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-18300 | Jadhav Theorem | Let $a, b, c$ be real numbers in arithmetic sequence.
Let the common difference of this arithmetic sequence be $d$.
Then:
:$b^2 - a c = d^2$ | {{begin-eqn}}
{{eqn | l = b^2 - a c
| r = b^2 - \paren {b + d} \paren {b - d}
| c = {{Defof|Arithmetic Sequence}}: $a + d = b$, $b + d = c$
}}
{{eqn | r = b^2 - \paren {b^2 - d^2}
| c = Difference of Two Squares
}}
{{eqn | r = d^2
| c =
}}
{{end-eqn}}
{{qed}} | Let $a, b, c$ be [[Definition:Real Number|real numbers]] in [[Definition:Arithmetic Sequence|arithmetic sequence]].
Let the [[Definition:Common Difference|common difference]] of this [[Definition:Arithmetic Sequence|arithmetic sequence]] be $d$.
Then:
:$b^2 - a c = d^2$ | {{begin-eqn}}
{{eqn | l = b^2 - a c
| r = b^2 - \paren {b + d} \paren {b - d}
| c = {{Defof|Arithmetic Sequence}}: $a + d = b$, $b + d = c$
}}
{{eqn | r = b^2 - \paren {b^2 - d^2}
| c = [[Difference of Two Squares]]
}}
{{eqn | r = d^2
| c =
}}
{{end-eqn}}
{{qed}} | Jadhav Theorem | https://proofwiki.org/wiki/Jadhav_Theorem | https://proofwiki.org/wiki/Jadhav_Theorem | [
"Arithmetic Sequences",
"Jadhav Theorem"
] | [
"Definition:Real Number",
"Definition:Arithmetic Sequence",
"Definition:Arithmetic Sequence/Common Difference",
"Definition:Arithmetic Sequence"
] | [
"Difference of Two Squares"
] |
proofwiki-18301 | Equivalence of Definitions of P-adic Norms | Let $p \in \N$ be a prime.
Let $\Q$ denote the rational numbers.
{{TFAE|def = P-adic Norm|view = $p$-adic norm on $\Q$}}
=== Definition 1 ===
{{:Definition:P-adic Norm/Rational Numbers/Definition 1}}
=== Definition 2 ===
{{:Definition:P-adic Norm/Rational Numbers/Definition 2}} | From Negative Powers of Group Elements, Definition 2 can be rewritten as:
:$\forall r \in \Q: \norm r_p = \begin {cases} 0 & : r = 0 \\ p^{-k} & : r = p^k \dfrac m n: k, m, n \in \Z, p \nmid m, n \end {cases}$
Hence if follows that Definition 1 and Definition 2 are equivalent if it is shown:
:$\forall r \in \Q_{\ne 0}:... | Let $p \in \N$ be a [[Definition:Prime Number|prime]].
Let $\Q$ denote the [[Definition:Rational Numbers|rational numbers]].
{{TFAE|def = P-adic Norm|view = $p$-adic norm on $\Q$}}
=== [[Definition:P-adic Norm/Rational Numbers/Definition 1|Definition 1]] ===
{{:Definition:P-adic Norm/Rational Numbers/Definition 1}}... | From [[Negative Powers of Group Elements]], [[Definition:P-adic Norm/Rational Numbers/Definition 2|Definition 2]] can be rewritten as:
:$\forall r \in \Q: \norm r_p = \begin {cases} 0 & : r = 0 \\ p^{-k} & : r = p^k \dfrac m n: k, m, n \in \Z, p \nmid m, n \end {cases}$
Hence if follows that [[Definition:P-adic Norm/... | Equivalence of Definitions of P-adic Norms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Norms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Norms | [
"Equivalence of Definitions of P-adic Norms",
"P-adic Norms"
] | [
"Definition:Prime Number",
"Definition:Rational Number",
"Definition:P-adic Norm/Rational Numbers/Definition 1",
"Definition:P-adic Norm/Rational Numbers/Definition 2"
] | [
"Powers of Group Elements/Negative Index",
"Definition:P-adic Norm/Rational Numbers/Definition 2",
"Definition:P-adic Norm/Rational Numbers/Definition 1",
"Definition:P-adic Norm/Rational Numbers/Definition 2"
] |
proofwiki-18302 | Equivalence of Definitions of P-adic Norms/Lemma 1 | :$\forall x \in Z_{\ne 0}: \map {\nu_p} x = k : x = p^k y : p \nmid y$ | Let $x \in \Z_{\ne 0}$.
By definition of the $p$-adic valuation:
:$\map {\nu_p} x = \sup \set {v \in \N: p^v \divides x}$
Let $\map {\nu_p} x = k$.
Then:
:$p^k \nmid x$
By definition of a divisor:
:$\exists y \in Z : x = p^k y$
{{AimForCont}}:
:$p \divides y$
By definition of a divisor:
:$\exists y' \in Z : y = p y'$
H... | :$\forall x \in Z_{\ne 0}: \map {\nu_p} x = k : x = p^k y : p \nmid y$ | Let $x \in \Z_{\ne 0}$.
By definition of the [[Definition:Restricted P-adic Valuation|$p$-adic valuation]]:
:$\map {\nu_p} x = \sup \set {v \in \N: p^v \divides x}$
Let $\map {\nu_p} x = k$.
Then:
:$p^k \nmid x$
By definition of a [[Definition:Divisor of Integer|divisor]]:
:$\exists y \in Z : x = p^k y$
{{AimForC... | Equivalence of Definitions of P-adic Norms/Lemma 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Norms/Lemma_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_P-adic_Norms/Lemma_1 | [
"Equivalence of Definitions of P-adic Norms"
] | [] | [
"Definition:P-adic Valuation/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Contradiction",
"Category:Equivalence of Definitions of P-adic Norms"
] |
proofwiki-18303 | Distributional Derivative of Absolute Value Function | Let $H: \R \to \closedint 0 1$ be the Heaviside step function.
Let $\size x$ be the absolute value of $x$.
Let $T_{\size x}$ be the Schwartz distribution associated with $\size x$.
Then the distributional derivative of $T_{\size x}$ is $T_{2 H - 1}$ | {{begin-eqn}}
{{eqn | l = \dfrac {\d \size x} {\d x}
| r = <nowiki>\begin{cases}
1 & : x > 0 \\
-1 & : x < 0
\end{cases} </nowiki>
}}
{{eqn | r = -1 + <nowiki>\begin{cases}
2 & : x > 0 \\
0 & : x < 0
\end{cas... | Let $H: \R \to \closedint 0 1$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $\size x$ be the [[Definition:Absolute Value|absolute value]] of $x$.
Let $T_{\size x}$ be the [[Definition:Schwartz Distribution|Schwartz distribution]] associated with $\size x$.
Then the [[Definition:Distrib... | {{begin-eqn}}
{{eqn | l = \dfrac {\d \size x} {\d x}
| r = <nowiki>\begin{cases}
1 & : x > 0 \\
-1 & : x < 0
\end{cases} </nowiki>
}}
{{eqn | r = -1 + <nowiki>\begin{cases}
2 & : x > 0 \\
0 & : x < 0
\end{cas... | Distributional Derivative of Absolute Value Function | https://proofwiki.org/wiki/Distributional_Derivative_of_Absolute_Value_Function | https://proofwiki.org/wiki/Distributional_Derivative_of_Absolute_Value_Function | [
"Absolute Value Function",
"Distributional Derivatives"
] | [
"Definition:Heaviside Step Function",
"Definition:Absolute Value",
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative"
] | [
"Jump Rule"
] |
proofwiki-18304 | Moment Generating Function of Geometric Distribution/Formulation 1 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the moment generating function $M_X$ of $X$ is given by:
:$\map {M_X} t = \dfrac {1 - p} {1 - p e^t}$
for $t < -\map \ln p$, and is undefined otherwise. | From the definition of the geometric distribution, $X$ has probability mass function:
:$\map \Pr {X = k} = \paren {1 - p} p^k$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect {e^{t X} } = \sum_{k \mathop = 0}^\infty \map \Pr {X = k} e^{k t}$
So:
{{begin-eqn}}
{{eqn | l = \sum_{k \matho... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the [[Definition:Moment Generating Function|moment generating function]] $M_X$ of $X$ is given by:
:$\map {M_X} t = \dfrac {1 - p} {1 - p e^t}$
for $t < -\map \ln p$, and is undefined otherwise. | From the definition of the [[Definition:Geometric Distribution|geometric distribution]], $X$ has [[Definition:Probability Mass Function|probability mass function]]:
:$\map \Pr {X = k} = \paren {1 - p} p^k$
From the definition of a [[Definition:Moment Generating Function|moment generating function]]:
:$\ds \map {M_X}... | Moment Generating Function of Geometric Distribution/Formulation 1 | https://proofwiki.org/wiki/Moment_Generating_Function_of_Geometric_Distribution/Formulation_1 | https://proofwiki.org/wiki/Moment_Generating_Function_of_Geometric_Distribution/Formulation_1 | [
"Moment Generating Function of Geometric Distribution"
] | [
"Definition:Moment Generating Function"
] | [
"Definition:Geometric Distribution",
"Definition:Probability Mass Function",
"Definition:Moment Generating Function",
"Sum of Infinite Geometric Sequence",
"Logarithm of Power",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-18305 | Moment Generating Function of Geometric Distribution/Formulation 2 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
Then the moment generating function $M_X$ of $X$ is given by:
:$\map {M_X} t = \dfrac p {1 - \paren {1 - p} e^t}$
for $t < -\map \ln {1 - p}$, and is undefined otherwise. | From the definition of the geometric distribution, $X$ has probability mass function:
:$\map \Pr {X = k} = p \paren {1 - p}^k$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect {e^{t X} } = \sum_{k \mathop = 0}^\infty \map \Pr {X = k} e^{k t}$
So:
{{begin-eqn}}
{{eqn | l = \sum_{k \matho... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
Then the [[Definition:Moment Generating Function|moment generating function]] $M_X$ of $X$ is given by:
:$\map {M_X} t = \dfrac p {1 - \paren {1 - p} e^t}$
for $t < -\map \ln {1 - p}$, and is undefined otherwise. | From the definition of the [[Definition:Geometric Distribution|geometric distribution]], $X$ has [[Definition:Probability Mass Function|probability mass function]]:
:$\map \Pr {X = k} = p \paren {1 - p}^k$
From the definition of a [[Definition:Moment Generating Function|moment generating function]]:
:$\ds \map {M_X}... | Moment Generating Function of Geometric Distribution/Formulation 2 | https://proofwiki.org/wiki/Moment_Generating_Function_of_Geometric_Distribution/Formulation_2 | https://proofwiki.org/wiki/Moment_Generating_Function_of_Geometric_Distribution/Formulation_2 | [
"Moment Generating Function of Geometric Distribution"
] | [
"Definition:Moment Generating Function"
] | [
"Definition:Geometric Distribution",
"Definition:Probability Mass Function",
"Definition:Moment Generating Function",
"Sum of Infinite Geometric Sequence",
"Logarithm of Power",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-18306 | Expectation of Geometric Distribution/Formulation 1 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the expectation of $X$ is given by:
:$\expect X = \dfrac p {1 - p}$ | From the Probability Generating Function of Geometric Distribution:
:$\map {\Pi_X} s = \dfrac q {1 - p s}$
where $q = 1 - p$.
From Expectation of Discrete Random Variable from PGF:
:$\expect X = \map {\Pi'_X} 1$
We have:
{{begin-eqn}}
{{eqn | l = \map {\Pi'_X} s
| r = \map {\frac \d {\d s} } {\frac q {1 - p s} }
... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the [[Definition:Expectation|expectation]] of $X$ is given by:
:$\expect X = \dfrac p {1 - p}$ | From the [[Probability Generating Function of Geometric Distribution]]:
:$\map {\Pi_X} s = \dfrac q {1 - p s}$
where $q = 1 - p$.
From [[Expectation of Discrete Random Variable from PGF]]:
:$\expect X = \map {\Pi'_X} 1$
We have:
{{begin-eqn}}
{{eqn | l = \map {\Pi'_X} s
| r = \map {\frac \d {\d s} } {\fra... | Expectation of Geometric Distribution/Formulation 1/Proof 2 | https://proofwiki.org/wiki/Expectation_of_Geometric_Distribution/Formulation_1 | https://proofwiki.org/wiki/Expectation_of_Geometric_Distribution/Formulation_1/Proof_2 | [
"Expectation of Geometric Distribution"
] | [
"Definition:Expectation"
] | [
"Probability Generating Function of Geometric Distribution",
"Expectation of Discrete Random Variable from PGF",
"Derivatives of PGF of Geometric Distribution"
] |
proofwiki-18307 | Expectation of Geometric Distribution/Formulation 1 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the expectation of $X$ is given by:
:$\expect X = \dfrac p {1 - p}$ | From the definition of expectation:
:$\ds \expect X = \sum_{x \mathop \in \Omega_X} x \map \Pr {X = x}$
Then
{{begin-eqn}}
{{eqn | l = \expect X
| r = \sum_{k \mathop \in \N} k p^k \paren {1 - p}
| c = {{Defof|Geometric Distribution}}
}}
{{eqn | r = \sum_{k \mathop \ge 1} k p^k \paren {1 - p}
| c = as... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the [[Definition:Expectation|expectation]] of $X$ is given by:
:$\expect X = \dfrac p {1 - p}$ | From the definition of [[Definition:Expectation|expectation]]:
:$\ds \expect X = \sum_{x \mathop \in \Omega_X} x \map \Pr {X = x}$
Then
{{begin-eqn}}
{{eqn | l = \expect X
| r = \sum_{k \mathop \in \N} k p^k \paren {1 - p}
| c = {{Defof|Geometric Distribution}}
}}
{{eqn | r = \sum_{k \mathop \ge 1} k p^k ... | Expectation of Geometric Distribution/Formulation 1/Proof 3 | https://proofwiki.org/wiki/Expectation_of_Geometric_Distribution/Formulation_1 | https://proofwiki.org/wiki/Expectation_of_Geometric_Distribution/Formulation_1/Proof_3 | [
"Expectation of Geometric Distribution"
] | [
"Definition:Expectation"
] | [
"Definition:Expectation",
"Real Multiplication Distributes over Addition",
"Ratio Test",
"Absolutely Convergent Series is Convergent/Real Numbers",
"Convergent Series can be Added Term by Term",
"Translation of Index Variable of Summation",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-18308 | Variance of Geometric Distribution/Formulation 1 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the variance of $X$ is given by:
:$\var X = \dfrac p {\paren {1-p}^2}$ | From Variance of Discrete Random Variable from PGF, we have:
:$\var X = \map {\Pi' '_X} 1 + \mu - \mu^2$
where $\mu = \map E x$ is the expectation of $X$.
From the Probability Generating Function of Geometric Distribution, we have:
:$\map {\Pi_X} s = \dfrac q {1 - ps}$
where $q = 1 - p$.
From Expectation of Geometric D... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \dfrac p {\paren {1-p}^2}$ | From [[Variance of Discrete Random Variable from PGF]], we have:
:$\var X = \map {\Pi' '_X} 1 + \mu - \mu^2$
where $\mu = \map E x$ is the [[Definition:Expectation|expectation]] of $X$.
From the [[Probability Generating Function of Geometric Distribution]], we have:
:$\map {\Pi_X} s = \dfrac q {1 - ps}$
where $q = 1 ... | Variance of Geometric Distribution/Formulation 1/Proof 2 | https://proofwiki.org/wiki/Variance_of_Geometric_Distribution/Formulation_1 | https://proofwiki.org/wiki/Variance_of_Geometric_Distribution/Formulation_1/Proof_2 | [
"Variance",
"Variance of Geometric Distribution"
] | [
"Definition:Variance"
] | [
"Variance of Discrete Random Variable from PGF",
"Definition:Expectation",
"Probability Generating Function of Geometric Distribution",
"Expectation of Geometric Distribution",
"Derivatives of PGF of Geometric Distribution"
] |
proofwiki-18309 | Distributional Derivative of Floor Function | Let $\floor x$ be the floor function.
Let $\map {\operatorname {III} } x$ be the Dirac comb.
Then the distributional derivative of $\floor x$ is $\map {\operatorname {III} } 0$. | By definition:
:$\floor x := \sup \set {m \in \Z: m \le x}$
Hence, $\forall m \in \Z : \forall x \in \openint m {m + 1}$ the floor function is constant.
Therefore:
:$\forall m \in \Z : \forall x \in \openint m {m + 1} : \dfrac {\d \floor x} {\d x} = 0$
Every $x \in \Z$ is a discontinuity of $\floor x$.
Hence, the jump ... | Let $\floor x$ be the [[Definition:Floor Function|floor function]].
Let $\map {\operatorname {III} } x$ be the [[Definition:Dirac Comb|Dirac comb]].
Then the [[Definition:Distributional Derivative|distributional derivative]] of $\floor x$ is $\map {\operatorname {III} } 0$. | By [[Definition:Floor Function/Definition 1|definition]]:
:$\floor x := \sup \set {m \in \Z: m \le x}$
Hence, $\forall m \in \Z : \forall x \in \openint m {m + 1}$ the [[Definition:Floor Function|floor function]] is [[Definition:Constant Function|constant]].
Therefore:
:$\forall m \in \Z : \forall x \in \openint m ... | Distributional Derivative of Floor Function | https://proofwiki.org/wiki/Distributional_Derivative_of_Floor_Function | https://proofwiki.org/wiki/Distributional_Derivative_of_Floor_Function | [
"Floor Function",
"Distributional Derivatives"
] | [
"Definition:Floor Function",
"Definition:Sampling Function",
"Definition:Distributional Derivative"
] | [
"Definition:Floor Function/Definition 1",
"Definition:Floor Function",
"Definition:Constant Mapping",
"Definition:Discontinuity (Real Analysis)/Jump",
"Jump Rule",
"Jump Rule"
] |
proofwiki-18310 | Variance of Geometric Distribution/Formulation 2 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
Then the variance of $X$ is given by:
:$\var X = \dfrac {1 - p} {p^2}$ | By Moment Generating Function of Geometric Distribution, the moment generating function of $X$ is given by:
:$\map {M_X} t = \dfrac p {1 - \paren {1 - p} e^t}$
for $t < -\map \ln {1 - p}$, and is undefined otherwise.
From Variance as Expectation of Square minus Square of Expectation:
:$\ds \var X = \expect {X^2} - \p... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
Then the [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \dfrac {1 - p} {p^2}$ | By [[Moment Generating Function of Geometric Distribution]], the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
:$\map {M_X} t = \dfrac p {1 - \paren {1 - p} e^t}$
for $t < -\map \ln {1 - p}$, and is undefined otherwise.
From [[Variance as Expectation of Square minus Squar... | Variance of Geometric Distribution/Formulation 2/Proof 2 | https://proofwiki.org/wiki/Variance_of_Geometric_Distribution/Formulation_2 | https://proofwiki.org/wiki/Variance_of_Geometric_Distribution/Formulation_2/Proof_2 | [
"Variance",
"Variance of Geometric Distribution"
] | [
"Definition:Variance"
] | [
"Moment Generating Function of Geometric Distribution",
"Definition:Moment Generating Function",
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Geometric Distribution/Formulation 2",
"Moment Generating Function of Geometric Distribution/Formulation 2/Examples/Second Moment"... |
proofwiki-18311 | Union of Union of Cartesian Product | Let $A$ and $B$ be sets such that $A \ne \O$ and $B \ne \O$.
Let the ordered pair $\tuple {a, b}$ be defined using the Kuratowski formalization:
:$\tuple {a, b} := \set {\set a, \set {a, b} }$
Then:
:$\ds \bigcup \bigcup \paren {A \times B} = A \cup B$
where:
:$\cup$ denotes union
:$\times$ denotes Cartesian product. | {{begin-eqn}}
{{eqn | l = \bigcup \bigcup \paren {A \times B}
| r = \bigcup \bigcup \set {\tuple {a, b}: a \in A, b \in B}
| c = {{Defof|Cartesian Product}}
}}
{{eqn | r = \bigcup \paren {\bigcup \set {\set {\set a, \set {a, b} }: a \in A, b \in B} }
| c = {{Defof|Kuratowski Formalization of Ordered P... | Let $A$ and $B$ be [[Definition:Set|sets]] such that $A \ne \O$ and $B \ne \O$.
Let the [[Definition:Ordered Pair|ordered pair]] $\tuple {a, b}$ be defined using the [[Definition:Kuratowski Formalization of Ordered Pair|Kuratowski formalization]]:
:$\tuple {a, b} := \set {\set a, \set {a, b} }$
Then:
:$\ds \bigcup \b... | {{begin-eqn}}
{{eqn | l = \bigcup \bigcup \paren {A \times B}
| r = \bigcup \bigcup \set {\tuple {a, b}: a \in A, b \in B}
| c = {{Defof|Cartesian Product}}
}}
{{eqn | r = \bigcup \paren {\bigcup \set {\set {\set a, \set {a, b} }: a \in A, b \in B} }
| c = {{Defof|Kuratowski Formalization of Ordered P... | Union of Union of Cartesian Product | https://proofwiki.org/wiki/Union_of_Union_of_Cartesian_Product | https://proofwiki.org/wiki/Union_of_Union_of_Cartesian_Product | [
"Set Union",
"Cartesian Product"
] | [
"Definition:Set",
"Definition:Ordered Pair",
"Definition:Ordered Pair/Kuratowski Formalization",
"Definition:Set Union",
"Definition:Cartesian Product"
] | [] |
proofwiki-18312 | Union of Union of Cartesian Product with Empty Factor | Let $A$ and $B$ be sets such that either $A = \O$ or $B = \O$.
Let the ordered pair $\tuple {a, b}$ be defined using the Kuratowski formalization:
:$\tuple {a, b} := \set {\set a, \set {a, b} }$
Then:
:$\ds \bigcup \bigcup \paren {A \times B} = A \cup B \iff A = B = \O$
where:
:$\cup$ denotes union
:$\times$ denotes Ca... | Let $A = \O$ or $B = \O$.
From Cartesian Product is Empty iff Factor is Empty:
:$A \times B = \O$
Hence from Union of Empty Set:
:$\ds \bigcup \bigcup \paren {A \times B} = \O$
However, from Union is Empty iff Sets are Empty:
:$A \cup B = \O \iff A = \O \text { and } B = \O$
The result follows.
{{qed}} | Let $A$ and $B$ be [[Definition:Set|sets]] such that either $A = \O$ or $B = \O$.
Let the [[Definition:Ordered Pair|ordered pair]] $\tuple {a, b}$ be defined using the [[Definition:Kuratowski Formalization of Ordered Pair|Kuratowski formalization]]:
:$\tuple {a, b} := \set {\set a, \set {a, b} }$
Then:
:$\ds \bigcup... | Let $A = \O$ or $B = \O$.
From [[Cartesian Product is Empty iff Factor is Empty]]:
:$A \times B = \O$
Hence from [[Union of Empty Set]]:
:$\ds \bigcup \bigcup \paren {A \times B} = \O$
However, from [[Union is Empty iff Sets are Empty]]:
:$A \cup B = \O \iff A = \O \text { and } B = \O$
The result follows.
{{qed}} | Union of Union of Cartesian Product with Empty Factor | https://proofwiki.org/wiki/Union_of_Union_of_Cartesian_Product_with_Empty_Factor | https://proofwiki.org/wiki/Union_of_Union_of_Cartesian_Product_with_Empty_Factor | [
"Set Union",
"Cartesian Product"
] | [
"Definition:Set",
"Definition:Ordered Pair",
"Definition:Ordered Pair/Kuratowski Formalization",
"Definition:Set Union",
"Definition:Cartesian Product",
"Definition:Empty Set",
"Definition:Empty Set"
] | [
"Cartesian Product is Empty iff Factor is Empty",
"Union of Empty Set",
"Union is Empty iff Sets are Empty"
] |
proofwiki-18313 | Skewness of Geometric Distribution/Formulation 2 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
Then the skewness of $X$ is given by:
:$\gamma_1 = \dfrac {2 - p} {\sqrt {1 - p} }$ | From the definition of skewness, we have:
:$\gamma_1 = \expect {\paren {\dfrac {X - \mu} \sigma}^3}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Geometric Distribution: Formulation 2, we have:
:$\mu = \dfrac {1 - p} p$
By Variance of Geometric Distribution:... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
Then the [[Definition:Skewness|skewness]] of $X$ is given by:
:$\gamma_1 = \dfrac {2 - p} {\sqrt {1 - p} }$ | From the definition of [[Definition:Skewness|skewness]], we have:
:$\gamma_1 = \expect {\paren {\dfrac {X - \mu} \sigma}^3}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Geometric Distribution/... | Skewness of Geometric Distribution/Formulation 2 | https://proofwiki.org/wiki/Skewness_of_Geometric_Distribution/Formulation_2 | https://proofwiki.org/wiki/Skewness_of_Geometric_Distribution/Formulation_2 | [
"Skewness of Geometric Distribution",
"Skewness"
] | [
"Definition:Skewness"
] | [
"Definition:Skewness",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Geometric Distribution/Formulation 2",
"Variance of Geometric Distribution/Formulation 2",
"Binomial Theorem/Examples/Cube of Difference",
"Expectation is Linear",
"Moment in terms of Moment Generating Fu... |
proofwiki-18314 | Skewness of Geometric Distribution/Formulation 1 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the skewness of $X$ is given by:
:$\gamma_1 = \dfrac {1 + p} {\sqrt p}$ | From the definition of skewness, we have:
:$\gamma_1 = \expect {\paren {\dfrac {X - \mu} \sigma}^3}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Geometric Distribution: Formulation 1, we have:
:$\mu = \dfrac p {1 - p}$
By Variance of Geometric Distribution:... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the [[Definition:Skewness|skewness]] of $X$ is given by:
:$\gamma_1 = \dfrac {1 + p} {\sqrt p}$ | From the definition of [[Definition:Skewness|skewness]], we have:
:$\gamma_1 = \expect {\paren {\dfrac {X - \mu} \sigma}^3}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Geometric Distribution/... | Skewness of Geometric Distribution/Formulation 1 | https://proofwiki.org/wiki/Skewness_of_Geometric_Distribution/Formulation_1 | https://proofwiki.org/wiki/Skewness_of_Geometric_Distribution/Formulation_1 | [
"Skewness of Geometric Distribution",
"Skewness"
] | [
"Definition:Skewness"
] | [
"Definition:Skewness",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Geometric Distribution/Formulation 1",
"Variance of Geometric Distribution/Formulation 1",
"Binomial Theorem/Examples/Cube of Difference",
"Expectation is Linear",
"Moment in terms of Moment Generating Fu... |
proofwiki-18315 | Excess Kurtosis of Geometric Distribution/Formulation 2 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = 6 + \dfrac {p^2} {1 - p}$ | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Geometric Distribution: Formulation 2, we have:
:$\mu = \dfrac {1 - p} p$
By Variance of Geometric Di... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = 6 + \dfrac {p^2} {1 - p}$ | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Geome... | Excess Kurtosis of Geometric Distribution/Formulation 2 | https://proofwiki.org/wiki/Excess_Kurtosis_of_Geometric_Distribution/Formulation_2 | https://proofwiki.org/wiki/Excess_Kurtosis_of_Geometric_Distribution/Formulation_2 | [
"Excess Kurtosis of Geometric Distribution"
] | [
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Geometric Distribution/Formulation 2",
"Variance of Geometric Distribution/Formulation 2",
"Kurtosis in terms of Non-Central Moments",
"Skewness of Geometric Distribution/Formulation 2",
"Definitio... |
proofwiki-18316 | Excess Kurtosis of Geometric Distribution/Formulation 1 | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = 6 + \dfrac {\paren {1 - p}^2} p$ | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Geometric Distribution: Formulation 1, we have:
:$\mu = \dfrac p {1 - p}$
By Variance of Geometric Di... | :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = 6 + \dfrac {\paren {1 - p}^2} p$ | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Geome... | Excess Kurtosis of Geometric Distribution/Formulation 1 | https://proofwiki.org/wiki/Excess_Kurtosis_of_Geometric_Distribution/Formulation_1 | https://proofwiki.org/wiki/Excess_Kurtosis_of_Geometric_Distribution/Formulation_1 | [
"Excess Kurtosis of Geometric Distribution"
] | [
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Geometric Distribution/Formulation 1",
"Variance of Geometric Distribution/Formulation 1",
"Kurtosis in terms of Non-Central Moments",
"Skewness of Geometric Distribution/Formulation 1",
"Definitio... |
proofwiki-18317 | Distributional Derivative of Heaviside Step Function times Cosine | Let $H$ be the Heaviside step function.
Let $\delta$ be the Dirac delta distribution.
Then in the distributional sense:
:$T'_{\map H x \cos x} = T_{- \map H x \sin x} + \delta$ | $x \stackrel f {\longrightarrow} \map H x \cos x$ is a continuously differentiable real function on $\R \setminus \set 0$ and has a discontinuity at $x = 0$.
By Differentiable Function as Distribution we have that:
:$T'_f = T_{f'}$
Moreover:
:$x < 0 \implies \paren { {\map H x} \map \cos x}' = 0$
:$x > 0 \implies \pare... | Let $H$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $\delta$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Then in the [[Definition:Distributional Derivative|distributional sense]]:
:$T'_{\map H x \cos x} = T_{- \map H x \sin x} + \delta$ | $x \stackrel f {\longrightarrow} \map H x \cos x$ is a [[Definition:Continuously Differentiable Real Function on Open Interval|continuously differentiable real function]] on $\R \setminus \set 0$ and has a [[Definition:Jump Discontinuity|discontinuity]] at $x = 0$.
By [[Differentiable Function as Distribution]] we hav... | Distributional Derivative of Heaviside Step Function times Cosine | https://proofwiki.org/wiki/Distributional_Derivative_of_Heaviside_Step_Function_times_Cosine | https://proofwiki.org/wiki/Distributional_Derivative_of_Heaviside_Step_Function_times_Cosine | [
"Examples of Distributional Derivatives",
"Piecewise Continuously Differentiable Functions"
] | [
"Definition:Heaviside Step Function",
"Definition:Dirac Delta Distribution",
"Definition:Distributional Derivative"
] | [
"Definition:Continuously Differentiable/Real Function/Open Interval",
"Definition:Discontinuity (Real Analysis)/Jump",
"Differentiable Function as Distribution",
"Jump Rule"
] |
proofwiki-18318 | Distributional Derivative of Heaviside Step Function times Sine | Let $H$ be the Heaviside step function.
Let $\delta$ be the Dirac delta distribution.
Then in the distributional sense:
:$T'_{\map H x \sin x} = T_{\map H x \cos x} $ | $x \stackrel f {\longrightarrow} \map H x \sin x$ is a continuously differentiable real function on $\R \setminus \set 0$ and possibly has a discontinuity at $x = 0$.
By Differentiable Function as Distribution we have that $T'_f = T_{f'}$.
Moreover:
:$x < 0 \implies \paren { {\map H x} \map \sin x}' = 0$.
:$x > 0 \imp... | Let $H$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $\delta$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Then in the [[Definition:Distributional Derivative|distributional sense]]:
:$T'_{\map H x \sin x} = T_{\map H x \cos x} $ | $x \stackrel f {\longrightarrow} \map H x \sin x$ is a [[Definition:Continuously Differentiable Real Function on Open Interval|continuously differentiable real function]] on $\R \setminus \set 0$ and possibly has a [[Definition:Discontinuous|discontinuity]] at $x = 0$.
By [[Differentiable Function as Distribution]] w... | Distributional Derivative of Heaviside Step Function times Sine | https://proofwiki.org/wiki/Distributional_Derivative_of_Heaviside_Step_Function_times_Sine | https://proofwiki.org/wiki/Distributional_Derivative_of_Heaviside_Step_Function_times_Sine | [
"Examples of Distributional Derivatives",
"Piecewise Continuously Differentiable Functions"
] | [
"Definition:Heaviside Step Function",
"Definition:Dirac Delta Distribution",
"Definition:Distributional Derivative"
] | [
"Definition:Continuously Differentiable/Real Function/Open Interval",
"Definition:Discontinuous",
"Differentiable Function as Distribution",
"Jump Rule"
] |
proofwiki-18319 | Moment Generating Function of Logistic Distribution | Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
for some $\mu \in \R, s \in \R_{> 0}$.
Then the moment generating function $M_X$ of $X$ is given by:
:$\map {M_X} t = \begin {cases} \map \exp {\mu t} \map \Beta {\paren {1 - s t}, \p... | From the definition of the logistic distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect { e^{t X} } = \int... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] which satisfies the [[Definition:Logistic Distribution|logistic distribution]]:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
for some $\mu \in \R, s \in \R_{> 0}$.
Then the [[Definition:Moment Generating Function|moment generat... | From the definition of the [[Definition:Logistic Distribution|logistic distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From the definiti... | Moment Generating Function of Logistic Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Logistic_Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Logistic_Distribution | [
"Moment Generating Functions",
"Logistic Distribution",
"Moment Generating Function of Logistic Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Moment Generating Function",
"Definition:Beta Function"
] | [
"Definition:Logistic Distribution",
"Definition:Probability Density Function",
"Definition:Moment Generating Function",
"Integration by Substitution",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Exponential Function/Corollary 1",
"Commutativity of Parameters of Bet... |
proofwiki-18320 | Fundamental Solution to 1D Laplace's Equation | Let $\ds \map f x = \frac {\size x} 2$ where $\size x$ is the absolute value function.
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Let $T_f \in \map {\DD'} \R$ be the Schwartz distribution associated with $f$.
Then $T_f$ is the fundamental solution to the $1$-dimensional Laplace's equation.
That is... | $x \stackrel f {\longrightarrow} \dfrac {\size x} 2$ is a continuously differentiable real function on $\R \setminus \set 0$ and possibly has a discontinuity at $x = 0$.
By Differentiable Function as Distribution we have that $T'_f = T_{f'}$.
Moreover:
:$x < 0 \implies \paren {\dfrac {\size x} 2}' = -\frac 1 2$
:$x > 0... | Let $\ds \map f x = \frac {\size x} 2$ where $\size x$ is the [[Definition:Absolute Value|absolute value function]].
Let $\delta \in \map {\DD'} \R$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Let $T_f \in \map {\DD'} \R$ be the [[Definition:Schwartz Distribution|Schwartz distribution]] a... | $x \stackrel f {\longrightarrow} \dfrac {\size x} 2$ is a [[Definition:Continuously Differentiable Real Function on Open Set|continuously differentiable real function]] on $\R \setminus \set 0$ and possibly has a [[Definition:Discontinuous|discontinuity]] at $x = 0$.
By [[Differentiable Function as Distribution]] we h... | Fundamental Solution to 1D Laplace's Equation | https://proofwiki.org/wiki/Fundamental_Solution_to_1D_Laplace's_Equation | https://proofwiki.org/wiki/Fundamental_Solution_to_1D_Laplace's_Equation | [
"Laplace's Equation",
"Examples of Fundamental Solutions"
] | [
"Definition:Absolute Value",
"Definition:Dirac Delta Distribution",
"Definition:Schwartz Distribution",
"Definition:Fundamental Solution",
"Definition:Dimension (Geometry)",
"Definition:Laplace's Equation",
"Definition:Distributional Derivative"
] | [
"Definition:Continuously Differentiable/Real Function/Open Set",
"Definition:Discontinuous",
"Differentiable Function as Distribution",
"Jump Rule",
"Definition:Continuously Differentiable/Real Function/Open Set",
"Definition:Discontinuous",
"Jump Rule",
"Definition:Distributional Derivative"
] |
proofwiki-18321 | Minimal Element need not be Unique | Let $\struct {S, \preccurlyeq}$ be an ordered set.
It is possible for $S$ to have more than one minimal element. | Proof by Counterexample:
Consider the set $S$ defined as:
:$S = \N \setminus \set {0, 1}$
That is, $S$ is the set of natural numbers with $0$ and $1$ removed.
Let $\preccurlyeq$ be the ordering on $S$ defined as:
:$\forall a, b \in S: a \preccurlyeq b \iff a \divides b$
where $a \divides b$ denotes that $a$ is a diviso... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
It is possible for $S$ to have more than one [[Definition:Minimal Element|minimal element]]. | [[Proof by Counterexample]]:
Consider the [[Definition:Set|set]] $S$ defined as:
:$S = \N \setminus \set {0, 1}$
That is, $S$ is the [[Definition:Natural Numbers|set of natural numbers]] with $0$ and $1$ removed.
Let $\preccurlyeq$ be the [[Definition:Ordering|ordering]] on $S$ defined as:
:$\forall a, b \in S: a \p... | Minimal Element need not be Unique | https://proofwiki.org/wiki/Minimal_Element_need_not_be_Unique | https://proofwiki.org/wiki/Minimal_Element_need_not_be_Unique | [
"Minimal Element need not be Unique",
"Minimal Elements"
] | [
"Definition:Ordered Set",
"Definition:Minimal/Element"
] | [
"Proof by Counterexample",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Ordering",
"Definition:Divisor (Algebra)/Integer",
"Divisor Relation on Positive Integers is Partial Ordering",
"Definition:Partially Ordered Set",
"Definition:Prime Number",
"Definition:Prime Number",
"Definiti... |
proofwiki-18322 | Maximal Element need not be Unique | Let $\struct {S, \preccurlyeq}$ be an ordered set.
It is possible for $S$ to have more than one maximal element. | Proof by Counterexample:
Consider the set $T$ defined as:
:$T = \set {0, 1}$
Let $S$ be defined as:
:$S := \powerset T \setminus T$
where $\powerset T$ denotes the power set of $T$.
That is:
:$S = \set {\O, \set 0, \set 1}$
Let $\preccurlyeq$ be the relation defined on $S$ as:
:$\forall a, b \in S: a \preccurlyeq b \if... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
It is possible for $S$ to have more than one [[Definition:Maximal Element|maximal element]]. | [[Proof by Counterexample]]:
Consider the [[Definition:Set|set]] $T$ defined as:
:$T = \set {0, 1}$
Let $S$ be defined as:
:$S := \powerset T \setminus T$
where $\powerset T$ denotes the [[Definition:Power Set|power set]] of $T$.
That is:
:$S = \set {\O, \set 0, \set 1}$
Let $\preccurlyeq$ be the [[Definition:Rela... | Maximal Element need not be Unique | https://proofwiki.org/wiki/Maximal_Element_need_not_be_Unique | https://proofwiki.org/wiki/Maximal_Element_need_not_be_Unique | [
"Maximal Element need not be Unique",
"Maximal Elements"
] | [
"Definition:Ordered Set",
"Definition:Maximal/Element"
] | [
"Proof by Counterexample",
"Definition:Set",
"Definition:Power Set",
"Definition:Relation",
"Definition:Subset",
"Subset Relation is Ordering",
"Definition:Ordered Set",
"Definition:Maximal/Element",
"Definition:Maximal/Element",
"Definition:Maximal/Element"
] |
proofwiki-18323 | Ordered Set may not have Minimal Element | Let $\struct {S, \preccurlyeq}$ be an ordered set.
It may be the case that $S$ has no minimal elements. | Let $\Q_{>0}$ denote the set of (strictly) positive rational numbers.
From Rational Numbers form Totally Ordered Field, the rational numbers $\Q$ are totally ordered by the usual ordering $\le$.
From Subset of Toset is Toset, $\Q_{>0}$ is also totally ordered by $\le$.
Thus $\struct {\Q_{>0}, \le}$ is an ordered set.
{... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
It may be the case that $S$ has no [[Definition:Minimal Element|minimal elements]]. | Let $\Q_{>0}$ denote the [[Definition:Set|set]] of [[Definition:Strictly Positive Rational Number|(strictly) positive rational numbers]].
From [[Rational Numbers form Totally Ordered Field]], the [[Definition:Rational Number|rational numbers]] $\Q$ are [[Definition:Totally Ordered Set|totally ordered]] by the [[Defini... | Ordered Set may not have Minimal Element | https://proofwiki.org/wiki/Ordered_Set_may_not_have_Minimal_Element | https://proofwiki.org/wiki/Ordered_Set_may_not_have_Minimal_Element | [
"Minimal Elements"
] | [
"Definition:Ordered Set",
"Definition:Minimal/Element"
] | [
"Definition:Set",
"Definition:Strictly Positive/Rational Number",
"Rational Numbers form Totally Ordered Field",
"Definition:Rational Number",
"Definition:Totally Ordered Set",
"Definition:Usual Ordering",
"Subset of Toset is Toset",
"Definition:Totally Ordered Set",
"Definition:Ordered Set",
"Def... |
proofwiki-18324 | Ordered Set may not have Maximal Element | Let $\struct {S, \preccurlyeq}$ be an ordered set.
It may be the case that $S$ has no maximal elements. | Consider the set $S$ defined as:
:$S = \N \setminus \set 0$
That is, $S$ is the set of natural numbers with $0$ removed.
Let $\preccurlyeq$ be the ordering on $S$ defined as:
:$\forall a, b \in S: a \preccurlyeq b \iff a \divides b$
where $a \divides b$ denotes that $a$ is a divisor of $b$.
From Divisor Relation on Pos... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
It may be the case that $S$ has no [[Definition:Maximal Element|maximal elements]]. | Consider the [[Definition:Set|set]] $S$ defined as:
:$S = \N \setminus \set 0$
That is, $S$ is the [[Definition:Natural Numbers|set of natural numbers]] with $0$ removed.
Let $\preccurlyeq$ be the [[Definition:Ordering|ordering]] on $S$ defined as:
:$\forall a, b \in S: a \preccurlyeq b \iff a \divides b$
where $a \d... | Ordered Set may not have Maximal Element | https://proofwiki.org/wiki/Ordered_Set_may_not_have_Maximal_Element | https://proofwiki.org/wiki/Ordered_Set_may_not_have_Maximal_Element | [
"Maximal Elements"
] | [
"Definition:Ordered Set",
"Definition:Maximal/Element"
] | [
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Ordering",
"Definition:Divisor (Algebra)/Integer",
"Divisor Relation on Positive Integers is Partial Ordering",
"Definition:Partially Ordered Set",
"Definition:Maximal/Element",
"Definition:Natural Numbers",
"Definition:Maximal/Element",
... |
proofwiki-18325 | Distributional Derivatives of Dirac Delta Distribution do not Vanish | Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Then for any $n \in \N$ the distributional derivative $\delta^{\paren n}$ does not vanish. | Let $\phi \in \map \DD \R$ be a test function such that $\map \phi 0 \ne 0$.
Then:
:$\forall n \in \N : \forall x \in \R : x^n \phi \in \map \DD \R$
By the definition of the distributional derivative:
{{begin-eqn}}
{{eqn | l = \map {\delta^{\paren n} } {x^n \phi}
| r = \paren {-1}^n \map \delta {\paren {x^n \phi}... | Let $\delta \in \map {\DD'} \R$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Then for any $n \in \N$ the [[Definition:Distributional Derivative|distributional derivative]] $\delta^{\paren n}$ does not vanish. | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]] such that $\map \phi 0 \ne 0$.
Then:
:$\forall n \in \N : \forall x \in \R : x^n \phi \in \map \DD \R$
By the definition of the [[Definition:Higher Distributional Derivative|distributional derivative]]:
{{begin-eqn}}
{{eqn | l = \map {\delta... | Distributional Derivatives of Dirac Delta Distribution do not Vanish | https://proofwiki.org/wiki/Distributional_Derivatives_of_Dirac_Delta_Distribution_do_not_Vanish | https://proofwiki.org/wiki/Distributional_Derivatives_of_Dirac_Delta_Distribution_do_not_Vanish | [
"Dirac Delta Distribution",
"Distributional Derivatives"
] | [
"Definition:Dirac Delta Distribution",
"Definition:Distributional Derivative"
] | [
"Definition:Test Function",
"Definition:Distributional Derivative/Higher Derivatives",
"Leibniz's Rule/One Variable",
"Nth Derivative of Nth Power"
] |
proofwiki-18326 | Expectation of Logistic Distribution | Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The expectation of $X$ is given by:
:$\expect X = \mu$ | From the definition of the logistic distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From the definition of the expected value of a continuous random variable:
:$\ds \expect X = \int_{... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] which satisfies the [[Definition:Logistic Distribution|logistic distribution]]:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The [[Definition:Expectation of Continuous Random Variable|expectation]] of $X$ is given by:
:$\expect ... | From the definition of the [[Definition:Logistic Distribution|logistic distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From the definiti... | Expectation of Logistic Distribution/Proof 1 | https://proofwiki.org/wiki/Expectation_of_Logistic_Distribution | https://proofwiki.org/wiki/Expectation_of_Logistic_Distribution/Proof_1 | [
"Logistic Distribution",
"Expectation",
"Expectation of Logistic Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Expectation/Continuous"
] | [
"Definition:Logistic Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Integration by Substitution",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Integral of Constant",
"Difference of Logarithms"
] |
proofwiki-18327 | Expectation of Logistic Distribution | Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The expectation of $X$ is given by:
:$\expect X = \mu$ | By Moment Generating Function of Logistic Distribution, the moment generating function of $X$ is given by:
:$\ds \map {M_X} t = \map \exp {\mu t} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u$
for $\size t < \dfrac 1 s$.
From Moment in terms of Moment Generating Function:
:$\expect X = \map { {M_X}'} 0... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] which satisfies the [[Definition:Logistic Distribution|logistic distribution]]:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The [[Definition:Expectation of Continuous Random Variable|expectation]] of $X$ is given by:
:$\expect ... | By [[Moment Generating Function of Logistic Distribution]], the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
:$\ds \map {M_X} t = \map \exp {\mu t} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u$
for $\size t < \dfrac 1 s$.
From [[Moment in terms of Moment G... | Expectation of Logistic Distribution/Proof 2 | https://proofwiki.org/wiki/Expectation_of_Logistic_Distribution | https://proofwiki.org/wiki/Expectation_of_Logistic_Distribution/Proof_2 | [
"Logistic Distribution",
"Expectation",
"Expectation of Logistic Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Expectation/Continuous"
] | [
"Moment Generating Function of Logistic Distribution",
"Definition:Moment Generating Function",
"Moment in terms of Moment Generating Function",
"Moment Generating Function of Logistic Distribution/Examples/First Moment",
"Integral of Constant/Definite"
] |
proofwiki-18328 | Ordered Set with Multiple Minimal Elements has no Smallest Element | Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $\struct {S, \preccurlyeq}$ have more than one minimal element.
Then $\struct {S, \preccurlyeq}$ has no smallest element. | Let $s$ and $t$ both be minimal elements of $\struct {S, \preccurlyeq}$ such that $s \ne t$.
Then by definition:
:$\forall x \in S: x \preccurlyeq s \implies s = x$
and:
:$\forall x \in S: x \preccurlyeq t \implies t = x$
{{AimForCont}} $S$ has a smallest element $m$.
Then by definition:
:$\forall y \in S: m \preccurly... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
Let $\struct {S, \preccurlyeq}$ have more than one [[Definition:Minimal Element|minimal element]].
Then $\struct {S, \preccurlyeq}$ has no [[Definition:Smallest Element|smallest element]]. | Let $s$ and $t$ both be [[Definition:Minimal Element|minimal elements]] of $\struct {S, \preccurlyeq}$ such that $s \ne t$.
Then by definition:
:$\forall x \in S: x \preccurlyeq s \implies s = x$
and:
:$\forall x \in S: x \preccurlyeq t \implies t = x$
{{AimForCont}} $S$ has a [[Definition:Smallest Element|smallest... | Ordered Set with Multiple Minimal Elements has no Smallest Element | https://proofwiki.org/wiki/Ordered_Set_with_Multiple_Minimal_Elements_has_no_Smallest_Element | https://proofwiki.org/wiki/Ordered_Set_with_Multiple_Minimal_Elements_has_no_Smallest_Element | [
"Minimal Elements",
"Smallest Elements"
] | [
"Definition:Ordered Set",
"Definition:Minimal/Element",
"Definition:Smallest Element"
] | [
"Definition:Minimal/Element",
"Definition:Smallest Element",
"Definition:Minimal/Element",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Smallest Element",
"Category:Minimal Elements",
"Category:Smallest Elements"
] |
proofwiki-18329 | Ordered Set with Multiple Maximal Elements has no Greatest Element | Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $\struct {S, \preccurlyeq}$ have more than one maximal element.
Then $\struct {S, \preccurlyeq}$ has no greatest element. | Let $s$ and $t$ both be maximal elements of $\struct {S, \preccurlyeq}$ such that $s \ne t$.
Then by definition:
:$\forall x \in S: s \preccurlyeq x \implies s = x$
and:
:$\forall x \in S: t \preccurlyeq x \implies t = x$
{{AimForCont}} $S$ has a greatest element $m$.
Then by definition:
:$\forall y \in S: y \preccurly... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
Let $\struct {S, \preccurlyeq}$ have more than one [[Definition:Maximal Element|maximal element]].
Then $\struct {S, \preccurlyeq}$ has no [[Definition:Greatest Element|greatest element]]. | Let $s$ and $t$ both be [[Definition:Maximal Element|maximal elements]] of $\struct {S, \preccurlyeq}$ such that $s \ne t$.
Then by definition:
:$\forall x \in S: s \preccurlyeq x \implies s = x$
and:
:$\forall x \in S: t \preccurlyeq x \implies t = x$
{{AimForCont}} $S$ has a [[Definition:Greatest Element|greatest... | Ordered Set with Multiple Maximal Elements has no Greatest Element | https://proofwiki.org/wiki/Ordered_Set_with_Multiple_Maximal_Elements_has_no_Greatest_Element | https://proofwiki.org/wiki/Ordered_Set_with_Multiple_Maximal_Elements_has_no_Greatest_Element | [
"Maximal Elements",
"Greatest Elements"
] | [
"Definition:Ordered Set",
"Definition:Maximal/Element",
"Definition:Greatest Element"
] | [
"Definition:Maximal/Element",
"Definition:Greatest Element",
"Definition:Maximal/Element",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Greatest Element",
"Category:Maximal Elements",
"Category:Greatest Elements"
] |
proofwiki-18330 | Unique Minimal Element may not be Smallest | Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $\struct {S, \preccurlyeq}$ have a unique minimal element.
Then it is not necessarily the case that $\struct {S, \preccurlyeq}$ has a smallest element. | Let $S$ be the set defined as:
:$S = \Z \cup \set m$
where:
:$\Z$ denotes the set of integers
:$m$ is an arbitrary object such that $m \ne \Z$.
Let $\preccurlyeq$ be the relation on $\Z$ defined as:
:$\forall a, b \in S: a \preccurlyeq b \iff \begin {cases} a \le b & : a, b \in \Z \\ a = m = b & : a, b \notin \Z \end{c... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
Let $\struct {S, \preccurlyeq}$ have a [[Definition:Unique|unique]] [[Definition:Minimal Element|minimal element]].
Then it is not necessarily the case that $\struct {S, \preccurlyeq}$ has a [[Definition:Smallest Element|smallest element]]... | Let $S$ be the [[Definition:Set|set]] defined as:
:$S = \Z \cup \set m$
where:
:$\Z$ denotes the [[Definition:Integer|set of integers]]
:$m$ is an arbitrary [[Definition:Object|object]] such that $m \ne \Z$.
Let $\preccurlyeq$ be the [[Definition:Relation|relation]] on $\Z$ defined as:
:$\forall a, b \in S: a \preccu... | Unique Minimal Element may not be Smallest | https://proofwiki.org/wiki/Unique_Minimal_Element_may_not_be_Smallest | https://proofwiki.org/wiki/Unique_Minimal_Element_may_not_be_Smallest | [
"Minimal Elements",
"Smallest Elements"
] | [
"Definition:Ordered Set",
"Definition:Unique",
"Definition:Minimal/Element",
"Definition:Smallest Element"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Object",
"Definition:Relation",
"Definition:Usual Ordering",
"Definition:Non-Comparable Elements",
"Definition:Ordered Set",
"Definition:Minimal/Element",
"Definition:Smallest Element",
"Definition:Minimal/Element",
"Definition:Minimal/Element"... |
proofwiki-18331 | Equivalence of Definitions of Well-Founded Relation | Let $\struct {S, \RR}$ be a Relational Structure.
{{TFAE|def = Well-Founded Relation}} | By definition of minimal element:
:$\forall y \in T: y \preceq x \implies x = y$ | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|Relational Structure]].
{{TFAE|def = Well-Founded Relation}} | By definition of [[Definition:Minimal Element|minimal element]]:
:$\forall y \in T: y \preceq x \implies x = y$ | Equivalence of Definitions of Well-Founded Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Well-Founded_Relation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Well-Founded_Relation | [
"Well-Founded Relations"
] | [
"Definition:Relational Structure"
] | [
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Minimal/Element"
] |
proofwiki-18332 | Strictly Minimal Element is Minimal Element | Let $\struct {S, \RR}$ be a relational structure.
Let $T \subseteq S$ be a subset of $S$.
Let $m \in T$ be a strictly minimal element of $T$ under $\RR$.
Then $m$ is a minimal element of $T$ under $\RR$. | Let $m \in T$ be a strictly minimal element of $T$ under $\RR$.
Then by definition:
:$\forall x \in T: \tuple {x, m} \notin \RR$
{{AimForCont}} $m$ is not a minimal element of $T$ under $\RR$.
Then:
:$\exists y \in T: \tuple {y, m} \in \RR$
such that $y \ne m$.
But this contradicts the assertion that $\tuple {y, m} \no... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $m \in T$ be a [[Definition:Strictly Minimal Element|strictly minimal element]] of $T$ under $\RR$.
Then $m$ is a [[Definition:Minimal Element|minimal element]] of ... | Let $m \in T$ be a [[Definition:Strictly Minimal Element|strictly minimal element]] of $T$ under $\RR$.
Then by definition:
:$\forall x \in T: \tuple {x, m} \notin \RR$
{{AimForCont}} $m$ is not a [[Definition:Minimal Element|minimal element]] of $T$ under $\RR$.
Then:
:$\exists y \in T: \tuple {y, m} \in \RR$
such ... | Strictly Minimal Element is Minimal Element | https://proofwiki.org/wiki/Strictly_Minimal_Element_is_Minimal_Element | https://proofwiki.org/wiki/Strictly_Minimal_Element_is_Minimal_Element | [
"Minimal Elements"
] | [
"Definition:Relational Structure",
"Definition:Subset",
"Definition:Strictly Minimal Element",
"Definition:Minimal/Element"
] | [
"Definition:Strictly Minimal Element",
"Definition:Minimal/Element",
"Definition:Contradiction",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Category:Minimal Elements"
] |
proofwiki-18333 | Strictly Maximal Element is Maximal Element | Let $\struct {S, \RR}$ be a relational structure.
Let $T \subseteq S$ be a subset of $S$.
Let $m \in T$ be a strictly maximal element of $T$ under $\RR$.
Then $m$ is a maximal element of $T$ under $\RR$. | Let $m \in T$ be a strictly maximal element of $T$ under $\RR$.
Then by definition:
:$\forall x \in T: \tuple {m, x} \notin \RR$
{{AimForCont}} $m$ is not a maximal element of $T$ under $\RR$.
Then:
:$\exists y \in T: \tuple {m, y} \in \RR$
such that $y \ne m$.
But this contradicts the assertion that $\tuple {m, y} \no... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $m \in T$ be a [[Definition:Strictly Maximal Element|strictly maximal element]] of $T$ under $\RR$.
Then $m$ is a [[Definition:Maximal Element|maximal element]] of ... | Let $m \in T$ be a [[Definition:Strictly Maximal Element|strictly maximal element]] of $T$ under $\RR$.
Then by definition:
:$\forall x \in T: \tuple {m, x} \notin \RR$
{{AimForCont}} $m$ is not a [[Definition:Maximal Element|maximal element]] of $T$ under $\RR$.
Then:
:$\exists y \in T: \tuple {m, y} \in \RR$
such ... | Strictly Maximal Element is Maximal Element | https://proofwiki.org/wiki/Strictly_Maximal_Element_is_Maximal_Element | https://proofwiki.org/wiki/Strictly_Maximal_Element_is_Maximal_Element | [
"Maximal Elements"
] | [
"Definition:Relational Structure",
"Definition:Subset",
"Definition:Strictly Maximal Element",
"Definition:Maximal/Element"
] | [
"Definition:Strictly Maximal Element",
"Definition:Maximal/Element",
"Definition:Contradiction",
"Definition:Maximal/Element",
"Definition:Maximal/Element",
"Category:Maximal Elements"
] |
proofwiki-18334 | Strictly Well-Founded Relation is Well-Founded | Let $\struct {S, \RR}$ be a relational structure.
Let $\RR$ be a strictly well-founded relation on $S$.
Then $\RR$ is a well-founded relation on $S$. | We have that $\RR$ is a strictly well-founded relation on $S$.
By definition:
:$\forall T: \paren {T \subseteq S \land T \ne \O} \implies \exists y \in T: \forall z \in T: \neg \paren {z \mathrel \RR y}$
It immediately follows that:
:$\forall T: \paren {T \subseteq S \land T \ne \O} \implies \exists y \in T: \forall z ... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $\RR$ be a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] on $S$.
Then $\RR$ is a [[Definition:Well-Founded Relation|well-founded relation]] on $S$. | We have that $\RR$ is a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] on $S$.
By definition:
:$\forall T: \paren {T \subseteq S \land T \ne \O} \implies \exists y \in T: \forall z \in T: \neg \paren {z \mathrel \RR y}$
It immediately follows that:
:$\forall T: \paren {T \subseteq S \l... | Strictly Well-Founded Relation is Well-Founded | https://proofwiki.org/wiki/Strictly_Well-Founded_Relation_is_Well-Founded | https://proofwiki.org/wiki/Strictly_Well-Founded_Relation_is_Well-Founded | [
"Well-Founded Relations"
] | [
"Definition:Relational Structure",
"Definition:Strictly Well-Founded Relation",
"Definition:Well-Founded Relation"
] | [
"Definition:Strictly Well-Founded Relation",
"Definition:Well-Founded Relation"
] |
proofwiki-18335 | Well-Founded Relation is not necessarily Ordering | Let $\struct {S, \RR}$ be a relational structure.
Let $\RR$ be a well-founded relation on $S$.
Then it is not necessarily the case that $\RR$ is also either an ordering or a strict ordering. | Proof by Counterexample:
Let $P$ be the set of all polynomials over $\R$ in one variable with real coefficients.
Let $\DD$ be a relation on $P$ defined as:
:$\forall p_0, p_1 \in P: \tuple {p_0, p_1} \in \DD$ {{iff}} $p_0$ is the derivative of $p_1$.
From Differentiation of Polynomials induces Well-Founded Relation, we... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $\RR$ be a [[Definition:Well-Founded Relation|well-founded relation]] on $S$.
Then it is not necessarily the case that $\RR$ is also either an [[Definition:Ordering|ordering]] or a [[Definition:Strict Ordering|strict ordering]]... | [[Proof by Counterexample]]:
Let $P$ be the [[Definition:Set|set]] of all [[Definition:Polynomial over Real Numbers|polynomials over $\R$]] [[Definition:Polynomial over Ring in One Variable|in one variable]] with [[Definition:Real Number|real]] [[Definition:Coefficient of Polynomial|coefficients]].
Let $\DD$ be a [[D... | Well-Founded Relation is not necessarily Ordering | https://proofwiki.org/wiki/Well-Founded_Relation_is_not_necessarily_Ordering | https://proofwiki.org/wiki/Well-Founded_Relation_is_not_necessarily_Ordering | [
"Well-Founded Relations",
"Orderings"
] | [
"Definition:Relational Structure",
"Definition:Well-Founded Relation",
"Definition:Ordering",
"Definition:Strict Ordering"
] | [
"Proof by Counterexample",
"Definition:Set",
"Definition:Polynomial/Real Numbers",
"Definition:Polynomial over Ring/One Variable",
"Definition:Real Number",
"Definition:Coefficient of Polynomial",
"Definition:Endorelation",
"Definition:Derivative",
"Differentiation of Polynomials induces Well-Founde... |
proofwiki-18336 | Well-Founded Relation has no Relational Loops | Let $\RR$ be a well-founded relation on $S$.
Let $x_1, x_2, \ldots, x_n \in S$.
Then:
:$\neg \paren {\paren {x_1 \mathrel \RR x_2} \land \paren {x_3 \mathrel \RR x_4} \land \cdots \land \paren {x_n \mathrel \RR x_1} }$
That is, there are no relational loops within $S$. | Since $x_1, x_2, \ldots, x_n \in S$, there exists a non-empty subset $T$ of $S$ such that:
:$T = \set {x_1, x_2, \ldots, x_n}$
By the definition of a well-founded relation:
:$(1): \quad \exists z \in T: \forall y \in T \setminus z: \neg y \mathrel \RR z$
{{AimForCont}} $\paren {x_1 \mathrel \RR x_2} \land \paren {x_2 \... | Let $\RR$ be a [[Definition:Well-Founded Relation|well-founded relation]] on $S$.
Let $x_1, x_2, \ldots, x_n \in S$.
Then:
:$\neg \paren {\paren {x_1 \mathrel \RR x_2} \land \paren {x_3 \mathrel \RR x_4} \land \cdots \land \paren {x_n \mathrel \RR x_1} }$
That is, there are no [[Definition:Relational Loop|relationa... | Since $x_1, x_2, \ldots, x_n \in S$, there exists a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] $T$ of $S$ such that:
:$T = \set {x_1, x_2, \ldots, x_n}$
By the definition of a [[Definition:Well-Founded Relation|well-founded relation]]:
:$(1): \quad \exists z \in T: \forall y \in T \setminus z:... | Well-Founded Relation has no Relational Loops | https://proofwiki.org/wiki/Well-Founded_Relation_has_no_Relational_Loops | https://proofwiki.org/wiki/Well-Founded_Relation_has_no_Relational_Loops | [
"Well-Founded Relations"
] | [
"Definition:Well-Founded Relation",
"Definition:Relational Loop"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Well-Founded Relation",
"Definition:Contradiction",
"Definition:Well-Founded Relation",
"Definition:Relational Loop"
] |
proofwiki-18337 | Infinite Sequence Property of Strictly Well-Founded Relation | Let $\struct {S, \RR}$ be a relational structure.
Then $\RR$ is a strictly well-founded relation {{iff}} there is no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$ | === Reverse Implication ===
{{:Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 1}}{{qed|lemma}} | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Then $\RR$ is a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] {{iff}} there is no [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such tha... | === [[Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 1|Reverse Implication]] ===
{{:Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 1}}{{qed|lemma}} | Infinite Sequence Property of Strictly Well-Founded Relation | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation | [
"Well-Founded Relations",
"Infinite Sequence Property of Strictly Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Strictly Well-Founded Relation",
"Definition:Sequence/Infinite Sequence",
"Definition:Element"
] | [
"Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 1"
] |
proofwiki-18338 | Infinite Sequence Property of Strictly Well-Founded Relation | Let $\struct {S, \RR}$ be a relational structure.
Then $\RR$ is a strictly well-founded relation {{iff}} there is no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$ | Suppose $\RR$ is not a strictly well-founded relation.
So by definition there exists a non-empty subset $T$ of $S$ which has no strictly minimal element.
Let $a \in T$.
Since $a$ is not strictly minimal in $T$, we can find $b \in T: b \mathrel \RR a$.
This holds for all $a \in T$.
Hence the restriction $\RR \restrictio... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Then $\RR$ is a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] {{iff}} there is no [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such tha... | Suppose $\RR$ is not a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]].
So by definition there exists a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] $T$ of $S$ which has no [[Definition:Strictly Minimal Element|strictly minimal element]].
Let $a \in T$.
Since $a$ i... | Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 1 | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation/Reverse_Implication/Proof_1 | [
"Well-Founded Relations",
"Infinite Sequence Property of Strictly Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Strictly Well-Founded Relation",
"Definition:Sequence/Infinite Sequence",
"Definition:Element"
] | [
"Definition:Strictly Well-Founded Relation",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Strictly Minimal Element",
"Definition:Strictly Minimal Element",
"Definition:Restriction/Relation",
"Definition:Right-Total Relation",
"Definition:Endorelation",
"Axiom:Axiom of Dependent Choic... |
proofwiki-18339 | Infinite Sequence Property of Strictly Well-Founded Relation | Let $\struct {S, \RR}$ be a relational structure.
Then $\RR$ is a strictly well-founded relation {{iff}} there is no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$ | Suppose $\RR$ is not a strictly well-founded relation.
Hence there exists $T \subseteq S$ such that $T$ has no strictly minimal element under $\RR$.
Let $a_0 \in T$.
We have that $a_0$ is not strictly minimal in $T$.
So:
:$\exists a_1 \in T: a_1 \mathrel \RR a_0$
Similarly, $a_1$ is not strictly minimal in $T$.
So:
:$\... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Then $\RR$ is a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] {{iff}} there is no [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such tha... | Suppose $\RR$ is not a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]].
Hence there exists $T \subseteq S$ such that $T$ has no [[Definition:Strictly Minimal Element|strictly minimal element]] under $\RR$.
Let $a_0 \in T$.
We have that $a_0$ is not [[Definition:Strictly Minimal Element|... | Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 2 | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation/Reverse_Implication/Proof_2 | [
"Well-Founded Relations",
"Infinite Sequence Property of Strictly Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Strictly Well-Founded Relation",
"Definition:Sequence/Infinite Sequence",
"Definition:Element"
] | [
"Definition:Strictly Well-Founded Relation",
"Definition:Strictly Minimal Element",
"Definition:Strictly Minimal Element",
"Definition:Strictly Minimal Element",
"Axiom:Axiom of Dependent Choice/Right-Total",
"Definition:Strictly Minimal Element",
"Definition:Right-Total Relation",
"Axiom:Axiom of Dep... |
proofwiki-18340 | Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication | Let $\struct {S, \RR}$ be a relational structure.
Let $\RR$ be such that there exists no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$
Then $\RR$ is a strictly well-founded relation. | Suppose $\RR$ is not a strictly well-founded relation.
So by definition there exists a non-empty subset $T$ of $S$ which has no strictly minimal element.
Let $a \in T$.
Since $a$ is not strictly minimal in $T$, we can find $b \in T: b \mathrel \RR a$.
This holds for all $a \in T$.
Hence the restriction $\RR \restrictio... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $\RR$ be such that there exists no [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$
Then $\RR$ is a [[Defi... | Suppose $\RR$ is not a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]].
So by definition there exists a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] $T$ of $S$ which has no [[Definition:Strictly Minimal Element|strictly minimal element]].
Let $a \in T$.
Since $a$ i... | Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 1 | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation/Reverse_Implication | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation/Reverse_Implication/Proof_1 | [
"Infinite Sequence Property of Strictly Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Sequence/Infinite Sequence",
"Definition:Element",
"Definition:Strictly Well-Founded Relation"
] | [
"Definition:Strictly Well-Founded Relation",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Strictly Minimal Element",
"Definition:Strictly Minimal Element",
"Definition:Restriction/Relation",
"Definition:Right-Total Relation",
"Definition:Endorelation",
"Axiom:Axiom of Dependent Choic... |
proofwiki-18341 | Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication | Let $\struct {S, \RR}$ be a relational structure.
Let $\RR$ be such that there exists no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$
Then $\RR$ is a strictly well-founded relation. | Suppose $\RR$ is not a strictly well-founded relation.
Hence there exists $T \subseteq S$ such that $T$ has no strictly minimal element under $\RR$.
Let $a_0 \in T$.
We have that $a_0$ is not strictly minimal in $T$.
So:
:$\exists a_1 \in T: a_1 \mathrel \RR a_0$
Similarly, $a_1$ is not strictly minimal in $T$.
So:
:$\... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $\RR$ be such that there exists no [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$
Then $\RR$ is a [[Defi... | Suppose $\RR$ is not a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]].
Hence there exists $T \subseteq S$ such that $T$ has no [[Definition:Strictly Minimal Element|strictly minimal element]] under $\RR$.
Let $a_0 \in T$.
We have that $a_0$ is not [[Definition:Strictly Minimal Element|... | Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 2 | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation/Reverse_Implication | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation/Reverse_Implication/Proof_2 | [
"Infinite Sequence Property of Strictly Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Sequence/Infinite Sequence",
"Definition:Element",
"Definition:Strictly Well-Founded Relation"
] | [
"Definition:Strictly Well-Founded Relation",
"Definition:Strictly Minimal Element",
"Definition:Strictly Minimal Element",
"Definition:Strictly Minimal Element",
"Axiom:Axiom of Dependent Choice/Right-Total",
"Definition:Strictly Minimal Element",
"Definition:Right-Total Relation",
"Axiom:Axiom of Dep... |
proofwiki-18342 | Infinite Sequence Property of Strictly Well-Founded Relation/Forward Implication | Let $\struct {S, \RR}$ be a relational structure.
Let $\RR$ be a strictly well-founded relation.
Then there exists no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$ | Let $\RR$ be a strictly well-founded relation.
{{AimForCont}} there exists an infinite sequence $\sequence {a_n}$ in $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$
Let $T = \set {a_0, a_1, a_2, \ldots}$.
Let $a_k \in T$ be a strictly minimal element of $T$.
That is:
:$\forall y \in T: y \not \mathrel \R... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $\RR$ be a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]].
Then there exists no [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such t... | Let $\RR$ be a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]].
{{AimForCont}} there exists an [[Definition:Sequence|infinite sequence]] $\sequence {a_n}$ in $S$ such that:
:$\forall n \in \N: a_{n + 1} \mathrel \RR a_n$
Let $T = \set {a_0, a_1, a_2, \ldots}$.
Let $a_k \in T$ be a [[Defi... | Infinite Sequence Property of Strictly Well-Founded Relation/Forward Implication | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation/Forward_Implication | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation/Forward_Implication | [
"Infinite Sequence Property of Strictly Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Strictly Well-Founded Relation",
"Definition:Sequence/Infinite Sequence",
"Definition:Element"
] | [
"Definition:Strictly Well-Founded Relation",
"Definition:Sequence",
"Definition:Strictly Minimal Element",
"Definition:Strictly Minimal Element",
"Proof by Contradiction",
"Definition:Sequence"
] |
proofwiki-18343 | Restriction of Well-Founded Relation is Well-Founded | Let $\struct {S, \RR}$ be a relational structure.
Let $\RR$ be a well-founded relation on $S$.
Let $T$ be a subset or subclass of $S$.
Let $\RR'$ be the restriction of $\RR$ to $T$.
Then $\preceq'$ is a well-founded relation on $T$. | Let $A$ be a non-empty subset of $T$.
By Subset Relation is Transitive, $A$ is a non-empty subset of $S$.
Since $\RR$ is a well-founded relation on $S$, $A$ has a minimal element $m$ under $\RR$.
Let $x \in A$.
Let $x \mathrel {\RR'} m$.
By the definition of restriction:
:$x \mathrel \RR m$
Thus by the definition of mi... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $\RR$ be a [[Definition:Well-Founded Relation|well-founded relation]] on $S$.
Let $T$ be a [[Definition:Subset|subset]] or [[Definition:Subclass|subclass]] of $S$.
Let $\RR'$ be the [[Definition:Restriction of Relation|restrict... | Let $A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $T$.
By [[Subset Relation is Transitive]], $A$ is a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Since $\RR$ is a [[Definition:Well-Founded Relation|well-founded relation]] on $S$, $A$ has a [[Definition... | Restriction of Well-Founded Relation is Well-Founded | https://proofwiki.org/wiki/Restriction_of_Well-Founded_Relation_is_Well-Founded | https://proofwiki.org/wiki/Restriction_of_Well-Founded_Relation_is_Well-Founded | [
"Well-Founded Relations"
] | [
"Definition:Relational Structure",
"Definition:Well-Founded Relation",
"Definition:Subset",
"Definition:Subclass",
"Definition:Restriction/Relation",
"Definition:Well-Founded Relation"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Subset Relation is Transitive",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Well-Founded Relation",
"Definition:Minimal/Element",
"Definition:Restriction/Relation",
"Definition:Minimal/Element",
"Definition:Element",
"Definition:... |
proofwiki-18344 | Divisor Relation on Positive Integers is Well-Founded Ordering | The divisor relation on $\Z_{>0}$ is a well-founded ordering. | Let $\struct {\Z_{>0}, \divides}$ denote the relational structure formed from the strictly positive integers $\Z_{>0}$ under the divisor relation $\divides$.
From Divisor Relation on Positive Integers is Partial Ordering, $\struct {\Z_{>0}, \divides}$ is a partially ordered set.
It remains to be shown that $\divides$ i... | The [[Definition:Divisor of Integer|divisor relation]] on $\Z_{>0}$ is a [[Definition:Well-Founded Ordering|well-founded ordering]]. | Let $\struct {\Z_{>0}, \divides}$ denote the [[Definition:Relational Structure|relational structure]] formed from the [[Definition:Strictly Positive Integer|strictly positive integers]] $\Z_{>0}$ under the [[Definition:Divisor of Integer|divisor relation]] $\divides$.
From [[Divisor Relation on Positive Integers is Pa... | Divisor Relation on Positive Integers is Well-Founded Ordering | https://proofwiki.org/wiki/Divisor_Relation_on_Positive_Integers_is_Well-Founded_Ordering | https://proofwiki.org/wiki/Divisor_Relation_on_Positive_Integers_is_Well-Founded_Ordering | [
"Divisors",
"Integers",
"Examples of Well-Founded Relations"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Well-Founded Ordered Set"
] | [
"Definition:Relational Structure",
"Definition:Strictly Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Divisor Relation on Positive Integers is Partial Ordering",
"Definition:Partially Ordered Set",
"Definition:Well-Founded Relation",
"Definition:Non-Empty Set",
"Definition:Element",
"D... |
proofwiki-18345 | Union of Indexed Family of Sets Equal to Union of Disjoint Sets/General Result | Let $I$ be a set which can be well-ordered by a well-ordering $\preccurlyeq$.
Let $\family {E_\alpha}_{\alpha \mathop \in I}$ be a countable indexed family of sets indexed by $I$ where at least two $E_\alpha$ are distinct.
Then there exists a countable indexed family of disjoint sets $\family {F_\alpha}_{\alpha \mathop... | Denote:
{{begin-eqn}}
{{eqn | l = E
| r = \bigcup_{\beta \mathop \in I} E_\beta
}}
{{eqn | l = F
| r = \bigcup_{\beta \mathop \in I} F_\beta
}}
{{end-eqn}}
where:
:$\ds F_\beta = E_\beta \setminus \paren {\bigcup_{\alpha \mathop \prec \beta} E_\alpha}$
We first show that $E = F$.
That $x \in E \implies x \i... | Let $I$ be a [[Definition:Set|set]] which can be [[Definition:Well-Ordered Set|well-ordered]] by a [[Definition:Well-Ordering|well-ordering]] $\preccurlyeq$.
Let $\family {E_\alpha}_{\alpha \mathop \in I}$ be a [[Definition:Countable Set|countable]] [[Definition:Indexed Family of Subsets|indexed family of sets]] [[Def... | Denote:
{{begin-eqn}}
{{eqn | l = E
| r = \bigcup_{\beta \mathop \in I} E_\beta
}}
{{eqn | l = F
| r = \bigcup_{\beta \mathop \in I} F_\beta
}}
{{end-eqn}}
where:
:$\ds F_\beta = E_\beta \setminus \paren {\bigcup_{\alpha \mathop \prec \beta} E_\alpha}$
We first show that $E = F$.
That $x \in E \impli... | Union of Indexed Family of Sets Equal to Union of Disjoint Sets/General Result | https://proofwiki.org/wiki/Union_of_Indexed_Family_of_Sets_Equal_to_Union_of_Disjoint_Sets/General_Result | https://proofwiki.org/wiki/Union_of_Indexed_Family_of_Sets_Equal_to_Union_of_Disjoint_Sets/General_Result | [
"Union of Indexed Family of Sets Equal to Union of Disjoint Sets"
] | [
"Definition:Set",
"Definition:Well-Ordered Set",
"Definition:Well-Ordering",
"Definition:Countable Set",
"Definition:Indexing Set/Family of Subsets",
"Definition:Indexing Set",
"Definition:Distinct",
"Definition:Countable Set",
"Definition:Pairwise Disjoint/Family",
"Definition:Disjoint Union (Set... | [
"Definition:Subset",
"Rule of Simplification",
"Definition:Set Equality",
"Definition:Pairwise Disjoint/Family",
"Well-Ordering Principle",
"Definition:Smallest Element",
"Definition:Distinct/Plural",
"Definition:Disjoint Sets",
"Definition:Disjoint Union (Set Theory)",
"Definition:Set",
"Defini... |
proofwiki-18346 | Subset of Well-Founded Relation is Well-Founded | Let $\struct {S, \RR}$ be a relational structure.
Let $\RR$ be a well-founded relation on $S$.
Let $\QQ$ be a subset of $\RR$.
Then $\QQ$ is also a well-founded relation on $S$. | {{AimForCont}} $\struct {S, \QQ}$ is not a well-founded set.
By Infinite Sequence Property of Well-Founded Relation there exists an infinite sequence $\sequence {x_n}$ in $S$ such that:
:$\forall n \in \N: \tuple {x_{n + 1}, x_n} \in \QQ \text { and } x_{n + 1} \ne x_n$
But then because $\QQ \subseteq \RR$, it follows ... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $\RR$ be a [[Definition:Well-Founded Relation|well-founded relation]] on $S$.
Let $\QQ$ be a [[Definition:Subset|subset]] of $\RR$.
Then $\QQ$ is also a [[Definition:Well-Founded Relation|well-founded relation]] on $S$. | {{AimForCont}} $\struct {S, \QQ}$ is not a [[Definition:Well-Founded Set|well-founded set]].
By [[Infinite Sequence Property of Well-Founded Relation]] there exists an [[Definition:Infinite Sequence|infinite sequence]] $\sequence {x_n}$ in $S$ such that:
:$\forall n \in \N: \tuple {x_{n + 1}, x_n} \in \QQ \text { and ... | Subset of Well-Founded Relation is Well-Founded | https://proofwiki.org/wiki/Subset_of_Well-Founded_Relation_is_Well-Founded | https://proofwiki.org/wiki/Subset_of_Well-Founded_Relation_is_Well-Founded | [
"Well-Founded Relations"
] | [
"Definition:Relational Structure",
"Definition:Well-Founded Relation",
"Definition:Subset",
"Definition:Well-Founded Relation"
] | [
"Definition:Well-Founded Set",
"Infinite Sequence Property of Well-Founded Relation",
"Definition:Sequence/Infinite Sequence",
"Infinite Sequence Property of Well-Founded Relation",
"Definition:Well-Founded Relation",
"Proof by Contradiction"
] |
proofwiki-18347 | Rank Function Property of Well-Founded Relation | Let $\struct {S, \RR}$ be a relational structure.
Let $\struct {T, \prec}$ be a strictly well-ordered set.
Let there exist a rank function $\operatorname {rk}: S \to T$, that is:
:$\forall x, y \in S: \paren {x \ne y \text { and } \tuple {x, y} \in \RR} \implies \map {\operatorname {rk} } x \prec \map {\operatorname {r... | {{AimForCont}} $\RR$ is not a well-founded relation.
From Infinite Sequence Property of Well-Founded Relation, there exists an infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: \paren {a_{n + 1} \mathrel \RR a_n} \text { and } \paren {a_{n + 1} \ne a_n}$
Let $A = \operatorname {rk} \... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $\struct {T, \prec}$ be a [[Definition:Strictly Well-Ordered Set|strictly well-ordered set]].
Let there exist a [[Definition:Rank Function for Relation|rank function]] $\operatorname {rk}: S \to T$, that is:
:$\forall x, y \in S... | {{AimForCont}} $\RR$ is not a [[Definition:Well-Founded Relation|well-founded relation]].
From [[Infinite Sequence Property of Well-Founded Relation]], there exists an [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such that:
:$\forall n \in \N: \paren {... | Rank Function Property of Well-Founded Relation/Proof | https://proofwiki.org/wiki/Rank_Function_Property_of_Well-Founded_Relation | https://proofwiki.org/wiki/Rank_Function_Property_of_Well-Founded_Relation/Proof | [
"Well-Founded Relations",
"Rank Functions",
"Rank Function Property of Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Strictly Well-Ordered Set",
"Definition:Rank Function for Relation",
"Definition:Well-Founded Relation"
] | [
"Definition:Well-Founded Relation",
"Infinite Sequence Property of Well-Founded Relation",
"Definition:Sequence/Infinite Sequence",
"Definition:Element",
"Definition:Image (Set Theory)/Mapping/Subset",
"Image is Subset of Codomain",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Defi... |
proofwiki-18348 | Set of Distributional Derivatives of Dirac Delta Distribution is Linearly Independent | Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Then for any $n \in \N$ the set of distributional derivatives $\set {\delta, \delta' \ldots, \delta^{\paren n} }$ is linearly independent in $\map {\DD'} \R$. | {{AimForCont}} there exist scalars $c_0, c_1, \ldots, c_n \in \R$ such that:
:$\ds \sum_{i \mathop = 0}^n c_i \delta^{\paren i} = \mathbf 0$
where $\mathbf 0 : \map \DD \R \to 0$ is the zero distribution.
Let $\phi \in \map \DD \R$ be a test function.
Let $\lambda \in \R_{\mathop > 0}$
Let $\map {\phi_\lambda} x := \ma... | Let $\delta \in \map {\DD'} \R$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Then for any $n \in \N$ the [[Definition:Set|set]] of [[Definition:Distributional Derivative|distributional derivatives]] $\set {\delta, \delta' \ldots, \delta^{\paren n} }$ is [[Definition:Linearly Independent|li... | {{AimForCont}} there exist [[Definition:Scalar|scalars]] $c_0, c_1, \ldots, c_n \in \R$ such that:
:$\ds \sum_{i \mathop = 0}^n c_i \delta^{\paren i} = \mathbf 0$
where $\mathbf 0 : \map \DD \R \to 0$ is the [[Definition:Zero Distribution|zero distribution]].
Let $\phi \in \map \DD \R$ be a [[Definition:Test Functio... | Set of Distributional Derivatives of Dirac Delta Distribution is Linearly Independent | https://proofwiki.org/wiki/Set_of_Distributional_Derivatives_of_Dirac_Delta_Distribution_is_Linearly_Independent | https://proofwiki.org/wiki/Set_of_Distributional_Derivatives_of_Dirac_Delta_Distribution_is_Linearly_Independent | [
"Distributional Derivatives",
"Dirac Delta Function"
] | [
"Definition:Dirac Delta Distribution",
"Definition:Set",
"Definition:Distributional Derivative",
"Definition:Linearly Independent"
] | [
"Definition:Scalar",
"Definition:Zero Mapping/Schwartz Distribution",
"Definition:Test Function",
"Test Function with Rescaled Argument is Test Function",
"Definition:Test Function",
"Leibniz's Rule/One Variable",
"Definition:Test Function",
"Definition:Contradiction"
] |
proofwiki-18349 | Derivatives of Moment Generating Function of Logistic Distribution | The $n$th derivative of $M_X$ is given by:
:$\ds {M_X}^{\paren n} = \map \exp {\mu t} \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \mu^{n - k} s^k \int_{\to 0}^{\to 1} \map {\ln^k} {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u$ | The proof proceeds by induction on $n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds {M_X}^{\paren n} = \map \exp {\mu t} \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \mu^{n - k} s^k \int_{\to 0}^{\to 1} \map {\ln^k} {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u$ | The [[Definition:Higher Derivative|$n$th derivative]] of $M_X$ is given by:
:$\ds {M_X}^{\paren n} = \map \exp {\mu t} \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \mu^{n - k} s^k \int_{\to 0}^{\to 1} \map {\ln^k} {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u$ | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds {M_X}^{\paren n} = \map \exp {\mu t} \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \mu^{n - k} s^k \int_{\to 0}^{\to 1} \map {\ln^k} {\dfrac... | Derivatives of Moment Generating Function of Logistic Distribution | https://proofwiki.org/wiki/Derivatives_of_Moment_Generating_Function_of_Logistic_Distribution | https://proofwiki.org/wiki/Derivatives_of_Moment_Generating_Function_of_Logistic_Distribution | [
"Moment Generating Function of Logistic Distribution"
] | [
"Definition:Derivative/Higher Derivatives/Higher Order"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-18350 | Representative of P-adic Number is Representative of Equivalence Class | Let $p$ be any prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $x \in \Q_p$.
Then for any sequence $\sequence{y_n}$ of the rational numbers $\Q$:
:$\sequence{y_n}$ is a representative of the $p$-adic number $x$ {{iff}} $\sequence{y_n}$ is a representative of the equivalence class $x... | By definition of the $p$-adic numbers:
:$\Q_p$ is quotient ring
By definition of a quotient ring:
:$\Q_p$ is a coset space
By definition of a coset space:
:Every $p$-adic number $x$ is an equivalence class
By definitions of a representative of a $p$-adic number and a representative of an equivalence class:
:for any seq... | Let $p$ be any [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $x \in \Q_p$.
Then for any [[Definition:Sequence|sequence]] $\sequence{y_n}$ of the [[Definition:Rational Number|rational numbers]] $\Q$:
:$\s... | By definition of the [[Definition:Field of P-adic Numbers|$p$-adic numbers]]:
:$\Q_p$ is [[Definition:Quotient Ring|quotient ring]]
By definition of a [[Definition:Quotient Ring|quotient ring]]:
:$\Q_p$ is a [[Definition:Coset Space|coset space]]
By definition of a [[Definition:Coset Space|coset space]]:
:Every [[Def... | Representative of P-adic Number is Representative of Equivalence Class | https://proofwiki.org/wiki/Representative_of_P-adic_Number_is_Representative_of_Equivalence_Class | https://proofwiki.org/wiki/Representative_of_P-adic_Number_is_Representative_of_Equivalence_Class | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Sequence",
"Definition:Rational Number",
"Definition:P-adic Number/Representative",
"Definition:P-adic Number",
"Definition:Equivalence Class/Representative",
"Definition:Equivalence Class"
] | [
"Definition:Field of P-adic Numbers",
"Definition:Quotient Ring",
"Definition:Quotient Ring",
"Definition:Coset Space",
"Definition:Coset Space",
"Definition:P-adic Number",
"Definition:Equivalence Class",
"Definition:P-adic Number/Representative",
"Definition:Equivalence Class/Representative",
"D... |
proofwiki-18351 | Distributional Partial Derivatives Commute | Let $T \in \map {\DD'} {\R^d}$ be a Schwartz distribution.
Then in the distributional sense:
:$\dfrac {\partial^2 T} {\partial x_i \partial x_j} = \dfrac {\partial^2 T} {\partial x_j \partial x_i}$
where:
:$i, j \in \N : 1 \le i, j \le d$ | Let $\phi \in \map \DD {\R^d}$ be a test function.
{{begin-eqn}}
{{eqn | l = \map {\dfrac {\partial^2 T} {\partial x_i \partial x_j} } \phi
| r = -\map {\dfrac {\partial T} {\partial x_j} } {\dfrac {\partial \phi} {\partial x_i} }
| c = {{Defof|Distributional Partial Derivative}}
}}
{{eqn | r = \map T {\dfr... | Let $T \in \map {\DD'} {\R^d}$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Then in the [[Definition:Distributional Derivative|distributional sense]]:
:$\dfrac {\partial^2 T} {\partial x_i \partial x_j} = \dfrac {\partial^2 T} {\partial x_j \partial x_i}$
where:
:$i, j \in \N : 1 \le i, j \le d... | Let $\phi \in \map \DD {\R^d}$ be a [[Definition:Test Function|test function]].
{{begin-eqn}}
{{eqn | l = \map {\dfrac {\partial^2 T} {\partial x_i \partial x_j} } \phi
| r = -\map {\dfrac {\partial T} {\partial x_j} } {\dfrac {\partial \phi} {\partial x_i} }
| c = {{Defof|Distributional Partial Derivative... | Distributional Partial Derivatives Commute | https://proofwiki.org/wiki/Distributional_Partial_Derivatives_Commute | https://proofwiki.org/wiki/Distributional_Partial_Derivatives_Commute | [
"Distributional Derivatives",
"Examples of Commutative Operations"
] | [
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative"
] | [
"Definition:Test Function",
"Clairaut's Theorem"
] |
proofwiki-18352 | Real Arctangent Function is Order Embedding into Reals | The real arctangent function $\arctan: \R \to \R$ is an order embedding on the set of real numbers under the usual ordering. | {{ProofWanted|straightforward but tedious. We should have Real Arctangent Function is Increasing for a start but we don't yet.}} | The [[Definition:Real Arctangent|real arctangent function]] $\arctan: \R \to \R$ is an [[Definition:Order Embedding|order embedding]] on the [[Definition:Real Number|set of real numbers]] under the [[Definition:Usual Ordering|usual ordering]]. | {{ProofWanted|straightforward but tedious. We should have [[Real Arctangent Function is Increasing]] for a start but we don't yet.}} | Real Arctangent Function is Order Embedding into Reals | https://proofwiki.org/wiki/Real_Arctangent_Function_is_Order_Embedding_into_Reals | https://proofwiki.org/wiki/Real_Arctangent_Function_is_Order_Embedding_into_Reals | [
"Examples of Order Embeddings"
] | [
"Definition:Inverse Tangent/Real/Arctangent",
"Definition:Order Embedding",
"Definition:Real Number",
"Definition:Usual Ordering"
] | [
"Real Arctangent Function is Increasing"
] |
proofwiki-18353 | Inclusion Mapping is Order Embedding | Let $\struct {S, \preccurlyeq_S}$ and $\struct {T, \preccurlyeq_T}$ be ordered sets such that $S \subseteq T$.
Let ${\preccurlyeq_S} = {\preccurlyeq_T} {\restriction_S}$ be the restriction of $\preccurlyeq_T$ to $S$.
Let $i_S: S \to T$ denote the inclusion mapping from $S$ to $T$:
:$\forall s \in S: \map {i_S} s = s$
T... | We have that Inclusion Mapping is Restriction of Identity.
Then we have that Identity Mapping is Order Isomorphism.
The result follows.
{{qed}} | Let $\struct {S, \preccurlyeq_S}$ and $\struct {T, \preccurlyeq_T}$ be [[Definition:Ordered Set|ordered sets]] such that $S \subseteq T$.
Let ${\preccurlyeq_S} = {\preccurlyeq_T} {\restriction_S}$ be the [[Definition:Restriction of Relation|restriction]] of $\preccurlyeq_T$ to $S$.
Let $i_S: S \to T$ denote the [[Def... | We have that [[Inclusion Mapping is Restriction of Identity]].
Then we have that [[Identity Mapping is Order Isomorphism]].
The result follows.
{{qed}} | Inclusion Mapping is Order Embedding | https://proofwiki.org/wiki/Inclusion_Mapping_is_Order_Embedding | https://proofwiki.org/wiki/Inclusion_Mapping_is_Order_Embedding | [
"Examples of Order Embeddings"
] | [
"Definition:Ordered Set",
"Definition:Restriction/Relation",
"Definition:Inclusion Mapping",
"Definition:Order Embedding"
] | [
"Inclusion Mapping is Restriction of Identity",
"Identity Mapping is Order Isomorphism"
] |
proofwiki-18354 | Non-Injective Mapping may be Strictly Order-Preserving and Order-Reversing | Let $\struct {S, \prec_1}$ and $\struct {T, \prec_2}$ be strictly ordered sets.
Let $\phi: S \to T$ be a mapping.
Let $\pi: S \to T$ be a mapping with the property that:
:$\forall x, y \in S: x \prec_1 y \iff \map \pi x \prec_2 \map \pi y$
Then it is not necessarily the case that $\pi$ is an injection. | Proof by Counterexample:
Let $S = \set {\O, \set a, \set b, \set {a, b} }$ and $T = \set {1, 2, 3}$.
Let $\prec_1$ be the proper subset relation:
:$\forall x, y \in S: x \prec_1 y \iff x \subsetneq y$
Let $\prec_2$ be the usual strict ordering on the integers $1, 2, 3$:
:$\forall x, y \in T: x \prec_2 y \iff x < y$
Let... | Let $\struct {S, \prec_1}$ and $\struct {T, \prec_2}$ be [[Definition:Strictly Ordered Set|strictly ordered sets]].
Let $\phi: S \to T$ be a [[Definition:Mapping|mapping]].
Let $\pi: S \to T$ be a [[Definition:Mapping|mapping]] with the property that:
:$\forall x, y \in S: x \prec_1 y \iff \map \pi x \prec_2 \map \pi... | [[Proof by Counterexample]]:
Let $S = \set {\O, \set a, \set b, \set {a, b} }$ and $T = \set {1, 2, 3}$.
Let $\prec_1$ be the [[Definition:Proper Subset|proper subset]] relation:
:$\forall x, y \in S: x \prec_1 y \iff x \subsetneq y$
Let $\prec_2$ be the [[Definition:Usual Ordering|usual]] [[Definition:Strict Orderi... | Non-Injective Mapping may be Strictly Order-Preserving and Order-Reversing | https://proofwiki.org/wiki/Non-Injective_Mapping_may_be_Strictly_Order-Preserving_and_Order-Reversing | https://proofwiki.org/wiki/Non-Injective_Mapping_may_be_Strictly_Order-Preserving_and_Order-Reversing | [
"Increasing Mappings"
] | [
"Definition:Strictly Ordered Set",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Injection"
] | [
"Proof by Counterexample",
"Definition:Proper Subset",
"Definition:Usual Ordering",
"Definition:Strict Ordering",
"Definition:Integer",
"Definition:Mapping",
"Definition:Injection"
] |
proofwiki-18355 | Strictly Order-Preserving and Order-Reversing Mapping on Strictly Totally Ordered Set is Injection | Let $\struct {S, \prec_1}$ and $\struct {T, \prec_2}$ be strictly totally ordered sets.
Let $\phi: S \to T$ be a mapping.
Let $\pi: S \to T$ be a mapping with the property that:
:$\forall x, y \in S: x \prec_1 y \iff \map \pi x \prec_2 \map \pi y$
Then $\pi$ is an injection. | {{AimForCont}} $\pi$ is not an injection.
Hence:
:$\exists x, y \in S: \map \pi x = \map \pi y$
As $S$ is strictly totally ordered:
:$x \prec_1 y$ or $y \prec_1 x$
{{WLOG}}, let $x \prec_1 y$.
Then we have:
:$\map \pi x = \map \pi y$
But {{hypothesis}}:
:$\map \pi x \prec_2 \map \pi y$
Because $\prec_2$ is a strict ord... | Let $\struct {S, \prec_1}$ and $\struct {T, \prec_2}$ be [[Definition:Strictly Totally Ordered Set|strictly totally ordered sets]].
Let $\phi: S \to T$ be a [[Definition:Mapping|mapping]].
Let $\pi: S \to T$ be a [[Definition:Mapping|mapping]] with the property that:
:$\forall x, y \in S: x \prec_1 y \iff \map \pi x ... | {{AimForCont}} $\pi$ is not an [[Definition:Injection|injection]].
Hence:
:$\exists x, y \in S: \map \pi x = \map \pi y$
As $S$ is [[Definition:Strictly Totally Ordered Set|strictly totally ordered]]:
:$x \prec_1 y$ or $y \prec_1 x$
{{WLOG}}, let $x \prec_1 y$.
Then we have:
:$\map \pi x = \map \pi y$
But {{hypoth... | Strictly Order-Preserving and Order-Reversing Mapping on Strictly Totally Ordered Set is Injection | https://proofwiki.org/wiki/Strictly_Order-Preserving_and_Order-Reversing_Mapping_on_Strictly_Totally_Ordered_Set_is_Injection | https://proofwiki.org/wiki/Strictly_Order-Preserving_and_Order-Reversing_Mapping_on_Strictly_Totally_Ordered_Set_is_Injection | [
"Increasing Mappings",
"Total Orderings",
"Strict Orderings"
] | [
"Definition:Strictly Totally Ordered Set",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Injection"
] | [
"Definition:Injection",
"Definition:Strictly Totally Ordered Set",
"Definition:Strict Ordering",
"Proof by Contradiction",
"Definition:Injection"
] |
proofwiki-18356 | Variance of Logistic Distribution | Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The variance of $X$ is given by:
:$\var X = \dfrac {s^2 \pi^2} 3$ | From the definition of the logistic distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From Variance as Expectation of Square minus Square of Expectation:
:$\ds \var X = \int_{-\infty}^\... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] which satisfies the [[Definition:Logistic Distribution|logistic distribution]]:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \dfrac {s^2 \pi^2} 3$ | From the definition of the [[Definition:Logistic Distribution|logistic distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From [[Variance a... | Variance of Logistic Distribution/Proof 1 | https://proofwiki.org/wiki/Variance_of_Logistic_Distribution | https://proofwiki.org/wiki/Variance_of_Logistic_Distribution/Proof_1 | [
"Variance",
"Logistic Distribution",
"Variance of Logistic Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Variance"
] | [
"Definition:Logistic Distribution",
"Definition:Probability Density Function",
"Variance as Expectation of Square minus Square of Expectation/Continuous",
"Integration by Substitution",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Integral of Constant/Definite",
"Difference of Lo... |
proofwiki-18357 | Variance of Logistic Distribution | Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The variance of $X$ is given by:
:$\var X = \dfrac {s^2 \pi^2} 3$ | By Moment Generating Function of Logistic Distribution, the moment generating function of $X$ is given by:
:$\ds \map {M_X} t = \map \exp {\mu t} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u$
for $\size t < \dfrac 1 s$.
From Variance as Expectation of Square minus Square of Expectation:
:$\ds \var X... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] which satisfies the [[Definition:Logistic Distribution|logistic distribution]]:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \dfrac {s^2 \pi^2} 3$ | By [[Moment Generating Function of Logistic Distribution]], the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
:$\ds \map {M_X} t = \map \exp {\mu t} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u$
for $\size t < \dfrac 1 s$.
From [[Variance as Expectation o... | Variance of Logistic Distribution/Proof 2 | https://proofwiki.org/wiki/Variance_of_Logistic_Distribution | https://proofwiki.org/wiki/Variance_of_Logistic_Distribution/Proof_2 | [
"Variance",
"Logistic Distribution",
"Variance of Logistic Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Variance"
] | [
"Moment Generating Function of Logistic Distribution",
"Definition:Moment Generating Function",
"Variance as Expectation of Square minus Square of Expectation/Continuous",
"Expectation of Logistic Distribution",
"Moment Generating Function of Logistic Distribution/Examples/Second Moment",
"Integral of Con... |
proofwiki-18358 | Variance of Logistic Distribution | Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The variance of $X$ is given by:
:$\var X = \dfrac {s^2 \pi^2} 3$ | From the definition of the logistic distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From Variance as Expectation of Square minus Square of Expectation:
:$\ds \var X = \int_{-\infty}^\... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] which satisfies the [[Definition:Logistic Distribution|logistic distribution]]:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
The [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \dfrac {s^2 \pi^2} 3$ | From the definition of the [[Definition:Logistic Distribution|logistic distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From [[Variance a... | Variance of Logistic Distribution/Proof 3 | https://proofwiki.org/wiki/Variance_of_Logistic_Distribution | https://proofwiki.org/wiki/Variance_of_Logistic_Distribution/Proof_3 | [
"Variance",
"Logistic Distribution",
"Variance of Logistic Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Variance"
] | [
"Definition:Logistic Distribution",
"Definition:Probability Density Function",
"Variance as Expectation of Square minus Square of Expectation/Continuous",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Sum of Infinite Geometric Sequence",
"Definition:Derivative/Real Function",
"Fubi... |
proofwiki-18359 | P-adic Numbers form Completion of Rational Numbers with P-adic Norm | Let $p$ be a prime number.
Let $\norm {\,\cdot\,}^\Q_p$ be the $p$-adic norm on the rationals $\Q$.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Then $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a completion of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$ | Let $\norm {\,\cdot\,}^\Q_p$ be the $p$-adic norm on the rationals $\Q$.
From $p$-adic Norm on Rational Numbers is Non-Archimedean Norm:
:$\struct{\Q, \norm {\,\cdot\,}^\Q_p}$ is a valued field with non-Archimedean norm $\norm {\,\cdot\,}_p$
By {{Defof|Field of P-adic Numbers|Field of $p$-adic Numbers}}:
:$\Q_p$ is th... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\norm {\,\cdot\,}^\Q_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Then $\struct {\... | Let $\norm {\,\cdot\,}^\Q_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]].
From [[P-adic Norm on Rational Numbers is Non-Archimedean Norm|$p$-adic Norm on Rational Numbers is Non-Archimedean Norm]]:
:$\struct{\Q, \norm {\,\cdot\,}^\Q_p}$ is a [[Definition:Valued... | P-adic Numbers form Completion of Rational Numbers with P-adic Norm | https://proofwiki.org/wiki/P-adic_Numbers_form_Completion_of_Rational_Numbers_with_P-adic_Norm | https://proofwiki.org/wiki/P-adic_Numbers_form_Completion_of_Rational_Numbers_with_P-adic_Norm | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Completion (Normed Division Ring)"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"P-adic Norm forms Non-Archimedean Valued Field/Rational Numbers",
"Definition:Valued Field",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Quotient Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Nu... |
proofwiki-18360 | Kurtosis in terms of Non-Central Moments | Let $X$ be a random variable with expectation $\mu$ and standard deviation $\sigma$.
Then the kurtosis $\alpha_4$ of $X$ is given by:
:$\ds \alpha_4 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4}$ | {{begin-eqn}}
{{eqn | l = \alpha_4
| r = \expect {\paren {\dfrac {X - \mu} \sigma}^4}
| c = {{Defof|Kurtosis}}
}}
{{eqn | r = \dfrac {\expect {X^4 - 4 X^3 \mu + 6 X^2 \mu^2 - 4 X \mu^3 + \mu^4} } {\sigma^4}
| c = Fourth Power of Difference
}}
{{eqn | r = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6... | Let $X$ be a [[Definition:Random Variable|random variable]] with [[Definition:Expectation|expectation]] $\mu$ and [[Definition:Standard Deviation|standard deviation]] $\sigma$.
Then the [[Definition:Kurtosis|kurtosis]] $\alpha_4$ of $X$ is given by:
:$\ds \alpha_4 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \m... | {{begin-eqn}}
{{eqn | l = \alpha_4
| r = \expect {\paren {\dfrac {X - \mu} \sigma}^4}
| c = {{Defof|Kurtosis}}
}}
{{eqn | r = \dfrac {\expect {X^4 - 4 X^3 \mu + 6 X^2 \mu^2 - 4 X \mu^3 + \mu^4} } {\sigma^4}
| c = [[Fourth Power of Difference]]
}}
{{eqn | r = \dfrac {\expect {X^4} - 4 \mu \expect {X^3}... | Kurtosis in terms of Non-Central Moments | https://proofwiki.org/wiki/Kurtosis_in_terms_of_Non-Central_Moments | https://proofwiki.org/wiki/Kurtosis_in_terms_of_Non-Central_Moments | [
"Kurtosis"
] | [
"Definition:Random Variable",
"Definition:Expectation",
"Definition:Standard Deviation",
"Definition:Kurtosis"
] | [
"Binomial Theorem/Examples/4th Power of Difference",
"Expectation is Linear",
"Category:Kurtosis"
] |
proofwiki-18361 | Mutual Order Embedding does not imply Order Isomorphism | Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be ordered sets.
Let it be possible for:
:$\struct {S_1, \preceq_1}$ to be embedded in $\struct {S_2, \preceq_2}$
:$\struct {S_2, \preceq_2}$ to be embedded in $\struct {S_1, \preceq_1}$.
Then it is not necessarily the case that $\struct {S_1, \preceq_1}$ an... | Consider the ordered structures:
:$\struct {S_1, \preceq_1} := \struct {\R, \le}$
:$\struct {S_2, \preceq_2} := \struct {\hointl {-\dfrac \pi 2} {\dfrac \pi 2}, \le}$
From Real Arctangent Function is Order Embedding into Reals, $\struct {S_1, \preceq_1}$ can be embedded into $\struct {S_2, \preceq_2}$.
From Inclusion M... | Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let it be possible for:
:$\struct {S_1, \preceq_1}$ to be [[Definition:Order Embedding|embedded]] in $\struct {S_2, \preceq_2}$
:$\struct {S_2, \preceq_2}$ to be [[Definition:Order Embedding|embedded]] in $\struct... | Consider the [[Definition:Ordered Structure|ordered structures]]:
:$\struct {S_1, \preceq_1} := \struct {\R, \le}$
:$\struct {S_2, \preceq_2} := \struct {\hointl {-\dfrac \pi 2} {\dfrac \pi 2}, \le}$
From [[Real Arctangent Function is Order Embedding into Reals]], $\struct {S_1, \preceq_1}$ can be embedded into $\st... | Mutual Order Embedding does not imply Order Isomorphism | https://proofwiki.org/wiki/Mutual_Order_Embedding_does_not_imply_Order_Isomorphism | https://proofwiki.org/wiki/Mutual_Order_Embedding_does_not_imply_Order_Isomorphism | [
"Order Embeddings",
"Order Isomorphisms"
] | [
"Definition:Ordered Set",
"Definition:Order Embedding",
"Definition:Order Embedding",
"Definition:Order Isomorphism/Isomorphic Sets"
] | [
"Definition:Ordered Structure",
"Real Arctangent Function is Order Embedding into Reals",
"Inclusion Mapping is Order Embedding",
"Definition:Order Isomorphism/Isomorphic Sets",
"Number of Maximal Elements is Order Property",
"Definition:Maximal/Element",
"Definition:Maximal/Element",
"Definition:Maxi... |
proofwiki-18362 | Number of Minimal Elements is Order Property | Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $\map m S$ be the number of minimal elements of $\struct {S, \preccurlyeq}$.
Then $\map m S$ is an order property. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be isomorphic ordered sets.
Hence let $\phi: S_1 \to S_2$ be an order isomorphism.
By definition of order property, we need to show that the number of minimal elements of $\struct {S_1, \preccurlyeq_1}$ is equal to the number of minimal elements of... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
Let $\map m S$ be the number of [[Definition:Minimal Element|minimal elements]] of $\struct {S, \preccurlyeq}$.
Then $\map m S$ is an [[Definition:Order Property|order property]]. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Isomorphic Ordered Sets|isomorphic ordered sets]].
Hence let $\phi: S_1 \to S_2$ be an [[Definition:Order Isomorphism|order isomorphism]].
By definition of [[Definition:Order Property|order property]], we need to show that the num... | Number of Minimal Elements is Order Property | https://proofwiki.org/wiki/Number_of_Minimal_Elements_is_Order_Property | https://proofwiki.org/wiki/Number_of_Minimal_Elements_is_Order_Property | [
"Examples of Order Properties",
"Minimal Elements"
] | [
"Definition:Ordered Set",
"Definition:Minimal/Element",
"Definition:Order Property"
] | [
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism",
"Definition:Order Property",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Order Isomorphism",
"Definition:Minimal/Element",
"Order Embedding is Injection",
"Defi... |
proofwiki-18363 | Number of Maximal Elements is Order Property | Let $\struct {S, \preccurlyeq}$ be an ordered set.
Let $\map M S$ be the number of maximal elements of $\struct {S, \preccurlyeq}$.
Then $\map M S$ is an order property. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be isomorphic ordered sets.
Hence let $\phi: S_1 \to S_2$ be an order isomorphism.
By definition of order property, we need to show that the number of maximal elements of $\struct {S_1, \preccurlyeq_1}$ is equal to the number of maximal elements of... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Set|ordered set]].
Let $\map M S$ be the number of [[Definition:Maximal Element|maximal elements]] of $\struct {S, \preccurlyeq}$.
Then $\map M S$ is an [[Definition:Order Property|order property]]. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Isomorphic Ordered Sets|isomorphic ordered sets]].
Hence let $\phi: S_1 \to S_2$ be an [[Definition:Order Isomorphism|order isomorphism]].
By definition of [[Definition:Order Property|order property]], we need to show that the num... | Number of Maximal Elements is Order Property | https://proofwiki.org/wiki/Number_of_Maximal_Elements_is_Order_Property | https://proofwiki.org/wiki/Number_of_Maximal_Elements_is_Order_Property | [
"Examples of Order Properties",
"Maximal Elements"
] | [
"Definition:Ordered Set",
"Definition:Maximal/Element",
"Definition:Order Property"
] | [
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism",
"Definition:Order Property",
"Definition:Maximal/Element",
"Definition:Maximal/Element",
"Definition:Maximal/Element",
"Definition:Order Isomorphism",
"Definition:Maximal/Element",
"Order Embedding is Injection",
"Defi... |
proofwiki-18364 | Densely Ordered is Order Property | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be isomorphic ordered sets.
Let $\struct {S_1, \preccurlyeq_1}$ be densely ordered.
Then $\struct {S_2, \preccurlyeq_2}$ is also densely ordered.
That is, the property of being densely ordered is an order property. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be isomorphic ordered sets.
Let $\struct {S_1, \preccurlyeq_1}$ be densely ordered.
Let $a, b \in S_2$ such that $a \prec b$.
Then as $\phi$ is an order isomorphism:
:$\exists p, q \in S_1: a = \map \phi p, b = \map \phi q$
Then by definition of de... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Isomorphic Ordered Sets|isomorphic ordered sets]].
Let $\struct {S_1, \preccurlyeq_1}$ be [[Definition:Densely Ordered|densely ordered]].
Then $\struct {S_2, \preccurlyeq_2}$ is also [[Definition:Densely Ordered|densely ordered]]... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Isomorphic Ordered Sets|isomorphic ordered sets]].
Let $\struct {S_1, \preccurlyeq_1}$ be [[Definition:Densely Ordered|densely ordered]].
Let $a, b \in S_2$ such that $a \prec b$.
Then as $\phi$ is an [[Definition:Order Isomorph... | Densely Ordered is Order Property | https://proofwiki.org/wiki/Densely_Ordered_is_Order_Property | https://proofwiki.org/wiki/Densely_Ordered_is_Order_Property | [
"Examples of Order Properties",
"Densely Ordered"
] | [
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Densely Ordered",
"Definition:Densely Ordered",
"Definition:Property",
"Definition:Densely Ordered",
"Definition:Order Property"
] | [
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Densely Ordered",
"Definition:Order Isomorphism",
"Definition:Densely Ordered",
"Definition:Order Isomorphism",
"Definition:Densely Ordered"
] |
proofwiki-18365 | Order Isomorphism is Symmetric | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\struct {S_1, \preccurlyeq_1}$ be isomorphic to $\struct {S_2, \preccurlyeq_2}$.
Then $\struct {S_2, \preccurlyeq_2}$ is isomorphic to $\struct {S_1, \preccurlyeq_1}$. | Let $\phi: S_1 \to S_2$ be an order isomorphism from $\struct {S_1, \preccurlyeq_1}$ to $\struct {S_2, \preccurlyeq_2}$.
From Inverse of Order Isomorphism is Order Isomorphism, $\phi^{-1}: S_2 \to S_1$ is an order isomorphism from $\struct {S_2, \preccurlyeq_2}$ to $\struct {S_1, \preccurlyeq_1}$.
The result follows.
{... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {S_1, \preccurlyeq_1}$ be [[Definition:Isomorphic Ordered Sets|isomorphic]] to $\struct {S_2, \preccurlyeq_2}$.
Then $\struct {S_2, \preccurlyeq_2}$ is [[Definition:Isomorphic Ordered Sets... | Let $\phi: S_1 \to S_2$ be an [[Definition:Order Isomorphism|order isomorphism]] from $\struct {S_1, \preccurlyeq_1}$ to $\struct {S_2, \preccurlyeq_2}$.
From [[Inverse of Order Isomorphism is Order Isomorphism]], $\phi^{-1}: S_2 \to S_1$ is an [[Definition:Order Isomorphism|order isomorphism]] from $\struct {S_2, \pr... | Order Isomorphism is Symmetric | https://proofwiki.org/wiki/Order_Isomorphism_is_Symmetric | https://proofwiki.org/wiki/Order_Isomorphism_is_Symmetric | [
"Order Isomorphisms",
"Symmetric Relations"
] | [
"Definition:Ordered Set",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism/Isomorphic Sets"
] | [
"Definition:Order Isomorphism",
"Inverse of Order Isomorphism is Order Isomorphism",
"Definition:Order Isomorphism"
] |
proofwiki-18366 | Order Isomorphism is Transitive | Let $\struct {S_1, \preccurlyeq_1}$, $\struct {S_2, \preccurlyeq_2}$ and $\struct {S_3, \preccurlyeq_3}$ be ordered sets.
Let $\struct {S_1, \preccurlyeq_1}$ be isomorphic to $\struct {S_2, \preccurlyeq_2}$.
Let $\struct {S_2, \preccurlyeq_2}$ be isomorphic to $\struct {S_3, \preccurlyeq_3}$.
Then $\struct {S_1, \precc... | Let $\phi: S_1 \to S_2$ be an order isomorphism from $\struct {S_1, \preccurlyeq_1}$ to $\struct {S_2, \preccurlyeq_2}$.
Let $\psi: S_2 \to S_3$ be an order isomorphism from $\struct {S_2, \preccurlyeq_2}$ to $\struct {S_3, \preccurlyeq_3}$.
From Composite of Order Isomorphisms is Order Isomorphism, $\psi \circ \phi: S... | Let $\struct {S_1, \preccurlyeq_1}$, $\struct {S_2, \preccurlyeq_2}$ and $\struct {S_3, \preccurlyeq_3}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {S_1, \preccurlyeq_1}$ be [[Definition:Isomorphic Ordered Sets|isomorphic]] to $\struct {S_2, \preccurlyeq_2}$.
Let $\struct {S_2, \preccurlyeq_2}$ be [[Def... | Let $\phi: S_1 \to S_2$ be an [[Definition:Order Isomorphism|order isomorphism]] from $\struct {S_1, \preccurlyeq_1}$ to $\struct {S_2, \preccurlyeq_2}$.
Let $\psi: S_2 \to S_3$ be an [[Definition:Order Isomorphism|order isomorphism]] from $\struct {S_2, \preccurlyeq_2}$ to $\struct {S_3, \preccurlyeq_3}$.
From [[Com... | Order Isomorphism is Transitive | https://proofwiki.org/wiki/Order_Isomorphism_is_Transitive | https://proofwiki.org/wiki/Order_Isomorphism_is_Transitive | [
"Order Isomorphisms",
"Examples of Transitive Relations"
] | [
"Definition:Ordered Set",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism/Isomorphic Sets"
] | [
"Definition:Order Isomorphism",
"Definition:Order Isomorphism",
"Composite of Order Isomorphisms is Order Isomorphism",
"Definition:Order Isomorphism"
] |
proofwiki-18367 | Collection of All Ordered Sets is not Set | Let $\mathrm {OS}$ denote the collection of all ordered sets.
Then $\mathrm {OS}$ is not a set. | Let $C$ be the collection of all singletons:
:$\set {x: \exists y: x = \set y}$
Define a mapping $\map f {\set y} = \RR$ where $\RR$ is a reflexive relation on $\set y$.
By Reflexive Relation on Singleton is Well-Ordering, $\RR$ is an ordering.
Thus:
:$f: C \to \mathrm {OS}$
By Equality of Ordered Pairs:
:$\map f {y_1}... | Let $\mathrm {OS}$ denote the [[Definition:Collection|collection]] of all [[Definition:Ordered Set|ordered sets]].
Then $\mathrm {OS}$ is not a [[Definition:Set|set]]. | Let $C$ be the [[Definition:Collection|collection]] of all [[Definition:Singleton|singletons]]:
:$\set {x: \exists y: x = \set y}$
Define a [[Definition:Mapping|mapping]] $\map f {\set y} = \RR$ where $\RR$ is a [[Definition:Reflexive Relation|reflexive relation]] on $\set y$.
By [[Reflexive Relation on Singleton is ... | Collection of All Ordered Sets is not Set | https://proofwiki.org/wiki/Collection_of_All_Ordered_Sets_is_not_Set | https://proofwiki.org/wiki/Collection_of_All_Ordered_Sets_is_not_Set | [
"Ordered Sets"
] | [
"Definition:Collection",
"Definition:Ordered Set",
"Definition:Set"
] | [
"Definition:Collection",
"Definition:Singleton",
"Definition:Mapping",
"Definition:Reflexive Relation",
"Reflexive Relation on Singleton is Well-Ordering",
"Definition:Ordering",
"Equality of Ordered Pairs",
"Definition:Injection",
"Definition:Cardinality",
"Definition:Collection",
"Definition:S... |
proofwiki-18368 | Order Sum of Well-Founded Orderings is Well-Founded Ordering | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\preccurlyeq_1$ and $\preccurlyeq_2$ be well-founded.
Then the order sum $\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is also well-... | Let $\struct {S, \preccurlyeq} := \struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$.
Let $T \subseteq S$ such that $T \ne \O$.
Let $T = T_1 \sqcup T_2$ where $T_1 \subseteq S_1$ and $T_2 \subseteq S_2$.
Let $T_1 \ne \O$.
Then:
:$\exists x \in T_1: \forall y \in T_1: y \preccurlyeq x \implies y = x$
Th... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\preccurlyeq_1$ and $\preccurlyeq_2$ be [[Definition:Well-Founded Ordering|well-founded]].
Then the [[Definition:Order Sum|order sum]] $\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurl... | Let $\struct {S, \preccurlyeq} := \struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$.
Let $T \subseteq S$ such that $T \ne \O$.
Let $T = T_1 \sqcup T_2$ where $T_1 \subseteq S_1$ and $T_2 \subseteq S_2$.
Let $T_1 \ne \O$.
Then:
:$\exists x \in T_1: \forall y \in T_1: y \preccurlyeq x \implies y = ... | Order Sum of Well-Founded Orderings is Well-Founded Ordering | https://proofwiki.org/wiki/Order_Sum_of_Well-Founded_Orderings_is_Well-Founded_Ordering | https://proofwiki.org/wiki/Order_Sum_of_Well-Founded_Orderings_is_Well-Founded_Ordering | [
"Order Sums",
"Well-Founded Relations"
] | [
"Definition:Ordered Set",
"Definition:Well-Founded Ordered Set",
"Definition:Order Sum",
"Definition:Well-Founded Ordered Set"
] | [
"Definition:Minimal/Element",
"Definition:Order Sum",
"Definition:Order Sum",
"Definition:Vacuous Truth",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Element",
"Definition:Minimal/Element",
"Definition:Order Sum",
"Definition:Minimal/Element",
"Definition:Minimal/Elem... |
proofwiki-18369 | Order Sum of Totally Ordered Sets is Totally Ordered | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be totally ordered sets.
Then the order sum $\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is also a totally ordered set. | Let $\struct {S, \preccurlyeq} := \struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$.
From Order Sum of Ordered Sets is Ordered, $\struct {S, \preccurlyeq}$ is an ordered set.
It remains to be shown that $\tuple {a, b}$ and $\tuple {c, d}$ are comparable for all $\tuple {a, b}, \tuple {c, d} \in S$.
Le... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Totally Ordered Set|totally ordered sets]].
Then the [[Definition:Order Sum|order sum]] $\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is... | Let $\struct {S, \preccurlyeq} := \struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$.
From [[Order Sum of Ordered Sets is Ordered]], $\struct {S, \preccurlyeq}$ is an [[Definition:Ordered Set|ordered set]].
It remains to be shown that $\tuple {a, b}$ and $\tuple {c, d}$ are [[Definition:Comparable E... | Order Sum of Totally Ordered Sets is Totally Ordered | https://proofwiki.org/wiki/Order_Sum_of_Totally_Ordered_Sets_is_Totally_Ordered | https://proofwiki.org/wiki/Order_Sum_of_Totally_Ordered_Sets_is_Totally_Ordered | [
"Order Sums",
"Total Orderings"
] | [
"Definition:Totally Ordered Set",
"Definition:Order Sum",
"Definition:Totally Ordered Set"
] | [
"Order Sum of Ordered Sets is Ordered",
"Definition:Ordered Set",
"Definition:Comparable Elements",
"Definition:Total Ordering",
"Definition:Comparable Elements",
"Definition:Comparable Elements",
"Definition:Total Ordering",
"Definition:Comparable Elements",
"Definition:Comparable Elements",
"Def... |
proofwiki-18370 | Order Sum of Ordered Sets is Ordered | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Then the order sum $\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is also an ordered set. | Let $\struct {S, \preccurlyeq} := \struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$.
By definition:
::$\forall \tuple {a, b}, \tuple {c, d} \in S: \tuple {a, b} \preccurlyeq \tuple {c, d} \iff \begin {cases} b = 0 \text { and } d = 1 \\ b = d = 0 \text { and } a \preccurlyeq_1 c \\ b = d = 1 \text { a... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Then the [[Definition:Order Sum|order sum]] $\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is also an [[Defin... | Let $\struct {S, \preccurlyeq} := \struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$.
By definition:
::$\forall \tuple {a, b}, \tuple {c, d} \in S: \tuple {a, b} \preccurlyeq \tuple {c, d} \iff \begin {cases} b = 0 \text { and } d = 1 \\ b = d = 0 \text { and } a \preccurlyeq_1 c \\ b = d = 1 \text ... | Order Sum of Ordered Sets is Ordered | https://proofwiki.org/wiki/Order_Sum_of_Ordered_Sets_is_Ordered | https://proofwiki.org/wiki/Order_Sum_of_Ordered_Sets_is_Ordered | [
"Order Sums"
] | [
"Definition:Ordered Set",
"Definition:Order Sum",
"Definition:Ordered Set"
] | [
"Definition:Ordering",
"Definition:Ordering"
] |
proofwiki-18371 | Order Isomorphism is Preserved by Order Sum | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ be ordered sets such that:
:$\struct {S_1, \preccurlyeq_1}$ is isomorphic to $\struct {T_1, \preccurlyeq_{1'} }$
:$\struct {S_2, \preccurlyeq_2}$ is is... | {{ProofWanted|tedium city or what}} | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ be [[Definition:Ordered Set|ordered sets]] such that:
:$\struct {S_1, \preccurlyeq_1}$ is [[Definition:Isomorphic Ordered ... | {{ProofWanted|tedium city or what}} | Order Isomorphism is Preserved by Order Sum | https://proofwiki.org/wiki/Order_Isomorphism_is_Preserved_by_Order_Sum | https://proofwiki.org/wiki/Order_Isomorphism_is_Preserved_by_Order_Sum | [
"Order Sums",
"Order Isomorphisms"
] | [
"Definition:Ordered Set",
"Definition:Ordered Set",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Sum",
"Definition:Order Isomorphism/Isomorphic Sets"
] | [] |
proofwiki-18372 | Mixed Partial Derivative of Heaviside Step Function | Let $\tuple {x, y} \stackrel u {\longrightarrow} \map u {x, y}: \R^2 \to \R$ be the Heaviside step function.
Let $u := T_u$ be the Schwartz distribution associated with $u$.
Let $\delta_{\tuple {0, 0} } \in \map {\DD'} {\R^2}$ be the Dirac delta distribution.
Then in the distributional sense:
:$\dfrac {\partial^2 u} {\... | Let $\phi \in \map \DD {\R^2}$ be a test function with support on $\openint 0 a^2 := \openint 0 a \times \openint 0 a$ where $\times$ is the Cartesian product and $a > 0$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\dfrac {\partial^2 u}{\partial x \partial y} } \phi
| r = - \map {\dfrac {\partial u}{\partial y} } {\df... | Let $\tuple {x, y} \stackrel u {\longrightarrow} \map u {x, y}: \R^2 \to \R$ be the [[Definition:Heaviside Step Function/Two Variables|Heaviside step function]].
Let $u := T_u$ be the [[Definition:Schwartz Distribution|Schwartz distribution]] [[Differentiable Function as Distribution|associated]] with $u$.
Let $\delt... | Let $\phi \in \map \DD {\R^2}$ be a [[Definition:Test Function|test function]] with [[Definition:Support of Schwartz Distribution|support]] on $\openint 0 a^2 := \openint 0 a \times \openint 0 a$ where $\times$ is the [[Definition:Cartesian Product|Cartesian product]] and $a > 0$.
Then:
{{begin-eqn}}
{{eqn | l = \map... | Mixed Partial Derivative of Heaviside Step Function | https://proofwiki.org/wiki/Mixed_Partial_Derivative_of_Heaviside_Step_Function | https://proofwiki.org/wiki/Mixed_Partial_Derivative_of_Heaviside_Step_Function | [
"Heaviside Step Function",
"Examples of Distributional Derivatives"
] | [
"Definition:Heaviside Step Function/Two Variables",
"Definition:Schwartz Distribution",
"Differentiable Function as Distribution",
"Definition:Dirac Delta Distribution",
"Definition:Distributional Derivative"
] | [
"Definition:Test Function",
"Definition:Support of Schwartz Distribution",
"Definition:Cartesian Product",
"Clairaut's Theorem",
"Definite Integral of Partial Derivative"
] |
proofwiki-18373 | Order Type Addition is Well-Defined Operation | The addition operation on order types is well-defined. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ be ordered sets such that:
:$\struct {S_1, \preccurlyeq_1}$ is isomorphic to $\struct {T_1, \preccurlyeq_{1'} }$
:$\struct {S_2, \preccurlyeq_2}$ is is... | The [[Definition:Addition of Order Types|addition operation]] on [[Definition:Order Type|order types]] is [[Definition:Well-Defined Operation|well-defined]]. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ be [[Definition:Ordered Set|ordered sets]] such that:
:$\struct {S_1, \preccurlyeq_1}$ is [[Definition:Isomorphic Ordered ... | Order Type Addition is Well-Defined Operation | https://proofwiki.org/wiki/Order_Type_Addition_is_Well-Defined_Operation | https://proofwiki.org/wiki/Order_Type_Addition_is_Well-Defined_Operation | [
"Order Types",
"Addition",
"Order Sums"
] | [
"Definition:Addition of Order Types",
"Definition:Order Type",
"Definition:Well-Defined/Operation"
] | [
"Definition:Ordered Set",
"Definition:Ordered Set",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Type",
"Definition:Order Type",
"Order Isomorphism is Preserved by Order Sum"
] |
proofwiki-18374 | Dual of Well-Ordering is not necessarily Well-Ordering | Let $\struct {S, \preccurlyeq}$ be a well-ordered set.
Then its dual $\struct {S, \preccurlyeq}$ is not necessarily also a well-ordered set. | Consider the ordered structure $\struct {\N, \le}$.
From the Well-Ordering Principle, $\struct {\N, \le}$ is a well-ordered set..
Consider the dual $\struct {\N, \ge}$ of $\struct {\N, \le}$.
Let this be expressed as:
:$\struct {\N, \preccurlyeq} := \struct {\N, \ge}$
so as to enhance the clarification of the nature of... | Let $\struct {S, \preccurlyeq}$ be a [[Definition:Well-Ordered Set|well-ordered set]].
Then its [[Definition:Dual Ordered Set|dual]] $\struct {S, \preccurlyeq}$ is not necessarily also a [[Definition:Well-Ordered Set|well-ordered set]]. | Consider the [[Definition:Ordered Structure|ordered structure]] $\struct {\N, \le}$.
From the [[Well-Ordering Principle]], $\struct {\N, \le}$ is a [[Definition:Well-Ordered Set|well-ordered set]]..
Consider the [[Definition:Dual Ordered Set|dual]] $\struct {\N, \ge}$ of $\struct {\N, \le}$.
Let this be expressed as... | Dual of Well-Ordering is not necessarily Well-Ordering | https://proofwiki.org/wiki/Dual_of_Well-Ordering_is_not_necessarily_Well-Ordering | https://proofwiki.org/wiki/Dual_of_Well-Ordering_is_not_necessarily_Well-Ordering | [
"Well-Orderings",
"Dual Orderings"
] | [
"Definition:Well-Ordered Set",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Well-Ordered Set"
] | [
"Definition:Ordered Structure",
"Well-Ordering Principle",
"Definition:Well-Ordered Set",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Ordering",
"Dual of Total Ordering is Total Ordering",
"Definition:Totally Ordered Set",
"Definition:Well-Ordered Set",
"Definition:Minimal/Element",
"... |
proofwiki-18375 | Order Types of Duals of Isomorphic Sets are Equal | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let:
:$\map \ot {S_1, \preccurlyeq_1} = \map \ot {S_2, \preccurlyeq_2}$
where $\ot$ denotes the order type operator.
Let $\struct {S_1, \succcurlyeq_1}$ and $\struct {S_2, \succcurlyeq_2}$ denote the dual ordered sets of $\struct {... | {{ProofWanted|when it doesn't look quite so tedious}} | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let:
:$\map \ot {S_1, \preccurlyeq_1} = \map \ot {S_2, \preccurlyeq_2}$
where $\ot$ denotes the [[Definition:Order Type|order type operator]].
Let $\struct {S_1, \succcurlyeq_1}$ and $\struct {S_2, \suc... | {{ProofWanted|when it doesn't look quite so tedious}} | Order Types of Duals of Isomorphic Sets are Equal | https://proofwiki.org/wiki/Order_Types_of_Duals_of_Isomorphic_Sets_are_Equal | https://proofwiki.org/wiki/Order_Types_of_Duals_of_Isomorphic_Sets_are_Equal | [
"Order Types",
"Dual Orderings"
] | [
"Definition:Ordered Set",
"Definition:Order Type",
"Definition:Dual Ordering/Dual Ordered Set"
] | [] |
proofwiki-18376 | Dual of Order Type is Well-Defined Mapping | The dual operation on order types is a well-defined mapping. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\struct {S_1, \succcurlyeq_1}$ and $\struct {S_2, \succcurlyeq_2}$ denote the dual ordered sets of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$.
Let $\struct {S_1, \preccurlyeq_1} \cong \struct {S_2, \pr... | The [[Definition:Dual of Order Type|dual operation]] on [[Definition:Order Type|order types]] is a [[Definition:Well-Defined Mapping|well-defined mapping]]. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {S_1, \succcurlyeq_1}$ and $\struct {S_2, \succcurlyeq_2}$ denote the [[Definition:Dual Ordered Set|dual ordered sets]] of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$... | Dual of Order Type is Well-Defined Mapping | https://proofwiki.org/wiki/Dual_of_Order_Type_is_Well-Defined_Mapping | https://proofwiki.org/wiki/Dual_of_Order_Type_is_Well-Defined_Mapping | [
"Dual Orderings"
] | [
"Definition:Dual of Order Type",
"Definition:Order Type",
"Definition:Well-Defined/Mapping"
] | [
"Definition:Ordered Set",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Order Isomorphism",
"Definition:Order Type",
"Definition:Order Type",
"Definition:Dual of Order Type",
"Order Types of Duals of Isomorphic Sets are Equal",
"Definition:Well-Defined/Mapping"
] |
proofwiki-18377 | Raw Moment of Pareto Distribution | Let $X$ be a continuous random variable with the Pareto distribution with $a, b \in \R_{> 0}$.
Let $n$ be a strictly positive integer.
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = \begin {cases} \dfrac {a b^n} {a - n} & n < a \\ \text {does not exist} & n \ge a \end {cases}$ | From the definition of the Pareto distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {a b^a} {x^{a + 1} }$
Where $\Img X \in \hointr b \infty$.
From the definition of the expected value of a continuous random variable:
:$\ds \expect {X^n} = \int_b^\infty x^n \map {f_X} x \rd x$
First take $a >... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Pareto Distribution|Pareto distribution]] with $a, b \in \R_{> 0}$.
Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Then the $n$th [[Definition:Raw Moment|raw moment]] $\expect {X... | From the definition of the [[Definition:Pareto Distribution|Pareto distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {a b^a} {x^{a + 1} }$
Where $\Img X \in \hointr b \infty$.
From the definition of the [[Definition:Expectation of Continuous Ra... | Raw Moment of Pareto Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Pareto_Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Pareto_Distribution | [
"Pareto Distribution",
"Raw Moments"
] | [
"Definition:Random Variable/Continuous",
"Definition:Pareto Distribution",
"Definition:Strictly Positive/Integer",
"Definition:Raw Moment"
] | [
"Definition:Pareto Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Primitive of Power",
"Primitive of Reciprocal",
"Logarithm Tends to Infinity",
"Primitive of Power",
"Category:Pareto Distribution",
"Category:Raw Moments"
] |
proofwiki-18378 | Expectation of Pareto Distribution | Let $X$ be a continuous random variable with the Pareto distribution with $a, b \in \R_{> 0}$.
The expectation of $X$ is given by:
:$\expect X = \begin {cases} \dfrac {a b} {a - 1} & 1 < a \\ \text {does not exist} & 1 \ge a \end {cases}$ | From Raw Moment of Pareto Distribution, we have:
The $n$th raw moment $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = \begin {cases} \dfrac {a b^n} {a - n} & n < a \\ \text {does not exist} & n \ge a \end {cases}$
Therefore, for $n = 1$ we have:
:$\expect X = \begin {cases} \dfrac {a b} {a - 1} & 1 < a \\ \text ... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Pareto Distribution|Pareto distribution]] with $a, b \in \R_{> 0}$.
The [[Definition:Expectation of Continuous Random Variable|expectation]] of $X$ is given by:
:$\expect X = \begin {cases} \dfrac {a b} {a - 1} & 1... | From [[Raw Moment of Pareto Distribution]], we have:
The $n$th [[Definition:Raw Moment|raw moment]] $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = \begin {cases} \dfrac {a b^n} {a - n} & n < a \\ \text {does not exist} & n \ge a \end {cases}$
Therefore, for $n = 1$ we have:
:$\expect X = \begin {cases} \df... | Expectation of Pareto Distribution | https://proofwiki.org/wiki/Expectation_of_Pareto_Distribution | https://proofwiki.org/wiki/Expectation_of_Pareto_Distribution | [
"Pareto Distribution",
"Expectation"
] | [
"Definition:Random Variable/Continuous",
"Definition:Pareto Distribution",
"Definition:Expectation/Continuous"
] | [
"Raw Moment of Pareto Distribution",
"Definition:Raw Moment",
"Category:Pareto Distribution",
"Category:Expectation"
] |
proofwiki-18379 | Variance of Pareto Distribution | Let $X$ be a continuous random variable with the Pareto distribution with $a, b \in \R_{> 0}$.
Then the variance of $X$ is given by:
:$\var X = \begin {cases} \dfrac {a b^2 } {\paren {a - 2} \paren {a - 1}^2 } & 2 < a \\ \text {does not exist} & 2 \ge a \end {cases}$ | By Variance as Expectation of Square minus Square of Expectation, we have:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By Expectation of Pareto Distribution, we have:
:$\expect X = \begin {cases} \dfrac {a b} {a - 1} & 1 < a \\ \text {does not exist} & 1 \ge a \end {cases}$
From Raw Moment of Pareto Distribution... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Pareto Distribution|Pareto distribution]] with $a, b \in \R_{> 0}$.
Then the [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \begin {cases} \dfrac {a b^2 } {\paren {a - 2} \paren {a - 1}^2 } & 2 < ... | By [[Variance as Expectation of Square minus Square of Expectation]], we have:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By [[Expectation of Pareto Distribution]], we have:
:$\expect X = \begin {cases} \dfrac {a b} {a - 1} & 1 < a \\ \text {does not exist} & 1 \ge a \end {cases}$
From [[Raw Moment of Paret... | Variance of Pareto Distribution | https://proofwiki.org/wiki/Variance_of_Pareto_Distribution | https://proofwiki.org/wiki/Variance_of_Pareto_Distribution | [
"Variance",
"Pareto Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Pareto Distribution",
"Definition:Variance"
] | [
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Pareto Distribution",
"Raw Moment of Pareto Distribution",
"Definition:Raw Moment",
"Category:Variance",
"Category:Pareto Distribution"
] |
proofwiki-18380 | Ordered Set with Order Type of Natural Numbers plus Dual has Minimum Element | Let $\struct {S, \preccurlyeq}$ be an ordered structure such that:
:$\map \ot {S, \preccurlyeq} = \omega + \omega^*$
where:
:$\ot$ denotes order type
:$\omega$ denotes the order type of the natural numbers $\N$
:$\omega^*$ denotes the dual of $\omega$
:$+$ denotes addition of order types.
Then $\struct {S, \preccurlyeq... | By definition of order type addition:
:$\struct {S, \preccurlyeq}$ is isomorphic to $\struct {\N, \le} \oplus \struct {\N, \ge}$
where:
:$\cong$ denotes order isomorphism
:$\oplus$ denotes order sum.
By the Well-Ordering Principle, $\struct {\N, \le}$ has a smallest element.
By definition of order sum, every element of... | Let $\struct {S, \preccurlyeq}$ be an [[Definition:Ordered Structure|ordered structure]] such that:
:$\map \ot {S, \preccurlyeq} = \omega + \omega^*$
where:
:$\ot$ denotes [[Definition:Order Type|order type]]
:$\omega$ denotes the [[Definition:Order Type of Natural Numbers|order type]] of the [[Definition:Natural Numbe... | By definition of [[Definition:Addition of Order Types|order type addition]]:
:$\struct {S, \preccurlyeq}$ is [[Definition:Isomorphic Ordered Sets|isomorphic]] to $\struct {\N, \le} \oplus \struct {\N, \ge}$
where:
:$\cong$ denotes [[Definition:Order Isomorphism|order isomorphism]]
:$\oplus$ denotes [[Definition:Order S... | Ordered Set with Order Type of Natural Numbers plus Dual has Minimum Element | https://proofwiki.org/wiki/Ordered_Set_with_Order_Type_of_Natural_Numbers_plus_Dual_has_Minimum_Element | https://proofwiki.org/wiki/Ordered_Set_with_Order_Type_of_Natural_Numbers_plus_Dual_has_Minimum_Element | [
"Examples of Order Types"
] | [
"Definition:Ordered Structure",
"Definition:Order Type",
"Definition:Order Type of Natural Numbers",
"Definition:Natural Numbers",
"Definition:Dual of Order Type",
"Definition:Addition of Order Types",
"Definition:Smallest Element"
] | [
"Definition:Addition of Order Types",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism",
"Definition:Order Sum",
"Well-Ordering Principle",
"Definition:Smallest Element",
"Definition:Order Sum",
"Definition:Element",
"Definition:Precede",
"Definition:Element",
"Defin... |
proofwiki-18381 | Order Type Addition is not Commutative | The operation of order type addition is not commutative. | Consider the order type $\omega^* + \omega$, where:
:$\omega$ denotes the order type of the natural numbers $\N$
:$\omega^*$ denotes the dual of $\omega$
:$+$ denotes addition of order types.
From Order Type of Integers under Usual Ordering, this is the order type of $\struct {\Z, \le}$, the set of integers under the u... | The [[Definition:Binary Operation|operation]] of [[Definition:Addition of Order Types|order type addition]] is not [[Definition:Commutative Operation|commutative]]. | Consider the [[Definition:Order Type|order type]] $\omega^* + \omega$, where:
:$\omega$ denotes the [[Definition:Order Type of Natural Numbers|order type]] of the [[Definition:Natural Numbers|natural numbers]] $\N$
:$\omega^*$ denotes the [[Definition:Dual of Order Type|dual]] of $\omega$
:$+$ denotes [[Definition:Addi... | Order Type Addition is not Commutative | https://proofwiki.org/wiki/Order_Type_Addition_is_not_Commutative | https://proofwiki.org/wiki/Order_Type_Addition_is_not_Commutative | [
"Order Types",
"Examples of Commutative Operations"
] | [
"Definition:Operation/Binary Operation",
"Definition:Addition of Order Types",
"Definition:Commutative/Operation"
] | [
"Definition:Order Type",
"Definition:Order Type of Natural Numbers",
"Definition:Natural Numbers",
"Definition:Dual of Order Type",
"Definition:Addition of Order Types",
"Order Type of Integers under Usual Ordering",
"Definition:Order Type",
"Definition:Integer",
"Definition:Usual Ordering",
"Defi... |
proofwiki-18382 | Skewness of Pareto Distribution | Let $X$ be a continuous random variable with the Pareto distribution with $a, b \in \R_{> 0}$.
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \begin {cases} \paren {\sqrt {\dfrac {\paren {a - 2} } a } } \paren { \dfrac {2 \paren {a + 1} } {a - 3} } & 3 < a \\ \text {does not exist} & 3 \ge a \end {cases}... | From Skewness in terms of Non-Central Moments, we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Pareto Distribution we have:
:$\mu = \dfrac {a b } {\paren {a - 1} }$
By Variance o... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Pareto Distribution|Pareto distribution]] with $a, b \in \R_{> 0}$.
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \begin {cases} \paren {\sqrt {\dfrac {\paren {a - 2} } a } }... | From [[Skewness in terms of Non-Central Moments]], we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Pareto Distribu... | Skewness of Pareto Distribution | https://proofwiki.org/wiki/Skewness_of_Pareto_Distribution | https://proofwiki.org/wiki/Skewness_of_Pareto_Distribution | [
"Skewness",
"Pareto Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Pareto Distribution",
"Definition:Skewness"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Pareto Distribution",
"Variance of Pareto Distribution",
"Raw Moment of Pareto Distribution",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Category:Skewness"... |
proofwiki-18383 | Simple Order Product of Pair of Ordered Sets is Ordered Set | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ denote the '''simple (order) product''' of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$.
Then $\struct {S_1, \preccurlyeq_1} \otimes... | Let $\struct {T, \preccurlyeq_s} := \struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$.
By definition of simple product:
:$T := S_1 \times S_2$ where $\times$ denotes Cartesian product
:$\forall \tuple {a, b}, \tuple {c, d} \in T: \tuple {a, b} \preccurlyeq_s \tuple {c, d} \iff a \preccurlyeq_1 c \t... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ denote the '''[[Definition:Simple Order Product|simple (order) product]]''' of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_... | Let $\struct {T, \preccurlyeq_s} := \struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$.
By definition of [[Definition:Simple Order Product|simple product]]:
:$T := S_1 \times S_2$ where $\times$ denotes [[Definition:Cartesian Product|Cartesian product]]
:$\forall \tuple {a, b}, \tuple {c, d} \in... | Simple Order Product of Pair of Ordered Sets is Ordered Set | https://proofwiki.org/wiki/Simple_Order_Product_of_Pair_of_Ordered_Sets_is_Ordered_Set | https://proofwiki.org/wiki/Simple_Order_Product_of_Pair_of_Ordered_Sets_is_Ordered_Set | [
"Simple Order Product"
] | [
"Definition:Ordered Set",
"Definition:Simple Order Product",
"Definition:Ordered Set"
] | [
"Definition:Simple Order Product",
"Definition:Cartesian Product",
"Definition:Ordering",
"Definition:Cartesian Product",
"Definition:Ordering",
"Definition:Simple Order Product",
"Definition:Simple Order Product",
"Definition:Ordering",
"Definition:Simple Order Product",
"Definition:Simple Order ... |
proofwiki-18384 | Simple Order Product of Totally Ordered Sets may not be Totally Ordered | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be totally ordered sets.
Let $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ denote the '''simple (order) product''' of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$.
Then it is not necessarily the case... | From Simple Order Product of Pair of Ordered Sets is Ordered Set, we do have that $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ is an ordered set.
Let us take the simple product of the ordered set that is the natural numbers under the usual ordering with itself:
:$\struct {\N, \le} \otimes^s \... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Totally Ordered Set|totally ordered sets]].
Let $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ denote the '''[[Definition:Simple Order Product|simple (order) product]]''' of $\struct {S_1, \preccurlyeq_1}$ ... | From [[Simple Order Product of Pair of Ordered Sets is Ordered Set]], we do have that $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ is an [[Definition:Ordered Set|ordered set]].
Let us take the [[Definition:Simple Order Product|simple product]] of the [[Definition:Ordered Set|ordered set]] th... | Simple Order Product of Totally Ordered Sets may not be Totally Ordered | https://proofwiki.org/wiki/Simple_Order_Product_of_Totally_Ordered_Sets_may_not_be_Totally_Ordered | https://proofwiki.org/wiki/Simple_Order_Product_of_Totally_Ordered_Sets_may_not_be_Totally_Ordered | [
"Simple Order Product",
"Total Orderings"
] | [
"Definition:Totally Ordered Set",
"Definition:Simple Order Product",
"Definition:Totally Ordered Set"
] | [
"Simple Order Product of Pair of Ordered Sets is Ordered Set",
"Definition:Ordered Set",
"Definition:Simple Order Product",
"Definition:Ordered Set",
"Definition:Natural Numbers",
"Definition:Usual Ordering",
"Well-Ordering Principle",
"Definition:Well-Ordered Set",
"Definition:Totally Ordered Set",... |
proofwiki-18385 | Antilexicographic Order is Ordering | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\preccurlyeq_a$ be the lexicographic order on $S_1 \times S_2$''':
:$\tuple {x_1, x_2} \preccurlyeq_a \tuple {y_1, y_2} \iff \paren {x_2 \prec_2 y_2} \lor \paren {x_2 = y_2 \land x_1 \preccurlyeq_1 y_1}$
Then $\preccurlyeq_a$ ... | In the following, $\tuple {x_1, x_2}, \tuple {y_1, y_2}, \tuple {z_1, z_2} \in S_1 \times S_2$.
Checking in turn each of the criteria for an ordering: | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\preccurlyeq_a$ be the [[Definition:Lexicographic Order|lexicographic order on $S_1 \times S_2$]]''':
:$\tuple {x_1, x_2} \preccurlyeq_a \tuple {y_1, y_2} \iff \paren {x_2 \prec_2 y_2} \lor \paren {... | In the following, $\tuple {x_1, x_2}, \tuple {y_1, y_2}, \tuple {z_1, z_2} \in S_1 \times S_2$.
Checking in turn each of the criteria for an [[Definition:Ordering|ordering]]: | Antilexicographic Order is Ordering | https://proofwiki.org/wiki/Antilexicographic_Order_is_Ordering | https://proofwiki.org/wiki/Antilexicographic_Order_is_Ordering | [
"Antilexicographic Order"
] | [
"Definition:Ordered Set",
"Definition:Lexicographic Order",
"Definition:Ordering"
] | [
"Definition:Ordering",
"Definition:Ordering",
"Definition:Ordering",
"Definition:Ordering",
"Definition:Ordering",
"Definition:Ordering"
] |
proofwiki-18386 | Moment Generating Function of Pareto Distribution | Let $X$ be a continuous random variable with a Pareto distribution with parameters a and b for $a, b \in \R_{> 0}$.
Then the moment generating function $M_X$ of $X$ is given by:
:$\map {M_X} t = \begin {cases} a \paren {-b t}^a \map \Gamma {-a, -b t} & t < 0 \\ 1 & t = 0 \\ \text {does not exist} & t > 0 \end {cases}... | From the definition of the Pareto distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {a b^a} {x^{a + 1} }$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect { e^{t X} } = \int_b^\infty e^{t x} \map {f_X} x \rd x$
First take $t < 0$.
Then:
:$\ds \map {M_X} t = a ... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with a [[Definition:Pareto Distribution|Pareto distribution with parameters a and b]] for $a, b \in \R_{> 0}$.
Then the [[Definition:Moment Generating Function|moment generating function]] $M_X$ of $X$ is given by:
:$\map {M_X} t = \... | From the definition of the [[Definition:Pareto Distribution|Pareto distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {a b^a} {x^{a + 1} }$
From the definition of a [[Definition:Moment Generating Function|moment generating function]]:
:$\ds \map ... | Moment Generating Function of Pareto Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Pareto_Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Pareto_Distribution | [
"Moment Generating Functions",
"Pareto Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Pareto Distribution",
"Definition:Moment Generating Function"
] | [
"Definition:Pareto Distribution",
"Definition:Probability Density Function",
"Definition:Moment Generating Function",
"Power Rule for Derivatives",
"Primitive of Power",
"Fundamental Theorem of Calculus",
"Primitive of Exponential of a x over Power of x",
"Exponential Dominates Polynomial",
"Categor... |
proofwiki-18387 | Skewness of Logistic Distribution | Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
Then the skewness $\gamma_1$ of $X$ is equal to $0$. | From Skewness in terms of Non-Central Moments, we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Logistic Distribution we have:
:$\mu = \mu$
By Variance of Logistic Distribution we... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] which satisfies the [[Definition:Logistic Distribution|logistic distribution]]:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is equal to $0$. | From [[Skewness in terms of Non-Central Moments]], we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Logistic Distri... | Skewness of Logistic Distribution | https://proofwiki.org/wiki/Skewness_of_Logistic_Distribution | https://proofwiki.org/wiki/Skewness_of_Logistic_Distribution | [
"Skewness",
"Logistic Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Logistic Distribution",
"Definition:Skewness"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Logistic Distribution",
"Variance of Logistic Distribution",
"Moment in terms of Moment Generating Function",
"Definition:Moment Generating Function",
"Derivatives of Moment Generatin... |
proofwiki-18388 | Antilexicographic Product on Pair of Well-Ordered Sets is Well-Ordered | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be well-ordered sets.
Let $S_1 \otimes^a S_2 = \struct {S_1 \times S_2, \preccurlyeq_a}$ be the antilexicographic product of $S_1$ and $S_2$.
Then $\struct {S_1 \times S_2, \preccurlyeq_a}$ is itself a well-ordered set. | By definition, a well-ordered set is a totally ordered set which is well-founded.
From Antilexicographic Product of Totally Ordered Sets is Totally Ordered, we have that $\preccurlyeq_a$ is a totally ordered set.
It remains to be shown that $\preccurlyeq_a$ is a well-founded relation.
Let $T = S_1 \times S_2$.
Let $A$ ... | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Well-Ordered Set|well-ordered sets]].
Let $S_1 \otimes^a S_2 = \struct {S_1 \times S_2, \preccurlyeq_a}$ be the [[Definition:Antilexicographic Order|antilexicographic product]] of $S_1$ and $S_2$.
Then $\struct {S_1 \times S_2, \... | By definition, a [[Definition:Well-Ordered Set|well-ordered set]] is a [[Definition:Totally Ordered Set|totally ordered set]] which is [[Definition:Well-Founded Set|well-founded]].
From [[Antilexicographic Product of Totally Ordered Sets is Totally Ordered]], we have that $\preccurlyeq_a$ is a [[Definition:Totally Ord... | Antilexicographic Product on Pair of Well-Ordered Sets is Well-Ordered | https://proofwiki.org/wiki/Antilexicographic_Product_on_Pair_of_Well-Ordered_Sets_is_Well-Ordered | https://proofwiki.org/wiki/Antilexicographic_Product_on_Pair_of_Well-Ordered_Sets_is_Well-Ordered | [
"Well-Orderings",
"Antilexicographic Order"
] | [
"Definition:Well-Ordered Set",
"Definition:Antilexicographic Order",
"Definition:Well-Ordered Set"
] | [
"Definition:Well-Ordered Set",
"Definition:Totally Ordered Set",
"Definition:Well-Founded Set",
"Antilexicographic Product of Totally Ordered Sets is Totally Ordered",
"Definition:Totally Ordered Set",
"Definition:Well-Founded Relation",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:S... |
proofwiki-18389 | Order Type Multiplication is Well-Defined Operation | The multiplication operation on order types is well-defined. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ be ordered sets such that:
:$\struct {S_1, \preccurlyeq_1}$ is isomorphic to $\struct {T_1, \preccurlyeq_{1'} }$
:$\struct {S_2, \preccurlyeq_2}$ is is... | The [[Definition:Multiplication of Order Types|multiplication operation]] on [[Definition:Order Type|order types]] is [[Definition:Well-Defined Operation|well-defined]]. | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ be [[Definition:Ordered Set|ordered sets]] such that:
:$\struct {S_1, \preccurlyeq_1}$ is [[Definition:Isomorphic Ordered ... | Order Type Multiplication is Well-Defined Operation | https://proofwiki.org/wiki/Order_Type_Multiplication_is_Well-Defined_Operation | https://proofwiki.org/wiki/Order_Type_Multiplication_is_Well-Defined_Operation | [
"Order Types",
"Multiplication",
"Order Products"
] | [
"Definition:Multiplication of Order Types",
"Definition:Order Type",
"Definition:Well-Defined/Operation"
] | [
"Definition:Ordered Set",
"Definition:Ordered Set",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Type",
"Definition:Order Type",
"Order Isomorphism is Preserved by Antilexicographic Order"
] |
proofwiki-18390 | Order Isomorphism is Preserved by Antilexicographic Order | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ be ordered sets such that:
:$\struct {S_1, \preccurlyeq_1}$ is isomorphic to $\struct {T_1, \preccurlyeq_{1'} }$
:$\struct {S_2, \preccurlyeq_2}$ is is... | {{ProofWanted|tedious}} | Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\struct {T_1, \preccurlyeq_{1'} }$ and $\struct {T_2, \preccurlyeq_{2'} }$ be [[Definition:Ordered Set|ordered sets]] such that:
:$\struct {S_1, \preccurlyeq_1}$ is [[Definition:Isomorphic Ordered ... | {{ProofWanted|tedious}} | Order Isomorphism is Preserved by Antilexicographic Order | https://proofwiki.org/wiki/Order_Isomorphism_is_Preserved_by_Antilexicographic_Order | https://proofwiki.org/wiki/Order_Isomorphism_is_Preserved_by_Antilexicographic_Order | [
"Antilexicographic Order",
"Order Isomorphisms"
] | [
"Definition:Ordered Set",
"Definition:Ordered Set",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Order Isomorphism/Isomorphic Sets",
"Definition:Antilexicographic Order",
"Definition:Order Isomorphism/Isomorphic Sets"
] | [] |
proofwiki-18391 | Order Type Addition is Associative | Let $\alpha$, $\beta$ and $\gamma$ be order types of ordered sets.
Then:
:$\paren {\alpha + \beta} + \gamma = \alpha + \paren {\beta + \gamma}$
where $+$ denotes order type addition. | Let $\struct {S_1, \preccurlyeq_1}$, $\struct {S_2, \preccurlyeq_2}$ and $\struct {S_3, \preccurlyeq_3}$ be ordered structures such that:
{{begin-eqn}}
{{eqn | l = \map \ot {S_1, \preccurlyeq_1}
| r = \alpha
}}
{{eqn | l = \map \ot {S_2, \preccurlyeq_2}
| r = \beta
}}
{{eqn | l = \map \ot {S_3, \preccurlyeq... | Let $\alpha$, $\beta$ and $\gamma$ be [[Definition:Order Type|order types]] of [[Definition:Ordered Set|ordered sets]].
Then:
:$\paren {\alpha + \beta} + \gamma = \alpha + \paren {\beta + \gamma}$
where $+$ denotes [[Definition:Addition of Order Types|order type addition]]. | Let $\struct {S_1, \preccurlyeq_1}$, $\struct {S_2, \preccurlyeq_2}$ and $\struct {S_3, \preccurlyeq_3}$ be [[Definition:Ordered Structure|ordered structures]] such that:
{{begin-eqn}}
{{eqn | l = \map \ot {S_1, \preccurlyeq_1}
| r = \alpha
}}
{{eqn | l = \map \ot {S_2, \preccurlyeq_2}
| r = \beta
}}
{{eqn... | Order Type Addition is Associative | https://proofwiki.org/wiki/Order_Type_Addition_is_Associative | https://proofwiki.org/wiki/Order_Type_Addition_is_Associative | [
"Order Sums",
"Examples of Associative Operations"
] | [
"Definition:Order Type",
"Definition:Ordered Set",
"Definition:Addition of Order Types"
] | [
"Definition:Ordered Structure",
"Definition:Order Type",
"Definition:Order Type",
"Definition:Order Sum",
"Definition:Order Isomorphism"
] |
proofwiki-18392 | Lexicographic Product of Family of Ordered Sets is Ordered Set | Let $\struct {I, \preceq}$ be a well-ordered set.
For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be an ordered set.
Let $\ds D = \prod_{i \mathop \in I} S_i$ be the Cartesian product of the family $\family {\struct {S_i, \preccurlyeq_i} }_{i \mathop \in I}$ indexed by $I$.
Let $\preccurlyeq_D$ be the lexicogra... | For a subset $J$ of $I$, let $D_J$ be defined as:
:$\ds D_J = \prod_{i \mathop \in J} S_i$
Similarly, let $\preccurlyeq_J$ be the lexicographic order on $D_J$:
:$\forall u, v \in D_J: u \preccurlyeq_J v \iff \begin {cases} u = v \\ \map u i \preccurlyeq_i \map v i & \text {for the $\preceq$-smallest $i \in J$ such that... | Let $\struct {I, \preceq}$ be a [[Definition:Well-Ordered Set|well-ordered set]].
For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be an [[Definition:Ordered Set|ordered set]].
Let $\ds D = \prod_{i \mathop \in I} S_i$ be the [[Definition:Cartesian Product of Family|Cartesian product]] of the [[Definition:Inde... | For a [[Definition:Subset|subset]] $J$ of $I$, let $D_J$ be defined as:
:$\ds D_J = \prod_{i \mathop \in J} S_i$
Similarly, let $\preccurlyeq_J$ be the [[Definition:Lexicographic Order on Family|lexicographic order]] on $D_J$:
:$\forall u, v \in D_J: u \preccurlyeq_J v \iff \begin {cases} u = v \\ \map u i \preccurl... | Lexicographic Product of Family of Ordered Sets is Ordered Set | https://proofwiki.org/wiki/Lexicographic_Product_of_Family_of_Ordered_Sets_is_Ordered_Set | https://proofwiki.org/wiki/Lexicographic_Product_of_Family_of_Ordered_Sets_is_Ordered_Set | [
"Lexicographic Order"
] | [
"Definition:Well-Ordered Set",
"Definition:Ordered Set",
"Definition:Cartesian Product/Family of Sets",
"Definition:Indexing Set/Family",
"Definition:Indexing Set",
"Definition:Lexicographic Order/Family",
"Definition:Ordered Set"
] | [
"Definition:Subset",
"Definition:Lexicographic Order/Family",
"Definition:Subset",
"Definition:Ordered Set",
"Definition:Smallest Element",
"Definition:Ordered Set",
"Definition:Ordered Set",
"Definition:Lexicographic Order",
"Lexicographic Order is Ordering",
"Definition:Ordered Set",
"Principl... |
proofwiki-18393 | Lexicographic Order of Family of Totally Ordered Sets is Totally Ordered Set | Let $\struct {I, \preceq}$ be a well-ordered set.
For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be a totally ordered set.
Let $\ds D = \prod_{i \mathop \in I} S_i$ be the Cartesian product of the family $\family {\struct {S_i, \preccurlyeq_i} }_{i \mathop \in I}$ indexed by $I$.
Let $\preccurlyeq_D$ be the le... | From Lexicographic Product of Family of Ordered Sets is Ordered Set, $\struct {D, \preccurlyeq_D}$ is an ordered set.
It remains to be shown that $\struct {D, \preccurlyeq_D}$ is totally ordered.
Let $p, q \in D$.
If $p = q$, then $p \preccurlyeq_D q$ by the definition of ordering.
If $p \ne q$, then the set $M = \set ... | Let $\struct {I, \preceq}$ be a [[Definition:Well-Ordered Set|well-ordered set]].
For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $\ds D = \prod_{i \mathop \in I} S_i$ be the [[Definition:Cartesian Product of Family|Cartesian product]] of the [[... | From [[Lexicographic Product of Family of Ordered Sets is Ordered Set]], $\struct {D, \preccurlyeq_D}$ is an [[Definition:Ordered Set|ordered set]].
It remains to be shown that $\struct {D, \preccurlyeq_D}$ is [[Definition:Totally Ordered Set|totally ordered]].
Let $p, q \in D$.
If $p = q$, then $p \preccurlyeq_D q... | Lexicographic Order of Family of Totally Ordered Sets is Totally Ordered Set | https://proofwiki.org/wiki/Lexicographic_Order_of_Family_of_Totally_Ordered_Sets_is_Totally_Ordered_Set | https://proofwiki.org/wiki/Lexicographic_Order_of_Family_of_Totally_Ordered_Sets_is_Totally_Ordered_Set | [
"Lexicographic Order",
"Total Orderings"
] | [
"Definition:Well-Ordered Set",
"Definition:Totally Ordered Set",
"Definition:Cartesian Product/Family of Sets",
"Definition:Indexing Set/Family",
"Definition:Indexing Set",
"Definition:Lexicographic Order/Family",
"Definition:Totally Ordered Set"
] | [
"Lexicographic Product of Family of Ordered Sets is Ordered Set",
"Definition:Ordered Set",
"Definition:Totally Ordered Set",
"Definition:Ordering",
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Well-Ordered Set",
"Definition:Smallest Element",
"Definition:Total Ordering",
"Definition:... |
proofwiki-18394 | Lexicographic Order of Family of Well-Ordered Sets is not necessarily Well-Ordered | Let $\struct {I, \preceq}$ be a well-ordered set.
For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be a well-ordered set.
Let $\ds D = \prod_{i \mathop \in I} S_i$ be the Cartesian product of the family $\family {\struct {S_i, \preccurlyeq_i} }_{i \mathop \in I}$ indexed by $I$.
Let $\preccurlyeq_D$ be the lexic... | Proof by Counterexample:
Let $\struct {I, \preceq} = \struct {\N, \le}$.
Let $\struct {S_i, \preccurlyeq_i} = \struct {\N, \le}$ for all $i \in \N$.
Let $n \in \N$.
Let $\ds u_n \in \prod_{i \mathop \in \N} \N$ be the mapping defined as:
:$\map {u_n} i = \begin {cases} 0 & : i \le n \\ 1 & : i > n \end {cases}$
It is t... | Let $\struct {I, \preceq}$ be a [[Definition:Well-Ordered Set|well-ordered set]].
For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be a [[Definition:Well-Ordered Set|well-ordered set]].
Let $\ds D = \prod_{i \mathop \in I} S_i$ be the [[Definition:Cartesian Product of Family|Cartesian product]] of the [[Defini... | [[Proof by Counterexample]]:
Let $\struct {I, \preceq} = \struct {\N, \le}$.
Let $\struct {S_i, \preccurlyeq_i} = \struct {\N, \le}$ for all $i \in \N$.
Let $n \in \N$.
Let $\ds u_n \in \prod_{i \mathop \in \N} \N$ be the [[Definition:Mapping|mapping]] defined as:
:$\map {u_n} i = \begin {cases} 0 & : i \le n \\ ... | Lexicographic Order of Family of Well-Ordered Sets is not necessarily Well-Ordered | https://proofwiki.org/wiki/Lexicographic_Order_of_Family_of_Well-Ordered_Sets_is_not_necessarily_Well-Ordered | https://proofwiki.org/wiki/Lexicographic_Order_of_Family_of_Well-Ordered_Sets_is_not_necessarily_Well-Ordered | [
"Lexicographic Order",
"Well-Orderings"
] | [
"Definition:Well-Ordered Set",
"Definition:Well-Ordered Set",
"Definition:Cartesian Product/Family of Sets",
"Definition:Indexing Set/Family",
"Definition:Indexing Set",
"Definition:Lexicographic Order/Family",
"Definition:Well-Ordered Set"
] | [
"Proof by Counterexample",
"Definition:Mapping",
"Definition:Sequence/Infinite Sequence",
"Definition:Decreasing/Sequence",
"Definition:Sequence",
"Definition:Element",
"Definition:Lexicographic Order/Family",
"Definition:Term of Sequence",
"Definition:Non-Empty Set",
"Definition:Subset",
"Defin... |
proofwiki-18395 | Ordering is Preordering | Let $S$ be a set.
Let $\RR$ be an ordering on $S$.
Then $\RR$ is also a preordering on $S$. | By definition of ordering:
{{:Definition:Ordering/Definition 1}}
By definition of preordering:
{{:Definition:Preordering/Definition 1}}
Thus an ordering is a preordering which is antisymmetric.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\RR$ be an [[Definition:Ordering|ordering]] on $S$.
Then $\RR$ is also a [[Definition:Preordering|preordering]] on $S$. | By definition of [[Definition:Ordering/Definition 1|ordering]]:
{{:Definition:Ordering/Definition 1}}
By definition of [[Definition:Preordering/Definition 1|preordering]]:
{{:Definition:Preordering/Definition 1}}
Thus an [[Definition:Ordering|ordering]] is a [[Definition:Preordering|preordering]] which is [[Definitio... | Ordering is Preordering | https://proofwiki.org/wiki/Ordering_is_Preordering | https://proofwiki.org/wiki/Ordering_is_Preordering | [
"Preorderings",
"Orderings"
] | [
"Definition:Set",
"Definition:Ordering",
"Definition:Preordering"
] | [
"Definition:Ordering/Definition 1",
"Definition:Preordering/Definition 1",
"Definition:Ordering",
"Definition:Preordering",
"Definition:Antisymmetric Relation"
] |
proofwiki-18396 | Preordering is not necessarily Ordering | Let $S$ be a set.
Let $\RR$ be a preordering on $S$.
Then it is not necessarily the case that $\RR$ is also an ordering on $S$. | Consider the relation $\RR$ on the powerset of the natural numbers:
:$\forall a, b \in \powerset \N: a \mathrel \RR b \iff a \setminus b \text { is finite}$
where $\setminus$ denotes set difference.
It is demonstrated in Preordering Example: Finite Set Difference on Natural Numbers that;
:$\RR$ is a preordering on $\po... | Let $S$ be a [[Definition:Set|set]].
Let $\RR$ be a [[Definition:Preordering|preordering]] on $S$.
Then it is not necessarily the case that $\RR$ is also an [[Definition:Ordering|ordering]] on $S$. | Consider the [[Definition:Relation|relation]] $\RR$ on the [[Definition:Powerset|powerset]] of the [[Definition:Natural Numbers|natural numbers]]:
:$\forall a, b \in \powerset \N: a \mathrel \RR b \iff a \setminus b \text { is finite}$
where $\setminus$ denotes [[Definition:Set Difference|set difference]].
It is dem... | Preordering is not necessarily Ordering | https://proofwiki.org/wiki/Preordering_is_not_necessarily_Ordering | https://proofwiki.org/wiki/Preordering_is_not_necessarily_Ordering | [
"Preorderings",
"Orderings"
] | [
"Definition:Set",
"Definition:Preordering",
"Definition:Ordering"
] | [
"Definition:Relation",
"Definition:Power Set",
"Definition:Natural Numbers",
"Definition:Set Difference",
"Preordering/Examples/Finite Set Difference on Natural Numbers",
"Definition:Preordering",
"Definition:Ordering"
] |
proofwiki-18397 | Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain | Let $\tuple {X, d}$ be a metric space.
Let $\tuple {Y, d'}$ be a complete metric space.
Let $A \subseteq X$.
Let $f : A \to Y$ be a uniformly continuous function.
Then there exists a unique continuous function $g : A^- \to Y$ such that:
:$\map g a = \map f a$
for all $a \in A$, where $A^-$ denotes the topological clos... | === Existence ===
Note that if $A$ is closed, then from Set is Closed iff Equals Topological Closure, we have:
:$A^- = A$
So taking $g = f$ suffices in this case.
Suppose now that $A$ is not closed. | Let $\tuple {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $\tuple {Y, d'}$ be a [[Definition:Complete Metric Space|complete metric space]].
Let $A \subseteq X$.
Let $f : A \to Y$ be a [[Definition:Uniformly Continuous Mapping (Metric Spaces)|uniformly continuous function]].
Then there exists a unique... | === Existence ===
Note that if $A$ is [[Definition:Closed Set of Metric Space|closed]], then from [[Set is Closed iff Equals Topological Closure]], we have:
:$A^- = A$
So taking $g = f$ suffices in this case.
Suppose now that $A$ is not [[Definition:Closed Set of Metric Space|closed]]. | Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain | https://proofwiki.org/wiki/Uniformly_Continuous_Function_to_Complete_Metric_Space_has_Unique_Continuous_Extension_to_Closure_of_Domain | https://proofwiki.org/wiki/Uniformly_Continuous_Function_to_Complete_Metric_Space_has_Unique_Continuous_Extension_to_Closure_of_Domain | [
"Uniformly Continuous Functions",
"Complete Metric Spaces",
"Set Closures",
"Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain"
] | [
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Uniform Continuity/Metric Space",
"Definition:Continuous Function",
"Definition:Closure (Topology)",
"Definition:Uniform Continuity/Metric Space"
] | [
"Definition:Closed Set/Metric Space",
"Set is Closed iff Equals Topological Closure",
"Definition:Closed Set/Metric Space"
] |
proofwiki-18398 | Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain/Lemma 1 | Let $\sequence {a_n}$ be a sequence in $A$ convergent to $a \in A^-$.
Then $\sequence {\map f {a_n} }$ converges. | From Set is Closed iff Equals Topological Closure, $A^- \setminus A$ is non-empty.
Let $a \in A^- \setminus A$.
From Point in Closure of Subset of Metric Space iff Limit of Sequence:
:there exists a sequence $\sequence {a_n}$ in $A$ converging to $a$.
Consider now the sequence $\sequence {\map f {a_n} }$ in $Y$.
Note... | Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence]] in $A$ [[Definition:Convergent Sequence (Metric Space)|convergent]] to $a \in A^-$.
Then $\sequence {\map f {a_n} }$ [[Definition:Convergent Sequence (Metric Space)|converges]]. | From [[Set is Closed iff Equals Topological Closure]], $A^- \setminus A$ is [[Definition:Non-Empty Set|non-empty]].
Let $a \in A^- \setminus A$.
From [[Point in Closure of Subset of Metric Space iff Limit of Sequence]]:
:there exists a [[Definition:Sequence|sequence]] $\sequence {a_n}$ in $A$ [[Definition:Converge... | Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain/Lemma 1 | https://proofwiki.org/wiki/Uniformly_Continuous_Function_to_Complete_Metric_Space_has_Unique_Continuous_Extension_to_Closure_of_Domain/Lemma_1 | https://proofwiki.org/wiki/Uniformly_Continuous_Function_to_Complete_Metric_Space_has_Unique_Continuous_Extension_to_Closure_of_Domain/Lemma_1 | [
"Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Convergent Sequence/Metric Space"
] | [
"Set is Closed iff Equals Topological Closure",
"Definition:Non-Empty Set",
"Point in Closure of Subset of Metric Space iff Limit of Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence",
"Definition:Sequence",
"Definition:Complete Metric Space",
"Definition:Cauchy Sequence",
"Definitio... |
proofwiki-18399 | Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain/Lemma 2 | Let $\sequence {a_n}$ be a convergent sequence in $A$.
Then the limit of $\sequence {\map f {a_n} }$ is dependent only on the limit of $\sequence {a_n}$.
That is, there exists a function $L : A^- \to Y$ such that:
:$\ds \lim_{n \mathop \to \infty} \map f {a_n} = \map L {\lim_{n \mathop \to \infty} a_n}$
for every co... | Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $A$ such that $a_n \to a$ and $b_n \to a$, with:
:$\map f {a_n} \to L_1$
and:
:$\map f {b_n} \to L_2$
We have, by the Triangle Inequality:
:$\map {d'} {\map f {a_n}, L_2} \le \map {d'} {\map f {a_n}, \map f {b_n} } + \map {d'} {\map f {b_n}, L_2}$
We want to ... | Let $\sequence {a_n}$ be a [[Definition:Convergent Sequence (Metric Space)|convergent sequence]] in $A$.
Then the [[Definition:Limit of Sequence in Metric Space|limit]] of $\sequence {\map f {a_n} }$ is dependent only on the [[Definition:Limit of Sequence in Metric Space|limit]] of $\sequence {a_n}$.
That is, ther... | Let $\sequence {a_n}$ and $\sequence {b_n}$ be [[Definition:Sequence|sequences]] in $A$ such that $a_n \to a$ and $b_n \to a$, with:
:$\map f {a_n} \to L_1$
and:
:$\map f {b_n} \to L_2$
We have, by the [[Triangle Inequality]]:
:$\map {d'} {\map f {a_n}, L_2} \le \map {d'} {\map f {a_n}, \map f {b_n} } + \map {d'}... | Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain/Lemma 2 | https://proofwiki.org/wiki/Uniformly_Continuous_Function_to_Complete_Metric_Space_has_Unique_Continuous_Extension_to_Closure_of_Domain/Lemma_2 | https://proofwiki.org/wiki/Uniformly_Continuous_Function_to_Complete_Metric_Space_has_Unique_Continuous_Extension_to_Closure_of_Domain/Lemma_2 | [
"Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain"
] | [
"Definition:Convergent Sequence/Metric Space",
"Definition:Limit of Sequence/Metric Space",
"Definition:Limit of Sequence/Metric Space",
"Definition:Function",
"Definition:Convergent Sequence/Metric Space"
] | [
"Definition:Sequence",
"Triangle Inequality",
"Definition:Uniform Continuity/Metric Space",
"Triangle Inequality",
"Convergent Sequence in Metric Space has Unique Limit",
"Definition:Sequence",
"Definition:Convergent Sequence/Metric Space",
"Definition:Limit of Sequence/Metric Space",
"Category:Unif... |
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