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-8400 | Mapping Images are Disjoint only if Domains are Disjoint | Let $S$ and $T$ be sets.
Let:
:$f \sqbrk S \cap f \sqbrk T = \O$
where $f \sqbrk S$ denotes the image set of $S$.
Then:
:$S \cap T = \O$ | From Image of Intersection under Mapping:
:$f \sqbrk {S \cap T} \subseteq f \sqbrk S \cap f \sqbrk T$
From Empty Set is Subset of All Sets:
:$f \sqbrk {S \cap T} = \O$
From Image of Subset under Mapping is Subset of Image:
:$S \cap T = \O$
{{qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let:
:$f \sqbrk S \cap f \sqbrk T = \O$
where $f \sqbrk S$ denotes the [[Definition:Image Set of Mapping|image set]] of $S$.
Then:
:$S \cap T = \O$ | From [[Image of Intersection under Mapping]]:
:$f \sqbrk {S \cap T} \subseteq f \sqbrk S \cap f \sqbrk T$
From [[Empty Set is Subset of All Sets]]:
:$f \sqbrk {S \cap T} = \O$
From [[Image of Subset under Mapping is Subset of Image]]:
:$S \cap T = \O$
{{qed}} | Mapping Images are Disjoint only if Domains are Disjoint | https://proofwiki.org/wiki/Mapping_Images_are_Disjoint_only_if_Domains_are_Disjoint | https://proofwiki.org/wiki/Mapping_Images_are_Disjoint_only_if_Domains_are_Disjoint | [
"Images",
"Disjoint Sets"
] | [
"Definition:Set",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Image of Intersection under Mapping",
"Empty Set is Subset of All Sets",
"Image of Subset under Mapping is Subset of Image"
] |
proofwiki-8401 | Image of Relation is Domain of Inverse Relation | Let $\RR \subseteq S \times T$ be a relation.
Let $\RR^{-1} \subseteq T \times S$ be the inverse of $\RR$.
Then:
:$\Img \RR = \Dom {\RR^{-1} }$
That is, the image of a relation is the domain of its inverse. | By definition:
{{begin-eqn}}
{{eqn | l = \Img \RR
| o = :=
| r = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}
}}
{{eqn | l = \Dom {\RR^{-1} }
| o = :=
| r = \set {t \in T: \exists s \in S: \tuple {t, s} \in \RR^{-1} }
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r... | Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]].
Let $\RR^{-1} \subseteq T \times S$ be the [[Definition:Inverse Relation|inverse of $\RR$]].
Then:
:$\Img \RR = \Dom {\RR^{-1} }$
That is, the [[Definition:Image of Relation|image]] of a [[Definition:Relation|relation]] is the [[Definition:Domain... | By definition:
{{begin-eqn}}
{{eqn | l = \Img \RR
| o = :=
| r = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}
}}
{{eqn | l = \Dom {\RR^{-1} }
| o = :=
| r = \set {t \in T: \exists s \in S: \tuple {t, s} \in \RR^{-1} }
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = x
| o = \in
|... | Image of Relation is Domain of Inverse Relation | https://proofwiki.org/wiki/Image_of_Relation_is_Domain_of_Inverse_Relation | https://proofwiki.org/wiki/Image_of_Relation_is_Domain_of_Inverse_Relation | [
"Relation Theory",
"Inverse Relations"
] | [
"Definition:Relation",
"Definition:Inverse Relation",
"Definition:Image (Set Theory)/Relation/Relation",
"Definition:Relation",
"Definition:Domain (Set Theory)/Relation",
"Definition:Inverse Relation"
] | [] |
proofwiki-8402 | Domain of Relation is Image of Inverse Relation | Let $\RR \subseteq S \times T$ be a relation.
Let $\RR^{-1} \subseteq T \times S$ be the inverse of $\RR$.
Then:
:$\Dom \RR = \Img {\RR^{-1} }$
That is, the domain of a relation is the image of its inverse. | By definition:
{{begin-eqn}}
{{eqn | l = \Dom \RR
| o = :=
| r = \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}
}}
{{eqn | l = \Img {\RR^{-1} }
| o = :=
| r = \set {s \in S: \exists T \in T: \tuple {t, s} \in \RR^{-1} }
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r... | Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]].
Let $\RR^{-1} \subseteq T \times S$ be the [[Definition:Inverse Relation|inverse of $\RR$]].
Then:
:$\Dom \RR = \Img {\RR^{-1} }$
That is, the [[Definition:Domain of Relation|domain]] of a [[Definition:Relation|relation]] is the [[Definition:Imag... | By definition:
{{begin-eqn}}
{{eqn | l = \Dom \RR
| o = :=
| r = \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}
}}
{{eqn | l = \Img {\RR^{-1} }
| o = :=
| r = \set {s \in S: \exists T \in T: \tuple {t, s} \in \RR^{-1} }
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = x
| o = \in
|... | Domain of Relation is Image of Inverse Relation | https://proofwiki.org/wiki/Domain_of_Relation_is_Image_of_Inverse_Relation | https://proofwiki.org/wiki/Domain_of_Relation_is_Image_of_Inverse_Relation | [
"Relation Theory",
"Inverse Relations"
] | [
"Definition:Relation",
"Definition:Inverse Relation",
"Definition:Domain (Set Theory)/Relation",
"Definition:Relation",
"Definition:Image (Set Theory)/Relation/Relation",
"Definition:Inverse Relation"
] | [] |
proofwiki-8403 | Element in Preimage of Image under Mapping | Let $f: S \to T$ be a mapping.
Let $f^{-1} \sqbrk {\map f x}$ denote the preimage of $\map f x$ under $f$.
Then:
:$\forall x \in S: x \in f^{-1} \sqbrk {\map f x}$ | A mapping is by definition a left-total relation.
Therefore Preimage of Image under Left-Total Relation is Superset applies:
:$A \subseteq S \implies A \subseteq f^{-1} \sqbrk {f \sqbrk A}$
Thus:
:$\set x \subseteq S \implies \set x \subseteq f^{-1} \sqbrk {f \sqbrk A}$
Hence the result.
{{qed}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^{-1} \sqbrk {\map f x}$ denote the [[Definition:Preimage of Subset under Mapping|preimage]] of $\map f x$ under $f$.
Then:
:$\forall x \in S: x \in f^{-1} \sqbrk {\map f x}$ | A [[Definition:Mapping|mapping]] is by definition a [[Definition:Left-Total Relation|left-total relation]].
Therefore [[Preimage of Image under Left-Total Relation is Superset]] applies:
:$A \subseteq S \implies A \subseteq f^{-1} \sqbrk {f \sqbrk A}$
Thus:
:$\set x \subseteq S \implies \set x \subseteq f^{-1} \sqbrk... | Element in Preimage of Image under Mapping | https://proofwiki.org/wiki/Element_in_Preimage_of_Image_under_Mapping | https://proofwiki.org/wiki/Element_in_Preimage_of_Image_under_Mapping | [
"Preimages under Mappings"
] | [
"Definition:Mapping",
"Definition:Preimage/Mapping/Subset"
] | [
"Definition:Mapping",
"Definition:Left-Total Relation",
"Preimage of Image under Left-Total Relation is Superset"
] |
proofwiki-8404 | Element in Image of Preimage under Mapping | Let $f: S \to T$ be a mapping.
Then:
:$\forall y \in T: \in f \sqbrk {f^{-1} \sqbrk y} = \set y$ | A mapping is by definition a relation.
Therefore {{Corollary|Image of Preimage under Mapping}} applies:
:$B \subseteq \Img S \implies \paren {f \circ f^{-1} } \sqbrk B = B$
Thus:
:$\set y \subseteq T \implies f^{-1} \sqbrk {f \sqbrk {\set y} } = \set y$
Hence the result.
{{qed}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
:$\forall y \in T: \in f \sqbrk {f^{-1} \sqbrk y} = \set y$ | A [[Definition:Mapping|mapping]] is by definition a [[Definition:Relation|relation]].
Therefore {{Corollary|Image of Preimage under Mapping}} applies:
:$B \subseteq \Img S \implies \paren {f \circ f^{-1} } \sqbrk B = B$
Thus:
:$\set y \subseteq T \implies f^{-1} \sqbrk {f \sqbrk {\set y} } = \set y$
Hence the result... | Element in Image of Preimage under Mapping | https://proofwiki.org/wiki/Element_in_Image_of_Preimage_under_Mapping | https://proofwiki.org/wiki/Element_in_Image_of_Preimage_under_Mapping | [
"Mapping Theory"
] | [
"Definition:Mapping"
] | [
"Definition:Mapping",
"Definition:Relation"
] |
proofwiki-8405 | Image of Set Difference under Injection | Let $f: S \to T$ be a mapping.
Let $S_1$ and $S_2$ be subsets of $S$.
Let $S_1 \setminus S_2$ denote the set difference between $S_1$ and $S_2$.
Then:
:$\forall S_1, S_2 \subseteq S: f \sqbrk {S_1} \setminus f \sqbrk {S_2} = f \sqbrk {S_1 \setminus S_2}$
{{iff}} $f$ is an injection. | An injection is a type of one-to-one relation, and therefore also a one-to-many relation.
Therefore One-to-Many Image of Set Difference applies:
:$\RR \sqbrk {S_1} \setminus \RR \sqbrk {S_2} = \RR \sqbrk {S_1 \setminus S_2}$
{{iff}} $\RR$ is one-to-many.
We have that $f$ is a mapping and therefore a many-to-one relatio... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $S_1$ and $S_2$ be [[Definition:Subset|subsets]] of $S$.
Let $S_1 \setminus S_2$ denote the [[Definition:Set Difference|set difference]] between $S_1$ and $S_2$.
Then:
:$\forall S_1, S_2 \subseteq S: f \sqbrk {S_1} \setminus f \sqbrk {S_2} = f \sqbrk {S_1 \s... | An [[Definition:Injection|injection]] is a type of [[Definition:One-to-One Relation|one-to-one relation]], and therefore also a [[Definition:One-to-Many Relation|one-to-many relation]].
Therefore [[One-to-Many Image of Set Difference]] applies:
:$\RR \sqbrk {S_1} \setminus \RR \sqbrk {S_2} = \RR \sqbrk {S_1 \setminus... | Image of Set Difference under Injection | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Injection | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Injection | [
"Set Difference",
"Injections"
] | [
"Definition:Mapping",
"Definition:Subset",
"Definition:Set Difference",
"Definition:Injection"
] | [
"Definition:Injection",
"Definition:One-to-One Relation",
"Definition:One-to-Many Relation",
"One-to-Many Image of Set Difference",
"Definition:One-to-Many Relation",
"Definition:Mapping",
"Definition:Many-to-One Relation",
"Definition:One-to-Many Relation",
"Definition:Injection",
"Definition:Inj... |
proofwiki-8406 | De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union | :$\ds \map \complement {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \map \complement {S_i}$ | {{begin-eqn}}
{{eqn | l = \map \complement {\bigcup_{i \mathop \in I} S_i}
| r = \mathbb U \setminus \paren {\bigcup_{i \mathop \in I} S_i}
| c = {{Defof|Set Complement}}
}}
{{eqn | r = \bigcap_{i \mathop \in I} \paren {\mathbb U \setminus S_i}
| c = De Morgan's Laws for Set Difference: Difference wit... | :$\ds \map \complement {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \map \complement {S_i}$ | {{begin-eqn}}
{{eqn | l = \map \complement {\bigcup_{i \mathop \in I} S_i}
| r = \mathbb U \setminus \paren {\bigcup_{i \mathop \in I} S_i}
| c = {{Defof|Set Complement}}
}}
{{eqn | r = \bigcap_{i \mathop \in I} \paren {\mathbb U \setminus S_i}
| c = [[De Morgan's Laws (Set Theory)/Set Difference/Fami... | De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Family_of_Sets/Complement_of_Union | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Family_of_Sets/Complement_of_Union | [
"De Morgan's Laws",
"Indexed Families"
] | [] | [
"De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Union"
] |
proofwiki-8407 | De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection | :$\ds \map \complement {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map \complement {S_i}$ | {{begin-eqn}}
{{eqn | l = \map \complement {\bigcap_{i \mathop \in I} S_i}
| r = \mathbb U \setminus \paren {\bigcap_{i \mathop \in I} S_i}
| c = {{Defof|Set Complement}}
}}
{{eqn | r = \bigcup_{i \mathop \in I} \paren {\mathbb U \setminus S_i}
| c = De Morgan's Laws for Set Difference: Difference wit... | :$\ds \map \complement {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map \complement {S_i}$ | {{begin-eqn}}
{{eqn | l = \map \complement {\bigcap_{i \mathop \in I} S_i}
| r = \mathbb U \setminus \paren {\bigcap_{i \mathop \in I} S_i}
| c = {{Defof|Set Complement}}
}}
{{eqn | r = \bigcup_{i \mathop \in I} \paren {\mathbb U \setminus S_i}
| c = [[De Morgan's Laws (Set Theory)/Set Difference/Fami... | De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Family_of_Sets/Complement_of_Intersection | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Family_of_Sets/Complement_of_Intersection | [
"De Morgan's Laws",
"Indexed Families"
] | [] | [
"De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Intersection"
] |
proofwiki-8408 | Limit of Sets Exists iff Limit Inferior contains Limit Superior | Let $\Bbb S = \set {E_n : n \in \N}$ be a sequence of sets.
Then $\Bbb S$ converges to a limit {{iff}}:
:$\ds \limsup_{n \mathop \to \infty} E_n \subseteq \liminf_{n \mathop \to \infty}E_n$ | === Sufficient Condition ===
Let $\Bbb S$ converge to a limit.
Then by definition:
:$\ds \limsup_{n \mathop \to \infty} E_n = \liminf_{n \mathop \to \infty} E_n$
and so by definition of set equality:
:$\ds \limsup_{n \mathop \to \infty} E_n \subseteq \liminf_{n \mathop \to \infty}E_n$
{{qed|lemma}} | Let $\Bbb S = \set {E_n : n \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]].
Then $\Bbb S$ [[Definition:Limit of Sets|converges to a limit]] {{iff}}:
:$\ds \limsup_{n \mathop \to \infty} E_n \subseteq \liminf_{n \mathop \to \infty}E_n$ | === Sufficient Condition ===
Let $\Bbb S$ [[Definition:Limit of Sets|converge to a limit]].
Then by definition:
:$\ds \limsup_{n \mathop \to \infty} E_n = \liminf_{n \mathop \to \infty} E_n$
and so by definition of [[Definition:Set Equality/Definition 2|set equality]]:
:$\ds \limsup_{n \mathop \to \infty} E_n \subset... | Limit of Sets Exists iff Limit Inferior contains Limit Superior | https://proofwiki.org/wiki/Limit_of_Sets_Exists_iff_Limit_Inferior_contains_Limit_Superior | https://proofwiki.org/wiki/Limit_of_Sets_Exists_iff_Limit_Inferior_contains_Limit_Superior | [
"Limits Superior of Set Sequences",
"Limits Inferior of Set Sequences"
] | [
"Definition:Sequence",
"Definition:Set",
"Definition:Limit of Sets"
] | [
"Definition:Limit of Sets",
"Definition:Set Equality/Definition 2",
"Definition:Limit of Sets",
"Definition:Set Equality/Definition 2",
"Definition:Limit of Sets"
] |
proofwiki-8409 | Closure of Intersection and Symmetric Difference imply Closure of Set Difference | Let $\RR$ be a system of sets such that for all $A, B \in \RR$:
:$(1): \quad A \cap B \in \RR$
:$(2): \quad A \symdif B \in \RR$
where $\cap$ denotes set intersection and $\symdif$ denotes set symmetric difference.
Then:
:$\forall A, B \in \RR: A \setminus B \in \RR$
where $\setminus$ denotes set difference. | Let $A, B \in \RR$.
From Set Difference as Symmetric Difference with Intersection:
:$A \symdif \paren {A \cap B} = A \setminus B$
By hypothesis:
:$A \cap B \in \RR$
and:
:$A \symdif \paren {A \cap B} \in \RR$
and so:
:$A \setminus B \in \RR$
{{qed}}
Category:Set Intersection
Category:Set Difference
Category:Symmetric D... | Let $\RR$ be a [[Definition:System of Sets|system of sets]] such that for all $A, B \in \RR$:
:$(1): \quad A \cap B \in \RR$
:$(2): \quad A \symdif B \in \RR$
where $\cap$ denotes [[Definition:Set Intersection|set intersection]] and $\symdif$ denotes [[Definition:Symmetric Difference|set symmetric difference]].
Then:... | Let $A, B \in \RR$.
From [[Set Difference as Symmetric Difference with Intersection]]:
:$A \symdif \paren {A \cap B} = A \setminus B$
By hypothesis:
:$A \cap B \in \RR$
and:
:$A \symdif \paren {A \cap B} \in \RR$
and so:
:$A \setminus B \in \RR$
{{qed}}
[[Category:Set Intersection]]
[[Category:Set Difference]]
[[Ca... | Closure of Intersection and Symmetric Difference imply Closure of Set Difference | https://proofwiki.org/wiki/Closure_of_Intersection_and_Symmetric_Difference_imply_Closure_of_Set_Difference | https://proofwiki.org/wiki/Closure_of_Intersection_and_Symmetric_Difference_imply_Closure_of_Set_Difference | [
"Set Intersection",
"Set Difference",
"Symmetric Difference",
"Set Systems"
] | [
"Definition:Set of Sets",
"Definition:Set Intersection",
"Definition:Symmetric Difference",
"Definition:Set Difference"
] | [
"Set Difference as Symmetric Difference with Intersection",
"Category:Set Intersection",
"Category:Set Difference",
"Category:Symmetric Difference",
"Category:Set Systems"
] |
proofwiki-8410 | Closure of Intersection and Symmetric Difference imply Closure of Union | Let $\R R$ be a system of sets such that for all $A, B \in \RR$:
:$(1): \quad A \cap B \in \RR$
:$(2): \quad A \symdif B \in \RR$
where $\cap$ denotes set intersection and $\symdif$ denotes set symmetric difference.
Then:
:$\forall A, B \in \RR: A \cup B \in \RR$
where $\cup$ denotes set union. | Let $A, B \in \RR$.
From Union as Symmetric Difference with Intersection:
:$\paren {A \symdif B} \symdif \paren {A \cap B} = A \cup B$
By hypothesis:
:$A \cap B \in \RR$
and:
:$\paren {A \symdif B} \symdif \paren {A \cap B} \in \RR$
and so:
:$A \cup B \in \RR$
{{qed}}
Category:Set Intersection
Category:Set Union
Catego... | Let $\R R$ be a [[Definition:System of Sets|system of sets]] such that for all $A, B \in \RR$:
:$(1): \quad A \cap B \in \RR$
:$(2): \quad A \symdif B \in \RR$
where $\cap$ denotes [[Definition:Set Intersection|set intersection]] and $\symdif$ denotes [[Definition:Symmetric Difference|set symmetric difference]].
Then... | Let $A, B \in \RR$.
From [[Union as Symmetric Difference with Intersection]]:
:$\paren {A \symdif B} \symdif \paren {A \cap B} = A \cup B$
By hypothesis:
:$A \cap B \in \RR$
and:
:$\paren {A \symdif B} \symdif \paren {A \cap B} \in \RR$
and so:
:$A \cup B \in \RR$
{{qed}}
[[Category:Set Intersection]]
[[Category:Se... | Closure of Intersection and Symmetric Difference imply Closure of Union | https://proofwiki.org/wiki/Closure_of_Intersection_and_Symmetric_Difference_imply_Closure_of_Union | https://proofwiki.org/wiki/Closure_of_Intersection_and_Symmetric_Difference_imply_Closure_of_Union | [
"Set Intersection",
"Set Union",
"Symmetric Difference",
"Set Systems"
] | [
"Definition:Set of Sets",
"Definition:Set Intersection",
"Definition:Symmetric Difference",
"Definition:Set Union"
] | [
"Union as Symmetric Difference with Intersection",
"Category:Set Intersection",
"Category:Set Union",
"Category:Symmetric Difference",
"Category:Set Systems"
] |
proofwiki-8411 | Empty Set and Set form Algebra of Sets | Let $S$ be any non-empty set.
Then $\set {S, \O}$ is (trivially) an algebra of sets, where $S$ is the unit. | From Set Union is Idempotent:
:$S \cup S = S$
and
:$\O \cup \O = \O$
Then from Union with Empty Set:
:$S \cup \O = S$
So $\set {S, \O}$ is closed under union.
From Relative Complement of Empty Set:
:$\relcomp S \O = S$
and from Relative Complement with Self is Empty Set:
:$\relcomp S S = \O$
so $\set {S, \O}$ is closed... | Let $S$ be any [[Definition:Non-Empty Set|non-empty set]].
Then $\set {S, \O}$ is (trivially) an [[Definition:Algebra of Sets|algebra of sets]], where $S$ is the [[Definition:Unit of System of Sets|unit]]. | From [[Set Union is Idempotent]]:
:$S \cup S = S$
and
:$\O \cup \O = \O$
Then from [[Union with Empty Set]]:
:$S \cup \O = S$
So $\set {S, \O}$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Set Union|union]].
From [[Relative Complement of Empty Set]]:
:$\relcomp S \O = S$
and from [[Relat... | Empty Set and Set form Algebra of Sets | https://proofwiki.org/wiki/Empty_Set_and_Set_form_Algebra_of_Sets | https://proofwiki.org/wiki/Empty_Set_and_Set_form_Algebra_of_Sets | [
"Algebras of Sets"
] | [
"Definition:Non-Empty Set",
"Definition:Algebra of Sets",
"Definition:Unit of System of Sets"
] | [
"Set Union is Idempotent",
"Union with Empty Set",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union",
"Relative Complement of Empty Set",
"Relative Complement with Self is Empty Set",
"Definition:Closed under Mapping",
"Definition:Set Complement",
"Definition:Algebr... |
proofwiki-8412 | Closure of Union and Complement imply Closure of Set Difference | Let $\RR$ be a system of sets on a universe $\mathbb U$ such that for all $A, B \in \RR$:
:$(1): \quad A \cup B \in \RR$
:$(2): \quad \map \complement A \in \RR$
where $\cup$ denotes set union and $\complement$ denotes complement (relative to $\mathbb U$).
Then:
:$\forall A, B \in \RR: A \setminus B \in \RR$
where $\se... | Let $A, B \in \RR$.
{{begin-eqn}}
{{eqn | l = A \setminus B
| r = A \cap \map \complement B
| c = Set Difference as Intersection with Complement
}}
{{eqn | r = \map \complement {\map \complement A \cup B}
| c = De Morgan's Laws: Complement of Intersection
}}
{{end-eqn}}
As both set union and complemen... | Let $\RR$ be a [[Definition:System of Sets|system of sets]] on a [[Definition:Universal Set|universe]] $\mathbb U$ such that for all $A, B \in \RR$:
:$(1): \quad A \cup B \in \RR$
:$(2): \quad \map \complement A \in \RR$
where $\cup$ denotes [[Definition:Set Union|set union]] and $\complement$ denotes [[Definition:Set ... | Let $A, B \in \RR$.
{{begin-eqn}}
{{eqn | l = A \setminus B
| r = A \cap \map \complement B
| c = [[Set Difference as Intersection with Complement]]
}}
{{eqn | r = \map \complement {\map \complement A \cup B}
| c = [[De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection|De Morgan's ... | Closure of Union and Complement imply Closure of Set Difference | https://proofwiki.org/wiki/Closure_of_Union_and_Complement_imply_Closure_of_Set_Difference | https://proofwiki.org/wiki/Closure_of_Union_and_Complement_imply_Closure_of_Set_Difference | [
"Set Union",
"Set Difference",
"Set Complement",
"Set Systems"
] | [
"Definition:Set of Sets",
"Definition:Universal Set",
"Definition:Set Union",
"Definition:Set Complement",
"Definition:Set Difference"
] | [
"Set Difference as Intersection with Complement",
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection",
"Definition:Set Union",
"Definition:Set Complement",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-8413 | Complement of Limit Inferior is Limit Superior of Complements | Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of sets.
Then:
:$\ds \map \complement {\liminf_{n \mathop \to \infty} \ E_n} = \limsup_{n \mathop \to \infty} \ \map \complement {E_n}$
where $\liminf$ and $\limsup$ denote the limit inferior and limit superior, respectively. | {{begin-eqn}}
{{eqn | l = \map \complement {\liminf_{n \mathop \to \infty} \ E_n}
| r = \map \complement {\bigcup_{n \mathop = 0}^\infty \bigcap_{i \mathop = n}^\infty E_n}
| c = {{Defof|Limit Inferior of Sequence of Sets|index = 1}}
}}
{{eqn | r = \bigcap_{n \mathop = 0}^\infty \map \complement {\bigcap_{i... | Let $\sequence {E_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]].
Then:
:$\ds \map \complement {\liminf_{n \mathop \to \infty} \ E_n} = \limsup_{n \mathop \to \infty} \ \map \complement {E_n}$
where $\liminf$ and $\limsup$ denote the [[Definition:Limit Inferior of Sequence of... | {{begin-eqn}}
{{eqn | l = \map \complement {\liminf_{n \mathop \to \infty} \ E_n}
| r = \map \complement {\bigcup_{n \mathop = 0}^\infty \bigcap_{i \mathop = n}^\infty E_n}
| c = {{Defof|Limit Inferior of Sequence of Sets|index = 1}}
}}
{{eqn | r = \bigcap_{n \mathop = 0}^\infty \map \complement {\bigcap_{i... | Complement of Limit Inferior is Limit Superior of Complements | https://proofwiki.org/wiki/Complement_of_Limit_Inferior_is_Limit_Superior_of_Complements | https://proofwiki.org/wiki/Complement_of_Limit_Inferior_is_Limit_Superior_of_Complements | [
"Limits Superior of Set Sequences",
"Limits Inferior of Set Sequences"
] | [
"Definition:Sequence",
"Definition:Set",
"Definition:Limit Inferior of Sequence of Sets",
"Definition:Limit Superior of Sequence of Sets"
] | [
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union",
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection"
] |
proofwiki-8414 | Complement of Limit Superior is Limit Inferior of Complements | Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of sets.
Then:
:$\ds \map \complement {\limsup_{n \mathop \to \infty} \ E_n} = \liminf_{n \mathop \to \infty} \ \map \complement {E_n}$
where $\limsup$ and $\liminf$ denote the limit superior and limit inferior, respectively. | {{begin-eqn}}
{{eqn | l = \map \complement {\limsup_{n \mathop \to \infty} \ E_n}
| r = \map \complement {\bigcap_{n \mathop = 0}^\infty \bigcup_{i \mathop = n}^\infty E_n}
| c = {{Defof|Limit Superior of Sequence of Sets|index = 1}}
}}
{{eqn | r = \bigcup_{n \mathop = 0}^\infty \map \complement {\bigcup_{i... | Let $\sequence {E_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]].
Then:
:$\ds \map \complement {\limsup_{n \mathop \to \infty} \ E_n} = \liminf_{n \mathop \to \infty} \ \map \complement {E_n}$
where $\limsup$ and $\liminf$ denote the [[Definition:Limit Superior of Sequence of... | {{begin-eqn}}
{{eqn | l = \map \complement {\limsup_{n \mathop \to \infty} \ E_n}
| r = \map \complement {\bigcap_{n \mathop = 0}^\infty \bigcup_{i \mathop = n}^\infty E_n}
| c = {{Defof|Limit Superior of Sequence of Sets|index = 1}}
}}
{{eqn | r = \bigcup_{n \mathop = 0}^\infty \map \complement {\bigcup_{i... | Complement of Limit Superior is Limit Inferior of Complements | https://proofwiki.org/wiki/Complement_of_Limit_Superior_is_Limit_Inferior_of_Complements | https://proofwiki.org/wiki/Complement_of_Limit_Superior_is_Limit_Inferior_of_Complements | [
"Limits Superior of Set Sequences",
"Limits Inferior of Set Sequences"
] | [
"Definition:Sequence",
"Definition:Set",
"Definition:Limit Superior of Sequence of Sets",
"Definition:Limit Inferior of Sequence of Sets"
] | [
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection",
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union"
] |
proofwiki-8415 | Sigma-Ring is Closed under Countable Intersections | Let $\RR$ be a $\sigma$-ring.
Let $\sequence {A_n}_{n \mathop \in \N} \in \RR$ be a sequence of sets in $\RR$.
Then:
:$\ds \bigcap_{n \mathop = 1}^\infty A_n \in \RR$ | {{begin-eqn}}
{{eqn | q = \forall n \in \N_{>0}
| l = A_1, A_n \in \RR
| o = \leadsto
| r = A_1 \setminus A_n \in \RR
| c = Axiom $(\text {SR} 2)$ for $\sigma$-rings
}}
{{eqn | o = \leadsto
| r = \bigcup_{n \mathop = 2}^\infty \paren {A_1 \setminus A_n} \in \RR
| c = Axiom $(\text {S... | Let $\RR$ be a [[Definition:Sigma-Ring|$\sigma$-ring]].
Let $\sequence {A_n}_{n \mathop \in \N} \in \RR$ be a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]] in $\RR$.
Then:
:$\ds \bigcap_{n \mathop = 1}^\infty A_n \in \RR$ | {{begin-eqn}}
{{eqn | q = \forall n \in \N_{>0}
| l = A_1, A_n \in \RR
| o = \leadsto
| r = A_1 \setminus A_n \in \RR
| c = Axiom $(\text {SR} 2)$ for [[Definition:Sigma-Ring|$\sigma$-rings]]
}}
{{eqn | o = \leadsto
| r = \bigcup_{n \mathop = 2}^\infty \paren {A_1 \setminus A_n} \in \RR
... | Sigma-Ring is Closed under Countable Intersections | https://proofwiki.org/wiki/Sigma-Ring_is_Closed_under_Countable_Intersections | https://proofwiki.org/wiki/Sigma-Ring_is_Closed_under_Countable_Intersections | [
"Sigma-Rings"
] | [
"Definition:Sigma-Ring",
"Definition:Sequence",
"Definition:Set"
] | [
"Definition:Sigma-Ring",
"Definition:Sigma-Ring",
"Definition:Sigma-Ring",
"De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Intersection",
"Set Difference with Set Difference"
] |
proofwiki-8416 | Pappus's Hexagon Theorem | Let $A, B, C$ be a set of collinear points.
Let $a, b, c$ be another set of collinear points.
Let $X, Y, Z$ be the points of intersection of each of the straight lines $Ab$ and $aB$, $Ac$ and $aC$, and $Bc$ and $bC$.
Then $X, Y, Z$ are collinear points. | 300px
The notation has been changed to match the source.
Let $ACE$ be collinear and $BFD$ also be collinear.
Join $ABCDEF$ in order.
The points where opposite sides of the hexagon cut each other are $NLM$:
* $AB$ and $DE$ cross at $L$
* $BC$ and $EF$ cross at $N$
* $CD$ and $FA$ cross at $M$
$NLM$ are to be proved coll... | Let $A, B, C$ be a [[Definition:Set|set]] of [[Definition:Collinear Points|collinear points]].
Let $a, b, c$ be another [[Definition:Set|set]] of [[Definition:Collinear Points|collinear points]].
Let $X, Y, Z$ be the [[Definition:Intersection (Geometry)|points of intersection]] of each of the [[Definition:Straight Li... | [[File:PappusHexagonTheorem-1.png|300px]]
The notation has been changed to match the source.
Let $ACE$ be [[Definition:Collinear Points|collinear]] and $BFD$ also be [[Definition:Collinear Points|collinear]].
Join $ABCDEF$ in order.
The [[Definition:Point|points]] where [[Definition:Opposite Sides|opposite sides]] ... | Pappus's Hexagon Theorem/Proof 1 | https://proofwiki.org/wiki/Pappus's_Hexagon_Theorem | https://proofwiki.org/wiki/Pappus's_Hexagon_Theorem/Proof_1 | [
"Pappus's Hexagon Theorem",
"Pappus's Theorems",
"Euclidean Geometry",
"Projective Geometry"
] | [
"Definition:Set",
"Definition:Collinear/Points",
"Definition:Set",
"Definition:Collinear/Points",
"Definition:Intersection (Geometry)",
"Definition:Line/Straight Line",
"Definition:Collinear/Points"
] | [
"File:PappusHexagonTheorem-1.png",
"Definition:Collinear/Points",
"Definition:Collinear/Points",
"Definition:Point",
"Definition:Polygon/Opposite",
"Definition:Hexagon",
"Definition:Collinear/Points",
"Menelaus's Theorem",
"Definition:Transversal (Geometry)",
"Menelaus's Theorem",
"Menelaus's Th... |
proofwiki-8417 | Plane contains Infinite Number of Lines | A plane contains an infinite number of distinct lines. | A plane contains an infinite number of points.
Not all these points are collinear.
Let $A$, $B$ and $C$ be points in a plane $P$.
From Propositions of Incidence: Line in Plane, any two of these points determine a line.
Consider the lines $AB$, $AC$ and $BC$, all of which are distinct.
Let $X$ be one of the infinite num... | A [[Definition:Plane|plane]] contains an [[Definition:Infinite Set|infinite number]] of [[Definition:Distinct Elements|distinct]] [[Definition:Straight Line|lines]]. | A [[Definition:Plane|plane]] contains an [[Definition:Infinite Set|infinite number]] of [[Definition:Point|points]].
Not all these [[Definition:Point|points]] are [[Definition:Collinear Points|collinear]].
Let $A$, $B$ and $C$ be [[Definition:Point|points]] in a [[Definition:Plane|plane]] $P$.
From [[Axiom:Propositi... | Plane contains Infinite Number of Lines | https://proofwiki.org/wiki/Plane_contains_Infinite_Number_of_Lines | https://proofwiki.org/wiki/Plane_contains_Infinite_Number_of_Lines | [
"Projective Geometry"
] | [
"Definition:Plane Surface",
"Definition:Infinite Set",
"Definition:Distinct/Plural",
"Definition:Line/Straight Line"
] | [
"Definition:Plane Surface",
"Definition:Infinite Set",
"Definition:Point",
"Definition:Point",
"Definition:Collinear/Points",
"Definition:Point",
"Definition:Plane Surface",
"Axiom:Propositions of Incidence/Line in Plane",
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Line/Str... |
proofwiki-8418 | Two Planes have Line in Common | Two distinct planes have exactly one (straight) line in common. | Take two distinct lines in plane $1$.
From Propositions of Incidence: Plane and Line, they each meet plane $2$ in one point each, say at $A$ and $B$.
Thus $A$ and $B$ both lie in both planes.
Thus the line defined by $A$ and $B$ lies in both planes.
{{qed}}
{{Handwaving}} | Two [[Definition:Distinct Elements|distinct]] [[Definition:Plane|planes]] have exactly one [[Definition:Straight Line|(straight) line]] in common. | Take two [[Definition:Distinct Elements|distinct]] [[Definition:Straight Line|lines]] in [[Definition:Plane|plane]] $1$.
From [[Axiom:Propositions of Incidence/Plane and Line|Propositions of Incidence: Plane and Line]], they each meet [[Definition:Plane|plane]] $2$ in one point each, say at $A$ and $B$.
Thus $A$ and ... | Two Planes have Line in Common | https://proofwiki.org/wiki/Two_Planes_have_Line_in_Common | https://proofwiki.org/wiki/Two_Planes_have_Line_in_Common | [
"Projective Geometry"
] | [
"Definition:Distinct/Plural",
"Definition:Plane Surface",
"Definition:Line/Straight Line"
] | [
"Definition:Distinct/Plural",
"Definition:Line/Straight Line",
"Definition:Plane Surface",
"Axiom:Propositions of Incidence/Plane and Line",
"Definition:Plane Surface",
"Definition:Plane Surface",
"Definition:Line/Straight Line",
"Definition:Plane Surface"
] |
proofwiki-8419 | Three Non-Collinear Planes have One Point in Common | Three planes which are not collinear have exactly one point in all three planes. | Let $A$, $B$ and $C$ be the three planes in question.
From Two Planes have Line in Common, $A$ and $B$ share a line, $p$ say.
From Propositions of Incidence: Plane and Line, $p$ meets $C$ in one point.
{{qed}} | Three [[Definition:Plane|planes]] which are not [[Definition:Collinear Planes|collinear]] have exactly one [[Definition:Point|point]] in all three [[Definition:Plane|planes]]. | Let $A$, $B$ and $C$ be the three [[Definition:Plane|planes]] in question.
From [[Two Planes have Line in Common]], $A$ and $B$ share a [[Definition:Straight Line|line]], $p$ say.
From [[Axiom:Propositions of Incidence/Plane and Line|Propositions of Incidence: Plane and Line]], $p$ meets $C$ in one [[Definition:Point... | Three Non-Collinear Planes have One Point in Common | https://proofwiki.org/wiki/Three_Non-Collinear_Planes_have_One_Point_in_Common | https://proofwiki.org/wiki/Three_Non-Collinear_Planes_have_One_Point_in_Common | [
"Projective Geometry"
] | [
"Definition:Plane Surface",
"Definition:Collinear/Planes",
"Definition:Point",
"Definition:Plane Surface"
] | [
"Definition:Plane Surface",
"Two Planes have Line in Common",
"Definition:Line/Straight Line",
"Axiom:Propositions of Incidence/Plane and Line",
"Definition:Point"
] |
proofwiki-8420 | Desargues' Theorem | Let $\triangle ABC$ and $\triangle A'B'C'$ be triangles lying in the same or different planes.
Let the lines $AA'$, $BB'$ and $CC'$ intersect in the point $O$.
Then:
:$BC$ meets $B'C'$ in $L$
:$CA$ meets $C'A'$ in $M$
:$AB$ meets $A'B'$ in $N$
where $L, M, N$ are collinear. | :500px
Let $\triangle ABC$ and $\triangle A'B'C'$ be in different planes $\pi$ and $\pi'$ respectively.
Since $BB'$ and $CC'$ intersect in $O$, it follows that $B$, $B'$, $C$ and $C'$ lie in a plane.
Thus $BC$ must meet $B'C'$ in a point $L$.
By the same argument, $CA$ meets $C'A'$ in a point $M$ and $AB$ meets $A'B'$ ... | Let $\triangle ABC$ and $\triangle A'B'C'$ be [[Definition:Triangle (Geometry)|triangles]] lying in the same or different [[Definition:Plane|planes]].
Let the [[Definition:Straight Line|lines]] $AA'$, $BB'$ and $CC'$ [[Definition:Intersection (Geometry)|intersect]] in the [[Definition:Point|point]] $O$.
Then:
:$BC$ ... | :[[File:DesarguesTheorem.png|500px]]
Let $\triangle ABC$ and $\triangle A'B'C'$ be in different [[Definition:Plane|planes]] $\pi$ and $\pi'$ respectively.
Since $BB'$ and $CC'$ [[Definition:Intersection (Geometry)|intersect]] in $O$, it follows that $B$, $B'$, $C$ and $C'$ lie in a [[Definition:Plane|plane]].
Thus ... | Desargues' Theorem | https://proofwiki.org/wiki/Desargues'_Theorem | https://proofwiki.org/wiki/Desargues'_Theorem | [
"Desargues' Theorem",
"Projective Geometry"
] | [
"Definition:Triangle (Geometry)",
"Definition:Plane Surface",
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Definition:Point",
"Definition:Collinear/Points"
] | [
"File:DesarguesTheorem.png",
"Definition:Plane Surface",
"Definition:Intersection (Geometry)",
"Definition:Plane Surface",
"Definition:Point",
"Definition:Point",
"Definition:Point",
"Definition:Point",
"Definition:Plane Surface",
"Two Planes have Line in Common",
"Definition:Collinear/Points",
... |
proofwiki-8421 | Union of Mappings which Agree is Mapping | Let $A, B, Y$ be sets.
Let $f: A \to Y$ and $g: B \to Y$ be mappings.
Let $X = A \cup B$.
Let $f$ and $g$ agree on $A \cap B$.
Then $f \cup g: X \to Y$ is a mapping. | By definition, $f \cup g$ is a relation whose domain is $X = A \cup B$.
Let $\tuple {x, y_1} \in f \cup g$ and $\tuple {x, y_2} \in f \cup g$.
At least one of the following must be true:
:$(1): \quad \tuple {x, y_1} \in f, \tuple {x, y_2} \in f$
:$(2): \quad \tuple {x, y_1} \in g, \tuple {x, y_2} \in g$
:$(3): \quad \t... | Let $A, B, Y$ be [[Definition:Set|sets]].
Let $f: A \to Y$ and $g: B \to Y$ be [[Definition:Mapping|mappings]].
Let $X = A \cup B$.
Let $f$ and $g$ [[Definition:Agreement of Mappings|agree]] on $A \cap B$.
Then $f \cup g: X \to Y$ is a [[Definition:Mapping|mapping]]. | By definition, $f \cup g$ is a [[Definition:Relation|relation]] whose [[Definition:Domain of Relation|domain]] is $X = A \cup B$.
Let $\tuple {x, y_1} \in f \cup g$ and $\tuple {x, y_2} \in f \cup g$.
At least one of the following must be true:
:$(1): \quad \tuple {x, y_1} \in f, \tuple {x, y_2} \in f$
:$(2): \quad \... | Union of Mappings which Agree is Mapping | https://proofwiki.org/wiki/Union_of_Mappings_which_Agree_is_Mapping | https://proofwiki.org/wiki/Union_of_Mappings_which_Agree_is_Mapping | [
"Mapping Theory",
"Set Union"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Agreement/Mappings",
"Definition:Mapping"
] | [
"Definition:Relation",
"Definition:Domain (Set Theory)/Relation",
"Definition:Mapping",
"Definition:By Hypothesis",
"Definition:Mapping"
] |
proofwiki-8422 | Domain of Composite Mapping | Let $S_1, S_2, S_3$ be sets.
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings.
Let $f_2 \circ f_1: S_1 \to S_3$ be the composite mapping of $f_1$ and $f_2$.
Then:
:$\Dom {f_1} = \Dom {f_2 \circ f_1}$
where $\Dom {f_1}$ denotes the domain of $f_1$. | By definition of composition of mappings:
:$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \tuple {x, y} \in f_1 \land \tuple {y, z} \in f_2}$
{{explain|Not quite what that definition says any more}}
Let $x \in \Dom {f_2 \circ f_1}$.
Then:
:$\exists z \in S_3: \tuple {x, z} \in S_1 \times S... | Let $S_1, S_2, S_3$ be [[Definition:Set|sets]].
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be [[Definition:Mapping|mappings]].
Let $f_2 \circ f_1: S_1 \to S_3$ be the [[Definition:Composition of Mappings|composite mapping]] of $f_1$ and $f_2$.
Then:
:$\Dom {f_1} = \Dom {f_2 \circ f_1}$
where $\Dom {f_1}$ denotes... | By definition of [[Definition:Composition of Mappings|composition of mappings]]:
:$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \tuple {x, y} \in f_1 \land \tuple {y, z} \in f_2}$
{{explain|Not quite what that definition says any more}}
Let $x \in \Dom {f_2 \circ f_1}$.
Then:
:$\exists... | Domain of Composite Mapping | https://proofwiki.org/wiki/Domain_of_Composite_Mapping | https://proofwiki.org/wiki/Domain_of_Composite_Mapping | [
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Domain (Set Theory)/Mapping"
] | [
"Definition:Composition of Mappings",
"Definition:Subset",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Subset",
"Definition:Set Equality/Definition 2"
] |
proofwiki-8423 | Identity Mapping is Idempotent | Let $S$ be a set.
Let $I_S: S \to S$ be the identity mapping on $S$.
Then $I_S$ is idempotent:
:$I_S \circ I_S = I_S$ | From Identity Mapping is Left Identity:
:$I_S \circ f = f$
for all mappings $f: S \to S$.
From Identity Mapping is Right Identity:
:$f \circ I_S = f$
for all mappings $f: S \to S$.
Substituting $I_S$ for $f$ in either one and the result follows.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $I_S: S \to S$ be the [[Definition:Identity Mapping|identity mapping]] on $S$.
Then $I_S$ is [[Definition:Idempotent Mapping|idempotent]]:
:$I_S \circ I_S = I_S$ | From [[Identity Mapping is Left Identity]]:
:$I_S \circ f = f$
for all [[Definition:Mapping|mappings]] $f: S \to S$.
From [[Identity Mapping is Right Identity]]:
:$f \circ I_S = f$
for all [[Definition:Mapping|mappings]] $f: S \to S$.
Substituting $I_S$ for $f$ in either one and the result follows.
{{qed}} | Identity Mapping is Idempotent | https://proofwiki.org/wiki/Identity_Mapping_is_Idempotent | https://proofwiki.org/wiki/Identity_Mapping_is_Idempotent | [
"Identity Mappings"
] | [
"Definition:Set",
"Definition:Identity Mapping",
"Definition:Idempotence/Mapping"
] | [
"Identity Mapping is Left Identity",
"Definition:Mapping",
"Identity Mapping is Right Identity",
"Definition:Mapping"
] |
proofwiki-8424 | Union of Functions Theorem | Let $X$ be a set.
Let $\sequence {X_i: i \in \N}$ be an exhausting sequence of sets on $X$.
For each $i \in \N$, let $g_i: X_i \to Y$ be a mapping such that:
:$g_{i + 1} \restriction X_i = g_i$
where $g_{i + 1} \restriction X_i$ denotes the restriction of $g_{i + 1}$ to $g_i$.
Then:
:$\ds \bigcup \set {g_i: i \in \N}$
... | By definition, $\ds g = \bigcup \set {g_i: i \in \N}$ is a relation whose domain is $X$.
{{AimForCont}} $g$ is not a mapping.
Then for some $x \in X$ and $i, h \in \N$:
:$(1): \quad x \in X_i, \map {g_i} x \ne \map {g_{i + h} } x$
Let $k \in \N$ be the smallest such that:
:$\map {g_i} x \ne g_{i + k}$
where $x$ and $i$... | Let $X$ be a [[Definition:Set|set]].
Let $\sequence {X_i: i \in \N}$ be an [[Definition:Exhausting Sequence of Sets|exhausting sequence of sets]] on $X$.
For each $i \in \N$, let $g_i: X_i \to Y$ be a [[Definition:Mapping|mapping]] such that:
:$g_{i + 1} \restriction X_i = g_i$
where $g_{i + 1} \restriction X_i$ deno... | By definition, $\ds g = \bigcup \set {g_i: i \in \N}$ is a [[Definition:Relation|relation]] whose [[Definition:Domain of Relation|domain]] is $X$.
{{AimForCont}} $g$ is not a [[Definition:Mapping|mapping]].
Then for some $x \in X$ and $i, h \in \N$:
:$(1): \quad x \in X_i, \map {g_i} x \ne \map {g_{i + h} } x$
Let $... | Union of Functions Theorem | https://proofwiki.org/wiki/Union_of_Functions_Theorem | https://proofwiki.org/wiki/Union_of_Functions_Theorem | [
"Mapping Theory",
"Named Theorems"
] | [
"Definition:Set",
"Definition:Exhausting Sequence of Sets",
"Definition:Mapping",
"Definition:Restriction/Mapping",
"Definition:Mapping"
] | [
"Definition:Relation",
"Definition:Domain (Set Theory)/Relation",
"Definition:Mapping",
"Proof by Contradiction",
"Definition:Mapping",
"Definition:False",
"Definition:Mapping"
] |
proofwiki-8425 | Inductive Definition of Sequence | Let $X$ be a set.
Let $h \in \N$.
Let $a_i \in X$ for all $i \in \set {1, 2, \ldots, h}$.
Let $S$ be the set of all finite sequences whose codomains are in $X$.
Let $G: S \to X$ be a mapping.
Then there is a unique sequence $f$ whose codomain is in $X$ such that:
:$f_i = \begin{cases} a_i & : i \in \set {1, 2, \ldots, ... | {{finish|tedious}} | Let $X$ be a [[Definition:Set|set]].
Let $h \in \N$.
Let $a_i \in X$ for all $i \in \set {1, 2, \ldots, h}$.
Let $S$ be the [[Definition:Set|set]] of all [[Definition:Finite Sequence|finite sequences]] whose [[Definition:Codomain|codomains]] are in $X$.
Let $G: S \to X$ be a [[Definition:Mapping|mapping]].
Then t... | {{finish|tedious}} | Inductive Definition of Sequence | https://proofwiki.org/wiki/Inductive_Definition_of_Sequence | https://proofwiki.org/wiki/Inductive_Definition_of_Sequence | [
"Mapping Theory"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Finite Sequence",
"Definition:Codomain",
"Definition:Mapping",
"Definition:Unique",
"Definition:Sequence",
"Definition:Codomain"
] | [] |
proofwiki-8426 | Model of Root of Propositional Tableau is Model of Branch | Let $\struct {T, \mathbf H, \Phi}$ be a propositional tableau.
Let $v: \LL_0 \to \set {\T, \F}$ be a boolean interpretation such that:
:$v \models_{\mathrm{BI}} \mathbf H$
that is, such that $v$ is a model for the root $\mathbf H$ of $T$.
Then there exists a branch $\Gamma$ of $T$ such that:
:$v \models_{\mathrm{BI}} \... | We will find it convenient to reason with the constructive definition of a propositional tableau.
First, let us reduce to the case of a finite propositional tableau.
Suppose the result were to hold for these finite tableaus.
Let $T$ be an infinite propositional tableau.
Suppose that no finite branch $\Gamma$ of $T$ sat... | Let $\struct {T, \mathbf H, \Phi}$ be a [[Definition:Propositional Tableau|propositional tableau]].
Let $v: \LL_0 \to \set {\T, \F}$ be a [[Definition:Boolean Interpretation|boolean interpretation]] such that:
:$v \models_{\mathrm{BI}} \mathbf H$
that is, such that $v$ is a [[Definition:Model (Boolean Interpretation... | We will find it convenient to reason with the [[Definition:Propositional Tableau/Construction|constructive definition of a propositional tableau]].
First, let us reduce to the case of a finite [[Definition:Propositional Tableau|propositional tableau]].
Suppose the result were to hold for these finite [[Definition:Pr... | Model of Root of Propositional Tableau is Model of Branch | https://proofwiki.org/wiki/Model_of_Root_of_Propositional_Tableau_is_Model_of_Branch | https://proofwiki.org/wiki/Model_of_Root_of_Propositional_Tableau_is_Model_of_Branch | [
"Propositional Tableaux"
] | [
"Definition:Propositional Tableau",
"Definition:Boolean Interpretation",
"Definition:Model (Boolean Interpretations)",
"Definition:Labeled Tree for Propositional Logic/Hypothesis Set",
"Definition:Rooted Tree/Branch",
"Definition:Image (Set Theory)/Mapping/Subset"
] | [
"Definition:Propositional Tableau/Construction",
"Definition:Propositional Tableau",
"Definition:Propositional Tableau",
"Definition:Propositional Tableau",
"Definition:Rooted Tree/Branch/Finite",
"Definition:Exhausting Sequence of Sets",
"Definition:Labeled Tree for Propositional Logic",
"Definition:... |
proofwiki-8427 | Antireflexive and Transitive Relation is Antisymmetric | Let $\RR \subseteq S \times S$ be a relation which is not null.
Let $\RR$ be antireflexive and transitive.
Then $\RR$ is also antisymmetric. | Let $\RR \subseteq S \times S$ be antireflexive and transitive.
From Antireflexive and Transitive Relation is Asymmetric it follows that $\RR$ is asymmetric.
The result follows from Asymmetric Relation is Antisymmetric.
{{qed}} | Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] which is not [[Definition:Null Relation|null]].
Let $\RR$ be [[Definition:Antireflexive Relation|antireflexive]] and [[Definition:Transitive Relation|transitive]].
Then $\RR$ is also [[Definition:Antisymmetric Relation|antisymmetric]]. | Let $\RR \subseteq S \times S$ be [[Definition:Antireflexive Relation|antireflexive]] and [[Definition:Transitive Relation|transitive]].
From [[Antireflexive and Transitive Relation is Asymmetric]] it follows that $\RR$ is [[Definition:Asymmetric Relation|asymmetric]].
The result follows from [[Asymmetric Relation is... | Antireflexive and Transitive Relation is Antisymmetric | https://proofwiki.org/wiki/Antireflexive_and_Transitive_Relation_is_Antisymmetric | https://proofwiki.org/wiki/Antireflexive_and_Transitive_Relation_is_Antisymmetric | [
"Antireflexive Relations",
"Antisymmetric Relations",
"Transitive Relations"
] | [
"Definition:Relation",
"Definition:Null Relation",
"Definition:Antireflexive Relation",
"Definition:Transitive Relation",
"Definition:Antisymmetric Relation"
] | [
"Definition:Antireflexive Relation",
"Definition:Transitive Relation",
"Antireflexive and Transitive Relation is Asymmetric",
"Definition:Asymmetric Relation",
"Asymmetric Relation is Antisymmetric"
] |
proofwiki-8428 | Tableau Confutation implies Unsatisfiable | Let $\mathbf H$ be a collection of WFFs of propositional logic.
Suppose there exists a tableau confutation of $\mathbf H$.
Then $\mathbf H$ is unsatisfiable for boolean interpretations. | Let $\struct {T, \mathbf H, \Phi}$ be a tableau confutation of $\mathbf H$.
Suppose that $v$ were a boolean interpretation model for $\mathbf H$, that is:
:$v \models_{\mathrm{BI}} \mathbf H$
By Model of Root of Propositional Tableau is Model of Branch, it follows that:
:$v \models_{\mathrm{BI}} \Phi \sqbrk \Gamma$
for... | Let $\mathbf H$ be a collection of [[Definition:WFF of Propositional Logic|WFFs of propositional logic]].
Suppose there exists a [[Definition:Tableau Confutation|tableau confutation]] of $\mathbf H$.
Then $\mathbf H$ is [[Definition:Unsatisfiable|unsatisfiable]] for [[Definition:Boolean Interpretation|boolean interp... | Let $\struct {T, \mathbf H, \Phi}$ be a [[Definition:Tableau Confutation|tableau confutation]] of $\mathbf H$.
Suppose that $v$ were a [[Definition:Boolean Interpretation|boolean interpretation]] [[Definition:Model (Boolean Interpretations)|model]] for $\mathbf H$, that is:
:$v \models_{\mathrm{BI}} \mathbf H$
By [[... | Tableau Confutation implies Unsatisfiable | https://proofwiki.org/wiki/Tableau_Confutation_implies_Unsatisfiable | https://proofwiki.org/wiki/Tableau_Confutation_implies_Unsatisfiable | [
"Propositional Tableaux"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Tableau Confutation",
"Definition:Unsatisfiable",
"Definition:Boolean Interpretation"
] | [
"Definition:Tableau Confutation",
"Definition:Boolean Interpretation",
"Definition:Model (Boolean Interpretations)",
"Model of Root of Propositional Tableau is Model of Branch",
"Definition:Rooted Tree/Branch",
"Definition:Tableau Confutation",
"Definition:Language of Propositional Logic/Formal Grammar/... |
proofwiki-8429 | Finished Branch Lemma | Let $\Gamma$ be a finished branch of a propositional tableau $\struct {T, \mathbf H, \Phi}$.
Let $v$ be a boolean interpretation such that:
:$v \models_{\mathrm{BI}} \mathbf A$ for every basic WFF $\mathbf A$ that occurs along $\Gamma$.
Then:
:$v \models_{\mathrm{BI} } \Phi \sqbrk \Gamma$
where $\Phi \sqbrk \Gamma$ is ... | The proof appeals to the Principle of Structural Induction, applied to the statement:
:If $\mathbf C$ occurs along $\Gamma$, then $v \models_{\mathrm{BI}} \mathbf C$.
When $\mathbf C$ is basic, the result holds per assumption.
Suppose $\mathbf C$ is not basic.
It is seen that one of the propositional tableau constructi... | Let $\Gamma$ be a [[Definition:Finished Branch of Propositional Tableau|finished branch]] of a [[Definition:Propositional Tableau|propositional tableau]] $\struct {T, \mathbf H, \Phi}$.
Let $v$ be a [[Definition:Boolean Interpretation|boolean interpretation]] such that:
:$v \models_{\mathrm{BI}} \mathbf A$ for every ... | The proof appeals to the [[Principle of Structural Induction]], applied to the [[Definition:Statement|statement]]:
:If $\mathbf C$ [[Definition:Occurrence along Branch|occurs]] along $\Gamma$, then $v \models_{\mathrm{BI}} \mathbf C$.
When $\mathbf C$ is [[Definition:Basic WFF|basic]], the result holds per assumptio... | Finished Branch Lemma | https://proofwiki.org/wiki/Finished_Branch_Lemma | https://proofwiki.org/wiki/Finished_Branch_Lemma | [
"Finished Branch Lemma",
"Propositional Tableaux"
] | [
"Definition:Finished Branch of Propositional Tableau",
"Definition:Propositional Tableau",
"Definition:Boolean Interpretation",
"Definition:Literal",
"Definition:Labeled Tree for Propositional Logic/Along a Branch",
"Definition:Image (Set Theory)/Mapping/Subset"
] | [
"Principle of Structural Induction",
"Definition:Statement",
"Definition:Labeled Tree for Propositional Logic/Along a Branch",
"Definition:Literal",
"Definition:Literal",
"Definition:Propositional Tableau/Construction",
"Definition:Used WFF",
"Double Negation/Formulation 1/Proof by Truth Table",
"De... |
proofwiki-8430 | Diagonal Relation is Serial | Let $S$ be a set.
Let $\Delta_S$ be the diagonal relation on $S$.
Then $\Delta_S$ is a serial relation. | By Diagonal Relation is Equivalence it follows {{afortiori}} that $\Delta_S$ is reflexive.
The result follows from Reflexive Relation is Serial.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\Delta_S$ be the [[Definition:Diagonal Relation|diagonal relation]] on $S$.
Then $\Delta_S$ is a [[Definition:Serial Relation|serial relation]]. | By [[Diagonal Relation is Equivalence]] it follows {{afortiori}} that $\Delta_S$ is [[Definition:Reflexive Relation|reflexive]].
The result follows from [[Reflexive Relation is Serial]].
{{qed}} | Diagonal Relation is Serial | https://proofwiki.org/wiki/Diagonal_Relation_is_Serial | https://proofwiki.org/wiki/Diagonal_Relation_is_Serial | [
"Serial Relations"
] | [
"Definition:Set",
"Definition:Diagonal Relation",
"Definition:Serial Relation"
] | [
"Diagonal Relation is Equivalence",
"Definition:Reflexive Relation",
"Reflexive Relation is Serial"
] |
proofwiki-8431 | Serial Relation is not Null | Let $S$ be a set such that $S \ne \O$.
Let $\RR$ be a serial relation on $S$.
Then $\RR$ is not a null relation. | As $S$ is non-empty set:
:$\exists x: x \in S$
As $\RR$ be a serial relation on $S$:
:$\exists y \in S: \tuple {x, y} \in \RR$
That is:
:$\RR \ne \O$
Hence the result by definition of null relation.
{{qed}} | Let $S$ be a [[Definition:Set|set]] such that $S \ne \O$.
Let $\RR$ be a [[Definition:Serial Relation|serial relation]] on $S$.
Then $\RR$ is not a [[Definition:Null Relation|null relation]]. | As $S$ is [[Definition:Non-Empty Set|non-empty set]]:
:$\exists x: x \in S$
As $\RR$ be a [[Definition:Serial Relation|serial relation]] on $S$:
:$\exists y \in S: \tuple {x, y} \in \RR$
That is:
:$\RR \ne \O$
Hence the result by definition of [[Definition:Null Relation|null relation]].
{{qed}} | Serial Relation is not Null | https://proofwiki.org/wiki/Serial_Relation_is_not_Null | https://proofwiki.org/wiki/Serial_Relation_is_not_Null | [
"Serial Relations",
"Null Relation"
] | [
"Definition:Set",
"Definition:Serial Relation",
"Definition:Null Relation"
] | [
"Definition:Non-Empty Set",
"Definition:Serial Relation",
"Definition:Null Relation"
] |
proofwiki-8432 | Transitive Relation is Antireflexive iff Asymmetric | Let $\RR \subseteq S \times S$ be a relation which is not null.
Let $\RR$ be transitive.
Then $\RR$ is antireflexive {{iff}} $\RR$ is asymmetric. | === Necessary Condition ===
Let $\RR \subseteq S \times S$ be antireflexive.
Then by Antireflexive and Transitive Relation is Asymmetric it follows that $\RR$ is asymmetric.
{{qed|lemma}} | Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] which is not [[Definition:Null Relation|null]].
Let $\RR$ be [[Definition:Transitive Relation|transitive]].
Then $\RR$ is [[Definition:Antireflexive Relation|antireflexive]] {{iff}} $\RR$ is [[Definition:Asymmetric Relation|asymmetric]]. | === Necessary Condition ===
Let $\RR \subseteq S \times S$ be [[Definition:Antireflexive Relation|antireflexive]].
Then by [[Antireflexive and Transitive Relation is Asymmetric]] it follows that $\RR$ is [[Definition:Asymmetric Relation|asymmetric]].
{{qed|lemma}} | Transitive Relation is Antireflexive iff Asymmetric | https://proofwiki.org/wiki/Transitive_Relation_is_Antireflexive_iff_Asymmetric | https://proofwiki.org/wiki/Transitive_Relation_is_Antireflexive_iff_Asymmetric | [
"Antireflexive Relations",
"Asymmetric Relations",
"Transitive Relations"
] | [
"Definition:Relation",
"Definition:Null Relation",
"Definition:Transitive Relation",
"Definition:Antireflexive Relation",
"Definition:Asymmetric Relation"
] | [
"Definition:Antireflexive Relation",
"Antireflexive and Transitive Relation is Asymmetric",
"Definition:Asymmetric Relation",
"Definition:Asymmetric Relation",
"Definition:Antireflexive Relation"
] |
proofwiki-8433 | Symmetric and Antisymmetric Relation is Transitive | Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a relation in $S$ which is both symmetric and antisymmetric.
Then $\RR$ is transitive. | Let $\tuple {x, y}, \tuple {y, z} \in \RR$.
By Relation is Symmetric and Antisymmetric iff Coreflexive:
:$x = y, y = z$
and so trivially:
:$\tuple {x, z} = \tuple {x, x} \in \RR$
Thus $\RR$ is transitive.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] in $S$ which is both [[Definition:Symmetric Relation|symmetric]] and [[Definition:Antisymmetric Relation|antisymmetric]].
Then $\RR$ is [[Definition:Transitive Relation|transitive]]. | Let $\tuple {x, y}, \tuple {y, z} \in \RR$.
By [[Relation is Symmetric and Antisymmetric iff Coreflexive]]:
:$x = y, y = z$
and so trivially:
:$\tuple {x, z} = \tuple {x, x} \in \RR$
Thus $\RR$ is [[Definition:Transitive Relation|transitive]].
{{qed}} | Symmetric and Antisymmetric Relation is Transitive | https://proofwiki.org/wiki/Symmetric_and_Antisymmetric_Relation_is_Transitive | https://proofwiki.org/wiki/Symmetric_and_Antisymmetric_Relation_is_Transitive | [
"Symmetric Relations",
"Antisymmetric Relations",
"Transitive Relations"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Symmetric Relation",
"Definition:Antisymmetric Relation",
"Definition:Transitive Relation"
] | [
"Relation is Symmetric and Antisymmetric iff Coreflexive",
"Definition:Transitive Relation"
] |
proofwiki-8434 | Relation is Reflexive Symmetric and Antisymmetric iff Diagonal Relation | Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a relation in $S$.
Then:
:$\RR$ is reflexive, symmetric and antisymmetric
{{iff}}:
:$\RR$ is the diagonal relation $\Delta_S$. | === Necessary Condition ===
Let $\RR$ is reflexive, symmetric and antisymmetric.
By definition of reflexive:
:$\Delta_S \subseteq \RR$
From Relation is Symmetric and Antisymmetric iff Coreflexive:
:$\RR \subseteq \Delta_S$
By definition of set equality:
:$\RR = \Delta_S$
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] in $S$.
Then:
:$\RR$ is [[Definition:Reflexive Relation|reflexive]], [[Definition:Symmetric Relation|symmetric]] and [[Definition:Antisymmetric Relation|antisymmetric]]
{{iff}}:
:$\RR$ is the [[Definition:Diagon... | === Necessary Condition ===
Let $\RR$ is [[Definition:Reflexive Relation|reflexive]], [[Definition:Symmetric Relation|symmetric]] and [[Definition:Antisymmetric Relation|antisymmetric]].
By definition of [[Definition:Reflexive Relation/Definition 2|reflexive]]:
:$\Delta_S \subseteq \RR$
From [[Relation is Symmetric ... | Relation is Reflexive Symmetric and Antisymmetric iff Diagonal Relation | https://proofwiki.org/wiki/Relation_is_Reflexive_Symmetric_and_Antisymmetric_iff_Diagonal_Relation | https://proofwiki.org/wiki/Relation_is_Reflexive_Symmetric_and_Antisymmetric_iff_Diagonal_Relation | [
"Reflexive Relations",
"Symmetric Relations",
"Antisymmetric Relations",
"Diagonal Relation"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Antisymmetric Relation",
"Definition:Diagonal Relation"
] | [
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Antisymmetric Relation",
"Definition:Reflexive Relation/Definition 2",
"Relation is Symmetric and Antisymmetric iff Coreflexive",
"Definition:Set Equality/Definition 2",
"Definition:Reflexive Relation",
"Relation is Symmetri... |
proofwiki-8435 | Relation is Reflexive and Coreflexive iff Diagonal | Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a relation on $S$.
Then $\RR$ is reflexive and coreflexive {{iff}}:
: $\RR = \Delta_S$
where $\Delta_S$ is the diagonal relation. | === Necessary Condition ===
Let $\RR \subseteq S \times S$ be reflexive and coreflexive.
Then:
{{begin-eqn}}
{{eqn | l = \RR
| o = \supseteq
| r = \Delta_S
| c = {{Defof|Reflexive Relation|index = 2}}
}}
{{eqn | l = \RR
| o = \subseteq
| r = \Delta_S
| c = {{Defof|Coreflexive Relatio... | Let $S$ be a [[Definition:Set|set]].
Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] on $S$.
Then $\RR$ is [[Definition:Reflexive Relation|reflexive]] and [[Definition:Coreflexive Relation|coreflexive]] {{iff}}:
: $\RR = \Delta_S$
where $\Delta_S$ is the [[Definition:Diagonal Relation|diagonal re... | === Necessary Condition ===
Let $\RR \subseteq S \times S$ be [[Definition:Reflexive Relation|reflexive]] and [[Definition:Coreflexive Relation|coreflexive]].
Then:
{{begin-eqn}}
{{eqn | l = \RR
| o = \supseteq
| r = \Delta_S
| c = {{Defof|Reflexive Relation|index = 2}}
}}
{{eqn | l = \RR
| o ... | Relation is Reflexive and Coreflexive iff Diagonal | https://proofwiki.org/wiki/Relation_is_Reflexive_and_Coreflexive_iff_Diagonal | https://proofwiki.org/wiki/Relation_is_Reflexive_and_Coreflexive_iff_Diagonal | [
"Reflexive Relations",
"Coreflexive Relations"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Reflexive Relation",
"Definition:Coreflexive Relation",
"Definition:Diagonal Relation"
] | [
"Definition:Reflexive Relation",
"Definition:Coreflexive Relation"
] |
proofwiki-8436 | Symmetric Preordering is Equivalence Relation | Let $\RR \subseteq S \times S$ be a preordering on a set $S$.
Let $\RR$ also be symmetric.
Then $\RR$ is an equivalence relation on $S$. | By definition, a preordering on $S$ is a relation on $S$ which is:
:$(1): \quad$ reflexive
and:
:$(2): \quad$ transitive.
Thus $\RR$ is a relation on $S$ which is reflexive, transitive and symmetric.
Thus by definition $\RR$ is an equivalence relation on $S$.
{{qed}} | Let $\RR \subseteq S \times S$ be a [[Definition:Preordering|preordering]] on a [[Definition:Set|set]] $S$.
Let $\RR$ also be [[Definition:Symmetric Relation|symmetric]].
Then $\RR$ is an [[Definition:Equivalence Relation|equivalence relation]] on $S$. | By definition, a [[Definition:Preordering|preordering]] on $S$ is a [[Definition:Endorelation|relation]] on $S$ which is:
:$(1): \quad$ [[Definition:Reflexive Relation|reflexive]]
and:
:$(2): \quad$ [[Definition:Transitive Relation|transitive]].
Thus $\RR$ is a [[Definition:Endorelation|relation]] on $S$ which is [[D... | Symmetric Preordering is Equivalence Relation | https://proofwiki.org/wiki/Symmetric_Preordering_is_Equivalence_Relation | https://proofwiki.org/wiki/Symmetric_Preordering_is_Equivalence_Relation | [
"Preorder Theory",
"Equivalence Relations",
"Symmetric Relations"
] | [
"Definition:Preordering",
"Definition:Set",
"Definition:Symmetric Relation",
"Definition:Equivalence Relation"
] | [
"Definition:Preordering",
"Definition:Endorelation",
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Definition:Endorelation",
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Definition:Symmetric Relation",
"Definition:Equivalence Relation"
] |
proofwiki-8437 | Antisymmetric Preordering is Ordering | Let $\RR \subseteq S \times S$ be a preordering on a set $S$.
Let $\RR$ also be antisymmetric.
Then $\RR$ is an ordering on $S$. | By definition, a preordering on $S$ is a relation on $S$ which is:
:$(1): \quad$ reflexive
and:
:$(2): \quad$ transitive.
Thus $\RR$ is a relation on $S$ which is reflexive, transitive and antisymmetric.
Thus by definition $\RR$ is an ordering on $S$.
{{qed}} | Let $\RR \subseteq S \times S$ be a [[Definition:Preordering|preordering]] on a [[Definition:Set|set]] $S$.
Let $\RR$ also be [[Definition:Antisymmetric Relation|antisymmetric]].
Then $\RR$ is an [[Definition:Ordering|ordering]] on $S$. | By definition, a [[Definition:Preordering|preordering]] on $S$ is a [[Definition:Endorelation|relation]] on $S$ which is:
:$(1): \quad$ [[Definition:Reflexive Relation|reflexive]]
and:
:$(2): \quad$ [[Definition:Transitive Relation|transitive]].
Thus $\RR$ is a [[Definition:Endorelation|relation]] on $S$ which is [[D... | Antisymmetric Preordering is Ordering | https://proofwiki.org/wiki/Antisymmetric_Preordering_is_Ordering | https://proofwiki.org/wiki/Antisymmetric_Preordering_is_Ordering | [
"Preorderings",
"Orderings",
"Asymmetric Relations"
] | [
"Definition:Preordering",
"Definition:Set",
"Definition:Antisymmetric Relation",
"Definition:Ordering"
] | [
"Definition:Preordering",
"Definition:Endorelation",
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Definition:Endorelation",
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Definition:Antisymmetric Relation",
"Definition:Ordering"
] |
proofwiki-8438 | Relation is Antisymmetric and Reflexive iff Intersection with Inverse equals Diagonal Relation | Let $\RR \subseteq S \times S$ be a relation on a set $S$.
Then $\RR$ is both antisymmetric and reflexive {{iff}}:
:$\RR \cap \RR^{-1} = \Delta_S$
where $\Delta_S$ denotes the diagonal relation. | === Necessary Condition ===
Let $\RR$ be both antisymmetric and reflexive.
Then:
{{begin-eqn}}
{{eqn | l = \RR \cap \RR^{-1}
| o = \subseteq
| r = \Delta_S
| c = Relation is Antisymmetric iff Intersection with Inverse is Coreflexive
}}
{{eqn | l = \RR
| o = \supseteq
| r = \Delta_S
|... | Let $\RR \subseteq S \times S$ be a [[Definition:Endorelation|relation]] on a [[Definition:Set|set]] $S$.
Then $\RR$ is both [[Definition:Antisymmetric Relation|antisymmetric]] and [[Definition:Reflexive Relation|reflexive]] {{iff}}:
:$\RR \cap \RR^{-1} = \Delta_S$
where $\Delta_S$ denotes the [[Definition:Diagonal R... | === Necessary Condition ===
Let $\RR$ be both [[Definition:Antisymmetric Relation|antisymmetric]] and [[Definition:Reflexive Relation|reflexive]].
Then:
{{begin-eqn}}
{{eqn | l = \RR \cap \RR^{-1}
| o = \subseteq
| r = \Delta_S
| c = [[Relation is Antisymmetric iff Intersection with Inverse is Coref... | Relation is Antisymmetric and Reflexive iff Intersection with Inverse equals Diagonal Relation | https://proofwiki.org/wiki/Relation_is_Antisymmetric_and_Reflexive_iff_Intersection_with_Inverse_equals_Diagonal_Relation | https://proofwiki.org/wiki/Relation_is_Antisymmetric_and_Reflexive_iff_Intersection_with_Inverse_equals_Diagonal_Relation | [
"Reflexive Relations",
"Antisymmetric Relations",
"Inverse Relations",
"Diagonal Relation"
] | [
"Definition:Endorelation",
"Definition:Set",
"Definition:Antisymmetric Relation",
"Definition:Reflexive Relation",
"Definition:Diagonal Relation"
] | [
"Definition:Antisymmetric Relation",
"Definition:Reflexive Relation",
"Relation is Antisymmetric iff Intersection with Inverse is Coreflexive",
"Inverse of Reflexive Relation is Reflexive",
"Intersection is Largest Subset",
"Relation is Antisymmetric iff Intersection with Inverse is Coreflexive",
"Defin... |
proofwiki-8439 | Reflexive and Transitive Relation is Idempotent | Let $\RR \subseteq S \times S$ be a relation on a set $S$.
Let $\RR$ be both reflexive and transitive.
Then $\RR$ is idempotent, in the sense that:
:$\RR \circ \RR = \RR$
where $\circ$ denotes composition of relations. | Let $\RR$ be both reflexive and transitive.
By definition of transitive relation:
:$\RR \circ \RR \subseteq \RR$
Let $\tuple {x, y} \in \RR$.
By definition of reflexive relation:
:$\tuple {y, y} \in \RR$
By definition of composition of relations:
:$\tuple {x, y} \in \RR \circ \RR$
Hence:
:$\RR \subseteq \RR \circ \RR$... | Let $\RR \subseteq S \times S$ be a [[Definition:Endorelation|relation]] on a [[Definition:Set|set]] $S$.
Let $\RR$ be both [[Definition:Reflexive Relation|reflexive]] and [[Definition:Transitive Relation|transitive]].
Then $\RR$ is [[Definition:Idempotent Relation|idempotent]], in the sense that:
:$\RR \circ \RR =... | Let $\RR$ be both [[Definition:Reflexive Relation|reflexive]] and [[Definition:Transitive Relation|transitive]].
By definition of [[Definition:Transitive Relation/Definition 2|transitive relation]]:
:$\RR \circ \RR \subseteq \RR$
Let $\tuple {x, y} \in \RR$.
By definition of [[Definition:Reflexive Relation|reflex... | Reflexive and Transitive Relation is Idempotent | https://proofwiki.org/wiki/Reflexive_and_Transitive_Relation_is_Idempotent | https://proofwiki.org/wiki/Reflexive_and_Transitive_Relation_is_Idempotent | [
"Reflexive Relations",
"Transitive Relations",
"Idempotence"
] | [
"Definition:Endorelation",
"Definition:Set",
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Definition:Idempotence/Relation",
"Definition:Composition of Relations"
] | [
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Definition:Transitive Relation/Definition 2",
"Definition:Reflexive Relation",
"Definition:Composition of Relations",
"Definition:Set Equality/Definition 2"
] |
proofwiki-8440 | Equivalence of Definitions of Ordering/Proof 2 | The following definitions of ordering are equivalent: | === Definition 1 implies Definition 2 ===
Let $\RR$ be a relation on $S$ satisfying:
{{begin-axiom}}
{{axiom | n = 1
| lc= $\RR$ is reflexive
| q = \forall a \in S
| m = a \mathrel \RR a
}}
{{axiom | n = 2
| lc= $\RR$ is transitive
| q = \forall a, b, c \in S
| m = a \mat... | The following definitions of [[Definition:Ordering|ordering]] are equivalent: | === [[Definition:Ordering/Definition 1|Definition 1]] implies [[Definition:Ordering/Definition 2|Definition 2]] ===
Let $\RR$ be a [[Definition:Relation|relation]] on $S$ satisfying:
{{begin-axiom}}
{{axiom | n = 1
| lc= $\RR$ is [[Definition:Reflexive Relation|reflexive]]
| q = \forall a \in S
... | Equivalence of Definitions of Ordering/Proof 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Ordering/Proof_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Ordering/Proof_2 | [
"Equivalence of Definitions of Ordering"
] | [
"Definition:Ordering"
] | [
"Definition:Ordering/Definition 1",
"Definition:Ordering/Definition 2",
"Definition:Relation",
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Definition:Antisymmetric Relation",
"Definition:Transitive Relation",
"Definition:Reflexivity",
"Definition:Composition of Relations",
... |
proofwiki-8441 | Image of Domain of Relation is Image Set | Let $S$ and $T$ be sets.
Let $\RR \subseteq S \times T$ be a relation.
The image of the domain of $\RR$ is the image set of $\RR$:
:$\RR \sqbrk {\Dom \RR} = \Img \RR$
where $\Img \RR$ is the image of $\RR$. | Let $y \in \RR \sqbrk {\Dom \RR}$.
{{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \RR \sqbrk {\Dom \RR}
| c =
}}
{{eqn | ll= \leadsto
| q = \exists x \in \Dom \RR
| l = \tuple {x, y}
| o = \in
| r = \RR
| c = {{Defof|Image of Subset under Relation}}
}}
{{eqn | ll= \leadsto
... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]].
The [[Definition:Image of Subset under Relation|image]] of the [[Definition:Domain of Relation|domain]] of $\RR$ is the [[Definition:Image of Relation|image set of $\RR$]]:
:$\RR \sqbrk {\Dom \RR} = \Im... | Let $y \in \RR \sqbrk {\Dom \RR}$.
{{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \RR \sqbrk {\Dom \RR}
| c =
}}
{{eqn | ll= \leadsto
| q = \exists x \in \Dom \RR
| l = \tuple {x, y}
| o = \in
| r = \RR
| c = {{Defof|Image of Subset under Relation}}
}}
{{eqn | ll= \leadsto... | Image of Domain of Relation is Image Set | https://proofwiki.org/wiki/Image_of_Domain_of_Relation_is_Image_Set | https://proofwiki.org/wiki/Image_of_Domain_of_Relation_is_Image_Set | [
"Relation Theory"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Domain (Set Theory)/Relation",
"Definition:Image (Set Theory)/Relation/Relation",
"Definition:Image (Set Theory)/Relation/Relation"
] | [
"Definition:Set Equality",
"Category:Relation Theory"
] |
proofwiki-8442 | Condition for Relation to be Transitive and Antitransitive | Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a relation in $S$.
Then:
:$\RR$ is both transitive and antitransitive
{{iff}}:
:$\neg \paren {\exists x, y, z \in S: x \mathrel \RR y \land y \mathrel \RR z}$ | === Necessary Condition ===
Suppose $\neg \paren {\exists x, y, z \in S: x \mathrel \RR y \land y \mathrel \RR z}$.
Then $\RR$ is both transitive and antitransitive vacuously.
{{qed|lemma}} | Let $S$ be a [[Definition:Set|set]].
Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] in $S$.
Then:
:$\RR$ is both [[Definition:Transitive Relation|transitive]] and [[Definition:Antitransitive Relation|antitransitive]]
{{iff}}:
:$\neg \paren {\exists x, y, z \in S: x \mathrel \RR y \land y \mathr... | === Necessary Condition ===
Suppose $\neg \paren {\exists x, y, z \in S: x \mathrel \RR y \land y \mathrel \RR z}$.
Then $\RR$ is both [[Definition:Transitive Relation|transitive]] and [[Definition:Antitransitive Relation|antitransitive]] [[Definition:Vacuous Truth|vacuously]].
{{qed|lemma}} | Condition for Relation to be Transitive and Antitransitive | https://proofwiki.org/wiki/Condition_for_Relation_to_be_Transitive_and_Antitransitive | https://proofwiki.org/wiki/Condition_for_Relation_to_be_Transitive_and_Antitransitive | [
"Transitive Relations",
"Antitransitive Relations"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Transitive Relation",
"Definition:Antitransitive Relation"
] | [
"Definition:Transitive Relation",
"Definition:Antitransitive Relation",
"Definition:Vacuous Truth",
"Definition:Transitive Relation",
"Definition:Antitransitive Relation",
"Definition:Transitive Relation",
"Definition:Antitransitive Relation"
] |
proofwiki-8443 | Transitive and Antitransitive Relation is Asymmetric | Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a relation in $S$.
Let $\RR$ be both transitive and antitransitive.
Then $\RR$ is asymmetric. | Let $\tuple {x, y} \in \RR$ for some $x, y \in S$.
Then as $\RR$ is antitransitive:
:$\tuple {x, x} \notin \RR$
and so as $\RR$ is transitive and $\tuple {x, x} \notin \RR$:
:$\tuple {y, x} \notin \RR$
That is, $\RR$ is asymmetric.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation]] in $S$.
Let $\RR$ be both [[Definition:Transitive Relation|transitive]] and [[Definition:Antitransitive Relation|antitransitive]].
Then $\RR$ is [[Definition:Asymmetric Relation|asymmetric]]. | Let $\tuple {x, y} \in \RR$ for some $x, y \in S$.
Then as $\RR$ is [[Definition:Antitransitive Relation|antitransitive]]:
:$\tuple {x, x} \notin \RR$
and so as $\RR$ is [[Definition:Transitive Relation|transitive]] and $\tuple {x, x} \notin \RR$:
:$\tuple {y, x} \notin \RR$
That is, $\RR$ is [[Definition:Asymmetric... | Transitive and Antitransitive Relation is Asymmetric | https://proofwiki.org/wiki/Transitive_and_Antitransitive_Relation_is_Asymmetric | https://proofwiki.org/wiki/Transitive_and_Antitransitive_Relation_is_Asymmetric | [
"Asymmetric Relations",
"Transitive Relations",
"Antitransitive Relations"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Transitive Relation",
"Definition:Antitransitive Relation",
"Definition:Asymmetric Relation"
] | [
"Definition:Antitransitive Relation",
"Definition:Transitive Relation",
"Definition:Asymmetric Relation"
] |
proofwiki-8444 | Valid Syllogism in Figure I needs Affirmative Minor Premise and Universal Major Premise | Let $Q$ be a valid categorical syllogism in Figure $\text I$.
Then it is a necessary condition that:
:The major premise of $Q$ be a universal categorical statement
and
:The minor premise of $Q$ be an affirmative categorical statement. | Consider Figure $\text I$:
{{:Definition:First Figure of Categorical Syllogism}}
Let the major premise of $Q$ be denoted $\text{Maj}$.
Let the minor premise of $Q$ be denoted $\text{Min}$.
Let the conclusion of $Q$ be denoted $\text{C}$.
$M$ is:
:the subject of $\text{Maj}$
:the predicate of $\text{Min}$.
So, in order ... | Let $Q$ be a [[Definition:Valid Argument|valid]] [[Definition:Categorical Syllogism|categorical syllogism]] in [[Definition:First Figure of Categorical Syllogism|Figure $\text I$]].
Then it is a [[Definition:Necessary Condition|necessary condition]] that:
:The [[Definition:Major Premise of Syllogism|major premise]] of... | Consider [[Definition:First Figure of Categorical Syllogism|Figure $\text I$]]:
{{:Definition:First Figure of Categorical Syllogism}}
Let the [[Definition:Major Premise of Syllogism|major premise]] of $Q$ be denoted $\text{Maj}$.
Let the [[Definition:Minor Premise of Syllogism|minor premise]] of $Q$ be denoted $\tex... | Valid Syllogism in Figure I needs Affirmative Minor Premise and Universal Major Premise | https://proofwiki.org/wiki/Valid_Syllogism_in_Figure_I_needs_Affirmative_Minor_Premise_and_Universal_Major_Premise | https://proofwiki.org/wiki/Valid_Syllogism_in_Figure_I_needs_Affirmative_Minor_Premise_and_Universal_Major_Premise | [
"Categorical Syllogisms"
] | [
"Definition:Valid Argument",
"Definition:Categorical Syllogism",
"Definition:Figure of Categorical Syllogism/I",
"Definition:Conditional/Necessary Condition",
"Definition:Categorical Syllogism/Premises/Major Premise",
"Definition:Universal Categorical Statement",
"Definition:Categorical Syllogism/Premis... | [
"Definition:Figure of Categorical Syllogism/I",
"Definition:Categorical Syllogism/Premises/Major Premise",
"Definition:Categorical Syllogism/Premises/Minor Premise",
"Definition:Categorical Syllogism/Conclusion",
"Definition:Categorical Statement/Subject",
"Definition:Categorical Statement/Predicate",
"... |
proofwiki-8445 | Valid Syllogism in Figure II needs Negative Conclusion and Universal Major Premise | Let $Q$ be a valid categorical syllogism in Figure $\text{II}$.
Then it is a necessary condition that:
:The major premise of $Q$ be a universal categorical statement
and
:The conclusion of $Q$ be a negative categorical statement. | Consider Figure $\text{II}$:
{{:Definition:Figure of Categorical Syllogism/II}}
Let the major premise of $Q$ be denoted $\text{Maj}$.
Let the minor premise of $Q$ be denoted $\text{Min}$.
Let the conclusion of $Q$ be denoted $\text{C}$.
$M$ is:
:the predicate of $\text{Maj}$
:the predicate of $\text{Min}$.
So, in order... | Let $Q$ be a [[Definition:Valid Argument|valid]] [[Definition:Categorical Syllogism|categorical syllogism]] in [[Definition:Second Figure of Categorical Syllogism|Figure $\text{II}$]].
Then it is a [[Definition:Necessary Condition|necessary condition]] that:
:The [[Definition:Major Premise of Syllogism|major premise]]... | Consider [[Definition:Second Figure of Categorical Syllogism|Figure $\text{II}$]]:
{{:Definition:Figure of Categorical Syllogism/II}}
Let the [[Definition:Major Premise of Syllogism|major premise]] of $Q$ be denoted $\text{Maj}$.
Let the [[Definition:Minor Premise of Syllogism|minor premise]] of $Q$ be denoted $\tex... | Valid Syllogism in Figure II needs Negative Conclusion and Universal Major Premise | https://proofwiki.org/wiki/Valid_Syllogism_in_Figure_II_needs_Negative_Conclusion_and_Universal_Major_Premise | https://proofwiki.org/wiki/Valid_Syllogism_in_Figure_II_needs_Negative_Conclusion_and_Universal_Major_Premise | [
"Categorical Syllogisms"
] | [
"Definition:Valid Argument",
"Definition:Categorical Syllogism",
"Definition:Figure of Categorical Syllogism/II",
"Definition:Conditional/Necessary Condition",
"Definition:Categorical Syllogism/Premises/Major Premise",
"Definition:Universal Categorical Statement",
"Definition:Categorical Syllogism/Concl... | [
"Definition:Figure of Categorical Syllogism/II",
"Definition:Categorical Syllogism/Premises/Major Premise",
"Definition:Categorical Syllogism/Premises/Minor Premise",
"Definition:Categorical Syllogism/Conclusion",
"Definition:Categorical Statement/Predicate",
"Definition:Categorical Statement/Predicate",
... |
proofwiki-8446 | Valid Syllogism in Figure III needs Particular Conclusion and if Negative then Negative Major Premise | Let $Q$ be a valid categorical syllogism in Figure $\text {III}$.
Then it is a necessary condition that:
:The conclusion of $Q$ be a particular categorical statement
and:
:If the conclusion of $Q$ be a negative categorical statement, then so is the major premise of $Q$. | Consider Figure $\text {III}$:
{{:Definition:Figure of Categorical Syllogism/III}}
Let the major premise of $Q$ be denoted $\text{Maj}$.
Let the minor premise of $Q$ be denoted $\text{Min}$.
Let the conclusion of $Q$ be denoted $\text{C}$.
$M$ is:
:the subject of $\text{Maj}$
:the subject of $\text{Min}$.
So, in order ... | Let $Q$ be a [[Definition:Valid Argument|valid]] [[Definition:Categorical Syllogism|categorical syllogism]] in [[Definition:Third Figure of Categorical Syllogism|Figure $\text {III}$]].
Then it is a [[Definition:Necessary Condition|necessary condition]] that:
:The [[Definition:Conclusion of Syllogism|conclusion]] of $... | Consider [[Definition:Third Figure of Categorical Syllogism|Figure $\text {III}$]]:
{{:Definition:Figure of Categorical Syllogism/III}}
Let the [[Definition:Major Premise of Syllogism|major premise]] of $Q$ be denoted $\text{Maj}$.
Let the [[Definition:Minor Premise of Syllogism|minor premise]] of $Q$ be denoted $\t... | Valid Syllogism in Figure III needs Particular Conclusion and if Negative then Negative Major Premise | https://proofwiki.org/wiki/Valid_Syllogism_in_Figure_III_needs_Particular_Conclusion_and_if_Negative_then_Negative_Major_Premise | https://proofwiki.org/wiki/Valid_Syllogism_in_Figure_III_needs_Particular_Conclusion_and_if_Negative_then_Negative_Major_Premise | [
"Categorical Syllogisms"
] | [
"Definition:Valid Argument",
"Definition:Categorical Syllogism",
"Definition:Figure of Categorical Syllogism/III",
"Definition:Conditional/Necessary Condition",
"Definition:Categorical Syllogism/Conclusion",
"Definition:Particular Categorical Statement",
"Definition:Categorical Syllogism/Conclusion",
... | [
"Definition:Figure of Categorical Syllogism/III",
"Definition:Categorical Syllogism/Premises/Major Premise",
"Definition:Categorical Syllogism/Premises/Minor Premise",
"Definition:Categorical Syllogism/Conclusion",
"Definition:Categorical Statement/Subject",
"Definition:Categorical Statement/Subject",
"... |
proofwiki-8447 | Valid Syllogisms in Figure IV | Let $Q$ be a valid categorical syllogism in Figure $\text {IV}$.
Then it is a necessary condition that:
:$(1): \quad$ Either:
:: the major premise of $Q$ be a negative categorical statement
:or:
:: the minor premise of $Q$ be a universal categorical statement
:or both.
:$(2): \quad$ If the conclusion of $Q$ be a negati... | Consider Figure $\text {IV}$:
{{:Definition:Figure of Categorical Syllogism/IV}}
Let the major premise of $Q$ be denoted $\text{Maj}$.
Let the minor premise of $Q$ be denoted $\text{Min}$.
Let the conclusion of $Q$ be denoted $\text{C}$.
$M$ is:
:the predicate of $\text{Maj}$
:the subject of $\text{Min}$.
We have:
:Mid... | Let $Q$ be a [[Definition:Valid Argument|valid]] [[Definition:Categorical Syllogism|categorical syllogism]] in [[Definition:Fourth Figure of Categorical Syllogism|Figure $\text {IV}$]].
Then it is a [[Definition:Necessary Condition|necessary condition]] that:
:$(1): \quad$ Either:
:: the [[Definition:Major Premise of... | Consider [[Definition:Fourth Figure of Categorical Syllogism|Figure $\text {IV}$]]:
{{:Definition:Figure of Categorical Syllogism/IV}}
Let the [[Definition:Major Premise of Syllogism|major premise]] of $Q$ be denoted $\text{Maj}$.
Let the [[Definition:Minor Premise of Syllogism|minor premise]] of $Q$ be denoted $\te... | Valid Syllogisms in Figure IV | https://proofwiki.org/wiki/Valid_Syllogisms_in_Figure_IV | https://proofwiki.org/wiki/Valid_Syllogisms_in_Figure_IV | [
"Categorical Syllogisms"
] | [
"Definition:Valid Argument",
"Definition:Categorical Syllogism",
"Definition:Figure of Categorical Syllogism/IV",
"Definition:Conditional/Necessary Condition",
"Definition:Categorical Syllogism/Premises/Major Premise",
"Definition:Negative Categorical Statement",
"Definition:Categorical Syllogism/Premis... | [
"Definition:Figure of Categorical Syllogism/IV",
"Definition:Categorical Syllogism/Premises/Major Premise",
"Definition:Categorical Syllogism/Premises/Minor Premise",
"Definition:Categorical Syllogism/Conclusion",
"Definition:Categorical Statement/Predicate",
"Definition:Categorical Statement/Subject",
... |
proofwiki-8448 | Elimination of all but 24 Categorical Syllogisms as Invalid | Of the $256$ different types of categorical syllogism, all but $24$ can be identified as invalid.
These are the $24$ patterns which may still be valid:
:$\begin{array}{rl}
\text{I} & AAA \\
\text{I} & AII \\
\text{I} & EAE \\
\text{I} & EIO \\
\text{I} & AAI \\
\text{I} & EAO \\
\end{array}
\qquad
\begin{array}{rl}
\te... | From Elimination of all but 48 Categorical Syllogisms as Invalid there are $12$ possible patterns of categorical syllogism per figure:
:$\begin{array}{cccccc}
AAA & AAI & AEE & AEO & AII & AOO \\
EAE & EAO & EIO & IAI & IEO & OAO \\
\end{array}$ | Of the $256$ different types of [[Definition:Categorical Syllogism|categorical syllogism]], all but $24$ can be identified as [[Definition:Invalid Argument|invalid]].
These are the $24$ patterns which may still be [[Definition:Valid Argument|valid]]:
:$\begin{array}{rl}
\text{I} & AAA \\
\text{I} & AII \\
\text{I} &... | From [[Elimination of all but 48 Categorical Syllogisms as Invalid]] there are $12$ possible patterns of [[Definition:Categorical Syllogism|categorical syllogism]] per [[Definition:Figure of Categorical Syllogism|figure]]:
:$\begin{array}{cccccc}
AAA & AAI & AEE & AEO & AII & AOO \\
EAE & EAO & EIO & IAI & IEO & OAO \... | Elimination of all but 24 Categorical Syllogisms as Invalid | https://proofwiki.org/wiki/Elimination_of_all_but_24_Categorical_Syllogisms_as_Invalid | https://proofwiki.org/wiki/Elimination_of_all_but_24_Categorical_Syllogisms_as_Invalid | [
"Categorical Syllogisms"
] | [
"Definition:Categorical Syllogism",
"Definition:Invalid Argument",
"Definition:Valid Argument"
] | [
"Elimination of all but 48 Categorical Syllogisms as Invalid",
"Definition:Categorical Syllogism",
"Definition:Figure of Categorical Syllogism",
"Definition:Categorical Syllogism"
] |
proofwiki-8449 | Extension of Contradictory Branch is Contradictory | Let $T$ be a propositional tableau.
Let $\Gamma$ be a contradictory branch of $T$.
Let $\Gamma'$ be an extension of $\Gamma$.
Then $\Gamma'$ is also contradictory. | Since $\Gamma$ is contradictory, there is some WFF $\mathbf A$ such that both $\mathbf A$ and $\neg \mathbf A$ occur along $\Gamma$.
Since $\Gamma'$ is an extension of $\Gamma$, $\mathbf A$ and $\neg \mathbf A$ also occur along $\Gamma'$.
Hence $\Gamma'$ is contradictory.
{{qed}}
Category:Propositional Tableaux
50yd024... | Let $T$ be a [[Definition:Propositional Tableau|propositional tableau]].
Let $\Gamma$ be a [[Definition:Contradictory Branch|contradictory branch]] of $T$.
Let $\Gamma'$ be an [[Definition:Extension of Branch of Propositional Tableau|extension]] of $\Gamma$.
Then $\Gamma'$ is also [[Definition:Contradictory Branch|... | Since $\Gamma$ is [[Definition:Contradictory Branch|contradictory]], there is some [[Definition:WFF of Propositional Logic|WFF]] $\mathbf A$ such that both $\mathbf A$ and $\neg \mathbf A$ [[Definition:Occurrence along Branch|occur]] along $\Gamma$.
Since $\Gamma'$ is an [[Definition:Extension of Branch of Proposition... | Extension of Contradictory Branch is Contradictory | https://proofwiki.org/wiki/Extension_of_Contradictory_Branch_is_Contradictory | https://proofwiki.org/wiki/Extension_of_Contradictory_Branch_is_Contradictory | [
"Propositional Tableaux"
] | [
"Definition:Propositional Tableau",
"Definition:Contradictory/Branch",
"Definition:Extension of Branch of Propositional Tableau",
"Definition:Contradictory/Branch"
] | [
"Definition:Contradictory/Branch",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Labeled Tree for Propositional Logic/Along a Branch",
"Definition:Extension of Branch of Propositional Tableau",
"Definition:Labeled Tree for Propositional Logic/Along a Branch",
"Definition:Con... |
proofwiki-8450 | Finished Propositional Tableau has Finished Branch or is Confutation | Let $\struct {T, \mathbf H, \Phi}$ be a finished propositional tableau.
Then one of the following holds:
:$T$ has a finished branch
:$T$ is a tableau confutation. | Suppose $T$ has no finished branch.
Then since $T$ is finished, every branch of $T$ is contradictory.
Hence $T$ is a tableau confutation.
{{qed}} | Let $\struct {T, \mathbf H, \Phi}$ be a [[Definition:Finished Propositional Tableau|finished]] [[Definition:Propositional Tableau|propositional tableau]].
Then one of the following holds:
:$T$ has a [[Definition:Finished Branch of Propositional Tableau|finished branch]]
:$T$ is a [[Definition:Tableau Confutation|tab... | Suppose $T$ has no [[Definition:Finished Branch of Propositional Tableau|finished branch]].
Then since $T$ is [[Definition:Finished Propositional Tableau|finished]], every [[Definition:Branch (Graph Theory)|branch]] of $T$ is [[Definition:Contradictory Branch|contradictory]].
Hence $T$ is a [[Definition:Tableau Conf... | Finished Propositional Tableau has Finished Branch or is Confutation | https://proofwiki.org/wiki/Finished_Propositional_Tableau_has_Finished_Branch_or_is_Confutation | https://proofwiki.org/wiki/Finished_Propositional_Tableau_has_Finished_Branch_or_is_Confutation | [
"Propositional Tableaux"
] | [
"Definition:Finished Propositional Tableau",
"Definition:Propositional Tableau",
"Definition:Finished Branch of Propositional Tableau",
"Definition:Tableau Confutation"
] | [
"Definition:Finished Branch of Propositional Tableau",
"Definition:Finished Propositional Tableau",
"Definition:Rooted Tree/Branch",
"Definition:Contradictory/Branch",
"Definition:Tableau Confutation"
] |
proofwiki-8451 | Alternating Harmonic Series is Conditionally Convergent | The alternating harmonic series:
:$\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \cdots$
is conditionally convergent. | Note first that:
:$\ds \sum_{n \mathop = 1}^\infty \size {\frac {\paren {-1}^\paren {n - 1} } n} = \sum_{n \mathop = 1}^\infty \frac 1 n$
which is divergent by Harmonic Series is Divergent.
Hence by definition $\ds \sum_{n \mathop = 1}^\infty \size {\frac {\paren {-1}^\paren {n - 1} } n}$ is not '''absolutely convergen... | The [[Definition:Alternating Harmonic Series|alternating harmonic series]]:
:$\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \cdots$
is [[Definition:Conditionally Convergent Series|conditionally convergent]]. | Note first that:
:$\ds \sum_{n \mathop = 1}^\infty \size {\frac {\paren {-1}^\paren {n - 1} } n} = \sum_{n \mathop = 1}^\infty \frac 1 n$
which is [[Definition:Divergent Series|divergent]] by [[Harmonic Series is Divergent]].
Hence by definition $\ds \sum_{n \mathop = 1}^\infty \size {\frac {\paren {-1}^\paren {n - 1... | Alternating Harmonic Series is Conditionally Convergent | https://proofwiki.org/wiki/Alternating_Harmonic_Series_is_Conditionally_Convergent | https://proofwiki.org/wiki/Alternating_Harmonic_Series_is_Conditionally_Convergent | [
"Alternating Series",
"Series",
"Newton-Mercator Series"
] | [
"Definition:Mercator's Constant",
"Definition:Conditionally Convergent Series"
] | [
"Definition:Divergent Series",
"Harmonic Series is Divergent",
"Definition:Absolutely Convergent Series/Real Numbers",
"Definition:Basic Null Sequence",
"Reciprocal Sequence is Strictly Decreasing",
"Alternating Series Test"
] |
proofwiki-8452 | Manipulation of Absolutely Convergent Series/Permutation | If $\pi: \N \to \N$ is a permutation of $N$, then:
:$\ds \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty a_{\map \pi n}$ | Let $\epsilon > 0$.
From Tail of Convergent Series tends to Zero, it follows that there exists $N \in \N$ such that:
:$\ds \sum_{n \mathop = N}^\infty \size {a_n} < \epsilon$
By definition, a permutation is bijective.
Hence we can find $M \in \N$ such that:
:$\set {1, \ldots, N - 1} \subseteq \set {\map \pi 1, \ldots, ... | If $\pi: \N \to \N$ is a [[Definition:Permutation|permutation]] of $N$, then:
:$\ds \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty a_{\map \pi n}$ | Let $\epsilon > 0$.
From [[Tail of Convergent Series tends to Zero]], it follows that there exists $N \in \N$ such that:
:$\ds \sum_{n \mathop = N}^\infty \size {a_n} < \epsilon$
By definition, a [[Definition:Permutation|permutation]] is [[Definition:Bijection|bijective]].
Hence we can find $M \in \N$ such that:
:$... | Manipulation of Absolutely Convergent Series/Permutation | https://proofwiki.org/wiki/Manipulation_of_Absolutely_Convergent_Series/Permutation | https://proofwiki.org/wiki/Manipulation_of_Absolutely_Convergent_Series/Permutation | [
"Convergence",
"Series"
] | [
"Definition:Permutation"
] | [
"Tail of Convergent Series tends to Zero",
"Definition:Permutation",
"Definition:Bijection",
"Definition:Characteristic Function (Set Theory)/Set",
"Triangle Inequality",
"Definition:Convergent Series"
] |
proofwiki-8453 | Manipulation of Absolutely Convergent Series/Characteristic Function | Let $A \subseteq \N$.
Then:
:$\ds \sum_{n \mathop = 1}^\infty a_n \map {\chi_A} n = \sum_{n \mathop \in A} a_n$
where $\chi_A$ is the characteristic function of $A$. | For all $N \in \N$, we have:
:$\ds \sum_{n \mathop = 1}^N \size {a_n \map {\chi_A} n} \le \sum_{n \mathop = 1}^N \size {a_n} \le \sum_{n \mathop = 1}^\infty \size {a_n}$
It follows that:
:$\ds \sum_{n \mathop = 1}^\infty \size {a_n \map {\chi_A} n} \le \sum_{n \mathop = 1}^\infty \size {a_n}$
Then $\ds \sum_{n \mathop ... | Let $A \subseteq \N$.
Then:
:$\ds \sum_{n \mathop = 1}^\infty a_n \map {\chi_A} n = \sum_{n \mathop \in A} a_n$
where $\chi_A$ is the [[Definition:Characteristic Function of Set|characteristic function]] of $A$. | For all $N \in \N$, we have:
:$\ds \sum_{n \mathop = 1}^N \size {a_n \map {\chi_A} n} \le \sum_{n \mathop = 1}^N \size {a_n} \le \sum_{n \mathop = 1}^\infty \size {a_n}$
It follows that:
:$\ds \sum_{n \mathop = 1}^\infty \size {a_n \map {\chi_A} n} \le \sum_{n \mathop = 1}^\infty \size {a_n}$
Then $\ds \sum_{n \math... | Manipulation of Absolutely Convergent Series/Characteristic Function | https://proofwiki.org/wiki/Manipulation_of_Absolutely_Convergent_Series/Characteristic_Function | https://proofwiki.org/wiki/Manipulation_of_Absolutely_Convergent_Series/Characteristic_Function | [
"Series",
"Convergence"
] | [
"Definition:Characteristic Function (Set Theory)/Set"
] | [
"Definition:Absolutely Convergent Series",
"Manipulation of Absolutely Convergent Series/Permutation",
"Definition:Series",
"Category:Series",
"Category:Convergence"
] |
proofwiki-8454 | Manipulation of Absolutely Convergent Series/Scale Factor | Let $c \in \R$, or $c \in \C$.
Then:
:$\ds c \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty c a_n$ | {{begin-eqn}}
{{eqn | l = c \sum_{n \mathop = 1}^\infty a_n
| r = c \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n
}}
{{eqn | r = \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N c a_n
| c = Multiple Rule for Sequences
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty c a_n
}}
{{end-eqn}}
{{qed}}
Categor... | Let $c \in \R$, or $c \in \C$.
Then:
:$\ds c \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty c a_n$ | {{begin-eqn}}
{{eqn | l = c \sum_{n \mathop = 1}^\infty a_n
| r = c \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n
}}
{{eqn | r = \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N c a_n
| c = [[Multiple Rule for Sequences]]
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty c a_n
}}
{{end-eqn}}
{{qed}}
[[... | Manipulation of Absolutely Convergent Series/Scale Factor | https://proofwiki.org/wiki/Manipulation_of_Absolutely_Convergent_Series/Scale_Factor | https://proofwiki.org/wiki/Manipulation_of_Absolutely_Convergent_Series/Scale_Factor | [
"Series",
"Convergence"
] | [] | [
"Combination Theorem for Sequences/Multiple Rule",
"Category:Series",
"Category:Convergence"
] |
proofwiki-8455 | Binomial Coefficient with Self minus One | :$\forall n \in \N_{>0}: \dbinom n {n - 1} = n$ | The case where $n = 1$ can be taken separately.
From Binomial Coefficient with Zero:
:$\dbinom 1 0 = 1$
demonstrating that the result holds for $n = 1$.
Let $n \in \N: n > 1$.
From the definition of binomial coefficients:
:$\dbinom n {n - 1} = \dfrac {n!} {\paren {n - 1}! \paren {n - \paren {n - 1} }!} = \dfrac {n!} {\... | :$\forall n \in \N_{>0}: \dbinom n {n - 1} = n$ | The case where $n = 1$ can be taken separately.
From [[Binomial Coefficient with Zero]]:
:$\dbinom 1 0 = 1$
demonstrating that the result holds for $n = 1$.
Let $n \in \N: n > 1$.
From the [[Definition:Binomial Coefficient|definition of binomial coefficients]]:
:$\dbinom n {n - 1} = \dfrac {n!} {\paren {n - 1}! \p... | Binomial Coefficient with Self minus One/Proof 1 | https://proofwiki.org/wiki/Binomial_Coefficient_with_Self_minus_One | https://proofwiki.org/wiki/Binomial_Coefficient_with_Self_minus_One/Proof_1 | [
"Examples of Binomial Coefficients",
"Binomial Coefficient with Self minus One"
] | [] | [
"Binomial Coefficient with Zero",
"Definition:Binomial Coefficient",
"Definition:Factorial"
] |
proofwiki-8456 | Binomial Coefficient with Self minus One | :$\forall n \in \N_{>0}: \dbinom n {n - 1} = n$ | From Cardinality of Set of Subsets, $\dbinom n {n - 1}$ is the number of combination of things taken $n - 1$ at a time.
Choosing $n - 1$ things from $n$ is the same thing as choosing which $1$ of the elements to be left out.
There are $n$ different choices for that $1$ element.
Therefore there are $n$ ways to choose $n... | :$\forall n \in \N_{>0}: \dbinom n {n - 1} = n$ | From [[Cardinality of Set of Subsets]], $\dbinom n {n - 1}$ is the number of combination of things taken $n - 1$ at a time.
Choosing $n - 1$ things from $n$ is the same thing as choosing which $1$ of the elements to be left out.
There are $n$ different choices for that $1$ element.
Therefore there are $n$ ways to ch... | Binomial Coefficient with Self minus One/Proof 2 | https://proofwiki.org/wiki/Binomial_Coefficient_with_Self_minus_One | https://proofwiki.org/wiki/Binomial_Coefficient_with_Self_minus_One/Proof_2 | [
"Examples of Binomial Coefficients",
"Binomial Coefficient with Self minus One"
] | [] | [
"Cardinality of Set of Subsets"
] |
proofwiki-8457 | Definite Integral of Uniformly Convergent Series of Continuous Functions | Let $\sequence {f_n}$ be a sequence of real functions.
Let each of $\sequence {f_n}$ be continuous on the interval $\hointr a b$.
{{explain|Investigation needed as to whether there is a mistake in {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition}} -- ... | Define $\map {S_N} x = \ds \sum_{n \mathop = 1}^N \map {f_n} x$.
We have:
{{begin-eqn}}
{{eqn | l = \size {\int_a^b \map f x \rd x - \sum_{n \mathop = 1}^N \int_a^b \map {f_n} x \rd x}
| r = \size {\int_a^b \paren {\map f x - \map {S_N} x} \rd x}
| c =
}}
{{eqn | o = \le
| r = \paren {b - a} \sup_{x ... | Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]].
Let each of $\sequence {f_n}$ be [[Definition:Continuous on Interval|continuous]] on the [[Definition:Half-Open Real Interval|interval]] $\hointr a b$.
{{explain|Investigation needed as to whether there is a mi... | Define $\map {S_N} x = \ds \sum_{n \mathop = 1}^N \map {f_n} x$.
We have:
{{begin-eqn}}
{{eqn | l = \size {\int_a^b \map f x \rd x - \sum_{n \mathop = 1}^N \int_a^b \map {f_n} x \rd x}
| r = \size {\int_a^b \paren {\map f x - \map {S_N} x} \rd x}
| c =
}}
{{eqn | o = \le
| r = \paren {b - a} \sup_{x... | Definite Integral of Uniformly Convergent Series of Continuous Functions | https://proofwiki.org/wiki/Definite_Integral_of_Uniformly_Convergent_Series_of_Continuous_Functions | https://proofwiki.org/wiki/Definite_Integral_of_Uniformly_Convergent_Series_of_Continuous_Functions | [
"Integral Calculus",
"Convergence",
"Continuity",
"Series"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Half-Open",
"Definition:Series",
"Definition:Uniform Convergence"
] | [] |
proofwiki-8458 | Power Series Converges Uniformly within Radius of Convergence | Let $\ds S := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$.
Let $R$ be the radius of convergence of $S$.
Let $\rho \in \R$ such that $0 \le \rho < R$.
Then $S$ is uniformly convergent on $D = \set {x: \size {x - \xi} \le \rho}$. | We shall make use of the Weierstrass M-Test to prove this result.
To begin with, for each $n \in N$, define for $x \in D$:
:$\map {f_n} x = a_n \paren {x - \xi}^n$
We have:
{{begin-eqn}}
{{eqn | l = \size {\map {f_n} x}
| r = \size {a_n \paren {x - \xi}^n}
}}
{{eqn | o = \le
| r = \size {a_n \rho^n}
|... | Let $\ds S := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a [[Definition:Power Series|power series]] about a point $\xi$.
Let $R$ be the [[Definition:Radius of Convergence|radius of convergence]] of $S$.
Let $\rho \in \R$ such that $0 \le \rho < R$.
Then $S$ is [[Definition:Uniform Convergence|uniformly ... | We shall make use of the [[Weierstrass M-Test]] to prove this result.
To begin with, for each $n \in N$, define for $x \in D$:
:$\map {f_n} x = a_n \paren {x - \xi}^n$
We have:
{{begin-eqn}}
{{eqn | l = \size {\map {f_n} x}
| r = \size {a_n \paren {x - \xi}^n}
}}
{{eqn | o = \le
| r = \size {a_n \rho^n}
... | Power Series Converges Uniformly within Radius of Convergence | https://proofwiki.org/wiki/Power_Series_Converges_Uniformly_within_Radius_of_Convergence | https://proofwiki.org/wiki/Power_Series_Converges_Uniformly_within_Radius_of_Convergence | [
"Power Series"
] | [
"Definition:Power Series",
"Definition:Radius of Convergence",
"Definition:Uniform Convergence"
] | [
"Weierstrass M-Test",
"Definition:Upper Bound of Mapping/Real-Valued",
"Definition:Supremum of Set",
"Weierstrass M-Test",
"Definition:Convergent Series",
"Definition:Radius of Convergence",
"Definition:Interval of Convergence",
"Definition:Convergent Series",
"Existence of Interval of Convergence o... |
proofwiki-8459 | Power Series Converges to Continuous Function | Let $\ds \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$.
Let $R$ be the radius of convergence of $S$.
Then $\map f x$ is a continuous function on $\set {x: \size {x - \xi} < R}$. | Let $\rho \in \R$ such that $0 \le \rho < R$.
From Power Series Converges Uniformly within Radius of Convergence, $\map f x$ is uniformly convergent on $\set {x: \size {x - \xi} \le \rho}$.
From Real Polynomial Function is Continuous, each of $\map {f_n} x = a_n x^n$ is a continuous function of $x$.
The result follows ... | Let $\ds \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a [[Definition:Power Series|power series]] about a point $\xi$.
Let $R$ be the [[Definition:Radius of Convergence|radius of convergence]] of $S$.
Then $\map f x$ is a [[Definition:Continuous Function|continuous function]] on $\set {x: \size ... | Let $\rho \in \R$ such that $0 \le \rho < R$.
From [[Power Series Converges Uniformly within Radius of Convergence]], $\map f x$ is [[Definition:Uniform Convergence|uniformly convergent]] on $\set {x: \size {x - \xi} \le \rho}$.
From [[Real Polynomial Function is Continuous]], each of $\map {f_n} x = a_n x^n$ is a [[... | Power Series Converges to Continuous Function | https://proofwiki.org/wiki/Power_Series_Converges_to_Continuous_Function | https://proofwiki.org/wiki/Power_Series_Converges_to_Continuous_Function | [
"Power Series"
] | [
"Definition:Power Series",
"Definition:Radius of Convergence",
"Definition:Continuous Function"
] | [
"Power Series Converges Uniformly within Radius of Convergence",
"Definition:Uniform Convergence",
"Real Polynomial Function is Continuous",
"Definition:Continuous Function",
"Uniformly Convergent Series of Continuous Functions is Continuous"
] |
proofwiki-8460 | Power Series is Termwise Integrable within Radius of Convergence | Let $\ds \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$.
Let $R$ be the radius of convergence of $S$.
Then:
:$\ds \int_a^b \map f x \rd x = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \rd x = \sum_{n \mathop = 0}^\infty a_n \frac {x^{n + 1} } {n + 1}$ | Let $\rho \in \R$ such that $0 \le \rho < R$.
From Power Series Converges Uniformly within Radius of Convergence, $\map f x$ is uniformly convergent on $\set {x: \size {x - \xi} \le \rho}$.
From Real Polynomial Function is Continuous, each of $\map {f_n} x = a_n x^n$ is a continuous function of $x$.
Then from Definite ... | Let $\ds \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a [[Definition:Power Series|power series]] about a point $\xi$.
Let $R$ be the [[Definition:Radius of Convergence|radius of convergence]] of $S$.
Then:
:$\ds \int_a^b \map f x \rd x = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \rd x = \sum... | Let $\rho \in \R$ such that $0 \le \rho < R$.
From [[Power Series Converges Uniformly within Radius of Convergence]], $\map f x$ is [[Definition:Uniform Convergence|uniformly convergent]] on $\set {x: \size {x - \xi} \le \rho}$.
From [[Real Polynomial Function is Continuous]], each of $\map {f_n} x = a_n x^n$ is a [[... | Power Series is Termwise Integrable within Radius of Convergence | https://proofwiki.org/wiki/Power_Series_is_Termwise_Integrable_within_Radius_of_Convergence | https://proofwiki.org/wiki/Power_Series_is_Termwise_Integrable_within_Radius_of_Convergence | [
"Power Series"
] | [
"Definition:Power Series",
"Definition:Radius of Convergence"
] | [
"Power Series Converges Uniformly within Radius of Convergence",
"Definition:Uniform Convergence",
"Real Polynomial Function is Continuous",
"Definition:Continuous Function",
"Definite Integral of Uniformly Convergent Series of Continuous Functions",
"Integral of Power"
] |
proofwiki-8461 | Power Series is Termwise Differentiable within Radius of Convergence | Let $\ds \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$.
Let $R$ be the radius of convergence of the series.
Then:
:$\ds \frac \d {\d x} \map f x = \sum_{n \mathop = 0}^\infty \frac \d {\d x} a_n \paren {x - \xi}^n = \sum_{n \mathop = 1}^\infty n a_n \paren {x - \... | Let $\rho \in \R$ such that $0 \le \rho < R$.
From Power Series Converges Uniformly within Radius of Convergence, $\map f x$ is uniformly convergent on $\set {x: \size {x - \xi} \le \rho}$.
From Real Polynomial Function is Continuous, each of $\map {f_n} x = a_n \paren {x - \xi}^n$ is a continuous function of $x$.
From... | Let $\ds \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a [[Definition:Power Series|power series]] about a point $\xi$.
Let $R$ be the [[Definition:Radius of Convergence|radius of convergence]] of the series.
Then:
:$\ds \frac \d {\d x} \map f x = \sum_{n \mathop = 0}^\infty \frac \d {\d x} a_n \... | Let $\rho \in \R$ such that $0 \le \rho < R$.
From [[Power Series Converges Uniformly within Radius of Convergence]], $\map f x$ is [[Definition:Uniform Convergence|uniformly convergent]] on $\set {x: \size {x - \xi} \le \rho}$.
From [[Real Polynomial Function is Continuous]], each of $\map {f_n} x = a_n \paren {x - ... | Power Series is Termwise Differentiable within Radius of Convergence | https://proofwiki.org/wiki/Power_Series_is_Termwise_Differentiable_within_Radius_of_Convergence | https://proofwiki.org/wiki/Power_Series_is_Termwise_Differentiable_within_Radius_of_Convergence | [
"Power Series"
] | [
"Definition:Power Series",
"Definition:Radius of Convergence"
] | [
"Power Series Converges Uniformly within Radius of Convergence",
"Definition:Uniform Convergence",
"Real Polynomial Function is Continuous",
"Definition:Continuous Function",
"Power Rule for Derivatives",
"Real Polynomial Function is Continuous",
"Definition:Continuous Function",
"Derivative of Unifor... |
proofwiki-8462 | Power Series Expansion for Logarithm of 1 + x | The Newton-Mercator series defines the natural logarithm function as a power series expansion:
{{begin-eqn}}
{{eqn | l = \map \ln {1 + x}
| r = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n
}}
{{eqn | r = x - \frac {x^2} 2 + \frac {x^3} 3 - \frac {x^4} 4 + \cdots
}}
{{end-eqn}}
valid for all $x \... | From Sum of Infinite Geometric Sequence, putting $-x$ for $x$:
:$(1): \quad \ds \sum_{n \mathop = 0}^\infty \paren {-x}^n = \frac 1 {1 + x}$
for $-1 < x < 1$.
From Power Series Converges Uniformly within Radius of Convergence, $(1)$ is uniformly convergent on every closed interval within the interval $\openint {-1} 1$.... | The [[Definition:Newton-Mercator Series|Newton-Mercator series]] defines the [[Definition:Natural Logarithm|natural logarithm function]] as a [[Definition:Power Series|power series expansion]]:
{{begin-eqn}}
{{eqn | l = \map \ln {1 + x}
| r = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n
}}
{{eq... | From [[Sum of Infinite Geometric Sequence]], putting $-x$ for $x$:
:$(1): \quad \ds \sum_{n \mathop = 0}^\infty \paren {-x}^n = \frac 1 {1 + x}$
for $-1 < x < 1$.
From [[Power Series Converges Uniformly within Radius of Convergence]], $(1)$ is [[Definition:Uniform Convergence|uniformly convergent]] on every [[Definit... | Power Series Expansion for Logarithm of 1 + x/Proof 1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_1_+_x/Proof_1 | [
"Power Series Expansion for Logarithm of 1 + x",
"Examples of Power Series",
"Logarithms",
"Newton-Mercator Series",
"Taylor Series"
] | [
"Definition:Newton-Mercator Series",
"Definition:Natural Logarithm",
"Definition:Power Series"
] | [
"Sum of Infinite Geometric Sequence",
"Power Series Converges Uniformly within Radius of Convergence",
"Definition:Uniform Convergence",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Open",
"Power Series is Termwise Integrable within Radius of Convergence",
"Definition:Primitive (Calculus... |
proofwiki-8463 | Newton-Mercator Series/Examples/2 | The Newton-Mercator Series for $x = 1$ converges to the natural logarithm of $2$:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n
| r = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb
| c =
}}
{{eqn | r = \ln 2
| c =
}}
{{end-eqn}}
This real number is known... | From the definition of the Newton-Mercator Series:
{{begin-eqn}}
{{eqn | l = \map \ln {1 + x}
| r = x - \dfrac {x^2} 2 + \dfrac {x^3} 3 - \dfrac {x^4} 4 + \cdots
| c =
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n x^n
| c =
}}
{{end-eqn}}
This is valid for $-1 < x \le 1$.... | The [[Definition:Newton-Mercator Series|Newton-Mercator Series]] for $x = 1$ converges to the [[Definition:Natural Logarithm|natural logarithm]] of $2$:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n
| r = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb
| c =
}}... | From the definition of the [[Definition:Newton-Mercator Series|Newton-Mercator Series]]:
{{begin-eqn}}
{{eqn | l = \map \ln {1 + x}
| r = x - \dfrac {x^2} 2 + \dfrac {x^3} 3 - \dfrac {x^4} 4 + \cdots
| c =
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n x^n
| c =
}}
{{end-... | Newton-Mercator Series/Examples/2 | https://proofwiki.org/wiki/Newton-Mercator_Series/Examples/2 | https://proofwiki.org/wiki/Newton-Mercator_Series/Examples/2 | [
"Examples of Power Series",
"Logarithms",
"Reciprocals",
"Newton-Mercator Series"
] | [
"Definition:Newton-Mercator Series",
"Definition:Natural Logarithm",
"Definition:Real Number",
"Definition:Mercator's Constant"
] | [
"Definition:Newton-Mercator Series",
"Alternating Harmonic Series is Conditionally Convergent"
] |
proofwiki-8464 | Power Series Expansion for Sine Function | The sine function has the power series expansion:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + \cdots
| c =
}}
{{end-eqn}}
valid for all $x \... | From Derivative of Sine Function:
:$\dfrac \d {\d x} \sin x = \cos x$
From Derivative of Cosine Function:
:$\dfrac \d {\d x} \cos x = -\sin x$
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac {\d^2} {\d x^2} \sin x
| r = -\sin x
| c =
}}
{{eqn | l = \dfrac {\d^3} {\d x^3} \sin x
| r = -\cos x
| c =
}}
... | The [[Definition:Sine Function|sine function]] has the [[Definition:Power Series|power series expansion]]:
{{begin-eqn}}
{{eqn | l = \sin x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7... | From [[Derivative of Sine Function]]:
:$\dfrac \d {\d x} \sin x = \cos x$
From [[Derivative of Cosine Function]]:
:$\dfrac \d {\d x} \cos x = -\sin x$
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac {\d^2} {\d x^2} \sin x
| r = -\sin x
| c =
}}
{{eqn | l = \dfrac {\d^3} {\d x^3} \sin x
| r = -\cos x
... | Power Series Expansion for Sine Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Sine_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Sine_Function | [
"Sine Function",
"Examples of Power Series",
"Taylor Series"
] | [
"Definition:Sine",
"Definition:Power Series"
] | [
"Derivative of Sine Function",
"Derivative of Cosine Function",
"Definition:Maclaurin Series",
"Sine of Zero is Zero",
"Cosine of Zero is One",
"Series of Power over Factorial Converges",
"Definition:Convergent Series"
] |
proofwiki-8465 | Power Series Expansion for Cosine Function | The cosine function has the power series expansion:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots
| c =
}}
{{end-eqn}}
valid for all $x \in \R$... | From Derivative of Cosine Function:
:$\dfrac \d {\d x} \cos x = -\sin x$
From Derivative of Sine Function:
:$\dfrac \d {\d x} \sin x = \cos x$
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac {\d^2} {\d x^2} \cos x
| r = -\cos x
| c =
}}
{{eqn | l = \dfrac {\d^3} {\d x^3} \cos x
| r = \sin x
| c =
}}
{... | The [[Definition:Cosine Function|cosine function]] has the [[Definition:Power Series|power series expansion]]:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} +... | From [[Derivative of Cosine Function]]:
:$\dfrac \d {\d x} \cos x = -\sin x$
From [[Derivative of Sine Function]]:
:$\dfrac \d {\d x} \sin x = \cos x$
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac {\d^2} {\d x^2} \cos x
| r = -\cos x
| c =
}}
{{eqn | l = \dfrac {\d^3} {\d x^3} \cos x
| r = \sin x
... | Power Series Expansion for Cosine Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cosine_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cosine_Function | [
"Cosine Function",
"Examples of Power Series",
"Taylor Series"
] | [
"Definition:Cosine",
"Definition:Power Series"
] | [
"Derivative of Cosine Function",
"Derivative of Sine Function",
"Definition:Maclaurin Series",
"Sine of Zero is Zero",
"Cosine of Zero is One",
"Series of Power over Factorial Converges",
"Definition:Convergent Series"
] |
proofwiki-8466 | Spacing Limit Theorem | Let $X_{\paren i}$ be the $i$th ordered statistic of $N$ samples from a continuous random distribution with density function $\map {f_X} x$.
Then the spacing between the ordered statistics given $X_{\paren i}$ converges in distribution to exponential for sufficiently large sampling according to:
:$N \paren {X_{\paren {... | Given $i$ and $N$, the ordered statistic $X_{\paren i}$ has the probability density function:
:$\map {f_{X_{\paren i} } } {x \mid i, N} = \dfrac {N!} {\paren {i - 1}! \paren {N - i}!} \map {F_X} x^{i - 1} \paren {1 - \map {F_X} x}^{N - i} \map {f_X} x$
where $\map {F_X} x$ is the cumulative distribution function of $X$... | Let $X_{\paren i}$ be the $i$th [[Definition:Ordered Statistic|ordered statistic]] of $N$ samples from a [[Definition:Continuous Random Variable|continuous random]] distribution with [[Definition:Probability Density Function|density function]] $\map {f_X} x$.
Then the spacing between the ordered statistics [[Definitio... | [[Definition:Conditional Probability|Given]] $i$ and $N$, the ordered statistic $X_{\paren i}$ has the probability density function:
:$\map {f_{X_{\paren i} } } {x \mid i, N} = \dfrac {N!} {\paren {i - 1}! \paren {N - i}!} \map {F_X} x^{i - 1} \paren {1 - \map {F_X} x}^{N - i} \map {f_X} x$
where $\map {F_X} x$ is the... | Spacing Limit Theorem | https://proofwiki.org/wiki/Spacing_Limit_Theorem | https://proofwiki.org/wiki/Spacing_Limit_Theorem | [
"Probability Theory",
"Named Theorems"
] | [
"Definition:Ordered Statistic",
"Definition:Random Variable/Continuous",
"Definition:Probability Density Function",
"Definition:Conditional Probability",
"Definition:Convergence",
"Definition:Exponential Distribution"
] | [
"Definition:Conditional Probability",
"Definition:Cumulative Distribution Function",
"Definition:Conditional Probability",
"Definition:Conditional Probability",
"Taylor's Theorem",
"Definition:Exponential Function/Real/Limit of Sequence",
"Category:Probability Theory",
"Category:Named Theorems"
] |
proofwiki-8467 | Rational Number Space is Topological Space | Let $\struct {\Q, \tau_d}$ be the rational number space formed by the rational numbers $\Q$ under the usual (Euclidean) topology $\tau_d$.
Then $\tau_d$ forms a topology. | From Rational Numbers form Metric Space we have that $\Q$ is a metric space under the Euclidean metric.
From Metric Induces Topology, it follows that the Euclidean topology forms a topology on $\Q$.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] formed by the [[Definition:Rational Number|rational numbers]] $\Q$ under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Then $\tau_d$ forms a [[Definition:Topology|topology]]. | From [[Rational Numbers form Metric Space]] we have that $\Q$ is a [[Definition:Metric Space|metric space]] under the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]].
From [[Metric Induces Topology]], it follows that the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] form... | Rational Number Space is Topological Space/Proof 1 | https://proofwiki.org/wiki/Rational_Number_Space_is_Topological_Space | https://proofwiki.org/wiki/Rational_Number_Space_is_Topological_Space/Proof_1 | [
"Rational Number Space is Topological Space",
"Rational Number Space"
] | [
"Definition:Rational Number Space",
"Definition:Rational Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Topology"
] | [
"Rational Numbers form Metric Space",
"Definition:Metric Space",
"Definition:Euclidean Metric/Real Number Line",
"Metric Induces Topology",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Topology"
] |
proofwiki-8468 | Rational Number Space is Topological Space | Let $\struct {\Q, \tau_d}$ be the rational number space formed by the rational numbers $\Q$ under the usual (Euclidean) topology $\tau_d$.
Then $\tau_d$ forms a topology. | Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$.
By definition of rational numbers, $\Q \subseteq \R$.
From Topological Subspace is Topological Space we have that $\struct {\Q, \tau_d}$ is a topological space.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] formed by the [[Definition:Rational Number|rational numbers]] $\Q$ under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Then $\tau_d$ forms a [[Definition:Topology|topology]]. | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number space]] $\R$ under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
By definition of [[Definition:Rational Number|rational numbers]], $\Q \subseteq \R$.
From [[Topological Subspace is Topological Space]] ... | Rational Number Space is Topological Space/Proof 2 | https://proofwiki.org/wiki/Rational_Number_Space_is_Topological_Space | https://proofwiki.org/wiki/Rational_Number_Space_is_Topological_Space/Proof_2 | [
"Rational Number Space is Topological Space",
"Rational Number Space"
] | [
"Definition:Rational Number Space",
"Definition:Rational Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Topology"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Rational Number",
"Topological Subspace is Topological Space",
"Definition:Topological Space"
] |
proofwiki-8469 | Irrational Number Space is Topological Space | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space formed by the irrational numbers $\R \setminus \Q$ under the usual (Euclidean) topology $\tau_d$.
Then $\tau_d$ forms a topology. | Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$.
By definition of irrational numbers, $\R \setminus \Q \subseteq \R$.
From Topological Subspace is Topological Space we have that $\struct {\R \setminus \Q, \tau_d}$ is a topology.
{{qed}} | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] formed by the [[Definition:Irrational Number|irrational numbers]] $\R \setminus \Q$ under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Then $\tau_d$ forms a [[... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number space]] $\R$ under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
By definition of [[Definition:Irrational Number|irrational numbers]], $\R \setminus \Q \subseteq \R$.
From [[Topological Subspace is Top... | Irrational Number Space is Topological Space | https://proofwiki.org/wiki/Irrational_Number_Space_is_Topological_Space | https://proofwiki.org/wiki/Irrational_Number_Space_is_Topological_Space | [
"Irrational Number Space"
] | [
"Definition:Irrational Number Space",
"Definition:Irrational Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Topology"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Irrational Number",
"Topological Subspace is Topological Space",
"Definition:Topology"
] |
proofwiki-8470 | Union of Closures of Singleton Rationals is Rational Space | Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$.
Let $B_\alpha$ denote the singleton containing the rational number $\alpha$.
Then the union of the closures in the set of real numbers $\R$ of all $B_\alpha$ is $\Q$:
:$\ds \bigcup_{\alpha \mathop \in \Q} \map \cl {B_... | Let $\alpha \in \Q$.
By Real Number is Closed in Real Number Line, $B_\alpha = \set \alpha$ is closed in $\R$.
From Closed Set Equals its Closure, it follows that:
:$B_\alpha = \map \cl {B_\alpha}$
Hence the result.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Let $B_\alpha$ denote the [[Definition:Singleton|singleton]] containing the [[Definition:Rational Number|rational number]] $\... | Let $\alpha \in \Q$.
By [[Real Number is Closed in Real Number Line]], $B_\alpha = \set \alpha$ is [[Definition:Closed Set (Topology)|closed]] in $\R$.
From [[Closed Set Equals its Closure]], it follows that:
:$B_\alpha = \map \cl {B_\alpha}$
Hence the result.
{{qed}} | Union of Closures of Singleton Rationals is Rational Space | https://proofwiki.org/wiki/Union_of_Closures_of_Singleton_Rationals_is_Rational_Space | https://proofwiki.org/wiki/Union_of_Closures_of_Singleton_Rationals_is_Rational_Space | [
"Rational Number Space",
"Singletons",
"Examples of Set Closures"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Singleton",
"Definition:Rational Number",
"Definition:Set Union",
"Definition:Closure (Topology)",
"Definition:Set",
"Definition:Real Number"
] | [
"Real Number is Closed in Real Number Line",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure"
] |
proofwiki-8471 | Closure of Union of Singleton Rationals is Real Number Line | Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$.
Let $B_\alpha$ be the singleton containing the rational number $\alpha$.
Then the closure in the set of real numbers $\R$ of the union of all $B_\alpha$ is $\R$ itself:
:$\ds \map \cl {\bigcup_{\alpha \mathop \in \Q} ... | By definition:
:$B_\alpha = \set \alpha$
Thus:
:$\ds \bigcup_{\alpha \mathop \in \Q} B_\alpha = \Q$
The result follows from Closure of Rational Numbers is Real Numbers.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Let $B_\alpha$ be the [[Definition:Singleton|singleton]] containing the [[Definition:Rational Number|rational number]] $\alph... | By definition:
:$B_\alpha = \set \alpha$
Thus:
:$\ds \bigcup_{\alpha \mathop \in \Q} B_\alpha = \Q$
The result follows from [[Closure of Rational Numbers is Real Numbers]].
{{qed}} | Closure of Union of Singleton Rationals is Real Number Line | https://proofwiki.org/wiki/Closure_of_Union_of_Singleton_Rationals_is_Real_Number_Line | https://proofwiki.org/wiki/Closure_of_Union_of_Singleton_Rationals_is_Real_Number_Line | [
"Rational Number Space",
"Real Number Line with Euclidean Topology",
"Singletons",
"Examples of Set Closures"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Singleton",
"Definition:Rational Number",
"Definition:Closure (Topology)",
"Definition:Set",
"Definition:Real Number",
"Definition:Set Union"
] | [
"Closure of Rational Numbers is Real Numbers"
] |
proofwiki-8472 | Closure of Rational Numbers is Real Numbers | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\struct {\Q, \tau_d}$ be the rational number space under the same topology.
Then:
:$\Q^- = \R$
where $\Q^-$ denotes the closure of $\Q$. | From Rationals are Everywhere Dense in Topological Space of Reals, $\Q$ is everywhere dense in $\R$.
It follows by definition of everywhere dense that $\Q^- = \R$.
{{qed}}
Category:Rational Number Space
Category:Real Number Line with Euclidean Topology
Category:Set Closures
h1q0jst1ieie6aoa4ptu2s188m4f779 | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|same topology]].
T... | From [[Rationals are Everywhere Dense in Topological Space of Reals]], $\Q$ is [[Definition:Everywhere Dense|everywhere dense]] in $\R$.
It follows by definition of [[Definition:Everywhere Dense|everywhere dense]] that $\Q^- = \R$.
{{qed}}
[[Category:Rational Number Space]]
[[Category:Real Number Line with Euclidean ... | Closure of Rational Numbers is Real Numbers | https://proofwiki.org/wiki/Closure_of_Rational_Numbers_is_Real_Numbers | https://proofwiki.org/wiki/Closure_of_Rational_Numbers_is_Real_Numbers | [
"Rational Number Space",
"Real Number Line with Euclidean Topology",
"Set Closures"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Closure (Topology)"
] | [
"Rational Numbers are Everywhere Dense in Set of Real Numbers/Topology",
"Definition:Everywhere Dense",
"Definition:Everywhere Dense",
"Category:Rational Number Space",
"Category:Real Number Line with Euclidean Topology",
"Category:Set Closures"
] |
proofwiki-8473 | Exterior of Union of Singleton Rationals is Empty | Let $B_\alpha$ be the singleton containing the rational number $\alpha$.
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology $\tau_d$.
Then the exterior in $\struct {\R, \tau_d}$ of the union of all $B_\alpha$ is the empty set:
:$\ds \paren {\bigcup_{\alpha \mathop \in \Q} B_\alpha}^e... | By definition:
:$B_\alpha = \set \alpha$
Thus:
:$\ds \bigcup_{\alpha \mathop \in \Q} B_\alpha = \Q$
By definition, the exterior of $\Q$ is the complement of the closure of $\Q$ in $\R$.
By Closure of Rational Numbers is Real Numbers:
:$\Q^- = \R$
By Relative Complement with Self is Empty Set:
:$\relcomp \R \R = \O$
Hen... | Let $B_\alpha$ be the [[Definition:Singleton|singleton]] containing the [[Definition:Rational Number|rational number]] $\alpha$.
Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]] $\tau_d$.
Then the [[Definition:Exterior (Top... | By definition:
:$B_\alpha = \set \alpha$
Thus:
:$\ds \bigcup_{\alpha \mathop \in \Q} B_\alpha = \Q$
By definition, the [[Definition:Exterior (Topology)|exterior]] of $\Q$ is the [[Definition:Set Complement|complement]] of the [[Definition:Closure (Topology)|closure]] of $\Q$ in $\R$.
By [[Closure of Rational Numbers... | Exterior of Union of Singleton Rationals is Empty | https://proofwiki.org/wiki/Exterior_of_Union_of_Singleton_Rationals_is_Empty | https://proofwiki.org/wiki/Exterior_of_Union_of_Singleton_Rationals_is_Empty | [
"Rational Number Space",
"Singletons",
"Examples of Set Exteriors"
] | [
"Definition:Singleton",
"Definition:Rational Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Exterior (Topology)",
"Definition:Set Union",
"Definition:Empty Set"
] | [
"Definition:Exterior (Topology)",
"Definition:Set Complement",
"Definition:Closure (Topology)",
"Closure of Rational Numbers is Real Numbers",
"Relative Complement with Self is Empty Set"
] |
proofwiki-8474 | Real Number is Closed in Real Number Line | Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology.
Let $\alpha \in \R$ be a real number.
Then $\set \alpha$ is closed in $\struct {\R, \tau}$. | From Open Sets in Real Number Line, the set:
:$S := \openint \gets \alpha \cup \openint \alpha \to$
is open in $\R$.
Thus by definition of closed, its complement relative to $\R$:
:$\R \setminus S = \set \alpha$
is closed in $\R$.
{{qed}}
Category:Real Number Line with Euclidean Topology
Category:Closed Sets
h6rshhni5f... | Let $\struct {\R, \tau}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\alpha \in \R$ be a [[Definition:Real Number|real number]].
Then $\set \alpha$ is [[Definition:Closed Set (Topology)|closed]] in $\struct {\R, \tau}$. | From [[Open Sets in Real Number Line]], the [[Definition:Set|set]]:
:$S := \openint \gets \alpha \cup \openint \alpha \to$
is [[Definition:Open Set (Topology)|open]] in $\R$.
Thus by definition of [[Definition:Closed Set (Topology)|closed]], its [[Definition:Relative Complement|complement relative to $\R$]]:
:$\R \set... | Real Number is Closed in Real Number Line | https://proofwiki.org/wiki/Real_Number_is_Closed_in_Real_Number_Line | https://proofwiki.org/wiki/Real_Number_is_Closed_in_Real_Number_Line | [
"Real Number Line with Euclidean Topology",
"Closed Sets"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Number",
"Definition:Closed Set/Topology"
] | [
"Open Sets in Real Number Line",
"Definition:Set",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Category:Real Number Line with Euclidean Topology",
"Category:Closed Sets"
] |
proofwiki-8475 | Intersection of Exteriors of Singleton Rationals is Irrationals | Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$.
Let $B_\alpha$ be the singleton containing the rational number $\alpha$.
Then:
:$\ds \bigcap_{\alpha \mathop \in \Q} B_\alpha^e = \R \setminus \Q$
where $B_\alpha^e$ denotes the exterior of $B_\alpha$ in $\R$. | {{begin-eqn}}
{{eqn | l = \bigcap_{\alpha \mathop \in \Q} B_\alpha^e
| r = \bigcap_{\alpha \mathop \in \Q} \R \setminus B_\alpha^-
| c = {{Defof|Exterior (Topology)|index = 1|Exterior}}
}}
{{eqn | r = \bigcap_{\alpha \mathop \in \Q} \R \setminus B_\alpha
| c = Real Number is Closed in Real Number Line... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Let $B_\alpha$ be the [[Definition:Singleton|singleton]] containing the [[Definition:Rational Number|rational number]] $\alph... | {{begin-eqn}}
{{eqn | l = \bigcap_{\alpha \mathop \in \Q} B_\alpha^e
| r = \bigcap_{\alpha \mathop \in \Q} \R \setminus B_\alpha^-
| c = {{Defof|Exterior (Topology)|index = 1|Exterior}}
}}
{{eqn | r = \bigcap_{\alpha \mathop \in \Q} \R \setminus B_\alpha
| c = [[Real Number is Closed in Real Number Li... | Intersection of Exteriors of Singleton Rationals is Irrationals | https://proofwiki.org/wiki/Intersection_of_Exteriors_of_Singleton_Rationals_is_Irrationals | https://proofwiki.org/wiki/Intersection_of_Exteriors_of_Singleton_Rationals_is_Irrationals | [
"Rational Number Space",
"Singletons",
"Examples of Set Exteriors"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Singleton",
"Definition:Rational Number",
"Definition:Exterior (Topology)"
] | [
"Real Number is Closed in Real Number Line",
"De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union"
] |
proofwiki-8476 | Rational Numbers form F-Sigma Set in Reals | Let $\Q$ be the set of rational numbers.
Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology.
Then $\Q$ is a $F_\sigma$ set in $\R$. | Let $\alpha \in \Q$ be a rational number.
From Real Number is Closed in Real Number Line, $\set \alpha$ is a closed set of $\R$.
From Rational Numbers are Countably Infinite, $\ds \bigcup_{\alpha \mathop \in \Q} \set \alpha$ is a countable union.
Thus $\Q = \ds \bigcup_{\alpha \mathop \in \Q} \set \alpha$ is a countabl... | Let $\Q$ be the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]].
Let $\struct {\R, \tau}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\Q$ is a [[Definition:F-Sigma Set|$F_\sigma$ set]] in $\R$. | Let $\alpha \in \Q$ be a [[Definition:Rational Number|rational number]].
From [[Real Number is Closed in Real Number Line]], $\set \alpha$ is a [[Definition:Closed Set (Topology)|closed set]] of $\R$.
From [[Rational Numbers are Countably Infinite]], $\ds \bigcup_{\alpha \mathop \in \Q} \set \alpha$ is a [[Definition... | Rational Numbers form F-Sigma Set in Reals | https://proofwiki.org/wiki/Rational_Numbers_form_F-Sigma_Set_in_Reals | https://proofwiki.org/wiki/Rational_Numbers_form_F-Sigma_Set_in_Reals | [
"Rational Numbers",
"Real Number Line with Euclidean Topology",
"Examples of F-Sigma Sets"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:F-Sigma Set"
] | [
"Definition:Rational Number",
"Real Number is Closed in Real Number Line",
"Definition:Closed Set/Topology",
"Rational Numbers are Countably Infinite",
"Definition:Set Union/Countable Union",
"Definition:Set Union/Countable Union",
"Definition:Closed Set/Topology",
"Definition:F-Sigma Set"
] |
proofwiki-8477 | Set of Rational Numbers is not Closed in Reals | Let $\Q$ be the set of rational numbers.
Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.
Then $\Q$ is not closed in $\R$. | Let $\alpha \in \R \setminus \Q$.
Let $I := \openint a b$ be an open interval in $\R$ such that $\alpha \in I$.
By Between two Real Numbers exists Rational Number:
:$\exists \beta \in \Q: \beta \in I$.
Thus $I$ contains elements of $\Q$ and so $\R \setminus \Q$ is not open in $\R$.
Thus by definition, $\Q$ is not close... | Let $\Q$ be the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]].
Let $\struct {\R, \tau}$ denote the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\Q$ is not [[Definition:Closed Set (Topology)|closed]] in $\R$. | Let $\alpha \in \R \setminus \Q$.
Let $I := \openint a b$ be an [[Definition:Open Real Interval|open interval]] in $\R$ such that $\alpha \in I$.
By [[Between two Real Numbers exists Rational Number]]:
:$\exists \beta \in \Q: \beta \in I$.
Thus $I$ contains elements of $\Q$ and so $\R \setminus \Q$ is not [[Definiti... | Set of Rational Numbers is not Closed in Reals | https://proofwiki.org/wiki/Set_of_Rational_Numbers_is_not_Closed_in_Reals | https://proofwiki.org/wiki/Set_of_Rational_Numbers_is_not_Closed_in_Reals | [
"Rational Numbers",
"Real Number Line with Euclidean Topology"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Closed Set/Topology"
] | [
"Definition:Real Interval/Open",
"Between two Real Numbers exists Rational Number",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology"
] |
proofwiki-8478 | Set of Rational Numbers is not G-Delta Set in Reals | Let $\Q$ be the set of rational numbers.
Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.
Then $\Q$ is not a $G_\delta$ set in $\R$. | {{AimForCont}} $\Q$ is a $G_\delta$ set in $\R$.
Let $\Q = \ds \bigcap_{i \mathop \in \N} V_i$.
Since Rational Numbers are Countably Infinite, there exists an enumeration of $\Q$.
Write $\Q = \set {q_i}_{i \mathop \in \N}$.
Define $U_i = V_i \setminus \set {q_i}$.
We show that $U_i$ is dense in $\R$.
:Let $A \subseteq ... | Let $\Q$ be the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]].
Let $\struct {\R, \tau}$ denote the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\Q$ is not a [[Definition:G-Delta Set|$G_\delta$ set]] in $\R$. | {{AimForCont}} $\Q$ is a [[Definition:G-Delta Set|$G_\delta$ set]] in $\R$.
Let $\Q = \ds \bigcap_{i \mathop \in \N} V_i$.
Since [[Rational Numbers are Countably Infinite]], there exists an [[Definition:Enumeration|enumeration]] of $\Q$.
Write $\Q = \set {q_i}_{i \mathop \in \N}$.
Define $U_i = V_i \setminus \set {... | Set of Rational Numbers is not G-Delta Set in Reals | https://proofwiki.org/wiki/Set_of_Rational_Numbers_is_not_G-Delta_Set_in_Reals | https://proofwiki.org/wiki/Set_of_Rational_Numbers_is_not_G-Delta_Set_in_Reals | [
"Rational Numbers",
"Real Number Line with Euclidean Topology",
"Examples of G-Delta Sets"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:G-Delta Set"
] | [
"Definition:G-Delta Set",
"Rational Numbers are Countably Infinite",
"Definition:Enumeration",
"Definition:Everywhere Dense",
"Definition:Open Set/Topology",
"Basis for Euclidean Topology on Real Number Line",
"Definition:Real Interval/Open",
"Between two Real Numbers exists Rational Number",
"Betwe... |
proofwiki-8479 | Irrational Numbers form G-Delta Set in Reals | Let $\R \setminus \Q$ denote the set of irrational numbers.
Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.
Then $\R \setminus \Q$ forms a $G_\delta$ set in $\R$. | {{begin-eqn}}
{{eqn | l = \Q
| r = \bigcup_{\alpha \mathop \in \Q} \set \alpha
| c =
}}
{{eqn | ll= \leadsto
| l = \R \setminus \Q
| r = \R \setminus \bigcup_{\alpha \mathop \in \Q} \set \alpha
| c =
}}
{{eqn | r = \bigcap_{\alpha \mathop \in \Q} \paren {\R \setminus \set \alpha}
|... | Let $\R \setminus \Q$ denote the [[Definition:Set|set]] of [[Definition:Irrational Number|irrational numbers]].
Let $\struct {\R, \tau}$ denote the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\R \setminus \Q$ forms a [[Definition:G-Delta Set|$G_... | {{begin-eqn}}
{{eqn | l = \Q
| r = \bigcup_{\alpha \mathop \in \Q} \set \alpha
| c =
}}
{{eqn | ll= \leadsto
| l = \R \setminus \Q
| r = \R \setminus \bigcup_{\alpha \mathop \in \Q} \set \alpha
| c =
}}
{{eqn | r = \bigcap_{\alpha \mathop \in \Q} \paren {\R \setminus \set \alpha}
|... | Irrational Numbers form G-Delta Set in Reals | https://proofwiki.org/wiki/Irrational_Numbers_form_G-Delta_Set_in_Reals | https://proofwiki.org/wiki/Irrational_Numbers_form_G-Delta_Set_in_Reals | [
"Irrational Numbers",
"Real Number Line with Euclidean Topology",
"Examples of G-Delta Sets"
] | [
"Definition:Set",
"Definition:Irrational Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:G-Delta Set"
] | [
"De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union",
"Rational Numbers are Countably Infinite"
] |
proofwiki-8480 | Eigenvalue of Matrix Power | Let $A$ be a square matrix.
Let $\lambda$ be an eigenvalue of $A$ and $\mathbf v$ be the corresponding eigenvector.
Then:
:$A^n \mathbf v = \lambda^n \mathbf v$
holds for each positive integer $n$.
Here $A^n$ is the $n$th power of $A$. | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$A^n \mathbf v = \lambda^n \mathbf v$ | Let $A$ be a [[Definition:Square Matrix|square matrix]].
Let $\lambda$ be an [[Definition:Eigenvalue of Square Matrix|eigenvalue]] of $A$ and $\mathbf v$ be the corresponding [[Definition:Eigenvector of Square Matrix|eigenvector]].
Then:
:$A^n \mathbf v = \lambda^n \mathbf v$
holds for each [[Definition:Positive In... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$A^n \mathbf v = \lambda^n \mathbf v$ | Eigenvalue of Matrix Power | https://proofwiki.org/wiki/Eigenvalue_of_Matrix_Power | https://proofwiki.org/wiki/Eigenvalue_of_Matrix_Power | [
"Matrix Algebra",
"Eigenvalues",
"Proofs by Induction"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Eigenvalue/Square Matrix",
"Definition:Eigenvector/Square Matrix",
"Definition:Positive/Integer",
"Definition:Power of Matrix"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-8481 | Closure of Intersection of Rationals and Irrationals is Empty Set | Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.
Let $\Q$ be the set of rational numbers.
Then:
:$\paren {\Q \cap \paren {\R \setminus \Q} }^- = \O$
where:
:$\R \setminus \Q$ denotes the set of irrational numbers
:$\paren {\Q \cap \paren {\R \setminus \Q} }^-$ denotes the closur... | From Set Difference Intersection with Second Set is Empty Set:
:$\Q \cap \paren {\R \setminus \Q} = \O$
By Empty Set is Closed in Topological Space, $\O$ is closed in $\R$.
From Closed Set Equals its Closure:
:$\O^- = \O$
Hence the result.
{{qed}} | Let $\struct {\R, \tau}$ denote the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\Q$ be the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]].
Then:
:$\paren {\Q \cap \paren {\R \setminus \Q} }^- = \O$
where:
:$\R \setminus... | From [[Set Difference Intersection with Second Set is Empty Set]]:
:$\Q \cap \paren {\R \setminus \Q} = \O$
By [[Empty Set is Closed in Topological Space]], $\O$ is [[Definition:Closed Set (Topology)|closed]] in $\R$.
From [[Closed Set Equals its Closure]]:
:$\O^- = \O$
Hence the result.
{{qed}} | Closure of Intersection of Rationals and Irrationals is Empty Set | https://proofwiki.org/wiki/Closure_of_Intersection_of_Rationals_and_Irrationals_is_Empty_Set | https://proofwiki.org/wiki/Closure_of_Intersection_of_Rationals_and_Irrationals_is_Empty_Set | [
"Rational Numbers",
"Irrational Numbers",
"Real Number Line with Euclidean Topology",
"Examples of Set Closures"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Set",
"Definition:Rational Number",
"Definition:Set",
"Definition:Irrational Number",
"Definition:Closure (Topology)"
] | [
"Set Difference Intersection with Second Set is Empty Set",
"Empty Set is Closed/Topological Space",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure"
] |
proofwiki-8482 | Intersection of Closures of Rationals and Irrationals is Reals | Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology.
Let $\Q$ be the set of rational numbers.
Then:
:$\Q^- \cap \paren {\R \setminus \Q}^- = \R$
where:
:$\R \setminus \Q$ denotes the set of irrational numbers
:$\Q^-$ denotes the closure of $\Q$. | From Closure of Rational Numbers is Real Numbers:
:$\Q^- = \R$
From Closure of Irrational Numbers is Real Numbers:
:$\paren {\R \setminus \Q}^- = \R$
The result follows from Set Intersection is Idempotent.
{{qed}} | Let $\struct {\R, \tau}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\Q$ be the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]].
Then:
:$\Q^- \cap \paren {\R \setminus \Q}^- = \R$
where:
:$\R \setminus \Q$ denotes... | From [[Closure of Rational Numbers is Real Numbers]]:
:$\Q^- = \R$
From [[Closure of Irrational Numbers is Real Numbers]]:
:$\paren {\R \setminus \Q}^- = \R$
The result follows from [[Set Intersection is Idempotent]].
{{qed}} | Intersection of Closures of Rationals and Irrationals is Reals | https://proofwiki.org/wiki/Intersection_of_Closures_of_Rationals_and_Irrationals_is_Reals | https://proofwiki.org/wiki/Intersection_of_Closures_of_Rationals_and_Irrationals_is_Reals | [
"Rational Numbers",
"Irrational Numbers",
"Real Number Line with Euclidean Topology",
"Examples of Set Closures"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Set",
"Definition:Rational Number",
"Definition:Set",
"Definition:Irrational Number",
"Definition:Closure (Topology)"
] | [
"Closure of Rational Numbers is Real Numbers",
"Closure of Irrational Numbers is Real Numbers",
"Set Intersection is Idempotent"
] |
proofwiki-8483 | Irrational Numbers form Metric Space | Let $\mathbb I = \R \setminus \Q$ be the set of all irrational numbers.
Let $d: \mathbb I \times \mathbb I \to \R$ be defined as:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the absolute value of $x$.
Then $d$ is a metric on $\mathbb I$ and so $\struct {\mathbb I, d}$ is a metric space. | From the definition of absolute value:
:$\size {x_1 - x_2} = \sqrt {\paren {x_1 - x_2}^2}$
This is the Euclidean metric.
in Euclidean Metric on Real Vector Space is Metric this is shown to be a metric.
{{qed}} | Let $\mathbb I = \R \setminus \Q$ be the [[Definition:Set|set]] of all [[Definition:Irrational Number|irrational numbers]].
Let $d: \mathbb I \times \mathbb I \to \R$ be defined as:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the [[Definition:Absolute Value|absolute value]] of $x$.
Then $d$ is a [[... | From the definition of [[Definition:Absolute Value|absolute value]]:
:$\size {x_1 - x_2} = \sqrt {\paren {x_1 - x_2}^2}$
This is the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]].
in [[Euclidean Metric on Real Vector Space is Metric]] this is shown to be a [[Definition:Metric|metric]].
{{qed}} | Irrational Numbers form Metric Space | https://proofwiki.org/wiki/Irrational_Numbers_form_Metric_Space | https://proofwiki.org/wiki/Irrational_Numbers_form_Metric_Space | [
"Irrational Number Space",
"Irrational Numbers",
"Examples of Euclidean Spaces",
"Examples of Metric Spaces"
] | [
"Definition:Set",
"Definition:Irrational Number",
"Definition:Absolute Value",
"Definition:Metric Space/Metric",
"Definition:Metric Space"
] | [
"Definition:Absolute Value",
"Definition:Euclidean Metric/Real Number Line",
"Euclidean Metric on Real Vector Space is Metric",
"Definition:Metric Space/Metric"
] |
proofwiki-8484 | Rational Number Space is Completely Normal | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is a completely normal space. | From Euclidean Space is Complete Metric Space, a Euclidean space is a metric space.
From Metric Space is Completely Normal it follows that $\struct {\Q, \tau_d}$ is a completely normal space.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is a [[Definition:Completely Normal Space|completely normal space]]. | From [[Euclidean Space is Complete Metric Space]], a [[Definition:Euclidean Space|Euclidean space]] is a [[Definition:Metric Space|metric space]].
From [[Metric Space is Completely Normal]] it follows that $\struct {\Q, \tau_d}$ is a [[Definition:Completely Normal Space|completely normal space]].
{{qed}} | Rational Number Space is Completely Normal | https://proofwiki.org/wiki/Rational_Number_Space_is_Completely_Normal | https://proofwiki.org/wiki/Rational_Number_Space_is_Completely_Normal | [
"Rational Number Space",
"Examples of Completely Normal Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Completely Normal Space"
] | [
"Euclidean Space is Complete Metric Space",
"Definition:Euclidean Space",
"Definition:Metric Space",
"Metric Space is Completely Normal",
"Definition:Completely Normal Space"
] |
proofwiki-8485 | Irrational Number Space is Completely Normal | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is a completely normal space. | From Euclidean Space is Complete Metric Space, a Euclidean space is a metric space.
Hence in particular $\struct {\R, \tau_d}$ is a metric space.
From Subspace of Metric Space is Metric Space, it follows that $\struct {\R \setminus \Q, \tau_d}$ is likewise a metric space.
From Metric Space is Completely Normal it follo... | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is a [[Definition:Completely Normal Space|completely normal space]]. | From [[Euclidean Space is Complete Metric Space]], a [[Definition:Euclidean Space|Euclidean space]] is a [[Definition:Metric Space|metric space]].
Hence in particular $\struct {\R, \tau_d}$ is a [[Definition:Metric Space|metric space]].
From [[Subspace of Metric Space is Metric Space]], it follows that $\struct {\R \... | Irrational Number Space is Completely Normal | https://proofwiki.org/wiki/Irrational_Number_Space_is_Completely_Normal | https://proofwiki.org/wiki/Irrational_Number_Space_is_Completely_Normal | [
"Irrational Number Space",
"Examples of Completely Normal Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Completely Normal Space"
] | [
"Euclidean Space is Complete Metric Space",
"Definition:Euclidean Space",
"Definition:Metric Space",
"Definition:Metric Space",
"Subspace of Metric Space is Metric Space",
"Definition:Metric Space",
"Metric Space is Completely Normal",
"Definition:Completely Normal Space"
] |
proofwiki-8486 | Rational Number Space is Paracompact | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is paracompact. | From Rational Numbers form Metric Space, $\struct {\Q, \tau_d}$ is a metric space.
The result follows from Metric Space is Paracompact.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is [[Definition:Paracompact Space|paracompact]]. | From [[Rational Numbers form Metric Space]], $\struct {\Q, \tau_d}$ is a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Paracompact]].
{{qed}} | Rational Number Space is Paracompact | https://proofwiki.org/wiki/Rational_Number_Space_is_Paracompact | https://proofwiki.org/wiki/Rational_Number_Space_is_Paracompact | [
"Rational Number Space",
"Examples of Paracompact Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Paracompact Space"
] | [
"Rational Numbers form Metric Space",
"Definition:Metric Space",
"Metric Space is Paracompact"
] |
proofwiki-8487 | Irrational Number Space is Paracompact | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is paracompact. | From Euclidean Space is Complete Metric Space, a Euclidean space is a metric space.
The result follows from Metric Space is Paracompact.
{{qed}} | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is [[Definition:Paracompact Space|paracompact]]. | From [[Euclidean Space is Complete Metric Space]], a [[Definition:Euclidean Space|Euclidean space]] is a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Paracompact]].
{{qed}} | Irrational Number Space is Paracompact | https://proofwiki.org/wiki/Irrational_Number_Space_is_Paracompact | https://proofwiki.org/wiki/Irrational_Number_Space_is_Paracompact | [
"Irrational Number Space",
"Examples of Paracompact Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Paracompact Space"
] | [
"Euclidean Space is Complete Metric Space",
"Definition:Euclidean Space",
"Definition:Metric Space",
"Metric Space is Paracompact"
] |
proofwiki-8488 | Euclidean Plus Metric is Metric | Let $\R$ be the set of real numbers.
Let $d: \R \times \R \to \R$ be the Euclidean plus metric:
:$\map d {x, y} := \size {x - y} + \ds \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j}} - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$
Then $d$ is indeed a me... | Recall that $\set {r_j}_{j \mathop \in \N}$ is an enumeration of the rational numbers $\Q$.
Also, we note that:
{{begin-eqn}}
{{eqn | o =
| r = \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {x - r_j} } } }
|... | Let $\R$ be the [[Definition:Real Number|set of real numbers]].
Let $d: \R \times \R \to \R$ be the [[Definition:Euclidean Plus Metric|Euclidean plus metric]]:
:$\map d {x, y} := \size {x - y} + \ds \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j}} - \max_{j \math... | Recall that $\set {r_j}_{j \mathop \in \N}$ is an [[Definition:Countably Infinite Enumeration|enumeration]] of the [[Definition:Rational Numbers|rational numbers]] $\Q$.
Also, we note that:
{{begin-eqn}}
{{eqn | o =
| r = \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {... | Euclidean Plus Metric is Metric | https://proofwiki.org/wiki/Euclidean_Plus_Metric_is_Metric | https://proofwiki.org/wiki/Euclidean_Plus_Metric_is_Metric | [
"Euclidean Plus Metric"
] | [
"Definition:Real Number",
"Definition:Euclidean Plus Metric",
"Definition:Metric Space/Metric"
] | [
"Definition:Enumeration/Countably Infinite",
"Definition:Rational Number",
"Sum of Infinite Geometric Sequence/Corollary 1",
"Definition:Convergent Series/Number Field",
"Axiom:Metric Space Axioms",
"Definition:Metric Space/Metric"
] |
proofwiki-8489 | Open Ball in Euclidean Plus Metric is Subset of Equivalent Ball in Euclidean Metric | Let $\R$ be the set of real numbers.
Let $d: \R \times \R \to \R$ be the Euclidean plus metric:
:$\map d {x, y} := \size {x - y} + \ds \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$
Let $d': \R \times \R ... | Let $p \in \R$.
Let $\epsilon \in \R_{>0}$.
Let $x \in \map {B_\epsilon} {p; d}$.
Then:
{{begin-eqn}}
{{eqn | l = \map d {x, p}
| o = <
| r = \epsilon
| c =
}}
{{eqn | ll= \leadsto
| l = \epsilon
| o = >
| r = \size {x - y} + \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\... | Let $\R$ be the [[Definition:Real Number|set of real numbers]].
Let $d: \R \times \R \to \R$ be the [[Definition:Euclidean Plus Metric|Euclidean plus metric]]:
:$\map d {x, y} := \size {x - y} + \ds \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mat... | Let $p \in \R$.
Let $\epsilon \in \R_{>0}$.
Let $x \in \map {B_\epsilon} {p; d}$.
Then:
{{begin-eqn}}
{{eqn | l = \map d {x, p}
| o = <
| r = \epsilon
| c =
}}
{{eqn | ll= \leadsto
| l = \epsilon
| o = >
| r = \size {x - y} + \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \si... | Open Ball in Euclidean Plus Metric is Subset of Equivalent Ball in Euclidean Metric | https://proofwiki.org/wiki/Open_Ball_in_Euclidean_Plus_Metric_is_Subset_of_Equivalent_Ball_in_Euclidean_Metric | https://proofwiki.org/wiki/Open_Ball_in_Euclidean_Plus_Metric_is_Subset_of_Equivalent_Ball_in_Euclidean_Metric | [
"Euclidean Plus Metric",
"Examples of Open Balls"
] | [
"Definition:Real Number",
"Definition:Euclidean Plus Metric",
"Definition:Euclidean Metric/Real Number Plane",
"Definition:Strictly Positive/Real Number",
"Definition:Open Ball",
"Definition:Open Ball"
] | [] |
proofwiki-8490 | Irrational Number Space is Completely Metrizable | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is completely metrizable. | Let $\set {r_i}$ be an enumeration of rational numbers.
Let $d: \R \times \R \to \R$ be the Euclidean plus metric:
:$\ds \map d {x, y} := \size {x - y} + \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$
Let... | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is [[Definition:Completely Metrizable Space|completely metrizable]]. | Let $\set {r_i}$ be an [[Definition:Countably Infinite Enumeration|enumeration]] of [[Definition:Rational Number|rational numbers]].
Let $d: \R \times \R \to \R$ be the [[Definition:Euclidean Plus Metric|Euclidean plus metric]]:
:$\ds \map d {x, y} := \size {x - y} + \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \s... | Irrational Number Space is Completely Metrizable | https://proofwiki.org/wiki/Irrational_Number_Space_is_Completely_Metrizable | https://proofwiki.org/wiki/Irrational_Number_Space_is_Completely_Metrizable | [
"Irrational Number Space",
"Examples of Complete Metric Spaces",
"Examples of Completely Metrizable Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Completely Metrizable Space"
] | [
"Definition:Enumeration/Countably Infinite",
"Definition:Rational Number",
"Definition:Euclidean Plus Metric",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Rational Number",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Seque... |
proofwiki-8491 | Irrational Number Space is Non-Meager | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is non-meager. | From Irrational Number Space is Completely Metrizable, $\struct {\R \setminus \Q, d}$ is a complete metric space.
From the Baire Category Theorem, a complete metric space is also a Baire space.
The result follows from Baire Space is Non-Meager.
{{qed}} | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is [[Definition:Non-Meager Space|non-meager]]. | From [[Irrational Number Space is Completely Metrizable]], $\struct {\R \setminus \Q, d}$ is a [[Definition:Complete Metric Space|complete metric space]].
From the [[Baire Category Theorem]], a [[Definition:Complete Metric Space|complete metric space]] is also a [[Definition:Baire Space (Topology)|Baire space]].
The ... | Irrational Number Space is Non-Meager | https://proofwiki.org/wiki/Irrational_Number_Space_is_Non-Meager | https://proofwiki.org/wiki/Irrational_Number_Space_is_Non-Meager | [
"Irrational Number Space",
"Examples of Non-Meager Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Meager Space/Non-Meager"
] | [
"Irrational Number Space is Completely Metrizable",
"Definition:Complete Metric Space",
"Baire Category Theorem",
"Definition:Complete Metric Space",
"Definition:Baire Space (Topology)",
"Baire Space is Non-Meager"
] |
proofwiki-8492 | Rational Number Space is Meager | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is meager. | From Rational Numbers are Countably Infinite, $\Q$ is a countable union of singleton subsets.
From Singleton Set is Nowhere Dense in Rational Space, each of those singleton subsets is nowhere dense in $\struct {\Q, \tau_d}$.
The result follows from definition of meager.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is [[Definition:Meager Space|meager]]. | From [[Rational Numbers are Countably Infinite]], $\Q$ is a [[Definition:Countable Union|countable union]] of [[Definition:Singleton|singleton]] [[Definition:Subset|subsets]].
From [[Singleton Set is Nowhere Dense in Rational Space]], each of those [[Definition:Singleton|singleton]] [[Definition:Subset|subsets]] is [[... | Rational Number Space is Meager | https://proofwiki.org/wiki/Rational_Number_Space_is_Meager | https://proofwiki.org/wiki/Rational_Number_Space_is_Meager | [
"Rational Number Space",
"Examples of Meager Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Meager Space"
] | [
"Rational Numbers are Countably Infinite",
"Definition:Set Union/Countable Union",
"Definition:Singleton",
"Definition:Subset",
"Singleton Set is Nowhere Dense in Rational Space",
"Definition:Singleton",
"Definition:Subset",
"Definition:Nowhere Dense",
"Definition:Meager Space"
] |
proofwiki-8493 | Rational Number Space is Separable | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is separable. | From Rational Numbers are Countably Infinite, $\Q$ is itself countable.
The result follows by Countable Space is Separable.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is [[Definition:Separable Space|separable]]. | From [[Rational Numbers are Countably Infinite]], $\Q$ is itself [[Definition:Countable Set|countable]].
The result follows by [[Countable Space is Separable]].
{{qed}} | Rational Number Space is Separable | https://proofwiki.org/wiki/Rational_Number_Space_is_Separable | https://proofwiki.org/wiki/Rational_Number_Space_is_Separable | [
"Rational Number Space",
"Examples of Separable Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Separable Space"
] | [
"Rational Numbers are Countably Infinite",
"Definition:Countable Set",
"Countable Space is Separable"
] |
proofwiki-8494 | Underlying Set of Topological Space is Everywhere Dense | Let $T = \struct {S, \tau}$ be a topological space.
Then the underlying set $S$ of $T$ is everywhere dense in $T$. | From Underlying Set of Topological Space is Closed, $S$ is closed in $T$.
From Closed Set Equals its Closure, $S = S^-$.
The result follows from definition of everywhere dense.
{{qed}}
Category:Denseness
9a0db4iczcdfb57dc5v1qkct58ghh81 | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then the [[Definition:Underlying Set of Topological Space|underlying set]] $S$ of $T$ is [[Definition:Everywhere Dense|everywhere dense]] in $T$. | From [[Underlying Set of Topological Space is Closed]], $S$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
From [[Closed Set Equals its Closure]], $S = S^-$.
The result follows from definition of [[Definition:Everywhere Dense|everywhere dense]].
{{qed}}
[[Category:Denseness]]
9a0db4iczcdfb57dc5v1qkct58ghh81 | Underlying Set of Topological Space is Everywhere Dense | https://proofwiki.org/wiki/Underlying_Set_of_Topological_Space_is_Everywhere_Dense | https://proofwiki.org/wiki/Underlying_Set_of_Topological_Space_is_Everywhere_Dense | [
"Denseness"
] | [
"Definition:Topological Space",
"Definition:Underlying Set/Topological Space",
"Definition:Everywhere Dense"
] | [
"Underlying Set of Topological Space is Closed",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Definition:Everywhere Dense",
"Category:Denseness"
] |
proofwiki-8495 | Irrational Number Space is Separable | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is separable. | Let $S$ be the set defined as:
:$S = \set {\pi + q: q \in \Q}$
From Rational Numbers are Countably Infinite, $\Q$ is countable.
Therefore $S$ is also countable.
From $\pi$ is Irrational:
:$\pi \in \R \setminus \Q$
It follows from Rational Addition is Closed that:
:$\forall q \in \Q: \pi + q \in \R \setminus \Q$
and so:... | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is [[Definition:Separable Space|separable]]. | Let $S$ be the [[Definition:Set|set]] defined as:
:$S = \set {\pi + q: q \in \Q}$
From [[Rational Numbers are Countably Infinite]], $\Q$ is [[Definition:Countable Set|countable]].
Therefore $S$ is also [[Definition:Countable Set|countable]].
From [[Pi is Irrational|$\pi$ is Irrational]]:
:$\pi \in \R \setminus \Q$
... | Irrational Number Space is Separable | https://proofwiki.org/wiki/Irrational_Number_Space_is_Separable | https://proofwiki.org/wiki/Irrational_Number_Space_is_Separable | [
"Irrational Number Space",
"Examples of Separable Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Separable Space"
] | [
"Definition:Set",
"Rational Numbers are Countably Infinite",
"Definition:Countable Set",
"Definition:Countable Set",
"Pi is Irrational",
"Rational Addition is Closed",
"Rationals plus Irrational are Everywhere Dense in Irrationals",
"Definition:Everywhere Dense",
"Definition:Countable Set",
"Defin... |
proofwiki-8496 | Rational Number Space is Second-Countable | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is second-countable. | From Rational Numbers form Metric Space, $\struct {\Q, \tau_d}$ is a metric space.
From Rational Number Space is Separable, $\struct {\Q, \tau_d}$ is a separable space.
The result follows from Separable Metric Space is Second-Countable.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is [[Definition:Second-Countable Space|second-countable]]. | From [[Rational Numbers form Metric Space]], $\struct {\Q, \tau_d}$ is a [[Definition:Metric Space|metric space]].
From [[Rational Number Space is Separable]], $\struct {\Q, \tau_d}$ is a [[Definition:Separable Space|separable space]].
The result follows from [[Separable Metric Space is Second-Countable]].
{{qed}} | Rational Number Space is Second-Countable | https://proofwiki.org/wiki/Rational_Number_Space_is_Second-Countable | https://proofwiki.org/wiki/Rational_Number_Space_is_Second-Countable | [
"Rational Number Space",
"Examples of Second-Countable Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Second-Countable Space"
] | [
"Rational Numbers form Metric Space",
"Definition:Metric Space",
"Rational Number Space is Separable",
"Definition:Separable Space",
"Separable Metric Space is Second-Countable"
] |
proofwiki-8497 | Irrational Number Space is Second-Countable | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is second-countable. | From Irrational Numbers form Metric Space, $\struct {\R \setminus \Q, \tau_d}$ is a metric space.
From Irrational Number Space is Separable, $\struct {\R \setminus \Q, \tau_d}$ is a separable space.
The result follows from Separable Metric Space is Second-Countable.
{{qed}} | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is [[Definition:Second-Countable Space|second-countable]]. | From [[Irrational Numbers form Metric Space]], $\struct {\R \setminus \Q, \tau_d}$ is a [[Definition:Metric Space|metric space]].
From [[Irrational Number Space is Separable]], $\struct {\R \setminus \Q, \tau_d}$ is a [[Definition:Separable Space|separable space]].
The result follows from [[Separable Metric Space is ... | Irrational Number Space is Second-Countable | https://proofwiki.org/wiki/Irrational_Number_Space_is_Second-Countable | https://proofwiki.org/wiki/Irrational_Number_Space_is_Second-Countable | [
"Irrational Number Space",
"Examples of Second-Countable Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Second-Countable Space"
] | [
"Irrational Numbers form Metric Space",
"Definition:Metric Space",
"Irrational Number Space is Separable",
"Definition:Separable Space",
"Separable Metric Space is Second-Countable"
] |
proofwiki-8498 | Compact Set of Rational Numbers is Nowhere Dense | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Let $S \subseteq \Q$ be a compact set of $\Q$.
Then $S$ is nowhere dense in $\Q$. | By Compact Subspace of Hausdorff Space is Closed, $S$ is closed in $\Q$.
By Set is Closed iff Equals Topological Closure, $S = S^-$.
{{AimForCont}} $S$ is not nowhere dense in $\Q$.
Then $S^-$ contains some non-empty open set.
From Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of ... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Let $S \subseteq \Q$ be a [[Definition:Compact Topological Subspace|compact set]] of $\Q$.
Then $S$ is [[Definition:Nowhere Dense|n... | By [[Compact Subspace of Hausdorff Space is Closed]], $S$ is [[Definition:Closed Set (Topology)|closed]] in $\Q$.
By [[Set is Closed iff Equals Topological Closure]], $S = S^-$.
{{AimForCont}} $S$ is not [[Definition:Nowhere Dense|nowhere dense]] in $\Q$.
Then $S^-$ contains some [[Definition:Non-Empty Set|non-empt... | Compact Set of Rational Numbers is Nowhere Dense | https://proofwiki.org/wiki/Compact_Set_of_Rational_Numbers_is_Nowhere_Dense | https://proofwiki.org/wiki/Compact_Set_of_Rational_Numbers_is_Nowhere_Dense | [
"Rational Number Space",
"Compact Topological Spaces",
"Examples of Nowhere Dense"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Compact Topological Space/Subspace",
"Definition:Nowhere Dense"
] | [
"Compact Subspace of Hausdorff Space is Closed",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Definition:Nowhere Dense",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Basis for Euclidean Topology on Real Number Line",
"Definition:Real Interval/Open"... |
proofwiki-8499 | Rational Number Space is not Locally Compact Hausdorff Space | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is not a locally compact Hausdorff Space. | For $\struct {\Q, \tau_d}$ to be a locally compact Hausdorff Space, it is required that every point of $\Q$ has a compact neighborhood.
Let $x \in \Q$.
Let $N \subseteq \Q$ be a neighborhood of $x$.
Then:
:$\exists U \in \tau: x \in U \subseteq N \subseteq \Q$.
{{AimForCont}} $N$ is compact.
By Compact Set of Rational ... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is not a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff Space]]. | For $\struct {\Q, \tau_d}$ to be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff Space]], it is required that every [[Definition:Point of Set|point]] of $\Q$ has a [[Definition:Compact Topological Subspace|compact]] [[Definition:Neighborhood of Point|neighborhood]].
Let $x \in \Q$.
Let $N \s... | Rational Number Space is not Locally Compact Hausdorff Space | https://proofwiki.org/wiki/Rational_Number_Space_is_not_Locally_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Rational_Number_Space_is_not_Locally_Compact_Hausdorff_Space | [
"Rational Number Space",
"Examples of Locally Compact Hausdorff Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Locally Compact Hausdorff Space"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Element",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Compact Topological Space/Subspace",
"Compact Set of Rational Numbers is Nowhere Dense"... |
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