S \to T$ be defined as:
:$\map f 0 = 2$
:$\map f 1 = 2$
Let $A \subseteq S$ be defined as:
:$A = \set 0$
Then we have:
:$f \sqbrk A = \set 2$
but:
:$f^{-1} \circ f \sqbrk A = f^{-1} \sqbrk 2 = \set {0, 1}$
That is:
:$A \subseteq \paren {f^... | Subset of Domain is Subset of Preimage of Image/Equality does Not Necessarily Hold | https://proofwiki.org/wiki/Subset_of_Domain_is_Subset_of_Preimage_of_Image/Equality_does_Not_Necessarily_Hold | https://proofwiki.org/wiki/Subset_of_Domain_is_Subset_of_Preimage_of_Image/Equality_does_Not_Necessarily_Hold | [
"Subset of Domain is Subset of Preimage of Image"
] | [] | [
"Proof by Counterexample"
] |
proofwiki-19154 | Number times Recurring Part of Reciprocal gives 9-Repdigit/Generalization | Let $M$ be an arbitrary integer.
Then:
:$M \equiv \sqbrk {mmm \dots m} \pmod {10^c}$
for some positive integer $c$, {{iff}}:
:$M \times n \equiv -1 \pmod {10^c}$
In other words, the last $c$ digits of $M$ coincide with that of $\sqbrk {mmm \dots m}$ {{iff}} the last $c$ digits of $M \times n$ are all $9$s. | $\sqbrk {mmm \dots m}$ can be expressed as:
:$\ds \sum_{k \mathop = 0}^{K - 1} m 10^{k d}$
for some sufficiently large $K > \dfrac c d$. | Let $M$ be an arbitrary [[Definition:Integer|integer]].
Then:
:$M \equiv \sqbrk {mmm \dots m} \pmod {10^c}$
for some [[Definition:Positive Integer|positive integer]] $c$, {{iff}}:
:$M \times n \equiv -1 \pmod {10^c}$
In other words, the last $c$ [[Definition:Digit|digits]] of $M$ coincide with that of $\sqbrk {mmm \... | $\sqbrk {mmm \dots m}$ can be expressed as:
:$\ds \sum_{k \mathop = 0}^{K - 1} m 10^{k d}$
for some sufficiently large $K > \dfrac c d$. | Number times Recurring Part of Reciprocal gives 9-Repdigit/Generalization | https://proofwiki.org/wiki/Number_times_Recurring_Part_of_Reciprocal_gives_9-Repdigit/Generalization | https://proofwiki.org/wiki/Number_times_Recurring_Part_of_Reciprocal_gives_9-Repdigit/Generalization | [
"Number times Recurring Part of Reciprocal gives 9-Repdigit"
] | [
"Definition:Integer",
"Definition:Positive/Integer",
"Definition:Digit",
"Definition:Digit"
] | [] |
proofwiki-19155 | Midy's Theorem | Let $p$ be a prime.
Let $a \in \set {1, 2, \ldots, p - 1}$ and $b > 1$ be integers.
Let $N$ be the recurring part of the expansion of $\dfrac a p$ in base $b$.
Let $\alpha$ be the period of recurrence of $N$.
Let $\alpha = c k$ for (strictly) positive integers $c > 1$ and $k$.
Then $N$ is divisible by $b^k - 1$.
Moreov... | === $N$ is divisible by $b^k - 1$ ===
By definition of recurrence, we have:
:$\dfrac a p b^\alpha = N + \dfrac a p$
Moreover, by definition of period of recurrence, $\alpha$ is the smallest positive integer for which this is true.
Rearranging, we obtain:
:$\dfrac a p \paren {b^\alpha - 1} = N$
In particular, $a \paren ... | Let $p$ be a [[Definition:Prime Number|prime]].
Let $a \in \set {1, 2, \ldots, p - 1}$ and $b > 1$ be [[Definition:Integer|integers]].
Let $N$ be the [[Definition:Recurring Part of Recurring Basis Expansion|recurring part]] of the expansion of $\dfrac a p$ in base $b$.
Let $\alpha$ be the [[Definition:Period of Recu... | === $N$ is divisible by $b^k - 1$ ===
By definition of [[Definition:Recurrence of Basis Expansion|recurrence]], we have:
:$\dfrac a p b^\alpha = N + \dfrac a p$
Moreover, by definition of [[Definition:Period of Recurrence|period of recurrence]], $\alpha$ is the smallest [[Definition:Strictly Positive Integer|positiv... | Midy's Theorem | https://proofwiki.org/wiki/Midy's_Theorem | https://proofwiki.org/wiki/Midy's_Theorem | [
"Midy's Theorem",
"Basis Expansions"
] | [
"Definition:Prime Number",
"Definition:Integer",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Strictly Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Basis Expansion/Recurrence",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Strictly Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Integer",
"Definition:Prime Number",
"Definition:Strictly Positive/Integer",
"Definition:Int... |
proofwiki-19156 | Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup | Let $G$ be a group.
Let $H$ and $K$ be normal subgroups of $G$.
Let $H \subseteq K$.
Then $H$ is a normal subgroup of $K$. | {{begin-eqn}}
{{eqn | q = \forall g \in G
| l = g K
| r = K g
| c = {{Defof|Normal Subgroup}}
}}
{{eqn | q = \forall g \in G
| l = g H
| r = H g
| c = {{Defof|Normal Subgroup}}
}}
{{eqn | ll= \leadsto
| q = \forall g \in K
| l = g H
| r = H g
| c = as $K \subs... | Let $G$ be a [[Definition:Group|group]].
Let $H$ and $K$ be [[Definition:Normal Subgroup|normal subgroups]] of $G$.
Let $H \subseteq K$.
Then $H$ is a [[Definition:Normal Subgroup|normal subgroup]] of $K$. | {{begin-eqn}}
{{eqn | q = \forall g \in G
| l = g K
| r = K g
| c = {{Defof|Normal Subgroup}}
}}
{{eqn | q = \forall g \in G
| l = g H
| r = H g
| c = {{Defof|Normal Subgroup}}
}}
{{eqn | ll= \leadsto
| q = \forall g \in K
| l = g H
| r = H g
| c = as $K \subs... | Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup | https://proofwiki.org/wiki/Normal_Subgroup_which_is_Subset_of_Normal_Subgroup_is_Normal_in_Subgroup | https://proofwiki.org/wiki/Normal_Subgroup_which_is_Subset_of_Normal_Subgroup_is_Normal_in_Subgroup | [
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Normal Subgroup"
] |
proofwiki-19157 | Exist Term in Arithmetic Sequence Divisible by Number | Let $\sequence {a_k}$ be an $n$-term arithmetic sequence in $\Z$ defined by:
:$a_k = a_0 + k d$
for $k = 0, 1, 2, \ldots, n - 1$.
Let $b$ be a (strictly) positive integer such that $b$ and $d$ are coprime and $b \le n$.
Then there exists a term in $\sequence {a_k}$ that is divisible by $b$. | We claim that at least one of the first $b$ terms:
:$a_0, a_0 + d, a_0 + 2 d, \dots, a_0 + \paren {b - 1} d$
is divisible by $b$.
Let $S$ be the set of remainders of each term after division by $b$.
$S$ can take on values of $0 \le r < b$.
If $0 \in S$ then the proof is complete.
{{AimForCont}} that $0 \ne S$.
Since th... | Let $\sequence {a_k}$ be an $n$-[[Definition:Term of Sequence|term]] [[Definition:Arithmetic Sequence|arithmetic sequence]] in $\Z$ defined by:
:$a_k = a_0 + k d$
for $k = 0, 1, 2, \ldots, n - 1$.
Let $b$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]] such that $b$ and $d$ are [[Definition:C... | We claim that at least one of the first $b$ [[Definition:Term of Sequence|terms]]:
:$a_0, a_0 + d, a_0 + 2 d, \dots, a_0 + \paren {b - 1} d$
is [[Definition:Divisor of Integer|divisible]] by $b$.
Let $S$ be the [[Definition:Set|set]] of [[Definition:Remainder|remainders]] of each [[Definition:Term of Sequence|term]] a... | Exist Term in Arithmetic Sequence Divisible by Number | https://proofwiki.org/wiki/Exist_Term_in_Arithmetic_Sequence_Divisible_by_Number | https://proofwiki.org/wiki/Exist_Term_in_Arithmetic_Sequence_Divisible_by_Number | [
"Arithmetic Sequences",
"Divisibility"
] | [
"Definition:Term of Sequence",
"Definition:Arithmetic Sequence",
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Term of Sequence",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Term of Sequence",
"Definition:Divisor (Algebra)/Integer",
"Definition:Set",
"Definition:Remainder",
"Definition:Term of Sequence",
"Definition:Integer Division",
"Definition:Term of Sequence",
"Definition:Remainder",
"Dirichlet's Box Principle/Corollary",
"Definition:Term of Sequence"... |
proofwiki-19158 | Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence | Let $\struct {A, \oplus}$ be an algebraic structure.
Let $\RR$ be a congruence relation on $\struct {A, \oplus}$.
Let $\SS$ be a congruence relation on the quotient structure $\struct {A / \RR, \oplus_\RR}$ defined by $\RR$.
Let $\TT$ be the relation on $A$ defined as:
:$\forall x, y \in A: x \mathrel \TT y \iff \eqcla... | Recall that by definition $\RR$ and $\SS$ are {{afortiori}} equivalence relations.
First it is demonstrated that $\TT$ is an equivalence relation.
Checking in turn each of the criteria for equivalence: | Let $\struct {A, \oplus}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\RR$ be a [[Definition:Congruence Relation|congruence relation]] on $\struct {A, \oplus}$.
Let $\SS$ be a [[Definition:Congruence Relation|congruence relation]] on the [[Definition:Quotient Structure|quotient structure]] $\st... | Recall that by definition $\RR$ and $\SS$ are {{afortiori}} [[Definition:Equivalence Relation|equivalence relations]].
First it is demonstrated that $\TT$ is an [[Definition:Equivalence Relation|equivalence relation]].
Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence | https://proofwiki.org/wiki/Equivalence_Relation_induced_by_Congruence_Relation_on_Quotient_Structure_is_Congruence | https://proofwiki.org/wiki/Equivalence_Relation_induced_by_Congruence_Relation_on_Quotient_Structure_is_Congruence | [
"Congruence Relations",
"Quotient Structures"
] | [
"Definition:Algebraic Structure",
"Definition:Congruence Relation",
"Definition:Congruence Relation",
"Definition:Quotient Structure",
"Definition:Relation",
"Definition:Congruence Relation",
"Definition:Unique",
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Quotient Epimorphism"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-19159 | Existence of Niven Number for Any Sum of Digits | Let $b, s$ be integers such that $b > 1$ and $s > 0$.
Then there exists a Niven Number in base $b$ with sum of digits $s$. | Consider the prime factorization of $b$:
:$b = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$
where $a_1, a_2, \dots, a_k \ge 1$.
Write:
:$s = p_1^{c_1} p_2^{c_2} \dots p_k^{c_k} t$
where $b$ and $t$ are coprime.
Let $c = \max \set {c_1, c_2, \dots, c_k}$.
We will show that the number:
:$n = b^c \paren {b^{\map \phi t} + b^{2 \m... | Let $b, s$ be [[Definition:Integer|integers]] such that $b > 1$ and $s > 0$.
Then there exists a [[Definition:Niven Number|Niven Number]] in [[Definition:Number Base|base $b$]] with [[Definition:Integer Addition|sum]] of [[Definition:Digit|digits]] $s$. | Consider the [[Definition:Prime Factorization|prime factorization]] of $b$:
:$b = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$
where $a_1, a_2, \dots, a_k \ge 1$.
Write:
:$s = p_1^{c_1} p_2^{c_2} \dots p_k^{c_k} t$
where $b$ and $t$ are [[Definition:Coprime Integers|coprime]].
Let $c = \max \set {c_1, c_2, \dots, c_k}$.
We... | Existence of Niven Number for Any Sum of Digits | https://proofwiki.org/wiki/Existence_of_Niven_Number_for_Any_Sum_of_Digits | https://proofwiki.org/wiki/Existence_of_Niven_Number_for_Any_Sum_of_Digits | [
"Harshad Numbers"
] | [
"Definition:Integer",
"Definition:Niven Number",
"Definition:Number Base",
"Definition:Addition/Integers",
"Definition:Digit"
] | [
"Definition:Prime Decomposition",
"Definition:Coprime/Integers",
"Definition:Niven Number",
"Definition:Number Base",
"Definition:Euler Phi Function",
"Definition:Addition/Integers",
"Definition:Distinct",
"Definition:Power (Algebra)/Integer",
"Basis Representation Theorem",
"Basis Representation ... |
proofwiki-19160 | Divisibility of Common Difference of Arithmetic Sequence of Primes | If $n$ terms of an arithmetic sequence are primes, then the common difference must be divisible by all primes less than $n$. | {{WLOG}} suppose the arithmetic sequence is increasing.
We also disregard the trivial case of zero common difference (do note that the theorem also holds in this case).
The proof proceeds by induction: | If $n$ terms of an [[Definition:Arithmetic Sequence|arithmetic sequence]] are [[Definition:Prime Number|primes]], then the [[Definition:Common Difference|common difference]] must be [[Definition:Divisor of Integer|divisible]] by all [[Definition:Prime Number|primes]] less than $n$. | {{WLOG}} suppose the [[Definition:Arithmetic Sequence|arithmetic sequence]] is [[Definition:Increasing Sequence|increasing]].
We also disregard the trivial case of zero [[Definition:Common Difference|common difference]] (do note that the theorem also holds in this case).
The proof proceeds by [[Definition:Principle ... | Divisibility of Common Difference of Arithmetic Sequence of Primes | https://proofwiki.org/wiki/Divisibility_of_Common_Difference_of_Arithmetic_Sequence_of_Primes | https://proofwiki.org/wiki/Divisibility_of_Common_Difference_of_Arithmetic_Sequence_of_Primes | [] | [
"Definition:Arithmetic Sequence",
"Definition:Prime Number",
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number"
] | [
"Definition:Arithmetic Sequence",
"Definition:Increasing/Sequence",
"Definition:Arithmetic Sequence/Common Difference",
"Principle of Mathematical Induction",
"Definition:Arithmetic Sequence",
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Arithmetic Sequence",
"Definition:Arithmetic ... |
proofwiki-19161 | Quotient Theorem for Group Homomorphisms/Corollary 2 | Let $\struct {G, \odot}$ and $\struct {H, *}$ be groups whose identities are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a group epimorphism.
Let $K$ be the kernel of $\phi$.
Let $N$ be a normal subgroup of $G$.
Let $q_N: G \to G / N$ denote the quotient epimorphism from $G$ to the quotient group $G / N$.
Then... | From {{Corollary|Quotient Theorem for Group Homomorphisms|1}}:
:$N \subseteq K$
{{iff}}:
:there exists a group homomorphism $\psi: G / N \to H$ such that $\phi = \psi \circ q_N$
From Surjection if Composite is Surjection, it follows that the group homomorphism $\psi$ is a surjection.
Hence by definition, $\psi$ is an e... | Let $\struct {G, \odot}$ and $\struct {H, *}$ be [[Definition:Group|groups]] whose [[Definition:Identity Element|identities]] are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a [[Definition:Group Epimorphism|group epimorphism]].
Let $K$ be the [[Definition:Kernel of Group Homomorphism|kernel]] of $\phi$.
Let... | From {{Corollary|Quotient Theorem for Group Homomorphisms|1}}:
:$N \subseteq K$
{{iff}}:
:there exists a [[Definition:Group Homomorphism|group homomorphism]] $\psi: G / N \to H$ such that $\phi = \psi \circ q_N$
From [[Surjection if Composite is Surjection]], it follows that the [[Definition:Group Homomorphism|group ... | Quotient Theorem for Group Homomorphisms/Corollary 2/Proof 1 | https://proofwiki.org/wiki/Quotient_Theorem_for_Group_Homomorphisms/Corollary_2 | https://proofwiki.org/wiki/Quotient_Theorem_for_Group_Homomorphisms/Corollary_2/Proof_1 | [
"Group Epimorphisms",
"Quotient Groups",
"Normal Subgroups",
"Quotient Theorem for Group Homomorphisms"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group Epimorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Normal Subgroup",
"Definition:Quotient Epimorphism/Group",
"Definition:Quotient Group",
"Definition:Group Epimorphism"
] | [
"Definition:Group Homomorphism",
"Surjection if Composite is Surjection",
"Definition:Group Homomorphism",
"Definition:Surjection",
"Definition:Quotient Epimorphism/Group"
] |
proofwiki-19162 | Quotient Theorem for Group Homomorphisms/Corollary 2 | Let $\struct {G, \odot}$ and $\struct {H, *}$ be groups whose identities are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a group epimorphism.
Let $K$ be the kernel of $\phi$.
Let $N$ be a normal subgroup of $G$.
Let $q_N: G \to G / N$ denote the quotient epimorphism from $G$ to the quotient group $G / N$.
Then... | Let $e$ be the identity element of $G$.
Let $\RR$ be the congruence relation defined by $N$ in $G$.
Let $\RR_\phi$ be the equivalence relation induced by $\phi$.
From Condition for Existence of Epimorphism from Quotient Structure to Epimorphic Image:
:there exists an epimorphism $\psi$ from $G / N$ to $H$ which satisfi... | Let $\struct {G, \odot}$ and $\struct {H, *}$ be [[Definition:Group|groups]] whose [[Definition:Identity Element|identities]] are $e_G$ and $e_H$ respectively.
Let $\phi: G \to H$ be a [[Definition:Group Epimorphism|group epimorphism]].
Let $K$ be the [[Definition:Kernel of Group Homomorphism|kernel]] of $\phi$.
Let... | Let $e$ be the [[Definition:Identity Element|identity element]] of $G$.
Let $\RR$ be the [[Definition:Congruence Modulo Subgroup|congruence relation defined by $N$]] in $G$.
Let $\RR_\phi$ be the [[Definition:Equivalence Relation Induced by Mapping|equivalence relation induced by $\phi$]].
From [[Condition for Exi... | Quotient Theorem for Group Homomorphisms/Corollary 2/Proof 2 | https://proofwiki.org/wiki/Quotient_Theorem_for_Group_Homomorphisms/Corollary_2 | https://proofwiki.org/wiki/Quotient_Theorem_for_Group_Homomorphisms/Corollary_2/Proof_2 | [
"Group Epimorphisms",
"Quotient Groups",
"Normal Subgroups",
"Quotient Theorem for Group Homomorphisms"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group Epimorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Normal Subgroup",
"Definition:Quotient Epimorphism/Group",
"Definition:Quotient Group",
"Definition:Group Epimorphism"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Congruence Modulo Subgroup",
"Definition:Equivalence Relation Induced by Mapping",
"Condition for Existence of Epimorphism from Quotient Structure to Epimorphic Image",
"Definition:Epimorphism (Abstract Algebra)"
] |
proofwiki-19163 | Arzelà-Ascoli Theorem | Let $X$ be a compact Hausdorff space.
Let $Y$ be a metric space.
Let $\CC$ be the set of all continuous functions $X \to Y$.
Consider $\CC$ as a metric space with the supremum metric:
:$\ds \map {d_\infty} {f, g} := \sup_{x \mathop \in X} \map d {\map f x, \map g x}$
Then a set $F \subseteq \CC$ is relatively compact {... | === Pointwise properties imply relative compactness ===
By Sequentially Compact Metric Space is Compact, it suffices to show that every sequence in $F$ has a uniformly convergent subsequence.
Note that the limit need not lie in $F$.
For the proof, we will first construct a set $P \subseteq X$ which we will use like a c... | Let $X$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $Y$ be a [[Definition:Metric Space|metric space]].
Let $\CC$ be the set of all [[Definition:Continuous Mapping (Topology)|continuous functions]] $X \to Y$.
Consider $\CC$ as a [[Definition:Metric Space|m... | === Pointwise properties imply relative compactness ===
By [[Sequentially Compact Metric Space is Compact]], it suffices to show that every sequence in $F$ has a [[Definition:Uniform Convergence|uniformly convergent]] [[Definition:Subsequence|subsequence]].
Note that the [[Definition:Limit of Sequence in Metric Space... | Arzelà-Ascoli Theorem | https://proofwiki.org/wiki/Arzelà-Ascoli_Theorem | https://proofwiki.org/wiki/Arzelà-Ascoli_Theorem | [
"Hausdorff Spaces",
"Metric Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Metric Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Metric Space",
"Definition:Supremum Metric",
"Definition:Relatively Compact Subspace",
"Definition:Pointwise Equicontinuous",
"Definition:Pointwise Relati... | [
"Sequentially Compact Metric Space is Compact",
"Definition:Uniform Convergence",
"Definition:Subsequence",
"Definition:Limit of Sequence/Metric Space",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Subsequence"
] |
proofwiki-19164 | Condition for Existence of Epimorphism from Quotient Structure to Epimorphic Image | Let $\struct {A, \odot}$ and $\struct {B, \otimes}$ be algebraic structures.
Let $\RR$ be a congruence relation on $\struct {A, \odot}$.
Let $f: \struct {A, \odot} \to \struct {B, \otimes}$ be an epimorphism.
Let $\struct {A / \RR, \odot_\RR}$ denote the quotient structure defined by $\RR$.
Let $q_\RR: A \to A / \RR$ d... | === Necessary Condition ===
Let $\RR \subseteq \RR_f$.
Recall the definition of $\RR_f$:
:$\forall x, y \in A: x \mathrel {\RR_f} y \iff \map f x = \map f y$
Let us define $g: \struct {A / \RR, \odot_\RR} \to \struct {B, \otimes}$ as:
:$\forall \eqclass x \RR \in A / \RR: \map g {\eqclass x \RR} = \map f x$
We show th... | Let $\struct {A, \odot}$ and $\struct {B, \otimes}$ be [[Definition:Algebraic Structure|algebraic structures]].
Let $\RR$ be a [[Definition:Congruence Relation|congruence relation]] on $\struct {A, \odot}$.
Let $f: \struct {A, \odot} \to \struct {B, \otimes}$ be an [[Definition:Epimorphism (Abstract Algebra)|epimorph... | === Necessary Condition ===
Let $\RR \subseteq \RR_f$.
Recall the [[Definition:Equivalence Relation Induced by Mapping|definition of $\RR_f$]]:
:$\forall x, y \in A: x \mathrel {\RR_f} y \iff \map f x = \map f y$
Let us define $g: \struct {A / \RR, \odot_\RR} \to \struct {B, \otimes}$ as:
:$\forall \eqclass x \RR ... | Condition for Existence of Epimorphism from Quotient Structure to Epimorphic Image | https://proofwiki.org/wiki/Condition_for_Existence_of_Epimorphism_from_Quotient_Structure_to_Epimorphic_Image | https://proofwiki.org/wiki/Condition_for_Existence_of_Epimorphism_from_Quotient_Structure_to_Epimorphic_Image | [
"Quotient Structures",
"Congruence Relations",
"Epimorphisms (Abstract Algebra)"
] | [
"Definition:Algebraic Structure",
"Definition:Congruence Relation",
"Definition:Epimorphism (Abstract Algebra)",
"Definition:Quotient Structure",
"Definition:Quotient Mapping",
"Definition:Equivalence Class",
"Definition:Epimorphism (Abstract Algebra)",
"Definition:Equivalence Relation Induced by Mapp... | [
"Definition:Equivalence Relation Induced by Mapping",
"Definition:Well-Defined/Mapping",
"Definition:Equivalence Class",
"Definition:Well-Defined/Mapping",
"Quotient Structure is Well-Defined",
"Definition:Epimorphism (Abstract Algebra)",
"Definition:Well-Defined/Mapping",
"Definition:Homomorphism (Ab... |
proofwiki-19165 | Group Epimorphism Preserves Subgroups | Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be groups.
Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a group epimorphism.
Then:
:$H \le G_1 \implies \phi \sqbrk H \le G_2$
where:
:$\phi \sqbrk H$ denotes the image of $H$ under $\phi$
:$\le$ denotes subgroup.
That is, group epimorphism preserves subgrou... | By definition, $\phi$ is {{afortiori}} a group homomorphism.
The result then follows from Group Homomorphism Preserves Subgroups.
{{qed}} | Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a [[Definition:Group Epimorphism|group epimorphism]].
Then:
:$H \le G_1 \implies \phi \sqbrk H \le G_2$
where:
:$\phi \sqbrk H$ denotes the [[Definition:Image of Subset under Map... | By definition, $\phi$ is {{afortiori}} a [[Definition:Group Homomorphism|group homomorphism]].
The result then follows from [[Group Homomorphism Preserves Subgroups]].
{{qed}} | Group Epimorphism Preserves Subgroups | https://proofwiki.org/wiki/Group_Epimorphism_Preserves_Subgroups | https://proofwiki.org/wiki/Group_Epimorphism_Preserves_Subgroups | [
"Group Epimorphisms",
"Subgroups"
] | [
"Definition:Group",
"Definition:Group Epimorphism",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Subgroup",
"Definition:Group Epimorphism",
"Definition:Subgroup"
] | [
"Definition:Group Homomorphism",
"Group Homomorphism Preserves Subgroups"
] |
proofwiki-19166 | Preimage of Image of Subgroup under Group Epimorphism | Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be groups.
Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a group epimorphism.
Let $K = \map \ker \phi$ denote the kernel of $\phi$.
Then:
:$\phi^{-1} \sqbrk {\phi \sqbrk H} = H \circ K$
where:
:$\phi \sqbrk H$ denotes the image of $H$ under $\phi$
:$\phi^{-1}... | Let $e_2$ be the identity element of $\struct {G_2, *}$.
Let $x \in \phi^{-1} \sqbrk {\phi \sqbrk H}$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \phi^{-1} \sqbrk {\phi \sqbrk H}
| c =
}}
{{eqn | ll= \leadsto
| l = \map \phi x
| o = \in
| r = \phi \sqbrk H
| c = {{Defof... | Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a [[Definition:Group Epimorphism|group epimorphism]].
Let $K = \map \ker \phi$ denote the [[Definition:Kernel of Group Homomorphism|kernel]] of $\phi$.
Then:
:$\phi^{-1} \sqbrk ... | Let $e_2$ be the [[Definition:Identity Element|identity element]] of $\struct {G_2, *}$.
Let $x \in \phi^{-1} \sqbrk {\phi \sqbrk H}$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \phi^{-1} \sqbrk {\phi \sqbrk H}
| c =
}}
{{eqn | ll= \leadsto
| l = \map \phi x
| o = \in
| r =... | Preimage of Image of Subgroup under Group Epimorphism | https://proofwiki.org/wiki/Preimage_of_Image_of_Subgroup_under_Group_Epimorphism | https://proofwiki.org/wiki/Preimage_of_Image_of_Subgroup_under_Group_Epimorphism | [
"Group Epimorphisms",
"Subgroups"
] | [
"Definition:Group",
"Definition:Group Epimorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Preimage/Mapping/Subset",
"Definition:Subset Product"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group Homomorphism",
"Kernel is Normal Subgroup of Domain",
"Definition:Group Homomorphism",
"Definition:Set Equality/Definition 2"
] |
proofwiki-19167 | Preimage of Subgroup under Epimorphism is Subgroup | Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be groups.
Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a group epimorphism.
Let $H$ be a subgroup of $\struct {G_2, *}$.
Then:
:$\phi^{-1} \sqbrk H$ is a subgroup of $\struct {G_1, \circ}$
where $\phi^{-1} \sqbrk H$ denotes the preimage of $H$ under $\phi$. | Let $H$ be a subgroup of $\struct {G_2, *}$.
First note that from Null Relation is Mapping iff Domain is Empty Set:
:$\phi^{-1} \sqbrk H = \O \implies H = \O$
But $H \ne \O$.
Hence $\phi^{-1} \sqbrk H$ is not empty.
Next, let $x, y \in \phi^{-1} \sqbrk H$.
Then:
:$\exists h_1, h_2 \in H: h_1 = \map \phi x, h_2 = \map \... | Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a [[Definition:Group Epimorphism|group epimorphism]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $\struct {G_2, *}$.
Then:
:$\phi^{-1} \sqbrk H$ is a [[Definition:Subgroup|... | Let $H$ be a [[Definition:Subgroup|subgroup]] of $\struct {G_2, *}$.
First note that from [[Null Relation is Mapping iff Domain is Empty Set]]:
:$\phi^{-1} \sqbrk H = \O \implies H = \O$
But $H \ne \O$.
Hence $\phi^{-1} \sqbrk H$ is not [[Definition:Empty Set|empty]].
Next, let $x, y \in \phi^{-1} \sqbrk H$.
Then... | Preimage of Subgroup under Epimorphism is Subgroup | https://proofwiki.org/wiki/Preimage_of_Subgroup_under_Epimorphism_is_Subgroup | https://proofwiki.org/wiki/Preimage_of_Subgroup_under_Epimorphism_is_Subgroup | [
"Group Epimorphisms",
"Subgroups"
] | [
"Definition:Group",
"Definition:Group Epimorphism",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Preimage/Mapping/Subset"
] | [
"Definition:Subgroup",
"Null Relation is Mapping iff Domain is Empty Set",
"Definition:Empty Set",
"Definition:Group Homomorphism",
"Definition:Subgroup",
"One-Step Subgroup Test",
"Definition:Subgroup"
] |
proofwiki-19168 | Preimage of Normal Subgroup under Epimorphism is Normal Subgroup | Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be groups.
Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a group epimorphism.
Let $H$ be a normal subgroup of $\struct {G_2, *}$.
Then:
:$\phi^{-1} \sqbrk H$ is a normal subgroup of $\struct {G_1, \circ}$
where $\phi^{-1} \sqbrk H$ denotes the preimage of $H$... | Recall that as $\phi$ is an group epimorphism, it is also {{afortiori}} a group Homomorphism.
Hence:
:$\forall a, b \in G_1: \map \phi {a \circ b} = \map \phi a * \map \phi b$
Let $H$ be a normal subgroup of $\struct {G_2, *}$.
From Preimage of Subgroup under Epimorphism is Subgroup:
:$\phi^{-1} \sqbrk H$ is a subgroup... | Let $\struct {G_1, \circ}$ and $\struct {G_2, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ be a [[Definition:Group Epimorphism|group epimorphism]].
Let $H$ be a [[Definition:Normal Subgroup|normal subgroup]] of $\struct {G_2, *}$.
Then:
:$\phi^{-1} \sqbrk H$ is a [[Defini... | Recall that as $\phi$ is an [[Definition:Group Epimorphism|group epimorphism]], it is also {{afortiori}} a [[Definition:Group Homomorphism|group Homomorphism]].
Hence:
:$\forall a, b \in G_1: \map \phi {a \circ b} = \map \phi a * \map \phi b$
Let $H$ be a [[Definition:Normal Subgroup|normal subgroup]] of $\struct {G... | Preimage of Normal Subgroup under Epimorphism is Normal Subgroup | https://proofwiki.org/wiki/Preimage_of_Normal_Subgroup_under_Epimorphism_is_Normal_Subgroup | https://proofwiki.org/wiki/Preimage_of_Normal_Subgroup_under_Epimorphism_is_Normal_Subgroup | [
"Group Epimorphisms",
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Group Epimorphism",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Definition:Preimage/Mapping/Subset"
] | [
"Definition:Group Epimorphism",
"Definition:Group Homomorphism",
"Definition:Normal Subgroup",
"Preimage of Subgroup under Epimorphism is Subgroup",
"Definition:Subgroup",
"Definition:Normal Subgroup",
"Definition:Normal Subgroup",
"Definition:Preimage/Mapping/Subset",
"Definition:Normal Subgroup",
... |
proofwiki-19169 | Unique Epimorphism from Quotient Group to Quotient Group | Let $\struct {G_1, \circledcirc}$ and $\struct {G_2, \circledast}$ be groups.
Let $\phi: \struct {G_1, \circledcirc} \to \struct {G_2, \circledast}$ be a group epimorphism.
Let $K$ be the kernel of $\phi$.
Let $H_1$ be a normal subgroup of $\struct {G_1, \circledcirc}$.
Let $H_2 := \phi \sqbrk {H_1}$ be the image of $H... | First we note that by Group Epimorphism Preserves Normal Subgroups, $H_2$ is indeed a normal subgroup of $\struct {G_2, \circledast}$.
Hence $G_2 / H_2$ and $q_{H_2}: G_2 \to G_2 / H_2$ are appropriately defined.
Let $\RR$ be the congruence relation induced on $\struct {G_1, \circledcirc}$ by $H$.
Let $q_\RR$ be the qu... | Let $\struct {G_1, \circledcirc}$ and $\struct {G_2, \circledast}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G_1, \circledcirc} \to \struct {G_2, \circledast}$ be a [[Definition:Group Epimorphism|group epimorphism]].
Let $K$ be the [[Definition:Kernel of Group Homomorphism|kernel]] of $\phi$.
Let $H_1$ be... | First we note that by [[Group Epimorphism Preserves Normal Subgroups]], $H_2$ is indeed a [[Definition:Normal Subgroup|normal subgroup]] of $\struct {G_2, \circledast}$.
Hence $G_2 / H_2$ and $q_{H_2}: G_2 \to G_2 / H_2$ are appropriately defined.
Let $\RR$ be the [[Definition:Congruence Modulo Subgroup|congruence re... | Unique Epimorphism from Quotient Group to Quotient Group | https://proofwiki.org/wiki/Unique_Epimorphism_from_Quotient_Group_to_Quotient_Group | https://proofwiki.org/wiki/Unique_Epimorphism_from_Quotient_Group_to_Quotient_Group | [
"Group Epimorphisms",
"Quotient Groups"
] | [
"Definition:Group",
"Definition:Group Epimorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Normal Subgroup",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Unique",
"Definition:Group Epimorphism",
"Definition:Quotient Group",
"Definition:Quotient Mapping",
"Definition:... | [
"Group Epimorphism Preserves Normal Subgroups",
"Definition:Normal Subgroup",
"Definition:Congruence Modulo Subgroup",
"Definition:Quotient Mapping",
"Definition:Group Epimorphism",
"Quotient Epimorphism is Epimorphism/Group",
"Definition:Group Epimorphism",
"Composite of Group Epimorphisms is Epimorp... |
proofwiki-19170 | Condition for Group Endomorphism to Commute with All Inner Automorphisms | Let $G$ be a group.
Let $\phi: G \to G$ be an endomorphism on $G$.
Let $\phi$ be such that:
:$\forall a \in G: \kappa_a \circ \phi = \phi \circ \kappa_a$
where:
:$\kappa_a$ denotes the inner automorphism of $G$ given by $a$
:$\circ$ denotes composition of mappings.
Then:
:$H = \set {x \in G: \map \phi {\map \phi x} = \... | We have for all $a, g \in G$:
{{begin-eqn}}
{{eqn | l = a \map \phi g a^{-1}
| r = \map {\kappa_a} {\map \phi g}
| c = {{Defof|Inner Automorphism}}
}}
{{eqn | r = \map {\paren{\kappa_a \circ \phi} } g
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {\paren{\phi \circ \kappa_a} } g
}}
{{eqn... | Let $G$ be a [[Definition:Group|group]].
Let $\phi: G \to G$ be an [[Definition:Group Endomorphism|endomorphism]] on $G$.
Let $\phi$ be such that:
:$\forall a \in G: \kappa_a \circ \phi = \phi \circ \kappa_a$
where:
:$\kappa_a$ denotes the [[Definition:Inner Automorphism|inner automorphism of $G$ given by $a$]]
:$\ci... | We have for all $a, g \in G$:
{{begin-eqn}}
{{eqn | l = a \map \phi g a^{-1}
| r = \map {\kappa_a} {\map \phi g}
| c = {{Defof|Inner Automorphism}}
}}
{{eqn | r = \map {\paren{\kappa_a \circ \phi} } g
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \map {\paren{\phi \circ \kappa_a} } g
}}
{{eqn... | Condition for Group Endomorphism to Commute with All Inner Automorphisms | https://proofwiki.org/wiki/Condition_for_Group_Endomorphism_to_Commute_with_All_Inner_Automorphisms | https://proofwiki.org/wiki/Condition_for_Group_Endomorphism_to_Commute_with_All_Inner_Automorphisms | [
"Group Endomorphisms",
"Inner Automorphisms"
] | [
"Definition:Group",
"Definition:Group Endomorphism",
"Definition:Inner Automorphism",
"Definition:Composition of Mappings",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Abelian Group"
] | [
"Definition:Normal Subgroup/Definition 3",
"Definition:Normal Subgroup/Definition 3",
"Definition:Quotient Group",
"Definition:Abelian Group",
"Cosets are Equal iff Product with Inverse in Subgroup",
"Group Homomorphism Preserves Inverses",
"Group Homomorphism Preserves Inverses"
] |
proofwiki-19171 | External Direct Product Associativity/Sufficient Condition | Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\circ_1$ and $\circ_2$ be associative.
Then $\circ$ is also associative. | Let $\circ_1$ and $\circ_2$ be associative.
{{begin-eqn}}
{{eqn | l = \paren {\tuple {s_1, t_1} \circ \tuple {s_2, t_2} } \circ \tuple {s_3, t_3}
| r = \tuple {\paren {s_1 \circ_1 s_2} \circ_1 s_3, \paren {t_1 \circ_2 t_2} \circ_2 t_3}
| c = {{Defof|Operation Induced by Direct Product}}
}}
{{eqn | r = \tup... | Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of the two [[Definition:Algebraic Structure with One Operation|algebraic structures]] $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\circ_1$ and $\circ_2$ be [[Definition:Associative Operation|associative]... | Let $\circ_1$ and $\circ_2$ be [[Definition:Associative Operation|associative]].
{{begin-eqn}}
{{eqn | l = \paren {\tuple {s_1, t_1} \circ \tuple {s_2, t_2} } \circ \tuple {s_3, t_3}
| r = \tuple {\paren {s_1 \circ_1 s_2} \circ_1 s_3, \paren {t_1 \circ_2 t_2} \circ_2 t_3}
| c = {{Defof|Operation Induced b... | External Direct Product Associativity/Sufficient Condition | https://proofwiki.org/wiki/External_Direct_Product_Associativity/Sufficient_Condition | https://proofwiki.org/wiki/External_Direct_Product_Associativity/Sufficient_Condition | [
"External Direct Product Associativity"
] | [
"Definition:External Direct Product",
"Definition:Algebraic Structure/One Operation",
"Definition:Associative Operation",
"Definition:Associative Operation"
] | [
"Definition:Associative Operation",
"Definition:Associative Operation"
] |
proofwiki-19172 | External Direct Product Associativity/Necessary Condition | Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\circ$ be associative.
Then $\circ_1$ and $\circ_2$ are both also associative. | Let $\circ$ be associative.
{{AimForCont}} it is not the case that both $\circ_1$ and $\circ_2$ are associative.
{{WLOG}}, suppose $\circ_1$ is not associative.
Hence:
{{begin-eqn}}
{{eqn | q = \exists s_1, s_2, s_3 \in S
| l = \paren {s_1 \circ_1 s_2} \circ_1 s_3
| o = \ne
| r = s_1 \circ_1 \paren {... | Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of the two [[Definition:Algebraic Structure with One Operation|algebraic structures]] $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\circ$ be [[Definition:Associative Operation|associative]].
Then $\circ... | Let $\circ$ be [[Definition:Associative Operation|associative]].
{{AimForCont}} it is not the case that both $\circ_1$ and $\circ_2$ are [[Definition:Associative Operation|associative]].
{{WLOG}}, suppose $\circ_1$ is not [[Definition:Associative Operation|associative]].
Hence:
{{begin-eqn}}
{{eqn | q = \exists s_... | External Direct Product Associativity/Necessary Condition | https://proofwiki.org/wiki/External_Direct_Product_Associativity/Necessary_Condition | https://proofwiki.org/wiki/External_Direct_Product_Associativity/Necessary_Condition | [
"External Direct Product Associativity"
] | [
"Definition:External Direct Product",
"Definition:Algebraic Structure/One Operation",
"Definition:Associative Operation",
"Definition:Associative Operation"
] | [
"Definition:Associative Operation",
"Definition:Associative Operation",
"Definition:Associative Operation",
"Definition:Associative Operation",
"Equality of Ordered Pairs",
"Definition:Contradiction",
"Definition:Associative Operation",
"Definition:Associative Operation",
"Proof by Contradiction",
... |
proofwiki-19173 | External Direct Product Commutativity/Sufficient Condition | Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\circ_1$ and $\circ_2$ be commutative operations.
Then $\circ$ is also a commutative operation. | Let $\circ_1$ and $\circ_2$ be commutative operations.
{{begin-eqn}}
{{eqn | l = \tuple {s_1, t_1} \circ \tuple {s_2, t_2}
| r = \tuple {s_1 \circ_1 s_2, t_1 \circ_2 t_2}
| c = {{Defof|Operation Induced by Direct Product}}
}}
{{eqn | r = \tuple {s_2 \circ_1 s_1, t_2 \circ_2 t_1}
| c = {{Defof|Commutat... | Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of the two [[Definition:Algebraic Structure with One Operation|algebraic structures]] $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\circ_1$ and $\circ_2$ be [[Definition:Commutative Operation|commutative ... | Let $\circ_1$ and $\circ_2$ be [[Definition:Commutative Operation|commutative operations]].
{{begin-eqn}}
{{eqn | l = \tuple {s_1, t_1} \circ \tuple {s_2, t_2}
| r = \tuple {s_1 \circ_1 s_2, t_1 \circ_2 t_2}
| c = {{Defof|Operation Induced by Direct Product}}
}}
{{eqn | r = \tuple {s_2 \circ_1 s_1, t_2 \ci... | External Direct Product Commutativity/Sufficient Condition | https://proofwiki.org/wiki/External_Direct_Product_Commutativity/Sufficient_Condition | https://proofwiki.org/wiki/External_Direct_Product_Commutativity/Sufficient_Condition | [
"External Direct Product Commutativity"
] | [
"Definition:External Direct Product",
"Definition:Algebraic Structure/One Operation",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation"
] | [
"Definition:Commutative/Operation",
"Definition:Commutative/Operation"
] |
proofwiki-19174 | External Direct Product Commutativity/Necessary Condition | Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\circ$ be commutative.
Then $\circ_1$ and $\circ_2$ are both also commutative. | Let $\circ$ be commutative.
{{AimForCont}} it is not the case that both $\circ_1$ and $\circ_2$ are commutative.
{{WLOG}}, suppose $\circ_1$ is not commutative.
Hence:
{{begin-eqn}}
{{eqn | q = \exists s_1, s_2 \in S
| l = s_1 \circ_1 s_2
| o = \ne
| r = s_2 \circ_1 s_1
| c = as $\circ_1$ is no... | Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of the two [[Definition:Algebraic Structure with One Operation|algebraic structures]] $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\circ$ be [[Definition:Commutative Operation|commutative]].
Then $\circ... | Let $\circ$ be [[Definition:Commutative Operation|commutative]].
{{AimForCont}} it is not the case that both $\circ_1$ and $\circ_2$ are [[Definition:Commutative Operation|commutative]].
{{WLOG}}, suppose $\circ_1$ is not [[Definition:Commutative Operation|commutative]].
Hence:
{{begin-eqn}}
{{eqn | q = \exists s_... | External Direct Product Commutativity/Necessary Condition | https://proofwiki.org/wiki/External_Direct_Product_Commutativity/Necessary_Condition | https://proofwiki.org/wiki/External_Direct_Product_Commutativity/Necessary_Condition | [
"External Direct Product Commutativity"
] | [
"Definition:External Direct Product",
"Definition:Algebraic Structure/One Operation",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation"
] | [
"Definition:Commutative/Operation",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation",
"Equality of Ordered Pairs",
"Definition:Contradiction",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation",
"Proof by Contradiction",
... |
proofwiki-19175 | External Direct Product Identity/Sufficient Condition | Let $\struct {S \times T, \circ}$ be the external direct product of two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let:
:$\struct {S, \circ_1}$ have identity element $e_S$
and:
:$\struct {T, \circ_2}$ have identity element $e_T$.
Then $\tuple {e_S, e_T}$ is the identity element for $\struct... | Let $e_S$ and $e_T$ be the identity elements of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ respectively.
{{begin-eqn}}
{{eqn | l = \tuple {s, t} \circ \tuple {e_S, e_T}
| r = \tuple {s \circ_1 e_S, t \circ_2 e_T}
| c =
}}
{{eqn | r = \tuple {s, t}
| c =
}}
{{eqn | l = \tuple {e_S, e_T} \circ ... | Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of two [[Definition:Algebraic Structure with One Operation|algebraic structures]] $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let:
:$\struct {S, \circ_1}$ have [[Definition:Identity Element|identity element]]... | Let $e_S$ and $e_T$ be the [[Definition:Identity Element|identity elements]] of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ respectively.
{{begin-eqn}}
{{eqn | l = \tuple {s, t} \circ \tuple {e_S, e_T}
| r = \tuple {s \circ_1 e_S, t \circ_2 e_T}
| c =
}}
{{eqn | r = \tuple {s, t}
| c =
}}
{{e... | External Direct Product Identity/Sufficient Condition | https://proofwiki.org/wiki/External_Direct_Product_Identity/Sufficient_Condition | https://proofwiki.org/wiki/External_Direct_Product_Identity/Sufficient_Condition | [
"External Direct Product Identity"
] | [
"Definition:External Direct Product",
"Definition:Algebraic Structure/One Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-19176 | External Direct Product Identity/Necessary Condition | Let $\struct {S \times T, \circ}$ be the external direct product of two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\struct {S \times T, \circ}$ have an identity element $\tuple {e_S, e_T}$.
Then:
:$\struct {S, \circ_1}$ has an identity element $e_S$
and:
:$\struct {T, \circ_2}$ has an ... | Let $\tuple {e_S, e_T}$ be an identity of $\struct {S \times T, \circ}$.
Then we have:
{{begin-eqn}}
{{eqn | q = \forall \tuple {s, t} \in S \times T
| l = \tuple {s, t} \circ \tuple {e_S, e_T}
| r = \tuple {s, t}
| c = {{Defof|Identity Element}}
}}
{{eqn | ll= \leadsto
| l = \tuple {s \circ_1 e... | Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of two [[Definition:Algebraic Structure with One Operation|algebraic structures]] $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\struct {S \times T, \circ}$ have an [[Definition:Identity Element|identity ... | Let $\tuple {e_S, e_T}$ be an [[Definition:Identity Element|identity]] of $\struct {S \times T, \circ}$.
Then we have:
{{begin-eqn}}
{{eqn | q = \forall \tuple {s, t} \in S \times T
| l = \tuple {s, t} \circ \tuple {e_S, e_T}
| r = \tuple {s, t}
| c = {{Defof|Identity Element}}
}}
{{eqn | ll= \leads... | External Direct Product Identity/Necessary Condition | https://proofwiki.org/wiki/External_Direct_Product_Identity/Necessary_Condition | https://proofwiki.org/wiki/External_Direct_Product_Identity/Necessary_Condition | [
"External Direct Product Identity"
] | [
"Definition:External Direct Product",
"Definition:Algebraic Structure/One Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Equality of Ordered Pairs",
"Equality of Ordered Pairs",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-19177 | External Direct Product Inverses/Sufficient Condition | Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let:
:$s^{-1}$ be an inverse of $s \in \struct {S, \circ_1}$
and:
:$t^{-1}$ be an inverse of $t \in \struct {T, \circ_2}$.
Then $\tuple {s^{-1}, t^{-1} }$ is an inverse of ... | Let:
:$e_S$ be the identity for $\struct {S, \circ_1}$
and:
:$e_T$ be the identity for $\struct {T, \circ_2}$.
Also let:
:$s^{-1}$ be the inverse of $s \in \struct {S, \circ_1}$
and
:$t^{-1}$ be the inverse of $t \in \struct {T, \circ_2}$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {s, t} \circ \tuple {s^{-1}, t^{-1} }
... | Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of the two [[Definition:Algebraic Structure with One Operation|algebraic structures]] $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let:
:$s^{-1}$ be an [[Definition:Inverse Element|inverse]] of $s \in \struct ... | Let:
:$e_S$ be the [[Definition:Identity Element|identity]] for $\struct {S, \circ_1}$
and:
:$e_T$ be the [[Definition:Identity Element|identity]] for $\struct {T, \circ_2}$.
Also let:
:$s^{-1}$ be the [[Definition:Inverse Element|inverse]] of $s \in \struct {S, \circ_1}$
and
:$t^{-1}$ be the [[Definition:Inverse Elem... | External Direct Product Inverses/Sufficient Condition | https://proofwiki.org/wiki/External_Direct_Product_Inverses/Sufficient_Condition | https://proofwiki.org/wiki/External_Direct_Product_Inverses/Sufficient_Condition | [
"External Direct Product Inverses"
] | [
"Definition:External Direct Product",
"Definition:Algebraic Structure/One Operation",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Inverse (Abstract Algebra)/Inverse"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Inverse (Abstract Algebra)/Inverse"
] |
proofwiki-19178 | External Direct Product Inverses/Necessary Condition | Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\tuple {s^{-1}, t^{-1} }$ be an inverse of $\tuple {s, t} \in \struct {S \times T, \circ}$.
Then:
:$s^{-1}$ is an inverse of $s \in \struct {S, \circ_1}$
and:
:$t^{-1}... | Let $\tuple {e_S, e_T}$ be the identity element of $\struct {S \times T, \circ}$.
Let $\tuple {s^{-1}, t^{-1} }$ be an inverse element of $\tuple {s, t} \in \struct {S \times T, \circ}$.
Then we have:
{{begin-eqn}}
{{eqn | l = \tuple {s, t} \circ \tuple {s^{-1}, t^{-1} }
| r = \tuple {e_S, e_T}
| c = {{Defo... | Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of the two [[Definition:Algebraic Structure with One Operation|algebraic structures]] $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\tuple {s^{-1}, t^{-1} }$ be an [[Definition:Inverse Element|inverse]] of... | Let $\tuple {e_S, e_T}$ be the [[Definition:Identity Element|identity element]] of $\struct {S \times T, \circ}$.
Let $\tuple {s^{-1}, t^{-1} }$ be an [[Definition:Inverse Element|inverse element]] of $\tuple {s, t} \in \struct {S \times T, \circ}$.
Then we have:
{{begin-eqn}}
{{eqn | l = \tuple {s, t} \circ \tuple ... | External Direct Product Inverses/Necessary Condition | https://proofwiki.org/wiki/External_Direct_Product_Inverses/Necessary_Condition | https://proofwiki.org/wiki/External_Direct_Product_Inverses/Necessary_Condition | [
"External Direct Product Identity"
] | [
"Definition:External Direct Product",
"Definition:Algebraic Structure/One Operation",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Inverse (Abstract Algebra)/Inverse"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Equality of Ordered Pairs",
"Equality of Ordered Pairs",
"Definition:Inverse (Abstract Algebra)/Inverse"
] |
proofwiki-19179 | External Direct Product Closure/Sufficient Condition | Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures.
Let $\struct {S \times T, \circ}$ be the external direct product of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be closed.
Then $\struct {S \times T, \circ}$ is also closed. | Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be closed.
Let $\tuple {s_1, t_1} \in S \times T$ and $\tuple {s_2, t_2} \in S \times T$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {s_1, t_1} \circ \tuple {s_2, t_2}
| r = \tuple {s_1 \circ_1 s_2, t_1 \circ_2 t_2}
| c = {{Defof|External Direct Product}}
}}... | Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be [[Definition:Algebraic Structure|algebraic structures]].
Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\struct {S, \circ_1}$ and $\struct {T, \c... | Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be [[Definition:Closed Algebraic Structure|closed]].
Let $\tuple {s_1, t_1} \in S \times T$ and $\tuple {s_2, t_2} \in S \times T$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {s_1, t_1} \circ \tuple {s_2, t_2}
| r = \tuple {s_1 \circ_1 s_2, t_1 \circ_2 t_2}
... | External Direct Product Closure/Sufficient Condition | https://proofwiki.org/wiki/External_Direct_Product_Closure/Sufficient_Condition | https://proofwiki.org/wiki/External_Direct_Product_Closure/Sufficient_Condition | [
"External Direct Product Closure"
] | [
"Definition:Algebraic Structure",
"Definition:External Direct Product",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-19180 | External Direct Product Closure/Necessary Condition | Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures.
Let $\struct {S \times T, \circ}$ be the external direct product of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\struct {S \times T, \circ}$ be a closed algebraic structure.
Then $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$... | Let $\struct {S \times T, \circ}$ be closed.
Let $s_1, s_2 \in S$ and $t_1, t_2 \in T$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {s_1 \circ_1 s_2, t_1 \circ_2 t_2}
| r = \tuple {s_1, t_1} \circ \tuple {s_2, t_2}
| c = {{Defof|External Direct Product}}
}}
{{eqn | o = \in
| r = S \times T
| c = as $... | Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be [[Definition:Algebraic Structure|algebraic structures]].
Let $\struct {S \times T, \circ}$ be the [[Definition:External Direct Product|external direct product]] of $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.
Let $\struct {S \times T, \circ}$ be a [[Defi... | Let $\struct {S \times T, \circ}$ be [[Definition:Closed Algebraic Structure|closed]].
Let $s_1, s_2 \in S$ and $t_1, t_2 \in T$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {s_1 \circ_1 s_2, t_1 \circ_2 t_2}
| r = \tuple {s_1, t_1} \circ \tuple {s_2, t_2}
| c = {{Defof|External Direct Product}}
}}
{{eqn | o... | External Direct Product Closure/Necessary Condition | https://proofwiki.org/wiki/External_Direct_Product_Closure/Necessary_Condition | https://proofwiki.org/wiki/External_Direct_Product_Closure/Necessary_Condition | [
"External Direct Product Closure"
] | [
"Definition:Algebraic Structure",
"Definition:External Direct Product",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-19181 | Green's Identities | Let $\struct {M, g}$ be a compact Riemannian manifold with or without boundary $\partial M$.
Let $\hat g$ be the induced metric on $\partial M$.
Let $\map {C^\infty} M$ be the smooth function space.
Let $u, v \in \map {C^\infty} M$ be smooth real functions on $M$.
Let $\rd V_g$ be the Riemannian volume form.
Let $\nabl... | {{ProofWanted}}
{{Namedfor|George Green|cat = Green}} | Let $\struct {M, g}$ be a [[Definition:Compact Manifold|compact]] [[Definition:Riemannian Manifold|Riemannian manifold]] with or without [[Definition:Boundary (Topology)|boundary]] $\partial M$.
Let $\hat g$ be the [[Definition:Induced Metric on Submanifold|induced metric]] on $\partial M$.
Let $\map {C^\infty} M$ be... | {{ProofWanted}}
{{Namedfor|George Green|cat = Green}} | Green's Identities | https://proofwiki.org/wiki/Green's_Identities | https://proofwiki.org/wiki/Green's_Identities | [
"Riemannian Geometry"
] | [
"Definition:Compact Manifold",
"Definition:Riemannian Manifold",
"Definition:Boundary (Topology)",
"Definition:Induced Metric on Submanifold",
"Definition:Space of Continuous Functions of Differentiability Class k",
"Definition:Smooth Mapping",
"Definition:Riemannian Volume Form",
"Definition:Laplacia... | [] |
proofwiki-19182 | Factors of Internal Direct Product of Subsemigroups are Normal Subgroups | Let $\struct {G, \odot}$ be a group.
Let $\struct {H, \odot_H}$ and $\struct {K, \odot_K}$ be subsemigroups of $\struct {G, \odot}$.
Let $\struct {G, \odot}$ be the internal direct product of $\struct {H, \odot_H}$ and $\struct {K, \odot_K}$.
Then $\struct {H, \odot_H}$ and $\struct {K, \odot_K}$ are normal subgroups o... | Let $\struct {G, \odot}$ be the internal direct product of $\struct {H, \odot_H}$ and $\struct {K, \odot_K}$.
Let $e$ denote the identity element of $\struct {G, \odot}$.
By definition of internal direct product, the mapping $\phi: H \times K \to G$ defined as:
:$\forall h \in H, k \in K: \map \phi {h, k} = h \odot k$
... | Let $\struct {G, \odot}$ be a [[Definition:Group|group]].
Let $\struct {H, \odot_H}$ and $\struct {K, \odot_K}$ be [[Definition:Subsemigroup|subsemigroups]] of $\struct {G, \odot}$.
Let $\struct {G, \odot}$ be the [[Definition:Internal Direct Product|internal direct product]] of $\struct {H, \odot_H}$ and $\struct {K... | Let $\struct {G, \odot}$ be the [[Definition:Internal Direct Product|internal direct product]] of $\struct {H, \odot_H}$ and $\struct {K, \odot_K}$.
Let $e$ denote the [[Definition:Identity Element|identity element]] of $\struct {G, \odot}$.
By definition of [[Definition:Internal Direct Product|internal direct produ... | Factors of Internal Direct Product of Subsemigroups are Normal Subgroups | https://proofwiki.org/wiki/Factors_of_Internal_Direct_Product_of_Subsemigroups_are_Normal_Subgroups | https://proofwiki.org/wiki/Factors_of_Internal_Direct_Product_of_Subsemigroups_are_Normal_Subgroups | [
"Internal Direct Products",
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Subsemigroup",
"Definition:Internal Direct Product",
"Definition:Normal Subgroup"
] | [
"Definition:Internal Direct Product",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Internal Direct Product",
"Definition:Mapping",
"Definition:Isomorphism (Abstract Algebra)",
"Definition:External Direct Product",
"Definition:Operation Induced by Direct Product",
"Condition... |
proofwiki-19183 | Integer as Difference between Two Squares/Formulation 1 | Let $n$ be a positive integer.
Then $n$ can be expressed as:
:$n = a^2 - b^2$
{{iff}} $n$ has at least two distinct divisors of the same parity that multiply to $n$. | {{begin-eqn}}
{{eqn | l = n
| r = a^2 - b^2
| c =
}}
{{eqn | r = \paren {a + b} \paren {a - b}
| c = Difference of Two Squares
}}
{{end-eqn}}
Thus $n = p q$ where:
{{begin-eqn}}
{{eqn | n = 1
| l = p
| r = \paren {a + b}
| c =
}}
{{eqn | n = 2
| l = q
| r = \paren {a - ... | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Then $n$ can be expressed as:
:$n = a^2 - b^2$
{{iff}} $n$ has at least two [[Definition:Distinct|distinct]] [[Definition:Divisor of Integer|divisors]] of the same [[Definition:Parity|parity]] that [[Definition:Integer Multiplication|multiply]] to $n$. | {{begin-eqn}}
{{eqn | l = n
| r = a^2 - b^2
| c =
}}
{{eqn | r = \paren {a + b} \paren {a - b}
| c = [[Difference of Two Squares]]
}}
{{end-eqn}}
Thus $n = p q$ where:
{{begin-eqn}}
{{eqn | n = 1
| l = p
| r = \paren {a + b}
| c =
}}
{{eqn | n = 2
| l = q
| r = \pare... | Integer as Difference between Two Squares/Formulation 1 | https://proofwiki.org/wiki/Integer_as_Difference_between_Two_Squares/Formulation_1 | https://proofwiki.org/wiki/Integer_as_Difference_between_Two_Squares/Formulation_1 | [
"Square Numbers",
"Difference of Two Squares"
] | [
"Definition:Positive/Integer",
"Definition:Distinct",
"Definition:Divisor (Algebra)/Integer",
"Definition:Parity",
"Definition:Multiplication/Integers"
] | [
"Difference of Two Squares",
"Definition:Integer",
"Definition:Distinct",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Category:Square Numbers",
"Category:Difference of Two Squares"
] |
proofwiki-19184 | Integer as Difference between Two Squares/Formulation 2 | Any integer can be expressed as the difference of two squares {{iff}} that integer is NOT $n \equiv 2 \pmod 4$ | : Each integer will be in one of the 4 sets of residue classes modulo $4$:
: $n \equiv 0 \pmod 4$
: $n \equiv 1 \pmod 4$
: $n \equiv 2 \pmod 4$
: $n \equiv 3 \pmod 4$
{{begin-eqn}}
{{eqn | l = 4x
| r = \paren{x + 1 }^2 - \paren{x - 1 }^2
| c =
}}
{{eqn | l = 4x + 1
| r = \paren{2x + 1 }^2 - 4x^2
... | Any [[Definition:Integer|integer]] can be expressed as the difference of two squares {{iff}} that [[Definition:Integer|integer]] is NOT $n \equiv 2 \pmod 4$ | : Each [[Definition:Integer|integer]] will be in one of the [[Definition:Set of Residue Classes| 4 sets of residue classes modulo $4$]]:
: $n \equiv 0 \pmod 4$
: $n \equiv 1 \pmod 4$
: $n \equiv 2 \pmod 4$
: $n \equiv 3 \pmod 4$
{{begin-eqn}}
{{eqn | l = 4x
| r = \paren{x + 1 }^2 - \paren{x - 1 }^2
| c =
... | Integer as Difference between Two Squares/Formulation 2 | https://proofwiki.org/wiki/Integer_as_Difference_between_Two_Squares/Formulation_2 | https://proofwiki.org/wiki/Integer_as_Difference_between_Two_Squares/Formulation_2 | [
"Square Numbers",
"Difference of Two Squares"
] | [
"Definition:Integer",
"Definition:Integer"
] | [
"Definition:Integer",
"Definition:Set of Residue Classes",
"Definition:Integer",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Residue Class",
"Definition:Square Number",
"Category:Square Numbers",
"Category:Difference of Two Squares"
] |
proofwiki-19185 | Integer as Sums and Differences of Consecutive Squares | Every integer $k$ can be represented in infinitely many ways in the form:
:$k = \pm 1^2 \pm 2^2 \pm 3^3 \pm \dots \pm m^2$
for some (strictly) positive integer $m$ and some choice of signs $+$ or $-$. | First we notice that:
{{begin-eqn}}
{{eqn | o =
| r = \paren {m + 1}^2 - \paren {m + 2}^2 - \paren {m + 3}^2 + \paren {m + 4}^2
}}
{{eqn | r = \paren {-1} \paren {m + 1 + m + 2} + \paren {m + 3 + m + 4}
| c = Difference of Two Squares
}}
{{eqn | r = 4
}}
{{end-eqn}}
Now we prove a weaker form of the theorem... | Every [[Definition:Integer|integer]] $k$ can be represented in [[Definition:Infinitely Many|infinitely many]] ways in the form:
:$k = \pm 1^2 \pm 2^2 \pm 3^3 \pm \dots \pm m^2$
for some [[Definition:Strictly Positive Integer|(strictly) positive integer]] $m$ and some choice of signs $+$ or $-$. | First we notice that:
{{begin-eqn}}
{{eqn | o =
| r = \paren {m + 1}^2 - \paren {m + 2}^2 - \paren {m + 3}^2 + \paren {m + 4}^2
}}
{{eqn | r = \paren {-1} \paren {m + 1 + m + 2} + \paren {m + 3 + m + 4}
| c = [[Difference of Two Squares]]
}}
{{eqn | r = 4
}}
{{end-eqn}}
Now we prove a weaker form of the t... | Integer as Sums and Differences of Consecutive Squares | https://proofwiki.org/wiki/Integer_as_Sums_and_Differences_of_Consecutive_Squares | https://proofwiki.org/wiki/Integer_as_Sums_and_Differences_of_Consecutive_Squares | [
"Squares",
"Integer as Sums and Differences of Consecutive Squares"
] | [
"Definition:Integer",
"Definition:Infinite Set",
"Definition:Strictly Positive/Integer"
] | [
"Difference of Two Squares",
"Principle of Mathematical Induction",
"Definition:Positive/Integer",
"Definition:Strictly Positive/Integer",
"Definition:Positive/Integer",
"Definition:Strictly Positive/Integer",
"Definition:Strictly Positive/Integer",
"Definition:Positive/Integer",
"Definition:Positiv... |
proofwiki-19186 | Condition for Mapping between Structure and Cartesian Product of Substructures to be Bijection | Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.
Let $\struct {A, \circ {\restriction_A} }$ and $\struct {B, \circ {\restriction_B} }$ be closed algebraic substructures of $\struct {S, \circ}$, where $\circ {\restriction_A}$ and $\circ {\restriction_B}$ are the operations induced by the restrictio... | First we establish that from Set of Finite Subsets under Induced Operation is Closed:
:$A \times B \subseteq S$
Thus:
:$\forall \tuple {a, b} \in A \times B: \exists s \in S: s = a \circ b = \map \phi {a, b}$
Thus $\phi$ is indeed a mapping. | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure with $1$ operation]].
Let $\struct {A, \circ {\restriction_A} }$ and $\struct {B, \circ {\restriction_B} }$ be [[Definition:Closed Algebraic Structure|closed]] [[Definition:Algebraic Substructure|algebraic substructu... | First we establish that from [[Set of Finite Subsets under Induced Operation is Closed]]:
:$A \times B \subseteq S$
Thus:
:$\forall \tuple {a, b} \in A \times B: \exists s \in S: s = a \circ b = \map \phi {a, b}$
Thus $\phi$ is indeed a [[Definition:Mapping|mapping]]. | Condition for Mapping between Structure and Cartesian Product of Substructures to be Bijection | https://proofwiki.org/wiki/Condition_for_Mapping_between_Structure_and_Cartesian_Product_of_Substructures_to_be_Bijection | https://proofwiki.org/wiki/Condition_for_Mapping_between_Structure_and_Cartesian_Product_of_Substructures_to_be_Bijection | [
"Internal Direct Products"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Algebraic Substructure",
"Definition:Operation Induced by Restriction",
"Definition:Mapping",
"Definition:Cartesian Product",
"Definition:Bijection",
"Definition:Unique"
] | [
"Set of Finite Subsets under Induced Operation is Closed",
"Definition:Mapping"
] |
proofwiki-19187 | Mapping on Cartesian Product of Substructures is Restriction of Operation | Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.
Let $\struct {A, \circ {\restriction_A} }, \struct {B, \circ {\restriction_B} }$ be closed algebraic substructures of $\struct {S, \circ}$, where $\circ {\restriction_A}$ and $\circ {\restriction_B}$ are the operations induced by the restrictions of... | {{Proofread}}
Suppose the mapping $\phi: A \times B \to S$ is defined as:
:$\forall \tuple {a, b} \in A \times B: \map \phi {a, b} = a \circ b$
where $A \times B$ denotes the Cartesian product of $A$ and $B$.
Then:
{{begin-eqn}}
{{eqn | q = \forall a \in A, b \in B
| l = a \circ b
| r = \map \phi {a, b}
... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure with $1$ operation]].
Let $\struct {A, \circ {\restriction_A} }, \struct {B, \circ {\restriction_B} }$ be [[Definition:Closed Algebraic Structure|closed]] [[Definition:Algebraic Substructure|algebraic substructures]]... | {{Proofread}}
Suppose the [[Definition:Mapping|mapping]] $\phi: A \times B \to S$ is defined as:
:$\forall \tuple {a, b} \in A \times B: \map \phi {a, b} = a \circ b$
where $A \times B$ denotes the [[Definition:Cartesian Product|Cartesian product]] of $A$ and $B$.
Then:
{{begin-eqn}}
{{eqn | q = \forall a \in A, b \... | Mapping on Cartesian Product of Substructures is Restriction of Operation | https://proofwiki.org/wiki/Mapping_on_Cartesian_Product_of_Substructures_is_Restriction_of_Operation | https://proofwiki.org/wiki/Mapping_on_Cartesian_Product_of_Substructures_is_Restriction_of_Operation | [
"Internal Direct Products"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Algebraic Substructure",
"Definition:Operation Induced by Restriction",
"Definition:Mapping",
"Definition:Cartesian Product",
"Definition:Restriction/Operation",
"Definition:Operati... | [
"Definition:Mapping",
"Definition:Cartesian Product"
] |
proofwiki-19188 | Codomain of Internal Direct Isomorphism is Subset Product of Factors | Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.
Let $\struct {A, \circ {\restriction_A} }$ and $\struct {B, \circ {\restriction_B} }$ be closed algebraic substructures of $\struct {S, \circ}$, where $\circ {\restriction_A}$ and $\circ {\restriction_B}$ are the operations induced by the restrictio... | First we establish that from Set of Finite Subsets under Induced Operation is Closed:
:$A \times B \subseteq S$
From Condition for Mapping between Structure and Cartesian Product of Substructures to be Bijection:
:$\phi$ is a bijection
{{iff}}:
:for all $s \in S$: there exists a unique $\tuple {a, b} \in A \times B$ su... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure with $1$ operation]].
Let $\struct {A, \circ {\restriction_A} }$ and $\struct {B, \circ {\restriction_B} }$ be [[Definition:Closed Algebraic Structure|closed]] [[Definition:Algebraic Substructure|algebraic substructu... | First we establish that from [[Set of Finite Subsets under Induced Operation is Closed]]:
:$A \times B \subseteq S$
From [[Condition for Mapping between Structure and Cartesian Product of Substructures to be Bijection]]:
:$\phi$ is a [[Definition:Bijection|bijection]]
{{iff}}:
:for all $s \in S$: there exists a [[Def... | Codomain of Internal Direct Isomorphism is Subset Product of Factors | https://proofwiki.org/wiki/Codomain_of_Internal_Direct_Isomorphism_is_Subset_Product_of_Factors | https://proofwiki.org/wiki/Codomain_of_Internal_Direct_Isomorphism_is_Subset_Product_of_Factors | [
"Subset Products",
"Internal Direct Products"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Algebraic Substructure",
"Definition:Operation Induced by Restriction",
"Definition:Internal Direct Product",
"Definition:Subset Product"
] | [
"Set of Finite Subsets under Induced Operation is Closed",
"Condition for Mapping between Structure and Cartesian Product of Substructures to be Bijection",
"Definition:Bijection",
"Definition:Unique",
"Definition:Internal Direct Product",
"Definition:Mapping",
"Definition:Isomorphism (Abstract Algebra)... |
proofwiki-19189 | Left Operation is Entropic | The left operation is entropic:
:$\forall a, b, c, d: \paren {a \gets b} \gets \paren {c \gets d} = \paren {a \gets c} \gets \paren {b \gets d}$ | {{begin-eqn}}
{{eqn | l = \paren {a \gets b} \gets \paren {c \gets d}
| r = a \gets c
| c = {{Defof|Left Operation}}
}}
{{eqn | r = a
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {a \gets c} \gets \paren {b \gets d}
| r = a \gets b
| c = {{Defof|Left Operation}}
}}
{{eqn | r = a
... | The [[Definition:Left Operation|left operation]] is [[Definition:Entropic Operation|entropic]]:
:$\forall a, b, c, d: \paren {a \gets b} \gets \paren {c \gets d} = \paren {a \gets c} \gets \paren {b \gets d}$ | {{begin-eqn}}
{{eqn | l = \paren {a \gets b} \gets \paren {c \gets d}
| r = a \gets c
| c = {{Defof|Left Operation}}
}}
{{eqn | r = a
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {a \gets c} \gets \paren {b \gets d}
| r = a \gets b
| c = {{Defof|Left Operation}}
}}
{{eqn | r = ... | Left Operation is Entropic | https://proofwiki.org/wiki/Left_Operation_is_Entropic | https://proofwiki.org/wiki/Left_Operation_is_Entropic | [
"Left Operation",
"Examples of Entropic Operations"
] | [
"Definition:Left Operation",
"Definition:Entropic Operation"
] | [] |
proofwiki-19190 | Right Operation is Entropic | The right operation is entropic:
:$\forall a, b, c, d: \paren {a \to b} \to \paren {c \to d} = \paren {a \to c} \to \paren {b \to d}$ | {{begin-eqn}}
{{eqn | l = \paren {a \to b} \to \paren {c \to d}
| r = b \to d
| c = {{Defof|Right Operation}}
}}
{{eqn | r = d
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {a \to c} \to \paren {b \to d}
| r = c \to d
| c = {{Defof|Right Operation}}
}}
{{eqn | r = d
| c =
}... | The [[Definition:Right Operation|right operation]] is [[Definition:Entropic Operation|entropic]]:
:$\forall a, b, c, d: \paren {a \to b} \to \paren {c \to d} = \paren {a \to c} \to \paren {b \to d}$ | {{begin-eqn}}
{{eqn | l = \paren {a \to b} \to \paren {c \to d}
| r = b \to d
| c = {{Defof|Right Operation}}
}}
{{eqn | r = d
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {a \to c} \to \paren {b \to d}
| r = c \to d
| c = {{Defof|Right Operation}}
}}
{{eqn | r = d
| c = ... | Right Operation is Entropic | https://proofwiki.org/wiki/Right_Operation_is_Entropic | https://proofwiki.org/wiki/Right_Operation_is_Entropic | [
"Right Operation",
"Examples of Entropic Operations"
] | [
"Definition:Right Operation",
"Definition:Entropic Operation"
] | [] |
proofwiki-19191 | Entropic Structure with Identity is Commutative Monoid | Let $\struct {S, \odot}$ be a magma.
Let $\struct {S, \odot}$ be an entropic structure:
:$\forall a, b, c, d \in S: \paren {a \odot b} \odot \paren {c \odot d} = \paren {a \odot c} \odot \paren {b \odot d}$
Let $\struct {S, \odot}$ have an identity element $e$.
Then $\struct {S, \odot}$ is a commutative monoid. | We have that $\struct {S, \odot}$ is a magma.
Thus {{afortiori}} $\struct {S, \odot}$ is closed, and {{MonoidAxiom|0}} is fulfilled.
Then:
{{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \odot b} \odot c
| r = \paren {a \odot b} \odot \paren {e \odot c}
| c = {{Defof|Identity Element}}... | Let $\struct {S, \odot}$ be a [[Definition:Magma|magma]].
Let $\struct {S, \odot}$ be an [[Definition:Entropic Structure|entropic structure]]:
:$\forall a, b, c, d \in S: \paren {a \odot b} \odot \paren {c \odot d} = \paren {a \odot c} \odot \paren {b \odot d}$
Let $\struct {S, \odot}$ have an [[Definition:Identity ... | We have that $\struct {S, \odot}$ is a [[Definition:Magma|magma]].
Thus {{afortiori}} $\struct {S, \odot}$ is [[Definition:Closed Algebraic Structure|closed]], and {{MonoidAxiom|0}} is fulfilled.
Then:
{{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \odot b} \odot c
| r = \paren {a \odot... | Entropic Structure with Identity is Commutative Monoid | https://proofwiki.org/wiki/Entropic_Structure_with_Identity_is_Commutative_Monoid | https://proofwiki.org/wiki/Entropic_Structure_with_Identity_is_Commutative_Monoid | [
"Entropic Structures",
"Commutative Monoids"
] | [
"Definition:Magma",
"Definition:Entropic Structure",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutative Monoid"
] | [
"Definition:Magma",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Associative Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Monoid",
"Definition:Commutative Monoid"
] |
proofwiki-19192 | Linear Combination Operator on Real Numbers is Entropic | Let $a, b \in \R$ be real numbers.
Let $\odot$ be the operation on $\R$ defined as:
:$\forall x, y \in \R: x \odot y := a x + b y$
Then $\odot$ is an entropic operation. | {{begin-eqn}}
{{eqn | q = \forall p, q, r, s \in \R
| l = \paren {p \odot q} \odot \paren {r \odot s}
| r = a \paren {a p + b q} + b \paren {a r + b s}
| c = Definition of $\odot$
}}
{{eqn | r = a^2 p + a b q + b a r + b^2 s
| c = Real Multiplication Distributes over Addition
}}
{{eqn | r = a^2 ... | Let $a, b \in \R$ be [[Definition:Real Number|real numbers]].
Let $\odot$ be the [[Definition:Binary Operation|operation]] on $\R$ defined as:
:$\forall x, y \in \R: x \odot y := a x + b y$
Then $\odot$ is an [[Definition:Entropic Operation|entropic operation]]. | {{begin-eqn}}
{{eqn | q = \forall p, q, r, s \in \R
| l = \paren {p \odot q} \odot \paren {r \odot s}
| r = a \paren {a p + b q} + b \paren {a r + b s}
| c = Definition of $\odot$
}}
{{eqn | r = a^2 p + a b q + b a r + b^2 s
| c = [[Real Multiplication Distributes over Addition]]
}}
{{eqn | r = ... | Linear Combination Operator on Real Numbers is Entropic | https://proofwiki.org/wiki/Linear_Combination_Operator_on_Real_Numbers_is_Entropic | https://proofwiki.org/wiki/Linear_Combination_Operator_on_Real_Numbers_is_Entropic | [
"Examples of Entropic Operations",
"Linear Combinations",
"Real Numbers"
] | [
"Definition:Real Number",
"Definition:Operation/Binary Operation",
"Definition:Entropic Operation"
] | [
"Real Multiplication Distributes over Addition",
"Real Multiplication is Commutative",
"Real Addition is Commutative",
"Real Multiplication Distributes over Addition",
"Real Multiplication Distributes over Addition"
] |
proofwiki-19193 | Set of Closed Subsets of Power Structure of Entropic Structure is Closed | Let $\struct {S, \odot}$ be a magma.
Let $\struct {S, \odot}$ be an entropic structure.
Let $\struct {\powerset S, \odot_\PP}$ be the power structure of $\struct {S, \odot}$.
Let $\TT$ be the set of all submagmas of $\struct {S, \odot}$.
Then the algebraic structure $\struct {\TT, \odot_\PP}$ is a submagma of $\struct ... | Recall the definition of subset product:
:$A \odot_\PP B = \set {a \odot b: \tuple {a, b} \in A \times B}$
First we show that:
:$\forall A, B \in \TT: A \odot_\PP B \in \TT$
Let $A$ and $B$ be arbitrary elements of $\TT$.
Let $a$ and $c$ be arbitrary elements of $A$.
Let $b$ and $d$ be arbitrary elements of $B$.
Then w... | Let $\struct {S, \odot}$ be a [[Definition:Magma|magma]].
Let $\struct {S, \odot}$ be an [[Definition:Entropic Structure|entropic structure]].
Let $\struct {\powerset S, \odot_\PP}$ be the [[Definition:Power Structure|power structure]] of $\struct {S, \odot}$.
Let $\TT$ be the [[Definition:Set|set]] of all [[Defini... | Recall the definition of [[Definition:Subset Product|subset product]]:
:$A \odot_\PP B = \set {a \odot b: \tuple {a, b} \in A \times B}$
First we show that:
:$\forall A, B \in \TT: A \odot_\PP B \in \TT$
Let $A$ and $B$ be arbitrary [[Definition:Element|elements]] of $\TT$.
Let $a$ and $c$ be arbitrary [[Definition... | Set of Closed Subsets of Power Structure of Entropic Structure is Closed | https://proofwiki.org/wiki/Set_of_Closed_Subsets_of_Power_Structure_of_Entropic_Structure_is_Closed | https://proofwiki.org/wiki/Set_of_Closed_Subsets_of_Power_Structure_of_Entropic_Structure_is_Closed | [
"Entropic Structures",
"Power Structures"
] | [
"Definition:Magma",
"Definition:Entropic Structure",
"Definition:Power Structure",
"Definition:Set",
"Definition:Submagma",
"Definition:Algebraic Structure",
"Definition:Submagma",
"Definition:Entropic Structure"
] | [
"Definition:Subset Product",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abs... |
proofwiki-19194 | Substructure of Entropic Structure is Entropic | Let $\struct {S, \odot}$ be an entropic structure:
:$\forall a, b, c, d \in S: \paren {a \odot b} \odot \paren {c \odot d} = \paren {a \odot c} \odot \paren {b \odot d}$
Let $\struct {T, \odot_T}$ be a substructure of $\struct {S, \odot}$.
Then $\struct {T, \odot_T}$ is also an entropic structure. | {{begin-eqn}}
{{eqn | q = \forall a, b, c, d \in T
| o =
| r = \paren {a \odot_T b} \odot_T \paren {c \odot_T d}
| c =
}}
{{eqn | r = \paren {a \odot b} \odot \paren {c \odot d}
| c = {{Defof|Operation Induced by Restriction}}
}}
{{eqn | r = \paren {a \odot c} \odot \paren {b \odot d}
| ... | Let $\struct {S, \odot}$ be an [[Definition:Entropic Structure|entropic structure]]:
:$\forall a, b, c, d \in S: \paren {a \odot b} \odot \paren {c \odot d} = \paren {a \odot c} \odot \paren {b \odot d}$
Let $\struct {T, \odot_T}$ be a [[Definition:Algebraic Substructure|substructure]] of $\struct {S, \odot}$.
Then ... | {{begin-eqn}}
{{eqn | q = \forall a, b, c, d \in T
| o =
| r = \paren {a \odot_T b} \odot_T \paren {c \odot_T d}
| c =
}}
{{eqn | r = \paren {a \odot b} \odot \paren {c \odot d}
| c = {{Defof|Operation Induced by Restriction}}
}}
{{eqn | r = \paren {a \odot c} \odot \paren {b \odot d}
| ... | Substructure of Entropic Structure is Entropic | https://proofwiki.org/wiki/Substructure_of_Entropic_Structure_is_Entropic | https://proofwiki.org/wiki/Substructure_of_Entropic_Structure_is_Entropic | [
"Entropic Structures"
] | [
"Definition:Entropic Structure",
"Definition:Algebraic Substructure",
"Definition:Entropic Structure"
] | [
"Definition:Entropic Structure",
"Category:Entropic Structures"
] |
proofwiki-19195 | Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product | Let $\struct {G, \odot}$ be a group
Let $\struct {H, \odot}$ and $\struct {K, \odot}$ be normal subgroups of $\struct {G, \odot}$.
Then:
:$\struct {G, \odot}$ is the internal group direct product of $\struct {H, \odot}$ and $\struct {K, \odot}$
{{iff}}
:the restriction of the quotient epimorphism $q_H$ to $K$ is an iso... | === Sufficient Condition ===
{{:Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product/Sufficient Condition}}{{qed|lemma}} | Let $\struct {G, \odot}$ be a [[Definition:Group|group]]
Let $\struct {H, \odot}$ and $\struct {K, \odot}$ be [[Definition:Normal Subgroup|normal subgroups]] of $\struct {G, \odot}$.
Then:
:$\struct {G, \odot}$ is the [[Definition:Internal Group Direct Product|internal group direct product]] of $\struct {H, \odot}$ a... | === [[Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product/Sufficient Condition|Sufficient Condition]] ===
{{:Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product/Sufficient Condition}}{{qed|lemma}} | Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product | https://proofwiki.org/wiki/Quotient_Epimorphism_Condition_for_Normal_Subgroup_Product_to_be_Internal_Group_Direct_Product | https://proofwiki.org/wiki/Quotient_Epimorphism_Condition_for_Normal_Subgroup_Product_to_be_Internal_Group_Direct_Product | [
"Internal Group Direct Products",
"Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Internal Group Direct Product",
"Definition:Restriction/Mapping",
"Definition:Quotient Epimorphism/Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Quotient Group",
"Definition:Restriction/Mapping",
"Defi... | [
"Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product/Sufficient Condition"
] |
proofwiki-19196 | Restriction of Homomorphism is Homomorphism | Let $\struct {S, \circ}$ and $\struct {T, \odot}$ be algebraic structures.
Let $\phi: S \to T$ be a homomorphism.
Let $A \subseteq S$ be a subset of $S$.
Then the restriction of $\phi$ to $A \times \Img A$ is also a homomorphism. | {{begin-eqn}}
{{eqn | q = \forall x, y \in A
| l = \map \phi {x \circ{\restriction_{A} } y}
| r = \map \phi {x \circ y}
| c = {{Defof|Restriction of Operation}}
}}
{{eqn | r = \map \phi x \odot \map \phi y
| c = {{Defof|Homomorphism (Abstract Algebra)}}
}}
{{eqn | r = \map \phi x \odot {\restric... | Let $\struct {S, \circ}$ and $\struct {T, \odot}$ be [[Definition:Algebraic Structure|algebraic structures]].
Let $\phi: S \to T$ be a [[Definition:Homomorphism (Abstract Algebra)|homomorphism]].
Let $A \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Then the [[Definition:Restriction of Mapping|restriction]]... | {{begin-eqn}}
{{eqn | q = \forall x, y \in A
| l = \map \phi {x \circ{\restriction_{A} } y}
| r = \map \phi {x \circ y}
| c = {{Defof|Restriction of Operation}}
}}
{{eqn | r = \map \phi x \odot \map \phi y
| c = {{Defof|Homomorphism (Abstract Algebra)}}
}}
{{eqn | r = \map \phi x \odot {\restric... | Restriction of Homomorphism is Homomorphism | https://proofwiki.org/wiki/Restriction_of_Homomorphism_is_Homomorphism | https://proofwiki.org/wiki/Restriction_of_Homomorphism_is_Homomorphism | [
"Homomorphisms (Abstract Algebra)",
"Restrictions"
] | [
"Definition:Algebraic Structure",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Subset",
"Definition:Restriction/Mapping",
"Definition:Homomorphism (Abstract Algebra)"
] | [
"Category:Homomorphisms (Abstract Algebra)",
"Category:Restrictions"
] |
proofwiki-19197 | Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product/Sufficient Condition | Let $\struct {G, \odot}$ be a group
Let $\struct {H, \odot}$ and $\struct {K, \odot}$ be normal subgroups of $\struct {G, \odot}$.
Let $\struct {G, \odot}$ be the internal group direct product of $\struct {H, \odot}$ and $\struct {K, \odot}$.
Then:
:the restriction of the quotient epimorphism $q_H$ to $K$ is an isomorp... | Recall that the quotient epimorphism $q_H: G \to G / H$ is defined as:
:$\forall x \in G: \map {q_H} x = x \odot H$
From Restriction of Homomorphism is Homomorphism:
:the restriction of $q_H$ to $K$ is a homomorphism.
From Lagrange's Theorem (Group Theory):
:$\card {G / H} = \card K$
Then we have:
{{begin-eqn}}
{{eqn |... | Let $\struct {G, \odot}$ be a [[Definition:Group|group]]
Let $\struct {H, \odot}$ and $\struct {K, \odot}$ be [[Definition:Normal Subgroup|normal subgroups]] of $\struct {G, \odot}$.
Let $\struct {G, \odot}$ be the [[Definition:Internal Group Direct Product|internal group direct product]] of $\struct {H, \odot}$ and... | Recall that the [[Definition:Quotient Group Epimorphism|quotient epimorphism]] $q_H: G \to G / H$ is defined as:
:$\forall x \in G: \map {q_H} x = x \odot H$
From [[Restriction of Homomorphism is Homomorphism]]:
:the [[Definition:Restriction of Mapping|restriction]] of $q_H$ to $K$ is a [[Definition:Group Homomorphis... | Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product/Sufficient Condition | https://proofwiki.org/wiki/Quotient_Epimorphism_Condition_for_Normal_Subgroup_Product_to_be_Internal_Group_Direct_Product/Sufficient_Condition | https://proofwiki.org/wiki/Quotient_Epimorphism_Condition_for_Normal_Subgroup_Product_to_be_Internal_Group_Direct_Product/Sufficient_Condition | [
"Quotient Epimorphism Condition for Normal Subgroup Product to be Internal Group Direct Product"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Internal Group Direct Product",
"Definition:Restriction/Mapping",
"Definition:Quotient Epimorphism/Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Quotient Group",
"Definition:Restriction/Mapping",
"Defi... | [
"Definition:Quotient Epimorphism/Group",
"Restriction of Homomorphism is Homomorphism",
"Definition:Restriction/Mapping",
"Definition:Group Homomorphism",
"Lagrange's Theorem (Group Theory)",
"Definition:Normal Subgroup",
"Definition:Internal Group Direct Product",
"Definition:Unique",
"Definition:R... |
proofwiki-19198 | Composition of Mappings is Right Distributive over Pointwise Operation | Let $A$ and $B$ be sets.
Let $\struct {S, \odot}$ be an algebraic structure.
Let:
:$B^A$ denote the set of mappings from $A$ to $B$
:$S^B$ denote the set of mappings from $B$ to $S$.
Let $f, g \in S^B$ be mappings from $B$ to $S$.
Let $h \in B^A$ be a mapping from $A$ to $B$.
Then:
:$\paren {f \odot g} \circ h = \paren... | First we establish:
The domain of $h$ is $A$.
The codomain of $h$ is $B$
The domain of both $f$ and $g$ is $B$.
The codomain of both $f$ and $g$ is $S$.
Hence:
:the domain of $\paren {f \odot g} \circ h$ is $A$
:the Domain of $f \odot g$ is $B$
:the codomain of $f \odot g$ is $S$
:the codomain of $\paren {f \odot g} \c... | Let $A$ and $B$ be [[Definition:Set|sets]].
Let $\struct {S, \odot}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let:
:$B^A$ denote the [[Definition:Set of All Mappings|set of mappings]] from $A$ to $B$
:$S^B$ denote the [[Definition:Set of All Mappings|set of mappings]] from $B$ to $S$.
Let $f, g... | First we establish:
The [[Definition:Domain of Mapping|domain]] of $h$ is $A$.
The [[Definition:Codomain of Mapping|codomain]] of $h$ is $B$
The [[Definition:Domain of Mapping|domain]] of both $f$ and $g$ is $B$.
The [[Definition:Codomain of Mapping|codomain]] of both $f$ and $g$ is $S$.
Hence:
:the [[Definition:D... | Composition of Mappings is Right Distributive over Pointwise Operation | https://proofwiki.org/wiki/Composition_of_Mappings_is_Right_Distributive_over_Pointwise_Operation | https://proofwiki.org/wiki/Composition_of_Mappings_is_Right_Distributive_over_Pointwise_Operation | [
"Pointwise Operations",
"Composite Mappings",
"Examples of Distributive Operations"
] | [
"Definition:Set",
"Definition:Algebraic Structure",
"Definition:Set of All Mappings",
"Definition:Set of All Mappings",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Pointwise Operation",
"Definition:Composition of Mappings"
] | [
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping",
"D... |
proofwiki-19199 | Composition of Mappings is Left Distributive over Homomorphism of Pointwise Operation | Let $A$ be a set.
Let $\struct {S, \odot}$ and $\struct {T, \otimes}$ be algebraic structures.
Let:
:$S^A$ denote the set of mappings from $A$ to $S$.
Let $f$ be a homomorphism from $S$ to $T$.
Let $g, h \in S^A$ be mappings from $A$ to $S$.
Then:
:$f \circ \paren {g \odot h} = \paren {f \circ g} \otimes \paren {f \cir... | First we establish:
The domain of $g$ and $h$ is $A$.
The codomain of $g$ and $h$ is $S$.
The domain of $f$ is $S$.
The codomain of $f$ is $T$.
Hence:
:the Domain of $g \odot h$ is $A$
:the codomain of $g \odot h$ is $S$
:the domain of $f \circ \paren {g \odot h}$ is $A$
:the codomain of $f \circ \paren {g \odot h}$ is... | Let $A$ be a [[Definition:Set|set]].
Let $\struct {S, \odot}$ and $\struct {T, \otimes}$ be [[Definition:Algebraic Structure|algebraic structures]].
Let:
:$S^A$ denote the [[Definition:Set of All Mappings|set of mappings]] from $A$ to $S$.
Let $f$ be a [[Definition:Homomorphism (Abstract Algebra)|homomorphism]] fro... | First we establish:
The [[Definition:Domain of Mapping|domain]] of $g$ and $h$ is $A$.
The [[Definition:Codomain of Mapping|codomain]] of $g$ and $h$ is $S$.
The [[Definition:Domain of Mapping|domain]] of $f$ is $S$.
The [[Definition:Codomain of Mapping|codomain]] of $f$ is $T$.
Hence:
:the [[Definition:Domain of ... | Composition of Mappings is Left Distributive over Homomorphism of Pointwise Operation | https://proofwiki.org/wiki/Composition_of_Mappings_is_Left_Distributive_over_Homomorphism_of_Pointwise_Operation | https://proofwiki.org/wiki/Composition_of_Mappings_is_Left_Distributive_over_Homomorphism_of_Pointwise_Operation | [
"Pointwise Operations",
"Composite Mappings",
"Homomorphisms (Abstract Algebra)",
"Examples of Distributive Operations"
] | [
"Definition:Set",
"Definition:Algebraic Structure",
"Definition:Set of All Mappings",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Mapping",
"Definition:Pointwise Operation",
"Definition:Composition of Mappings"
] | [
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Domain (Set Theory)/Mapping",
"D... |
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