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-19300 | Power Set with Union and Superset Relation is Ordered Semigroup | Let $S$ be a set and let $\powerset S$ be its power set.
Let $\struct {\powerset S, \cup, \supseteq}$ be the ordered structure formed from the set union operation and superset relation.
Then $\struct {\powerset S, \cup, \supseteq}$ is an ordered semigroup. | From Power Set with Union is Commutative Monoid, $\struct {\powerset S, \cup}$ is {{afortiori}} a semigroup.
From Subset Relation is Ordering, $\struct {\powerset S, \subseteq}$ is an ordered set.
We have that $\supseteq$ is the dual to $\subseteq$.
Hence $\struct {\powerset S, \supseteq}$ is an ordered set.
It remains... | Let $S$ be a [[Definition:Set|set]] and let $\powerset S$ be its [[Definition:Power Set|power set]].
Let $\struct {\powerset S, \cup, \supseteq}$ be the [[Definition:Ordered Structure|ordered structure]] formed from the [[Definition:Set Union|set union operation]] and [[Definition:Superset|superset relation]].
Then ... | From [[Power Set with Union is Commutative Monoid]], $\struct {\powerset S, \cup}$ is {{afortiori}} a [[Definition:Semigroup|semigroup]].
From [[Subset Relation is Ordering]], $\struct {\powerset S, \subseteq}$ is an [[Definition:Ordered Set|ordered set]].
We have that $\supseteq$ is the [[Definition:Dual Ordering|du... | Power Set with Union and Superset Relation is Ordered Semigroup | https://proofwiki.org/wiki/Power_Set_with_Union_and_Superset_Relation_is_Ordered_Semigroup | https://proofwiki.org/wiki/Power_Set_with_Union_and_Superset_Relation_is_Ordered_Semigroup | [
"Examples of Ordered Semigroups",
"Power Set",
"Set Union",
"Subsets"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Ordered Structure",
"Definition:Set Union",
"Definition:Subset/Superset",
"Definition:Ordered Semigroup"
] | [
"Power Set with Union is Commutative Monoid",
"Definition:Semigroup",
"Subset Relation is Ordering",
"Definition:Ordered Set",
"Definition:Dual Ordering",
"Definition:Ordered Set",
"Definition:Relation Compatible with Operation",
"Set Union Preserves Subsets/Corollary",
"Union is Commutative"
] |
proofwiki-19301 | Hensel's Lemma/P-adic Integers/Lemma 4 | :$\tuple{d_0} \in S_1$ | Let the $p$-adic expansion for $\alpha_0$ be:
:$\ds \alpha_0 = \sum_{n \mathop = 0}^\infty d_n p^n$
We have:
{{begin-eqn}}
{{eqn | l = \alpha_0 - d_0
| r = \paren{\sum_{n \mathop = 0}^\infty d_n p^n} - d_0
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty d_n p^n
| c = Subtract first term from the sum
}}
{{eqn | ... | :$\tuple{d_0} \in S_1$ | Let the [[Definition:P-adic Expansion|$p$-adic expansion]] for $\alpha_0$ be:
:$\ds \alpha_0 = \sum_{n \mathop = 0}^\infty d_n p^n$
We have:
{{begin-eqn}}
{{eqn | l = \alpha_0 - d_0
| r = \paren{\sum_{n \mathop = 0}^\infty d_n p^n} - d_0
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty d_n p^n
| c = Subtract fi... | Hensel's Lemma/P-adic Integers/Lemma 4 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_4 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_4 | [
"Hensel's Lemma"
] | [] | [
"Definition:P-adic Expansion",
"Polynomials of Congruent Ring Elements are Congruent",
"Category:Hensel's Lemma"
] |
proofwiki-19302 | Hensel's Lemma/P-adic Integers/Lemma 5 | Let:
:$\tuple{b_0, b_1, \ldots, b_{k-1}} \in S_k$.
Then there exists a unique $p$-adic digit $\map b {b_0, b_1, \ldots, b_{k-1}}$:
:$\tuple{b_0, b_1, \ldots, b_{k-1}, \map b {b_0, b_1, \ldots, b_{k-1}}} \in S_{k+1}$ | === Lemma 6 ===
{{:Hensel's Lemma/P-adic Integers/Lemma 6}}{{qed|lemma}} | Let:
:$\tuple{b_0, b_1, \ldots, b_{k-1}} \in S_k$.
Then there exists a [[Definition:Unique|unique]] [[Definition:P-adic Digit|$p$-adic digit]] $\map b {b_0, b_1, \ldots, b_{k-1}}$:
:$\tuple{b_0, b_1, \ldots, b_{k-1}, \map b {b_0, b_1, \ldots, b_{k-1}}} \in S_{k+1}$ | === [[Hensel's Lemma/P-adic Integers/Lemma 6|Lemma 6]] ===
{{:Hensel's Lemma/P-adic Integers/Lemma 6}}{{qed|lemma}} | Hensel's Lemma/P-adic Integers/Lemma 5 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_5 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_5 | [
"Hensel's Lemma"
] | [
"Definition:Unique",
"Definition:P-adic Digit"
] | [
"Hensel's Lemma/P-adic Integers/Lemma 6"
] |
proofwiki-19303 | Hensel's Lemma/P-adic Integers/Lemma 2 | :$\alpha \equiv \alpha_0 \pmod {p\Z_p}$ | From Ideals of $p$-adic Integers:
:$p \Z_p$ is an ideal of the ring $\Z_p$
By definition of congruence modulo ideal:
:$\forall k \in \N: a_k - \alpha_0 \in p \Z_p$
By definition of $p$-adic expansion:
:$\alpha = \ds \lim_{k \mathop \to \infty} a_k$
From Sum Rule for Sequences in Normed Division Ring:
:$\alpha - \alpha_... | :$\alpha \equiv \alpha_0 \pmod {p\Z_p}$ | From [[Ideals of P-adic Integers|Ideals of $p$-adic Integers]]:
:$p \Z_p$ is an [[Definition:Ideal of Ring|ideal]] of the [[Definition:Ring|ring]] $\Z_p$
By definition of [[Definition:Congruence Modulo Ideal|congruence modulo ideal]]:
:$\forall k \in \N: a_k - \alpha_0 \in p \Z_p$
By definition of [[Definition:P-adic... | Hensel's Lemma/P-adic Integers/Lemma 2 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_2 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_2 | [
"Hensel's Lemma"
] | [] | [
"Ideals of P-adic Integers",
"Definition:Ideal of Ring",
"Definition:Ring",
"Definition:Congruence Modulo Ideal",
"Definition:P-adic Expansion",
"Combination Theorem for Sequences/Normed Division Ring/Sum Rule",
"Closed Subgroups of P-adic Integers",
"Definition:Closed Set/Metric Space",
"Definition... |
proofwiki-19304 | Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup | Let $\struct {S, \circ}$ be a semigroup.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Let $\struct {\powerset S, \circ_\PP, \subseteq}$ be the ordered structure formed from $\struct {\powerset S, \circ_\PP}$ and the subset relation.
Then $\struct {\powerset S, \circ_\PP, \subse... | From Power Structure of Semigroup is Semigroup, $\struct {\powerset S, \circ_\PP}$ is a semigroup.
From Subset Relation is Ordering, $\struct {\powerset S, \subseteq}$ is an ordered set.
It remains to be shown that $\subseteq$ is compatible with $\circ_\PP$.
This is demonstrated directly in Subset Relation is Compatibl... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $\struct {\powerset S, \circ_\PP}$ be the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$.
Let $\struct {\powerset S, \circ_\PP, \subseteq}$ be the [[Definition:Ordered Structure|ordered structure]] formed from $\struct {\po... | From [[Power Structure of Semigroup is Semigroup]], $\struct {\powerset S, \circ_\PP}$ is a [[Definition:Semigroup|semigroup]].
From [[Subset Relation is Ordering]], $\struct {\powerset S, \subseteq}$ is an [[Definition:Ordered Set|ordered set]].
It remains to be shown that $\subseteq$ is [[Definition:Relation Compa... | Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup | https://proofwiki.org/wiki/Power_Structure_of_Semigroup_Ordered_by_Subsets_is_Ordered_Semigroup | https://proofwiki.org/wiki/Power_Structure_of_Semigroup_Ordered_by_Subsets_is_Ordered_Semigroup | [
"Examples of Ordered Semigroups",
"Power Structures"
] | [
"Definition:Semigroup",
"Definition:Power Structure",
"Definition:Ordered Structure",
"Definition:Subset",
"Definition:Ordered Semigroup"
] | [
"Power Structure of Semigroup is Semigroup",
"Definition:Semigroup",
"Subset Relation is Ordering",
"Definition:Ordered Set",
"Definition:Relation Compatible with Operation",
"Subset Relation is Compatible with Subset Product"
] |
proofwiki-19305 | Power Structure of Semigroup Ordered by Supersets is Ordered Semigroup | Let $\struct {S, \circ}$ be a semigroup.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Let $\struct {\powerset S, \circ_\PP, \supseteq}$ be the ordered structure formed from $\struct {\powerset S, \circ_\PP}$ and the superset relation.
Then $\struct {\powerset S, \circ_\PP, \sup... | From Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup, $\struct {\powerset S, \circ_\PP, \subseteq}$ is an ordered semigroup.
The result then follows from Dual of Ordered Semigroup is Ordered Semigroup.
{{qed}} | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $\struct {\powerset S, \circ_\PP}$ be the [[Definition:Power Structure|power structure]] of $\struct {S, \circ}$.
Let $\struct {\powerset S, \circ_\PP, \supseteq}$ be the [[Definition:Ordered Structure|ordered structure]] formed from $\struct {\po... | From [[Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup]], $\struct {\powerset S, \circ_\PP, \subseteq}$ is an [[Definition:Ordered Semigroup|ordered semigroup]].
The result then follows from [[Dual of Ordered Semigroup is Ordered Semigroup]].
{{qed}} | Power Structure of Semigroup Ordered by Supersets is Ordered Semigroup | https://proofwiki.org/wiki/Power_Structure_of_Semigroup_Ordered_by_Supersets_is_Ordered_Semigroup | https://proofwiki.org/wiki/Power_Structure_of_Semigroup_Ordered_by_Supersets_is_Ordered_Semigroup | [
"Examples of Ordered Semigroups",
"Power Structures"
] | [
"Definition:Semigroup",
"Definition:Power Structure",
"Definition:Ordered Structure",
"Definition:Subset/Superset",
"Definition:Ordered Semigroup"
] | [
"Power Structure of Semigroup Ordered by Subsets is Ordered Semigroup",
"Definition:Ordered Semigroup",
"Dual of Ordered Semigroup is Ordered Semigroup"
] |
proofwiki-19306 | Complement Operation between Union and Intersection Power Structures is Isomorphism | Let $S$ be a set and let $\powerset S$ be its power set.
Let $\struct {\powerset S, \cap, \subseteq}$ be the ordered semigroup formed from the set intersection operation and subset relation.
Let $\struct {\powerset S, \cap, \supseteq}$ be the ordered semigroup formed from the set intersection operation and superset rel... | From:
:Power Set with Intersection and Subset Relation is Ordered Semigroup
:Power Set with Intersection and Superset Relation is Ordered Semigroup
:Power Set with Union and Subset Relation is Ordered Semigroup
:Power Set with Union and Superset Relation is Ordered Semigroup
each of the given ordered structures is inde... | Let $S$ be a [[Definition:Set|set]] and let $\powerset S$ be its [[Definition:Power Set|power set]].
Let $\struct {\powerset S, \cap, \subseteq}$ be the [[Definition:Ordered Semigroup|ordered semigroup]] formed from the [[Definition:Set Intersection|set intersection operation]] and [[Definition:Subset|subset relation]... | From:
:[[Power Set with Intersection and Subset Relation is Ordered Semigroup]]
:[[Power Set with Intersection and Superset Relation is Ordered Semigroup]]
:[[Power Set with Union and Subset Relation is Ordered Semigroup]]
:[[Power Set with Union and Superset Relation is Ordered Semigroup]]
each of the given [[Definit... | Complement Operation between Union and Intersection Power Structures is Isomorphism | https://proofwiki.org/wiki/Complement_Operation_between_Union_and_Intersection_Power_Structures_is_Isomorphism | https://proofwiki.org/wiki/Complement_Operation_between_Union_and_Intersection_Power_Structures_is_Isomorphism | [
"Ordered Semigroup Isomorphisms",
"Power Structures"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Ordered Semigroup",
"Definition:Set Intersection",
"Definition:Subset",
"Definition:Ordered Semigroup",
"Definition:Set Intersection",
"Definition:Subset/Superset",
"Definition:Ordered Semigroup",
"Definition:Set Union",
"Definition:Subset",
... | [
"Power Set with Intersection and Subset Relation is Ordered Semigroup",
"Power Set with Intersection and Superset Relation is Ordered Semigroup",
"Power Set with Union and Subset Relation is Ordered Semigroup",
"Power Set with Union and Superset Relation is Ordered Semigroup",
"Definition:Ordered Structure"... |
proofwiki-19307 | Non-Zero Natural Numbers under Multiplication with Divisibility forms Ordered Semigroup | Let $\N_{>0}$ be the set of natural numbers without zero, that is, $\N_{>0} = \N \setminus \set 0$.
Let $\divides$ denote the divisibility relation on $\N_{>0}$:
:$\forall a, b \in \N_{>0}: a \divides b \iff \exists k \in \Z: k \times a = b$
where $\times$ denotes conventional integer multiplication.
The ordered struct... | First we note that from Non-Zero Natural Numbers under Multiplication form Commutative Semigroup, $\struct {\N_{>0}, \times}$ is a semigroup.
From Divisor Relation on Positive Integers is Partial Ordering, $\struct {\N_{>0}, \divides}$ is an ordered set.
It remains to be shown that $\divides$ is compatible with $\times... | Let $\N_{>0}$ be the set of [[Definition:Natural Numbers|natural numbers]] without [[Definition:Zero (Number)|zero]], that is, $\N_{>0} = \N \setminus \set 0$.
Let $\divides$ denote the [[Definition:Divisor of Integer|divisibility relation]] on $\N_{>0}$:
:$\forall a, b \in \N_{>0}: a \divides b \iff \exists k \in \Z:... | First we note that from [[Non-Zero Natural Numbers under Multiplication form Commutative Semigroup]], $\struct {\N_{>0}, \times}$ is a [[Definition:Semigroup|semigroup]].
From [[Divisor Relation on Positive Integers is Partial Ordering]], $\struct {\N_{>0}, \divides}$ is an [[Definition:Ordered Set|ordered set]].
It ... | Non-Zero Natural Numbers under Multiplication with Divisibility forms Ordered Semigroup | https://proofwiki.org/wiki/Non-Zero_Natural_Numbers_under_Multiplication_with_Divisibility_forms_Ordered_Semigroup | https://proofwiki.org/wiki/Non-Zero_Natural_Numbers_under_Multiplication_with_Divisibility_forms_Ordered_Semigroup | [
"Examples of Ordered Semigroups",
"Natural Number Multiplication",
"Divisibility"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Divisor (Algebra)/Integer",
"Definition:Multiplication/Integers",
"Definition:Ordered Structure",
"Definition:Ordered Semigroup"
] | [
"Non-Zero Natural Numbers under Multiplication form Commutative Semigroup",
"Definition:Semigroup",
"Divisor Relation on Positive Integers is Partial Ordering",
"Definition:Ordered Set",
"Definition:Relation Compatible with Operation"
] |
proofwiki-19308 | Suprema in Ordered Group | Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group.
Let $x, y, z \in G$ be arbitrary.
Let any one of the sets $\set {x, y}$, $\set {x \circ z, y \circ z}$ or $\set {z \circ x, z \circ y}$ admit a supremum.
Then all three sets admit a supremum, and:
{{begin-eqn}}
{{eqn | l = \sup \set {x \circ z, y \circ z}
... | First we recall that by definition of ordered group, $\preccurlyeq$ is compatible with $\circ$:
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = x \preccurlyeq y
| o = \implies
| r = x \circ z \preccurlyeq y \circ z
}}
{{eqn | ll= \land
| l = x \preccurlyeq y
| o = \implies
| r ... | Let $\struct {G, \circ, \preccurlyeq}$ be an [[Definition:Ordered Group|ordered group]].
Let $x, y, z \in G$ be arbitrary.
Let any one of the [[Definition:Set|sets]] $\set {x, y}$, $\set {x \circ z, y \circ z}$ or $\set {z \circ x, z \circ y}$ admit a [[Definition:Supremum of Set|supremum]].
Then all three [[Defini... | First we recall that by definition of [[Definition:Ordered Group|ordered group]], $\preccurlyeq$ is [[Definition:Relation Compatible with Operation|compatible]] with $\circ$:
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = x \preccurlyeq y
| o = \implies
| r = x \circ z \preccurlyeq y \circ z
}... | Suprema in Ordered Group | https://proofwiki.org/wiki/Suprema_in_Ordered_Group | https://proofwiki.org/wiki/Suprema_in_Ordered_Group | [
"Ordered Groups",
"Suprema"
] | [
"Definition:Ordered Group",
"Definition:Set",
"Definition:Supremum of Set",
"Definition:Set",
"Definition:Supremum of Set"
] | [
"Definition:Ordered Group",
"Definition:Relation Compatible with Operation",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Group",
"Definiti... |
proofwiki-19309 | Raw Moment of Weibull Distribution | Let $X$ be a continuous random variable with the Weibull distribution with $\alpha, \beta \in \R_{> 0}$.
Let $n$ be a strictly positive integer.
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = \beta^n \map \Gamma {1 + \dfrac n \alpha}$
where $\Gamma$ is the Gamma function. | From the definition of the Weibull distribution, $X$ has probability density function:
:$\map {f_X} x = \alpha \beta^{-\alpha} x^{\alpha - 1} e^{-\paren {\frac x \beta}^\alpha}$
where $\Img X = \R_{\ge 0}$.
From the definition of the expected value of a continuous random variable:
:$\ds \expect {X^n} = \int_0^\infty x... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Weibull Distribution|Weibull distribution]] with $\alpha, \beta \in \R_{> 0}$.
Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Then the $n$th [[Definition:Raw Moment|raw moment]] ... | From the definition of the [[Definition:Weibull Distribution|Weibull distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \alpha \beta^{-\alpha} x^{\alpha - 1} e^{-\paren {\frac x \beta}^\alpha}$
where $\Img X = \R_{\ge 0}$.
From the definition of the [[... | Raw Moment of Weibull Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Weibull_Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Weibull_Distribution | [
"Weibull Distribution",
"Raw Moments"
] | [
"Definition:Random Variable/Continuous",
"Definition:Weibull Distribution",
"Definition:Strictly Positive/Integer",
"Definition:Raw Moment",
"Definition:Gamma Function"
] | [
"Definition:Weibull Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Derivative of Composite Function",
"Linear Combination of Integrals/Definite",
"Category:Weibull Distribution",
"Category:Raw Moments"
] |
proofwiki-19310 | Expectation of Weibull Distribution | Let $X$ be a continuous random variable with the Weibull distribution with $\alpha, \beta \in \R_{> 0}$.
The expectation of $X$ is given by:
:$\expect X = \beta \, \map \Gamma {1 + \dfrac 1 \alpha}$
where $\Gamma$ is the Gamma function. | From Raw Moment of Weibull Distribution, we have:
The $n$th raw moment $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = \beta^n \map \Gamma {1 + \dfrac n \alpha}$
Therefore, for $n = 1$ we have:
:$\expect X = \beta^1 \map \Gamma {1 + \dfrac 1 \alpha}$
Hence the result.
{{qed}}
Category:Weibull Distribution
Catego... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Weibull Distribution|Weibull distribution]] with $\alpha, \beta \in \R_{> 0}$.
The [[Definition:Expectation of Continuous Random Variable|expectation]] of $X$ is given by:
:$\expect X = \beta \, \map \Gamma {1 + \d... | From [[Raw Moment of Weibull Distribution]], we have:
The $n$th [[Definition:Raw Moment|raw moment]] $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = \beta^n \map \Gamma {1 + \dfrac n \alpha}$
Therefore, for $n = 1$ we have:
:$\expect X = \beta^1 \map \Gamma {1 + \dfrac 1 \alpha}$
Hence the result.
{{qed}}
... | Expectation of Weibull Distribution | https://proofwiki.org/wiki/Expectation_of_Weibull_Distribution | https://proofwiki.org/wiki/Expectation_of_Weibull_Distribution | [
"Weibull Distribution",
"Expectation"
] | [
"Definition:Random Variable/Continuous",
"Definition:Weibull Distribution",
"Definition:Expectation/Continuous",
"Definition:Gamma Function"
] | [
"Raw Moment of Weibull Distribution",
"Definition:Raw Moment",
"Category:Weibull Distribution",
"Category:Expectation"
] |
proofwiki-19311 | Variance of Weibull Distribution | Let $X$ be a continuous random variable with the Weibull distribution with $\alpha, \beta \in \R_{> 0}$.
Then the variance of $X$ is given by:
:$\var X = \beta^2 \paren {\map \Gamma {1 + \dfrac 2 \alpha} - \paren {\map \Gamma {1 + \dfrac 1 \alpha} }^2}$
where $\Gamma$ is the Gamma function. | By Variance as Expectation of Square minus Square of Expectation, we have:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By Expectation of Weibull Distribution, we have:
:$\expect X = \beta \, \map \Gamma {1 + \dfrac 1 \alpha}$
From Raw Moment of Weibull Distribution, we have:
The $n$th raw moment $\expect {X^n}$ ... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Weibull Distribution|Weibull distribution]] with $\alpha, \beta \in \R_{> 0}$.
Then the [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \beta^2 \paren {\map \Gamma {1 + \dfrac 2 \alpha} - \paren {\... | By [[Variance as Expectation of Square minus Square of Expectation]], we have:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By [[Expectation of Weibull Distribution]], we have:
:$\expect X = \beta \, \map \Gamma {1 + \dfrac 1 \alpha}$
From [[Raw Moment of Weibull Distribution]], we have:
The $n$th [[Definiti... | Variance of Weibull Distribution | https://proofwiki.org/wiki/Variance_of_Weibull_Distribution | https://proofwiki.org/wiki/Variance_of_Weibull_Distribution | [
"Variance",
"Weibull Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Weibull Distribution",
"Definition:Variance",
"Definition:Gamma Function"
] | [
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Weibull Distribution",
"Raw Moment of Weibull Distribution",
"Definition:Raw Moment",
"Category:Variance",
"Category:Weibull Distribution"
] |
proofwiki-19312 | Skewness of Weibull Distribution | Let $X$ be a continuous random variable with the Weibull distribution with $\alpha, \beta \in \R_{> 0}$.
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac {\map \Gamma {1 + \dfrac 3 \alpha} - 3 \map \Gamma {1 + \dfrac 1 \alpha} \map \Gamma {1 + \dfrac 2 \alpha} + 2 \map \Gamma {1 + \dfrac 1 \alpha}^... | From Skewness in terms of Non-Central Moments, we have:
:$\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 Weibull Distribution we have:
:$\mu = \beta \, \map \Gamma {1 + \dfrac 1 \alpha}$
B... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Weibull Distribution|Weibull distribution]] with $\alpha, \beta \in \R_{> 0}$.
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac {\map \Gamma {1 + \dfrac 3 \alpha} - 3 \ma... | From [[Skewness in terms of Non-Central Moments]], we have:
:$\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 Weibull Distrib... | Skewness of Weibull Distribution | https://proofwiki.org/wiki/Skewness_of_Weibull_Distribution | https://proofwiki.org/wiki/Skewness_of_Weibull_Distribution | [
"Skewness",
"Weibull Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Weibull Distribution",
"Definition:Skewness",
"Definition:Gamma Function"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Weibull Distribution",
"Variance of Weibull Distribution",
"Raw Moment of Weibull Distribution",
"Category:Skewness",
"Category:Weibull Distribution"
] |
proofwiki-19313 | Excess Kurtosis of Weibull Distribution | Let $X$ be a continuous random variable with the Weibull distribution with $\alpha, \beta \in \R_{> 0}$.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac {\map \Gamma {1 + \dfrac 4 \alpha} - 4 \map \Gamma {1 + \dfrac 1 \alpha} \map \Gamma {1 + \dfrac 3 \alpha} + 12 \paren {\map \Gamma {1 + ... | From Kurtosis in terms of Non-Central Moments, we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Weibull Distribution we have:
:$\mu = \beta \, \m... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Weibull Distribution|Weibull distribution]] with $\alpha, \beta \in \R_{> 0}$.
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac {\map \Gamma {1 + \dfrac 4 ... | From [[Kurtosis in terms of Non-Central Moments]], we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
... | Excess Kurtosis of Weibull Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Weibull_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Weibull_Distribution | [
"Kurtosis",
"Weibull Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Weibull Distribution",
"Definition:Excess Kurtosis",
"Definition:Gamma Function"
] | [
"Kurtosis in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Weibull Distribution",
"Variance of Weibull Distribution",
"Raw Moment of Weibull Distribution",
"Square of Sum",
"Category:Kurtosis",
"Category:Weibull Distribution"
] |
proofwiki-19314 | Infima in Ordered Group | Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group.
Let $x, y, z \in G$ be arbitrary.
Let any one of the sets $\set {x, y}$, $\set {x \circ z, y \circ z}$ or $\set {z \circ x, z \circ y}$ admit an infimum.
Then all three sets admit an infimum, and:
{{begin-eqn}}
{{eqn | l = \inf \set {x \circ z, y \circ z}
... | First we recall that by definition of ordered group, $\preccurlyeq$ is compatible with $\circ$:
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = x \preccurlyeq y
| o = \implies
| r = x \circ z \preccurlyeq y \circ z
}}
{{eqn | ll= \land
| l = x \preccurlyeq y
| o = \implies
| r ... | Let $\struct {G, \circ, \preccurlyeq}$ be an [[Definition:Ordered Group|ordered group]].
Let $x, y, z \in G$ be arbitrary.
Let any one of the [[Definition:Set|sets]] $\set {x, y}$, $\set {x \circ z, y \circ z}$ or $\set {z \circ x, z \circ y}$ admit an [[Definition:Infimum of Set|infimum]].
Then all three [[Definit... | First we recall that by definition of [[Definition:Ordered Group|ordered group]], $\preccurlyeq$ is [[Definition:Relation Compatible with Operation|compatible]] with $\circ$:
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in G
| l = x \preccurlyeq y
| o = \implies
| r = x \circ z \preccurlyeq y \circ z
}... | Infima in Ordered Group | https://proofwiki.org/wiki/Infima_in_Ordered_Group | https://proofwiki.org/wiki/Infima_in_Ordered_Group | [
"Ordered Groups",
"Infima"
] | [
"Definition:Ordered Group",
"Definition:Set",
"Definition:Infimum of Set",
"Definition:Set",
"Definition:Infimum of Set"
] | [
"Definition:Ordered Group",
"Definition:Relation Compatible with Operation",
"Definition:Infimum of Set",
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Group",
"Definition... |
proofwiki-19315 | Inverse of Infimum in Ordered Group is Supremum of Inverses | Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group.
Let $x, y \in G$.
Then:
:$\set {x, y}$ admits an infimum in $G$
{{iff}}:
:$\set {x^{-1}, y^{-1} }$ admits a supremum in $G$
in which case:
:$\paren {\inf \set {x, y} }^{-1} = \sup \set {x^{-1}, y^{-1} }$ | === Sufficient Condition ===
Let $\set {x, y}$ admits an infimum $c$ in $G$.
Then:
{{begin-eqn}}
{{eqn | l = c
| o = \preccurlyeq
| r = x
| c = as $c$ is a lower bound of $\set {x, y}$
}}
{{eqn | lo= \land
| l = c
| o = \preccurlyeq
| r = y
| c =
}}
{{eqn | ll= \leadsto
... | Let $\struct {G, \circ, \preccurlyeq}$ be an [[Definition:Ordered Group|ordered group]].
Let $x, y \in G$.
Then:
:$\set {x, y}$ admits an [[Definition:Infimum of Set|infimum]] in $G$
{{iff}}:
:$\set {x^{-1}, y^{-1} }$ admits a [[Definition:Supremum of Set|supremum]] in $G$
in which case:
:$\paren {\inf \set {x, y} ... | === Sufficient Condition ===
Let $\set {x, y}$ admits an [[Definition:Infimum of Set|infimum]] $c$ in $G$.
Then:
{{begin-eqn}}
{{eqn | l = c
| o = \preccurlyeq
| r = x
| c = as $c$ is a [[Definition:Lower Bound of Set|lower bound]] of $\set {x, y}$
}}
{{eqn | lo= \land
| l = c
| o = \pr... | Inverse of Infimum in Ordered Group is Supremum of Inverses | https://proofwiki.org/wiki/Inverse_of_Infimum_in_Ordered_Group_is_Supremum_of_Inverses | https://proofwiki.org/wiki/Inverse_of_Infimum_in_Ordered_Group_is_Supremum_of_Inverses | [
"Ordered Groups",
"Suprema",
"Infima"
] | [
"Definition:Ordered Group",
"Definition:Infimum of Set",
"Definition:Supremum of Set"
] | [
"Definition:Infimum of Set",
"Definition:Lower Bound of Set",
"Inversion Mapping Reverses Ordering in Ordered Group",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Inversion Mapping Reverses Ordering in Ordered Group",
"Definition:Lower Bound of Set",
"Definition:Infimum of Set",
... |
proofwiki-19316 | Inverse of Supremum in Ordered Group is Infimum of Inverses | Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group.
Let $x, y \in G$.
Then:
:$\set {x, y}$ admits a supremum in $G$
{{iff}}:
:$\set {x^{-1}, y^{-1} }$ admits an infimum in $G$
in which case:
:$\paren {\sup \set {x, y} }^{-1} = \inf \set {x^{-1}, y^{-1} }$ | Let:
{{begin-eqn}}
{{eqn | l = a
| r = x^{-1}
}}
{{eqn | l = b
| r = y^{-1}
}}
{{end-eqn}}
Recall from Inverse of Group Inverse:
{{begin-eqn}}
{{eqn | l = a^{-1}
| r = x
}}
{{eqn | l = b^{-1}
| r = y
}}
{{end-eqn}}
From Inverse of Infimum in Ordered Group is Supremum of Inverses:
Then:
:$\set {a... | Let $\struct {G, \circ, \preccurlyeq}$ be an [[Definition:Ordered Group|ordered group]].
Let $x, y \in G$.
Then:
:$\set {x, y}$ admits a [[Definition:Supremum of Set|supremum]] in $G$
{{iff}}:
:$\set {x^{-1}, y^{-1} }$ admits an [[Definition:Infimum of Set|infimum]] in $G$
in which case:
:$\paren {\sup \set {x, y} ... | Let:
{{begin-eqn}}
{{eqn | l = a
| r = x^{-1}
}}
{{eqn | l = b
| r = y^{-1}
}}
{{end-eqn}}
Recall from [[Inverse of Group Inverse]]:
{{begin-eqn}}
{{eqn | l = a^{-1}
| r = x
}}
{{eqn | l = b^{-1}
| r = y
}}
{{end-eqn}}
From [[Inverse of Infimum in Ordered Group is Supremum of Inverses]]:
... | Inverse of Supremum in Ordered Group is Infimum of Inverses | https://proofwiki.org/wiki/Inverse_of_Supremum_in_Ordered_Group_is_Infimum_of_Inverses | https://proofwiki.org/wiki/Inverse_of_Supremum_in_Ordered_Group_is_Infimum_of_Inverses | [
"Ordered Groups",
"Suprema",
"Infima"
] | [
"Definition:Ordered Group",
"Definition:Supremum of Set",
"Definition:Infimum of Set"
] | [
"Inverse of Group Inverse",
"Inverse of Infimum in Ordered Group is Supremum of Inverses",
"Definition:Infimum of Set",
"Definition:Supremum of Set",
"Definition:Infimum of Set",
"Definition:Supremum of Set"
] |
proofwiki-19317 | Product of Supremum and Infimum in Lattice-Ordered Group | Let $\struct {G, \odot}$ be a group.
Let $\preccurlyeq$ be a lattice ordering on $G$.
Let $x, y \in G$ such that $x$ commutes with $y$.
Then:
:$\sup \set {x, y} \odot \inf \set {x, y} = x \odot y$ | First we show that $x$ commutes with $\sup \set {x, y}$.
Indeed:
{{begin-eqn}}
{{eqn | l = x \odot \sup \set {x, y}
| r = \sup \set {x \odot x, x \odot y}
| c = Suprema in Ordered Group
}}
{{eqn | r = \sup \set {x \odot x, y \odot x}
| c = as $x$ and $y$ commute
}}
{{eqn | r = \sup \set {x, y} \odot x... | Let $\struct {G, \odot}$ be a [[Definition:Group|group]].
Let $\preccurlyeq$ be a [[Definition:Lattice Ordering|lattice ordering]] on $G$.
Let $x, y \in G$ such that $x$ [[Definition:Commuting Elements|commutes]] with $y$.
Then:
:$\sup \set {x, y} \odot \inf \set {x, y} = x \odot y$ | First we show that $x$ [[Definition:Commuting Elements|commutes]] with $\sup \set {x, y}$.
Indeed:
{{begin-eqn}}
{{eqn | l = x \odot \sup \set {x, y}
| r = \sup \set {x \odot x, x \odot y}
| c = [[Suprema in Ordered Group]]
}}
{{eqn | r = \sup \set {x \odot x, y \odot x}
| c = as $x$ and $y$ [[Defin... | Product of Supremum and Infimum in Lattice-Ordered Group | https://proofwiki.org/wiki/Product_of_Supremum_and_Infimum_in_Lattice-Ordered_Group | https://proofwiki.org/wiki/Product_of_Supremum_and_Infimum_in_Lattice-Ordered_Group | [
"Ordered Groups",
"Lattice Theory",
"Suprema",
"Infima"
] | [
"Definition:Group",
"Definition:Lattice Ordering",
"Definition:Commutative/Elements"
] | [
"Definition:Commutative/Elements",
"Suprema in Ordered Group",
"Definition:Commutative/Elements",
"Suprema in Ordered Group",
"Definition:Commutative/Elements",
"Infima in Ordered Group",
"Definition:Commutative/Elements",
"Definition:Commutative/Elements",
"Inverse of Supremum in Ordered Group is I... |
proofwiki-19318 | Ordered Group whose Doubletons with Identity admit Supremum is Lattice | Let $\struct {G, \odot, \preccurlyeq}$ be an ordered group whose identity element is $e$.
Let $\preccurlyeq$ be such that for all $x \in G$, $\set {x, e}$ admits a supremum.
Then $\preccurlyeq$ is a lattice ordering. | Let $x, y \in G$ be arbitrary.
We have that $y = x \odot z$ for some $z \in G$.
From Suprema in Ordered Group:
:$\set {z, e}$ admits a supremum {{iff}} $\set {x \odot z, x \odot e}$ admits a supremum.
It follows that:
:$\set {x, y}$ admits a supremum.
As $x$ and $y$ are arbitrary, it follows that:
:$\forall x, y \in G:... | Let $\struct {G, \odot, \preccurlyeq}$ be an [[Definition:Ordered Group|ordered group]] whose [[Definition:Identity Element|identity element]] is $e$.
Let $\preccurlyeq$ be such that for all $x \in G$, $\set {x, e}$ admits a [[Definition:Supremum of Set|supremum]].
Then $\preccurlyeq$ is a [[Definition:Lattice Order... | Let $x, y \in G$ be arbitrary.
We have that $y = x \odot z$ for some $z \in G$.
From [[Suprema in Ordered Group]]:
:$\set {z, e}$ admits a [[Definition:Supremum of Set|supremum]] {{iff}} $\set {x \odot z, x \odot e}$ admits a [[Definition:Supremum of Set|supremum]].
It follows that:
:$\set {x, y}$ admits a [[Defini... | Ordered Group whose Doubletons with Identity admit Supremum is Lattice | https://proofwiki.org/wiki/Ordered_Group_whose_Doubletons_with_Identity_admit_Supremum_is_Lattice | https://proofwiki.org/wiki/Ordered_Group_whose_Doubletons_with_Identity_admit_Supremum_is_Lattice | [
"Ordered Groups",
"Lattice Theory",
"Suprema"
] | [
"Definition:Ordered Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Supremum of Set",
"Definition:Lattice Ordering"
] | [
"Suprema in Ordered Group",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Inverse of Supremum in Ordered Group is Infimum of Inverses",
"Definition:Infimum of Set",
"Definition:Infimum of Set",
... |
proofwiki-19319 | Conditions for Ordering in Ordered Group to be Directed | Let $\struct {G, \odot, \preccurlyeq}$ be an ordered group whose identity element is $e$.
Then:
:$\preccurlyeq$ is a directed ordering
{{iff}}:
:for every $x \in G$ there exist $y, z \in G$ such that $e \preccurlyeq y$, $e \preccurlyeq z$ and $x = y \odot z^{-1}$. | === Sufficient Condition ===
Let $\preccurlyeq$ be a directed ordering.
By definition of directed ordering:
:$\forall x, z \in G: \exists y \in G: x \preccurlyeq y$ and $z \preccurlyeq y$
Let $x \in G$ be arbitrary.
{{begin-eqn}}
{{eqn | q = \exists y \in G
| l = x
| o = \preccurlyeq
| r = y
| c... | Let $\struct {G, \odot, \preccurlyeq}$ be an [[Definition:Ordered Group|ordered group]] whose [[Definition:Identity Element|identity element]] is $e$.
Then:
:$\preccurlyeq$ is a [[Definition:Directed Ordering|directed ordering]]
{{iff}}:
:for every $x \in G$ there exist $y, z \in G$ such that $e \preccurlyeq y$, $e \... | === Sufficient Condition ===
Let $\preccurlyeq$ be a [[Definition:Directed Ordering|directed ordering]].
By definition of [[Definition:Directed Ordering|directed ordering]]:
:$\forall x, z \in G: \exists y \in G: x \preccurlyeq y$ and $z \preccurlyeq y$
Let $x \in G$ be arbitrary.
{{begin-eqn}}
{{eqn | q = \exists... | Conditions for Ordering in Ordered Group to be Directed | https://proofwiki.org/wiki/Conditions_for_Ordering_in_Ordered_Group_to_be_Directed | https://proofwiki.org/wiki/Conditions_for_Ordering_in_Ordered_Group_to_be_Directed | [
"Ordered Groups",
"Directed Orderings"
] | [
"Definition:Ordered Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Directed Ordering"
] | [
"Definition:Directed Ordering",
"Definition:Directed Ordering",
"Inverse of Group Product",
"Definition:Directed Ordering"
] |
proofwiki-19320 | Raw Moment of Log Normal Distribution | Let $X$ be a continuous random variable with the Log Normal distribution with $\mu \in \R, \sigma \in \R_{>0}$.
Let $n$ be a strictly positive integer.
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = \exp {\paren {n\mu + \dfrac {\sigma^2 n^2} 2} }$ | From the definition of the Log Normal distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} x} \map \exp {-\dfrac {\paren {\map \ln x - \mu}^2} {2 \sigma^2} }$
where $\Img X = \R_{>0}$.
From the definition of the expected value of a continuous random variable:
:$\ds \expec... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Log Normal Distribution|Log Normal distribution]] with $\mu \in \R, \sigma \in \R_{>0}$.
Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Then the $n$th [[Definition:Raw Moment|raw... | From the definition of the [[Definition:Log Normal Distribution|Log Normal distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} x} \map \exp {-\dfrac {\paren {\map \ln x - \mu}^2} {2 \sigma^2} }$
where $\Img X = \R_{>0}$.
F... | Raw Moment of Log Normal Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Log_Normal_Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Log_Normal_Distribution | [
"Log Normal Distribution",
"Raw Moments"
] | [
"Definition:Random Variable/Continuous",
"Definition:Log Normal Distribution",
"Definition:Strictly Positive/Integer",
"Definition:Raw Moment"
] | [
"Definition:Log Normal Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Derivative of Composite Function",
"Linear Combination of Integrals/Definite",
"Square of Sum",
"Gaussian Integral",
"Category:Log Normal Distribution",
"Category:Raw Moments"
] |
proofwiki-19321 | Expectation of Log Normal Distribution | Let $X$ be a continuous random variable with the Log Normal distribution with $\mu \in \R, \sigma \in \R_{> 0}$.
The expectation of $X$ is given by:
:$\expect X = \exp {\paren {\mu + \dfrac {\sigma^2 } 2 } }$ | From Raw Moment of Log Normal Distribution, we have:
The $n$th raw moment $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = \exp {\paren {n\mu + \dfrac {\sigma^2 n^2 } 2 } }$
Therefore, for $n = 1$ we have:
:$\expect X = \exp {\paren {\mu + \dfrac {\sigma^2 } 2 } }$
Hence the result.
{{qed}}
Category:Log Normal Di... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Log Normal Distribution|Log Normal distribution]] with $\mu \in \R, \sigma \in \R_{> 0}$.
The [[Definition:Expectation of Continuous Random Variable|expectation]] of $X$ is given by:
:$\expect X = \exp {\paren {\mu... | From [[Raw Moment of Log Normal Distribution]], we have:
The $n$th [[Definition:Raw Moment|raw moment]] $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = \exp {\paren {n\mu + \dfrac {\sigma^2 n^2 } 2 } }$
Therefore, for $n = 1$ we have:
:$\expect X = \exp {\paren {\mu + \dfrac {\sigma^2 } 2 } }$
Hence the res... | Expectation of Log Normal Distribution | https://proofwiki.org/wiki/Expectation_of_Log_Normal_Distribution | https://proofwiki.org/wiki/Expectation_of_Log_Normal_Distribution | [
"Log Normal Distribution",
"Expectation"
] | [
"Definition:Random Variable/Continuous",
"Definition:Log Normal Distribution",
"Definition:Expectation/Continuous"
] | [
"Raw Moment of Log Normal Distribution",
"Definition:Raw Moment",
"Category:Log Normal Distribution",
"Category:Expectation"
] |
proofwiki-19322 | Variance of Log Normal Distribution | Let $X$ be a continuous random variable with the Log Normal distribution with $\mu \in \R, \sigma \in \R_{> 0}$.
Then the variance of $X$ is given by:
:$\var X = \map \exp {2 \mu + \sigma^2} \paren {\map \exp {\sigma^2} - 1}$ | By Variance as Expectation of Square minus Square of Expectation:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By Expectation of Log Normal Distribution:
:$\expect X = \map \exp {\mu + \dfrac {\sigma^2} 2}$
From Raw Moment of Log Normal Distribution:
The $n$th raw moment $\expect {X^n}$ of $X$ is given by:
:$\expe... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Log Normal Distribution|Log Normal distribution]] with $\mu \in \R, \sigma \in \R_{> 0}$.
Then the [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \map \exp {2 \mu + \sigma^2} \paren {\map \exp {\s... | By [[Variance as Expectation of Square minus Square of Expectation]]:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
By [[Expectation of Log Normal Distribution]]:
:$\expect X = \map \exp {\mu + \dfrac {\sigma^2} 2}$
From [[Raw Moment of Log Normal Distribution]]:
The $n$th [[Definition:Raw Moment|raw moment]] $... | Variance of Log Normal Distribution | https://proofwiki.org/wiki/Variance_of_Log_Normal_Distribution | https://proofwiki.org/wiki/Variance_of_Log_Normal_Distribution | [
"Variance",
"Log Normal Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Log Normal Distribution",
"Definition:Variance"
] | [
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Log Normal Distribution",
"Raw Moment of Log Normal Distribution",
"Definition:Raw Moment",
"Exponent Combination Laws/Power of Power",
"Category:Variance",
"Category:Log Normal Distribution"
] |
proofwiki-19323 | Composition of Dirac Delta Distribution with Function with Simple Zero | Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Let $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ be a delta sequence.
Let $f : \R \to \R$ be a real function with a simple zero at $x_0$.
Let $f$ be strictly monotone.
Let $\phi \in \map \DD \R$ be a test function.
Then in the distributional sense i... | Suppose $\map {f'} {x_0} > 0$.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \int_{-\infty}^{\infty} \map {\delta_n} {\map f x} \map \phi x \rd x
| r = \lim_{n \mathop \to \infty} \int_{-\infty}^{\infty} \map {\delta_n} y \frac {\map \phi {\map x y} } {\map {f'} {\map x y} } \rd y
| c = Deriva... | Let $\delta \in \map {\DD'} \R$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Let $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ be a [[Definition:Delta Sequence|delta sequence]].
Let $f : \R \to \R$ be a [[Definition:Real Function|real function]] with a [[Definition:Simple Zero|simple ... | Suppose $\map {f'} {x_0} > 0$.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \int_{-\infty}^{\infty} \map {\delta_n} {\map f x} \map \phi x \rd x
| r = \lim_{n \mathop \to \infty} \int_{-\infty}^{\infty} \map {\delta_n} y \frac {\map \phi {\map x y} } {\map {f'} {\map x y} } \rd y
| c = [[De... | Composition of Dirac Delta Distribution with Function with Simple Zero/Proof 1 | https://proofwiki.org/wiki/Composition_of_Dirac_Delta_Distribution_with_Function_with_Simple_Zero | https://proofwiki.org/wiki/Composition_of_Dirac_Delta_Distribution_with_Function_with_Simple_Zero/Proof_1 | [
"Composition of Dirac Delta Distribution with Function with Simple Zero",
"Dirac Delta Distribution",
"Delta Sequence"
] | [
"Definition:Dirac Delta Distribution",
"Definition:Delta Sequence",
"Definition:Real Function",
"Definition:Order of Zero/Simple Zero",
"Definition:Strictly Monotone/Real Function",
"Definition:Test Function",
"Definition:Schwartz Distribution"
] | [
"Derivative of Inverse Function",
"Integration by Substitution/Definite Integral",
"Derivative of Inverse Function",
"Integration by Substitution/Definite Integral"
] |
proofwiki-19324 | Composition of Dirac Delta Distribution with Function with Simple Zero | Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Let $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ be a delta sequence.
Let $f : \R \to \R$ be a real function with a simple zero at $x_0$.
Let $f$ be strictly monotone.
Let $\phi \in \map \DD \R$ be a test function.
Then in the distributional sense i... | Let $H$ be the Heaviside step function.
Let $T \in \map \DD \R$ be a Schwartz distribution associated with $\map H {\map f x}$:
:$T = T_{\map H {\map f x} }$
We have that:
{{begin-eqn}}
{{eqn | l = \map H {\map f x}
| r = \begin {cases} \map H {x - x_0} & : \forall x \in \R : \map {f'} x > 0 \\ 1 - \map H {x - x... | Let $\delta \in \map {\DD'} \R$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Let $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ be a [[Definition:Delta Sequence|delta sequence]].
Let $f : \R \to \R$ be a [[Definition:Real Function|real function]] with a [[Definition:Simple Zero|simple ... | Let $H$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $T \in \map \DD \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]] associated with $\map H {\map f x}$:
:$T = T_{\map H {\map f x} }$
We have that:
{{begin-eqn}}
{{eqn | l = \map H {\map f x}
| r = \begin {cas... | Composition of Dirac Delta Distribution with Function with Simple Zero/Proof 2 | https://proofwiki.org/wiki/Composition_of_Dirac_Delta_Distribution_with_Function_with_Simple_Zero | https://proofwiki.org/wiki/Composition_of_Dirac_Delta_Distribution_with_Function_with_Simple_Zero/Proof_2 | [
"Composition of Dirac Delta Distribution with Function with Simple Zero",
"Dirac Delta Distribution",
"Delta Sequence"
] | [
"Definition:Dirac Delta Distribution",
"Definition:Delta Sequence",
"Definition:Real Function",
"Definition:Order of Zero/Simple Zero",
"Definition:Strictly Monotone/Real Function",
"Definition:Test Function",
"Definition:Schwartz Distribution"
] | [
"Definition:Heaviside Step Function",
"Definition:Schwartz Distribution",
"Definition:Derivative/Real Function",
"Definition:Derivative/Real Function",
"Jump Rule",
"Definition:Composition of Mappings",
"Definition:Dirac Delta Distribution",
"Distributional Derivative of Heaviside Step Function",
"D... |
proofwiki-19325 | Skewness of Log Normal Distribution | Let $X$ be a continuous random variable with the Log Normal distribution with $\mu \in \R, \sigma \in \R_{> 0}$.
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \paren {\map \exp {\sigma^2} + 2} \sqrt {\paren {\map \exp {\sigma^2} - 1} }$ | 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 Log Normal Distribution:
:$\mu = \map \exp {\mu + \dfrac {\sigma^2} 2}$
By Variance ... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Log Normal Distribution|Log Normal distribution]] with $\mu \in \R, \sigma \in \R_{> 0}$.
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \paren {\map \exp {\sigma^2} + 2} \sqr... | From [[Skewness in terms of Non-Central Moments]], we have:
:$\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 Log Normal Dist... | Skewness of Log Normal Distribution | https://proofwiki.org/wiki/Skewness_of_Log_Normal_Distribution | https://proofwiki.org/wiki/Skewness_of_Log_Normal_Distribution | [
"Skewness",
"Log Normal Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Log Normal Distribution",
"Definition:Skewness"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Log Normal Distribution",
"Variance of Log Normal Distribution",
"Raw Moment of Log Normal Distribution",
"Exponent Combination Laws/Power of Power",
"Distributive Laws/Arithmetic",
... |
proofwiki-19326 | D'Ocagne's Identity | :$\forall m, n \in \Z: \paren {-1}^n F_{m - n} = F_m F_{n + 1} - F_n F_{n - 1}$ | {{begin-eqn}}
{{eqn | q =
| l = \paren {-1}^n F_i F_j
| r = F_{n + i} F_{n + j} - F_n F_{n + i + j}
| c = Vajda's Identity
}}
{{eqn | ll= \leadsto
| l = \paren {-1}^n F_{m - n} F_1
| r = F_{n + \paren {m - n} } F_{n + 1} - F_n F_{n + \paren {m - n} + 1}
| c = setting $i \gets m - n$... | :$\forall m, n \in \Z: \paren {-1}^n F_{m - n} = F_m F_{n + 1} - F_n F_{n - 1}$ | {{begin-eqn}}
{{eqn | q =
| l = \paren {-1}^n F_i F_j
| r = F_{n + i} F_{n + j} - F_n F_{n + i + j}
| c = [[Vajda's Identity]]
}}
{{eqn | ll= \leadsto
| l = \paren {-1}^n F_{m - n} F_1
| r = F_{n + \paren {m - n} } F_{n + 1} - F_n F_{n + \paren {m - n} + 1}
| c = setting $i \gets m ... | D'Ocagne's Identity | https://proofwiki.org/wiki/D'Ocagne's_Identity | https://proofwiki.org/wiki/D'Ocagne's_Identity | [
"Fibonacci Numbers"
] | [] | [
"Vajda's Identity",
"Category:Fibonacci Numbers"
] |
proofwiki-19327 | Dipper Relation is Equivalence Relation | Let $m \in \N$ be a natural number.
Let $n \in \N_{>0}$ be a non-zero natural number.
Let $\RR_{m, n}$ be the dipper relation on $\N$:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y... | First let it be noted that $\RR_{m, n}$ can be written as:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin {cases} x = y \\ m \le x, y \text { and } n \divides \size {x - y} \end {cases}$
where $\size {x - y}$ denotes the absolute value of $x - y$.
Checking in turn each of the criteria for equivalence: | Let $m \in \N$ be a [[Definition:Natural Number|natural number]].
Let $n \in \N_{>0}$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural number]].
Let $\RR_{m, n}$ be the [[Definition:Dipper Relation|dipper relation]] on $\N$:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin... | First let it be noted that $\RR_{m, n}$ can be written as:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin {cases} x = y \\ m \le x, y \text { and } n \divides \size {x - y} \end {cases}$
where $\size {x - y}$ denotes the [[Definition:Absolute Value|absolute value]] of $x - y$.
Checking in turn each o... | Dipper Relation is Equivalence Relation | https://proofwiki.org/wiki/Dipper_Relation_is_Equivalence_Relation | https://proofwiki.org/wiki/Dipper_Relation_is_Equivalence_Relation | [
"Dipper Relations",
"Examples of Equivalence Relations",
"Dipper Relation is Equivalence Relation"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Dipper Relation",
"Definition:Equivalence Relation"
] | [
"Definition:Absolute Value",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-19328 | Dipper Relation is Congruence for Addition | Let $m \in \N$ be a natural number.
Let $n \in \N_{>0}$ be a non-zero natural number.
Let $\RR_{m, n}$ be the dipper relation on $\N$:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y... | From Dipper Relation is Equivalence Relation we have that $\RR_{m, n}$ is an equivalence relation.
From Equivalence Relation is Congruence iff Compatible with Operation, it is sufficient to show that:
:$\forall x, y, z \in \N: x \mathrel {\RR_{m, n} } y \implies \paren {x + z} \mathrel {\RR_{m, n} } \paren {y + z}$
and... | Let $m \in \N$ be a [[Definition:Natural Number|natural number]].
Let $n \in \N_{>0}$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural number]].
Let $\RR_{m, n}$ be the [[Definition:Dipper Relation|dipper relation]] on $\N$:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin... | From [[Dipper Relation is Equivalence Relation]] we have that $\RR_{m, n}$ is an [[Definition:Equivalence Relation|equivalence relation]].
From [[Equivalence Relation is Congruence iff Compatible with Operation]], it is sufficient to show that:
:$\forall x, y, z \in \N: x \mathrel {\RR_{m, n} } y \implies \paren {x +... | Dipper Relation is Congruence for Addition | https://proofwiki.org/wiki/Dipper_Relation_is_Congruence_for_Addition | https://proofwiki.org/wiki/Dipper_Relation_is_Congruence_for_Addition | [
"Dipper Relations",
"Examples of Congruence Relations"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Dipper Relation",
"Definition:Congruence Relation",
"Definition:Addition/Natural Numbers"
] | [
"Dipper Relation is Equivalence Relation",
"Definition:Equivalence Relation",
"Equivalence Relation is Congruence iff Compatible with Operation",
"Natural Number Addition is Commutative",
"Definition:Symmetric Relation"
] |
proofwiki-19329 | Dipper Relation is Congruence for Multiplication | Let $m \in \N$ be a natural number.
Let $n \in \N_{>0}$ be a non-zero natural number.
Let $\RR_{m, n}$ be the dipper relation on $\N$:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y... | From Dipper Relation is Equivalence Relation we have that $\RR_{m, n}$ is an equivalence relation.
From Equivalence Relation is Congruence iff Compatible with Operation, it is sufficient to show that:
:$\forall x, y, z \in \N: x \mathrel {\RR_{m, n} } y \implies \paren {x z} \mathrel {\RR_{m, n} } \paren {y z}$
and:
:$... | Let $m \in \N$ be a [[Definition:Natural Number|natural number]].
Let $n \in \N_{>0}$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural number]].
Let $\RR_{m, n}$ be the [[Definition:Dipper Relation|dipper relation]] on $\N$:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin... | From [[Dipper Relation is Equivalence Relation]] we have that $\RR_{m, n}$ is an [[Definition:Equivalence Relation|equivalence relation]].
From [[Equivalence Relation is Congruence iff Compatible with Operation]], it is sufficient to show that:
:$\forall x, y, z \in \N: x \mathrel {\RR_{m, n} } y \implies \paren {x z... | Dipper Relation is Congruence for Multiplication | https://proofwiki.org/wiki/Dipper_Relation_is_Congruence_for_Multiplication | https://proofwiki.org/wiki/Dipper_Relation_is_Congruence_for_Multiplication | [
"Dipper Relations",
"Examples of Congruence Relations"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Dipper Relation",
"Definition:Congruence Relation",
"Definition:Multiplication/Natural Numbers"
] | [
"Dipper Relation is Equivalence Relation",
"Definition:Equivalence Relation",
"Equivalence Relation is Congruence iff Compatible with Operation",
"Natural Number Addition is Commutative",
"Definition:Symmetric Relation"
] |
proofwiki-19330 | Excess Kurtosis of Log Normal Distribution | Let $X$ be a continuous random variable with the Log Normal distribution with $\mu \in \R, \sigma \in \R_{> 0}$.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \map \exp {4 \sigma^2} + 2 \map \exp {3 \sigma^2} + 3 \map \exp {2 \sigma^2} - 6$ | From Kurtosis in terms of Non-Central Moments, we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Log Normal Distribution we have:
:$\mu = \map \ex... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Log Normal Distribution|Log Normal distribution]] with $\mu \in \R, \sigma \in \R_{> 0}$.
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \map \exp {4 \sigma^2} ... | From [[Kurtosis in terms of Non-Central Moments]], we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
... | Excess Kurtosis of Log Normal Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Log_Normal_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Log_Normal_Distribution | [
"Kurtosis",
"Log Normal Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Log Normal Distribution",
"Definition:Excess Kurtosis"
] | [
"Kurtosis in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Log Normal Distribution",
"Variance of Log Normal Distribution",
"Raw Moment of Log Normal Distribution",
"Exponent Combination Laws/Power of Power",
"Exponent Combination Laws/Produc... |
proofwiki-19331 | Restricted Dipper Relation is Equivalence Relation | Let $m, n \in \N_{>0}$ be non-zero natural numbers.
Let $\RR^*_{m, n}$ be the restricted dipper relation on $\N$:
:$\forall x, y \in \N_{>0}: x \mathrel {\RR^*_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y} \end {cases}... | By definition, $\RR^*_{m, n}$ is the restriction of the dipper relation $\RR_{m, n}$ to $\N_{>0}$.
We have from Dipper Relation is Equivalence Relation that $\RR_{m, n}$ is an equivalence relation.
The result follows from Restriction of Equivalence Relation is Equivalence.
{{qed}} | Let $m, n \in \N_{>0}$ be non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural numbers]].
Let $\RR^*_{m, n}$ be the [[Definition:Restricted Dipper Relation|restricted dipper relation]] on $\N$:
:$\forall x, y \in \N_{>0}: x \mathrel {\RR^*_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \tex... | By definition, $\RR^*_{m, n}$ is the [[Definition:Restriction of Relation|restriction]] of the [[Definition:Dipper Relation|dipper relation]] $\RR_{m, n}$ to $\N_{>0}$.
We have from [[Dipper Relation is Equivalence Relation]] that $\RR_{m, n}$ is an [[Definition:Equivalence Relation|equivalence relation]].
The result... | Restricted Dipper Relation is Equivalence Relation | https://proofwiki.org/wiki/Restricted_Dipper_Relation_is_Equivalence_Relation | https://proofwiki.org/wiki/Restricted_Dipper_Relation_is_Equivalence_Relation | [
"Restricted Dipper Relations",
"Examples of Equivalence Relations"
] | [
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Restricted Dipper Relation",
"Definition:Equivalence Relation"
] | [
"Definition:Restriction/Relation",
"Definition:Dipper Relation",
"Dipper Relation is Equivalence Relation",
"Definition:Equivalence Relation",
"Restriction of Equivalence Relation is Equivalence"
] |
proofwiki-19332 | Restricted Dipper Relation is Congruence for Multiplication | Let $m, n \in \N_{>0}$ be non-zero natural numbers.
Let $\RR^*_{m, n}$ be the restricted dipper relation on $\N$:
:$\forall x, y \in \N_{>0}: x \mathrel {\RR^*_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y} \end {cases}... | By definition, $\RR^*_{m, n}$ is the restriction of the dipper relation $\RR_{m, n}$ to $\N_{>0}$.
We have from Dipper Relation is Congruence for Multiplication that $\RR_{m, n}$ is a congruence relation for multiplication.
The result follows from Restriction of Congruence Relation is Congruence.
{{qed}} | Let $m, n \in \N_{>0}$ be non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural numbers]].
Let $\RR^*_{m, n}$ be the [[Definition:Restricted Dipper Relation|restricted dipper relation]] on $\N$:
:$\forall x, y \in \N_{>0}: x \mathrel {\RR^*_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \tex... | By definition, $\RR^*_{m, n}$ is the [[Definition:Restriction of Relation|restriction]] of the [[Definition:Dipper Relation|dipper relation]] $\RR_{m, n}$ to $\N_{>0}$.
We have from [[Dipper Relation is Congruence for Multiplication]] that $\RR_{m, n}$ is a [[Definition:Congruence Relation|congruence relation]] for [[... | Restricted Dipper Relation is Congruence for Multiplication | https://proofwiki.org/wiki/Restricted_Dipper_Relation_is_Congruence_for_Multiplication | https://proofwiki.org/wiki/Restricted_Dipper_Relation_is_Congruence_for_Multiplication | [
"Restricted Dipper Relations",
"Examples of Congruence Relations"
] | [
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Restricted Dipper Relation",
"Definition:Congruence Relation",
"Definition:Multiplication/Natural Numbers"
] | [
"Definition:Restriction/Relation",
"Definition:Dipper Relation",
"Dipper Relation is Congruence for Multiplication",
"Definition:Congruence Relation",
"Definition:Multiplication/Natural Numbers",
"Restriction of Congruence Relation is Congruence"
] |
proofwiki-19333 | Dipper Operation is Closed on Initial Segment | Let $m \in \N$ be a natural number.
Let $n \in \N_{>0}$ be a non-zero natural number.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
Let $+_{m, n}$ denote the '''dipper operation''' on $\N_{< \paren {m \mathop +... | Let $a + b < m$.
Then:
:$0 \le a +_{m, n} b < m$
and so:
:$a +_{m, n} b \in \N_{< \paren {m \mathop + n} }$
Let $a + b > m$.
Then:
:$a +_{m, n} b = a + b - k n$
where $k$ is the largest such that $m + k n < a + b$
Hence:
{{begin-eqn}}
{{eqn | q =
| l = m + k n
| o = \le
| m = a + b
| mo= <
... | Let $m \in \N$ be a [[Definition:Natural Number|natural number]].
Let $n \in \N_{>0}$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural number]].
Let $\N_{< \paren {m \mathop + n} }$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment]] of the [[Definitio... | Let $a + b < m$.
Then:
:$0 \le a +_{m, n} b < m$
and so:
:$a +_{m, n} b \in \N_{< \paren {m \mathop + n} }$
Let $a + b > m$.
Then:
:$a +_{m, n} b = a + b - k n$
where $k$ is the largest such that $m + k n < a + b$
Hence:
{{begin-eqn}}
{{eqn | q =
| l = m + k n
| o = \le
| m = a + b
| mo= <... | Dipper Operation is Closed on Initial Segment | https://proofwiki.org/wiki/Dipper_Operation_is_Closed_on_Initial_Segment | https://proofwiki.org/wiki/Dipper_Operation_is_Closed_on_Initial_Segment | [
"Dipper Operations",
"Algebraic Closure"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Natural Numbers",
"Definition:Dipper Operation",
"Definition:Integer",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Category:Dipper Operations",
"Category:Algebraic Closure"
] |
proofwiki-19334 | Restriction of Dipper Operation to Non-Zero Initial Segment is Closed | Let $m, n \in \N_{>0}$ be non-zero natural numbers.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
Let $+_{m, n}$ denote the '''dipper operation''' on $\N_{< \paren {m \mathop + n} }$:
:$\forall a, b \in \N_{< \... | From Dipper Operation is Closed on Initial Segment, $+_{m, n}$ is closed on $\N_{< \paren {m \mathop + n} }$.
That is:
:$\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, n} b < m + n$
It remains to be shown that:
:$\forall a, b \in \N^*_{< \paren {m \mathop + n} }: a +_{m, n} b > 0$
Let $a + b < m$.
Then as $a ... | Let $m, n \in \N_{>0}$ be non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural numbers]].
Let $\N_{< \paren {m \mathop + n} }$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment]] of the [[Definition:Natural Numbers|natural numbers]]:
:$\N_{< \paren {m \mathop + ... | From [[Dipper Operation is Closed on Initial Segment]], $+_{m, n}$ is [[Definition:Closed Operation|closed]] on $\N_{< \paren {m \mathop + n} }$.
That is:
:$\forall a, b \in \N_{< \paren {m \mathop + n} }: a +_{m, n} b < m + n$
It remains to be shown that:
:$\forall a, b \in \N^*_{< \paren {m \mathop + n} }: a +_{m, ... | Restriction of Dipper Operation to Non-Zero Initial Segment is Closed | https://proofwiki.org/wiki/Restriction_of_Dipper_Operation_to_Non-Zero_Initial_Segment_is_Closed | https://proofwiki.org/wiki/Restriction_of_Dipper_Operation_to_Non-Zero_Initial_Segment_is_Closed | [
"Restricted Dipper Operations",
"Dipper Operations",
"Algebraic Closure"
] | [
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Natural Numbers",
"Definition:Dipper Operation",
"Definition:Integer",
"Definition:Set",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Dipper Operation is Closed on Initial Segment",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Integer"
] |
proofwiki-19335 | Equivalence Classes of Dipper Relation | Let $m \in \N$ be a natural number.
Let $n \in \N_{>0}$ be a non-zero natural number.
Let $\RR_{m, n}$ be the dipper relation on $\N$:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin {cases} x = y \\ m \le x < y \text { and } n \divides \paren {y - x} \\ m \le y < x \text { and } n \divides \paren {x - y... | From Dipper Relation is Equivalence Relation we have that $\RR_{m, n}$ is indeed an equivalence relation.
Let $x < m$.
Then by definition:
:$y \in \eqclass x {\RR_{m, n} } \iff x = y$
and so:
:$\eqclass x {\RR_{m, n} } = \set x$
Let $m \le x$.
Let $x \mathrel {\RR_{m, n} } y$.
First we note that $m \le y$.
Let $x = y$.... | Let $m \in \N$ be a [[Definition:Natural Number|natural number]].
Let $n \in \N_{>0}$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural number]].
Let $\RR_{m, n}$ be the [[Definition:Dipper Relation|dipper relation]] on $\N$:
:$\forall x, y \in \N: x \mathrel {\RR_{m, n} } y \iff \begin... | From [[Dipper Relation is Equivalence Relation]] we have that $\RR_{m, n}$ is indeed an [[Definition:Equivalence Relation|equivalence relation]].
Let $x < m$.
Then by definition:
:$y \in \eqclass x {\RR_{m, n} } \iff x = y$
and so:
:$\eqclass x {\RR_{m, n} } = \set x$
Let $m \le x$.
Let $x \mathrel {\RR_{m, n} }... | Equivalence Classes of Dipper Relation | https://proofwiki.org/wiki/Equivalence_Classes_of_Dipper_Relation | https://proofwiki.org/wiki/Equivalence_Classes_of_Dipper_Relation | [
"Dipper Relations",
"Examples of Equivalence Classes"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Dipper Relation",
"Definition:Equivalence Class"
] | [
"Dipper Relation is Equivalence Relation",
"Definition:Equivalence Relation",
"Category:Dipper Relations",
"Category:Examples of Equivalence Classes"
] |
proofwiki-19336 | Restricted Dipper Operation is Associative | Let $m, n \in \N_{>0}$ be non-zero natural numbers.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
Let $\N^*_{< \paren {m \mathop + n} }$ denote the set defined as $\N_{< \paren {m \mathop + n} } \setminus \set ... | By definition, $+^*_{m, n}$ is the restriction of the dipper relation $+_{m, n}$ to $\N_{>0}$.
We have from Dipper Operation is Associative that $+_{m, n}$ is an associative operation.
The result follows from Restriction of Associative Operation is Associative.
{{qed}} | Let $m, n \in \N_{>0}$ be non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural numbers]].
Let $\N_{< \paren {m \mathop + n} }$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment]] of the [[Definition:Natural Numbers|natural numbers]]:
:$\N_{< \paren {m \mathop + ... | By definition, $+^*_{m, n}$ is the [[Definition:Restriction of Relation|restriction]] of the [[Definition:Dipper Relation|dipper relation]] $+_{m, n}$ to $\N_{>0}$.
We have from [[Dipper Operation is Associative]] that $+_{m, n}$ is an [[Definition:Associative Operation|associative operation]].
The result follows fro... | Restricted Dipper Operation is Associative | https://proofwiki.org/wiki/Restricted_Dipper_Operation_is_Associative | https://proofwiki.org/wiki/Restricted_Dipper_Operation_is_Associative | [
"Restricted Dipper Operations",
"Examples of Associative Operations"
] | [
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Restricted Dipper Operation",
"Definition:Associative Operation"
] | [
"Definition:Restriction/Relation",
"Definition:Dipper Relation",
"Dipper Operation is Associative",
"Definition:Associative Operation",
"Restriction of Associative Operation is Associative"
] |
proofwiki-19337 | Restricted Dipper Operation is Commutative | Let $m, n \in \N_{>0}$ be non-zero natural numbers.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
Let $\N^*_{< \paren {m \mathop + n} }$ denote the set defined as $\N_{< \paren {m \mathop + n} } \setminus \set ... | By definition, $+^*_{m, n}$ is the restriction of the dipper relation $+_{m, n}$ to $\N_{>0}$.
We have from Dipper Operation is Associative that $+_{m, n}$ is a commutative operation.
The result follows from Restriction of Commutative Operation is Commutative.
{{qed}} | Let $m, n \in \N_{>0}$ be non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural numbers]].
Let $\N_{< \paren {m \mathop + n} }$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment]] of the [[Definition:Natural Numbers|natural numbers]]:
:$\N_{< \paren {m \mathop + ... | By definition, $+^*_{m, n}$ is the [[Definition:Restriction of Relation|restriction]] of the [[Definition:Dipper Relation|dipper relation]] $+_{m, n}$ to $\N_{>0}$.
We have from [[Dipper Operation is Associative]] that $+_{m, n}$ is a [[Definition:Commutative Operation|commutative operation]].
The result follows from... | Restricted Dipper Operation is Commutative | https://proofwiki.org/wiki/Restricted_Dipper_Operation_is_Commutative | https://proofwiki.org/wiki/Restricted_Dipper_Operation_is_Commutative | [
"Restricted Dipper Operations",
"Examples of Commutative Operations"
] | [
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Restricted Dipper Operation",
"Definition:Commutative/Operation"
] | [
"Definition:Restriction/Relation",
"Definition:Dipper Relation",
"Dipper Operation is Associative",
"Definition:Commutative/Operation",
"Restriction of Commutative Operation is Commutative"
] |
proofwiki-19338 | Restriction of Canonical Surjection to Restricted Dipper Semigroup is Isomorphism | Let $m, n \in \N_{>0}$ be non-zero natural numbers.
Let $\N_{< \paren {m \mathop + n} }$ denote the initial segment of the natural numbers:
:$\N_{< \paren {m \mathop + n} } := \set {0, 1, \ldots, m + n - 1}$
Let $\N^*_{< \paren {m \mathop + n} }$ denote the set defined as $\N_{< \paren {m \mathop + n} } \setminus \set ... | {{ProofWanted|Formalise the notion that everything is a restriction of everything else}} | Let $m, n \in \N_{>0}$ be non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural numbers]].
Let $\N_{< \paren {m \mathop + n} }$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment]] of the [[Definition:Natural Numbers|natural numbers]]:
:$\N_{< \paren {m \mathop + ... | {{ProofWanted|Formalise the notion that everything is a restriction of everything else}} | Restriction of Canonical Surjection to Restricted Dipper Semigroup is Isomorphism | https://proofwiki.org/wiki/Restriction_of_Canonical_Surjection_to_Restricted_Dipper_Semigroup_is_Isomorphism | https://proofwiki.org/wiki/Restriction_of_Canonical_Surjection_to_Restricted_Dipper_Semigroup_is_Isomorphism | [
"Restricted Dipper Semigroups"
] | [
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Dipper Operation",
"Definition:Integer",
"Definition:Restricted Dipper Operation",
"Definition:Integer",
"Definition:Alg... | [] |
proofwiki-19339 | Excess Kurtosis of Hat-Check Distribution | Let $X$ be a discrete random variable with a Hat-Check distribution with parameter $n$. ($n \gt 3$)
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = 1$ | From Kurtosis in terms of Non-Central Moments, we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
We have, by Expectation of Hat-Check Distribution:
:$\expect X = n -... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with a [[Definition:Hat-Check Distribution|Hat-Check distribution with parameter $n$]]. ($n \gt 3$)
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = 1$ | From [[Kurtosis in terms of Non-Central Moments]], we have:
:$\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
... | Excess Kurtosis of Hat-Check Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Hat-Check_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Hat-Check_Distribution | [
"Kurtosis",
"Hat-Check Distribution"
] | [
"Definition:Random Variable/Discrete",
"Definition:Hat-Check Distribution",
"Definition:Excess Kurtosis"
] | [
"Kurtosis in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Hat-Check Distribution",
"Variance of Hat-Check Distribution",
"Skewness of Hat-Check Distribution",
"Hat-Check Distribution Gives Rise to Probability Mass Function",
"Hat-Check Distr... |
proofwiki-19340 | Inductive Semigroup whose Inductive Elements Commute is Commutative Semigroup | Let $\struct {S, \circ}$ be a semigroup.
Let there exist $\alpha, \beta \in S$ which fulfil the condition for $\struct {S, \circ}$ to be an inductive semigroup:
:the only subset of $S$ containing both $\alpha$ and $x \circ \beta$ whenever it contains $x$ is $S$ itself.
Let $\alpha$ and $\beta$ commute.
Then $\struct {S... | Suppose $\struct {S, \circ}$ is a semigroup.
Suppose there exist $\alpha, \beta \in S$ such that $\struct {S, \circ}$ is an inductive semigroup.
That is, suppose there exist $\alpha, \beta \in S$ such that the only subset of $S$ containing both $\alpha$ and $x \circ \beta$ whenever it contains $x$ is $S$ itself.
That ... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let there exist $\alpha, \beta \in S$ which fulfil the condition for $\struct {S, \circ}$ to be an [[Definition:Inductive Semigroup|inductive semigroup]]:
:the only [[Definition:Subset|subset]] of $S$ containing both $\alpha$ and $x \circ \beta$ wheneve... | Suppose $\struct {S, \circ}$ is a [[Definition:Semigroup|semigroup]].
Suppose there exist $\alpha, \beta \in S$ such that $\struct {S, \circ}$ is an [[Definition:Inductive Semigroup|inductive semigroup]].
That is, suppose there exist $\alpha, \beta \in S$ such that the only [[Definition:Subset|subset]] of $S$ contai... | Inductive Semigroup whose Inductive Elements Commute is Commutative Semigroup | https://proofwiki.org/wiki/Inductive_Semigroup_whose_Inductive_Elements_Commute_is_Commutative_Semigroup | https://proofwiki.org/wiki/Inductive_Semigroup_whose_Inductive_Elements_Commute_is_Commutative_Semigroup | [
"Inductive Semigroups",
"Commutative Semigroups"
] | [
"Definition:Semigroup",
"Definition:Inductive Semigroup",
"Definition:Subset",
"Definition:Commutative/Elements",
"Definition:Commutative Semigroup"
] | [
"Definition:Semigroup",
"Definition:Inductive Semigroup",
"Definition:Subset",
"Form of Elements of Inductive Semigroup",
"Definition:Element",
"Definition:Commutative Semigroup"
] |
proofwiki-19341 | Strictly Inductive Semigroup is Inductive Semigroup | Let $\struct {S, \circ}$ be a strictly inductive semigroup.
Then $\struct {S, \circ}$ is an inductive semigroup. | In accordance with our assertion, let $\struct {S, \circ}$ be a strictly inductive semigroup.
By definition of strictly inductive semigroup, there exists $\beta \in S$ such that the only subset of $S$ containing both $\beta$ and $x \circ \beta$ whenever it contains $x$ is $S$ itself.
That is:
:$\exists \beta \in S: \fo... | Let $\struct {S, \circ}$ be a [[Definition:Strictly Inductive Semigroup|strictly inductive semigroup]].
Then $\struct {S, \circ}$ is an [[Definition:Inductive Semigroup|inductive semigroup]]. | In accordance with our assertion, let $\struct {S, \circ}$ be a [[Definition:Strictly Inductive Semigroup|strictly inductive semigroup]].
By definition of [[Definition:Strictly Inductive Semigroup|strictly inductive semigroup]], there exists $\beta \in S$ such that the only [[Definition:Subset|subset]] of $S$ containi... | Strictly Inductive Semigroup is Inductive Semigroup | https://proofwiki.org/wiki/Strictly_Inductive_Semigroup_is_Inductive_Semigroup | https://proofwiki.org/wiki/Strictly_Inductive_Semigroup_is_Inductive_Semigroup | [
"Inductive Semigroups",
"Strictly Inductive Semigroups"
] | [
"Definition:Strictly Inductive Semigroup",
"Definition:Inductive Semigroup"
] | [
"Definition:Strictly Inductive Semigroup",
"Definition:Strictly Inductive Semigroup",
"Definition:Subset",
"Definition:Subset",
"Definition:Inductive Semigroup"
] |
proofwiki-19342 | Natural Numbers under Addition form Inductive but not Strictly Inductive Semigroup | Let $\struct {\N, +}$ denote the algebraic structure consisting of the set of natural numbers $\N$ under addition $+$.
Then $\struct {\N, +}$ forms an inductive semigroup, but not a strictly inductive semigroup | Recall the definition of inductive semigroup:
{{:Definition:Inductive Semigroup}}
Recall the definition of strictly inductive semigroup:
{{:Definition:Strictly Inductive Semigroup/Definition 1}}
The natural numbers $\N$ can be considered as a naturally ordered semigroup.
From Naturally Ordered Semigroup is Unique, $\st... | Let $\struct {\N, +}$ denote the [[Definition:Algebraic Structure with One Operation|algebraic structure]] consisting of the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ under [[Definition:Natural Number Addition|addition]] $+$.
Then $\struct {\N, +}$ forms an [[Definition:Inductive Se... | Recall the definition of [[Definition:Inductive Semigroup|inductive semigroup]]:
{{:Definition:Inductive Semigroup}}
Recall the definition of [[Definition:Strictly Inductive Semigroup|strictly inductive semigroup]]:
{{:Definition:Strictly Inductive Semigroup/Definition 1}}
The [[Definition:Natural Numbers|natural num... | Natural Numbers under Addition form Inductive but not Strictly Inductive Semigroup | https://proofwiki.org/wiki/Natural_Numbers_under_Addition_form_Inductive_but_not_Strictly_Inductive_Semigroup | https://proofwiki.org/wiki/Natural_Numbers_under_Addition_form_Inductive_but_not_Strictly_Inductive_Semigroup | [
"Natural Number Addition",
"Inductive Semigroups",
"Strictly Inductive Semigroups"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Addition/Natural Numbers",
"Definition:Inductive Semigroup",
"Definition:Strictly Inductive Semigroup"
] | [
"Definition:Inductive Semigroup",
"Definition:Strictly Inductive Semigroup",
"Definition:Natural Numbers",
"Definition:Naturally Ordered Semigroup",
"Naturally Ordered Semigroup is Unique",
"Definition:Unique",
"Definition:Ordered Semigroup Isomorphism",
"Naturally Ordered Semigroup forms Peano Struct... |
proofwiki-19343 | Ideals of P-adic Integers | Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Then the ideals of $\Z_p$ are the principal ideals:
:$\text a) \quad \set 0$
:$\text b) \quad \forall k \in \N: p^k \Z_p$ | Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ denote the $p$-adic valuation on the $p$-adic numbers.
Let $\Z_p^\times$ denote the $p$-adic units.
Let $I \ne \set 0$ be a non-null ideal of $\Z_p$.
Hence:
:$\exists j \in I : \map {\nu_p} j < \infty$
Let:
:$k = \inf \set {\map {\nu_p} i : i \in I}$
Hence:
:$k \le j < \inft... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
Then the [[Definition:Ideal of Ring|ideals]] of $\Z_p$ are the [[Definition:Principal Ideal of Ring|principal ideals]]:
:$\text a) \quad \set 0$
:$\text b) \quad \forall k \in \N: p^k \Z_p$ | Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ denote the [[Definition:P-adic Valuation on P-adic Numbers|$p$-adic valuation]] on the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p^\times$ denote the [[Definition:P-adic Unit|$p$-adic units]].
Let $I \ne \set 0$ be a [[Definition:Non-Null Ide... | Ideals of P-adic Integers | https://proofwiki.org/wiki/Ideals_of_P-adic_Integers | https://proofwiki.org/wiki/Ideals_of_P-adic_Integers | [
"P-adic Integers",
"Ideals of P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Ideal of Ring",
"Definition:Principal Ideal of Ring"
] | [
"Definition:P-adic Valuation/P-adic Numbers",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Unit",
"Definition:Non-Null Ideal",
"P-adic Number is Power of p Times P-adic Unit",
"P-adic Number is Power of p Times P-adic Unit"
] |
proofwiki-19344 | Closed Subgroups of P-adic Integers | Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Then the closed subgroups of $\Z_p$ are the principal ideals:
:$\text a) \quad \set 0$
:$\text b) \quad \forall k \in \N : p^k \Z_p$ | From Metric Space is Hausdorff:
:$\Z_p$ is a Hausdorff space
From Finite Subspace of Hausdorff Space is Closed:
:$\set 0$ is closed
From Cosets Form Local Basis of P-adic Number:
:$\forall k \in \N : p^k \Z_p = 0 + p^k \Z_p$ is closed
Hence the ideals:
:$\text a) \quad \set 0$
:$\text b) \quad \forall k \in \N : p^k \Z... | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
Then the [[Definition:Closed Set (Metric Space)|closed]] [[Definition:Subgroup|subgroups]] of $\Z_p$ are the [[Definition:Principal Ideal of Ring|principal ideals]]:
:$\text a) \quad \set 0$
:$\text b) \q... | From [[Metric Space is Hausdorff]]:
:$\Z_p$ is a [[Definition:Hausdorff Space|Hausdorff space]]
From [[Finite Subspace of Hausdorff Space is Closed]]:
:$\set 0$ is [[Definition:Closed Set (Metric Space)|closed]]
From [[Cosets Form Local Basis of P-adic Number]]:
:$\forall k \in \N : p^k \Z_p = 0 + p^k \Z_p$ is [[Defi... | Closed Subgroups of P-adic Integers | https://proofwiki.org/wiki/Closed_Subgroups_of_P-adic_Integers | https://proofwiki.org/wiki/Closed_Subgroups_of_P-adic_Integers | [
"P-adic Integers"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Closed Set/Metric Space",
"Definition:Subgroup",
"Definition:Principal Ideal of Ring"
] | [
"Metric Space is T2",
"Definition:T2 Space",
"Compact Subspace of Hausdorff Space is Closed/Corollary",
"Definition:Closed Set/Metric Space",
"Local Basis of P-adic Number/Cosets",
"Definition:Closed Set/Metric Space",
"Definition:Ideal of Ring",
"Definition:Closed Set/Metric Space",
"Definition:Sub... |
proofwiki-19345 | Ideals of P-adic Integers/Corollary | :$\Z_p$ is a principal ideal domain | From Ideals of P-adic Integers, all ideals of $\Z_p$ are the principal ideals:
:* $\quad\set 0$
:* $\quad\forall n \in \N : p^n \Z_p$
Hence $\Z_p$ is a principal ideal domain by definition.
{{qed}} | :$\Z_p$ is a [[Definition:Principal Ideal Domain|principal ideal domain]] | From [[Ideals of P-adic Integers]], all [[Definition:Ideal of Ring|ideals]] of $\Z_p$ are the [[Definition:Principal Ideal|principal ideals]]:
:* $\quad\set 0$
:* $\quad\forall n \in \N : p^n \Z_p$
Hence $\Z_p$ is a [[Definition:Principal Ideal Domain|principal ideal domain]] by definition.
{{qed}} | Ideals of P-adic Integers/Corollary | https://proofwiki.org/wiki/Ideals_of_P-adic_Integers/Corollary | https://proofwiki.org/wiki/Ideals_of_P-adic_Integers/Corollary | [
"Ideals of P-adic Integers"
] | [
"Definition:Principal Ideal Domain"
] | [
"Ideals of P-adic Integers",
"Definition:Ideal of Ring",
"Definition:Principal Ideal",
"Definition:Principal Ideal Domain"
] |
proofwiki-19346 | Form of Elements of Inductive Semigroup | Let $\struct {S, \circ}$ be an inductive semigroup.
Then the elements of $S$ are of the form:
:$\alpha \circ \beta \circ \beta \circ \cdots \circ \beta$ | Recall the definition of inductive semigroup:
{{:Definition:Inductive Semigroup}}
It follows from the definition that all elements of the form:
:$\alpha \circ \beta \circ \beta \circ \cdots \circ \beta$
are indeed elements of $S$.
{{AimForCont}} $x \in S$ is not of the above form.
Let $A \subseteq S$ be such that $\alp... | Let $\struct {S, \circ}$ be an [[Definition:Inductive Semigroup|inductive semigroup]].
Then the [[Definition:Element|elements]] of $S$ are of the form:
:$\alpha \circ \beta \circ \beta \circ \cdots \circ \beta$ | Recall the definition of [[Definition:Inductive Semigroup|inductive semigroup]]:
{{:Definition:Inductive Semigroup}}
It follows from the definition that all [[Definition:Element|elements]] of the form:
:$\alpha \circ \beta \circ \beta \circ \cdots \circ \beta$
are indeed [[Definition:Element|elements]] of $S$.
{{Aim... | Form of Elements of Inductive Semigroup | https://proofwiki.org/wiki/Form_of_Elements_of_Inductive_Semigroup | https://proofwiki.org/wiki/Form_of_Elements_of_Inductive_Semigroup | [
"Inductive Semigroups"
] | [
"Definition:Inductive Semigroup",
"Definition:Element"
] | [
"Definition:Inductive Semigroup",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Definition:Inductive Semigroup",
"Definition:Contradiction",
"Definition:Element",
"Category:Inductive Semigroups"
] |
proofwiki-19347 | Left Self-Distributive Operation with Right Identity is Idempotent | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ be left self-distributive.
Let $\struct {S, \circ}$ have a right identity.
Then $\circ$ is an idempotent operation. | Let the right identity of $\struct {S, \circ}$ be $e_R$.
We have:
{{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = a \circ \paren {b \circ c}
| r = \paren {a \circ b} \circ \paren {a \circ c}
| c = {{Defof|Left Self-Distributive Operation}}
}}
{{eqn | ll= \leadsto
| q = \forall a \in S
... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\circ$ be [[Definition:Left Self-Distributive Operation|left self-distributive]].
Let $\struct {S, \circ}$ have a [[Definition:Right Identity|right identity]].
Then $\circ$ is an [[Definition:Idempotent Operation|idempotent ... | Let the [[Definition:Right Identity|right identity]] of $\struct {S, \circ}$ be $e_R$.
We have:
{{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = a \circ \paren {b \circ c}
| r = \paren {a \circ b} \circ \paren {a \circ c}
| c = {{Defof|Left Self-Distributive Operation}}
}}
{{eqn | ll= \leadsto... | Left Self-Distributive Operation with Right Identity is Idempotent | https://proofwiki.org/wiki/Left_Self-Distributive_Operation_with_Right_Identity_is_Idempotent | https://proofwiki.org/wiki/Left_Self-Distributive_Operation_with_Right_Identity_is_Idempotent | [
"Idempotence",
"Self-Distributive Operations",
"Identity Elements"
] | [
"Definition:Algebraic Structure",
"Definition:Self-Distributive Operation/Left",
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Idempotence/Operation"
] | [
"Definition:Identity (Abstract Algebra)/Right Identity",
"Definition:Idempotence/Operation"
] |
proofwiki-19348 | Right Self-Distributive Operation with Left Identity is Idempotent | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ be right self-distributive.
Let $\struct {S, \circ}$ have a left identity.
Then $\circ$ is an idempotent operation. | Let the left identity of $\struct {S, \circ}$ be $e_L$.
We have:
{{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \circ b} \circ c
| r = \paren {a \circ c} \circ \paren {b \circ c}
| c = {{Defof|Right Self-Distributive Operation}}
}}
{{eqn | ll= \leadsto
| q = \forall c \in S
... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\circ$ be [[Definition:Right Self-Distributive Operation|right self-distributive]].
Let $\struct {S, \circ}$ have a [[Definition:Left Identity|left identity]].
Then $\circ$ is an [[Definition:Idempotent Operation|idempotent ... | Let the [[Definition:Left Identity|left identity]] of $\struct {S, \circ}$ be $e_L$.
We have:
{{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \circ b} \circ c
| r = \paren {a \circ c} \circ \paren {b \circ c}
| c = {{Defof|Right Self-Distributive Operation}}
}}
{{eqn | ll= \leadsto
... | Right Self-Distributive Operation with Left Identity is Idempotent | https://proofwiki.org/wiki/Right_Self-Distributive_Operation_with_Left_Identity_is_Idempotent | https://proofwiki.org/wiki/Right_Self-Distributive_Operation_with_Left_Identity_is_Idempotent | [
"Idempotence",
"Self-Distributive Operations",
"Identity Elements"
] | [
"Definition:Algebraic Structure",
"Definition:Self-Distributive Operation/Right",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Definition:Idempotence/Operation"
] | [
"Definition:Identity (Abstract Algebra)/Left Identity",
"Definition:Idempotence/Operation"
] |
proofwiki-19349 | Product on Left with Idempotent Element under Left Self-Distributive Operation is Idempotent | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ be left self-distributive.
Let $a \in S$ be an idempotent element of $\struct {S, \circ}$.
Then for all $b \in S$, $b \circ a$ is an idempotent element of $\struct {S, \circ}$ | Let $a \in S$ be an idempotent element of $\struct {S, \circ}$.
We have:
{{begin-eqn}}
{{eqn | q = \forall b \in S
| l = \paren {b \circ a} \circ \paren {b \circ a}
| r = b \circ \paren {a \circ a}
| c = {{Defof|Left Self-Distributive Operation}}
}}
{{eqn | r = b \circ a
| c = {{Defof|Idempotent... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\circ$ be [[Definition:Left Self-Distributive Operation|left self-distributive]].
Let $a \in S$ be an [[Definition:Idempotent Element|idempotent element]] of $\struct {S, \circ}$.
Then for all $b \in S$, $b \circ a$ is an [[... | Let $a \in S$ be an [[Definition:Idempotent Element|idempotent element]] of $\struct {S, \circ}$.
We have:
{{begin-eqn}}
{{eqn | q = \forall b \in S
| l = \paren {b \circ a} \circ \paren {b \circ a}
| r = b \circ \paren {a \circ a}
| c = {{Defof|Left Self-Distributive Operation}}
}}
{{eqn | r = b \c... | Product on Left with Idempotent Element under Left Self-Distributive Operation is Idempotent | https://proofwiki.org/wiki/Product_on_Left_with_Idempotent_Element_under_Left_Self-Distributive_Operation_is_Idempotent | https://proofwiki.org/wiki/Product_on_Left_with_Idempotent_Element_under_Left_Self-Distributive_Operation_is_Idempotent | [
"Idempotence",
"Self-Distributive Operations"
] | [
"Definition:Algebraic Structure",
"Definition:Self-Distributive Operation/Left",
"Definition:Idempotence/Element",
"Definition:Idempotence/Element"
] | [
"Definition:Idempotence/Element",
"Definition:Idempotence/Element"
] |
proofwiki-19350 | Product on Right with Idempotent Element under Right Self-Distributive Operation is Idempotent | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ be right self-distributive.
Let $a \in S$ be an idempotent element of $\struct {S, \circ}$.
Then for all $b \in S$, $a \circ b$ is an idempotent element of $\struct {S, \circ}$ | Let $a \in S$ be an idempotent element of $\struct {S, \circ}$.
We have:
{{begin-eqn}}
{{eqn | q = \forall b \in S
| l = \paren {a \circ b} \circ \paren {a \circ b}
| r = \paren {a \circ a} \circ b
| c = {{Defof|Right Self-Distributive Operation}}
}}
{{eqn | r = a \circ b
| c = {{Defof|Idempoten... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\circ$ be [[Definition:Right Self-Distributive Operation|right self-distributive]].
Let $a \in S$ be an [[Definition:Idempotent Element|idempotent element]] of $\struct {S, \circ}$.
Then for all $b \in S$, $a \circ b$ is an ... | Let $a \in S$ be an [[Definition:Idempotent Element|idempotent element]] of $\struct {S, \circ}$.
We have:
{{begin-eqn}}
{{eqn | q = \forall b \in S
| l = \paren {a \circ b} \circ \paren {a \circ b}
| r = \paren {a \circ a} \circ b
| c = {{Defof|Right Self-Distributive Operation}}
}}
{{eqn | r = a \... | Product on Right with Idempotent Element under Right Self-Distributive Operation is Idempotent | https://proofwiki.org/wiki/Product_on_Right_with_Idempotent_Element_under_Right_Self-Distributive_Operation_is_Idempotent | https://proofwiki.org/wiki/Product_on_Right_with_Idempotent_Element_under_Right_Self-Distributive_Operation_is_Idempotent | [
"Idempotence",
"Self-Distributive Operations"
] | [
"Definition:Algebraic Structure",
"Definition:Self-Distributive Operation/Right",
"Definition:Idempotence/Element",
"Definition:Idempotence/Element"
] | [
"Definition:Idempotence/Element",
"Definition:Idempotence/Element"
] |
proofwiki-19351 | Condition for Operation to be Left Distributive over Constant Operation | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\sqbrk c$ be the constant operation for some $c \in S$.
Then:
:$\circ$ is left distributive over $\sqbrk c$
{{iff}}:
:$\forall x \in S: x \circ c = c$ | === Sufficient Condition ===
Let $\circ$ be left distributive over $\sqbrk c$.
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in S
| l = c
| r = \paren {x \circ y} \sqbrk c \paren {x \circ z}
| c = {{Defof|Constant Operation}}
}}
{{eqn | r = x \circ \paren {y \sqbrk c z}
| c = {{Defof|Left Distribut... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\sqbrk c$ be the [[Definition:Constant Operation|constant operation]] for some $c \in S$.
Then:
:$\circ$ is [[Definition:Left Distributive Operation|left distributive]] over $\sqbrk c$
{{iff}}:
:$\forall x \in S: x \circ c = ... | === Sufficient Condition ===
Let $\circ$ be [[Definition:Left Distributive Operation|left distributive]] over $\sqbrk c$.
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in S
| l = c
| r = \paren {x \circ y} \sqbrk c \paren {x \circ z}
| c = {{Defof|Constant Operation}}
}}
{{eqn | r = x \circ \paren {y \... | Condition for Operation to be Left Distributive over Constant Operation | https://proofwiki.org/wiki/Condition_for_Operation_to_be_Left_Distributive_over_Constant_Operation | https://proofwiki.org/wiki/Condition_for_Operation_to_be_Left_Distributive_over_Constant_Operation | [
"Constant Operation",
"Distributive Operations"
] | [
"Definition:Algebraic Structure",
"Definition:Constant Operation",
"Definition:Distributive Operation/Left"
] | [
"Definition:Distributive Operation/Left",
"Definition:Distributive Operation/Left"
] |
proofwiki-19352 | Condition for Operation to be Right Distributive over Constant Operation | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\sqbrk c$ be the constant operation for some $c \in S$.
Then:
:$\circ$ is right distributive over $\sqbrk c$
{{iff}}:
:$\forall x \in S: c \circ x = c$ | === Sufficient Condition ===
Let $\circ$ be right distributive over $\sqbrk c$.
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in S
| l = c
| r = \paren {y \circ x} \sqbrk c \paren {z \circ x}
| c = {{Defof|Constant Operation}}
}}
{{eqn | r = \paren {y \sqbrk c z} \circ x
| c = {{Defof|Right Distrib... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\sqbrk c$ be the [[Definition:Constant Operation|constant operation]] for some $c \in S$.
Then:
:$\circ$ is [[Definition:Right Distributive Operation|right distributive]] over $\sqbrk c$
{{iff}}:
:$\forall x \in S: c \circ x ... | === Sufficient Condition ===
Let $\circ$ be [[Definition:RIght Distributive Operation|right distributive]] over $\sqbrk c$.
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in S
| l = c
| r = \paren {y \circ x} \sqbrk c \paren {z \circ x}
| c = {{Defof|Constant Operation}}
}}
{{eqn | r = \paren {y \sqbrk ... | Condition for Operation to be Right Distributive over Constant Operation | https://proofwiki.org/wiki/Condition_for_Operation_to_be_Right_Distributive_over_Constant_Operation | https://proofwiki.org/wiki/Condition_for_Operation_to_be_Right_Distributive_over_Constant_Operation | [
"Constant Operation",
"Distributive Operations"
] | [
"Definition:Algebraic Structure",
"Definition:Constant Operation",
"Definition:Distributive Operation/Right"
] | [
"Definition:RIght Distributive Operation"
] |
proofwiki-19353 | Condition for Constant Operation to be Distributive over Another Operation | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\sqbrk c$ denote the constant operation for some $c \in S$.
Then:
:$\sqbrk c$ is distributive over $\circ$
{{iff}}:
:$c \circ c = c$ | === Sufficient Condition ===
Let $\sqbrk c$ be distributive over $\circ$
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in S
| l = \paren {x \sqbrk c y} \circ \paren {x \sqbrk c z}
| r = c circ c
| c = {{Defof|Constant Operation}}
}}
{{eqn | q = \forall x, y, z \in S
| l = x \sqbrk c \paren {y \circ... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\sqbrk c$ denote the [[Definition:Constant Operation|constant operation]] for some $c \in S$.
Then:
:$\sqbrk c$ is [[Definition:Distributive Operation|distributive]] over $\circ$
{{iff}}:
:$c \circ c = c$ | === Sufficient Condition ===
Let $\sqbrk c$ be [[Definition:Distributive Operation|distributive]] over $\circ$
{{begin-eqn}}
{{eqn | q = \forall x, y, z \in S
| l = \paren {x \sqbrk c y} \circ \paren {x \sqbrk c z}
| r = c circ c
| c = {{Defof|Constant Operation}}
}}
{{eqn | q = \forall x, y, z \in ... | Condition for Constant Operation to be Distributive over Another Operation | https://proofwiki.org/wiki/Condition_for_Constant_Operation_to_be_Distributive_over_Another_Operation | https://proofwiki.org/wiki/Condition_for_Constant_Operation_to_be_Distributive_over_Another_Operation | [
"Constant Operation",
"Distributive Operations"
] | [
"Definition:Algebraic Structure",
"Definition:Constant Operation",
"Definition:Distributive Operation"
] | [
"Definition:Distributive Operation",
"Definition:Distributive Operation"
] |
proofwiki-19354 | Operation which is Left Distributive over Every Commutative Associative Operation is Right Operation | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ have the property that:
:for every arbitrary operation $*$ on $S$ which is both commutative and associative, $\circ$ is left distributive over $*$.
Then $\circ$ is the right operation $\to$:
:$\forall a, b \in S: a \to b = b$ | First recall from Right Operation is Left Distributive over All Operations that the right operation is indeed left distributive over all operations, whether commutative or associative.
Let $*$ be an arbitrary operation on $S$ which is both commutative and associative.
As asserted, let $\circ$ be left distributive over... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\circ$ have the property that:
:for every arbitrary [[Definition:Binary Operation|operation]] $*$ on $S$ which is both [[Definition:Commutative Operation|commutative]] and [[Definition:Associative Operation|associative]], $\cir... | First recall from [[Right Operation is Left Distributive over All Operations]] that the [[Definition:Right Operation|right operation]] is indeed [[Definition:Left Distributive Operation|left distributive]] over all [[Definition:Binary Operation|operations]], whether [[Definition:Commutative Operation|commutative]] or ... | Operation which is Left Distributive over Every Commutative Associative Operation is Right Operation | https://proofwiki.org/wiki/Operation_which_is_Left_Distributive_over_Every_Commutative_Associative_Operation_is_Right_Operation | https://proofwiki.org/wiki/Operation_which_is_Left_Distributive_over_Every_Commutative_Associative_Operation_is_Right_Operation | [
"Right Operation",
"Distributive Operations",
"Commutativity",
"Associativity"
] | [
"Definition:Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation",
"Definition:Associative Operation",
"Definition:Distributive Operation/Left",
"Definition:Right Operation"
] | [
"Right Operation is Left Distributive over All Operations",
"Definition:Right Operation",
"Definition:Distributive Operation/Left",
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation",
"Definition:Associative Operation",
"Definition:Operation/Binary Operation",
"Definition:Comm... |
proofwiki-19355 | Operation which is Right Distributive over Every Commutative Associative Operation is Left Operation | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ have the property that:
:for every arbitrary operation $*$ on $S$ which is both commutative and associative, $\circ$ is right distributive over $*$.
Then $\circ$ is the left operation $\gets$:
:$\forall a, b \in S: a \gets b = a$ | First recall from Left Operation is Right Distributive over All Operations that the left operation is indeed right distributive over all operations, whether commutative or associative.
Let $*$ be an arbitrary operation on $S$ which is both commutative and associative.
As asserted, let $\circ$ be right distributive over... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\circ$ have the property that:
:for every arbitrary [[Definition:Binary Operation|operation]] $*$ on $S$ which is both [[Definition:Commutative Operation|commutative]] and [[Definition:Associative Operation|associative]], $\cir... | First recall from [[Left Operation is Right Distributive over All Operations]] that the [[Definition:Left Operation|left operation]] is indeed [[Definition:Right Distributive Operation|right distributive]] over all [[Definition:Binary Operation|operations]], whether [[Definition:Commutative Operation|commutative]] or [... | Operation which is Right Distributive over Every Commutative Associative Operation is Left Operation | https://proofwiki.org/wiki/Operation_which_is_Right_Distributive_over_Every_Commutative_Associative_Operation_is_Left_Operation | https://proofwiki.org/wiki/Operation_which_is_Right_Distributive_over_Every_Commutative_Associative_Operation_is_Left_Operation | [
"Left Operation",
"Distributive Operations",
"Commutativity",
"Associativity"
] | [
"Definition:Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation",
"Definition:Associative Operation",
"Definition:Distributive Operation/Right",
"Definition:Left Operation"
] | [
"Left Operation is Right Distributive over All Operations",
"Definition:Left Operation",
"Definition:Distributive Operation/Right",
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation",
"Definition:Associative Operation",
"Definition:Operation/Binary Operation",
"Definition:Comm... |
proofwiki-19356 | Every Operation is Distributive over Right Operation | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\to$ denote the right operation on $S$.
Then $\circ$ is distributive over $\to$. | {{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = a \circ \paren {b \to c}
| r = a \circ c
| c = {{Defof|Right Operation}}
}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \circ b} \to \paren {a \circ c}
| r = a \circ c
| c = {{Defof|Right Operation}}
}}
{{eqn | ll= \leadsto
... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\to$ denote the [[Definition:Right Operation|right operation]] on $S$.
Then $\circ$ is [[Definition:Distributive Operation|distributive]] over $\to$. | {{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = a \circ \paren {b \to c}
| r = a \circ c
| c = {{Defof|Right Operation}}
}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \circ b} \to \paren {a \circ c}
| r = a \circ c
| c = {{Defof|Right Operation}}
}}
{{eqn | ll= \leadsto
... | Every Operation is Distributive over Right Operation | https://proofwiki.org/wiki/Every_Operation_is_Distributive_over_Right_Operation | https://proofwiki.org/wiki/Every_Operation_is_Distributive_over_Right_Operation | [
"Right Operation",
"Distributive Operations"
] | [
"Definition:Algebraic Structure",
"Definition:Right Operation",
"Definition:Distributive Operation"
] | [
"Definition:Distributive Operation"
] |
proofwiki-19357 | Every Operation is Distributive over Left Operation | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\gets$ denote the left operation on $S$.
Then $\circ$ is distributive over $\gets$. | {{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = a \circ \paren {b \gets c}
| r = a \circ b
| c = {{Defof|Left Operation}}
}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \circ b} \gets \paren {a \circ c}
| r = a \circ b
| c = {{Defof|Left Operation}}
}}
{{eqn | ll= \leadst... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\gets$ denote the [[Definition:Left Operation|left operation]] on $S$.
Then $\circ$ is [[Definition:Distributive Operation|distributive]] over $\gets$. | {{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = a \circ \paren {b \gets c}
| r = a \circ b
| c = {{Defof|Left Operation}}
}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \circ b} \gets \paren {a \circ c}
| r = a \circ b
| c = {{Defof|Left Operation}}
}}
{{eqn | ll= \leadst... | Every Operation is Distributive over Left Operation | https://proofwiki.org/wiki/Every_Operation_is_Distributive_over_Left_Operation | https://proofwiki.org/wiki/Every_Operation_is_Distributive_over_Left_Operation | [
"Left Operation",
"Distributive Operations"
] | [
"Definition:Algebraic Structure",
"Definition:Left Operation",
"Definition:Distributive Operation"
] | [
"Definition:Distributive Operation"
] |
proofwiki-19358 | Absolute Continuity of Measures is Transitive Relation | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$, $\nu$ and $\lambda$ be measures on $\struct {X, \Sigma}$ such that:
:$\mu$ is absolutely continuous {{WRT}} $\nu$
and:
:$\nu$ is absolutely continuous {{WRT}} $\lambda$.
That is:
:$\mu \ll \nu$
and:
:$\nu \ll \lambda$
Then:
:$\mu$ is absolutely continuous {{... | Let $A \in \Sigma$ be such that $\map \lambda A = 0$.
Then, since $\nu \ll \lambda$ we have $\map \nu A = 0$ from the definition of absolute continuity.
Since $\mu \ll \nu$ we similarly have $\map \mu A = 0$ again applying the definition of absolute continuity.
So, whenever $A \in \Sigma$ is such that $\map \lambda A ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$, $\nu$ and $\lambda$ be [[Definition:Measure (Measure Theory)|measures]] on $\struct {X, \Sigma}$ such that:
:$\mu$ is [[Definition:Absolutely Continuous Measure|absolutely continuous]] {{WRT}} $\nu$
and:
:$\nu$ is [[Definit... | Let $A \in \Sigma$ be such that $\map \lambda A = 0$.
Then, since $\nu \ll \lambda$ we have $\map \nu A = 0$ from the definition of [[Definition:Absolutely Continuous Measure|absolute continuity]].
Since $\mu \ll \nu$ we similarly have $\map \mu A = 0$ again applying the definition of [[Definition:Absolutely Continu... | Absolute Continuity of Measures is Transitive Relation | https://proofwiki.org/wiki/Absolute_Continuity_of_Measures_is_Transitive_Relation | https://proofwiki.org/wiki/Absolute_Continuity_of_Measures_is_Transitive_Relation | [
"Absolutely Continuous Measures",
"Transitive Relations"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Absolute Continuity/Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Transitive Relation"
] | [
"Definition:Absolute Continuity/Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Absolute Continuity/Measure",
"Category:Absolutely Continuous Measures",
"Category:Transitive Relations"
] |
proofwiki-19359 | P-adic Integers Form Integral Domain | Let $\Q_p$ be the $p$-adic numbers for some prime $p$.
Let $\Z_p$ be the $p$-adic integers.
Then:
:$\Z_p$ is an integral domain | From Field is Integral Domain:
:$\Q_p$ is an integral domain
From {{Corollary|P-adic Integers is Valuation Ring Induced by P-adic Norm}}:
:$\Z_p$ is a local ring
By the definition of local ring:
:$\Z_p$ is a ring with unity
From Subdomain Test:
:$\Z_p$ is an integral domain
{{qed}} | Let $\Q_p$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Then:
:$\Z_p$ is an [[Definition:Integral Domain|integral domain]] | From [[Field is Integral Domain]]:
:$\Q_p$ is an [[Definition:Integral Domain|integral domain]]
From {{Corollary|P-adic Integers is Valuation Ring Induced by P-adic Norm}}:
:$\Z_p$ is a [[Definition:Local Ring|local ring]]
By the definition of [[Definition:Local Ring|local ring]]:
:$\Z_p$ is a [[Definition:Ring with... | P-adic Integers Form Integral Domain | https://proofwiki.org/wiki/P-adic_Integers_Form_Integral_Domain | https://proofwiki.org/wiki/P-adic_Integers_Form_Integral_Domain | [
"P-adic Integers",
"Examples of Integral Domains"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:P-adic Integer",
"Definition:Integral Domain"
] | [
"Field is Integral Domain",
"Definition:Integral Domain",
"Definition:Local Ring",
"Definition:Local Ring",
"Definition:Ring with Unity",
"Subdomain Test",
"Definition:Integral Domain"
] |
proofwiki-19360 | Operation over which Every Commutative Associative Operation is Distributive is either Left or Right Operation/Lemma | Let $S$ be a set.
Let $A \subsetneqq S$ be a proper subset of $S$.
Let $a \in A$ and $b \in \relcomp S A$.
Let $\odot$ be the operation on $S$ defined as:
:$\forall x, y \in S: x \odot y = \begin {cases} a & : \set {x, y} \subseteq A \\ b & : \set {x, y} \not \subseteq A \end {cases}$
Then $\odot$ is commutative and as... | === Commutativity ===
Trivially:
{{begin-eqn}}
{{eqn | q = \forall x, y \in S
| l = y \odot x
| r = \begin {cases} a & : \set {y, x} \subseteq A \\ b & : \set {y, x} \not \subseteq A \end {cases}
| c = Definition of $\odot$
}}
{{eqn | r = \begin {cases} a & : \set {x, y} \subseteq A \\ b & : \set {x, ... | Let $S$ be a [[Definition:Set|set]].
Let $A \subsetneqq S$ be a [[Definition:Proper Subset|proper subset]] of $S$.
Let $a \in A$ and $b \in \relcomp S A$.
Let $\odot$ be the [[Definition:Binary Operation|operation]] on $S$ defined as:
:$\forall x, y \in S: x \odot y = \begin {cases} a & : \set {x, y} \subseteq A \\... | === Commutativity ===
Trivially:
{{begin-eqn}}
{{eqn | q = \forall x, y \in S
| l = y \odot x
| r = \begin {cases} a & : \set {y, x} \subseteq A \\ b & : \set {y, x} \not \subseteq A \end {cases}
| c = Definition of $\odot$
}}
{{eqn | r = \begin {cases} a & : \set {x, y} \subseteq A \\ b & : \set {x... | Operation over which Every Commutative Associative Operation is Distributive is either Left or Right Operation/Lemma | https://proofwiki.org/wiki/Operation_over_which_Every_Commutative_Associative_Operation_is_Distributive_is_either_Left_or_Right_Operation/Lemma | https://proofwiki.org/wiki/Operation_over_which_Every_Commutative_Associative_Operation_is_Distributive_is_either_Left_or_Right_Operation/Lemma | [
"Operation over which Every Commutative Associative Operation is Distributive is either Left or Right Operation"
] | [
"Definition:Set",
"Definition:Proper Subset",
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation",
"Definition:Associative Operation"
] | [] |
proofwiki-19361 | Operation over which Every Commutative Associative Operation is Distributive is either Left or Right Operation | Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ be such that every operation on $S$ which is both commutative and associative is distributive over $\circ$.
Then $\circ$ is either the left operation $\gets$ or the right operation $\to$. | Recall from:
:Every Operation is Distributive over Right Operation
and:
:Every Operation is Distributive over Left Operation
that if $\circ$ is either $\gets$ or $\to$, then every operation is distributive over it, whether commutative or associative. | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $\circ$ be such that every [[Definition:Binary Operation|operation]] on $S$ which is both [[Definition:Commutative Operation|commutative]] and [[Definition:Associative Operation|associative]] is [[Definition:Distributive Operati... | Recall from:
:[[Every Operation is Distributive over Right Operation]]
and:
:[[Every Operation is Distributive over Left Operation]]
that if $\circ$ is either $\gets$ or $\to$, then every [[Definition:Binary Operation|operation]] is [[Definition:Distributive Operation|distributive]] over it, whether [[Definition:Commu... | Operation over which Every Commutative Associative Operation is Distributive is either Left or Right Operation | https://proofwiki.org/wiki/Operation_over_which_Every_Commutative_Associative_Operation_is_Distributive_is_either_Left_or_Right_Operation | https://proofwiki.org/wiki/Operation_over_which_Every_Commutative_Associative_Operation_is_Distributive_is_either_Left_or_Right_Operation | [
"Left Operation",
"Right Operation",
"Commutativity",
"Associativity",
"Distributive Operations",
"Operation over which Every Commutative Associative Operation is Distributive is either Left or Right Operation"
] | [
"Definition:Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation",
"Definition:Associative Operation",
"Definition:Distributive Operation",
"Definition:Left Operation",
"Definition:Right Operation"
] | [
"Every Operation is Distributive over Right Operation",
"Every Operation is Distributive over Left Operation",
"Definition:Operation/Binary Operation",
"Definition:Distributive Operation",
"Definition:Commutative/Operation",
"Definition:Associative Operation",
"Definition:Distributive Operation",
"Def... |
proofwiki-19362 | Entropic Idempotent Structure is Self-Distributive | Let $\struct {S, \odot}$ be an algebraic structure such that $\odot$ is both idempotent and entropic.
Then $\struct {S, \odot}$ is a self-distributive structure. | {{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \odot b} \odot \paren {a \odot c}
| r = \paren {a \odot a} \odot \paren {b \odot c}
| c = {{Defof|Entropic Operation}}
}}
{{eqn | r = a \odot \paren {b \odot c}
| c = {{Defof|Idempotent Operation}}
}}
{{end-eqn}}
and:
{{begin-eqn}... | Let $\struct {S, \odot}$ be an [[Definition:Algebraic Structure|algebraic structure]] such that $\odot$ is both [[Definition:Idempotent Operation|idempotent]] and [[Definition:Entropic Operation|entropic]].
Then $\struct {S, \odot}$ is a [[Definition:Self-Distributive Structure|self-distributive structure]]. | {{begin-eqn}}
{{eqn | q = \forall a, b, c \in S
| l = \paren {a \odot b} \odot \paren {a \odot c}
| r = \paren {a \odot a} \odot \paren {b \odot c}
| c = {{Defof|Entropic Operation}}
}}
{{eqn | r = a \odot \paren {b \odot c}
| c = {{Defof|Idempotent Operation}}
}}
{{end-eqn}}
and:
{{begin-eq... | Entropic Idempotent Structure is Self-Distributive | https://proofwiki.org/wiki/Entropic_Idempotent_Structure_is_Self-Distributive | https://proofwiki.org/wiki/Entropic_Idempotent_Structure_is_Self-Distributive | [
"Idempotence",
"Entropic Structures",
"Self-Distributive Operations"
] | [
"Definition:Algebraic Structure",
"Definition:Idempotence/Operation",
"Definition:Entropic Operation",
"Definition:Self-Distributive Structure"
] | [
"Definition:Self-Distributive Structure"
] |
proofwiki-19363 | Set of Idempotent Elements of Entropic Structure is Closed | Let $\struct {S, \odot}$ be an algebraic structure such that $\odot$ is entropic.
Let $T \subseteq S$ be the set of idempotent elements of $S$.
Then $\struct {T, \odot}$ is closed. | {{begin-eqn}}
{{eqn | q = \forall a, b \in T
| l = \paren {a \odot b} \odot \paren {a \odot b}
| r = \paren {a \odot a} \odot \paren {b \odot b}
| c = {{Defof|Entropic Operation}}
}}
{{eqn | r = a \odot b
| c = {{Defof|Idempotent Operation}}
}}
{{eqn | ll= \leadsto
| l = a \odot b
... | Let $\struct {S, \odot}$ be an [[Definition:Algebraic Structure|algebraic structure]] such that $\odot$ is [[Definition:Entropic Operation|entropic]].
Let $T \subseteq S$ be the [[Definition:Set|set]] of [[Definition:Idempotent Element|idempotent elements]] of $S$.
Then $\struct {T, \odot}$ is [[Definition:Closed Al... | {{begin-eqn}}
{{eqn | q = \forall a, b \in T
| l = \paren {a \odot b} \odot \paren {a \odot b}
| r = \paren {a \odot a} \odot \paren {b \odot b}
| c = {{Defof|Entropic Operation}}
}}
{{eqn | r = a \odot b
| c = {{Defof|Idempotent Operation}}
}}
{{eqn | ll= \leadsto
| l = a \odot b
... | Set of Idempotent Elements of Entropic Structure is Closed | https://proofwiki.org/wiki/Set_of_Idempotent_Elements_of_Entropic_Structure_is_Closed | https://proofwiki.org/wiki/Set_of_Idempotent_Elements_of_Entropic_Structure_is_Closed | [
"Idempotence",
"Entropic Structures"
] | [
"Definition:Algebraic Structure",
"Definition:Entropic Operation",
"Definition:Set",
"Definition:Idempotence/Element",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [] |
proofwiki-19364 | Endomorphisms on Entropic Structure whose Pointwise Product is Identity Automorphism | Let $\struct {S, \odot}$ be an entropic structure.
Let $\alpha$ and $\beta$ be endomorphisms on $S$ such that:
:$\alpha \odot \beta$ is the identity automorphism on $S$
where $\alpha \odot \beta$ denotes the pointwise product of $\alpha$ with $\beta$:
:$\forall x \in S: \map {\paren {\alpha \odot \beta} } x = \map \alp... | Let $a, b, c, d \in S$ be arbitrary.
{{begin-eqn}}
{{eqn | q =
| l = a \otimes a
| r = \map \alpha a \odot \map \beta a
| c = Definition of $\otimes$
}}
{{eqn | r = \map {\paren {\alpha \odot \beta} } a
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = a
| c = {{Defof|Identity Mapping}}
... | Let $\struct {S, \odot}$ be an [[Definition:Entropic Structure|entropic structure]].
Let $\alpha$ and $\beta$ be [[Definition:Endomorphism|endomorphisms]] on $S$ such that:
:$\alpha \odot \beta$ is the [[Definition:Identity Mapping|identity]] [[Definition:Automorphism (Abstract Algebra)|automorphism]] on $S$
where $\a... | Let $a, b, c, d \in S$ be arbitrary.
{{begin-eqn}}
{{eqn | q =
| l = a \otimes a
| r = \map \alpha a \odot \map \beta a
| c = Definition of $\otimes$
}}
{{eqn | r = \map {\paren {\alpha \odot \beta} } a
| c = {{Defof|Pointwise Operation}}
}}
{{eqn | r = a
| c = {{Defof|Identity Mapping}... | Endomorphisms on Entropic Structure whose Pointwise Product is Identity Automorphism | https://proofwiki.org/wiki/Endomorphisms_on_Entropic_Structure_whose_Pointwise_Product_is_Identity_Automorphism | https://proofwiki.org/wiki/Endomorphisms_on_Entropic_Structure_whose_Pointwise_Product_is_Identity_Automorphism | [
"Examples of Idempotence",
"Examples of Entropic Operations",
"Examples of Self-Distributive Operations"
] | [
"Definition:Entropic Structure",
"Definition:Endomorphism",
"Definition:Identity Mapping",
"Definition:Automorphism (Abstract Algebra)",
"Definition:Pointwise Operation",
"Definition:Operation/Binary Operation",
"Definition:Entropic Structure",
"Definition:Idempotence/Operation",
"Definition:Self-Di... | [
"Definition:Idempotence/Operation"
] |
proofwiki-19365 | Naturally Ordered Semigroup is Unique/Isomorphism is Unique | Let $\struct {S, \circ, \preceq}$ and $\struct {S', \circ', \preceq'}$ be naturally ordered semigroups.
Let:
:$0'$ be the smallest element of $S'$
:$1'$ be the smallest element of $S' \setminus \set {0'} = S'^*$.
Then the isomorphism $g: \struct {S, \circ, \preceq} \to \struct {S', \circ', \preceq'}$ defined as:
:$\for... | Let $f: S \to S'$ be another isomorphism different from $g$.
{{AimForCont}} $\map f 1 \ne 1'$.
We show by induction that $1' \notin \Cdm f$.
{{finish}}
... Thus $1' \notin \Cdm f$ which is a contradiction.
Thus $\map f 1 = 1$ and it follows
{{finish}}
that $f = g$.
Thus the isomorphism $g$ is unique. | Let $\struct {S, \circ, \preceq}$ and $\struct {S', \circ', \preceq'}$ be [[Definition:Naturally Ordered Semigroup|naturally ordered semigroups]].
Let:
:$0'$ be the [[Definition:Smallest Element|smallest element]] of $S'$
:$1'$ be the [[Definition:Smallest Element|smallest element]] of $S' \setminus \set {0'} = S'^*$... | Let $f: S \to S'$ be another [[Definition:Isomorphism (Abstract Algebra)|isomorphism]] different from $g$.
{{AimForCont}} $\map f 1 \ne 1'$.
We show by [[Principle of Mathematical Induction|induction]] that $1' \notin \Cdm f$.
{{finish}}
... Thus $1' \notin \Cdm f$ which is a [[Definition:Contradiction|contradictio... | Naturally Ordered Semigroup is Unique/Isomorphism is Unique | https://proofwiki.org/wiki/Naturally_Ordered_Semigroup_is_Unique/Isomorphism_is_Unique | https://proofwiki.org/wiki/Naturally_Ordered_Semigroup_is_Unique/Isomorphism_is_Unique | [
"Naturally Ordered Semigroup is Unique"
] | [
"Definition:Naturally Ordered Semigroup",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:Isomorphism (Abstract Algebra)"
] | [
"Definition:Isomorphism (Abstract Algebra)",
"Principle of Mathematical Induction",
"Definition:Contradiction",
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Unique"
] |
proofwiki-19366 | Naturally Ordered Semigroup is Unique/Existence of Isomorphism | Let $\struct {S, \circ, \preceq}$ and $\struct {S', \circ', \preceq'}$ be naturally ordered semigroups.
Let:
:$0'$ be the smallest element of $S'$
:$1'$ be the smallest element of $S' \setminus \set {0'} = S'^*$.
Then the mapping $g: S \to S'$ defined as:
:$\forall a \in S: \map g a = \circ'^a 1'$
is an isomorphism fro... | Let $T' = \Cdm g$, that is, the codomain of $g$.
By Zero is Identity in Naturally Ordered Semigroup, $0'$ is the identity for $\circ'$.
Thus:
:$\map g 0 = \circ'^0 1' = 0'$
and so $0' \in T'$
Suppose $x' \in T'$.
Then:
:$\map g n = x'$
and so:
{{begin-eqn}}
{{eqn | l = x' \circ' 1'
| r = \circ'^n 1' \circ' 1'
... | Let $\struct {S, \circ, \preceq}$ and $\struct {S', \circ', \preceq'}$ be [[Definition:Naturally Ordered Semigroup|naturally ordered semigroups]].
Let:
:$0'$ be the [[Definition:Smallest Element|smallest element]] of $S'$
:$1'$ be the [[Definition:Smallest Element|smallest element]] of $S' \setminus \set {0'} = S'^*$... | Let $T' = \Cdm g$, that is, the [[Definition:Codomain of Mapping|codomain]] of $g$.
By [[Zero is Identity in Naturally Ordered Semigroup]], $0'$ is the [[Definition:Identity Element|identity]] for $\circ'$.
Thus:
:$\map g 0 = \circ'^0 1' = 0'$
and so $0' \in T'$
Suppose $x' \in T'$.
Then:
:$\map g n = x'$
and so:... | Naturally Ordered Semigroup is Unique/Existence of Isomorphism | https://proofwiki.org/wiki/Naturally_Ordered_Semigroup_is_Unique/Existence_of_Isomorphism | https://proofwiki.org/wiki/Naturally_Ordered_Semigroup_is_Unique/Existence_of_Isomorphism | [
"Naturally Ordered Semigroup is Unique"
] | [
"Definition:Naturally Ordered Semigroup",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:Mapping",
"Definition:Isomorphism (Abstract Algebra)"
] | [
"Definition:Codomain (Set Theory)/Mapping",
"Zero is Identity in Naturally Ordered Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Principle of Mathematical Induction/Naturally Ordered Semigroup",
"Definition:Surjection",
"Index Laws/Sum of Indices/Semigroup",
"Definition:Homom... |
proofwiki-19367 | Operations with Identities which Distribute over each other are Idempotent | Let $S$ be a set.
Let $\odot$ and $\otimes$ be operations on $S$ which have identity elements $e$ and $u$ respectively.
Let $\odot$ and $\otimes$ be distributive over each other.
Then $\odot$ and $\otimes$ are both idempotent. | {{begin-eqn}}
{{eqn | l = e \otimes \paren {u \odot e}
| r = \paren {e \otimes u} \odot \paren {e \otimes e}
| c = as $\otimes$ is distributive over $\odot$
}}
{{eqn | ll= \leadsto
| l = e \otimes u
| r = \paren {e \otimes u} \odot \paren {e \otimes e}
| c = as $e$ is the identity of $\odo... | Let $S$ be a set.
Let $\odot$ and $\otimes$ be [[Definition:Binary Operation|operations]] on $S$ which have [[Definition:Identity Element|identity elements]] $e$ and $u$ respectively.
Let $\odot$ and $\otimes$ be [[Definition:Distributive Operation|distributive]] over each other.
Then $\odot$ and $\otimes$ are both... | {{begin-eqn}}
{{eqn | l = e \otimes \paren {u \odot e}
| r = \paren {e \otimes u} \odot \paren {e \otimes e}
| c = as $\otimes$ is [[Definition:Distributive Operation|distributive]] over $\odot$
}}
{{eqn | ll= \leadsto
| l = e \otimes u
| r = \paren {e \otimes u} \odot \paren {e \otimes e}
... | Operations with Identities which Distribute over each other are Idempotent | https://proofwiki.org/wiki/Operations_with_Identities_which_Distribute_over_each_other_are_Idempotent | https://proofwiki.org/wiki/Operations_with_Identities_which_Distribute_over_each_other_are_Idempotent | [
"Idempotence",
"Distributive Operations"
] | [
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Distributive Operation",
"Definition:Idempotence/Operation"
] | [
"Definition:Distributive Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Distributive Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided... |
proofwiki-19368 | Condition for Algebraic Structure to be Self-Distributive Quasigroup | Let $\struct {S, \odot}$ be an algebraic structure.
Then:
:$\struct {S, \odot}$ is a self-distributive quasigroup
{{iff}}:
:for every $a \in S$, the left and right regular representations $\lambda_a$ and $\rho_a$ on $S$ are automorphisms of $\struct {S, \odot}$. | === Sufficient Condition ===
Let $\struct {S, \odot}$ be a self-distributive quasigroup.
By definition of quasigroup, we have that $\lambda_a$ and $\rho_a$ are both permutations on $S$.
It remains for the morphism property to be demonstrated.
Indeed:
{{begin-eqn}}
{{eqn | q = \forall x, y \in S
| l = \map {\lambd... | Let $\struct {S, \odot}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Then:
:$\struct {S, \odot}$ is a [[Definition:Self-Distributive Structure|self-distributive]] [[Definition:Quasigroup|quasigroup]]
{{iff}}:
:for every $a \in S$, the [[Definition:Regular Representations|left and right regular repres... | === Sufficient Condition ===
Let $\struct {S, \odot}$ be a [[Definition:Self-Distributive Structure|self-distributive]] [[Definition:Quasigroup|quasigroup]].
By definition of [[Definition:Quasigroup|quasigroup]], we have that $\lambda_a$ and $\rho_a$ are both [[Definition:Permutation|permutations]] on $S$.
It remai... | Condition for Algebraic Structure to be Self-Distributive Quasigroup | https://proofwiki.org/wiki/Condition_for_Algebraic_Structure_to_be_Self-Distributive_Quasigroup | https://proofwiki.org/wiki/Condition_for_Algebraic_Structure_to_be_Self-Distributive_Quasigroup | [
"Self-Distributive Operations",
"Quasigroups"
] | [
"Definition:Algebraic Structure",
"Definition:Self-Distributive Structure",
"Definition:Quasigroup",
"Definition:Regular Representations",
"Definition:Automorphism (Abstract Algebra)"
] | [
"Definition:Self-Distributive Structure",
"Definition:Quasigroup",
"Definition:Quasigroup",
"Definition:Permutation",
"Definition:Morphism Property",
"Definition:Automorphism (Abstract Algebra)",
"Definition:Automorphism (Abstract Algebra)",
"Definition:Permutation",
"Definition:Quasigroup",
"Defi... |
proofwiki-19369 | Self-Distributive Quasigroup is Idempotent | Let $\struct {S, \odot}$ be a self-distributive quasigroup.
Then $\odot$ is an idempotent operation. | Because $S$ is a quasigroup:
:$\forall a, b \in S: \exists ! x \in S: x \odot a = b$
:$\forall a, b \in S: \exists ! y \in S: a \odot y = b$
In particular:
:$\exists ! x \in S: x \circ \paren {a \odot b} = y$
for arbitrary $a, b \in S$.
Let $y = a \odot \paren {a \odot b}$.
Because $\struct {S, \odot}$ is self-distribu... | Let $\struct {S, \odot}$ be a [[Definition:Self-Distributive Structure|self-distributive]] [[Definition:Quasigroup|quasigroup]].
Then $\odot$ is an [[Definition:Idempotent Operation|idempotent operation]]. | Because $S$ is a [[Definition:Quasigroup|quasigroup]]:
:$\forall a, b \in S: \exists ! x \in S: x \odot a = b$
:$\forall a, b \in S: \exists ! y \in S: a \odot y = b$
In particular:
:$\exists ! x \in S: x \circ \paren {a \odot b} = y$
for arbitrary $a, b \in S$.
Let $y = a \odot \paren {a \odot b}$.
Because $\stru... | Self-Distributive Quasigroup is Idempotent | https://proofwiki.org/wiki/Self-Distributive_Quasigroup_is_Idempotent | https://proofwiki.org/wiki/Self-Distributive_Quasigroup_is_Idempotent | [
"Self-Distributive Operations",
"Quasigroups",
"Idempotence"
] | [
"Definition:Self-Distributive Structure",
"Definition:Quasigroup",
"Definition:Idempotence/Operation"
] | [
"Definition:Quasigroup",
"Definition:Self-Distributive Structure",
"Definition:Quasigroup"
] |
proofwiki-19370 | Self-Distributive Quasigroup with at least Two Elements is not Associative | Let $\struct {S, \odot}$ be a self-distributive quasigroup.
Let $S$ have at least $2$ elements.
Then $\odot$ is not an associative operation. | {{AimForCont}} $\odot$ is associative operation.
Let $a, b \in S$ such that $a \ne b$.
Then:
{{begin-eqn}}
{{eqn | q = \forall a, b \in S
| l = \paren {a \odot a} \odot \paren {a \odot b}
| r = a \odot \paren {a \odot b}
| c = {{Defof|Self-Distributive Structure}}
}}
{{eqn | r = \paren {a \odot a} \od... | Let $\struct {S, \odot}$ be a [[Definition:Self-Distributive Structure|self-distributive]] [[Definition:Quasigroup|quasigroup]].
Let $S$ have at least $2$ [[Definition:Element|elements]].
Then $\odot$ is not an [[Definition:Associative Operation|associative operation]]. | {{AimForCont}} $\odot$ is [[Definition:Associative Operation|associative operation]].
Let $a, b \in S$ such that $a \ne b$.
Then:
{{begin-eqn}}
{{eqn | q = \forall a, b \in S
| l = \paren {a \odot a} \odot \paren {a \odot b}
| r = a \odot \paren {a \odot b}
| c = {{Defof|Self-Distributive Structure... | Self-Distributive Quasigroup with at least Two Elements is not Associative | https://proofwiki.org/wiki/Self-Distributive_Quasigroup_with_at_least_Two_Elements_is_not_Associative | https://proofwiki.org/wiki/Self-Distributive_Quasigroup_with_at_least_Two_Elements_is_not_Associative | [
"Self-Distributive Operations",
"Quasigroups",
"Associativity"
] | [
"Definition:Self-Distributive Structure",
"Definition:Quasigroup",
"Definition:Element",
"Definition:Associative Operation"
] | [
"Definition:Associative Operation"
] |
proofwiki-19371 | Self-Distributive Quasigroup with at least Two Elements has no Identity | Let $\struct {S, \odot}$ be a self-distributive quasigroup.
Let $S$ have at least $2$ elements.
Then $\struct {S, \odot}$ has no identity element. | {{AimForCont}} $S$ has an identity element $e$ and another element $a$ such that $a \ne e$.
Recall the definition of quasigroup:
:$\forall a, b \in S: \exists ! x \in S: x \circ a = b$
That is:
We have:
{{begin-eqn}}
{{eqn | l = a \circ a
| r = a
| c = Self-Distributive Quasigroup is Idempotent
}}
{{eqn | l... | Let $\struct {S, \odot}$ be a [[Definition:Self-Distributive Structure|self-distributive]] [[Definition:Quasigroup|quasigroup]].
Let $S$ have at least $2$ [[Definition:Element|elements]].
Then $\struct {S, \odot}$ has no [[Definition:Identity Element|identity element]]. | {{AimForCont}} $S$ has an [[Definition:Identity Element|identity element]] $e$ and another [[Definition:Element|element]] $a$ such that $a \ne e$.
Recall the definition of [[Definition:Quasigroup|quasigroup]]:
:$\forall a, b \in S: \exists ! x \in S: x \circ a = b$
That is:
We have:
{{begin-eqn}}
{{eqn | l = a \... | Self-Distributive Quasigroup with at least Two Elements has no Identity | https://proofwiki.org/wiki/Self-Distributive_Quasigroup_with_at_least_Two_Elements_has_no_Identity | https://proofwiki.org/wiki/Self-Distributive_Quasigroup_with_at_least_Two_Elements_has_no_Identity | [
"Self-Distributive Operations",
"Quasigroups",
"Identity Elements"
] | [
"Definition:Self-Distributive Structure",
"Definition:Quasigroup",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Quasigroup",
"Self-Distributive Quasigroup is Idempotent",
"Definition:Contradiction",
"Definition:Quasigroup",
"Definition:Element"
] |
proofwiki-19372 | P-adic Number is Power of p Times P-adic Unit | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p^\times$ be the $p$-adic units.
Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ denote the $p$-adic valuaton on the $p$-adic numbers.
Let $a \in \Q_p$.
Then there exists $u \in \Z_p^\times$ such that:
:$a = p^{\map {\nu_... | From P-adic Number times P-adic Norm is P-adic Unit, there exists $n \in \Z$ such that:
:$p^n a \in \Z_p^\times$
where
:$p^n = \norm a_p$
We have:
{{begin-eqn}}
{{eqn | l = \map {\nu_p} a
| r = -\log_p \norm a_p
| c = {{Defof|P-adic Valuation on P-adic Numbers}}
}}
{{eqn | r = -\log_p p^n
}}
{{eqn | r = -n
... | Let $p$ be a [[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 $\Z_p^\times$ be the [[Definition:P-adic Unit|$p$-adic units]].
Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ denote the [[Definition:P-adic Valua... | From [[P-adic Number times P-adic Norm is P-adic Unit]], there exists $n \in \Z$ such that:
:$p^n a \in \Z_p^\times$
where
:$p^n = \norm a_p$
We have:
{{begin-eqn}}
{{eqn | l = \map {\nu_p} a
| r = -\log_p \norm a_p
| c = {{Defof|P-adic Valuation on P-adic Numbers}}
}}
{{eqn | r = -\log_p p^n
}}
{{eqn | r... | P-adic Number is Power of p Times P-adic Unit | https://proofwiki.org/wiki/P-adic_Number_is_Power_of_p_Times_P-adic_Unit | https://proofwiki.org/wiki/P-adic_Number_is_Power_of_p_Times_P-adic_Unit | [
"P-adic Number Theory",
"P-adic Units"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Unit",
"Definition:P-adic Valuation/P-adic Numbers",
"Definition:Valued Field of P-adic Numbers"
] | [
"P-adic Number times P-adic Norm is P-adic Unit",
"Category:P-adic Number Theory",
"Category:P-adic Units"
] |
proofwiki-19373 | Polynomials of Congruent Ring Elements are Congruent | Let $R$ be a commutative ring with unity.
Let $I$ be an ideal of $R$.
Let $x, y \in R$.
Let:
:$x \equiv y \pmod I$
where the notation indicates congruence modulo $I$.
Let $\map F X \in R \sqbrk X$ be a polynomial in one variable over $R$.
Then:
:$\ds \map F x \equiv \map F y \pmod I$ | Let $\map F X = \ds \sum_{k \mathop = 0}^r a_k X^k$ where $X$ is the indeterminate and $a_0, a_1, \ldots, a_r \in R$.
It has to be shown:
:$\ds \sum_{k \mathop = 0}^r a_k x^k \equiv \sum_{k \mathop = 0}^r a_k y^k \pmod I$
From Left Cosets are Equal iff Product with Inverse in Subgroup:
:$\forall a, b \in R: a \equiv b ... | Let $R$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $I$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
Let $x, y \in R$.
Let:
:$x \equiv y \pmod I$
where the notation indicates [[Definition:Congruence Modulo Ideal|congruence]] modulo $I$.
Let $\map F X \in R \sqbrk X$ be a [... | Let $\map F X = \ds \sum_{k \mathop = 0}^r a_k X^k$ where $X$ is the [[Definition:Indeterminate (Polynomial Theory)|indeterminate]] and $a_0, a_1, \ldots, a_r \in R$.
It has to be shown:
:$\ds \sum_{k \mathop = 0}^r a_k x^k \equiv \sum_{k \mathop = 0}^r a_k y^k \pmod I$
From [[Left Cosets are Equal iff Product with ... | Polynomials of Congruent Ring Elements are Congruent | https://proofwiki.org/wiki/Polynomials_of_Congruent_Ring_Elements_are_Congruent | https://proofwiki.org/wiki/Polynomials_of_Congruent_Ring_Elements_are_Congruent | [
"Polynomial Theory",
"Ideal Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Congruence Modulo Ideal",
"Definition:Polynomial over Ring"
] | [
"Definition:Polynomial Ring/Indeterminate",
"Left Cosets are Equal iff Product with Inverse in Subgroup",
"Quotient Ring is Ring/Quotient Ring Addition is Well-Defined",
"Quotient Ring is Ring/Quotient Ring Product is Well-Defined",
"Left Cosets are Equal iff Product with Inverse in Subgroup",
"Quotient R... |
proofwiki-19374 | Binary Relation is Subclass of Product of Domain with Range | Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation.
Let $\Dom \RR$ denote the domain of $\RR$.
Let $\Img \RR$ denote the image of $\RR$.
Then:
:$\RR$ is a subclass of $\Dom \RR \times \Img \RR$ | Let $\tuple {x, y} \in \RR$.
Then by definition of domain of $\RR$:
:$x \in \Dom \RR$
and by definition of image of $\RR$:
:$y \in \Img \RR$
Hence by definition of Cartesian product:
:$\tuple {x, y} \in \Dom \RR \times \Img \RR$
Hence the result by definition of subclass:
:$\RR \subseteq \paren {\Dom \RR \times \Img \R... | Let $V$ be a [[Definition:Basic Universe|basic universe]].
Let $\RR \subseteq V \times V$ be a [[Definition:Relation (Class Theory)|relation]].
Let $\Dom \RR$ denote the [[Definition:Domain of Relation (Class Theory)|domain]] of $\RR$.
Let $\Img \RR$ denote the [[Definition:Image of Relation (Class Theory)|image]] o... | Let $\tuple {x, y} \in \RR$.
Then by definition of [[Definition:Domain of Relation (Class Theory)|domain]] of $\RR$:
:$x \in \Dom \RR$
and by definition of [[Definition:Image of Relation (Class Theory)|image]] of $\RR$:
:$y \in \Img \RR$
Hence by definition of [[Definition:Cartesian Product (Class Theory)|Cartesian... | Binary Relation is Subclass of Product of Domain with Range | https://proofwiki.org/wiki/Binary_Relation_is_Subclass_of_Product_of_Domain_with_Range | https://proofwiki.org/wiki/Binary_Relation_is_Subclass_of_Product_of_Domain_with_Range | [
"Relation Theory"
] | [
"Definition:Basic Universe",
"Definition:Relation/Class Theory",
"Definition:Domain (Set Theory)/Relation/Class Theory",
"Definition:Image (Set Theory)/Relation/Relation/Class Theory",
"Definition:Subclass"
] | [
"Definition:Domain (Set Theory)/Relation/Class Theory",
"Definition:Image (Set Theory)/Relation/Relation/Class Theory",
"Definition:Cartesian Product/Class Theory",
"Definition:Subclass"
] |
proofwiki-19375 | Equivalent Statements for Transitive Set | Let $a$ be a set.
{{TFAE}}
:$(1): \quad \bigcup a \subseteq a$
:$(2): \quad a \subseteq \powerset a$
:$(3): \quad a$ is a transitive set | === $(1) \iff (3)$ ===
This is shown in Class is Transitive iff Union is Subclass.
{{qed|lemma}} | Let $a$ be a [[Definition:Set|set]].
{{TFAE}}
:$(1): \quad \bigcup a \subseteq a$
:$(2): \quad a \subseteq \powerset a$
:$(3): \quad a$ is a [[Definition:Transitive Set|transitive set]] | === $(1) \iff (3)$ ===
This is shown in [[Class is Transitive iff Union is Subclass]].
{{qed|lemma}} | Equivalent Statements for Transitive Set | https://proofwiki.org/wiki/Equivalent_Statements_for_Transitive_Set | https://proofwiki.org/wiki/Equivalent_Statements_for_Transitive_Set | [
"Transitive Classes"
] | [
"Definition:Set",
"Definition:Transitive Class"
] | [
"Class is Transitive iff Union is Subclass"
] |
proofwiki-19376 | Union of Sigma-Algebras may not be Sigma-Algebra | Let $X$ be a set.
Let $\struct {X, \Sigma_1}$ and $\struct {X, \Sigma_2}$ be $\sigma$-algebras.
Then $\Sigma_1 \cup \Sigma_2$ may not be a $\sigma$-algebra on $X$. | Let $X = \set {1, 2, 3}$.
Let:
:$\Sigma_1 = \set {\O, \set 1, \set {2, 3}, \set {1, 2, 3} }$
and:
:$\Sigma_2 = \set {\O, \set 2, \set {1, 3}, \set {1, 2, 3} }$
Then $\Sigma_1$ and $\Sigma_2$ are $\sigma$-algebras on $X$.
We have:
:$\Sigma_1 \cup \Sigma_2 = \set {\O, \set 1, \set 2, \set {1, 3}, \set {2, 3}, \set {1,... | Let $X$ be a [[Definition:Set|set]].
Let $\struct {X, \Sigma_1}$ and $\struct {X, \Sigma_2}$ be [[Definition:Sigma-Algebra|$\sigma$-algebras]].
Then $\Sigma_1 \cup \Sigma_2$ may not be a [[Definition:Sigma-Algebra|$\sigma$-algebra]] on $X$. | Let $X = \set {1, 2, 3}$.
Let:
:$\Sigma_1 = \set {\O, \set 1, \set {2, 3}, \set {1, 2, 3} }$
and:
:$\Sigma_2 = \set {\O, \set 2, \set {1, 3}, \set {1, 2, 3} }$
Then $\Sigma_1$ and $\Sigma_2$ are [[Definition:Sigma-Algebra|$\sigma$-algebras]] on $X$.
We have:
:$\Sigma_1 \cup \Sigma_2 = \set {\O, \set 1, \set 2... | Union of Sigma-Algebras may not be Sigma-Algebra | https://proofwiki.org/wiki/Union_of_Sigma-Algebras_may_not_be_Sigma-Algebra | https://proofwiki.org/wiki/Union_of_Sigma-Algebras_may_not_be_Sigma-Algebra | [
"Sigma-Algebras"
] | [
"Definition:Set",
"Definition:Sigma-Algebra",
"Definition:Sigma-Algebra"
] | [
"Definition:Sigma-Algebra",
"Definition:Sigma-Algebra",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union/Countable Union",
"Definition:Sigma-Algebra",
"Category:Sigma-Algebras"
] |
proofwiki-19377 | Characterization of Measures that Share Null Sets | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ and $\nu$ be $\sigma$-finite measures on $\struct {X, \Sigma}$.
{{TFAE}}
:$(1) \quad$ $\nu \ll \mu$ and $\mu \ll \nu$, where $\ll$ denotes absolute continuity
:$(2) \quad$ $A \in \Sigma$ is a $\mu$-null set {{iff}} it is a $\nu$-null set
:$(3) \quad$ there exis... | === $(1)$ implies $(2)$ ===
Suppose that $\nu \ll \mu$ and $\mu \ll \nu$.
Let $A \in \Sigma$ have $\map \mu A = 0$.
Then, since $\nu \ll \mu$, we have $\map \nu A = 0$ from the definition of absolute continuity.
So if a $\Sigma$-measurable set is $\mu$-null, it is $\nu$-null.
Similarly, let $B \in \Sigma$ have $\map \... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ and $\nu$ be [[Definition:Sigma-Finite Measure|$\sigma$-finite measures]] on $\struct {X, \Sigma}$.
{{TFAE}}
:$(1) \quad$ $\nu \ll \mu$ and $\mu \ll \nu$, where $\ll$ denotes [[Definition:Absolutely Continuous Measure|absolut... | === $(1)$ implies $(2)$ ===
Suppose that $\nu \ll \mu$ and $\mu \ll \nu$.
Let $A \in \Sigma$ have $\map \mu A = 0$.
Then, since $\nu \ll \mu$, we have $\map \nu A = 0$ from the definition of [[Definition:Absolutely Continuous Measure|absolute continuity]].
So if a [[Definition:Measurable Set|$\Sigma$-measurable se... | Characterization of Measures that Share Null Sets | https://proofwiki.org/wiki/Characterization_of_Measures_that_Share_Null_Sets | https://proofwiki.org/wiki/Characterization_of_Measures_that_Share_Null_Sets | [
"Absolutely Continuous Measures",
"Null Sets"
] | [
"Definition:Measurable Space",
"Definition:Sigma-Finite Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Null Set",
"Definition:Null Set",
"Definition:Measurable Function/Positive"
] | [
"Definition:Absolute Continuity/Measure",
"Definition:Measurable Set",
"Definition:Null Set",
"Definition:Null Set",
"Definition:Absolute Continuity/Measure",
"Definition:Measurable Set",
"Definition:Null Set",
"Definition:Null Set",
"Definition:Measurable Set",
"Definition:Null Set",
"Definitio... |
proofwiki-19378 | Principle of Recursive Definition/Strong Version | Let $\omega$ denote the natural numbers as defined by the von Neumann construction.
Let $A$ be a class.
Let $c \in A$.
Let $g: \omega \times A \to A$ be a mapping.
Then there exists exactly one mapping $f: \omega \to A$ such that:
:<nowiki>$\forall x \in \omega: \map f x = \begin{cases}
c & : x = \O \\
\map g {n, \map ... | From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is a minimally inductive class under the successor mapping.
First an '''admissible mapping''' is defined.
Let $n \in \omega$.
The mapping $h: n \to A$ is defined as an '''admissible mapping''' for $n$ {{iff}}:
:<nowiki>$\forall r \subsete... | Let $\omega$ denote the [[Definition:Natural Numbers|natural numbers]] as defined by the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]].
Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $c \in A$.
Let $g: \omega \times A \to A$ be a [[Definition:Mapping|mapping]].
Th... | From [[Von Neumann Construction of Natural Numbers is Minimally Inductive]], $\omega$ is a [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class]] under the [[Definition:Successor Mapping|successor mapping]].
First an '''admissible mapping''' is defined.
Let $n \in \omega$.
The [[De... | Principle of Recursive Definition/Strong Version | https://proofwiki.org/wiki/Principle_of_Recursive_Definition/Strong_Version | https://proofwiki.org/wiki/Principle_of_Recursive_Definition/Strong_Version | [
"Principle of Recursive Definition"
] | [
"Definition:Natural Numbers",
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Class (Class Theory)",
"Definition:Mapping",
"Definition:Unique",
"Definition:Mapping"
] | [
"Von Neumann Construction of Natural Numbers is Minimally Inductive",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Successor Mapping",
"Definition:Mapping",
"Principle of Finite Induction/Peano Structure",
"Definition:Mapping",
"Definition:Mapping",
"Principle of Finite In... |
proofwiki-19379 | Relation between Volume Forms of Conformally Related Metrics on Oriented Riemannian Manifold | Let $M$ be an $n$-dimensional oriented Riemannian manifold.
Let $g_1$ and $g_2$ be conformally related Riemannian metrics.
That is, suppose:
:$g_2 = f g_1$
where $f \in \map {C^\infty} M$ is a positive smooth real function.
Let $\rd V_{g_1}$ and $\rd V_{g_2}$ be Riemannian volume forms associated with $g_1$ and $g_2$.
... | Let $\tuple {x_1, \ldots, x_n}$ be a set of local oriented coordinates.
Let $g_{i j}$ be a local form of the metric $g_1$.
Then $f g_{i j}$ is the local form of the metric $g_2$.
{{begin-eqn}}
{{eqn | l = \rd V_{g_2}
| r = \sqrt {\map \det {f g_{i j} } } \rd x^1 \wedge \ldots \wedge \rd x^n
| c = {{Defof|Ri... | Let $M$ be an [[Definition:Riemannian Manifold/Dimension|$n$-dimensional]] [[Definition:Oriented Manifold|oriented]] [[Definition:Riemannian Manifold|Riemannian manifold]].
Let $g_1$ and $g_2$ be [[Definition:Pointwise Conformal Riemannian Metrics|conformally related]] [[Definition:Riemannian Metric|Riemannian metrics... | Let $\tuple {x_1, \ldots, x_n}$ be a [[Definition:Set|set]] of [[Definition:Local Coordinates|local]] [[Definition:Oriented Coordinates|oriented]] [[Definition:Coordinate|coordinates]].
Let $g_{i j}$ be a [[Definition:Local Metric|local form of the metric]] $g_1$.
Then $f g_{i j}$ is the [[Definition:Local Metric|loc... | Relation between Volume Forms of Conformally Related Metrics on Oriented Riemannian Manifold | https://proofwiki.org/wiki/Relation_between_Volume_Forms_of_Conformally_Related_Metrics_on_Oriented_Riemannian_Manifold | https://proofwiki.org/wiki/Relation_between_Volume_Forms_of_Conformally_Related_Metrics_on_Oriented_Riemannian_Manifold | [
"Riemannian Geometry"
] | [
"Definition:Riemannian Manifold/Dimension",
"Definition:Oriented Manifold",
"Definition:Riemannian Manifold",
"Definition:Pointwise Conformal Riemannian Metrics",
"Definition:Riemannian Metric",
"Definition:Positive/Real Number",
"Definition:Smooth Real Function",
"Definition:Riemannian Volume Form"
] | [
"Definition:Set",
"Definition:Local Coordinates",
"Definition:Oriented Coordinates",
"Definition:Coordinate",
"Definition:Local Metric",
"Definition:Local Metric"
] |
proofwiki-19380 | Hensel's Lemma/P-adic Integers/Lemma 6 | :$x \equiv 0 \pmod {p^k \Z_p} \implies \exists y \in \Z_p : x = y p^k$ | Let:
:$x \equiv 0 \pmod {p^k \Z_p}$
By definition of congruence modulo ideal:
:$x \in p^k \Z_p$
By definition of principal ideal:
:$\exists y \in \Z_p : x = y p^k$
{{qed}}
Category:Hensel's Lemma
gri91j3fizduk4zsihea7lqyawt7pkn | :$x \equiv 0 \pmod {p^k \Z_p} \implies \exists y \in \Z_p : x = y p^k$ | Let:
:$x \equiv 0 \pmod {p^k \Z_p}$
By definition of [[Definition:Congruence Modulo Ideal|congruence modulo ideal]]:
:$x \in p^k \Z_p$
By definition of [[Definition:Principal Ideal|principal ideal]]:
:$\exists y \in \Z_p : x = y p^k$
{{qed}}
[[Category:Hensel's Lemma]]
gri91j3fizduk4zsihea7lqyawt7pkn | Hensel's Lemma/P-adic Integers/Lemma 6 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_6 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_6 | [
"Hensel's Lemma"
] | [] | [
"Definition:Congruence Modulo Ideal",
"Definition:Principal Ideal",
"Category:Hensel's Lemma"
] |
proofwiki-19381 | Hensel's Lemma/P-adic Integers/Lemma 7 | :$x \in \Z_p \implies \exists y \in T : y p^k \equiv x p^k \pmod {p^{k+1}\Z_p}$ | Let $x \in \Z_p$.
From $p$-adic Integer is Limit of Unique $p$-adic Expansion, let:
:$x = \ds \sum_{n \mathop = 0}^\infty d_n p^n$
We have:
{{begin-eqn}}
{{eqn | l = x - d_0
| r = \paren{d_0 + d_1p + d_2p^2 + d_3p^3 + \ldots} - d_0
}}
{{eqn | r = \paren{d_1p + d_2p^2 + d_3p^3 + \ldots}
| c = $d_0$ terms can... | :$x \in \Z_p \implies \exists y \in T : y p^k \equiv x p^k \pmod {p^{k+1}\Z_p}$ | Let $x \in \Z_p$.
From [[P-adic Integer is Limit of Unique P-adic Expansion|$p$-adic Integer is Limit of Unique $p$-adic Expansion]], let:
:$x = \ds \sum_{n \mathop = 0}^\infty d_n p^n$
We have:
{{begin-eqn}}
{{eqn | l = x - d_0
| r = \paren{d_0 + d_1p + d_2p^2 + d_3p^3 + \ldots} - d_0
}}
{{eqn | r = \paren{d_... | Hensel's Lemma/P-adic Integers/Lemma 7 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_7 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_7 | [
"Hensel's Lemma"
] | [] | [
"P-adic Integer is Limit of Unique P-adic Expansion",
"Definition:P-adic Expansion",
"Definition:P-adic Integer",
"Category:Hensel's Lemma"
] |
proofwiki-19382 | Hensel's Lemma/P-adic Integers/Lemma 8 | :$x, y \in \Z_p \implies \map F {x + y p ^k} \equiv \map F x + y p^k \map {F'} x \pmod {p^{k+1}\Z_p}$ | Let $\ds \map F X = \sum_{j \mathop = 0}^r c_j X^j$ where $X$ is the indeterminate and $c_0, c_1, \ldots, c_r \in \Z_p$.
Then:
:$\ds \map {F'} X = \sum_{j \mathop = 1}^r j c_j X^j$
We have:
{{begin-eqn}}
{{eqn | l = \map F {x + yp^k}
| r = \sum_{j \mathop = 0}^r c_j \paren{x + yp^k}^j
| c = Definition of $\... | :$x, y \in \Z_p \implies \map F {x + y p ^k} \equiv \map F x + y p^k \map {F'} x \pmod {p^{k+1}\Z_p}$ | Let $\ds \map F X = \sum_{j \mathop = 0}^r c_j X^j$ where $X$ is the [[Definition:Indeterminate (Polynomial Theory)|indeterminate]] and $c_0, c_1, \ldots, c_r \in \Z_p$.
Then:
:$\ds \map {F'} X = \sum_{j \mathop = 1}^r j c_j X^j$
We have:
{{begin-eqn}}
{{eqn | l = \map F {x + yp^k}
| r = \sum_{j \mathop = 0}^r ... | Hensel's Lemma/P-adic Integers/Lemma 8 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_8 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_8 | [
"Hensel's Lemma"
] | [] | [
"Definition:Polynomial Ring/Indeterminate",
"Binomial Theorem",
"Category:Hensel's Lemma"
] |
proofwiki-19383 | Hensel's Lemma/P-adic Integers/Lemma 9 | :$\paren{c, d \in T : \tuple{b_0, b_1, \ldots, b_{k-1}, c}, \tuple{b_0, b_1, \ldots, b_{k-1}, d} \in S_{k+1}} \implies c = d$ | === Lemma 7 ===
{{:Hensel's Lemma/P-adic Integers/Lemma 7}}{{qed|lemma}} | :$\paren{c, d \in T : \tuple{b_0, b_1, \ldots, b_{k-1}, c}, \tuple{b_0, b_1, \ldots, b_{k-1}, d} \in S_{k+1}} \implies c = d$ | === [[Hensel's Lemma/P-adic Integers/Lemma 7|Lemma 7]] ===
{{:Hensel's Lemma/P-adic Integers/Lemma 7}}{{qed|lemma}} | Hensel's Lemma/P-adic Integers/Lemma 9 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_9 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_9 | [
"Hensel's Lemma"
] | [] | [
"Hensel's Lemma/P-adic Integers/Lemma 7"
] |
proofwiki-19384 | Poincaré Conjecture/Dimension 1 | Let $\Sigma^1$ be a smooth $1$-manifold.
Let $\Sigma^1$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_m \struct {\Sigma; \Z} = \Z$
Then $\Sigma^1$ is homeomorphic to the $1$-sphere $\Bbb S^1$.
{{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_m$}} | Follows from the Classification of Compact One-Manifolds. | Let $\Sigma^1$ be a smooth $1$-manifold.
Let $\Sigma^1$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_m \struct {\Sigma; \Z} = \Z$
Then $\Sigma^1$ is homeomorphic to the $1$-sphere $\Bbb S^1$.
{{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_m$}} | Follows from the [[Classification of Compact One-Manifolds]]. | Poincaré Conjecture/Dimension 1 | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_1 | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_1 | [
"Poincaré Conjecture"
] | [] | [
"Classification of Compact One-Manifolds"
] |
proofwiki-19385 | Poincaré Conjecture/Dimension 2 | Let $\Sigma^2$ be a smooth $2$-manifold.
Let $\Sigma^2$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_2 \struct {\Sigma; \Z} = \Z$
Then $\Sigma^2$ is homeomorphic to the $2$-sphere $\Bbb S^2$.
{{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_m$}} | Follows from the Classification of Compact Two-Manifolds. | Let $\Sigma^2$ be a smooth $2$-manifold.
Let $\Sigma^2$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_2 \struct {\Sigma; \Z} = \Z$
Then $\Sigma^2$ is homeomorphic to the $2$-sphere $\Bbb S^2$.
{{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_m$}} | Follows from the [[Classification of Compact Two-Manifolds]]. | Poincaré Conjecture/Dimension 2 | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_2 | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_2 | [
"Poincaré Conjecture",
"Poincaré Conjecture"
] | [] | [
"Classification of Compact Two-Manifolds"
] |
proofwiki-19386 | Poincaré Conjecture/Dimension 3 | Let $\Sigma^3$ be a smooth $3$-manifold.
Let $\Sigma^3$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_3 \struct {\Sigma; \Z} = \Z$
Then $\Sigma^3$ is homeomorphic to the $3$-sphere $\Bbb S^3$.
{{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_3$}} | Follows from Thurston's Geometrization Conjecture.
{{ProofWanted}} | Let $\Sigma^3$ be a smooth $3$-manifold.
Let $\Sigma^3$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_3 \struct {\Sigma; \Z} = \Z$
Then $\Sigma^3$ is homeomorphic to the $3$-sphere $\Bbb S^3$.
{{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_3$}} | Follows from [[Thurston's Geometrization Conjecture]].
{{ProofWanted}} | Poincaré Conjecture/Dimension 3 | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_3 | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_3 | [
"Poincaré Conjecture"
] | [] | [
"Thurston's Geometrization Conjecture"
] |
proofwiki-19387 | Poincaré Conjecture/Dimension 4 | Let $\Sigma^4$ be a smooth $4$-manifold.
Let $\Sigma^4$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_4 \struct {\Sigma; \Z} = \Z$
Then $\Sigma^4$ is homeomorphic to the $4$-sphere $\Bbb S^4$.
{{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_4$}} | Follows from $4$-dimensional Topological $h$-Cobordism Theorem.
{{ProofWanted}} | Let $\Sigma^4$ be a smooth $4$-manifold.
Let $\Sigma^4$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_4 \struct {\Sigma; \Z} = \Z$
Then $\Sigma^4$ is homeomorphic to the $4$-sphere $\Bbb S^4$.
{{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_4$}} | Follows from $4$-dimensional [[Topological h-Cobordism Theorem|Topological $h$-Cobordism Theorem]].
{{ProofWanted}} | Poincaré Conjecture/Dimension 4 | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_4 | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_4 | [
"Poincaré Conjecture"
] | [] | [
"Topological h-Cobordism Theorem"
] |
proofwiki-19388 | Poincaré Conjecture/Dimension 6 or Greater | Let $\Sigma^m$ be a smooth $m$-manifold where $m \ge 6$.
Let $\Sigma^m$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_m \struct {\Sigma; \Z} = \Z$
Then $\Sigma^m$ is homeomorphic to the $m$-sphere $\Bbb S^m$. | We can cut two small $m$-disks $D', D' '$ from $\Sigma$.
The remaining manifold, $\Sigma \setminus \paren {D' \cup D' '}$ is an h-cobordism between $\partial D'$ and $\partial D' '$.
These are just two copies of $\Bbb S^{m-1}$.
By the $h$-cobordism theorem, there exists a diffeomorphism:
:$\phi: \Sigma \setminus \paren... | Let $\Sigma^m$ be a smooth $m$-manifold where $m \ge 6$.
Let $\Sigma^m$ satisfy:
:$H_0 \struct {\Sigma; \Z} = 0$
and:
:$H_m \struct {\Sigma; \Z} = \Z$
Then $\Sigma^m$ is homeomorphic to the $m$-sphere $\Bbb S^m$. | We can cut two small $m$-disks $D', D' '$ from $\Sigma$.
The remaining manifold, $\Sigma \setminus \paren {D' \cup D' '}$ is an h-cobordism between $\partial D'$ and $\partial D' '$.
These are just two copies of $\Bbb S^{m-1}$.
By the $h$-cobordism theorem, there exists a diffeomorphism:
:$\phi: \Sigma \setminus \pa... | Poincaré Conjecture/Dimension 6 or Greater | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_6_or_Greater | https://proofwiki.org/wiki/Poincaré_Conjecture/Dimension_6_or_Greater | [
"Poincaré Conjecture"
] | [] | [] |
proofwiki-19389 | Addition in Minimally Inductive Set is Unique | Let $\omega$ be the minimally inductive set.
Let $A: \omega \times \omega \to \omega$ be the mapping defined as the addition operation:
:$\forall \tuple {x, y} \in \omega \times \omega: \map A {x, y} = \begin {cases} x & : y = 0 \\ \paren {\map A {x, r} }^+ & : y = r^+ \end {cases}$
where $r^+$ is the successor set of ... | Recall the Principle of Recursive Definition (Strong Version):
{{:Principle of Recursive Definition/Strong Version}}
From the von Neumann construction, this $\omega$ is exactly the minimally inductive set as defined in the problem statement.
Let $g: \omega \times \omega \to \omega$ be the mapping defined as:
:$\forall ... | Let $\omega$ be the [[Definition:Minimally Inductive Set|minimally inductive set]].
Let $A: \omega \times \omega \to \omega$ be the [[Definition:Mapping|mapping]] defined as the [[Definition:Addition in Minimally Inductive Set|addition operation]]:
:$\forall \tuple {x, y} \in \omega \times \omega: \map A {x, y} = \be... | Recall the [[Principle of Recursive Definition/Strong Version|Principle of Recursive Definition (Strong Version)]]:
{{:Principle of Recursive Definition/Strong Version}}
From the [[Definition:Von Neumann Construction of Natural Numbers|von Neumann construction]], this $\omega$ is exactly the [[Definition:Minimally Ind... | Addition in Minimally Inductive Set is Unique | https://proofwiki.org/wiki/Addition_in_Minimally_Inductive_Set_is_Unique | https://proofwiki.org/wiki/Addition_in_Minimally_Inductive_Set_is_Unique | [
"Natural Number Addition"
] | [
"Definition:Minimally Inductive Set",
"Definition:Mapping",
"Definition:Addition in Minimally Inductive Set",
"Definition:Successor Mapping/Successor Set",
"Definition:Unique"
] | [
"Principle of Recursive Definition/Strong Version",
"Definition:Natural Numbers/Von Neumann Construction",
"Definition:Minimally Inductive Set",
"Definition:Mapping",
"Principle of Recursive Definition/Strong Version",
"Definition:Unique",
"Definition:Mapping"
] |
proofwiki-19390 | Increasing Sequence of Sets forms Nest | Let $\sequence {a_n}$ be an increasing sequence of sets:
:$\forall k \in \N: S_k \subseteq S_{k + 1}$
Let $c = \set {a_1, a_1, \ldots, a_n, a_{n + 1}, \ldots}$ be the class of all terms of $\sequence {a_n}$.
Then $c$ is a nest. | Recall the definition of nest:
$c$ is a '''nest''' {{iff}}:
:$\forall x, y \in c: x \subseteq y$ or $y \subseteq x$
Let $a_i$ and $a_j$ be arbitrary elements of $c$.
From Ordering on Natural Numbers is Trichotomy, either $i < j$ or $i = j$ or $i > j$.
;Case $(1)$
Let $i = j$.
Then we have $a_i = a_j$ and so both $a_i \... | Let $\sequence {a_n}$ be an [[Definition:Increasing Sequence of Sets|increasing sequence of sets]]:
:$\forall k \in \N: S_k \subseteq S_{k + 1}$
Let $c = \set {a_1, a_1, \ldots, a_n, a_{n + 1}, \ldots}$ be the [[Definition:Class (Class Theory)|class]] of all [[Definition:Term of Sequence|terms]] of $\sequence {a_n}$.... | Recall the definition of [[Definition:Nest (Class Theory)|nest]]:
$c$ is a '''nest''' {{iff}}:
:$\forall x, y \in c: x \subseteq y$ or $y \subseteq x$
Let $a_i$ and $a_j$ be arbitrary [[Definition:Element of Class|elements]] of $c$.
From [[Ordering on Natural Numbers is Trichotomy]], either $i < j$ or $i = j$ or $i... | Increasing Sequence of Sets forms Nest | https://proofwiki.org/wiki/Increasing_Sequence_of_Sets_forms_Nest | https://proofwiki.org/wiki/Increasing_Sequence_of_Sets_forms_Nest | [
"Nests",
"Increasing Sequences"
] | [
"Definition:Increasing Sequence of Sets",
"Definition:Class (Class Theory)",
"Definition:Term of Sequence",
"Definition:Nest/Class Theory"
] | [
"Definition:Nest/Class Theory",
"Definition:Element/Class",
"Ordering on Natural Numbers is Trichotomy",
"Subset Relation is Transitive"
] |
proofwiki-19391 | Minimally Inductive Class under Slowly Progressing Mapping is Nest | Let $M$ be a class.
Let $g: M \to M$ be a slowly progressing mapping on $M$.
Let $M$ be a minimally inductive class under $g$.
Then $M$ is a nest. | By definition, a slowly progressing mapping is indeed a progressing mapping.
The result then follows from Minimally Inductive Class under Progressing Mapping induces Nest. | Let $M$ be a [[Definition:Class (Class Theory)|class]].
Let $g: M \to M$ be a [[Definition:Slowly Progressing Mapping|slowly progressing mapping]] on $M$.
Let $M$ be a [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class]] under $g$.
Then $M$ is a [[Definition:Nest (Class Theory)|n... | By definition, a [[Definition:Slowly Progressing Mapping|slowly progressing mapping]] is indeed a [[Definition:Progressing Mapping|progressing mapping]].
The result then follows from [[Minimally Inductive Class under Progressing Mapping induces Nest]]. | Minimally Inductive Class under Slowly Progressing Mapping is Nest/Proof 2 | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Slowly_Progressing_Mapping_is_Nest | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Slowly_Progressing_Mapping_is_Nest/Proof_2 | [
"Minimally Inductive Classes",
"Slowly Progressing Mappings",
"Nests",
"Minimally Inductive Class under Slowly Progressing Mapping is Nest"
] | [
"Definition:Class (Class Theory)",
"Definition:Slowly Progressing Mapping",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Nest/Class Theory"
] | [
"Definition:Slowly Progressing Mapping",
"Definition:Progressing Mapping",
"Minimally Inductive Class under Progressing Mapping induces Nest"
] |
proofwiki-19392 | Closed Form for Millin Series | The '''Millin series''' has the closed form expression:
:$\ds \sum_{n \mathop = 0}^\infty \frac 1 {F_{2^n} } = \frac {7 - \sqrt 5} 2$ | First we will prove that:
:$\ds \sum_{r \mathop = 0}^n \frac 1 {F_{2^r} } = 3 - \frac {F_{2^n - 1} } {F_{2^n} }$
for $n \ge 1$.
We see that:
:$\dfrac 1 {F_1} + \dfrac 1 {F_2} = 2 = 3 - \dfrac {F_1} {F_2}$
so the proposition holds for $n = 1$.
Suppose the proposition is true for $n = k$.
Then:
{{begin-eqn}}
{{eqn | l =... | The '''[[Definition:Millin Series|Millin series]]''' has the closed form expression:
:$\ds \sum_{n \mathop = 0}^\infty \frac 1 {F_{2^n} } = \frac {7 - \sqrt 5} 2$ | First we will prove that:
:$\ds \sum_{r \mathop = 0}^n \frac 1 {F_{2^r} } = 3 - \frac {F_{2^n - 1} } {F_{2^n} }$
for $n \ge 1$.
We see that:
:$\dfrac 1 {F_1} + \dfrac 1 {F_2} = 2 = 3 - \dfrac {F_1} {F_2}$
so the proposition holds for $n = 1$.
Suppose the proposition is true for $n = k$.
Then:
{{begin-eqn}}
{{e... | Closed Form for Millin Series/Proof 1 | https://proofwiki.org/wiki/Closed_Form_for_Millin_Series | https://proofwiki.org/wiki/Closed_Form_for_Millin_Series/Proof_1 | [
"Fibonacci Numbers",
"Closed Form for Millin Series"
] | [
"Definition:Millin Series"
] | [
"Definition:Fibonacci Number",
"Cassini's Identity",
"Definition:Fibonacci Number",
"Principle of Mathematical Induction",
"Ratio of Consecutive Fibonacci Numbers",
"Definition:Multiplication/Real Numbers"
] |
proofwiki-19393 | Closed Form for Millin Series | The '''Millin series''' has the closed form expression:
:$\ds \sum_{n \mathop = 0}^\infty \frac 1 {F_{2^n} } = \frac {7 - \sqrt 5} 2$ | Let:
:$\ds \map F x := \sum_{k \mathop = 1}^\infty \frac {x^{2^{k - 1} } } {F_{2^k} }$
:$\alpha := \dfrac {1 + \sqrt 5} 2$
:$\beta := \dfrac {1 - \sqrt 5} 2$
Note that for $\size x \le 1$ we have:
:$\size {\dfrac {x^{2^{k - 1} } } {F_{2^k} } } \le \dfrac 1 {F_{2^k} }$
Given that the Millin Series converges, by Comparis... | The '''[[Definition:Millin Series|Millin series]]''' has the closed form expression:
:$\ds \sum_{n \mathop = 0}^\infty \frac 1 {F_{2^n} } = \frac {7 - \sqrt 5} 2$ | Let:
:$\ds \map F x := \sum_{k \mathop = 1}^\infty \frac {x^{2^{k - 1} } } {F_{2^k} }$
:$\alpha := \dfrac {1 + \sqrt 5} 2$
:$\beta := \dfrac {1 - \sqrt 5} 2$
Note that for $\size x \le 1$ we have:
:$\size {\dfrac {x^{2^{k - 1} } } {F_{2^k} } } \le \dfrac 1 {F_{2^k} }$
Given that the [[Definition:Millin Series|Millin... | Closed Form for Millin Series/Proof 2 | https://proofwiki.org/wiki/Closed_Form_for_Millin_Series | https://proofwiki.org/wiki/Closed_Form_for_Millin_Series/Proof_2 | [
"Fibonacci Numbers",
"Closed Form for Millin Series"
] | [
"Definition:Millin Series"
] | [
"Definition:Millin Series",
"Definition:Convergent Series",
"Comparison Test",
"Definition:Convergent Series",
"Closed Form for Lucas Numbers",
"Fibonacci Number 2n equals Fibonacci Number n by Lucas Number n",
"Translation of Index Variable of Summation"
] |
proofwiki-19394 | Successor Mapping is Slowly Progressing | Let $V$ be a basic universe.
Let $s: V \to V$ denote the '''successor mapping''' on $V$:
:$\forall x \in V: \map s x := x \cup \set x$
Then $s$ is a slowly progressing mapping. | From Successor Mapping is Progressing we have that $s$ is a progressing mapping.
Then we have that:
:$\set x \notin x$
Thus:
:$\card {x \cup \set x} = \card x + 1$
Hence the result.
{{qed}} | Let $V$ be a [[Definition:Basic Universe|basic universe]].
Let $s: V \to V$ denote the '''[[Definition:Successor Mapping|successor mapping]]''' on $V$:
:$\forall x \in V: \map s x := x \cup \set x$
Then $s$ is a [[Definition:Slowly Progressing Mapping|slowly progressing mapping]]. | From [[Successor Mapping is Progressing]] we have that $s$ is a [[Definition:Progressing Mapping|progressing mapping]].
Then we have that:
:$\set x \notin x$
Thus:
:$\card {x \cup \set x} = \card x + 1$
Hence the result.
{{qed}} | Successor Mapping is Slowly Progressing | https://proofwiki.org/wiki/Successor_Mapping_is_Slowly_Progressing | https://proofwiki.org/wiki/Successor_Mapping_is_Slowly_Progressing | [
"Successor Mapping",
"Slowly Progressing Mappings"
] | [
"Definition:Basic Universe",
"Definition:Successor Mapping",
"Definition:Slowly Progressing Mapping"
] | [
"Successor Mapping is Progressing",
"Definition:Progressing Mapping"
] |
proofwiki-19395 | Subset Relation is Reflexive | Let $C$ be a class.
The subset relation $\subseteq$ on $C$ is a reflexive relation on $C$. | {{begin-eqn}}
{{eqn | q = \forall x \in C
| l = x
| o = \subseteq
| r = x
| c = Set is Subset of Itself
}}
{{eqn | ll= \leadsto
| q = \forall x \in C
| l = \tuple {x, x}
| o = \in
| r = \subseteq
| c =
}}
{{end-eqn}}
So $\subseteq$ is reflexive. | Let $C$ be a [[Definition:Class (Class Theory)|class]].
The [[Definition:Subset Relation|subset relation]] $\subseteq$ on $C$ is a [[Definition:Reflexive Relation|reflexive relation]] on $C$. | {{begin-eqn}}
{{eqn | q = \forall x \in C
| l = x
| o = \subseteq
| r = x
| c = [[Set is Subset of Itself]]
}}
{{eqn | ll= \leadsto
| q = \forall x \in C
| l = \tuple {x, x}
| o = \in
| r = \subseteq
| c =
}}
{{end-eqn}}
So $\subseteq$ is [[Definition:Reflexive Re... | Subset Relation is Reflexive | https://proofwiki.org/wiki/Subset_Relation_is_Reflexive | https://proofwiki.org/wiki/Subset_Relation_is_Reflexive | [
"Subset Relation",
"Examples of Reflexive Relations"
] | [
"Definition:Class (Class Theory)",
"Definition:Subset Relation",
"Definition:Reflexive Relation"
] | [
"Set is Subset of Itself",
"Definition:Reflexive Relation"
] |
proofwiki-19396 | Identity Mapping on Normed Vector Space is Bounded Linear Operator | Let $\Bbb F$ be a subfield of $\C$.
Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $I : X \to X$ be the identity mapping on $X$.
Then $I$ is a bounded linear operator.
Further:
Let $\map B X$, be the space of bounded linear operators on $X$.
Then:
:$\norm I_{\map B X} = 1$
where $\norm {\, \cdot \,}_{\... | Let $\lambda \in \Bbb F$ and $x, y \in X$.
Then, we have:
:$\map I {\lambda x + y} = \lambda x + y = \lambda I x + I y$
So $I$ is a linear operator.
Further, we have:
:$\norm {I x} = \norm x$
for each $x \in X$, so $I$ is bounded.
Further, for all $x \in X$ with $\norm x = 1$ we have that $\norm {I x} = 1$, so that:... | Let $\Bbb F$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $I : X \to X$ be the [[Definition:Identity Mapping|identity mapping]] on $X$.
Then $I$ is a [[Definition:Bounded Linear Operator|bounded linear operator]].
F... | Let $\lambda \in \Bbb F$ and $x, y \in X$.
Then, we have:
:$\map I {\lambda x + y} = \lambda x + y = \lambda I x + I y$
So $I$ is a [[Definition:Linear Operator|linear operator]].
Further, we have:
:$\norm {I x} = \norm x$
for each $x \in X$, so $I$ is [[Definition:Bounded Linear Operator|bounded]].
Further, ... | Identity Mapping on Normed Vector Space is Bounded Linear Operator | https://proofwiki.org/wiki/Identity_Mapping_on_Normed_Vector_Space_is_Bounded_Linear_Operator | https://proofwiki.org/wiki/Identity_Mapping_on_Normed_Vector_Space_is_Bounded_Linear_Operator | [
"Bounded Linear Operators"
] | [
"Definition:Subfield",
"Definition:Normed Vector Space",
"Definition:Identity Mapping",
"Definition:Bounded Linear Operator",
"Definition:Bounded Linear Transformation",
"Definition:Norm on Space of Bounded Linear Transformations"
] | [
"Definition:Linear Operator",
"Definition:Bounded Linear Operator",
"Category:Bounded Linear Operators"
] |
proofwiki-19397 | Hensel's Lemma/P-adic Integers/Lemma 10 | :$\forall x \in \Z_p: p^k x \equiv 0 \pmod {p^{k + 1} \Z_p} \implies x \equiv 0 \pmod {p \Z_p}$ | We have:
{{begin-eqn}}
{{eqn | l = p^k x
| o = \equiv
| r = 0
| rr= \pmod {p^{k + 1} \Z_p}
}}
{{eqn | ll= \leadsto
| l = p^k x
| o = \in
| r = p^{k + 1} \Z_p
| c = {{Defof|Congruence Modulo Ideal}}
}}
{{eqn | ll= \leadsto
| q = \exists y \in \Z_p
| l = p^k x
|... | :$\forall x \in \Z_p: p^k x \equiv 0 \pmod {p^{k + 1} \Z_p} \implies x \equiv 0 \pmod {p \Z_p}$ | We have:
{{begin-eqn}}
{{eqn | l = p^k x
| o = \equiv
| r = 0
| rr= \pmod {p^{k + 1} \Z_p}
}}
{{eqn | ll= \leadsto
| l = p^k x
| o = \in
| r = p^{k + 1} \Z_p
| c = {{Defof|Congruence Modulo Ideal}}
}}
{{eqn | ll= \leadsto
| q = \exists y \in \Z_p
| l = p^k x
... | Hensel's Lemma/P-adic Integers/Lemma 10 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_10 | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers/Lemma_10 | [
"Hensel's Lemma"
] | [] | [
"Category:Hensel's Lemma"
] |
proofwiki-19398 | Closure of product equals product of closures | Let $\set {X_i}_{i \mathop \in I}$ be an arbitrary family of topological spaces.
Let:
:$A_i \subset X_i$
for each $i \in I$.
Let $X = \ds \prod_{i \mathop \in I} X_i$ have the product topology.
Then:
:$\ds \prod_{i \mathop \in I} \overline{A_i} = \overline {\prod_{i \mathop \in I} A_i}$ | We proceed by double inclusion.
{{explain|what does "double inclusion" mean?}}
Suppose:
:$\ds \family {x_i}_{i \mathop \in I} \in \prod_{i \mathop \in I} \overline {A_i}$
This means:
:$x_i \in \overline {A_i}$
for each $i \in I$.
Let $U$ be an open set containing $\family {x_i}_{i \mathop \in I}$.
Since:
:$x_i \in \... | Let $\set {X_i}_{i \mathop \in I}$ be an arbitrary [[Definition:Set Union/Family of Sets|family]] of [[Definition:Topological Space|topological spaces]].
Let:
:$A_i \subset X_i$
for each $i \in I$.
Let $X = \ds \prod_{i \mathop \in I} X_i$ have the [[Definition:Product Topology|product topology]].
Then:
:$\ds \prod... | We proceed by double inclusion.
{{explain|what does "double inclusion" mean?}}
Suppose:
:$\ds \family {x_i}_{i \mathop \in I} \in \prod_{i \mathop \in I} \overline {A_i}$
This means:
:$x_i \in \overline {A_i}$
for each $i \in I$.
Let $U$ be an [[Definition:Open Set (Topology)|open set]] containing $\family {x_i}... | Closure of product equals product of closures | https://proofwiki.org/wiki/Closure_of_product_equals_product_of_closures | https://proofwiki.org/wiki/Closure_of_product_equals_product_of_closures | [] | [
"Definition:Set Union/Family of Sets",
"Definition:Topological Space",
"Definition:Product Topology"
] | [
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] |
proofwiki-19399 | Minimal Polynomial Exists | Let $L / K$ be a field extension.
Let $\alpha \in L$ be algebraic over $K$.
Then there exists a minimal polynomial $f \in K \sqbrk x$ for $\alpha$ over $K$. | According to definition 1 of minimal polynomial, we ought to find $f \in K \sqbrk x$ such that:
:$f \in K \sqbrk x$ is a monic polynomial of smallest degree such that $\map f \alpha = 0$
Since $\alpha$ is algebraic over $K$, there is some $f \in K \sqbrk x$ such that $\map f \alpha = 0$.
Therefore we can define:
:$n = ... | Let $L / K$ be a [[Definition:Field Extension|field extension]].
Let $\alpha \in L$ be [[Definition:Algebraic Element of Field Extension|algebraic]] over $K$.
Then there exists a [[Definition:Minimal Polynomial|minimal polynomial]] $f \in K \sqbrk x$ for $\alpha$ over $K$. | According to [[Definition:Minimal Polynomial/Definition 1|definition 1 of minimal polynomial]], we ought to find $f \in K \sqbrk x$ such that:
:$f \in K \sqbrk x$ is a [[Definition:Monic Polynomial|monic polynomial]] of [[Definition:Smallest Natural Number|smallest]] [[Definition:Degree of Polynomial|degree]] such tha... | Minimal Polynomial Exists/Proof 1 | https://proofwiki.org/wiki/Minimal_Polynomial_Exists | https://proofwiki.org/wiki/Minimal_Polynomial_Exists/Proof_1 | [
"Minimal Polynomial Exists",
"Minimal Polynomials"
] | [
"Definition:Field Extension",
"Definition:Algebraic Element of Field Extension",
"Definition:Minimal Polynomial"
] | [
"Definition:Minimal Polynomial/Definition 1",
"Definition:Monic Polynomial",
"Definition:Smallest Natural Number",
"Definition:Degree of Polynomial",
"Definition:Algebraic Element of Field Extension",
"Definition:Degree of Polynomial",
"Definition:Smallest Natural Number",
"Definition:Degree of Polyno... |
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