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-16900 | Minimally Inductive Class under Progressing Mapping induces Nest | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Then $M$ is a nest in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$ | A minimally inductive class under $g$ is the same thing as a minimally closed class under $g$ with respect to $\O$.
The result then follows by a direct application of Minimally Closed Class under Progressing Mapping induces Nest.
{{qed}} | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Then $M$ is a [[Definition:Nest (Class Theory)|nest]] in which:
:$\forall x, y \in M: \map g x \subseteq... | A [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class under $g$]] is the same thing as a [[Definition:Minimally Closed Class|minimally closed class under $g$ with respect to $\O$]].
The result then follows by a direct application of [[Minimally Closed Class under Progressing Mapping ... | Minimally Inductive Class under Progressing Mapping induces Nest/Proof 2 | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Progressing_Mapping_induces_Nest | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Progressing_Mapping_induces_Nest/Proof_2 | [
"Progressing Mappings",
"Minimally Inductive Classes",
"Minimally Inductive Class under Progressing Mapping induces Nest"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping",
"Definition:Nest/Class Theory"
] | [
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Minimally Closed Class",
"Minimally Closed Class under Progressing Mapping induces Nest"
] |
proofwiki-16901 | Sandwich Principle | Let $A$ be a class.
Let $g: A \to A$ be a mapping on $A$ such that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Then:
:$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
That is, there is no element $y$ of $A$ such that:
:$x \subset y \subset \map g x$
w... | From Minimally Inductive Class under Progressing Mapping induces Nest, we have that $M$ is a nest in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus the Sandwich Principle applies directly.
{{qed}} | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $g: A \to A$ be a [[Definition:Class Mapping|mapping]] on $A$ such that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Then:
:$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
That is, there... | From [[Minimally Inductive Class under Progressing Mapping induces Nest]], we have that $M$ is a [[Definition:Nest (Class Theory)|nest]] in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus the [[Sandwich Principle]] applies directly.
{{qed}} | Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle/Proof 1 | https://proofwiki.org/wiki/Sandwich_Principle | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Sandwich_Principle/Proof_1 | [
"Well-Orderings",
"Named Theorems",
"Sandwich Principle"
] | [
"Definition:Class (Class Theory)",
"Definition:Mapping/Class Theory",
"Definition:Element/Class",
"Definition:Proper Subset"
] | [
"Minimally Inductive Class under Progressing Mapping induces Nest",
"Definition:Nest/Class Theory",
"Sandwich Principle"
] |
proofwiki-16902 | Sandwich Principle | Let $A$ be a class.
Let $g: A \to A$ be a mapping on $A$ such that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Then:
:$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
That is, there is no element $y$ of $A$ such that:
:$x \subset y \subset \map g x$
w... | By definition of minimally inductive class, $M$ is minimally closed under $g$ with respect to $\O$.
The result is then seen to be a direct application of Sandwich Principle for Minimally Closed Class.
{{qed}} | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $g: A \to A$ be a [[Definition:Class Mapping|mapping]] on $A$ such that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Then:
:$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
That is, there... | By definition of [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class]], $M$ is [[Definition:Minimally Closed Class|minimally closed under $g$ with respect to $\O$]].
The result is then seen to be a direct application of [[Sandwich Principle for Minimally Closed Class]].
{{qed}} | Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle/Proof 2 | https://proofwiki.org/wiki/Sandwich_Principle | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Sandwich_Principle/Proof_2 | [
"Well-Orderings",
"Named Theorems",
"Sandwich Principle"
] | [
"Definition:Class (Class Theory)",
"Definition:Mapping/Class Theory",
"Definition:Element/Class",
"Definition:Proper Subset"
] | [
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Minimally Closed Class",
"Sandwich Principle for Minimally Closed Class"
] |
proofwiki-16903 | Sandwich Principle | Let $A$ be a class.
Let $g: A \to A$ be a mapping on $A$ such that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Then:
:$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
That is, there is no element $y$ of $A$ such that:
:$x \subset y \subset \map g x$
w... | We are given that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Let $x, y \in A$ such that:
:$x \subseteq y \subseteq \map g x$
{{AimForCont}} both $x \subset y$ and $y \subset \map g x$.
From $x \subset y$, it follows by definition of proper subset that:
:$\exists a \in y: a \notin x$
and s... | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $g: A \to A$ be a [[Definition:Class Mapping|mapping]] on $A$ such that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Then:
:$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
That is, there... | We are given that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Let $x, y \in A$ such that:
:$x \subseteq y \subseteq \map g x$
{{AimForCont}} both $x \subset y$ and $y \subset \map g x$.
From $x \subset y$, it follows by definition of [[Definition:Proper Subset|proper subset]] that:
:... | Sandwich Principle/Proof 1 | https://proofwiki.org/wiki/Sandwich_Principle | https://proofwiki.org/wiki/Sandwich_Principle/Proof_1 | [
"Well-Orderings",
"Named Theorems",
"Sandwich Principle"
] | [
"Definition:Class (Class Theory)",
"Definition:Mapping/Class Theory",
"Definition:Element/Class",
"Definition:Proper Subset"
] | [
"Definition:Proper Subset",
"Definition:Proper Subset",
"Definition:Contradiction",
"Proof by Contradiction"
] |
proofwiki-16904 | Sandwich Principle | Let $A$ be a class.
Let $g: A \to A$ be a mapping on $A$ such that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Then:
:$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
That is, there is no element $y$ of $A$ such that:
:$x \subset y \subset \map g x$
w... | We are given that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Let $x, y \in A$ such that:
:$x \subseteq y \subseteq \map g x$
Then either we have:
:$\map g x \subseteq y$ and $y \subseteq \map g x$
in which case, by definition of set equality:
:$y = \map g x$
or we have that:
:$x \subseteq... | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $g: A \to A$ be a [[Definition:Class Mapping|mapping]] on $A$ such that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Then:
:$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
That is, there... | We are given that:
:for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Let $x, y \in A$ such that:
:$x \subseteq y \subseteq \map g x$
Then either we have:
:$\map g x \subseteq y$ and $y \subseteq \map g x$
in which case, by definition of [[Definition:Set Equality|set equality]]:
:$y = \map g ... | Sandwich Principle/Proof 2 | https://proofwiki.org/wiki/Sandwich_Principle | https://proofwiki.org/wiki/Sandwich_Principle/Proof_2 | [
"Well-Orderings",
"Named Theorems",
"Sandwich Principle"
] | [
"Definition:Class (Class Theory)",
"Definition:Mapping/Class Theory",
"Definition:Element/Class",
"Definition:Proper Subset"
] | [
"Definition:Set Equality",
"Definition:Set Equality"
] |
proofwiki-16905 | Sandwich Principle/Corollary 1 | Let:
:$x \subset y$
where $\subset$ denotes a proper subset.
Then:
:$\map g x \subseteq y$ | Let $x \subset y$.
By hypothesis, either $\map g x \subseteq y$ or $y \subseteq x$.
But because $x \subset y$, it follows that $y \subseteq x$ cannot be the case.
Hence the result.
{{Qed}} | Let:
:$x \subset y$
where $\subset$ denotes a [[Definition:Proper Subset|proper subset]].
Then:
:$\map g x \subseteq y$ | Let $x \subset y$.
By hypothesis, either $\map g x \subseteq y$ or $y \subseteq x$.
But because $x \subset y$, it follows that $y \subseteq x$ cannot be the case.
Hence the result.
{{Qed}} | Sandwich Principle/Corollary 1 | https://proofwiki.org/wiki/Sandwich_Principle/Corollary_1 | https://proofwiki.org/wiki/Sandwich_Principle/Corollary_1 | [
"Sandwich Principle"
] | [
"Definition:Proper Subset"
] | [] |
proofwiki-16906 | Sandwich Principle/Corollary 2 | Let $g$ be a progressing mapping.
Let $x \subseteq y$.
Then:
:$\map g x \subseteq \map g y$ | Let $x \subseteq y$.
Suppose $x = y$.
Then $\map g x \subseteq \map g y$ and the result holds.
{{qed|lemma}}
Suppose that $x \ne y$.
Then $x \subset y$
It follows from {{Corollary|Sandwich Principle|1}} that:
:$\map g x \subseteq y$
As $g$ is a progressing mapping on $A$:
:$y \subseteq \map g y$
Hence by Subset Relatio... | Let $g$ be a [[Definition:Progressing Mapping|progressing mapping]].
Let $x \subseteq y$.
Then:
:$\map g x \subseteq \map g y$ | Let $x \subseteq y$.
Suppose $x = y$.
Then $\map g x \subseteq \map g y$ and the result holds.
{{qed|lemma}}
Suppose that $x \ne y$.
Then $x \subset y$
It follows from {{Corollary|Sandwich Principle|1}} that:
:$\map g x \subseteq y$
As $g$ is a [[Definition:Progressing Mapping|progressing mapping]] on $A$:
:$y \... | Sandwich Principle/Corollary 2 | https://proofwiki.org/wiki/Sandwich_Principle/Corollary_2 | https://proofwiki.org/wiki/Sandwich_Principle/Corollary_2 | [
"Sandwich Principle"
] | [
"Definition:Progressing Mapping"
] | [
"Definition:Progressing Mapping",
"Subset Relation is Transitive"
] |
proofwiki-16907 | Class under Progressing Mapping such that Elements are Sandwiched is Nest | Let $A$ be a class.
Let $g: A \to A$ be a progressing mapping on $A$ such that:
:$\forall x, y \in A: \map g x \subseteq y \lor y \subseteq x$
Then $A$ is a nest:
:$\forall x, y \in A: x \subseteq y \lor y \subseteq x$ | By definition of progressing mapping:
:$\forall x \in A: x \subseteq \map g x$
Thus by Subset Relation is Transitive:
:$\map g x \subseteq y \implies x \subseteq y$
and it follows that:
:$\forall x, y \in A: x \subseteq y \lor y \subseteq x$
Hence the result by definition of nest.
{{qed}} | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $g: A \to A$ be a [[Definition:Progressing Mapping|progressing mapping]] on $A$ such that:
:$\forall x, y \in A: \map g x \subseteq y \lor y \subseteq x$
Then $A$ is a [[Definition:Nest (Class Theory)|nest]]:
:$\forall x, y \in A: x \subseteq y \lor y \sub... | By definition of [[Definition:Progressing Mapping|progressing mapping]]:
:$\forall x \in A: x \subseteq \map g x$
Thus by [[Subset Relation is Transitive]]:
:$\map g x \subseteq y \implies x \subseteq y$
and it follows that:
:$\forall x, y \in A: x \subseteq y \lor y \subseteq x$
Hence the result by definition of [[... | Class under Progressing Mapping such that Elements are Sandwiched is Nest | https://proofwiki.org/wiki/Class_under_Progressing_Mapping_such_that_Elements_are_Sandwiched_is_Nest | https://proofwiki.org/wiki/Class_under_Progressing_Mapping_such_that_Elements_are_Sandwiched_is_Nest | [
"Progressing Mappings"
] | [
"Definition:Class (Class Theory)",
"Definition:Progressing Mapping",
"Definition:Nest/Class Theory"
] | [
"Definition:Progressing Mapping",
"Subset Relation is Transitive",
"Definition:Nest/Class Theory"
] |
proofwiki-16908 | Characteristics of Minimally Inductive Class under Progressing Mapping | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Then for all $x, y \in M$: | From Minimally Inductive Class under Progressing Mapping induces Nest, we have that $M$ is a nest in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus the Sandwich Principle applies directly.
{{qed}} | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Then for all $x, y \in M$: | From [[Minimally Inductive Class under Progressing Mapping induces Nest]], we have that $M$ is a [[Definition:Nest (Class Theory)|nest]] in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus the [[Sandwich Principle]] applies directly.
{{qed}} | Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle/Proof 1 | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Sandwich_Principle/Proof_1 | [
"Minimally Inductive Classes",
"Progressing Mappings",
"Characteristics of Minimally Inductive Class under Progressing Mapping"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping"
] | [
"Minimally Inductive Class under Progressing Mapping induces Nest",
"Definition:Nest/Class Theory",
"Sandwich Principle"
] |
proofwiki-16909 | Characteristics of Minimally Inductive Class under Progressing Mapping | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Then for all $x, y \in M$: | By definition of minimally inductive class, $M$ is minimally closed under $g$ with respect to $\O$.
The result is then seen to be a direct application of Sandwich Principle for Minimally Closed Class.
{{qed}} | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Then for all $x, y \in M$: | By definition of [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class]], $M$ is [[Definition:Minimally Closed Class|minimally closed under $g$ with respect to $\O$]].
The result is then seen to be a direct application of [[Sandwich Principle for Minimally Closed Class]].
{{qed}} | Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle/Proof 2 | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Sandwich_Principle/Proof_2 | [
"Minimally Inductive Classes",
"Progressing Mappings",
"Characteristics of Minimally Inductive Class under Progressing Mapping"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping"
] | [
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Minimally Closed Class",
"Sandwich Principle for Minimally Closed Class"
] |
proofwiki-16910 | Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle | :$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$ | From Minimally Inductive Class under Progressing Mapping induces Nest, we have that $M$ is a nest in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus the Sandwich Principle applies directly.
{{qed}} | :$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$ | From [[Minimally Inductive Class under Progressing Mapping induces Nest]], we have that $M$ is a [[Definition:Nest (Class Theory)|nest]] in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus the [[Sandwich Principle]] applies directly.
{{qed}} | Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle/Proof 1 | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Sandwich_Principle | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Sandwich_Principle/Proof_1 | [
"Sandwich Principle",
"Characteristics of Minimally Inductive Class under Progressing Mapping",
"Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle"
] | [] | [
"Minimally Inductive Class under Progressing Mapping induces Nest",
"Definition:Nest/Class Theory",
"Sandwich Principle"
] |
proofwiki-16911 | Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle | :$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$ | By definition of minimally inductive class, $M$ is minimally closed under $g$ with respect to $\O$.
The result is then seen to be a direct application of Sandwich Principle for Minimally Closed Class.
{{qed}} | :$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$ | By definition of [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class]], $M$ is [[Definition:Minimally Closed Class|minimally closed under $g$ with respect to $\O$]].
The result is then seen to be a direct application of [[Sandwich Principle for Minimally Closed Class]].
{{qed}} | Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle/Proof 2 | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Sandwich_Principle | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Sandwich_Principle/Proof_2 | [
"Sandwich Principle",
"Characteristics of Minimally Inductive Class under Progressing Mapping",
"Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle"
] | [] | [
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Minimally Closed Class",
"Sandwich Principle for Minimally Closed Class"
] |
proofwiki-16912 | Characteristics of Minimally Inductive Class under Progressing Mapping/Image of Proper Subset is Subset | :$x \subset y \implies \map g x \subseteq y$ | From Minimally Inductive Class under Progressing Mapping induces Nest, we have that $M$ is a nest in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus corollary $1$ of the Sandwich Principle applies directly.
{{qed}} | :$x \subset y \implies \map g x \subseteq y$ | From [[Minimally Inductive Class under Progressing Mapping induces Nest]], we have that $M$ is a [[Definition:Nest (Class Theory)|nest]] in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus [[Sandwich Principle/Corollary 1|corollary $1$ of the Sandwich Principle]] applies directly.
{{qed}} | Characteristics of Minimally Inductive Class under Progressing Mapping/Image of Proper Subset is Subset | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Image_of_Proper_Subset_is_Subset | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Image_of_Proper_Subset_is_Subset | [
"Characteristics of Minimally Inductive Class under Progressing Mapping"
] | [] | [
"Minimally Inductive Class under Progressing Mapping induces Nest",
"Definition:Nest/Class Theory",
"Sandwich Principle/Corollary 1"
] |
proofwiki-16913 | Characteristics of Minimally Inductive Class under Progressing Mapping/Mapping Preserves Subsets | :$x \subseteq y \implies \map g x \subseteq \map g y$ | From Minimally Inductive Class under Progressing Mapping induces Nest, we have that $M$ is a nest in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus corollary $2$ of the Sandwich Principle applies directly.
{{qed}} | :$x \subseteq y \implies \map g x \subseteq \map g y$ | From [[Minimally Inductive Class under Progressing Mapping induces Nest]], we have that $M$ is a [[Definition:Nest (Class Theory)|nest]] in which:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Thus [[Sandwich Principle/Corollary 2|corollary $2$ of the Sandwich Principle]] applies directly.
{{qed}} | Characteristics of Minimally Inductive Class under Progressing Mapping/Mapping Preserves Subsets | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Mapping_Preserves_Subsets | https://proofwiki.org/wiki/Characteristics_of_Minimally_Inductive_Class_under_Progressing_Mapping/Mapping_Preserves_Subsets | [
"Characteristics of Minimally Inductive Class under Progressing Mapping"
] | [] | [
"Minimally Inductive Class under Progressing Mapping induces Nest",
"Definition:Nest/Class Theory",
"Sandwich Principle/Corollary 2"
] |
proofwiki-16914 | Bounded Class is Set | Let $B$ be a class.
Let it be assumed that $B$ is a subclass of a basic universe $V$.
Let $B$ be bounded by a set $x$.
Then $B$ is itself a set. | By definition, every element of $B$ is a subset of $x$.
Then every element of $B$ is an element of the power set $\powerset x$ of $x$.
Thus $B$ is a subclass of $\powerset x$.
By the Axiom of Powers, $\powerset x$ is a set.
That is, $\powerset x$ is an element of $V$.
As $V$ is a swelled class, then by definition, then... | Let $B$ be a [[Definition:Class (Class Theory)|class]].
Let it be assumed that $B$ is a [[Definition:Subclass|subclass]] of a [[Definition:Basic Universe|basic universe]] $V$.
Let $B$ be [[Definition:Set Bounded by Set|bounded]] by a [[Definition:Set|set]] $x$.
Then $B$ is itself a [[Definition:Set|set]]. | By definition, every [[Definition:Element of Class|element]] of $B$ is a [[Definition:Subset|subset]] of $x$.
Then every [[Definition:Element of Class|element]] of $B$ is an [[Definition:Element|element]] of the [[Definition:Power Set|power set]] $\powerset x$ of $x$.
Thus $B$ is a [[Definition:Subclass|subclass]] of... | Bounded Class is Set | https://proofwiki.org/wiki/Bounded_Class_is_Set | https://proofwiki.org/wiki/Bounded_Class_is_Set | [
"Bounded Classes"
] | [
"Definition:Class (Class Theory)",
"Definition:Subclass",
"Definition:Basic Universe",
"Definition:Bounded Class/Bounded by Set",
"Definition:Set",
"Definition:Set"
] | [
"Definition:Element/Class",
"Definition:Subset",
"Definition:Element/Class",
"Definition:Element",
"Definition:Power Set",
"Definition:Subclass",
"Axiom:Axiom of Powers/Class Theory",
"Definition:Set",
"Definition:Element/Class",
"Definition:Swelled Class",
"Definition:Subclass",
"Definition:E... |
proofwiki-16915 | Non-Empty Bounded Subset of Minimally Inductive Class under Progressing Mapping has Greatest Element | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Then every non-empty bounded subset of $M$ has a greatest element. | Let the hypothesis be assumed.
The proof proceeds by general induction.
For all $x \in M$, let $\map P x$ be the proposition:
:Every non-empty subset of $M$ which is bounded by $x$ has a greatest element.
=== Basis for the Induction ===
Let $x = \O$.
The only non-empty subset of $M$ which is bounded by $\O$ is $\set \O... | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Then every [[Definition:Non-Empty Set|non-empty]] [[Definition:Bounded Subset of Class|bounded subset]] ... | Let the hypothesis be assumed.
The proof proceeds by [[Principle of General Induction|general induction]].
For all $x \in M$, let $\map P x$ be the [[Definition:Proposition|proposition]]:
:Every [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $M$ which is [[Definition:Set Bounded by Set|bounde... | Non-Empty Bounded Subset of Minimally Inductive Class under Progressing Mapping has Greatest Element/Proof 1 | https://proofwiki.org/wiki/Non-Empty_Bounded_Subset_of_Minimally_Inductive_Class_under_Progressing_Mapping_has_Greatest_Element | https://proofwiki.org/wiki/Non-Empty_Bounded_Subset_of_Minimally_Inductive_Class_under_Progressing_Mapping_has_Greatest_Element/Proof_1 | [
"Bounded Classes",
"Progressing Mappings",
"Minimally Inductive Classes",
"Non-Empty Bounded Subset of Minimally Inductive Class under Progressing Mapping has Greatest Element"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping",
"Definition:Non-Empty Set",
"Definition:Bounded Class/Bounded Subset of Class",
"Definition:Greatest Set by Set Inclusion/Class Theory"
] | [
"Principle of General Induction",
"Definition:Proposition",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Bounded Class/Bounded by Set",
"Definition:Greatest Set by Set Inclusion/Class Theory",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Bounded Class/Bounded by Set",... |
proofwiki-16916 | Projection is Injection iff Factor is Singleton/Family of Sets/Necessary Condition | Let $\family {S_i}_{i \mathop \in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set.
Let $S = \ds \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.
Let $\pr_j: S \to S_j$ be the $j$th projection on $S$.
Let $\pr_j$ be an injection.
Then $S_i$ is a ... | Let $\pr_j$ be an injection.
Then:
:$\forall x, y \in S: \map {\pr_j} x = \map {\pr_j} y \implies x = y$
We have that $\family {S_i}_{i \mathop \in I}$ is a non-empty family of non-empty sets
Hence, by the axiom of choice (formulation $2$), $S$ is non-empty.
Let $z \in S$.
By the definition of Cartesian product $S$:
:$... | Let $\family {S_i}_{i \mathop \in I}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Indexed Family of Sets|family]] of [[Definition: Non-Empty Set|non-empty sets]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $S = \ds \prod_{i \mathop \in I} S_i$ be the [[Definition:Cartesian Product... | Let $\pr_j$ be an [[Definition:Injection|injection]].
Then:
:$\forall x, y \in S: \map {\pr_j} x = \map {\pr_j} y \implies x = y$
We have that $\family {S_i}_{i \mathop \in I}$ is a [[Definition: Non-Empty Set|non-empty]] [[Definition:Indexed Family of Sets|family]] of [[Definition: Non-Empty Set|non-empty sets]]
H... | Projection is Injection iff Factor is Singleton/Family of Sets/Necessary Condition | https://proofwiki.org/wiki/Projection_is_Injection_iff_Factor_is_Singleton/Family_of_Sets/Necessary_Condition | https://proofwiki.org/wiki/Projection_is_Injection_iff_Factor_is_Singleton/Family_of_Sets/Necessary_Condition | [
"Projection is Injection iff Factor is Singleton"
] | [
"Definition:Non-Empty Set",
"Definition:Indexing Set/Family of Sets",
"Definition: Non-Empty Set",
"Definition:Indexing Set",
"Definition:Cartesian Product of Family ",
"Definition:Projection (Mapping Theory)",
"Definition:Injection",
"Definition:Singleton"
] | [
"Definition:Injection",
"Definition: Non-Empty Set",
"Definition:Indexing Set/Family of Sets",
"Definition: Non-Empty Set",
"Axiom:Axiom of Choice/Formulation 2",
"Definition:Non-Empty Set",
"Definition:Cartesian Product/Family of Sets"
] |
proofwiki-16917 | Projection is Injection iff Factor is Singleton/Family of Sets/Sufficient Condition | Let $\family {S_i}_{i \mathop \in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set.
Let $S = \ds \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.
Let $\pr_j: S \to S_j$ be the $j$th projection on $S$.
Let $S_i$ be a singleton for all $i \in I \se... | Let $S_i = \set {s_i}$ for all $i \in I \setminus \set {j}$.
Let $\map {\pr_j} x = \map {\pr_j} y = s_j$ for $x, y \in S$.
By definition of $j$th projection:
:$\map x j = \map {\pr_j} x = s_j$
:$\map y j = \map {\pr_j} y = s_j$
and so $\map x j = \map y j$.
By the definition of Cartesian product, for all $i \in I \setm... | Let $\family {S_i}_{i \mathop \in I}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Indexed Family of Sets|family]] of [[Definition: Non-Empty Set|non-empty sets]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $S = \ds \prod_{i \mathop \in I} S_i$ be the [[Definition:Cartesian Product... | Let $S_i = \set {s_i}$ for all $i \in I \setminus \set {j}$.
Let $\map {\pr_j} x = \map {\pr_j} y = s_j$ for $x, y \in S$.
By definition of [[Definition:Projection (Mapping Theory)|$j$th projection]]:
:$\map x j = \map {\pr_j} x = s_j$
:$\map y j = \map {\pr_j} y = s_j$
and so $\map x j = \map y j$.
By the defini... | Projection is Injection iff Factor is Singleton/Family of Sets/Sufficient Condition | https://proofwiki.org/wiki/Projection_is_Injection_iff_Factor_is_Singleton/Family_of_Sets/Sufficient_Condition | https://proofwiki.org/wiki/Projection_is_Injection_iff_Factor_is_Singleton/Family_of_Sets/Sufficient_Condition | [
"Projection is Injection iff Factor is Singleton"
] | [
"Definition:Non-Empty Set",
"Definition:Indexing Set/Family of Sets",
"Definition: Non-Empty Set",
"Definition:Indexing Set",
"Definition:Cartesian Product of Family ",
"Definition:Projection (Mapping Theory)",
"Definition:Singleton",
"Definition:Injection"
] | [
"Definition:Projection (Mapping Theory)",
"Definition:Cartesian Product/Family of Sets",
"Definition:Injection"
] |
proofwiki-16918 | Product Space of Subspaces is Subspace of Product Space | Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$.
Let $\family {\struct {Y_i, \upsil... | From Cartesian Product of Family of Subsets, $Y \subseteq X$.
Thus the topological subspace $T_Y$ is well-defined.
From Natural Basis of Product Topology, a (synthetic) basis for $T$ is:
:$\ds \BB_T = \set {\prod_{i \mathop \in I} U_i : U_i \in \tau_i, U_i = X_i \text{ for all but finitely many } i \in I}$
From Basis f... | Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the [[Definition:... | From [[Cartesian Product of Family of Subsets]], $Y \subseteq X$.
Thus the [[Definition:Topological Subspace|topological subspace]] $T_Y$ is [[Definition:Well-Defined|well-defined]].
From [[Natural Basis of Product Topology]], a [[Definition:Synthetic Basis|(synthetic) basis]] for $T$ is:
:$\ds \BB_T = \set {\prod_{... | Product Space of Subspaces is Subspace of Product Space | https://proofwiki.org/wiki/Product_Space_of_Subspaces_is_Subspace_of_Product_Space | https://proofwiki.org/wiki/Product_Space_of_Subspaces_is_Subspace_of_Product_Space | [
"Product Topology",
"Subsets"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Topological Subspace",
"Definition:Product Space (Topology)",
"Definition:Topological Subs... | [
"Cartesian Product of Subsets/Family of Subsets",
"Definition:Topological Subspace",
"Definition:Well-Defined",
"Natural Basis of Product Topology",
"Definition:Basis (Topology)/Synthetic Basis",
"Basis for Topological Subspace",
"Definition:Basis (Topology)/Synthetic Basis",
"Cartesian Product of Int... |
proofwiki-16919 | Difference of Complex Conjugates | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\overline z$ denote the complex conjugate of the complex number $z$.
Then:
:$\overline {z_1 - z_2} = \overline {z_1} - \overline {z_2}$ | Let $w = -z_2$.
Then:
{{begin-eqn}}
{{eqn | l = \overline {z_1 - z_2}
| r = \overline {z_1 + \paren {-z_2} }
| c = {{Defof|Complex Subtraction}}
}}
{{eqn | r = \overline {z_1 + w}
| c = Definition of $w$
}}
{{eqn | r = \overline {z_1} + \overline w
| c = Sum of Complex Conjugates
}}
{{eqn | r = ... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of the [[Definition:Complex Number|complex number]] $z$.
Then:
:$\overline {z_1 - z_2} = \overline {z_1} - \overline {z_2}$ | Let $w = -z_2$.
Then:
{{begin-eqn}}
{{eqn | l = \overline {z_1 - z_2}
| r = \overline {z_1 + \paren {-z_2} }
| c = {{Defof|Complex Subtraction}}
}}
{{eqn | r = \overline {z_1 + w}
| c = Definition of $w$
}}
{{eqn | r = \overline {z_1} + \overline w
| c = [[Sum of Complex Conjugates]]
}}
{{eqn ... | Difference of Complex Conjugates | https://proofwiki.org/wiki/Difference_of_Complex_Conjugates | https://proofwiki.org/wiki/Difference_of_Complex_Conjugates | [
"Complex Conjugates",
"Complex Subtraction"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number"
] | [
"Sum of Complex Conjugates"
] |
proofwiki-16920 | Nonzero Natural Number is Successor | Let $\N$ be the $0$-based natural numbers:
:$\N = \set {0, 1, 2, \ldots}$
Let $s: \N \to \N: \map s n = n + 1$ be the successor mapping.
Then:
:$\forall n \in \N \setminus \set 0 \paren {\exists m \in \N: \map s m = n}$ | The proof will proceed by the Principle of Finite Induction on $\N \setminus \set 0$. | Let $\N$ be the [[Definition:Natural Numbers|$0$-based natural numbers]]:
:$\N = \set {0, 1, 2, \ldots}$
Let $s: \N \to \N: \map s n = n + 1$ be the [[Definition:Successor Mapping|successor mapping]].
Then:
:$\forall n \in \N \setminus \set 0 \paren {\exists m \in \N: \map s m = n}$ | The proof will proceed by the [[Principle of Finite Induction]] on $\N \setminus \set 0$. | Nonzero Natural Number is Successor | https://proofwiki.org/wiki/Nonzero_Natural_Number_is_Successor | https://proofwiki.org/wiki/Nonzero_Natural_Number_is_Successor | [
"Proofs by Induction",
"Natural Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Successor Mapping"
] | [
"Principle of Finite Induction",
"Principle of Finite Induction"
] |
proofwiki-16921 | Determinant with Columns Transposed | If two columns of a matrix with determinant $D$ are transposed, its determinant becomes $-D$. | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $1 \le r < s \le n$.
Let $\mathbf B$ be $\mathbf A$ with columns $r$ and $s$ transposed.
Consider:
:the transpose $\mathbf A^\intercal$ of $\mathbf A$
:the transpose $\mathbf B^\intercal$ of ... | If two [[Definition:Column of Matrix|columns]] of a [[Definition:Matrix|matrix]] with [[Definition:Determinant of Matrix|determinant]] $D$ are [[Definition:Transposition|transposed]], its determinant becomes $-D$. | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix of order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $1 \le r < s \le n$.
Let $\mathbf B$ be $\mathbf A$ with [[Definition:Column of Matrix|columns]] $r$ and $s$ [[Definition... | Determinant with Columns Transposed | https://proofwiki.org/wiki/Determinant_with_Columns_Transposed | https://proofwiki.org/wiki/Determinant_with_Columns_Transposed | [
"Determinants"
] | [
"Definition:Matrix/Column",
"Definition:Matrix",
"Definition:Determinant/Matrix",
"Definition:Transposition"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Matrix/Column",
"Definition:Transposition",
"Definition:Transpose of Matrix",
"Definition:Transpose of Matrix",
"Definition:Matrix/Row",
"Definition:Transposition",
"Determinant with Rows Transposed",
"Determinant of T... |
proofwiki-16922 | Subspace of Product Space is Homeomorphic to Factor Space/Proof 1/Lemma 2 | :$\pr_i {\restriction_{Y_i} } = p_i$ | For all $y \in Y_i$:
{{begin-eqn}}
{{eqn | l = \map {\pr_i {\restriction_{Y_i} } } y
| r = \map {\pr_i} y
| c = {{Defof|Restriction of Mapping}}: $\pr_i {\restriction_{Y_i} } : Y_i \to X_i$
}}
{{eqn | r = y_i
| c = {{Defof|Projection}}: $\pr_i: X \to X_i$
}}
{{eqn | r = \map {p_i} y
| c = {{Defo... | :$\pr_i {\restriction_{Y_i} } = p_i$ | For all $y \in Y_i$:
{{begin-eqn}}
{{eqn | l = \map {\pr_i {\restriction_{Y_i} } } y
| r = \map {\pr_i} y
| c = {{Defof|Restriction of Mapping}}: $\pr_i {\restriction_{Y_i} } : Y_i \to X_i$
}}
{{eqn | r = y_i
| c = {{Defof|Projection}}: $\pr_i: X \to X_i$
}}
{{eqn | r = \map {p_i} y
| c = {{Def... | Subspace of Product Space is Homeomorphic to Factor Space/Proof 1/Lemma 2 | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_1/Lemma_2 | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_1/Lemma_2 | [
"Subspace of Product Space is Homeomorphic to Factor Space"
] | [] | [
"Equality of Mappings",
"Category:Subspace of Product Space is Homeomorphic to Factor Space"
] |
proofwiki-16923 | Subspace of Product Space is Homeomorphic to Factor Space/Proof 1/Lemma 1 | :$Y_i = \prod_{j \mathop \in I} Z_j$ | {{begin-eqn}}
{{eqn | r = x \in Y_i
| o =
}}
{{eqn | ll = \leadstoandfrom
| q = \forall j \in I
| l = x_j
| r = \begin {cases} z_j & j \ne i \\ x_i \in X_i & i = j \end {cases}
| c = Definition of $Y_i$
}}
{{eqn | ll = \leadstoandfrom
| q = \forall j \in I
| l = x_j
| o =... | :$Y_i = \prod_{j \mathop \in I} Z_j$ | {{begin-eqn}}
{{eqn | r = x \in Y_i
| o =
}}
{{eqn | ll = \leadstoandfrom
| q = \forall j \in I
| l = x_j
| r = \begin {cases} z_j & j \ne i \\ x_i \in X_i & i = j \end {cases}
| c = Definition of $Y_i$
}}
{{eqn | ll = \leadstoandfrom
| q = \forall j \in I
| l = x_j
| o =... | Subspace of Product Space is Homeomorphic to Factor Space/Proof 1/Lemma 1 | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_1/Lemma_1 | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_1/Lemma_1 | [
"Subspace of Product Space is Homeomorphic to Factor Space"
] | [] | [
"Definition:Set Equality",
"Category:Subspace of Product Space is Homeomorphic to Factor Space"
] |
proofwiki-16924 | Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Let $x$ be a fixed point of $g$.
Then $x$ is the greatest element of $M$. | Let $x$ be an element of $M$ such that $\map g x = x$.
From Empty Set is Subset of All Sets, we have that:
:$\O \subseteq x$
Suppose that $y \subseteq x$.
Then by Characteristics of Minimally Inductive Class under Progressing Mapping:
:$\map g y \subseteq \map g x$
But we have that $\map g x = x$.
Thus:
:$\map g y \sub... | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let $x$ be a [[Definition:Fixed Point|fixed point]] of $g$.
Then $x$ is the [[Definition:Greatest Set ... | Let $x$ be an [[Definition:Element of Class|element]] of $M$ such that $\map g x = x$.
From [[Empty Set is Subset of All Sets]], we have that:
:$\O \subseteq x$
Suppose that $y \subseteq x$.
Then by [[Characteristics of Minimally Inductive Class under Progressing Mapping]]:
:$\map g y \subseteq \map g x$
But we ha... | Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element/Proof 1 | https://proofwiki.org/wiki/Fixed_Point_of_Progressing_Mapping_on_Minimally_Inductive_Class_is_Greatest_Element | https://proofwiki.org/wiki/Fixed_Point_of_Progressing_Mapping_on_Minimally_Inductive_Class_is_Greatest_Element/Proof_1 | [
"Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element",
"Minimally Inductive Classes",
"Progressing Mappings"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping",
"Definition:Fixed Point",
"Definition:Greatest Set by Set Inclusion/Class Theory"
] | [
"Definition:Element/Class",
"Empty Set is Subset of All Sets",
"Characteristics of Minimally Inductive Class under Progressing Mapping",
"Principle of General Induction"
] |
proofwiki-16925 | Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Let $x$ be a fixed point of $g$.
Then $x$ is the greatest element of $M$. | A minimally inductive class under $g$ is the same thing as a minimally closed class under $g$ with respect to $\O$.
The result then follows by a direct application of Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element.
{{qed}} | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let $x$ be a [[Definition:Fixed Point|fixed point]] of $g$.
Then $x$ is the [[Definition:Greatest Set ... | A [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class under $g$]] is the same thing as a [[Definition:Minimally Closed Class|minimally closed class under $g$ with respect to $\O$]].
The result then follows by a direct application of [[Fixed Point of Progressing Mapping on Minimally C... | Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element/Proof 2 | https://proofwiki.org/wiki/Fixed_Point_of_Progressing_Mapping_on_Minimally_Inductive_Class_is_Greatest_Element | https://proofwiki.org/wiki/Fixed_Point_of_Progressing_Mapping_on_Minimally_Inductive_Class_is_Greatest_Element/Proof_2 | [
"Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element",
"Minimally Inductive Classes",
"Progressing Mappings"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping",
"Definition:Fixed Point",
"Definition:Greatest Set by Set Inclusion/Class Theory"
] | [
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Minimally Closed Class",
"Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element"
] |
proofwiki-16926 | Closed Class under Progressing Mapping Lemma | Let $N$ be a class which is closed under a progressing mapping $g$.
Let $g$ be such that:
:$\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$
:if $\map g x = x$, then $x$ is the greatest element of $N$.
Let the following hold:
:$A \subseteq N$ is a subclass of $N$
:$x \in N$ is an element of $N$
Let $x$ be:
... | Let the hypothesis be assumed.
Let $A$ be an arbitrary non-empty subclass of $N$.
Let $L$ be the class of all elements $y$ of $N$ such that $y$ is a proper subset of all elements of $A$.
Let $x$ be the greatest element of $L$.
It is to be shown that $\map g x$ is the smallest element of $A$.
We have that $x$ is a prope... | Let $N$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Closed Class under Mapping|closed]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let $g$ be such that:
:$\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$
:if $\map g x = x$, then $x$ is the [[Definition:Grea... | Let the hypothesis be assumed.
Let $A$ be an arbitrary [[Definition:Non-Empty Class|non-empty]] [[Definition:Subclass|subclass]] of $N$.
Let $L$ be the [[Definition:Class (Class Theory)|class]] of all [[Definition:Element of Class|elements]] $y$ of $N$ such that $y$ is a [[Definition:Proper Subset|proper subset]] of ... | Closed Class under Progressing Mapping Lemma | https://proofwiki.org/wiki/Closed_Class_under_Progressing_Mapping_Lemma | https://proofwiki.org/wiki/Closed_Class_under_Progressing_Mapping_Lemma | [
"Progressing Mappings",
"Closedness under Mappings"
] | [
"Definition:Class (Class Theory)",
"Definition:Closed under Mapping/Class Theory",
"Definition:Progressing Mapping",
"Definition:Greatest Set by Set Inclusion/Class Theory",
"Definition:Subclass",
"Definition:Element/Class",
"Definition:Proper Subset",
"Definition:Element/Class",
"Definition:Greates... | [
"Definition:Non-Empty Set/Class Theory",
"Definition:Subclass",
"Definition:Class (Class Theory)",
"Definition:Element/Class",
"Definition:Proper Subset",
"Definition:Element/Class",
"Definition:Greatest Set by Set Inclusion/Class Theory",
"Definition:Smallest Set by Set Inclusion/Class Theory",
"De... |
proofwiki-16927 | Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Let $x$ be a fixed point of $g$.
Then $M$ is well-ordered under the subset relation. | According to hypothesis, let $M$ be minimally inductive under $g$.
By Minimally Inductive Class under Progressing Mapping induces Nest:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
By Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element:
:if $x$ is a fixed point of $g$, ... | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let $x$ be a [[Definition:Fixed Point|fixed point]] of $g$.
Then $M$ is [[Definition:Well-Ordered Clas... | According to hypothesis, let $M$ be [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under $g$.
By [[Minimally Inductive Class under Progressing Mapping induces Nest]]:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
By [[Fixed Point of Progressing Mapping on Minimally... | Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation/Proof 1 | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Progressing_Mapping_is_Well-Ordered_under_Subset_Relation | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Progressing_Mapping_is_Well-Ordered_under_Subset_Relation/Proof_1 | [
"Minimally Inductive Classes",
"Progressing Mappings",
"Well-Orderings",
"Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping",
"Definition:Fixed Point",
"Definition:Well-Ordered Class under Subset Relation",
"Definition:Subset Relation"
] | [
"Definition:Minimally Inductive Class under General Mapping",
"Minimally Inductive Class under Progressing Mapping induces Nest",
"Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element",
"Definition:Fixed Point",
"Definition:Greatest Set by Set Inclusion/Class Theory",
"Close... |
proofwiki-16928 | Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation | Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Let $x$ be a fixed point of $g$.
Then $M$ is well-ordered under the subset relation. | A minimally inductive class under $g$ is the same thing as a minimally closed class under $g$ with respect to $\O$.
The result then follows by a direct application of Minimally Closed Class under Progressing Mapping is Well-Ordered.
{{qed}} | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Minimally Inductive Class under General Mapping|minimally inductive]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let $x$ be a [[Definition:Fixed Point|fixed point]] of $g$.
Then $M$ is [[Definition:Well-Ordered Clas... | A [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class under $g$]] is the same thing as a [[Definition:Minimally Closed Class|minimally closed class under $g$ with respect to $\O$]].
The result then follows by a direct application of [[Minimally Closed Class under Progressing Mapping ... | Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation/Proof 2 | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Progressing_Mapping_is_Well-Ordered_under_Subset_Relation | https://proofwiki.org/wiki/Minimally_Inductive_Class_under_Progressing_Mapping_is_Well-Ordered_under_Subset_Relation/Proof_2 | [
"Minimally Inductive Classes",
"Progressing Mappings",
"Well-Orderings",
"Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation"
] | [
"Definition:Class (Class Theory)",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Progressing Mapping",
"Definition:Fixed Point",
"Definition:Well-Ordered Class under Subset Relation",
"Definition:Subset Relation"
] | [
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Minimally Closed Class",
"Minimally Closed Class under Progressing Mapping is Well-Ordered"
] |
proofwiki-16929 | Minimally Closed Class under Progressing Mapping induces Nest | For all $x, y \in N$:
:either $\map g x \subseteq y$ or $y \subseteq x$
and $N$ forms a nest:
:$\forall x, y \in N: x \subseteq y$ or $y \subseteq x$ | Let $\RR$ be the relation on $N$ defined as:
:$\forall x, y \in N: \map \RR {x, y} \iff \map g x \subseteq y \lor y \subseteq x$
We are given that $g$ is a progressing mapping.
From the Progressing Function Lemma, we have that:
{{begin-axiom}}
{{axiom | n = 1
| q = \forall y \in \Dom g
| ml= \map \RR {y... | For all $x, y \in N$:
:either $\map g x \subseteq y$ or $y \subseteq x$
and $N$ forms a [[Definition:Nest (Class Theory)|nest]]:
:$\forall x, y \in N: x \subseteq y$ or $y \subseteq x$ | Let $\RR$ be the [[Definition:Relation (Class Theory)|relation]] on $N$ defined as:
:$\forall x, y \in N: \map \RR {x, y} \iff \map g x \subseteq y \lor y \subseteq x$
We are given that $g$ is a [[Definition:Progressing Mapping|progressing mapping]].
From the [[Progressing Function Lemma]], we have that:
{{begin-a... | Minimally Closed Class under Progressing Mapping induces Nest/Proof | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping_induces_Nest | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping_induces_Nest/Proof | [
"Minimally Closed Classes under Progressing Mapping"
] | [
"Definition:Nest/Class Theory"
] | [
"Definition:Relation/Class Theory",
"Definition:Progressing Mapping",
"Progressing Function Lemma",
"Double Induction Principle/Minimally Closed Class",
"Definition:Relation/Class Theory",
"Smallest Element of Minimally Closed Class under Progressing Mapping",
"Rule of Addition",
"Double Induction Pri... |
proofwiki-16930 | Minimally Closed Class under Progressing Mapping | Statement of Conditions:
{{:Minimally Closed Class under Progressing Mapping/Statement}}
Then the following results hold:
=== Minimally Closed Class under Progressing Mapping induces Nest ===
{{:Minimally Closed Class under Progressing Mapping induces Nest}}
=== Bounded Subset of Minimally Closed Class under Progressin... | Let the hypothesis be assumed.
The proof proceeds by general induction.
For all $x \in N$, let $\map P x$ be the proposition:
:Every subset of $N$ which is bounded by $x$ has a greatest element.
=== Basis for the Induction ===
Let $x = b$.
From Smallest Element of Minimally Closed Class under Progressing Mapping, $b$ i... | [[Minimally Closed Class under Progressing Mapping/Statement|Statement of Conditions]]:
{{:Minimally Closed Class under Progressing Mapping/Statement}}
Then the following results hold:
=== [[Minimally Closed Class under Progressing Mapping induces Nest]] ===
{{:Minimally Closed Class under Progressing Mapping induce... | Let the hypothesis be assumed.
The proof proceeds by [[Principle of General Induction for Minimally Closed Class|general induction]].
For all $x \in N$, let $\map P x$ be the [[Definition:Proposition|proposition]]:
:Every [[Definition:Subset|subset]] of $N$ which is [[Definition:Set Bounded by Set|bounded]] by $x$ h... | Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element/Proof | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping | https://proofwiki.org/wiki/Bounded_Subset_of_Minimally_Closed_Class_under_Progressing_Mapping_has_Greatest_Element/Proof | [
"Minimally Closed Classes",
"Progressing Mappings",
"Minimally Closed Classes under Progressing Mapping"
] | [
"Minimally Closed Class under Progressing Mapping/Statement",
"Minimally Closed Class under Progressing Mapping induces Nest",
"Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element",
"Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element",
"Minim... | [
"Principle of General Induction/Minimally Closed Class",
"Definition:Proposition",
"Definition:Subset",
"Definition:Bounded Class/Bounded by Set",
"Definition:Greatest Set by Set Inclusion/Class Theory",
"Smallest Element of Minimally Closed Class under Progressing Mapping",
"Definition:Smallest Set by ... |
proofwiki-16931 | Minimally Closed Class under Progressing Mapping | Statement of Conditions:
{{:Minimally Closed Class under Progressing Mapping/Statement}}
Then the following results hold:
=== Minimally Closed Class under Progressing Mapping induces Nest ===
{{:Minimally Closed Class under Progressing Mapping induces Nest}}
=== Bounded Subset of Minimally Closed Class under Progressin... | Let $\RR$ be the relation on $N$ defined as:
:$\forall x, y \in N: \map \RR {x, y} \iff \map g x \subseteq y \lor y \subseteq x$
We are given that $g$ is a progressing mapping.
From the Progressing Function Lemma, we have that:
{{begin-axiom}}
{{axiom | n = 1
| q = \forall y \in \Dom g
| ml= \map \RR {y... | [[Minimally Closed Class under Progressing Mapping/Statement|Statement of Conditions]]:
{{:Minimally Closed Class under Progressing Mapping/Statement}}
Then the following results hold:
=== [[Minimally Closed Class under Progressing Mapping induces Nest]] ===
{{:Minimally Closed Class under Progressing Mapping induce... | Let $\RR$ be the [[Definition:Relation (Class Theory)|relation]] on $N$ defined as:
:$\forall x, y \in N: \map \RR {x, y} \iff \map g x \subseteq y \lor y \subseteq x$
We are given that $g$ is a [[Definition:Progressing Mapping|progressing mapping]].
From the [[Progressing Function Lemma]], we have that:
{{begin-a... | Minimally Closed Class under Progressing Mapping induces Nest/Proof | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping_induces_Nest/Proof | [
"Minimally Closed Classes",
"Progressing Mappings",
"Minimally Closed Classes under Progressing Mapping"
] | [
"Minimally Closed Class under Progressing Mapping/Statement",
"Minimally Closed Class under Progressing Mapping induces Nest",
"Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element",
"Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element",
"Minim... | [
"Definition:Relation/Class Theory",
"Definition:Progressing Mapping",
"Progressing Function Lemma",
"Double Induction Principle/Minimally Closed Class",
"Definition:Relation/Class Theory",
"Smallest Element of Minimally Closed Class under Progressing Mapping",
"Rule of Addition",
"Double Induction Pri... |
proofwiki-16932 | Minimally Closed Class under Progressing Mapping | Statement of Conditions:
{{:Minimally Closed Class under Progressing Mapping/Statement}}
Then the following results hold:
=== Minimally Closed Class under Progressing Mapping induces Nest ===
{{:Minimally Closed Class under Progressing Mapping induces Nest}}
=== Bounded Subset of Minimally Closed Class under Progressin... | According to hypothesis, let $M$ be a minimally closed under $g$ with respect to $b$.
By Minimally Closed Class under Progressing Mapping induces Nest:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
By Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element:
:if $x$ is a fixed p... | [[Minimally Closed Class under Progressing Mapping/Statement|Statement of Conditions]]:
{{:Minimally Closed Class under Progressing Mapping/Statement}}
Then the following results hold:
=== [[Minimally Closed Class under Progressing Mapping induces Nest]] ===
{{:Minimally Closed Class under Progressing Mapping induce... | According to hypothesis, let $M$ be a [[Definition:Minimally Closed Class|minimally closed]] under $g$ with respect to $b$.
By [[Minimally Closed Class under Progressing Mapping induces Nest]]:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
By [[Fixed Point of Progressing Mapping on Minimally Closed C... | Minimally Closed Class under Progressing Mapping is Well-Ordered/Proof | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping_is_Well-Ordered/Proof | [
"Minimally Closed Classes",
"Progressing Mappings",
"Minimally Closed Classes under Progressing Mapping"
] | [
"Minimally Closed Class under Progressing Mapping/Statement",
"Minimally Closed Class under Progressing Mapping induces Nest",
"Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element",
"Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element",
"Minim... | [
"Definition:Minimally Closed Class",
"Minimally Closed Class under Progressing Mapping induces Nest",
"Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element",
"Definition:Fixed Point",
"Definition:Greatest Set by Set Inclusion/Class Theory",
"Closed Class under Progressing Mappi... |
proofwiki-16933 | Minimally Closed Class under Progressing Mapping | Statement of Conditions:
{{:Minimally Closed Class under Progressing Mapping/Statement}}
Then the following results hold:
=== Minimally Closed Class under Progressing Mapping induces Nest ===
{{:Minimally Closed Class under Progressing Mapping induces Nest}}
=== Bounded Subset of Minimally Closed Class under Progressin... | {{AimForCont}} $b$ is not the smallest element of $N$.
Then there exists $m \in N$ such that $b \nsubseteq m$.
In particular:
:$m \ne b$
Let $B$ be the subclass of $A$ defined as:
:$B = \set {x \in A: \paren {x = b} \lor \paren {\exists y \in B: x = \map g y} }$
This is a subclass of $A$ containing $b$ which is closed ... | [[Minimally Closed Class under Progressing Mapping/Statement|Statement of Conditions]]:
{{:Minimally Closed Class under Progressing Mapping/Statement}}
Then the following results hold:
=== [[Minimally Closed Class under Progressing Mapping induces Nest]] ===
{{:Minimally Closed Class under Progressing Mapping induce... | {{AimForCont}} $b$ is not the [[Definition:Smallest Set by Set Inclusion (Class Theory)|smallest element]] of $N$.
Then there exists $m \in N$ such that $b \nsubseteq m$.
In particular:
:$m \ne b$
Let $B$ be the [[Definition:Subclass|subclass]] of $A$ defined as:
:$B = \set {x \in A: \paren {x = b} \lor \paren {\ex... | Smallest Element of Minimally Closed Class under Progressing Mapping/Proof | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping | https://proofwiki.org/wiki/Smallest_Element_of_Minimally_Closed_Class_under_Progressing_Mapping/Proof | [
"Minimally Closed Classes",
"Progressing Mappings",
"Minimally Closed Classes under Progressing Mapping"
] | [
"Minimally Closed Class under Progressing Mapping/Statement",
"Minimally Closed Class under Progressing Mapping induces Nest",
"Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element",
"Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element",
"Minim... | [
"Definition:Smallest Set by Set Inclusion/Class Theory",
"Definition:Subclass",
"Definition:Subclass",
"Definition:Closed under Mapping/Class Theory",
"Definition:Progressing Mapping",
"Principle of General Induction/Minimally Closed Class",
"Definition:Proper Subclass",
"Definition:Closed under Mappi... |
proofwiki-16934 | Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element | Every bounded subset of $N$ has a greatest element. | Let the hypothesis be assumed.
The proof proceeds by general induction.
For all $x \in N$, let $\map P x$ be the proposition:
:Every subset of $N$ which is bounded by $x$ has a greatest element.
=== Basis for the Induction ===
Let $x = b$.
From Smallest Element of Minimally Closed Class under Progressing Mapping, $b$ i... | Every [[Definition:Bounded Subset of Class|bounded subset]] of $N$ has a [[Definition:Greatest Set by Set Inclusion (Class Theory)|greatest element]]. | Let the hypothesis be assumed.
The proof proceeds by [[Principle of General Induction for Minimally Closed Class|general induction]].
For all $x \in N$, let $\map P x$ be the [[Definition:Proposition|proposition]]:
:Every [[Definition:Subset|subset]] of $N$ which is [[Definition:Set Bounded by Set|bounded]] by $x$ h... | Bounded Subset of Minimally Closed Class under Progressing Mapping has Greatest Element/Proof | https://proofwiki.org/wiki/Bounded_Subset_of_Minimally_Closed_Class_under_Progressing_Mapping_has_Greatest_Element | https://proofwiki.org/wiki/Bounded_Subset_of_Minimally_Closed_Class_under_Progressing_Mapping_has_Greatest_Element/Proof | [
"Minimally Closed Classes under Progressing Mapping"
] | [
"Definition:Bounded Class/Bounded Subset of Class",
"Definition:Greatest Set by Set Inclusion/Class Theory"
] | [
"Principle of General Induction/Minimally Closed Class",
"Definition:Proposition",
"Definition:Subset",
"Definition:Bounded Class/Bounded by Set",
"Definition:Greatest Set by Set Inclusion/Class Theory",
"Smallest Element of Minimally Closed Class under Progressing Mapping",
"Definition:Smallest Set by ... |
proofwiki-16935 | Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element | $g$ has no fixed point, unless possibly the greatest element, if there is one. | Suppose $g$ has a fixed point.
Let $x$ be an element of $M$ such that $\map g x = x$.
We have from Smallest Element of Minimally Closed Class under Progressing Mapping that:
:$b \subseteq x$
Suppose that $y \subseteq x$.
Then by Image of Proper Subset under Progressing Mapping on Minimally Closed Class:
:$\map g y \sub... | $g$ has no [[Definition:Fixed Point|fixed point]], unless possibly the [[Definition:Greatest Set by Set Inclusion (Class Theory)|greatest element]], if there is one. | Suppose $g$ has a [[Definition:Fixed Point|fixed point]].
Let $x$ be an [[Definition:Element of Class|element]] of $M$ such that $\map g x = x$.
We have from [[Smallest Element of Minimally Closed Class under Progressing Mapping]] that:
:$b \subseteq x$
Suppose that $y \subseteq x$.
Then by [[Image of Proper Subse... | Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element/Proof | https://proofwiki.org/wiki/Fixed_Point_of_Progressing_Mapping_on_Minimally_Closed_Class_is_Greatest_Element | https://proofwiki.org/wiki/Fixed_Point_of_Progressing_Mapping_on_Minimally_Closed_Class_is_Greatest_Element/Proof | [
"Minimally Closed Classes under Progressing Mapping"
] | [
"Definition:Fixed Point",
"Definition:Greatest Set by Set Inclusion/Class Theory"
] | [
"Definition:Fixed Point",
"Definition:Element/Class",
"Smallest Element of Minimally Closed Class under Progressing Mapping",
"Image of Proper Subset under Progressing Mapping on Minimally Closed Class",
"Principle of General Induction/Minimally Closed Class"
] |
proofwiki-16936 | Minimally Closed Class under Progressing Mapping is Well-Ordered | $N$ is well-ordered under the subset relation. | According to hypothesis, let $M$ be a minimally closed under $g$ with respect to $b$.
By Minimally Closed Class under Progressing Mapping induces Nest:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
By Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element:
:if $x$ is a fixed p... | $N$ is [[Definition:Well-Ordered Class under Subset Relation|well-ordered]] under the [[Definition:Subset Relation|subset relation]]. | According to hypothesis, let $M$ be a [[Definition:Minimally Closed Class|minimally closed]] under $g$ with respect to $b$.
By [[Minimally Closed Class under Progressing Mapping induces Nest]]:
:$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
By [[Fixed Point of Progressing Mapping on Minimally Closed C... | Minimally Closed Class under Progressing Mapping is Well-Ordered/Proof | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping_is_Well-Ordered | https://proofwiki.org/wiki/Minimally_Closed_Class_under_Progressing_Mapping_is_Well-Ordered/Proof | [
"Minimally Closed Classes under Progressing Mapping"
] | [
"Definition:Well-Ordered Class under Subset Relation",
"Definition:Subset Relation"
] | [
"Definition:Minimally Closed Class",
"Minimally Closed Class under Progressing Mapping induces Nest",
"Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element",
"Definition:Fixed Point",
"Definition:Greatest Set by Set Inclusion/Class Theory",
"Closed Class under Progressing Mappi... |
proofwiki-16937 | Smallest Element of Minimally Closed Class under Progressing Mapping | $b$ is the smallest element of $N$. | {{AimForCont}} $b$ is not the smallest element of $N$.
Then there exists $m \in N$ such that $b \nsubseteq m$.
In particular:
:$m \ne b$
Let $B$ be the subclass of $A$ defined as:
:$B = \set {x \in A: \paren {x = b} \lor \paren {\exists y \in B: x = \map g y} }$
This is a subclass of $A$ containing $b$ which is closed ... | $b$ is the [[Definition:Smallest Set by Set Inclusion (Class Theory)|smallest element]] of $N$. | {{AimForCont}} $b$ is not the [[Definition:Smallest Set by Set Inclusion (Class Theory)|smallest element]] of $N$.
Then there exists $m \in N$ such that $b \nsubseteq m$.
In particular:
:$m \ne b$
Let $B$ be the [[Definition:Subclass|subclass]] of $A$ defined as:
:$B = \set {x \in A: \paren {x = b} \lor \paren {\ex... | Smallest Element of Minimally Closed Class under Progressing Mapping/Proof | https://proofwiki.org/wiki/Smallest_Element_of_Minimally_Closed_Class_under_Progressing_Mapping | https://proofwiki.org/wiki/Smallest_Element_of_Minimally_Closed_Class_under_Progressing_Mapping/Proof | [
"Minimally Closed Classes under Progressing Mapping"
] | [
"Definition:Smallest Set by Set Inclusion/Class Theory"
] | [
"Definition:Smallest Set by Set Inclusion/Class Theory",
"Definition:Subclass",
"Definition:Subclass",
"Definition:Closed under Mapping/Class Theory",
"Definition:Progressing Mapping",
"Principle of General Induction/Minimally Closed Class",
"Definition:Proper Subclass",
"Definition:Closed under Mappi... |
proofwiki-16938 | Equivalence of Definitions of Minimally Closed Class | Let $A$ be a class.
Let $g$ be a mapping on $A$.
{{TFAE|def = Minimally Closed Class|view = minimally closed class under $g$}} | Let it be given that $A$ is closed under $g$. | Let $A$ be a [[Definition:Class (Class Theory)|class]].
Let $g$ be a [[Definition:Class Mapping|mapping]] on $A$.
{{TFAE|def = Minimally Closed Class|view = minimally closed class under $g$}} | Let it be given that $A$ is [[Definition:Closed Class under Mapping|closed under $g$]]. | Equivalence of Definitions of Minimally Closed Class | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Minimally_Closed_Class | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Minimally_Closed_Class | [
"Minimally Closed Classes"
] | [
"Definition:Class (Class Theory)",
"Definition:Mapping/Class Theory"
] | [
"Definition:Closed under Mapping/Class Theory",
"Definition:Closed under Mapping/Class Theory",
"Definition:Closed under Mapping/Class Theory",
"Definition:Closed under Mapping/Class Theory",
"Definition:Closed under Mapping/Class Theory"
] |
proofwiki-16939 | Principle of General Induction/Minimally Closed Class | Let $M$ be a class.
Let $g: M \to M$ be a mapping on $M$.
Let $b \in M$ such that $M$ is minimally closed under $g$ with respect to $b$.
Let $P: M \to \set {\T, \F}$ be a propositional function on $M$.
Suppose that:
:$(1): \quad \map P b = \T$
:$(2): \quad \forall x \in M: \map P x = \T \implies \map P {\map g x} = \T$... | We are given that $M$ is minimally closed under $g$ with respect to $b$.
That is, $M$ is closed under $g$ with the extra property that $M$ has no proper class containing $b$ which is also closed under $g$.
Let $P$ be a propositional function on $M$ which has the properties specified:
:$(1): \quad \map P b = \T$
:$(2): ... | Let $M$ be a [[Definition:Class (Class Theory)|class]].
Let $g: M \to M$ be a [[Definition:Class Mapping|mapping]] on $M$.
Let $b \in M$ such that $M$ is [[Definition:Minimally Closed Class|minimally closed under $g$ with respect to $b$]].
Let $P: M \to \set {\T, \F}$ be a [[Definition:Propositional Function|propos... | We are given that $M$ is [[Definition:Minimally Closed Class|minimally closed under $g$ with respect to $b$]].
That is, $M$ is [[Definition:Closed Class under Mapping|closed under $g$]] with the extra property that $M$ has no [[Definition:Proper Class|proper class]] containing $b$ which is also [[Definition:Closed Cla... | Principle of General Induction/Minimally Closed Class | https://proofwiki.org/wiki/Principle_of_General_Induction/Minimally_Closed_Class | https://proofwiki.org/wiki/Principle_of_General_Induction/Minimally_Closed_Class | [
"Mathematical Induction",
"Proof Techniques",
"Principle of General Induction"
] | [
"Definition:Class (Class Theory)",
"Definition:Mapping/Class Theory",
"Definition:Minimally Closed Class",
"Definition:Propositional Function"
] | [
"Definition:Minimally Closed Class",
"Definition:Closed under Mapping/Class Theory",
"Definition:Class (Class Theory)/Proper Class",
"Definition:Closed under Mapping/Class Theory",
"Definition:Propositional Function",
"Definition:Subclass",
"Definition:Class (Class Theory)",
"Definition:Element/Class"... |
proofwiki-16940 | Double Induction Principle/Minimally Closed Class | Let $M$ be a class which is closed under a progressing mapping $g$.
Let $b$ be an element of $M$ such that $M$ is minimally closed under $g$ with respect to $b$.
Let $\RR$ be a relation on $M$ which satisfies:
{{begin-axiom}}
{{axiom | n = \text D_1
| q = \forall x \in M
| m = \map \RR {x, b}
}}
{{axiom... | The proof proceeds by general induction.
Let an element $x$ of $M$ be defined as:
:'''left normal''' with respect to $\RR$ {{iff}} $\map \RR {x, y}$ for all $y \in M$
:'''right normal''' with respect to $\RR$ {{iff}} $\map \RR {y, x}$ for all $y \in M$.
Let the hypothesis be assumed.
First we demonstrate a lemma: | Let $M$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Closed Class under Mapping|closed]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let $b$ be an [[Definition:Element of Class|element]] of $M$ such that $M$ is [[Definition:Minimally Closed Class|minimally closed under $... | The proof proceeds by [[Principle of General Induction for Minimally Closed Class|general induction]].
Let an [[Definition:Element of Class|element]] $x$ of $M$ be defined as:
:'''[[Definition:Left Normal Element of Relation|left normal]]''' with respect to $\RR$ {{iff}} $\map \RR {x, y}$ for all $y \in M$
:'''[[Defin... | Double Induction Principle/Minimally Closed Class | https://proofwiki.org/wiki/Double_Induction_Principle/Minimally_Closed_Class | https://proofwiki.org/wiki/Double_Induction_Principle/Minimally_Closed_Class | [
"Double Induction Principle"
] | [
"Definition:Class (Class Theory)",
"Definition:Closed under Mapping/Class Theory",
"Definition:Progressing Mapping",
"Definition:Element/Class",
"Definition:Minimally Closed Class",
"Definition:Relation/Class Theory"
] | [
"Principle of General Induction/Minimally Closed Class",
"Definition:Element/Class",
"Definition:Left Normal Element of Relation",
"Definition:Right Normal Element of Relation",
"Definition:Lemma",
"Principle of General Induction/Minimally Closed Class",
"Definition:Right Normal Element of Relation",
... |
proofwiki-16941 | Double Induction Principle/Minimally Closed Class/Lemma | Let $x$ be a right normal element of $M$ with respect to $\RR$.
Then $x$ is also a left normal element of $M$ with respect to $\RR$. | The proof proceeds by general induction.
Let $x \in M$ be right normal with respect to $\RR$
Let $\map P y$ be the proposition:
:$\map \RR {x, y}$ holds. | Let $x$ be a [[Definition:Right Normal Element of Relation|right normal element]] of $M$ with respect to $\RR$.
Then $x$ is also a [[Definition:Left Normal Element of Relation|left normal element]] of $M$ with respect to $\RR$. | The proof proceeds by [[Principle of General Induction for Minimally Closed Class|general induction]].
Let $x \in M$ be [[Definition:Right Normal Element of Relation|right normal]] with respect to $\RR$
Let $\map P y$ be the [[Definition:Proposition|proposition]]:
:$\map \RR {x, y}$ holds. | Double Induction Principle/Minimally Closed Class/Lemma | https://proofwiki.org/wiki/Double_Induction_Principle/Minimally_Closed_Class/Lemma | https://proofwiki.org/wiki/Double_Induction_Principle/Minimally_Closed_Class/Lemma | [
"Double Induction Principle"
] | [
"Definition:Right Normal Element of Relation",
"Definition:Left Normal Element of Relation"
] | [
"Principle of General Induction/Minimally Closed Class",
"Definition:Right Normal Element of Relation",
"Definition:Proposition",
"Definition:Right Normal Element of Relation",
"Principle of General Induction",
"Definition:Right Normal Element of Relation"
] |
proofwiki-16942 | Sandwich Principle for Minimally Closed Class | Let $N$ be a class which is closed under a progressing mapping $g$.
Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$.
Then for all $x, y \in N$:
:$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$ | From Minimally Closed Class under Progressing Mapping induces Nest, we have that $N$ is a nest in which:
:$\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$
Thus the Sandwich Principle applies directly.
{{qed}}
Category:Sandwich Principle
Category:Minimally Closed Classes
Category:Progressing Mappings
lt38fy... | Let $N$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Closed Class under Mapping|closed]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let $b$ be an [[Definition:Element of Class|element]] of $N$ such that $N$ is [[Definition:Minimally Closed Class|minimally closed under $... | From [[Minimally Closed Class under Progressing Mapping induces Nest]], we have that $N$ is a [[Definition:Nest (Class Theory)|nest]] in which:
:$\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$
Thus the [[Sandwich Principle]] applies directly.
{{qed}}
[[Category:Sandwich Principle]]
[[Category:Minimally ... | Sandwich Principle for Minimally Closed Class | https://proofwiki.org/wiki/Sandwich_Principle_for_Minimally_Closed_Class | https://proofwiki.org/wiki/Sandwich_Principle_for_Minimally_Closed_Class | [
"Sandwich Principle",
"Minimally Closed Classes",
"Progressing Mappings"
] | [
"Definition:Class (Class Theory)",
"Definition:Closed under Mapping/Class Theory",
"Definition:Progressing Mapping",
"Definition:Element/Class",
"Definition:Minimally Closed Class"
] | [
"Minimally Closed Class under Progressing Mapping induces Nest",
"Definition:Nest/Class Theory",
"Sandwich Principle",
"Category:Sandwich Principle",
"Category:Minimally Closed Classes",
"Category:Progressing Mappings"
] |
proofwiki-16943 | Image of Proper Subset under Progressing Mapping on Minimally Closed Class | Let $N$ be a class which is closed under a progressing mapping $g$.
Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$.
Then:
:$x \subset y \implies \map g x \subseteq y$ | From Minimally Closed Class under Progressing Mapping induces Nest, we have that $N$ is a nest in which:
:$\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$
Thus the corollary 1 of the Sandwich Principle applies directly.
{{qed}}
Category:Minimally Closed Classes under Progressing Mapping
nj5p8p3qi74wqw5b8x5... | Let $N$ be a [[Definition:Class (Class Theory)|class]] which is [[Definition:Closed Class under Mapping|closed]] under a [[Definition:Progressing Mapping|progressing mapping]] $g$.
Let $b$ be an [[Definition:Element of Class|element]] of $N$ such that $N$ is [[Definition:Minimally Closed Class|minimally closed under $... | From [[Minimally Closed Class under Progressing Mapping induces Nest]], we have that $N$ is a [[Definition:Nest (Class Theory)|nest]] in which:
:$\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$
Thus the [[Sandwich Principle/Corollary 1|corollary 1 of the Sandwich Principle]] applies directly.
{{qed}}
[[C... | Image of Proper Subset under Progressing Mapping on Minimally Closed Class | https://proofwiki.org/wiki/Image_of_Proper_Subset_under_Progressing_Mapping_on_Minimally_Closed_Class | https://proofwiki.org/wiki/Image_of_Proper_Subset_under_Progressing_Mapping_on_Minimally_Closed_Class | [
"Minimally Closed Classes under Progressing Mapping"
] | [
"Definition:Class (Class Theory)",
"Definition:Closed under Mapping/Class Theory",
"Definition:Progressing Mapping",
"Definition:Element/Class",
"Definition:Minimally Closed Class"
] | [
"Minimally Closed Class under Progressing Mapping induces Nest",
"Definition:Nest/Class Theory",
"Sandwich Principle/Corollary 1",
"Category:Minimally Closed Classes under Progressing Mapping"
] |
proofwiki-16944 | Even Impulse Pair is Fourier Transform of Cosine Function | Consider the (real) cosine function $\map \cos x: \R \to \R$.
:$\map f x = \map \cos {\pi x}$
Then:
{{begin-eqn}}
{{eqn | l = \map {\hat f} s
| r = \dfrac 1 2 \map \delta {s - \dfrac 1 2} + \dfrac 1 2 \map \delta {s + \dfrac 1 2}
| c =
}}
{{eqn | r = \map {\operatorname {II} } s
| c =
}}
{{end-eqn}}... | By the definition of a Fourier transform:
{{begin-eqn}}
{{eqn | l = \map {\hat f} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map f x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map \cos {\pi x} \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} \dfrac 1 2 \... | Consider the [[Definition:Real Cosine Function|(real) cosine function]] $\map \cos x: \R \to \R$.
:$\map f x = \map \cos {\pi x}$
Then:
{{begin-eqn}}
{{eqn | l = \map {\hat f} s
| r = \dfrac 1 2 \map \delta {s - \dfrac 1 2} + \dfrac 1 2 \map \delta {s + \dfrac 1 2}
| c =
}}
{{eqn | r = \map {\operatorna... | By the definition of a [[Definition:Fourier Transform of Real Function|Fourier transform]]:
{{begin-eqn}}
{{eqn | l = \map {\hat f} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map f x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map \cos {\pi x} \rd x
| c =
}}
{{eqn | r = ... | Even Impulse Pair is Fourier Transform of Cosine Function | https://proofwiki.org/wiki/Even_Impulse_Pair_is_Fourier_Transform_of_Cosine_Function | https://proofwiki.org/wiki/Even_Impulse_Pair_is_Fourier_Transform_of_Cosine_Function | [
"Cosine Function",
"Even Impulse Pair Function",
"Examples of Fourier Transforms"
] | [
"Definition:Cosine/Real Function",
"Definition:Fourier Transform/Real Function",
"Definition:Even Impulse Pair Function"
] | [
"Definition:Fourier Transform/Real Function",
"Euler's Cosine Identity",
"Linear Combination of Integrals/Definite",
"Fourier Transform of 1"
] |
proofwiki-16945 | Convolution of Real Function with Rectangle Function | Let $f: \R \to \R$ be a bounded piecewise continuous real function.
Consider the rectangle function $\Pi: \R \to \R$.
Then:
:$\forall x \in \R: \map \Pi x * \map f x = \ds \int_{x \mathop - \frac 1 2}^{x \mathop + \frac 1 2} \map f u \rd u$
where $*$ denotes the convolution integral. | {{MissingLinks}}
By definition of convolution integral:
:$\ds \map \Pi x * \map f x = \int_{-\infty}^\infty \map \Pi {x - t} \map f x \rd t$
The above integral exists because $f$ is discontinuous on a countable set (see Lebesgue-Vitali Theorem).
This is equal to:
:$\ds \int_{-\infty}^{x - \frac 1 2} \map \Pi {x - t} \... | Let $f: \R \to \R$ be a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Piecewise Continuous Function/Variant 3|piecewise continuous]] [[Definition:Real Function|real function]].
Consider the [[Definition:Rectangle Function|rectangle function]] $\Pi: \R \to \R$.
Then:
:$\forall x \in \R: \map \Pi x *... | {{MissingLinks}}
By definition of [[Definition:Convolution Integral|convolution integral]]:
:$\ds \map \Pi x * \map f x = \int_{-\infty}^\infty \map \Pi {x - t} \map f x \rd t$
The above integral exists because $f$ is discontinuous on a countable set (see [[Lebesgue-Vitali Theorem]]).
This is equal to:
:$\ds \int_{... | Convolution of Real Function with Rectangle Function | https://proofwiki.org/wiki/Convolution_of_Real_Function_with_Rectangle_Function | https://proofwiki.org/wiki/Convolution_of_Real_Function_with_Rectangle_Function | [
"Rectangle Function",
"Convolution Integrals"
] | [
"Definition:Bounded Mapping/Real-Valued",
"Definition:Piecewise Continuous Function/Variant 3",
"Definition:Real Function",
"Definition:Rectangle Function",
"Definition:Convolution Integral"
] | [
"Definition:Convolution Integral",
"Lebesgue-Vitali Theorem",
"Definition:Rectangle Function"
] |
proofwiki-16946 | Repeated Fourier Transform of Even Function | Let $f: \R \to \R$ be an even real function which is Lebesgue integrable.
Let $\ds \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the Fourier transform of $f$.
Let $\ds \map \FF {\map F s} = \map g t = \int_{-\infty}^\infty e^{-2 \pi i t s} \map F s \rd s$ be the Fourier tran... | {{begin-eqn}}
{{eqn | l = \map g t
| r = \map f {-t}
| c = Repeated Fourier Transform of Real Function
}}
{{eqn | r = \map f t
| c = {{Defof|Even Function}}
}}
{{end-eqn}}
{{qed}} | Let $f: \R \to \R$ be an [[Definition:Even Function|even]] [[Definition:Real Function|real function]] which is [[Definition:Lebesgue Integrable Function|Lebesgue integrable]].
Let $\ds \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the [[Definition:Fourier Transform of Real ... | {{begin-eqn}}
{{eqn | l = \map g t
| r = \map f {-t}
| c = [[Repeated Fourier Transform of Real Function]]
}}
{{eqn | r = \map f t
| c = {{Defof|Even Function}}
}}
{{end-eqn}}
{{qed}} | Repeated Fourier Transform of Even Function | https://proofwiki.org/wiki/Repeated_Fourier_Transform_of_Even_Function | https://proofwiki.org/wiki/Repeated_Fourier_Transform_of_Even_Function | [
"Fourier Transforms"
] | [
"Definition:Even Function",
"Definition:Real Function",
"Definition:Integrable Function/Lebesgue",
"Definition:Fourier Transform/Real Function",
"Definition:Fourier Transform/Real Function"
] | [
"Repeated Fourier Transform of Real Function"
] |
proofwiki-16947 | Repeated Fourier Transform of Odd Function | Let $f: \R \to \R$ be an odd real function which is Lebesgue integrable.
Let $\ds \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the Fourier transform of $f$.
Let $\ds \map \FF {\map F s} = \map g t = \int_{-\infty}^\infty e^{-2 \pi i t s} \map F s \rd s$ be the Fourier trans... | {{begin-eqn}}
{{eqn | l = \map g t
| r = \map f {-t}
| c = Repeated Fourier Transform of Real Function
}}
{{eqn | r = -\map f t
| c = {{Defof|Odd Function}}
}}
{{end-eqn}}
{{qed}} | Let $f: \R \to \R$ be an [[Definition:Odd Function|odd]] [[Definition:Real Function|real function]] which is [[Definition:Lebesgue Integrable Function|Lebesgue integrable]].
Let $\ds \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the [[Definition:Fourier Transform of Real Fu... | {{begin-eqn}}
{{eqn | l = \map g t
| r = \map f {-t}
| c = [[Repeated Fourier Transform of Real Function]]
}}
{{eqn | r = -\map f t
| c = {{Defof|Odd Function}}
}}
{{end-eqn}}
{{qed}} | Repeated Fourier Transform of Odd Function | https://proofwiki.org/wiki/Repeated_Fourier_Transform_of_Odd_Function | https://proofwiki.org/wiki/Repeated_Fourier_Transform_of_Odd_Function | [
"Fourier Transforms"
] | [
"Definition:Odd Function",
"Definition:Real Function",
"Definition:Integrable Function/Lebesgue",
"Definition:Fourier Transform/Real Function",
"Definition:Fourier Transform/Real Function"
] | [
"Repeated Fourier Transform of Real Function"
] |
proofwiki-16948 | Fourier's Theorem/Integral Form | Let $f: \R \to \R$ be a real function which satisfies the Dirichlet conditions on $\R$.
Then:
:$\dfrac {\map f {t^+} + \map f {t^-} } 2 = \ds \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i t s} \map f t \rd t} \rd s$
where:
:$\map f {t^+}$ and $\map f {t^-}$ denote the limit from above ... | {{ProofWanted}}
{{Namedfor|Joseph Fourier|cat = Fourier}} | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] which satisfies the [[Definition:Dirichlet Conditions|Dirichlet conditions]] on $\R$.
Then:
:$\dfrac {\map f {t^+} + \map f {t^-} } 2 = \ds \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i t s} \map f t \rd t} \rd s$
whe... | {{ProofWanted}}
{{Namedfor|Joseph Fourier|cat = Fourier}} | Fourier's Theorem/Integral Form | https://proofwiki.org/wiki/Fourier's_Theorem/Integral_Form | https://proofwiki.org/wiki/Fourier's_Theorem/Integral_Form | [
"Fourier Transforms"
] | [
"Definition:Real Function",
"Definition:Dirichlet Conditions",
"Definition:Limit of Real Function/Right",
"Definition:Limit of Real Function/Left"
] | [] |
proofwiki-16949 | Fourier's Theorem/Integral Form/Continuous Point | Let $f$ be continuous at $t \in \R$.
Then:
:$\ds \map f t = \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i t s} \map f t \rd t} \rd s$ | At a point of continuity we have:
{{begin-eqn}}
{{eqn | l = \dfrac {\map f {t^+} + \map f {t^-} } 2
| r = \dfrac {\map f t + \map f t} 2
| c = as $\map f t = \map f {t^+} = \map f {t^-}$ at a point of continuity
}}
{{eqn | r = \dfrac {2 \map f t} 2
| c =
}}
{{eqn | r = \map f t
| c =
}}
{{end-... | Let $f$ be [[Definition:Continuous Real Function at Point|continuous]] at $t \in \R$.
Then:
:$\ds \map f t = \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i t s} \map f t \rd t} \rd s$ | At a [[Definition:Continuous Real Function at Point|point of continuity]] we have:
{{begin-eqn}}
{{eqn | l = \dfrac {\map f {t^+} + \map f {t^-} } 2
| r = \dfrac {\map f t + \map f t} 2
| c = as $\map f t = \map f {t^+} = \map f {t^-}$ at a [[Definition:Continuous Real Function at Point|point of continuity... | Fourier's Theorem/Integral Form/Continuous Point | https://proofwiki.org/wiki/Fourier's_Theorem/Integral_Form/Continuous_Point | https://proofwiki.org/wiki/Fourier's_Theorem/Integral_Form/Continuous_Point | [
"Fourier Transforms"
] | [
"Definition:Continuous Real Function/Point"
] | [
"Definition:Continuous Real Function/Point",
"Definition:Continuous Real Function/Point",
"Fourier's Theorem/Integral Form"
] |
proofwiki-16950 | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Lemma 1 | :$\map {p_i^\to} {\map {\pr_k^\gets} {V_k} \cap Y_i}$ is open in $\struct{X_i, \tau_i}$ | We have that $p_i$ is a bijection from the lemmas:
:$p_i$ is an injection
:$p_i$ is a surjection
Let $x \in X_i$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {p_i^\to} {\map {\pr_k^\gets} {V_k} \cap Y_i}
}}
{{eqn | ll= \leadstoandfrom
| l = \map {p_i^{-1} } x
| o = \in
| r = \... | :$\map {p_i^\to} {\map {\pr_k^\gets} {V_k} \cap Y_i}$ is [[Definition:Open Set (Topology)|open]] in $\struct{X_i, \tau_i}$ | We have that $p_i$ is a [[Definition:Bijection|bijection]] from the [[Definition:Lemma|lemmas]]:
:[[Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Injection|$p_i$ is an injection]]
:[[Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Surjection|$p_i$ is a surjection]]
Let $x \in X... | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Lemma 1 | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Lemma_1 | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Lemma_1 | [
"Subspace of Product Space is Homeomorphic to Factor Space"
] | [
"Definition:Open Set/Topology"
] | [
"Definition:Bijection",
"Definition:Lemma",
"Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Injection",
"Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Surjection"
] |
proofwiki-16951 | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Injection | Let $\family {X_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set.
Let $\ds X = \prod_{i \mathop \in I} X_i$ be the Cartesian product of $\family {X_i}_{i \mathop \in I}$.
Let $z \in X$.
Let $i \in I$.
Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$.
Let $p_i = \pr_i... | Let $x, y \in Y_i$.
Then for all $j \in I \setminus \set i$:
:$x_j = z_j = y_j$
Let $\map {p_i} x = \map {p_i} y$.
Then:
:$x_i = y_i$
Thus:
:$x = y$
It follows that $p_i$ is an injection by definition. | Let $\family {X_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Set|sets]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds X = \prod_{i \mathop \in I} X_i$ be the [[Definition:Cartesian Product of Family|Cartesian product]] of $\family {X_i}_{i \mathop \in I}$.... | Let $x, y \in Y_i$.
Then for all $j \in I \setminus \set i$:
:$x_j = z_j = y_j$
Let $\map {p_i} x = \map {p_i} y$.
Then:
:$x_i = y_i$
Thus:
:$x = y$
It follows that $p_i$ is an [[Definition:Injection|injection]] by definition. | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Injection | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Injection | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Injection | [
"Subspace of Product Space is Homeomorphic to Factor Space"
] | [
"Definition:Indexing Set/Family",
"Definition:Set",
"Definition:Indexing Set",
"Definition:Cartesian Product/Family of Sets",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Injection"
] | [
"Definition:Injection"
] |
proofwiki-16952 | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Surjection | Let $\family {X_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set.
Let $\ds X = \prod_{i \mathop \in I} X_i$ be the Cartesian product of $\family {X_i}_{i \mathop \in I}$.
Let $z \in X$.
Let $i \in I$.
Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$.
Let $p_i = \pr_i... | Let $x \in X_i$.
Let $y \in Y_i$ be defined by:
:<nowiki>$\forall j \in I: y_j = \begin{cases}
z_j & j \ne i \\
x & j = i
\end{cases}$</nowiki>
Then:
:$\map {p_i} y = y_i = x$
It follows that $p_i$ is a surjection by definition. | Let $\family {X_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Set|sets]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds X = \prod_{i \mathop \in I} X_i$ be the [[Definition:Cartesian Product of Family|Cartesian product]] of $\family {X_i}_{i \mathop \in I}$.... | Let $x \in X_i$.
Let $y \in Y_i$ be defined by:
:<nowiki>$\forall j \in I: y_j = \begin{cases}
z_j & j \ne i \\
x & j = i
\end{cases}$</nowiki>
Then:
:$\map {p_i} y = y_i = x$
It follows that $p_i$ is a [[Definition:Surjection|surjection]] by definition. | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Surjection | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Surjection | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Surjection | [
"Subspace of Product Space is Homeomorphic to Factor Space"
] | [
"Definition:Indexing Set/Family",
"Definition:Set",
"Definition:Indexing Set",
"Definition:Cartesian Product/Family of Sets",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Surjection"
] | [
"Definition:Surjection"
] |
proofwiki-16953 | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Continuous Mapping | Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.
Let $\ds \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$.
Let $z \in X$.
Let $i \in I$.
Let $Y_... | Let $V \in \tau_i$.
Let $\ds U = \prod_{i \mathop \in I} U_i$ where:
:$U_j = \begin{cases} X_j & j \ne i \\ V & j = i \end{cases}$
From Natural Basis of Product Topology, $U$ is an element of the the natural basis.
By definition of the product topology $\tau$ on the product space $\struct {X, \tau}$ the natural basis i... | Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the [[Definition:Prod... | Let $V \in \tau_i$.
Let $\ds U = \prod_{i \mathop \in I} U_i$ where:
:$U_j = \begin{cases} X_j & j \ne i \\ V & j = i \end{cases}$
From [[Natural Basis of Product Topology]], $U$ is an [[Definition:Element|element]] of the the [[Definition:Natural Basis of Product Topology|natural basis]].
By definition of the [[De... | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Continuous Mapping | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Continuous_Mapping | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Continuous_Mapping | [
"Subspace of Product Space is Homeomorphic to Factor Space"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:Topological Subspace",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Continuous Mapping"
] | [
"Natural Basis of Product Topology",
"Definition:Element",
"Definition:Product Topology/Natural Basis",
"Definition:Product Topology",
"Definition:Product Space (Topology)",
"Definition:Product Topology/Natural Basis",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Product Topology",
"De... |
proofwiki-16954 | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Open Mapping | Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.
Let $\ds \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$.
Let $z \in X$.
Let $i \in I$.
Let $Y_... | Let $U \in \upsilon_i$.
Let $x \in \map {p_i^\to} U$.
Then by definition of the direct image mapping:
:$\exists y \in U : x = \map {p_i} y$
By the definition of the subspace topology:
:$\exists U' \in \tau: U = U' \cap Y_i$
For all $k \in I$ let $\pr_k$ denote the projection from $X$ to $X_k$.
By definition of the natu... | Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the [[Definition:Prod... | Let $U \in \upsilon_i$.
Let $x \in \map {p_i^\to} U$.
Then by definition of the [[Definition:Direct Image Mapping of Mapping|direct image mapping]]:
:$\exists y \in U : x = \map {p_i} y$
By the definition of the [[Definition:Subspace Topology|subspace topology]]:
:$\exists U' \in \tau: U = U' \cap Y_i$
For all $k ... | Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Open Mapping | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Open_Mapping | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Proof_2/Open_Mapping | [
"Subspace of Product Space is Homeomorphic to Factor Space"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:Topological Subspace",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Open Mapping"
] | [
"Definition:Direct Image Mapping/Mapping",
"Definition:Topological Subspace",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Product Topology/Natural Basis",
"Definition:Product Topology",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Direct Image Mapping/Mapping",
... |
proofwiki-16955 | Exponential Distribution in terms of Continuous Uniform Distribution | Let $X \sim \mathrm U \hointl 0 1$ where $\mathrm U \hointl 0 1$ is the continuous uniform distribution on $\hointl 0 1$.
Let $\beta$ be a positive real number.
Then:
:$-\beta \ln X \sim \Exponential \lambda$
where $\Exponential \cdot$ is the exponential distribution. | Let $Y \sim \Exponential \lambda$.
We aim to show that:
:$\map \Pr {Y < -\beta \ln x} = \map \Pr {X > x}$
for all $x \in \hointl 0 1$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y < -\beta \ln x}
| r = \frac 1 \beta \int_0^{-\beta \ln x} \map \exp {-\frac u \beta} \rd u
| c = {{Defof|Exponential Distribution}}
}}
{... | Let $X \sim \mathrm U \hointl 0 1$ where $\mathrm U \hointl 0 1$ is the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]] on $\hointl 0 1$.
Let $\beta$ be a [[Definition:Positive Real Number|positive real number]].
Then:
:$-\beta \ln X \sim \Exponential \lambda$
where $\Exponential \c... | Let $Y \sim \Exponential \lambda$.
We aim to show that:
:$\map \Pr {Y < -\beta \ln x} = \map \Pr {X > x}$
for all $x \in \hointl 0 1$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y < -\beta \ln x}
| r = \frac 1 \beta \int_0^{-\beta \ln x} \map \exp {-\frac u \beta} \rd u
| c = {{Defof|Exponential Distribution}}... | Exponential Distribution in terms of Continuous Uniform Distribution | https://proofwiki.org/wiki/Exponential_Distribution_in_terms_of_Continuous_Uniform_Distribution | https://proofwiki.org/wiki/Exponential_Distribution_in_terms_of_Continuous_Uniform_Distribution | [
"Exponential Distribution",
"Continuous Uniform Distribution"
] | [
"Definition:Uniform Distribution/Continuous",
"Definition:Positive/Real Number",
"Definition:Exponential Distribution"
] | [
"Primitive of Exponential of a x",
"Exponential of Zero",
"Primitive of Constant",
"Category:Exponential Distribution",
"Category:Continuous Uniform Distribution"
] |
proofwiki-16956 | Power of Random Variable with Continuous Uniform Distribution has Beta Distribution | Let $X \sim \ContinuousUniform 0 1$ where $\ContinuousUniform 0 1$ is the continuous uniform distribution on $\closedint 0 1$.
Let $n$ be a positive real number.
Then:
:$X^n \sim \BetaDist {\dfrac 1 n} 1$
where $\operatorname {Beta}$ is the beta distribution. | Let:
:$Y \sim \BetaDist {\dfrac 1 n} 1$
We aim to show that:
:$\map \Pr {Y < x^n} = \map \Pr {X < x}$
for all $x \in \closedint 0 1$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y < x^n}
| r = \int_0^{x^n} \frac 1 {\map \Beta {\frac 1 n, 1} } u^{\frac 1 n - 1} \paren {1 - u}^{1 - 1} \rd u
| c = {{Defof|Beta Distri... | Let $X \sim \ContinuousUniform 0 1$ where $\ContinuousUniform 0 1$ is the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]] on $\closedint 0 1$.
Let $n$ be a [[Definition:Positive Real Number|positive real number]].
Then:
:$X^n \sim \BetaDist {\dfrac 1 n} 1$
where $\operatorname {Beta... | Let:
:$Y \sim \BetaDist {\dfrac 1 n} 1$
We aim to show that:
:$\map \Pr {Y < x^n} = \map \Pr {X < x}$
for all $x \in \closedint 0 1$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Pr {Y < x^n}
| r = \int_0^{x^n} \frac 1 {\map \Beta {\frac 1 n, 1} } u^{\frac 1 n - 1} \paren {1 - u}^{1 - 1} \rd u
| c = {{Defof|Beta ... | Power of Random Variable with Continuous Uniform Distribution has Beta Distribution | https://proofwiki.org/wiki/Power_of_Random_Variable_with_Continuous_Uniform_Distribution_has_Beta_Distribution | https://proofwiki.org/wiki/Power_of_Random_Variable_with_Continuous_Uniform_Distribution_has_Beta_Distribution | [
"Continuous Uniform Distribution",
"Beta Distribution"
] | [
"Definition:Uniform Distribution/Continuous",
"Definition:Positive/Real Number",
"Definition:Beta Distribution"
] | [
"Primitive of Power",
"Gamma Difference Equation",
"Primitive of Constant",
"Category:Continuous Uniform Distribution",
"Category:Beta Distribution"
] |
proofwiki-16957 | Expectation of Non-Negative Random Variable is Non-Negative | Let $X$ be a random variable.
Let $\map \Pr {X \ge 0} = 1$.
Then:
:$\expect X \ge 0$
where $\expect X$ denotes the expectation of $X$. | === Discrete Random Variable ===
{{:Expectation of Non-Negative Random Variable is Non-Negative/Discrete}} | Let $X$ be a [[Definition:Random Variable|random variable]].
Let $\map \Pr {X \ge 0} = 1$.
Then:
:$\expect X \ge 0$
where $\expect X$ denotes the [[Definition:Expectation|expectation]] of $X$. | === [[Expectation of Non-Negative Random Variable is Non-Negative/Discrete|Discrete Random Variable]] ===
{{:Expectation of Non-Negative Random Variable is Non-Negative/Discrete}} | Expectation of Non-Negative Random Variable is Non-Negative | https://proofwiki.org/wiki/Expectation_of_Non-Negative_Random_Variable_is_Non-Negative | https://proofwiki.org/wiki/Expectation_of_Non-Negative_Random_Variable_is_Non-Negative | [
"Expectation",
"Expectation of Non-Negative Random Variable is Non-Negative"
] | [
"Definition:Random Variable",
"Definition:Expectation"
] | [
"Expectation of Non-Negative Random Variable is Non-Negative/Discrete"
] |
proofwiki-16958 | Expectation of Non-Negative Random Variable is Non-Negative/Discrete | Let $X$ be a discrete random variable.
Let $\map \Pr {X \ge 0} = 1$.
Then:
:$\expect X \ge 0$
where $\expect X$ denotes the expectation of $X$. | Let $\map \supp X$ be the support of $X$.
Note that since $X$ is discrete, its sample space and hence support is countable.
Therefore, there exists some sequence $\sequence {x_i}_{i \mathop \in I}$ such that:
:$\map \supp X = \set {x_i \mid i \in I}$
for some $I \subseteq \N$.
By the definition of a sample space, w... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]].
Let $\map \Pr {X \ge 0} = 1$.
Then:
:$\expect X \ge 0$
where $\expect X$ denotes the [[Definition:Expectation of Discrete Random Variable|expectation]] of $X$. | Let $\map \supp X$ be the [[Definition:Support of Random Variable/Discrete|support]] of $X$.
Note that since $X$ is [[Definition:Discrete Random Variable|discrete]], its [[Definition:Sample Space|sample space]] and hence support is [[Definition:Countable Set|countable]].
Therefore, there exists some [[Definition:Se... | Expectation of Non-Negative Random Variable is Non-Negative/Discrete | https://proofwiki.org/wiki/Expectation_of_Non-Negative_Random_Variable_is_Non-Negative/Discrete | https://proofwiki.org/wiki/Expectation_of_Non-Negative_Random_Variable_is_Non-Negative/Discrete | [
"Expectation of Non-Negative Random Variable is Non-Negative"
] | [
"Definition:Random Variable/Discrete",
"Definition:Expectation/Discrete"
] | [
"Definition:Support of Random Variable/Discrete",
"Definition:Random Variable/Discrete",
"Definition:Sample Space",
"Definition:Countable Set",
"Definition:Sequence",
"Definition:Sample Space",
"Definition:Element",
"Definition:Positive/Real Number",
"Definition:Expectation/Discrete"
] |
proofwiki-16959 | Expectation of Almost Surely Constant Random Variable | Let $X$ be an almost surely constant random variable.
That is, there exists some $c \in \R$ such that:
:$\map \Pr {X = c} = 1$
Then:
:$\expect X = c$ | Note that since $\map \Pr {X = c} = 1$, we have $\map \Pr {X \ne c} = 0$ from Probability of Event not Occurring.
Therefore:
:$\map {\mathrm {supp} } X = \set c$
{{MissingLinks|supp}}
We therefore have:
{{begin-eqn}}
{{eqn | l = \expect X
| r = \sum_{x \mathop \in \map {\mathrm {supp} } X} x \map \Pr {X = x}
| c = {... | Let $X$ be an [[Definition:Almost Surely Constant Random Variable|almost surely constant random variable]].
That is, there exists some $c \in \R$ such that:
:$\map \Pr {X = c} = 1$
Then:
:$\expect X = c$ | Note that since $\map \Pr {X = c} = 1$, we have $\map \Pr {X \ne c} = 0$ from [[Probability of Event not Occurring]].
Therefore:
:$\map {\mathrm {supp} } X = \set c$
{{MissingLinks|supp}}
We therefore have:
{{begin-eqn}}
{{eqn | l = \expect X
| r = \sum_{x \mathop \in \map {\mathrm {supp} } X} x \map \Pr {X = x}... | Expectation of Almost Surely Constant Random Variable | https://proofwiki.org/wiki/Expectation_of_Almost_Surely_Constant_Random_Variable | https://proofwiki.org/wiki/Expectation_of_Almost_Surely_Constant_Random_Variable | [
"Expectation"
] | [
"Definition:Almost Surely Constant Random Variable"
] | [
"Probability of Event not Occurring",
"Category:Expectation"
] |
proofwiki-16960 | Expectation Preserves Inequality | Let $X$, $Y$ be random variables.
Let $\map \Pr {X \ge Y} = 1$.
Then:
:$\expect X \ge \expect Y$ | Note that we have:
:$\map \Pr {X - Y \ge 0} = 1$
From Expectation of Non-Negative Random Variable is Non-Negative, we then have:
:$\expect {X - Y} \ge 0$
From Sum of Expectations of Independent Trials, we have:
:$\expect X + \expect {-Y} \ge 0$
From Expectation of Linear Transformation of Random Variable, we have:
:... | Let $X$, $Y$ be [[Definition:Random Variable|random variables]].
Let $\map \Pr {X \ge Y} = 1$.
Then:
:$\expect X \ge \expect Y$ | Note that we have:
:$\map \Pr {X - Y \ge 0} = 1$
From [[Expectation of Non-Negative Random Variable is Non-Negative]], we then have:
:$\expect {X - Y} \ge 0$
From [[Sum of Expectations of Independent Trials]], we have:
:$\expect X + \expect {-Y} \ge 0$
From [[Expectation of Linear Transformation of Random Vari... | Expectation Preserves Inequality | https://proofwiki.org/wiki/Expectation_Preserves_Inequality | https://proofwiki.org/wiki/Expectation_Preserves_Inequality | [
"Expectation"
] | [
"Definition:Random Variable"
] | [
"Expectation of Non-Negative Random Variable is Non-Negative",
"Sum of Expectations of Independent Trials",
"Expectation of Linear Transformation of Random Variable",
"Category:Expectation"
] |
proofwiki-16961 | Random Variable has Zero Variance iff Almost Surely Constant | Let $X$ be a random variable such that $\expect {X^2}$ exists.
Then $\var X = 0$ {{iff}} there exists $c \in \R$ with $\map \Pr {X = c} = 1$.
That is, $X$ is almost surely constant. | === Sufficient Condition ===
Suppose that there exists some $c \in \R$ with $\map \Pr {X = c} = 1$.
From Expectation of Almost Surely Constant Random Variable:
:$\expect X = c$
Let $\map \supp X$ be the support of $X$.
Since $\map \Pr {X = c} = 1$, we have:
:$\map \supp X = \set c$
We therefore have:
{{begin-eqn}}
... | Let $X$ be a [[Definition:Random Variable|random variable]] such that $\expect {X^2}$ exists.
Then $\var X = 0$ {{iff}} there exists $c \in \R$ with $\map \Pr {X = c} = 1$.
That is, $X$ is [[Definition:Almost Surely Constant Random Variable|almost surely constant]]. | === Sufficient Condition ===
Suppose that there exists some $c \in \R$ with $\map \Pr {X = c} = 1$.
From [[Expectation of Almost Surely Constant Random Variable]]:
:$\expect X = c$
Let $\map \supp X$ be the [[Definition:Support of Random Variable|support]] of $X$.
Since $\map \Pr {X = c} = 1$, we have:
:$\map ... | Random Variable has Zero Variance iff Almost Surely Constant | https://proofwiki.org/wiki/Random_Variable_has_Zero_Variance_iff_Almost_Surely_Constant | https://proofwiki.org/wiki/Random_Variable_has_Zero_Variance_iff_Almost_Surely_Constant | [
"Variance"
] | [
"Definition:Random Variable",
"Definition:Almost Surely Constant Random Variable"
] | [
"Expectation of Almost Surely Constant Random Variable",
"Definition:Support of Random Variable",
"Variance as Expectation of Square minus Square of Expectation"
] |
proofwiki-16962 | Covariance of Random Variable with Itself | Let $X$ be a random variable.
Then $\cov {X, X} = \var X$. | We have:
{{begin-eqn}}
{{eqn | l = \cov {X, X}
| r = \expect {\paren {X - \expect X} \paren {X - \expect X} }
| c = {{Defof|Covariance}}
}}
{{eqn | r = \expect {\paren {X - \expect X}^2}
}}
{{eqn | r = \var X
| c = {{Defof|Variance}}
}}
{{end-eqn}}
{{qed}}
Category:Covariance
Category:Variance
ir6l7arixypl4c30xq378... | Let $X$ be a [[Definition:Random Variable|random variable]].
Then $\cov {X, X} = \var X$. | We have:
{{begin-eqn}}
{{eqn | l = \cov {X, X}
| r = \expect {\paren {X - \expect X} \paren {X - \expect X} }
| c = {{Defof|Covariance}}
}}
{{eqn | r = \expect {\paren {X - \expect X}^2}
}}
{{eqn | r = \var X
| c = {{Defof|Variance}}
}}
{{end-eqn}}
{{qed}}
[[Category:Covariance]]
[[Category:Variance]]
ir6l7arixyp... | Covariance of Random Variable with Itself | https://proofwiki.org/wiki/Covariance_of_Random_Variable_with_Itself | https://proofwiki.org/wiki/Covariance_of_Random_Variable_with_Itself | [
"Covariance",
"Variance"
] | [
"Definition:Random Variable"
] | [
"Category:Covariance",
"Category:Variance"
] |
proofwiki-16963 | Covariance is Symmetric | Let $X$ and $Y$ be random variables.
Suppose the covariance $\cov {X, Y}$ exists.
Then $\cov {X, Y} = \cov {Y, X}$. | {{begin-eqn}}
{{eqn | l = \cov {X, Y}
| r = \expect {\paren {X - \expect X} \paren {Y - \expect Y} }
| c = {{Defof|Covariance}}
}}
{{eqn | r = \expect {\paren {Y - \expect Y} \paren {X - \expect X} }
| c = Real Multiplication is Commutative
}}
{{eqn | r = \cov {Y, X}
| c = {{Defof|Covariance}}
}}
{{end-eqn}}
{{qed}... | Let $X$ and $Y$ be [[Definition:Random Variable|random variables]].
Suppose the [[Definition:Covariance|covariance]] $\cov {X, Y}$ exists.
Then $\cov {X, Y} = \cov {Y, X}$. | {{begin-eqn}}
{{eqn | l = \cov {X, Y}
| r = \expect {\paren {X - \expect X} \paren {Y - \expect Y} }
| c = {{Defof|Covariance}}
}}
{{eqn | r = \expect {\paren {Y - \expect Y} \paren {X - \expect X} }
| c = [[Real Multiplication is Commutative]]
}}
{{eqn | r = \cov {Y, X}
| c = {{Defof|Covariance}}
}}
{{end-eqn}}
{{... | Covariance is Symmetric | https://proofwiki.org/wiki/Covariance_is_Symmetric | https://proofwiki.org/wiki/Covariance_is_Symmetric | [
"Covariance"
] | [
"Definition:Random Variable",
"Definition:Covariance"
] | [
"Real Multiplication is Commutative",
"Category:Covariance"
] |
proofwiki-16964 | Covariance of Linear Combination of Random Variables with Another | Let $X, Y, Z$ be random variables.
Let $a, b$ be real numbers.
Then:
:$\cov {a X + b Y, Z} = a \cov {X, Z} + b \cov {Y, Z}$ | {{begin-eqn}}
{{eqn | l = \cov {a X + b Y, Z}
| r = \expect {\paren {a X + b Y} Z} - \expect {a X + b Y} \expect Z
| c = Covariance as Expectation of Product minus Product of Expectations
}}
{{eqn | r = a \expect {X Z} + b \expect {Y Z} - \paren {a \expect X + b \expect Y} \expect Z
| c = Expectation ... | Let $X, Y, Z$ be [[Definition:Random Variable|random variables]].
Let $a, b$ be [[Definition:Real Number|real numbers]].
Then:
:$\cov {a X + b Y, Z} = a \cov {X, Z} + b \cov {Y, Z}$ | {{begin-eqn}}
{{eqn | l = \cov {a X + b Y, Z}
| r = \expect {\paren {a X + b Y} Z} - \expect {a X + b Y} \expect Z
| c = [[Covariance as Expectation of Product minus Product of Expectations]]
}}
{{eqn | r = a \expect {X Z} + b \expect {Y Z} - \paren {a \expect X + b \expect Y} \expect Z
| c = [[Expect... | Covariance of Linear Combination of Random Variables with Another | https://proofwiki.org/wiki/Covariance_of_Linear_Combination_of_Random_Variables_with_Another | https://proofwiki.org/wiki/Covariance_of_Linear_Combination_of_Random_Variables_with_Another | [
"Covariance"
] | [
"Definition:Random Variable",
"Definition:Real Number"
] | [
"Covariance as Expectation of Product minus Product of Expectations",
"Expectation is Linear",
"Covariance as Expectation of Product minus Product of Expectations",
"Category:Covariance"
] |
proofwiki-16965 | Covariance of Sums of Random Variables | Let $n$ be a strictly positive integer.
Let $\sequence {X_i}_{1 \mathop \le i \mathop \le n}$, $\sequence {Y_j}_{1 \mathop \le j \mathop \le n}$ be sequences of random variables.
Then:
:$\ds \cov {\sum_{i \mathop = 1}^n X_i, \sum_{j \mathop = 1}^n Y_j} = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \cov {X_i, Y_j}$ | {{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \cov {X_i, Y_j}
| r = \sum_{i \mathop = 1}^n \paren {\sum_{j \mathop = 1}^n \cov {Y_j, X_i} }
| c = Covariance is Symmetric
}}
{{eqn | r = \sum_{i \mathop = 1}^n \cov {\sum_{j \mathop = 1}^n Y_j, X_i}
| c = Covariance of Sums of Random Variables... | Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\sequence {X_i}_{1 \mathop \le i \mathop \le n}$, $\sequence {Y_j}_{1 \mathop \le j \mathop \le n}$ be [[Definition:Sequence|sequences]] of [[Definition:Random Variable|random variables]].
Then:
:$\ds \cov {\sum_{i \mathop = 1}^n... | {{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \cov {X_i, Y_j}
| r = \sum_{i \mathop = 1}^n \paren {\sum_{j \mathop = 1}^n \cov {Y_j, X_i} }
| c = [[Covariance is Symmetric]]
}}
{{eqn | r = \sum_{i \mathop = 1}^n \cov {\sum_{j \mathop = 1}^n Y_j, X_i}
| c = [[Covariance of Sums of Random Var... | Covariance of Sums of Random Variables | https://proofwiki.org/wiki/Covariance_of_Sums_of_Random_Variables | https://proofwiki.org/wiki/Covariance_of_Sums_of_Random_Variables | [
"Covariance",
"Covariance of Sums of Random Variables"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Sequence",
"Definition:Random Variable"
] | [
"Covariance is Symmetric",
"Covariance of Sums of Random Variables/Lemma",
"Covariance is Symmetric",
"Covariance of Sums of Random Variables/Lemma",
"Category:Covariance",
"Category:Covariance of Sums of Random Variables"
] |
proofwiki-16966 | Covariance of Sums of Random Variables/Lemma | Let $n$ be a strictly positive integer.
Let $\sequence {X_i}_{1 \mathop \le i \mathop \le n}$ be a sequence of random variables.
Let $Y$ be a random variable.
Then:
:$\ds \cov {\sum_{i \mathop = 1}^n X_i, Y} = \sum_{i \mathop = 1}^n \cov {X_i, Y}$ | Proof by induction:
For all $n \in \N$, let $\map P n$ be the proposition:
:$\ds \cov {\sum_{i \mathop = 1}^n X_i, Y} = \sum_{i \mathop = 1}^n \cov {X_i, Y}$ | Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\sequence {X_i}_{1 \mathop \le i \mathop \le n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Random Variable|random variables]].
Let $Y$ be a [[Definition:Random Variable|random variable]].
Then:
:$\ds \cov {\sum_{i \m... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N$, let $\map P n$ be the proposition:
:$\ds \cov {\sum_{i \mathop = 1}^n X_i, Y} = \sum_{i \mathop = 1}^n \cov {X_i, Y}$ | Covariance of Sums of Random Variables/Lemma | https://proofwiki.org/wiki/Covariance_of_Sums_of_Random_Variables/Lemma | https://proofwiki.org/wiki/Covariance_of_Sums_of_Random_Variables/Lemma | [
"Covariance",
"Covariance of Sums of Random Variables"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Sequence",
"Definition:Random Variable",
"Definition:Random Variable"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-16967 | Expectation of Linear Transformation of Random Variable | Let $X$ be a random variable.
Let $a, b$ be real numbers.
Let $\expect X$ denote the expectation of $X$.
Then we have:
:$\expect {a X + b} = a \expect X + b$
if that expectation exists. | === Discrete Random Variable ===
{{:Expectation of Linear Transformation of Random Variable/Discrete}} | Let $X$ be a [[Definition:Random Variable|random variable]].
Let $a, b$ be [[Definition:Real Number|real numbers]].
Let $\expect X$ denote the [[Definition:Expectation|expectation]] of $X$.
Then we have:
:$\expect {a X + b} = a \expect X + b$
if that [[Definition:Expectation|expectation]] exists. | === [[Expectation of Linear Transformation of Random Variable/Discrete|Discrete Random Variable]] ===
{{:Expectation of Linear Transformation of Random Variable/Discrete}} | Expectation of Linear Transformation of Random Variable | https://proofwiki.org/wiki/Expectation_of_Linear_Transformation_of_Random_Variable | https://proofwiki.org/wiki/Expectation_of_Linear_Transformation_of_Random_Variable | [
"Expectation",
"Expectation of Linear Transformation of Random Variable"
] | [
"Definition:Random Variable",
"Definition:Real Number",
"Definition:Expectation",
"Definition:Expectation"
] | [
"Expectation of Linear Transformation of Random Variable/Discrete"
] |
proofwiki-16968 | Expectation of Linear Transformation of Random Variable/Discrete | Let $X$ be a discrete random variable.
Let $a, b$ be real numbers.
Then we have:
:$\expect {a X + b} = a \expect X + b$
where $\expect X$ denotes the expectation of $X$. | We have:
{{begin-eqn}}
{{eqn | l = \expect {a X + b}
| r = \sum_{x \mathop \in \Img X} \paren {a x + b} \map \Pr {X = x}
| c = Expectation of Function of Discrete Random Variable
}}
{{eqn | r = a \sum_{x \mathop \in \Img X} x \map \Pr {X = x} + b \sum_{x \mathop \in \Img X} \map \Pr {X = x}
}}
{{eqn | r = a \expect ... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]].
Let $a, b$ be [[Definition:Real Number|real numbers]].
Then we have:
:$\expect {a X + b} = a \expect X + b$
where $\expect X$ denotes the [[Definition:Expectation of Discrete Random Variable|expectation]] of $X$. | We have:
{{begin-eqn}}
{{eqn | l = \expect {a X + b}
| r = \sum_{x \mathop \in \Img X} \paren {a x + b} \map \Pr {X = x}
| c = [[Expectation of Function of Discrete Random Variable]]
}}
{{eqn | r = a \sum_{x \mathop \in \Img X} x \map \Pr {X = x} + b \sum_{x \mathop \in \Img X} \map \Pr {X = x}
}}
{{eqn | r = a \ex... | Expectation of Linear Transformation of Random Variable/Discrete | https://proofwiki.org/wiki/Expectation_of_Linear_Transformation_of_Random_Variable/Discrete | https://proofwiki.org/wiki/Expectation_of_Linear_Transformation_of_Random_Variable/Discrete | [
"Expectation of Linear Transformation of Random Variable"
] | [
"Definition:Random Variable/Discrete",
"Definition:Real Number",
"Definition:Expectation/Discrete"
] | [
"Expectation of Function of Discrete Random Variable"
] |
proofwiki-16969 | Expectation of Linear Transformation of Random Variable/Continuous | Let $X$ be a continuous random variable.
Let $a, b$ be real numbers.
Then we have:
:$\expect {a X + b} = a \expect X + b$
where $\expect X$ denotes the expectation of $X$. | Let $\map \supp X$ be the support of $X$.
Let $f_X : \map \supp X \to \R$ be the probability density function of $X$.
{{questionable|What if the density $f_X$ does not exist?}}
Then:
{{begin-eqn}}
{{eqn | l = \expect {a X + b}
| r = \int_{x \mathop \in \map \supp X} \paren {a x + b} \map {f_X} x \rd x
| c = Expectat... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]].
Let $a, b$ be [[Definition:Real Number|real numbers]].
Then we have:
:$\expect {a X + b} = a \expect X + b$
where $\expect X$ denotes the [[Definition:Expectation|expectation]] of $X$. | Let $\map \supp X$ be the [[Definition:Support of Random Variable|support]] of $X$.
Let $f_X : \map \supp X \to \R$ be the [[Definition:Probability Density Function|probability density function]] of $X$.
{{questionable|What if the density $f_X$ does not exist?}}
Then:
{{begin-eqn}}
{{eqn | l = \expect {a X + b}
| r... | Expectation of Linear Transformation of Random Variable/Continuous | https://proofwiki.org/wiki/Expectation_of_Linear_Transformation_of_Random_Variable/Continuous | https://proofwiki.org/wiki/Expectation_of_Linear_Transformation_of_Random_Variable/Continuous | [
"Expectation of Linear Transformation of Random Variable"
] | [
"Definition:Random Variable/Continuous",
"Definition:Real Number",
"Definition:Expectation"
] | [
"Definition:Support of Random Variable",
"Definition:Probability Density Function",
"Expectation of Function of Continuous Random Variable",
"Linear Combination of Integrals/Definite"
] |
proofwiki-16970 | Cauchy's Mean Theorem/Proof of Equality Condition | Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive.
Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$.
Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$.
Then:
:$A_n = G_n$
{{iff}}:
:$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$
That is, {{iff}} all terms are equal.
Then... | === Necessary Condition ===
Let:
:$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j = x$
Then:
{{begin-eqn}}
{{eqn | l = A_n
| r = \dfrac 1 n \sum_{j \mathop = 1}^n x
| c =
}}
{{eqn | r = \dfrac 1 n n x
| c =
}}
{{eqn | r = x
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = G_n
| r = \p... | Let $x_1, x_2, \ldots, x_n \in \R$ be [[Definition:Real Number|real numbers]] which are all [[Definition:Positive Real Number|positive]].
Let $A_n$ be the [[Definition:Arithmetic Mean|arithmetic mean]] of $x_1, x_2, \ldots, x_n$.
Let $G_n$ be the [[Definition:Geometric Mean|geometric mean]] of $x_1, x_2, \ldots, x_n$... | === Necessary Condition ===
Let:
:$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j = x$
Then:
{{begin-eqn}}
{{eqn | l = A_n
| r = \dfrac 1 n \sum_{j \mathop = 1}^n x
| c =
}}
{{eqn | r = \dfrac 1 n n x
| c =
}}
{{eqn | r = x
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = G_n
| ... | Cauchy's Mean Theorem/Proof of Equality Condition | https://proofwiki.org/wiki/Cauchy's_Mean_Theorem/Proof_of_Equality_Condition | https://proofwiki.org/wiki/Cauchy's_Mean_Theorem/Proof_of_Equality_Condition | [
"Cauchy's Mean Theorem"
] | [
"Definition:Real Number",
"Definition:Positive/Real Number",
"Definition:Arithmetic Mean",
"Definition:Geometric Mean",
"Definition:Term of Sequence"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Forward-Backward Induction",
"Definition:Induction Hypothesis",
"Cauchy's Mean Theorem/Proof of Equality Condition",
"Definition:Induction Step",
"Cauchy's Mean Theorem/Proof of Equality Condition",... |
proofwiki-16971 | Variance of Linear Transformation of Random Variable | Let $X$ be a random variable.
Let $a, b$ be real numbers.
Then we have:
:$\var {a X + b} = a^2 \var X$
where $\var X$ denotes the variance of $X$. | We have:
{{begin-eqn}}
{{eqn | l = \var {a X + b}
| r = \expect {\paren {a X + b - \expect {a X + b} }^2}
| c = {{Defof|Variance}}
}}
{{eqn | r = \expect {\paren {a X + b - a \expect X - b}^2}
| c = Expectation of Linear Transformation of Random Variable
}}
{{eqn | r = \expect {a^2 \paren {X - \expect X}^2}
}}
{{eqn... | Let $X$ be a [[Definition:Random Variable|random variable]].
Let $a, b$ be [[Definition:Real Number|real numbers]].
Then we have:
:$\var {a X + b} = a^2 \var X$
where $\var X$ denotes the [[Definition:Variance|variance]] of $X$. | We have:
{{begin-eqn}}
{{eqn | l = \var {a X + b}
| r = \expect {\paren {a X + b - \expect {a X + b} }^2}
| c = {{Defof|Variance}}
}}
{{eqn | r = \expect {\paren {a X + b - a \expect X - b}^2}
| c = [[Expectation of Linear Transformation of Random Variable]]
}}
{{eqn | r = \expect {a^2 \paren {X - \expect X}^2}
}}
... | Variance of Linear Transformation of Random Variable | https://proofwiki.org/wiki/Variance_of_Linear_Transformation_of_Random_Variable | https://proofwiki.org/wiki/Variance_of_Linear_Transformation_of_Random_Variable | [
"Variance"
] | [
"Definition:Random Variable",
"Definition:Real Number",
"Definition:Variance"
] | [
"Expectation of Linear Transformation of Random Variable",
"Expectation of Linear Transformation of Random Variable",
"Category:Variance"
] |
proofwiki-16972 | Fourier Series/Square Wave | 600pxthumbrightSquare Wave and $9$th Approximation
Let $\map S x$ be the square wave defined on the real numbers $\R$ as:
:$\forall x \in \R: \map S x = \begin {cases}
1 & : x \in \openint 0 l \\
-1 & : x \in \openint {-l} 0 \\
\map S {x + 2 l} & : x < -l \\
\map S {x - 2 l} & : x > +l \end {cases}$
Then its Fourier se... | Let $\map f x$ be the function defined as:
:$\forall x \in \openint {-l} l: \begin{cases} -1 & : -l < x < 0 \\ 1 & : 0 < x < l \end {cases}$
By inspection we see that $\map f x$ is an odd function.
Hence from Fourier Series for Odd Function over Symmetric Range we can express $f$ by a half-range Fourier sine series:
:$... | [[File:Square-wave-Fourier-series.png|600px|thumb|right|Square Wave and $9$th Approximation]]
Let $\map S x$ be the [[Definition:Square Wave|square wave]] defined on the [[Definition:Real Number|real numbers]] $\R$ as:
:$\forall x \in \R: \map S x = \begin {cases}
1 & : x \in \openint 0 l \\
-1 & : x \in \openint {-l... | Let $\map f x$ be the [[Definition:Real Function|function]] defined as:
:$\forall x \in \openint {-l} l: \begin{cases} -1 & : -l < x < 0 \\ 1 & : 0 < x < l \end {cases}$
By inspection we see that $\map f x$ is an [[Definition:Odd Function|odd function]].
Hence from [[Fourier Series for Odd Function over Symmetric Ra... | Fourier Series/Square Wave | https://proofwiki.org/wiki/Fourier_Series/Square_Wave | https://proofwiki.org/wiki/Fourier_Series/Square_Wave | [
"Square Waves",
"Fourier Series for Square Wave"
] | [
"File:Square-wave-Fourier-series.png",
"Definition:Square Wave",
"Definition:Real Number",
"Definition:Fourier Series"
] | [
"Definition:Real Function",
"Definition:Odd Function",
"Fourier Series for Odd Function over Symmetric Range",
"Definition:Half-Range Fourier Sine Series",
"Definition:Real Interval/Open",
"Primitive of Sine Function/Corollary",
"Cosine of Zero is One",
"Cosine of Integer Multiple of Pi"
] |
proofwiki-16973 | Fourier Series/Triangle Wave | 600pxthumbrightTriangle Wave and $9$th Approximation
<onlyinclude>
Let $\map T x$ be the triangle wave defined on the real numbers $\R$ as:
:<nowiki>$\forall x \in \R: \map T x = \begin {cases}
\size x & : x \in \closedint {-l} l \\
\map T {x + 2 l} & : x < -l \\
\map T {x - 2 l} & : x > +l \end {cases}$</nowiki>
where... | Let $\map f x: \openint {-l} l \to \R$ denote the absolute value function on the open interval $\openint {-l} l$:
:<nowiki>$\map f x = \size x = \begin{cases}
x & : x > 0 \\
0 & : x = 0 \\
-x & : x < 0
\end{cases}$</nowiki>
From Fourier Series for Absolute Value Function over Symmetric Range, $\map f x$ can immediately... | [[File:Triangle-wave-Fourier-series.png|600px|thumb|right|Triangle Wave and $9$th Approximation]]
<onlyinclude>
Let $\map T x$ be the [[Definition:Triangle Wave|triangle wave]] defined on the [[Definition:Real Number|real numbers]] $\R$ as:
:<nowiki>$\forall x \in \R: \map T x = \begin {cases}
\size x & : x \in \close... | Let $\map f x: \openint {-l} l \to \R$ denote the [[Definition:Absolute Value|absolute value function]] on the [[Definition:Open Real Interval|open interval $\openint {-l} l$]]:
:<nowiki>$\map f x = \size x = \begin{cases}
x & : x > 0 \\
0 & : x = 0 \\
-x & : x < 0
\end{cases}$</nowiki>
From [[Fourier Series for Abso... | Fourier Series/Triangle Wave | https://proofwiki.org/wiki/Fourier_Series/Triangle_Wave | https://proofwiki.org/wiki/Fourier_Series/Triangle_Wave | [
"Triangle Waves",
"Fourier Series for Triangle Wave"
] | [
"File:Triangle-wave-Fourier-series.png",
"Definition:Triangle Wave",
"Definition:Real Number",
"Definition:Given",
"Definition:Real Number",
"Definition:Constant",
"Definition:Absolute Value",
"Definition:Fourier Series"
] | [
"Definition:Absolute Value",
"Definition:Real Interval/Open",
"Fourier Series/Absolute Value Function over Symmetric Range"
] |
proofwiki-16974 | Fourier Series/Sawtooth Wave | 600pxthumbrightSawtooth Wave and $6$th Approximation
Let $\map S x$ be the sawtooth wave defined on the real numbers $\R$ as:
:$\forall x \in \R: \map S x = \begin {cases} x & : x \in \openint {-l} l \\ \map S {x + 2 l} & : x < -l \\ \map S {x - 2 l} & : x > +l \end {cases}$
where $l$ is a given real constant.
Then its... | Let $\map f x: \openint {-l} l \to \R$ denote the identity function on the open interval $\openint {-l} l$:
:$\map f x = x$
From Fourier Series for Identity Function over Symmetric Range, $\map f x$ can immediately be expressed as:
:$\ds \map f x \sim \frac {2 l} \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + ... | [[File:Sawtooth-wave-Fourier-series.png|600px|thumb|right|Sawtooth Wave and $6$th Approximation]]
Let $\map S x$ be the [[Definition:Sawtooth Wave|sawtooth wave]] defined on the [[Definition:Real Number|real numbers]] $\R$ as:
:$\forall x \in \R: \map S x = \begin {cases} x & : x \in \openint {-l} l \\ \map S {x + 2 ... | Let $\map f x: \openint {-l} l \to \R$ denote the [[Definition:Identity Mapping|identity function]] on the [[Definition:Open Real Interval|open interval $\openint {-l} l$]]:
:$\map f x = x$
From [[Fourier Series for Identity Function over Symmetric Range]], $\map f x$ can immediately be expressed as:
:$\ds \map f x ... | Fourier Series/Sawtooth Wave | https://proofwiki.org/wiki/Fourier_Series/Sawtooth_Wave | https://proofwiki.org/wiki/Fourier_Series/Sawtooth_Wave | [
"Sawtooth Waves",
"Fourier Series for Sawtooth Wave"
] | [
"File:Sawtooth-wave-Fourier-series.png",
"Definition:Sawtooth Wave",
"Definition:Real Number",
"Definition:Given",
"Definition:Real Number",
"Definition:Constant",
"Definition:Fourier Series"
] | [
"Definition:Identity Mapping",
"Definition:Real Interval/Open",
"Fourier Series/Identity Function over Symmetric Range"
] |
proofwiki-16975 | Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton/Lemma | :$f$ is a bijection. | === $f$ is an Injection ===
Let $x, y \in T_1$.
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
}}
{{eqn | ll = \leadstoandfrom
| l = \tuple {x, b}
| r = \tuple {y, b}
| c = Definition of $f$
}}
{{eqn | ll = \leadstoandfrom
| l = x
| r = y
| c = Equality of ordered pairs
}}
{... | :$f$ is a [[Definition:Bijection|bijection]]. | === $f$ is an Injection ===
Let $x, y \in T_1$.
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
}}
{{eqn | ll = \leadstoandfrom
| l = \tuple {x, b}
| r = \tuple {y, b}
| c = Definition of $f$
}}
{{eqn | ll = \leadstoandfrom
| l = x
| r = y
| c = [[Definition:Equality|Equali... | Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton/Lemma | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Product_with_Singleton/Lemma | https://proofwiki.org/wiki/Subspace_of_Product_Space_is_Homeomorphic_to_Factor_Space/Product_with_Singleton/Lemma | [
"Subspace of Product Space is Homeomorphic to Factor Space"
] | [
"Definition:Bijection"
] | [
"Definition:Equals",
"Definition:Ordered Pair",
"Definition:Injection"
] |
proofwiki-16976 | Standard Continuous Uniform Distribution in terms of Exponential Distribution | Let $X$ and $Y$ be independent random variables.
Let $\beta$ be a strictly positive real number.
Let $X$ and $Y$ be random samples from the exponential distribution with parameter $\beta$.
Then:
:$\dfrac X {X + Y} \sim \operatorname U \openint 0 1$
where $\operatorname U \openint 0 1$ is the uniform distribution on $\... | Note that the support of $\operatorname U \openint 0 1$ is $\openint 0 1$.
It is therefore sufficient to show that for $0 < z < 1$:
:$\map \Pr {\dfrac X {X + Y} \le z} = z$
Note that if $x, y > 0$ then:
:$0 < \dfrac x {x + y} < 1$
Note also that:
:$\dfrac x {x + y} \le z$
with $0 < z < 1$ is equivalent to:
:$x \le... | Let $X$ and $Y$ be [[Definition:Independent Random Variables|independent random variables]].
Let $\beta$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $X$ and $Y$ be [[Definition:Random Sample (Probability Theory)|random samples]] from the [[Definition:Exponential Distribution|e... | Note that the [[Definition:Support of Random Variable|support]] of $\operatorname U \openint 0 1$ is $\openint 0 1$.
It is therefore sufficient to show that for $0 < z < 1$:
:$\map \Pr {\dfrac X {X + Y} \le z} = z$
Note that if $x, y > 0$ then:
:$0 < \dfrac x {x + y} < 1$
Note also that:
:$\dfrac x {x + y} \le... | Standard Continuous Uniform Distribution in terms of Exponential Distribution | https://proofwiki.org/wiki/Standard_Continuous_Uniform_Distribution_in_terms_of_Exponential_Distribution | https://proofwiki.org/wiki/Standard_Continuous_Uniform_Distribution_in_terms_of_Exponential_Distribution | [
"Continuous Uniform Distribution",
"Exponential Distribution"
] | [
"Definition:Independent Random Variables",
"Definition:Strictly Positive/Real Number",
"Definition:Random Sample (Probability Theory)",
"Definition:Exponential Distribution",
"Definition:Uniform Distribution/Continuous"
] | [
"Definition:Support of Random Variable",
"Definition:Joint Probability Density Function",
"Definition:Probability Density Function",
"Condition for Independence from Joint Probability Density Function",
"Primitive of Exponential of a x",
"Exponential of Zero",
"Primitive of Exponential of a x",
"Expon... |
proofwiki-16977 | Inverse Image Mapping Induced by Projection | Let $\family {S_i}_{i \mathop \in I}$ be a family of sets.
Let $\ds S = \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.
For each $j \in I$, let $\pr_j: S \to S_j$ denote the $j$-th projection.
For each $j \in I$ let $\pr_j^\gets: \powerset {S_i} \to \powerset S$ denote the in... | Let $j \in I$.
Let $T \subseteq S_j$.
Let $\family {T_i}_{i \mathop \in I}$ be the family of sets defined by:
:$T_i = \begin {cases} T & : i = j \\ S_i & : i \ne j \end {cases}$
Then:
{{begin-eqn}}
{{eqn | l = \map {\pr_j^\gets} T
| r = \set {x \in S: \map {\pr_j} x \in T}
| c = {{Defof|Inverse Image Mappin... | Let $\family {S_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family of Sets|family of sets]].
Let $\ds S = \prod_{i \mathop \in I} S_i$ be the [[Definition:Cartesian Product of Family|Cartesian product]] of $\family {S_i}_{i \mathop \in I}$.
For each $j \in I$, let $\pr_j: S \to S_j$ denote the [[Definition:Projec... | Let $j \in I$.
Let $T \subseteq S_j$.
Let $\family {T_i}_{i \mathop \in I}$ be the [[Definition:Indexed Family of Sets|family of sets]] defined by:
:$T_i = \begin {cases} T & : i = j \\ S_i & : i \ne j \end {cases}$
Then:
{{begin-eqn}}
{{eqn | l = \map {\pr_j^\gets} T
| r = \set {x \in S: \map {\pr_j} x \in T}... | Inverse Image Mapping Induced by Projection | https://proofwiki.org/wiki/Inverse_Image_Mapping_Induced_by_Projection | https://proofwiki.org/wiki/Inverse_Image_Mapping_Induced_by_Projection | [
"Projections",
"Inverse Image Mappings"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Cartesian Product/Family of Sets",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Mapping"
] | [
"Definition:Indexing Set/Family of Sets",
"Category:Projections",
"Category:Inverse Image Mappings"
] |
proofwiki-16978 | Identity Function is Odd Function | Let $I_\R: \R \to \R$ denote the identity function on $\R$.
Then $I_\R$ is an odd function. | {{begin-eqn}}
{{eqn | l = \map {I_\R} {-x}
| r = -x
| c = {{Defof|Identity Function}}
}}
{{eqn | r = -\map {I_\R} x
| c =
}}
{{end-eqn}}
Hence the result by definition of odd function.
{{qed}}
Category:Identity Mappings
Category:Examples of Odd Functions
oxam7nnhno8vncvpprywkit3jqbco7m | Let $I_\R: \R \to \R$ denote the [[Definition:Identity Function|identity function]] on $\R$.
Then $I_\R$ is an [[Definition:Odd Function|odd function]]. | {{begin-eqn}}
{{eqn | l = \map {I_\R} {-x}
| r = -x
| c = {{Defof|Identity Function}}
}}
{{eqn | r = -\map {I_\R} x
| c =
}}
{{end-eqn}}
Hence the result by definition of [[Definition:Odd Function|odd function]].
{{qed}}
[[Category:Identity Mappings]]
[[Category:Examples of Odd Functions]]
oxam7nnh... | Identity Function is Odd Function | https://proofwiki.org/wiki/Identity_Function_is_Odd_Function | https://proofwiki.org/wiki/Identity_Function_is_Odd_Function | [
"Identity Mappings",
"Examples of Odd Functions"
] | [
"Definition:Identity Mapping",
"Definition:Odd Function"
] | [
"Definition:Odd Function",
"Category:Identity Mappings",
"Category:Examples of Odd Functions"
] |
proofwiki-16979 | Half-Range Fourier Series/Identity Function/Cosine | The half-range Fourier cosine series for $\map f x$ can be expressed as:
{{begin-eqn}}
{{eqn | l = \map f x
| o = \sim
| r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \dfrac {\paren {2 n + 1} \pi x} \lambda
| c =
}}
{{eqn | r = \frac \la... | By definition of half-range Fourier cosine series:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos \dfrac {n \pi x} \lambda$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \frac 2 \lambda \int_0^\lambda \map f x \cos \dfrac {n \pi x} \lambda \rd x$
Thus by definition of $f$:
{{begin-eqn}}
{{eqn | ... | The [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosine series]] for $\map f x$ can be expressed as:
{{begin-eqn}}
{{eqn | l = \map f x
| o = \sim
| r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \dfrac {\paren {2 n + 1} \pi... | By definition of [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosine series]]:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos \dfrac {n \pi x} \lambda$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \frac 2 \lambda \int_0^\lambda \map f x \cos \dfrac {n \pi x} \lambda \rd x... | Half-Range Fourier Series/Identity Function/Cosine | https://proofwiki.org/wiki/Half-Range_Fourier_Series/Identity_Function/Cosine | https://proofwiki.org/wiki/Half-Range_Fourier_Series/Identity_Function/Cosine | [
"Half-Range Fourier Series for Identity Function"
] | [
"Definition:Half-Range Fourier Cosine Series"
] | [
"Definition:Half-Range Fourier Cosine Series",
"Cosine of Zero is One",
"Primitive of Power",
"Primitive of x by Cosine of a x",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi",
"Definition:Even Integer",
"Definition:Odd Integer",
"Category:Half-Range Fourier Series for Identity... |
proofwiki-16980 | Half-Range Fourier Series/Identity Function/Sine | The half-range Fourier sine series for $\map f x$ can be expressed as:
{{begin-eqn}}
{{eqn | l = \map f x
| o = \sim
| r = \dfrac {2 \lambda} \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \frac {n \pi x} \lambda
| c =
}}
{{eqn | r = \dfrac {2 \lambda} \pi \paren {\sin \dfrac {\p... | By definition of half-range Fourier sine series:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 1}^\infty b_n \sin \dfrac {n \pi x} \lambda$
where for all $n \in \Z_{> 0}$:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 2 \lambda \int_0^\lambda \map f x \sin \dfrac {n \pi x} \lambda \rd x
| c =
}}
{{eqn | r =... | The [[Definition:Half-Range Fourier Sine Series|half-range Fourier sine series]] for $\map f x$ can be expressed as:
{{begin-eqn}}
{{eqn | l = \map f x
| o = \sim
| r = \dfrac {2 \lambda} \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \frac {n \pi x} \lambda
| c =
}}
{{eqn | r =... | By definition of [[Definition:Half-Range Fourier Sine Series|half-range Fourier sine series]]:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 1}^\infty b_n \sin \dfrac {n \pi x} \lambda$
where for all $n \in \Z_{> 0}$:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 2 \lambda \int_0^\lambda \map f x \sin \dfrac {... | Half-Range Fourier Series/Identity Function/Sine | https://proofwiki.org/wiki/Half-Range_Fourier_Series/Identity_Function/Sine | https://proofwiki.org/wiki/Half-Range_Fourier_Series/Identity_Function/Sine | [
"Half-Range Fourier Series for Identity Function"
] | [
"Definition:Half-Range Fourier Sine Series"
] | [
"Definition:Half-Range Fourier Sine Series",
"Primitive of x by Sine of a x",
"Fundamental Theorem of Calculus",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi",
"Category:Half-Range Fourier Series for Identity Function"
] |
proofwiki-16981 | Absolute Value Function is Even Function | Let $\size {\, \cdot \,} : \R \to \R$ denote the absolute value function on $\R$:
Then $\size {\, \cdot \,}$ is an even function. | Recall the definition of the absolute value function:
:<nowiki>$\size x = \begin{cases}
x & : x > 0 \\
0 & : x = 0 \\
-x & : x < 0
\end{cases}$</nowiki>
Testing the $3$ cases in turn:
{{begin-eqn}}
{{eqn | l = x
| o = >
| r = 0
}}
{{eqn | ll= \leadsto
| l = -x
| o = <
| r = 0
}}
{{eqn | ll... | Let $\size {\, \cdot \,} : \R \to \R$ denote the [[Definition:Absolute Value|absolute value function]] on $\R$:
Then $\size {\, \cdot \,}$ is an [[Definition:Even Function|even function]]. | Recall the definition of the [[Definition:Absolute Value|absolute value function]]:
:<nowiki>$\size x = \begin{cases}
x & : x > 0 \\
0 & : x = 0 \\
-x & : x < 0
\end{cases}$</nowiki>
Testing the $3$ cases in turn:
{{begin-eqn}}
{{eqn | l = x
| o = >
| r = 0
}}
{{eqn | ll= \leadsto
| l = -x
| ... | Absolute Value Function is Even Function | https://proofwiki.org/wiki/Absolute_Value_Function_is_Even_Function | https://proofwiki.org/wiki/Absolute_Value_Function_is_Even_Function | [
"Absolute Value Function",
"Examples of Even Functions"
] | [
"Definition:Absolute Value",
"Definition:Even Function"
] | [
"Definition:Absolute Value",
"Definition:Even Function",
"Category:Absolute Value Function",
"Category:Examples of Even Functions"
] |
proofwiki-16982 | Fourier Series/Absolute Value Function over Symmetric Range | Let $\lambda \in \R_{>0}$ be a strictly positive real number.
Let $\map f x: \openint {-\lambda} \lambda \to \R$ be the absolute value function on the open real interval $\openint {-\lambda} \lambda$:
:$\forall x \in \openint {-\lambda} \lambda: \map f x = \size x$
The Fourier series of $f$ over $\openint {-\lambda} \l... | From Absolute Value Function is Even Function, $\map f x$ is an even function.
By Fourier Series for Even Function over Symmetric Range, we have:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \frac 2 \lambda \int_0^\pi \map f x \cos \f... | Let $\lambda \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\map f x: \openint {-\lambda} \lambda \to \R$ be the [[Definition:Absolute Value|absolute value function]] on the [[Definition:Open Real Interval|open real interval]] $\openint {-\lambda} \lambda$:
:$\forall... | From [[Absolute Value Function is Even Function]], $\map f x$ is an [[Definition:Even Function|even function]].
By [[Fourier Series for Even Function over Symmetric Range]], we have:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \f... | Fourier Series/Absolute Value Function over Symmetric Range | https://proofwiki.org/wiki/Fourier_Series/Absolute_Value_Function_over_Symmetric_Range | https://proofwiki.org/wiki/Fourier_Series/Absolute_Value_Function_over_Symmetric_Range | [
"Absolute Value Function",
"Fourier Series for Absolute Value Function"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Absolute Value",
"Definition:Real Interval/Open",
"Definition:Fourier Series"
] | [
"Absolute Value Function is Even Function",
"Definition:Even Function",
"Fourier Series for Even Function over Symmetric Range",
"Half-Range Fourier Series/Identity Function/Cosine",
"Category:Absolute Value Function",
"Category:Fourier Series for Absolute Value Function"
] |
proofwiki-16983 | Versed Sine Function is Even | The versed sine is an even function:
:$\forall \theta \in \R: \map \vers {-\theta} = \vers \theta$ | {{begin-eqn}}
{{eqn | l = \map \vers {-\theta}
| r = 1 - \map \cos {-\theta}
| c = {{Defof|Versed Sine}}
}}
{{eqn | r = 1 - \cos \theta
| c = Cosine Function is Even
}}
{{eqn | r = \vers \theta
| c = {{Defof|Versed Sine}}
}}
{{end-eqn}}
{{qed}}
Category:Versed Sines
Category:Examples of Even Fun... | The [[Definition:Versed Sine|versed sine]] is an [[Definition:Even Function|even function]]:
:$\forall \theta \in \R: \map \vers {-\theta} = \vers \theta$ | {{begin-eqn}}
{{eqn | l = \map \vers {-\theta}
| r = 1 - \map \cos {-\theta}
| c = {{Defof|Versed Sine}}
}}
{{eqn | r = 1 - \cos \theta
| c = [[Cosine Function is Even]]
}}
{{eqn | r = \vers \theta
| c = {{Defof|Versed Sine}}
}}
{{end-eqn}}
{{qed}}
[[Category:Versed Sines]]
[[Category:Examples ... | Versed Sine Function is Even | https://proofwiki.org/wiki/Versed_Sine_Function_is_Even | https://proofwiki.org/wiki/Versed_Sine_Function_is_Even | [
"Versed Sines",
"Examples of Even Functions"
] | [
"Definition:Versed Sine",
"Definition:Even Function"
] | [
"Cosine Function is Even",
"Category:Versed Sines",
"Category:Examples of Even Functions"
] |
proofwiki-16984 | Cartesian Product is Empty iff Factor is Empty/Family of Sets | Let $I$ be an indexing set.
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.
Let $\ds S = \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.
Then:
:$S = \O$ {{iff}} $S_i = \O$ for some $i \in I$ | === Necessary Condition ===
By the axiom of choice, the contrapositive statement holds:
:if $S_i \ne \O$ for all $i \in I$ then $S \ne \O$
By the Rule of Transposition, the converse holds:
:if $S = \O$ then $S_i = \O$ for some $i \in I$
{{qed|lemma}} | Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {S_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family of Sets|family of sets indexed by $I$]].
Let $\ds S = \prod_{i \mathop \in I} S_i$ be the [[Definition:Cartesian Product of Family|Cartesian product]] of $\family {S_i}_{i \mathop \in I}$.
T... | === Necessary Condition ===
By the [[Axiom:Axiom of Choice|axiom of choice]], the [[Definition:Contrapositive Statement|contrapositive statement]] holds:
:if $S_i \ne \O$ for all $i \in I$ then $S \ne \O$
By the [[Rule of Transposition]], the [[Definition:Converse|converse]] holds:
:if $S = \O$ then $S_i = \O$ for s... | Cartesian Product is Empty iff Factor is Empty/Family of Sets | https://proofwiki.org/wiki/Cartesian_Product_is_Empty_iff_Factor_is_Empty/Family_of_Sets | https://proofwiki.org/wiki/Cartesian_Product_is_Empty_iff_Factor_is_Empty/Family_of_Sets | [
"Cartesian Product",
"Empty Set"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Cartesian Product/Family of Sets"
] | [
"Axiom:Axiom of Choice",
"Definition:Contrapositive Statement",
"Rule of Transposition",
"Definition:Converse"
] |
proofwiki-16985 | Derivative of P-Norm wrt P | Let $p \ge 1$ be a real number.
Let $\ell^p$ denote the $p$-sequence space.
Let $\mathbf x = \sequence {x_n} \in \ell^p$.
Let $\norm {\mathbf x}_p$ be a $p$-norm.
Suppose, $\norm {\mathbf x}_p \ne 0$.
Then:
:$\ds \dfrac \d {\d p} \norm {\mathbf x}_p = \frac {\norm {\mathbf x}_p} p \paren { \frac {\sum_{n \mathop = 0}^\... | We begin with the natural logarithm of $\norm {\mathbf x}_p$:
{{begin-eqn}}
{{eqn | l = \dfrac \d {\d p} \map \ln {\norm {\bf x}_p}
| r = \frac {\dfrac \d {\d p} \norm {\bf x}_p} {\norm {\bf x}_p}
}}
{{eqn | r = \map {\dfrac \d {\d p} } {\frac 1 p \map \ln {\sum_{n \mathop = 0}^\infty \size {x_n}^p} }
}}
{{eqn | ... | Let $p \ge 1$ be a [[Definition:Real Number|real number]].
Let $\ell^p$ denote the [[Definition:P-Sequence Space|$p$-sequence space]].
Let $\mathbf x = \sequence {x_n} \in \ell^p$.
Let $\norm {\mathbf x}_p$ be a [[Definition:P-Norm|$p$-norm]].
Suppose, $\norm {\mathbf x}_p \ne 0$.
Then:
:$\ds \dfrac \d {\d p} \n... | We begin with the [[Definition:Natural Logarithm|natural logarithm]] of $\norm {\mathbf x}_p$:
{{begin-eqn}}
{{eqn | l = \dfrac \d {\d p} \map \ln {\norm {\bf x}_p}
| r = \frac {\dfrac \d {\d p} \norm {\bf x}_p} {\norm {\bf x}_p}
}}
{{eqn | r = \map {\dfrac \d {\d p} } {\frac 1 p \map \ln {\sum_{n \mathop = 0}^\... | Derivative of P-Norm wrt P | https://proofwiki.org/wiki/Derivative_of_P-Norm_wrt_P | https://proofwiki.org/wiki/Derivative_of_P-Norm_wrt_P | [
"P-Norms"
] | [
"Definition:Real Number",
"Definition:P-Sequence Space",
"Definition:P-Norm"
] | [
"Definition:Natural Logarithm",
"Derivative of General Exponential Function",
"Definition:Multiplication/Real Numbers",
"Definition:Proof",
"Category:P-Norms"
] |
proofwiki-16986 | P-Norm of Real Sequence is Strictly Decreasing Function of P | Let $p \ge 1$ be a real number.
Let ${\ell^p}_\R$ denote the real $p$-sequence space.
Let $\mathbf x = \sequence {x_n} \in {\ell^p}_\R$.
Suppose $\mathbf x$ is not a sequence of zero elements.
Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$ where $p \ge 1$.
Then the mapping $p \to \norm {\mathbf x}_p$ is s... | {{begin-eqn}}
{{eqn | q = \forall i \in \N
| l = \sum_{n \mathop = 0}^\infty {\size {x_n} }
| o = \ge
| r = \size {x_i}
| c = {{EuclidCommonNotionLink|5}}: the whole is greater than the part
}}
{{eqn | ll= \leadsto
| q = \forall i \in \N
| l = \paren {\sum_{n \mathop = 0}^\infty {\s... | Let $p \ge 1$ be a [[Definition:Real Number|real number]].
Let ${\ell^p}_\R$ denote the [[Definition:Real P-Sequence Space|real $p$-sequence space]].
Let $\mathbf x = \sequence {x_n} \in {\ell^p}_\R$.
Suppose $\mathbf x$ is not a [[Definition:Real Sequence|sequence]] of [[Definition:Zero (Number)|zero]] [[Definition... | {{begin-eqn}}
{{eqn | q = \forall i \in \N
| l = \sum_{n \mathop = 0}^\infty {\size {x_n} }
| o = \ge
| r = \size {x_i}
| c = {{EuclidCommonNotionLink|5}}: the whole is greater than the part
}}
{{eqn | ll= \leadsto
| q = \forall i \in \N
| l = \paren {\sum_{n \mathop = 0}^\infty {\s... | P-Norm of Real Sequence is Strictly Decreasing Function of P | https://proofwiki.org/wiki/P-Norm_of_Real_Sequence_is_Strictly_Decreasing_Function_of_P | https://proofwiki.org/wiki/P-Norm_of_Real_Sequence_is_Strictly_Decreasing_Function_of_P | [
"P-Norms"
] | [
"Definition:Real Number",
"Definition:P-Sequence Space/Real",
"Definition:Real Sequence",
"Definition:Zero (Number)",
"Definition:Element",
"Definition:P-Norm/Real",
"Definition:Mapping",
"Definition:Strictly Decreasing"
] | [
"Definition:Real Sequence",
"Definition:Zero (Number)",
"Definition:Element",
"Definition:Real Sequence",
"Definition:Zero (Number)",
"Definition:Element",
"Definition:Multiplication/Real Numbers",
"Definition:Summation",
"Definition:Inequality",
"Derivative of P-Norm wrt P",
"Definition:Term of... |
proofwiki-16987 | Extension of Half-Range Fourier Sine Function to Symmetric Range | Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$.
Let $\map f x$ be represented by the half-range Fourier sine series $\map S x$:
:$\map f x \sim \map S x = \ds \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$
where for all $n \in \Z_{> 0}$:
:$b_n = \ds \frac 2 \lambda \int_0^... | It is apparent by inspection that:
:$(1): \quad g$ is an extension of $f$
:$(2): \quad g$ is an odd function.
Let $\map T x$ be the Fourier series representing $g$:
:$\map g x \sim \map T x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
w... | Let $\map f x$ be a [[Definition:Real Function|real function]] defined on the [[Definition:Real Interval|interval]] $\openint 0 \lambda$.
Let $\map f x$ be represented by the [[Definition:Half-Range Fourier Sine Series|half-range Fourier sine series]] $\map S x$:
:$\map f x \sim \map S x = \ds \sum_{n \mathop = 1}^\i... | It is apparent by inspection that:
:$(1): \quad g$ is an [[Definition:Extension of Mapping|extension]] of $f$
:$(2): \quad g$ is an [[Definition:Odd Function|odd function]].
Let $\map T x$ be the [[Definition:Fourier Series|Fourier series]] representing $g$:
:$\map g x \sim \map T x = \dfrac {a_0} 2 + \ds \sum_{n \m... | Extension of Half-Range Fourier Sine Function to Symmetric Range | https://proofwiki.org/wiki/Extension_of_Half-Range_Fourier_Sine_Function_to_Symmetric_Range | https://proofwiki.org/wiki/Extension_of_Half-Range_Fourier_Sine_Function_to_Symmetric_Range | [
"Half-Range Fourier Series"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Half-Range Fourier Sine Series",
"Definition:Extension of Mapping"
] | [
"Definition:Extension of Mapping",
"Definition:Odd Function",
"Definition:Fourier Series",
"Fourier Cosine Coefficients for Odd Function over Symmetric Range",
"Fourier Sine Coefficients for Odd Function over Symmetric Range",
"Category:Half-Range Fourier Series"
] |
proofwiki-16988 | Extension of Half-Range Fourier Cosine Function to Symmetric Range | Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$.
Let $\map f x$ be represented by the half-range Fourier cosine series $\map S x$:
:$\map f x \sim \map S x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \fra... | It is apparent by inspection that:
:$(1): \quad g$ is an extension of $f$
:$(2): \quad g$ is an even function.
Let $\map T x$ be the Fourier series representing $g$:
:$\map g x \sim \map T x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
... | Let $\map f x$ be a [[Definition:Real Function|real function]] defined on the [[Definition:Real Interval|interval]] $\openint 0 \lambda$.
Let $\map f x$ be represented by the [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosine series]] $\map S x$:
:$\map f x \sim \map S x = \dfrac {a_0} 2 + \ds \s... | It is apparent by inspection that:
:$(1): \quad g$ is an [[Definition:Extension of Mapping|extension]] of $f$
:$(2): \quad g$ is an [[Definition:Even Function|even function]].
Let $\map T x$ be the [[Definition:Fourier Series|Fourier series]] representing $g$:
:$\map g x \sim \map T x = \dfrac {a_0} 2 + \ds \sum_{n ... | Extension of Half-Range Fourier Cosine Function to Symmetric Range | https://proofwiki.org/wiki/Extension_of_Half-Range_Fourier_Cosine_Function_to_Symmetric_Range | https://proofwiki.org/wiki/Extension_of_Half-Range_Fourier_Cosine_Function_to_Symmetric_Range | [
"Half-Range Fourier Series"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Half-Range Fourier Cosine Series",
"Definition:Extension of Mapping",
"Definition:Even Function"
] | [
"Definition:Extension of Mapping",
"Definition:Even Function",
"Definition:Fourier Series",
"Fourier Sine Coefficients for Even Function over Symmetric Range",
"Fourier Cosine Coefficients for Even Function over Symmetric Range",
"Category:Half-Range Fourier Series"
] |
proofwiki-16989 | Primitive of x by Logarithm of x squared plus a squared | :$\ds \int x \map \ln {x^2 + a^2} \rd x = \frac {\paren {x^2 + a^2} \map \ln {x^2 + a^2} - x^2} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int x \map \ln {x^2 + a^2} \rd x
| r = \frac {x^2 \map \ln {x^2 + a^2} } 2 - \int \frac {x^3} {x^2 + a^2} \rd x + C
| c = Primitive of $x^m \map \ln {x^2 + a^2}$ with $m = 1$
}}
{{eqn | r = \frac {x^2 \map \ln {x^2 + a^2} } 2 - \paren {\frac {x^2} 2 - \frac {a^2} 2 \map \ln {x^2 +... | :$\ds \int x \map \ln {x^2 + a^2} \rd x = \frac {\paren {x^2 + a^2} \map \ln {x^2 + a^2} - x^2} 2 + C$ | {{begin-eqn}}
{{eqn | l = \int x \map \ln {x^2 + a^2} \rd x
| r = \frac {x^2 \map \ln {x^2 + a^2} } 2 - \int \frac {x^3} {x^2 + a^2} \rd x + C
| c = [[Primitive of Power of x by Logarithm of x squared plus a squared|Primitive of $x^m \map \ln {x^2 + a^2}$]] with $m = 1$
}}
{{eqn | r = \frac {x^2 \map \ln {x... | Primitive of x by Logarithm of x squared plus a squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_by_Logarithm_of_x_squared_plus_a_squared | https://proofwiki.org/wiki/Primitive_of_x_by_Logarithm_of_x_squared_plus_a_squared/Proof_1 | [
"Primitives involving Logarithm Function",
"Primitives involving x squared plus a squared",
"Primitive of x by Logarithm of x squared plus a squared"
] | [] | [
"Primitive of Power of x by Logarithm of x squared plus a squared",
"Primitive of x cubed over x squared plus a squared"
] |
proofwiki-16990 | Equivalence of Definitions of Sets Separated by Neighborhoods | Let $T = \struct {S, \tau}$ be a topological space.
{{TFAE|def = Separated by Neighborhoods/Sets|view = Sets Separated by Neighborhoods}} | === Definition 1 implies Definition 2 ===
Let $A, B \subseteq S$ such that:
:$\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A \cap N_B = \O$
From Subsets of Disjoint Sets are Disjoint then:
:$U \cap V = \O$
Thus:
:$\exists U, V \in \tau: A \subseteq U, ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
{{TFAE|def = Separated by Neighborhoods/Sets|view = Sets Separated by Neighborhoods}} | === Definition 1 implies Definition 2 ===
Let $A, B \subseteq S$ such that:
:$\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A \cap N_B = \O$
From [[Subsets of Disjoint Sets are Disjoint]] then:
:$U \cap V = \O$
Thus:
:$\exists U, V \in \tau: A \subse... | Equivalence of Definitions of Sets Separated by Neighborhoods | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Sets_Separated_by_Neighborhoods | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Sets_Separated_by_Neighborhoods | [
"Separation Axioms"
] | [
"Definition:Topological Space"
] | [
"Subsets of Disjoint Sets are Disjoint"
] |
proofwiki-16991 | Equivalence of Definitions of Points Separated by Neighborhoods | Let $T = \struct {S, \tau}$ be a topological space.
{{TFAE|def = Points Separated by Neighborhoods}} | Let $x, y \in S$.
From Singleton of Element is Subset:
:$x$ and $y$ are separated as points by neighborhoods {{iff}} the singletons $\set x$ and $\set y$ are separated as sets by neighborhoods.
From Equivalence of Definitions of Sets Separated by Neighborhoods:
:the singletons $\set x$ and $\set y$ are separated as set... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
{{TFAE|def = Points Separated by Neighborhoods}} | Let $x, y \in S$.
From [[Singleton of Element is Subset]]:
:$x$ and $y$ are [[Definition:Points Separated by Neighborhoods|separated as points by neighborhoods]] {{iff}} the [[Definition:Singleton|singletons]] $\set x$ and $\set y$ are [[Definition:Sets Separated by Neighborhoods|separated as sets by neighborhoods]].... | Equivalence of Definitions of Points Separated by Neighborhoods | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Points_Separated_by_Neighborhoods | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Points_Separated_by_Neighborhoods | [
"Separation Axioms"
] | [
"Definition:Topological Space"
] | [
"Singleton of Element is Subset",
"Definition:Separated by Neighborhoods/Points",
"Definition:Singleton",
"Definition:Separated by Neighborhoods/Sets",
"Equivalence of Definitions of Sets Separated by Neighborhoods",
"Definition:Singleton",
"Definition:Separated by Neighborhoods/Sets",
"Definition:Sin... |
proofwiki-16992 | Product Space is T3 iff Factor Spaces are T3/Sufficient Condition | Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$.
Let $\ds T = \struct{S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$.
Let $T$ be a ... | Let $T$ be a $T_3$ space.
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the axiom of choice.
Let $\alpha \in I$.
From Subspace of Product Space is Homeomorphic to Factor Space, $\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to a subspace $T_\alpha$ of $T$.
From $T_3$ property is hereditary, $T_\alpha$ is $T_3$.
Fr... | Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] for $\alpha$ in some [[Definition:Indexing Set|indexing set]] $I$.
Let $\ds T = \struct{S... | Let $T$ be a [[Definition:T3 Space|$T_3$ space]].
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the [[Axiom:Axiom of Choice|axiom of choice]].
Let $\alpha \in I$.
From [[Subspace of Product Space is Homeomorphic to Factor Space]], $\struct {S_\alpha, \tau_\alpha}$ is [[Definition:Homeomorphism (Topological Spaces)... | Product Space is T3 iff Factor Spaces are T3/Sufficient Condition | https://proofwiki.org/wiki/Product_Space_is_T3_iff_Factor_Spaces_are_T3/Sufficient_Condition | https://proofwiki.org/wiki/Product_Space_is_T3_iff_Factor_Spaces_are_T3/Sufficient_Condition | [
"Product Space is T3 iff Factor Spaces are T3"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T3 Space",
"Definition:T3 Space"
] | [
"Definition:T3 Space",
"Axiom:Axiom of Choice",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subspace",
"T3 Property is Hereditary",
"T3 Property is Preserved under Homeomorphism",
"Definition:T3 Space"
] |
proofwiki-16993 | Product Space is T3 iff Factor Spaces are T3/Necessary Condition | Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$.
Let $\ds T = \struct {S, \tau} = \ds \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$.
For each... | For each $\alpha \in I$, let $\struct {S_\alpha, \tau_\alpha}$ be a $T_3$ space.
Let $U$ be open in $T$ and $x \in U$.
From Natural Basis of Product Topology, there exists $\ds U' = \prod_{\alpha \mathop \in I} U'_\alpha$ such that:
:for all $\alpha \in I : U_\alpha \in \tau_\alpha$
:$J = \set {\alpha \in I: U_\alpha \... | Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] for $\alpha$ in some [[Definition:Indexing Set|indexing set]] $I$.
Let $\ds T = \struct {... | For each $\alpha \in I$, let $\struct {S_\alpha, \tau_\alpha}$ be a [[Definition:T3 Space|$T_3$ space]].
Let $U$ be [[Definition:Open Set (Topology)|open]] in $T$ and $x \in U$.
From [[Natural Basis of Product Topology]], there exists $\ds U' = \prod_{\alpha \mathop \in I} U'_\alpha$ such that:
:for all $\alpha \in ... | Product Space is T3 iff Factor Spaces are T3/Necessary Condition | https://proofwiki.org/wiki/Product_Space_is_T3_iff_Factor_Spaces_are_T3/Necessary_Condition | https://proofwiki.org/wiki/Product_Space_is_T3_iff_Factor_Spaces_are_T3/Necessary_Condition | [
"Product Space is T3 iff Factor Spaces are T3"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T3 Space",
"Definition:T3 Space"
] | [
"Definition:T3 Space",
"Definition:Open Set/Topology",
"Natural Basis of Product Topology",
"Definition:Finite Set",
"Definition:T3 Space",
"Definition:Cartesian Product",
"Natural Basis of Product Topology",
"Definition:Open Set/Topology",
"Definition:Cartesian Product",
"Definition:Closed Set/To... |
proofwiki-16994 | Sum from -m to m of Sine of n + alpha of theta over n + alpha | For $0 < \theta < 2 \pi$:
:$\ds \sum_{n \mathop = -m}^m \dfrac {\sin \paren {n + \alpha} \theta} {n + \alpha} = \int_0^\theta \map \cos {\alpha \theta} \dfrac {\sin \paren {m + \frac 1 2} \theta \rd \theta} {\sin \frac 1 2 \theta}$ | We have:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = -m}^m e^{i \paren {n + \alpha} \theta}
| r = \sum_{n \mathop = -m}^m e^{i n \theta} e^{i \alpha \theta}
| c =
}}
{{eqn | r = e^{i \alpha \theta} e^{-i m \theta} \sum_{n \mathop = 0}^{2 m} e^{i n \theta}
| c =
}}
{{eqn | n = 1
| r = e^{i \alp... | For $0 < \theta < 2 \pi$:
:$\ds \sum_{n \mathop = -m}^m \dfrac {\sin \paren {n + \alpha} \theta} {n + \alpha} = \int_0^\theta \map \cos {\alpha \theta} \dfrac {\sin \paren {m + \frac 1 2} \theta \rd \theta} {\sin \frac 1 2 \theta}$ | We have:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = -m}^m e^{i \paren {n + \alpha} \theta}
| r = \sum_{n \mathop = -m}^m e^{i n \theta} e^{i \alpha \theta}
| c =
}}
{{eqn | r = e^{i \alpha \theta} e^{-i m \theta} \sum_{n \mathop = 0}^{2 m} e^{i n \theta}
| c =
}}
{{eqn | n = 1
| r = e^{i \al... | Sum from -m to m of Sine of n + alpha of theta over n + alpha | https://proofwiki.org/wiki/Sum_from_-m_to_m_of_Sine_of_n_+_alpha_of_theta_over_n_+_alpha | https://proofwiki.org/wiki/Sum_from_-m_to_m_of_Sine_of_n_+_alpha_of_theta_over_n_+_alpha | [
"Sine Function"
] | [] | [
"Sum of Geometric Sequence",
"Exponential of Sum",
"Euler's Sine Identity",
"Euler's Formula",
"Primitive of Cosine Function/Corollary",
"Sine of Zero is Zero",
"Linear Combination of Integrals/Definite"
] |
proofwiki-16995 | Sum over Integers of Sine of n + alpha of theta over n + alpha | Let $\alpha \in \R$ be a real number which is specifically not an integer.
For $0 < \theta < 2\pi$:
:$\ds \sum_{n \mathop \in \Z} \dfrac {\map \sin {n + \alpha} \theta} {n + \alpha} = \pi$ | First we establish the following, as they will be needed later.
{{begin-eqn}}
{{eqn | o =
| r = \map \sin {\alpha + n} \theta + \map \sin {\alpha - n} \theta
| c =
}}
{{eqn | r = 2 \map \sin {\dfrac {\paren {\alpha + n} \theta + \paren {\alpha - n} \theta} 2} \map \cos {\dfrac {\paren {\alpha + n} \theta ... | Let $\alpha \in \R$ be a [[Definition:Real Number|real number]] which is specifically not an [[Definition:Integer|integer]].
For $0 < \theta < 2\pi$:
:$\ds \sum_{n \mathop \in \Z} \dfrac {\map \sin {n + \alpha} \theta} {n + \alpha} = \pi$ | First we establish the following, as they will be needed later.
{{begin-eqn}}
{{eqn | o =
| r = \map \sin {\alpha + n} \theta + \map \sin {\alpha - n} \theta
| c =
}}
{{eqn | r = 2 \map \sin {\dfrac {\paren {\alpha + n} \theta + \paren {\alpha - n} \theta} 2} \map \cos {\dfrac {\paren {\alpha + n} \theta... | Sum over Integers of Sine of n + alpha of theta over n + alpha | https://proofwiki.org/wiki/Sum_over_Integers_of_Sine_of_n_+_alpha_of_theta_over_n_+_alpha | https://proofwiki.org/wiki/Sum_over_Integers_of_Sine_of_n_+_alpha_of_theta_over_n_+_alpha | [
"Sine Function"
] | [
"Definition:Real Number",
"Definition:Integer"
] | [
"Prosthaphaeresis Formulas/Sine plus Sine",
"Prosthaphaeresis Formulas/Sine minus Sine",
"Linear Combination of Indexed Summations",
"Mittag-Leffler Expansion for Cosecant Function/Real Domain",
"Zero Derivative implies Constant Function",
"Derivative of Sine Function/Corollary",
"Cosine of Sum",
"Lin... |
proofwiki-16996 | Mittag-Leffler Expansion for Cosecant Function/Real Domain | :$\pi \cosec \pi \alpha = \dfrac 1 \alpha + \ds 2 \sum_{n \mathop \ge 1} \paren {-1}^n \dfrac {\alpha} {\alpha^2 - n^2}$ | From Half-Range Fourier Cosine Series for $\cos \alpha x$ over $\openint 0 \pi$:
:$\ds \cos \alpha x \sim \frac {2 \alpha \sin \alpha \pi} \pi \paren {\frac 1 {2 \alpha^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n x} {\alpha^2 - n^2} }$
Setting $x = 0$:
{{begin-eqn}}
{{eqn | l = \cos 0
| r = \frac... | :$\pi \cosec \pi \alpha = \dfrac 1 \alpha + \ds 2 \sum_{n \mathop \ge 1} \paren {-1}^n \dfrac {\alpha} {\alpha^2 - n^2}$ | From [[Half-Range Fourier Cosine Series/Cosine of Non-Integer Multiple of x over 0 to Pi|Half-Range Fourier Cosine Series for $\cos \alpha x$ over $\openint 0 \pi$]]:
:$\ds \cos \alpha x \sim \frac {2 \alpha \sin \alpha \pi} \pi \paren {\frac 1 {2 \alpha^2} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {\cos n x} ... | Mittag-Leffler Expansion for Cosecant Function/Real Domain | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cosecant_Function/Real_Domain | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Cosecant_Function/Real_Domain | [
"Mittag-Leffler Expansion for Cosecant Function"
] | [] | [
"Half-Range Fourier Cosine Series/Cosine of Non-Integer Multiple of x over 0 to Pi",
"Cosine of Zero is One"
] |
proofwiki-16997 | Sum from -m to m of 1 minus Cosine of n + alpha of theta over n + alpha | For $0 < \theta < 2 \pi$:
:$\ds \sum_{n \mathop = -m}^m \dfrac {1 - \cos \paren {n + \alpha} \theta} {n + \alpha} = \int_0^\theta \map \sin {\alpha u} \dfrac {\sin \paren {m + \frac 1 2} u \rd u} {\sin \frac 1 2 u}$ | We have:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = -m}^m e^{i \paren {n + \alpha} \theta}
| r = \sum_{n \mathop = -m}^m e^{i n \theta} e^{i \alpha \theta}
| c =
}}
{{eqn | r = e^{i \alpha \theta} e^{-i m \theta} \sum_{n \mathop = 0}^{2 m} e^{i n \theta}
| c =
}}
{{eqn | n = 1
| r = e^{i \alp... | For $0 < \theta < 2 \pi$:
:$\ds \sum_{n \mathop = -m}^m \dfrac {1 - \cos \paren {n + \alpha} \theta} {n + \alpha} = \int_0^\theta \map \sin {\alpha u} \dfrac {\sin \paren {m + \frac 1 2} u \rd u} {\sin \frac 1 2 u}$ | We have:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = -m}^m e^{i \paren {n + \alpha} \theta}
| r = \sum_{n \mathop = -m}^m e^{i n \theta} e^{i \alpha \theta}
| c =
}}
{{eqn | r = e^{i \alpha \theta} e^{-i m \theta} \sum_{n \mathop = 0}^{2 m} e^{i n \theta}
| c =
}}
{{eqn | n = 1
| r = e^{i \al... | Sum from -m to m of 1 minus Cosine of n + alpha of theta over n + alpha | https://proofwiki.org/wiki/Sum_from_-m_to_m_of_1_minus_Cosine_of_n_+_alpha_of_theta_over_n_+_alpha | https://proofwiki.org/wiki/Sum_from_-m_to_m_of_1_minus_Cosine_of_n_+_alpha_of_theta_over_n_+_alpha | [
"Cosine Function"
] | [] | [
"Sum of Geometric Sequence",
"Exponential of Sum",
"Euler's Sine Identity",
"Euler's Formula",
"Primitive of Sine Function/Corollary",
"Cosine of Zero is One",
"Linear Combination of Integrals/Definite"
] |
proofwiki-16998 | Sum over Integers of Cosine of n + alpha of theta over n + alpha | Let $\alpha \in \R$ be a real number which is specifically not an integer.
For $0 \le \theta < 2 \pi$:
:$\ds \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \sum_{n \mathop \in \Z} \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}$ | First we establish the following, as they will be needed later.
{{begin-eqn}}
{{eqn | o =
| r = \cos \paren {\alpha + n} \theta + \cos \paren {\alpha - n} \theta
| c =
}}
{{eqn | r = 2 \map \cos {\dfrac {\paren {\alpha + n} \theta + \paren {\alpha - n} \theta} 2} \map \cos {\dfrac {\paren {\alpha + n} \th... | Let $\alpha \in \R$ be a [[Definition:Real Number|real number]] which is specifically not an [[Definition:Integer|integer]].
For $0 \le \theta < 2 \pi$:
:$\ds \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \sum_{n \mathop \in \Z} \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}$ | First we establish the following, as they will be needed later.
{{begin-eqn}}
{{eqn | o =
| r = \cos \paren {\alpha + n} \theta + \cos \paren {\alpha - n} \theta
| c =
}}
{{eqn | r = 2 \map \cos {\dfrac {\paren {\alpha + n} \theta + \paren {\alpha - n} \theta} 2} \map \cos {\dfrac {\paren {\alpha + n} \t... | Sum over Integers of Cosine of n + alpha of theta over n + alpha | https://proofwiki.org/wiki/Sum_over_Integers_of_Cosine_of_n_+_alpha_of_theta_over_n_+_alpha | https://proofwiki.org/wiki/Sum_over_Integers_of_Cosine_of_n_+_alpha_of_theta_over_n_+_alpha | [
"Cosine Function"
] | [
"Definition:Real Number",
"Definition:Integer"
] | [
"Prosthaphaeresis Formulas/Cosine plus Cosine",
"Prosthaphaeresis Formulas/Cosine minus Cosine",
"Linear Combination of Indexed Summations",
"Sine of Zero is Zero",
"Cosine of Zero is One",
"Zero Derivative implies Constant Function",
"Derivative of Cosine Function/Corollary",
"Sine of Sum",
"Linear... |
proofwiki-16999 | Mittag-Leffler Expansion for Real Cotangent Function | :$\ds \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \pi \cot \pi \alpha$ | Corollary of Sum over Integers of $\dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}$
{{ProofWanted|fill in details}}
{{qed}} | :$\ds \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \pi \cot \pi \alpha$ | Corollary of [[Sum over Integers of Cosine of n + alpha of theta over n + alpha|Sum over Integers of $\dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}$]]
{{ProofWanted|fill in details}}
{{qed}} | Mittag-Leffler Expansion for Real Cotangent Function/Proof 2 | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Real_Cotangent_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Real_Cotangent_Function/Proof_2 | [
"Mittag-Leffler Expansion for Cotangent Function"
] | [] | [
"Sum over Integers of Cosine of n + alpha of theta over n + alpha"
] |
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