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-10500 | Kuratowski's Closure-Complement Problem | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$ be a subset of $T$.
By successive applications of the operations of complement relative to $S$ and the closure, there can be as many as $14$ distinct subsets of $S$ (including $A$ itself). | Consider an arbitrary subset $A$ of a topological space $T = \struct {S, \tau}$.
To simplify the presentation:
:let $a$ be used to denote the operation of taking the complement of $A$ relative to $S$: $\map a A = S \setminus A$
:let $b$ be used to denote the operation of taking the closure of $A$ in $T$: $\map b A = A^... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq S$ be a [[Definition:Subset|subset]] of $T$.
By successive applications of the [[Definition:Unary Operation|operations]] of [[Definition:Relative Complement|complement relative to $S$]] and the [[Definition:Closure (... | Consider an arbitrary [[Definition:Subset|subset]] $A$ of a [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$.
To simplify the presentation:
:let $a$ be used to denote the [[Definition:Unary Operation|operation]] of taking the [[Definition:Relative Complement|complement of $A$ relative to $S$]... | Kuratowski's Closure-Complement Problem/Proof of Maximum | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Proof_of_Maximum | [
"Kuratowski's Closure-Complement Problem",
"Set Closures",
"Relative Complement",
"14"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Operation/Unary Operation",
"Definition:Relative Complement",
"Definition:Closure (Topology)",
"Definition:Distinct",
"Definition:Subset"
] | [
"Definition:Subset",
"Definition:Topological Space",
"Definition:Operation/Unary Operation",
"Definition:Relative Complement",
"Definition:Operation/Unary Operation",
"Definition:Closure (Topology)",
"Definition:Identity Mapping",
"Definition:Parenthesis",
"Relative Complement of Relative Complement... |
proofwiki-10501 | Kuratowski's Closure-Complement Problem/Complement | The complement of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A'
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 1
| c = {{Defof|Singleton}}
}}
{{eqn | o =
| ro= \cup
| r = \hointr 2 3
| c = {{Defof|Half-Ope... | For ease of analysis, let:
:$A_1 := \openint 0 1$
:$A_2 := \openint 1 2$
:$A_3 := \set 3$
:$A_4 := \Q \cap \openint 4 5$
Thus:
:$\ds A = \bigcup_{i \mathop = 1}^4 A_i$
By De Morgan's Laws:
:$\ds A' := \R \setminus A = \bigcap_{i \mathop = 1}^4 \paren {\R \setminus A_i}$
{{begin-eqn}}
{{eqn | l = \R \setminus A_1
... | The [[Definition:Relative Complement|complement of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A'
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 1
| c = {{Defof|Singleton}}
}}
{{eqn | o =
| ro= \cup
| r = \hoi... | For ease of analysis, let:
:$A_1 := \openint 0 1$
:$A_2 := \openint 1 2$
:$A_3 := \set 3$
:$A_4 := \Q \cap \openint 4 5$
Thus:
:$\ds A = \bigcup_{i \mathop = 1}^4 A_i$
By [[De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Intersection|De Morgan's Laws]]:
:$\ds A' := \R \setminus A = \b... | Kuratowski's Closure-Complement Problem/Complement | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Complement | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Complement | [
"Kuratowski's Closure-Complement Problem"
] | [
"Definition:Relative Complement",
"Definition:Real Interval/Half-Open",
"Definition:Irrational Number",
"File:Kuratowski-Closure-Complement-Theorem-Comp.png"
] | [
"De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Intersection",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection",
"Category:Kuratowski's Closure-Complement Problem"
] |
proofwiki-10502 | Kuratowski's Closure-Complement Problem/Interior | The interior of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^\circ
| r = \openint 0 1 \cup \openint 1 2
| c = Union of Adjacent Open Intervals
}}
{{end-eqn}}
:500px | From Interior equals Complement of Closure of Complement:
:$A^\circ = A^{\prime \, - \, \prime}$
From Kuratowski's Closure-Complement Problem: Closure of Complement:
{{begin-eqn}}
{{eqn | l = A^{\prime \, -}
| r = \hointl \gets 0
| c = Unbounded Closed Real Interval
}}
{{eqn | o =
| ro= \cup
| r... | The [[Definition:Interior (Topology)|interior of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^\circ
| r = \openint 0 1 \cup \openint 1 2
| c = [[Definition:Union of Adjacent Open Intervals|Union of Adjacent Open Intervals]]
}}
{{end-eqn}}
:[[File:Kuratowski-Closure-Complement-Theorem-Int.png|50... | From [[Interior equals Complement of Closure of Complement]]:
:$A^\circ = A^{\prime \, - \, \prime}$
From [[Kuratowski's Closure-Complement Problem/Closure of Complement|Kuratowski's Closure-Complement Problem: Closure of Complement]]:
{{begin-eqn}}
{{eqn | l = A^{\prime \, -}
| r = \hointl \gets 0
| c = ... | Kuratowski's Closure-Complement Problem/Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior | [
"Kuratowski's Closure-Complement Problem",
"Examples of Set Interiors"
] | [
"Definition:Interior (Topology)",
"Definition:Union of Adjacent Open Intervals",
"File:Kuratowski-Closure-Complement-Theorem-Int.png"
] | [
"Interior equals Complement of Closure of Complement",
"Kuratowski's Closure-Complement Problem/Closure of Complement",
"Definition:Real Interval/Unbounded Closed",
"Definition:Singleton",
"Definition:Real Interval/Unbounded Closed",
"Category:Kuratowski's Closure-Complement Problem",
"Category:Examples... |
proofwiki-10503 | Kuratowski's Closure-Complement Problem/Closure | The closure of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^-
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 3
| c = {{Defof|Singleton}}
}}
{{eqn | o =
| ro= \cup
| r = \closedint 4 5
| c = {{Defof|Closed Real Interv... | From Closure of Union of Adjacent Open Intervals:
:$\paren {\openint 0 1 \cup \openint 1 2}^- = \closedint 0 2$
From Real Number is Closed in Real Number Line:
:$\set 3$ is closed in $\R$
From Set is Closed iff Equals Topological Closure:
:$\set 3^- = \set 3$
From Closure of Rational Interval is Closed Real Interval:
:... | The [[Definition:Closure (Topology)|closure of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^-
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 3
| c = {{Defof|Singleton}}
}}
{{eqn | o =
| ro= \cup
| r = \closedint 4 5
... | From [[Closure of Union of Adjacent Open Intervals]]:
:$\paren {\openint 0 1 \cup \openint 1 2}^- = \closedint 0 2$
From [[Real Number is Closed in Real Number Line]]:
:$\set 3$ is [[Definition:Closed Set (Topology)|closed]] in $\R$
From [[Set is Closed iff Equals Topological Closure]]:
:$\set 3^- = \set 3$
From [... | Kuratowski's Closure-Complement Problem/Closure | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure | [
"Kuratowski's Closure-Complement Problem",
"Set Closures"
] | [
"Definition:Closure (Topology)",
"File:Kuratowski-Closure-Complement-Theorem-Clos.png"
] | [
"Closure of Union of Adjacent Open Intervals",
"Real Number is Closed in Real Number Line",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Closure of Rational Interval is Closed Real Interval",
"Closure of Finite Union equals Union of Closures",
"Category:Kuratowski's... |
proofwiki-10504 | Kuratowski's Closure-Complement Problem/Exterior | The exterior of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^e
| r = \openint \gets 0
| c = {{Defof|Unbounded Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 2 3 \cup \openint 3 4
| c = {{Defof|Union of Adjacent Open Intervals}}
}}
{{eqn | o =
| ro= \cup
|... | By definition, the exterior of $A$ in $\R$ can be defined either as:
:the complement of the closure of $A$ in $\R$: $A^{- \, \prime}$
or as:
:the interior of the complement of $A$ in $\R$: $A^{\prime \, \circ}$
From Kuratowski's Closure-Complement Problem: Closure:
{{begin-eqn}}
{{eqn | l = A^-
| r = \closedint 0... | The [[Definition:Exterior (Topology)|exterior of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^e
| r = \openint \gets 0
| c = {{Defof|Unbounded Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 2 3 \cup \openint 3 4
| c = {{Defof|Union of Adjacent Open Intervals}}
}}
... | By definition, the [[Definition:Exterior (Topology)|exterior of $A$ in $\R$]] can be defined either as:
:the [[Definition:Set Complement|complement]] of the [[Definition:Closure (Topology)|closure]] of $A$ in $\R$: $A^{- \, \prime}$
or as:
:the [[Definition:Interior (Topology)|interior]] of the [[Definition:Set Complem... | Kuratowski's Closure-Complement Problem/Exterior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Exterior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Exterior | [
"Kuratowski's Closure-Complement Problem",
"Set Exteriors"
] | [
"Definition:Exterior (Topology)",
"File:Kuratowski-Closure-Complement-Theorem-Ext.png"
] | [
"Definition:Exterior (Topology)",
"Definition:Set Complement",
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"Definition:Set Complement",
"Kuratowski's Closure-Complement Problem/Closure",
"Category:Kuratowski's Closure-Complement Problem",
"Category:Set Exteriors"
] |
proofwiki-10505 | Kuratowski's Closure-Complement Problem/Closure of Complement | The closure of the complement of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^{\prime \, -}
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 1
| c = {{Defof|Singleton}}
}}
{{eqn | o =
| ro= \cup
| r = \hointr 2 \t... | From Kuratowski's Closure-Complement Problem: Complement:
{{begin-eqn}}
{{eqn | l = A'
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 1
| c = {{Defof|Singleton}}
}}
{{eqn | o =
| ro= \cup
| r = \hointr 2 3
| c = {... | The [[Definition:Closure (Topology)|closure]] of the [[Definition:Relative Complement|complement of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^{\prime \, -}
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 1
| c = {{Defo... | From [[Kuratowski's Closure-Complement Problem/Complement|Kuratowski's Closure-Complement Problem: Complement]]:
{{begin-eqn}}
{{eqn | l = A'
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 1
| c = {{Defof|Singleton}}
}}
{{eqn | o ... | Kuratowski's Closure-Complement Problem/Closure of Complement | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure_of_Complement | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure_of_Complement | [
"Kuratowski's Closure-Complement Problem"
] | [
"Definition:Closure (Topology)",
"Definition:Relative Complement",
"File:Kuratowski-Closure-Complement-Theorem-ClosComp.png"
] | [
"Kuratowski's Closure-Complement Problem/Complement",
"Definition:Real Interval/Half-Open",
"Definition:Irrational Number",
"Real Number is Closed in Real Number Line",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topol... |
proofwiki-10506 | Kuratowski's Closure-Complement Problem/Closure of Interior | The closure of the interior of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^{\circ \, -}
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{end-eqn}}
:500px | From Kuratowski's Closure-Complement Problem: Interior:
:$A^\circ = \openint 0 1 \cup \openint 1 2$
From Closure of Union of Adjacent Open Intervals:
:$A^{\circ \, -} = \closedint 0 2$
{{qed}}
Category:Kuratowski's Closure-Complement Problem
Category:Examples of Set Interiors
Category:Set Closures
4gwoy8l3ag4cni02ik7jx... | The [[Definition:Closure (Topology)|closure]] of the [[Definition:Interior (Topology)|interior of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^{\circ \, -}
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{end-eqn}}
:[[File:Kuratowski-Closure-Complement-Theorem-ClosInt.png|500px]] | From [[Kuratowski's Closure-Complement Problem/Interior|Kuratowski's Closure-Complement Problem: Interior]]:
:$A^\circ = \openint 0 1 \cup \openint 1 2$
From [[Closure of Union of Adjacent Open Intervals]]:
:$A^{\circ \, -} = \closedint 0 2$
{{qed}}
[[Category:Kuratowski's Closure-Complement Problem]]
[[Category:Ex... | Kuratowski's Closure-Complement Problem/Closure of Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure_of_Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure_of_Interior | [
"Kuratowski's Closure-Complement Problem",
"Examples of Set Interiors",
"Set Closures"
] | [
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"File:Kuratowski-Closure-Complement-Theorem-ClosInt.png"
] | [
"Kuratowski's Closure-Complement Problem/Interior",
"Closure of Union of Adjacent Open Intervals",
"Category:Kuratowski's Closure-Complement Problem",
"Category:Examples of Set Interiors",
"Category:Set Closures"
] |
proofwiki-10507 | Kuratowski's Closure-Complement Problem/Interior of Closure | The interior of the closure of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 4 5
| c = {{Defof|Open Real Interval}}
}}
{{end-eqn}}
:500px | From Kuratowski's Closure-Complement Problem: Closure:
{{begin-eqn}}
{{eqn | l = A^-
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 3
| c = {{Defof|Singleton}}
}}
{{eqn | o =
| ro= \cup
| r = \closedint 4 5
| c = {{Defof|Clo... | The [[Definition:Interior (Topology)|interior]] of the [[Definition:Closure (Topology)|closure of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 4 5
| c = {{Defof|Open Real ... | From [[Kuratowski's Closure-Complement Problem/Closure|Kuratowski's Closure-Complement Problem: Closure]]:
{{begin-eqn}}
{{eqn | l = A^-
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \set 3
| c = {{Defof|Singleton}}
}}
{{eqn | o =
| ro= \c... | Kuratowski's Closure-Complement Problem/Interior of Closure | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior_of_Closure | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior_of_Closure | [
"Kuratowski's Closure-Complement Problem",
"Examples of Set Interiors",
"Set Closures"
] | [
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"File:Kuratowski-Closure-Complement-Theorem-IntClos.png"
] | [
"Kuratowski's Closure-Complement Problem/Closure",
"Interior of Closed Real Interval is Open Real Interval",
"Interior of Singleton in Real Number Line is Empty",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"Category:Kuratowski's Closure-Complement Problem",
"Category:Examples of ... |
proofwiki-10508 | Kuratowski's Closure-Complement Problem/Interior of Closure of Interior | The interior of the closure of the interior of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^{\circ \, - \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{end-eqn}}
:500px | From Kuratowski's Closure-Complement Problem: Closure of Interior:
{{begin-eqn}}
{{eqn | l = A^{\circ \, -}
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{end-eqn}}
From Interior of Closed Real Interval is Open Real Interval:
:$\closedint 0 2^\circ = \openint 0 2$
Hence the result.
{{qed}}
C... | The [[Definition:Interior (Topology)|interior]] of the [[Definition:Closure (Topology)|closure]] of the [[Definition:Interior (Topology)|interior of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^{\circ \, - \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{end-eqn}}
:[[File:Ku... | From [[Kuratowski's Closure-Complement Problem/Closure of Interior|Kuratowski's Closure-Complement Problem: Closure of Interior]]:
{{begin-eqn}}
{{eqn | l = A^{\circ \, -}
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{end-eqn}}
From [[Interior of Closed Real Interval is Open Real Interva... | Kuratowski's Closure-Complement Problem/Interior of Closure of Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior_of_Closure_of_Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior_of_Closure_of_Interior | [
"Kuratowski's Closure-Complement Problem",
"Examples of Set Interiors",
"Set Closures"
] | [
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"File:Kuratowski-Closure-Complement-Theorem-IntClosInt.png"
] | [
"Kuratowski's Closure-Complement Problem/Closure of Interior",
"Interior of Closed Real Interval is Open Real Interval",
"Category:Kuratowski's Closure-Complement Problem",
"Category:Examples of Set Interiors",
"Category:Set Closures"
] |
proofwiki-10509 | Kuratowski's Closure-Complement Problem/Interior of Complement of Interior | The interior of the complement of the interior of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^{\circ \, \prime \, \circ}
| r = \openint \gets 0
| c = {{Defof|Unbounded Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 2 \to
| c = {{Defof|Unbounded Open Real Interval}}
... | From Complement of Interior equals Closure of Complement:
:$A^{\circ \, \prime} = A^{\prime \, -}$
From Kuratowski's Closure-Complement Problem: Closure of Complement:
{{begin-eqn}}
{{eqn | l = A^{\prime \, -}
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn | o =
| ro= \c... | The [[Definition:Interior (Topology)|interior]] of the [[Definition:Relative Complement|complement]] of the [[Definition:Interior (Topology)|interior of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^{\circ \, \prime \, \circ}
| r = \openint \gets 0
| c = {{Defof|Unbounded Open Real Interval}}
}}
{... | From [[Complement of Interior equals Closure of Complement]]:
:$A^{\circ \, \prime} = A^{\prime \, -}$
From [[Kuratowski's Closure-Complement Problem/Closure of Complement|Kuratowski's Closure-Complement Problem: Closure of Complement]]:
{{begin-eqn}}
{{eqn | l = A^{\prime \, -}
| r = \hointl \gets 0
| c... | Kuratowski's Closure-Complement Problem/Interior of Complement of Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior_of_Complement_of_Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior_of_Complement_of_Interior | [
"Kuratowski's Closure-Complement Problem",
"Set Interiors",
"Set Complement"
] | [
"Definition:Interior (Topology)",
"Definition:Relative Complement",
"Definition:Interior (Topology)",
"File:Kuratowski-Closure-Complement-Theorem-IntCompInt.png"
] | [
"Complement of Interior equals Closure of Complement",
"Kuratowski's Closure-Complement Problem/Closure of Complement",
"Interior of Closed Real Interval is Open Real Interval",
"Interior of Singleton in Real Number Line is Empty",
"Definition:Interior (Topology)",
"Definition:Relative Complement",
"Def... |
proofwiki-10510 | Kuratowski's Closure-Complement Problem/Closure of Interior of Complement | The closure of the interior of the complement of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^{\prime \, \circ \, -}
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \closedint 2 4
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | o... | By definition, the exterior is the interior of the complement.
Hence from Kuratowski's Closure-Complement Problem: Exterior:
{{begin-eqn}}
{{eqn | l = A^e
| r = \openint \gets 0
| c = {{Defof|Unbounded Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 2 3 \cup \openint 3 4
| c ... | The [[Definition:Closure (Topology)|closure]] of the [[Definition:Interior (Topology)|interior]] of the [[Definition:Relative Complement|complement of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^{\prime \, \circ \, -}
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn ... | By definition, the [[Definition:Exterior (Topology)|exterior]] is the [[Definition:Interior (Topology)|interior]] of the [[Definition:Set Complement|complement]].
Hence from [[Kuratowski's Closure-Complement Problem/Exterior|Kuratowski's Closure-Complement Problem: Exterior]]:
{{begin-eqn}}
{{eqn | l = A^e
| r ... | Kuratowski's Closure-Complement Problem/Closure of Interior of Complement | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure_of_Interior_of_Complement | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure_of_Interior_of_Complement | [
"Kuratowski's Closure-Complement Problem",
"Examples of Set Interiors",
"Set Closures",
"Set Complement"
] | [
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"Definition:Relative Complement",
"File:Kuratowski-Closure-Complement-Theorem-ClosIntComp.png"
] | [
"Definition:Exterior (Topology)",
"Definition:Interior (Topology)",
"Definition:Set Complement",
"Kuratowski's Closure-Complement Problem/Exterior",
"Closure of Open Real Interval is Closed Real Interval",
"Closure of Union of Adjacent Open Intervals",
"Closure of Finite Union equals Union of Closures",... |
proofwiki-10511 | Kuratowski's Closure-Complement Problem/Closure of Interior of Closure | The closure of the interior of the closure of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ \, -}
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \closedint 4 5
| c = {{Defof|Closed Real Interval}}
}}
{{end-eqn}}
:500px | From Kuratowski's Closure-Complement Problem: Interior of Closure:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 4 5
| c = {{Defof|Open Real Interval}}
}}
{{end-eqn}}
From Closure of Open Real Interva... | The [[Definition:Closure (Topology)|closure]] of the [[Definition:Interior (Topology)|interior]] of the [[Definition:Closure (Topology)|closure of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ \, -}
| r = \closedint 0 2
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | o =
| ro= \c... | From [[Kuratowski's Closure-Complement Problem/Interior of Closure|Kuratowski's Closure-Complement Problem: Interior of Closure]]:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 4 5
| c = {{Defof|Open... | Kuratowski's Closure-Complement Problem/Closure of Interior of Closure | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure_of_Interior_of_Closure | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Closure_of_Interior_of_Closure | [
"Kuratowski's Closure-Complement Problem",
"Examples of Set Interiors",
"Set Closures"
] | [
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"File:Kuratowski-Closure-Complement-Theorem-ClosIntClos.png"
] | [
"Kuratowski's Closure-Complement Problem/Interior of Closure",
"Closure of Open Real Interval is Closed Real Interval",
"Closure of Finite Union equals Union of Closures",
"Category:Kuratowski's Closure-Complement Problem",
"Category:Examples of Set Interiors",
"Category:Set Closures"
] |
proofwiki-10512 | Kuratowski's Closure-Complement Problem/Interior of Complement of Interior of Closure | The interior of the complement of the interior of the closure of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ \, \prime \, \circ}
| r = \openint \gets 0
| c = {{Defof|Unbounded Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 2 4
| c = {{Defof|Open Real Int... | From Kuratowski's Closure-Complement Problem: Interior of Closure:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 4 5
| c = {{Defof|Open Real Interval}}
}}
{{end-eqn}}
By inspection:
{{begin-eqn}}
{{eq... | The [[Definition:Interior (Topology)|interior]] of the [[Definition:Relative Complement|complement]] of the [[Definition:Interior (Topology)|interior]] of the [[Definition:Closure (Topology)|closure of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ \, \prime \, \circ}
| r = \openint \gets 0
... | From [[Kuratowski's Closure-Complement Problem/Interior of Closure|Kuratowski's Closure-Complement Problem: Interior of Closure]]:
{{begin-eqn}}
{{eqn | l = A^{- \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \openint 4 5
| c = {{Defof|Open... | Kuratowski's Closure-Complement Problem/Interior of Complement of Interior of Closure | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior_of_Complement_of_Interior_of_Closure | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Interior_of_Complement_of_Interior_of_Closure | [
"Kuratowski's Closure-Complement Problem"
] | [
"Definition:Interior (Topology)",
"Definition:Relative Complement",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"File:Kuratowski-Closure-Complement-Theorem-IntCompIntClos.png"
] | [
"Kuratowski's Closure-Complement Problem/Interior of Closure",
"Interior of Closed Real Interval is Open Real Interval",
"Definition:Interior (Topology)",
"Definition:Relative Complement",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"Category:Kuratowski's Closure-Complement Proble... |
proofwiki-10513 | Kuratowski's Closure-Complement Problem/Complement of Interior of Closure of Interior | The complement of the interior of the closure of the interior of $A$ in $\R$ is given by:
{{begin-eqn}}
{{eqn | l = A^{\circ \, - \, \circ \, \prime}
| r = \hointl \gets 0
| c = {{Defof|Unbounded Closed Real Interval}}
}}
{{eqn | o =
| ro= \cup
| r = \hointr 2 \to
| c = {{Defof|Unbounded C... | From Kuratowski's Closure-Complement Problem: Interior of Closure of Interior:
{{begin-eqn}}
{{eqn | l = A^{\circ \, - \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{end-eqn}}
The result follows by inspection.
{{qed}}
Category:Kuratowski's Closure-Complement Problem
26ctbparebwc0llvkfj... | The [[Definition:Relative Complement|complement]] of the [[Definition:Interior (Topology)|interior]] of the [[Definition:Closure (Topology)|closure]] of the [[Definition:Interior (Topology)|interior of $A$ in $\R$]] is given by:
{{begin-eqn}}
{{eqn | l = A^{\circ \, - \, \circ \, \prime}
| r = \hointl \gets 0
... | From [[Kuratowski's Closure-Complement Problem/Interior of Closure of Interior|Kuratowski's Closure-Complement Problem: Interior of Closure of Interior]]:
{{begin-eqn}}
{{eqn | l = A^{\circ \, - \, \circ}
| r = \openint 0 2
| c = {{Defof|Open Real Interval}}
}}
{{end-eqn}}
The result follows by inspectio... | Kuratowski's Closure-Complement Problem/Complement of Interior of Closure of Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Complement_of_Interior_of_Closure_of_Interior | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Complement_of_Interior_of_Closure_of_Interior | [
"Kuratowski's Closure-Complement Problem"
] | [
"Definition:Relative Complement",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"File:Kuratowski-Closure-Complement-Theorem-CompIntClosInt.png"
] | [
"Kuratowski's Closure-Complement Problem/Interior of Closure of Interior",
"Category:Kuratowski's Closure-Complement Problem"
] |
proofwiki-10514 | Sum of Integrals on Adjacent Intervals for Integrable Functions | Let $f$ be a real function which is Darboux integrable on any closed interval $\mathbb I$.
Let $a, b, c \in \mathbb I$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | === Lemma ===
{{:Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma}}{{qed|lemma}}
{{WLOG}}, assume $a < b$.
First let $a < c < b$.
Let $P_1$ and $P_2$ be any subdivisions of $\closedint a c$ and $\closedint c b$ respectively.
Then $P = P_1 \cup P_2$ is a subdivision of $\closedint a b$.
From the def... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Darboux Integrable Function|Darboux integrable]] on any [[Definition:Closed Real Interval|closed interval]] $\mathbb I$.
Let $a, b, c \in \mathbb I$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | === [[Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma|Lemma]] ===
{{:Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma}}{{qed|lemma}}
{{WLOG}}, assume $a < b$.
First let $a < c < b$.
Let $P_1$ and $P_2$ be any [[Definition:Subdivision of Interval|subdivisions]] of $\closedi... | Sum of Integrals on Adjacent Intervals for Integrable Functions | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Darboux Integrable Function",
"Definition:Real Interval/Closed"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma",
"Definition:Subdivision of Interval",
"Definition:Subdivision of Interval",
"Definition:Upper Darboux Sum",
"Definition:Lower Darboux Sum",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Definition:Definite Inte... |
proofwiki-10515 | Sum of Integrals on Adjacent Intervals for Integrable Functions | Let $f$ be a real function which is Darboux integrable on any closed interval $\mathbb I$.
Let $a, b, c \in \mathbb I$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | This is an instance of Lower Sum of Refinement.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Darboux Integrable Function|Darboux integrable]] on any [[Definition:Closed Real Interval|closed interval]] $\mathbb I$.
Let $a, b, c \in \mathbb I$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | This is an instance of [[Lower Sum of Refinement]].
{{qed}} | Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma/Proof 1 | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions/Lemma/Proof_1 | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Darboux Integrable Function",
"Definition:Real Interval/Closed"
] | [
"Lower Sum of Refinement"
] |
proofwiki-10516 | Sum of Integrals on Adjacent Intervals for Integrable Functions | Let $f$ be a real function which is Darboux integrable on any closed interval $\mathbb I$.
Let $a, b, c \in \mathbb I$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | Let $P = \set {x_0, x_1, \ldots, x_n}$.
Suppose that:
:$c \in P$
Then:
:$Q = P$
We have:
:$\map L P \ge \map L P$
:$\leadsto \map L Q \ge \map L P$ as $Q = P$
This finishes the proof for this case.
The only other possibility for $c$ is:
:$x_{j-1} < c < x_j$
where $1 \le j \le n$.
Let $m_i$ be the infimum of $f$ on the ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Darboux Integrable Function|Darboux integrable]] on any [[Definition:Closed Real Interval|closed interval]] $\mathbb I$.
Let $a, b, c \in \mathbb I$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | Let $P = \set {x_0, x_1, \ldots, x_n}$.
Suppose that:
:$c \in P$
Then:
:$Q = P$
We have:
:$\map L P \ge \map L P$
:$\leadsto \map L Q \ge \map L P$ as $Q = P$
This finishes the proof for this case.
The only other possibility for $c$ is:
:$x_{j-1} < c < x_j$
where $1 \le j \le n$.
Let $m_i$ be the [[Definition:... | Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma/Proof 2 | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions/Lemma/Proof_2 | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Darboux Integrable Function",
"Definition:Real Interval/Closed"
] | [
"Definition:Infimum of Set/Real Numbers",
"Definition:Real Interval/Closed",
"Definition:Infimum of Set/Real Numbers",
"Definition:Real Interval/Closed",
"Definition:Infimum of Set/Real Numbers",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Closed",
"Definition:Subset",
"Definition:L... |
proofwiki-10517 | Closure of Irrational Numbers is Real Numbers | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the same topology.
Then:
:$\paren {\R \setminus \Q}^- = \R$
where $\paren {\R \setminus \Q}^-$ denotes the closure of $\R \setminus \Q$. | From Irrationals are Everywhere Dense in Topological Space of Reals, $\R \setminus \Q$ is everywhere dense in $\R$.
It follows by definition of everywhere dense that $\paren {\R \setminus \Q}^- = \R$.
{{qed}}
Category:Irrational Number Space
Category:Real Number Line with Euclidean Topology
Category:Set Closures
2gq8yk... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
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|sam... | From [[Irrationals are Everywhere Dense in Topological Space of Reals]], $\R \setminus \Q$ is [[Definition:Everywhere Dense|everywhere dense]] in $\R$.
It follows by definition of [[Definition:Everywhere Dense|everywhere dense]] that $\paren {\R \setminus \Q}^- = \R$.
{{qed}}
[[Category:Irrational Number Space]]
[[Ca... | Closure of Irrational Numbers is Real Numbers | https://proofwiki.org/wiki/Closure_of_Irrational_Numbers_is_Real_Numbers | https://proofwiki.org/wiki/Closure_of_Irrational_Numbers_is_Real_Numbers | [
"Irrational Number Space",
"Real Number Line with Euclidean Topology",
"Set Closures"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Closure (Topology)"
] | [
"Irrationals are Everywhere Dense in Reals/Topology",
"Definition:Everywhere Dense",
"Definition:Everywhere Dense",
"Category:Irrational Number Space",
"Category:Real Number Line with Euclidean Topology",
"Category:Set Closures"
] |
proofwiki-10518 | Closure of Union of Adjacent Open Intervals | Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the union of the two adjacent open intervals:
:$A := \openint a b \cup \openint b c$
Then:
:$A^- = \closedint a c$
where:
:$A^-$ is the closure of $A$. | {{begin-eqn}}
{{eqn | l = A^-
| r = \paren {\openint a b \cup \openint b c}^-
| c =
}}
{{eqn | r = \openint a b^- \cup \openint b c^-
| c = Closure of Finite Union equals Union of Closures
}}
{{eqn | r = \closedint a b \cup \closedint b c
| c = Closure of Open Ball in Metric Space
}}
{{eqn | r ... | Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the [[Definition:Union of Adjacent Open Intervals|union of the two adjacent open intervals]]:
:$A := \openint a b \cup \openint b c$
Then:
:$A^- = \closedint a c$
where:
:$A^-$ is the [[Definition:Closure (Topology)|closure]] of $A$. | {{begin-eqn}}
{{eqn | l = A^-
| r = \paren {\openint a b \cup \openint b c}^-
| c =
}}
{{eqn | r = \openint a b^- \cup \openint b c^-
| c = [[Closure of Finite Union equals Union of Closures]]
}}
{{eqn | r = \closedint a b \cup \closedint b c
| c = [[Closure of Open Ball in Metric Space]]
}}
{{... | Closure of Union of Adjacent Open Intervals | https://proofwiki.org/wiki/Closure_of_Union_of_Adjacent_Open_Intervals | https://proofwiki.org/wiki/Closure_of_Union_of_Adjacent_Open_Intervals | [
"Union of Adjacent Open Intervals",
"Examples of Set Closures"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Closure (Topology)"
] | [
"Closure of Finite Union equals Union of Closures",
"Closure of Open Ball in Metric Space",
"Category:Union of Adjacent Open Intervals",
"Category:Examples of Set Closures"
] |
proofwiki-10519 | Interior equals Complement of Closure of Complement | Let $T$ be a topological space.
Let $H \subseteq T$.
Let $H^-$ denote the closure of $H$ and $H^\circ$ denote the interior of $H$.
Let $H^\prime$ denote the complement of $H$ in $T$:
:$H^\prime = T \setminus H$
Then:
:$H^\circ = H^{\prime \, - \, \prime}$ | {{begin-eqn}}
{{eqn | l = H^{\circ \, \prime}
| r = H^{\prime \, -}
| c = Complement of Interior equals Closure of Complement
}}
{{eqn | ll= \leadsto
| l = \paren {H^{\circ \, \prime} }^\prime
| r = \paren {H^{\prime \, -} }^\prime
| c =
}}
{{eqn | ll= \leadsto
| l = H^{\circ \, \pr... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq T$.
Let $H^-$ denote the [[Definition:Closure (Topology)|closure]] of $H$ and $H^\circ$ denote the [[Definition:Interior (Topology)|interior]] of $H$.
Let $H^\prime$ denote the [[Definition:Relative Complement|complement of $H$ in $T$]... | {{begin-eqn}}
{{eqn | l = H^{\circ \, \prime}
| r = H^{\prime \, -}
| c = [[Complement of Interior equals Closure of Complement]]
}}
{{eqn | ll= \leadsto
| l = \paren {H^{\circ \, \prime} }^\prime
| r = \paren {H^{\prime \, -} }^\prime
| c =
}}
{{eqn | ll= \leadsto
| l = H^{\circ \,... | Interior equals Complement of Closure of Complement | https://proofwiki.org/wiki/Interior_equals_Complement_of_Closure_of_Complement | https://proofwiki.org/wiki/Interior_equals_Complement_of_Closure_of_Complement | [
"Set Closures",
"Set Interiors"
] | [
"Definition:Topological Space",
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"Definition:Relative Complement"
] | [
"Complement of Interior equals Closure of Complement",
"Composition of Mappings is Associative",
"Relative Complement of Relative Complement"
] |
proofwiki-10520 | Closure of Half-Open Real Interval is Closed Real Interval | Let $\struct {\R, \tau_d}$ be the real number line under the usual (Euclidean) topology.
Let $H_1 = \hointl a b$ and $H_2 = \hointr a b$ be half-open intervals of $\R$.
Then the closure of both $H_1$ and $H_2$ in $\R$ are the closed interval $\closedint a b$. | For an arbitrary $H \subseteq \R$, let $H^-$ denote the closure of $H$ in $\R$.
By the definition of closure:
:$H^-$ is the smallest closed set of $\struct {\R, \tau_d}$ containing $H$ as a subset.
Let $A := \openint a b$ be the open interval between $a$ and $b$.
By definition of $H_1$ and $H_2$:
:$A \subseteq H_1$
and... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number line]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $H_1 = \hointl a b$ and $H_2 = \hointr a b$ be [[Definition:Half-Open Real Interval|half-open intervals]] of $\R$.
Then the [[Definition:... | For an arbitrary $H \subseteq \R$, let $H^-$ denote the [[Definition:Closure (Topology)|closure]] of $H$ in $\R$.
By the definition of [[Definition:Closure (Topology)/Definition 3|closure]]:
:$H^-$ is the [[Definition:Smallest Set by Set Inclusion|smallest]] [[Definition:Closed Set (Topology)|closed set]] of $\struct ... | Closure of Half-Open Real Interval is Closed Real Interval | https://proofwiki.org/wiki/Closure_of_Half-Open_Real_Interval_is_Closed_Real_Interval | https://proofwiki.org/wiki/Closure_of_Half-Open_Real_Interval_is_Closed_Real_Interval | [
"Real Intervals",
"Set Closures"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval/Half-Open",
"Definition:Closure (Topology)",
"Definition:Real Interval/Closed"
] | [
"Definition:Closure (Topology)",
"Definition:Closure (Topology)/Definition 3",
"Definition:Smallest Set by Set Inclusion",
"Definition:Closed Set/Topology",
"Definition:Subset",
"Definition:Real Interval/Open",
"Closure of Open Real Interval is Closed Real Interval",
"Definition:Closure (Topology)",
... |
proofwiki-10521 | Closure of Irrational Interval is Closed Real Interval | Let $\struct {\R, \tau_d}$ be the real numbers under the usual (Euclidean) topology.
Let $\struct{\R \setminus \Q, \tau_d}$ be the irrational number space under the same topology.
Let $a, b \in \R$ such that $a < b$.
Let $\Bbb I \subseteq \R$ be an interval of $\R$
Then the closure of the set:
:$\Bbb I \cap \paren {\R ... | Let $\Bbb I$ be an open real interval.
From Closure of Real Interval is Closed Real Interval:
:$\Bbb I^- = \closedint a b$
From Closure of Irrational Numbers is Real Numbers:
:$\paren {\R \setminus \Q}^- = \R$
From Closure of Intersection is Subset of Intersection of Closures:
:$\paren{\Bbb I \cap \paren{\R \setminus \... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number|real numbers]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $\struct{\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on... | Let $\Bbb I$ be an [[Definition:Open Real Interval|open real interval]].
From [[Closure of Real Interval is Closed Real Interval]]:
:$\Bbb I^- = \closedint a b$
From [[Closure of Irrational Numbers is Real Numbers]]:
:$\paren {\R \setminus \Q}^- = \R$
From [[Closure of Intersection is Subset of Intersection of Closu... | Closure of Irrational Interval is Closed Real Interval | https://proofwiki.org/wiki/Closure_of_Irrational_Interval_is_Closed_Real_Interval | https://proofwiki.org/wiki/Closure_of_Irrational_Interval_is_Closed_Real_Interval | [
"Real Intervals",
"Irrational Number Space",
"Set Closures"
] | [
"Definition:Real Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval",
"Definition:Closure (Topology)",
"Definition:Real Interval/Closed"
] | [
"Definition:Real Interval/Open",
"Closure of Real Interval is Closed Real Interval",
"Closure of Irrational Numbers is Real Numbers",
"Closure of Intersection is Subset of Intersection of Closures",
"Intersection with Subset is Subset",
"Irrationals are Everywhere Dense in Reals/Topology",
"Category:Rea... |
proofwiki-10522 | Closure of Rational Interval is Closed Real Interval | Let $\struct {\R, \tau_d}$ be the real numbers under the usual (Euclidean) topology.
Let $\struct {\Q, \tau_d}$ be the rational number space under the same topology.
Let $a, b \in \R$ such that $a < b$.
Let $\Bbb I \subseteq \R$ be an interval of $\R$
Then the closure of the set :
:$\Bbb I \cap \Q$
is the closed real i... | Let $\Bbb I$ be an open real interval.
From Closure of Real Interval is Closed Real Interval:
:$\Bbb I^- = \closedint a b$
From Closure of Rational Numbers is Real Numbers:
:$\Q^- = \R$
From Closure of Intersection is Subset of Intersection of Closures:
:$\paren{\Bbb I \cap \Q}^- \subseteq \Bbb I^- \cap \Q^-$
From Inte... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number|real numbers]] under the [[Definition:Euclidean Topology on Real Number Line|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 Lin... | Let $\Bbb I$ be an [[Definition:Open Real Interval|open real interval]].
From [[Closure of Real Interval is Closed Real Interval]]:
:$\Bbb I^- = \closedint a b$
From [[Closure of Rational Numbers is Real Numbers]]:
:$\Q^- = \R$
From [[Closure of Intersection is Subset of Intersection of Closures]]:
:$\paren{\Bbb I \... | Closure of Rational Interval is Closed Real Interval | https://proofwiki.org/wiki/Closure_of_Rational_Interval_is_Closed_Real_Interval | https://proofwiki.org/wiki/Closure_of_Rational_Interval_is_Closed_Real_Interval | [
"Real Intervals",
"Irrational Number Space"
] | [
"Definition:Real Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval",
"Definition:Closure (Topology)",
"Definition:Real Interval/Closed"
] | [
"Definition:Real Interval/Open",
"Closure of Real Interval is Closed Real Interval",
"Closure of Rational Numbers is Real Numbers",
"Closure of Intersection is Subset of Intersection of Closures",
"Intersection with Subset is Subset",
"Rational Numbers are Everywhere Dense in Set of Real Numbers/Topology"... |
proofwiki-10523 | Interior of Singleton in Real Number Line is Empty | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $a \in \R$ be a real number.
Then:
:$\set a^\circ = \O$
where $\set a^\circ$ denotes the interior of $\set a$ in $\R$. | {{begin-eqn}}
{{eqn | l = \set a^\circ
| r = \closedint a a^\circ
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | r = \openint a a
| c = Interior of Closed Real Interval is Open Real Interval
}}
{{eqn | r = \set {x \in \R: a < x < a}
| c = {{Defof|Open Real Interval}}
}}
{{eqn | r = \O
| c... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $a \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$\set a^\circ = \O$
where $\set a^\circ$ denotes the [[Definition:Interior (Topology)|interior]] of $\set a... | {{begin-eqn}}
{{eqn | l = \set a^\circ
| r = \closedint a a^\circ
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | r = \openint a a
| c = [[Interior of Closed Real Interval is Open Real Interval]]
}}
{{eqn | r = \set {x \in \R: a < x < a}
| c = {{Defof|Open Real Interval}}
}}
{{eqn | r = \O
... | Interior of Singleton in Real Number Line is Empty | https://proofwiki.org/wiki/Interior_of_Singleton_in_Real_Number_Line_is_Empty | https://proofwiki.org/wiki/Interior_of_Singleton_in_Real_Number_Line_is_Empty | [
"Examples of Set Interiors",
"Singletons",
"Real Number Line with Euclidean Topology"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Number",
"Definition:Interior (Topology)"
] | [
"Interior of Closed Real Interval is Open Real Interval",
"Category:Examples of Set Interiors",
"Category:Singletons",
"Category:Real Number Line with Euclidean Topology"
] |
proofwiki-10524 | Kuratowski's Closure-Complement Problem/Proof of Maximum | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$ be a subset of $T$.
By successive applications of the operations of complement relative to $S$ and the closure, there can be no more than $14$ distinct subsets of $S$ (including $A$ itself). | Consider an arbitrary subset $A$ of a topological space $T = \struct {S, \tau}$.
To simplify the presentation:
:let $a$ be used to denote the operation of taking the complement of $A$ relative to $S$: $\map a A = S \setminus A$
:let $b$ be used to denote the operation of taking the closure of $A$ in $T$: $\map b A = A^... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq S$ be a [[Definition:Subset|subset]] of $T$.
By successive applications of the [[Definition:Unary Operation|operations]] of [[Definition:Relative Complement|complement relative to $S$]] and the [[Definition:Topologic... | Consider an arbitrary [[Definition:Subset|subset]] $A$ of a [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$.
To simplify the presentation:
:let $a$ be used to denote the [[Definition:Unary Operation|operation]] of taking the [[Definition:Relative Complement|complement of $A$ relative to $S$]... | Kuratowski's Closure-Complement Problem/Proof of Maximum | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Proof_of_Maximum | https://proofwiki.org/wiki/Kuratowski's_Closure-Complement_Problem/Proof_of_Maximum | [
"Set Closures",
"Relative Complement",
"Kuratowski's Closure-Complement Problem"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Operation/Unary Operation",
"Definition:Relative Complement",
"Definition:Closure (Topology)",
"Definition:Distinct",
"Definition:Subset"
] | [
"Definition:Subset",
"Definition:Topological Space",
"Definition:Operation/Unary Operation",
"Definition:Relative Complement",
"Definition:Operation/Unary Operation",
"Definition:Closure (Topology)",
"Definition:Identity Mapping",
"Definition:Parenthesis",
"Relative Complement of Relative Complement... |
proofwiki-10525 | Closure of Complement of Closure is Regular Closed | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$ be a subset of $T$.
Let $A^-$ denote the closure of $A$ in $T$.
Let $A'$ denote the complement of $A$ in $S$: $A' = S \setminus A$.
Then $A^{-'-}$ is regular closed. | Let $A^\circ$ denote the interior of $A$.
From Set is Closed iff Equals Topological Closure, $A^-$ is closed in $T$.
Since $A^-$ is a closed set, $A^{-'}$ is open.
Therefore:
:$\forall x \in A^{-'}: \exists \epsilon \in \R_{>0}$
{{questionable|The above statement makes no sense in this context|prime.mover}}
Additionall... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq S$ be a [[Definition:Subset|subset]] of $T$.
Let $A^-$ denote the [[Definition:Topological Closure|closure]] of $A$ in $T$.
Let $A'$ denote the [[Definition:Relative Complement|complement]] of $A$ in $S$: $A' = S \s... | Let $A^\circ$ denote the [[Definition:Interior (Topology)|interior]] of $A$.
From [[Set is Closed iff Equals Topological Closure]], $A^-$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
Since $A^-$ is a [[Definition:Closed Set (Topology)|closed set]], $A^{-'}$ is [[Definition:Open Set (Topology)|open]].
There... | Closure of Complement of Closure is Regular Closed | https://proofwiki.org/wiki/Closure_of_Complement_of_Closure_is_Regular_Closed | https://proofwiki.org/wiki/Closure_of_Complement_of_Closure_is_Regular_Closed | [
"Regular Closed Sets"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Relative Complement",
"Definition:Regular Closed Set"
] | [
"Definition:Interior (Topology)",
"Set is Closed iff Equals Topological Closure",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Relative Complement",
"Definition:Limit Point/Topology/Set",
"Definition:Regular Closed Set",
"Category:Re... |
proofwiki-10526 | Topology induced by Usual Metric on Positive Integers is Discrete | Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.
Then the metric topology for $d$ is a discrete topology. | Let $\tau_d$ denote the metric topology for $d$.
Let $\epsilon \in \R_{>0}$ such that $\epsilon < 1$.
Let $a \in \Z_{>0}$.
Recall the definition of the open $\epsilon$-ball of $a$ in $\struct {\Z_{>0}, d}$:
:$\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$
But we have:
:$\forall x \in \Z_{>0}, x \ne a:... | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition:Euclidean Metric on Real Number Line|usual (Euclidean) metric]] on $\Z_{>0}$.
Then the [[Definition:Topology Induced by Metric|metric topolo... | Let $\tau_d$ denote the [[Definition:Topology Induced by Metric|metric topology]] for $d$.
Let $\epsilon \in \R_{>0}$ such that $\epsilon < 1$.
Let $a \in \Z_{>0}$.
Recall the definition of the [[Definition:Open Ball of Metric Space|open $\epsilon$-ball]] of $a$ in $\struct {\Z_{>0}, d}$:
:$\map {B_\epsilon} a := \s... | Topology induced by Usual Metric on Positive Integers is Discrete | https://proofwiki.org/wiki/Topology_induced_by_Usual_Metric_on_Positive_Integers_is_Discrete | https://proofwiki.org/wiki/Topology_induced_by_Usual_Metric_on_Positive_Integers_is_Discrete | [
"Discrete Topologies",
"Metric Spaces"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Topology Induced by Metric",
"Definition:Discrete Topology"
] | [
"Definition:Topology Induced by Metric",
"Definition:Open Ball",
"Basis for Discrete Topology",
"Definition:Discrete Topology"
] |
proofwiki-10527 | Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete | Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as:
:$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Then the metric topology for $\delta$ is a discrete topology. | Let $\tau_\delta$ denote the metric topology for $\delta$.
In Scaled Euclidean Metric is Metric it is demonstrated that $\delta$ is indeed a metric on $\Z_{>0}$.
Let $a \in \Z_{>0}$.
Recall the definition of the open $\epsilon$-ball of $a$ in $\struct {\Z_{>0}, \delta}$:
:$\map {B_\epsilon} a := \set {x \in A: \map \de... | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition:Metric|metric]] on $\Z_{>0}$ defined as:
:$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Then the [[Def... | Let $\tau_\delta$ denote the [[Definition:Topology Induced by Metric|metric topology]] for $\delta$.
In [[Scaled Euclidean Metric is Metric]] it is demonstrated that $\delta$ is indeed a [[Definition:Metric|metric]] on $\Z_{>0}$.
Let $a \in \Z_{>0}$.
Recall the definition of the [[Definition:Open Ball of Metric Spa... | Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete | https://proofwiki.org/wiki/Topology_induced_by_Scaled_Euclidean_Metric_on_Positive_Integers_is_Discrete | https://proofwiki.org/wiki/Topology_induced_by_Scaled_Euclidean_Metric_on_Positive_Integers_is_Discrete | [
"Discrete Topologies",
"Metric Spaces"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Metric Space/Metric",
"Definition:Topology Induced by Metric",
"Definition:Discrete Topology"
] | [
"Definition:Topology Induced by Metric",
"Scaled Euclidean Metric is Metric",
"Definition:Metric Space/Metric",
"Definition:Open Ball",
"Basis for Discrete Topology",
"Definition:Discrete Topology"
] |
proofwiki-10528 | Scaled Euclidean Metric is Metric | Let $\R_{>0}$ be the set of (strictly) positive integers.
Let $\delta: \R_{>0} \times \R_{>0} \to \R$ be the metric on $\R_{>0}$ defined as:
:$\forall x, y \in \R_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Then $\delta$ is a metric. | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map \delta {x, x}
| r = \dfrac {\size {x - x} } {x^2}
| c = Definition of $\delta$
}}
{{eqn | r = 0
| c = as $\size {x - x} = 0$
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} holds for $\delta$.
{{qed|lemma}} | Let $\R_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $\delta: \R_{>0} \times \R_{>0} \to \R$ be the [[Definition:Metric|metric]] on $\R_{>0}$ defined as:
:$\forall x, y \in \R_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Then $\delta$ ... | === Proof of {{Metric-space-axiom|1|nolink}} ===
{{begin-eqn}}
{{eqn | l = \map \delta {x, x}
| r = \dfrac {\size {x - x} } {x^2}
| c = Definition of $\delta$
}}
{{eqn | r = 0
| c = as $\size {x - x} = 0$
}}
{{end-eqn}}
So {{Metric-space-axiom|1}} holds for $\delta$.
{{qed|lemma}} | Scaled Euclidean Metric is Metric | https://proofwiki.org/wiki/Scaled_Euclidean_Metric_is_Metric | https://proofwiki.org/wiki/Scaled_Euclidean_Metric_is_Metric | [
"Scaled Euclidean Metric"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric"
] | [] |
proofwiki-10529 | Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Integers are Homeomorphic | Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as:
:$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Let $\tau_d$ d... | From Topology induced by Usual Metric on Positive Integers is Discrete:
:$\struct {\Z_{>0}, \tau_d}$ is a discrete space.
From Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete:
:$\struct {\Z_{>0}, \tau_\delta}$ is a discrete space.
Let $I_{\Z_{>0} }$ be the identity mapping from $\Z_{>0}$ to... | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition:Euclidean Metric on Real Number Line|usual (Euclidean) metric]] on $\Z_{>0}$.
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition... | From [[Topology induced by Usual Metric on Positive Integers is Discrete]]:
:$\struct {\Z_{>0}, \tau_d}$ is a [[Definition:Discrete Space|discrete space]].
From [[Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete]]:
:$\struct {\Z_{>0}, \tau_\delta}$ is a [[Definition:Discrete Space|discrete ... | Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Integers are Homeomorphic | https://proofwiki.org/wiki/Topologies_induced_by_Usual_Metric_and_Scaled_Euclidean_Metric_on_Positive_Integers_are_Homeomorphic | https://proofwiki.org/wiki/Topologies_induced_by_Usual_Metric_and_Scaled_Euclidean_Metric_on_Positive_Integers_are_Homeomorphic | [
"Discrete Topologies",
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Metric Space/Metric",
"Definition:Topology Induced by Metric",
"Definition:Topology Induced by Metric",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Topology induced by Usual Metric on Positive Integers is Discrete",
"Definition:Discrete Topology",
"Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete",
"Definition:Discrete Topology",
"Definition:Identity Mapping",
"Mapping from Discrete Space is Continuous",
"Definition:Con... |
proofwiki-10530 | Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant | Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.
Let $\sequence {x_n}$ be a Cauchy sequence in $\struct {\Z_{>0}, d}$.
Then:
:$\exists m, n \in \Z_{>0}: \forall r > n: x_r = m$
That is, $\sequence {x_n}$ is eventually constant... | Let $\sequence {x_n}$ be a Cauchy sequence in $\struct {\Z_{>0}, d}$.
By definition:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \map d {x_n, x_m} < \epsilon$
Let $\epsilon < 1$, say: $\epsilon = \dfrac 1 2$.
By the definition of $d$:
:$\forall m, n \in \N: x_m \ne x_n \implies \m... | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition:Euclidean Metric on Real Number Line|usual (Euclidean) metric]] on $\Z_{>0}$.
Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence (Metri... | Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence (Metric Space)|Cauchy sequence]] in $\struct {\Z_{>0}, d}$.
By definition:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \map d {x_n, x_m} < \epsilon$
Let $\epsilon < 1$, say: $\epsilon = \dfrac 1 2$.
By the definition of $d... | Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant | https://proofwiki.org/wiki/Cauchy_Sequence_in_Positive_Integers_under_Usual_Metric_is_eventually_Constant | https://proofwiki.org/wiki/Cauchy_Sequence_in_Positive_Integers_under_Usual_Metric_is_eventually_Constant | [
"Euclidean Metric",
"Cauchy Sequences"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Cauchy Sequence/Metric Space",
"Definition:Constant"
] | [
"Definition:Cauchy Sequence/Metric Space"
] |
proofwiki-10531 | Positive Integers under Usual Metric is Complete Metric Space | Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.
Then $\struct {\Z_{>0}, d}$ is a complete metric space. | Let $\sequence {x_n}$ be a Cauchy sequence in $\struct {\Z_{>0}, d}$.
From Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant:
:$\sequence {x_n}$ is a convergent sequence to some $n \in \Z_{>0}$.
Hence the result by definition of complete metric space.
{{qed}} | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition:Euclidean Metric on Real Number Line|usual (Euclidean) metric]] on $\Z_{>0}$.
Then $\struct {\Z_{>0}, d}$ is a [[Definition:Complete Metric ... | Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence (Metric Space)|Cauchy sequence]] in $\struct {\Z_{>0}, d}$.
From [[Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant]]:
:$\sequence {x_n}$ is a [[Definition:Convergent Sequence (Metric Space)|convergent sequence]] to some $n \in \Z_{>... | Positive Integers under Usual Metric is Complete Metric Space | https://proofwiki.org/wiki/Positive_Integers_under_Usual_Metric_is_Complete_Metric_Space | https://proofwiki.org/wiki/Positive_Integers_under_Usual_Metric_is_Complete_Metric_Space | [
"Euclidean Metric",
"Complete Metric Spaces"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Complete Metric Space"
] | [
"Definition:Cauchy Sequence/Metric Space",
"Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant",
"Definition:Convergent Sequence/Metric Space",
"Definition:Complete Metric Space"
] |
proofwiki-10532 | Cauchy Sequence in Positive Integers under Scaled Euclidean Metric | Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the scaled Euclidean metric on $\Z_{>0}$ defined as:
:$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
The sequence $\sequence {x_n}$ in $\Z_{>0}$ defined as:
:$\forall n \in \N: x_n = ... | For a general $x_m, x_n \in \sequence {x_n}$ as defined:
{{begin-eqn}}
{{eqn | l = \map \delta {x, y}
| r = \frac {\size {x_m - x_n} } {x_m x_n}
| c = Definition of $\delta$
}}
{{eqn | r = \size {\frac 1 {x_m} - \frac 1 {x_n} }
| c = algebra
}}
{{eqn | n = 1
| r = \size {\frac 1 m - \dfrac 1 n}
... | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition:Scaled Euclidean Metric|scaled Euclidean metric]] on $\Z_{>0}$ defined as:
:$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size... | For a general $x_m, x_n \in \sequence {x_n}$ as defined:
{{begin-eqn}}
{{eqn | l = \map \delta {x, y}
| r = \frac {\size {x_m - x_n} } {x_m x_n}
| c = Definition of $\delta$
}}
{{eqn | r = \size {\frac 1 {x_m} - \frac 1 {x_n} }
| c = algebra
}}
{{eqn | n = 1
| r = \size {\frac 1 m - \dfrac 1 n}... | Cauchy Sequence in Positive Integers under Scaled Euclidean Metric | https://proofwiki.org/wiki/Cauchy_Sequence_in_Positive_Integers_under_Scaled_Euclidean_Metric | https://proofwiki.org/wiki/Cauchy_Sequence_in_Positive_Integers_under_Scaled_Euclidean_Metric | [
"Scaled Euclidean Metric",
"Cauchy Sequences"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Scaled Euclidean Metric",
"Definition:Sequence",
"Definition:Cauchy Sequence/Metric Space"
] | [
"Axiom of Archimedes",
"Definition:Cauchy Sequence/Metric Space"
] |
proofwiki-10533 | Positive Integers under Scaled Euclidean Metric is not Complete Metric Space | Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the scaled Euclidean metric on $\Z_{>0}$ defined as:
:$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Then $\struct {\Z_{>0}, \delta}$ is not a complete metric space. | Consider the sequence $\sequence {x_n}$ in $\Z_{>0}$ defined as:
:$\forall n \in \N: x_n = n$
From Cauchy Sequence in Positive Integers under Scaled Euclidean Metric:
:$\sequence {x_n}$ is a Cauchy sequence in $\struct {\Z_{>0}, \delta}$.
But $\sequence {x_n}$ is not convergent to any $m \in \Z_{>0}$.
Hence the result,... | Let $\Z_{>0}$ be the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the [[Definition:Scaled Euclidean Metric|scaled Euclidean metric]] on $\Z_{>0}$ defined as:
:$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size... | Consider the [[Definition:Sequence|sequence]] $\sequence {x_n}$ in $\Z_{>0}$ defined as:
:$\forall n \in \N: x_n = n$
From [[Cauchy Sequence in Positive Integers under Scaled Euclidean Metric]]:
:$\sequence {x_n}$ is a [[Definition:Cauchy Sequence (Metric Space)|Cauchy sequence]] in $\struct {\Z_{>0}, \delta}$.
But $... | Positive Integers under Scaled Euclidean Metric is not Complete Metric Space | https://proofwiki.org/wiki/Positive_Integers_under_Scaled_Euclidean_Metric_is_not_Complete_Metric_Space | https://proofwiki.org/wiki/Positive_Integers_under_Scaled_Euclidean_Metric_is_not_Complete_Metric_Space | [
"Scaled Euclidean Metric",
"Complete Metric Spaces"
] | [
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Scaled Euclidean Metric",
"Definition:Complete Metric Space"
] | [
"Definition:Sequence",
"Cauchy Sequence in Positive Integers under Scaled Euclidean Metric",
"Definition:Cauchy Sequence/Metric Space",
"Definition:Convergent Sequence/Metric Space",
"Definition:Complete Metric Space"
] |
proofwiki-10534 | Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed | Let $\struct {\R^2, \tau_d}$ be the real number plane with the usual (Euclidean) topology.
Let $A \subseteq R^2$ be the set of all points defined as:
:$A := \set {\tuple {x, y} \in \R^2: x y \ge 1}$
Then $A$ is a closed set in $\struct {\R^2, d}$. | By definition, $\tau_d$ is the topology induced by the Euclidean metric $d$.
Consider the complement of $A$ in $\R^2$:
:$A' := \R^2 \setminus A$
Thus:
:$A := \set {\tuple {x, y} \in \R^2: x y < 1}$
Let $a = \tuple {x_a, y_a} \in A^2$.
Let $\epsilon = \size {1 - x_a y_a}$.
Then the open $\epsilon$-ball of $a$ in $\R^2$ ... | Let $\struct {\R^2, \tau_d}$ be the [[Definition:Real Number Plane with Euclidean Topology|real number plane with the usual (Euclidean) topology]].
Let $A \subseteq R^2$ be the [[Definition:Set|set]] of all points defined as:
:$A := \set {\tuple {x, y} \in \R^2: x y \ge 1}$
Then $A$ is a [[Definition:Closed Set (Top... | By definition, $\tau_d$ is the [[Definition:Topology Induced by Metric|topology induced]] by the [[Definition:Euclidean Metric on Real Number Plane|Euclidean metric]] $d$.
Consider the [[Definition:Relative Complement|complement of $A$ in $\R^2$]]:
:$A' := \R^2 \setminus A$
Thus:
:$A := \set {\tuple {x, y} \in \R^2: ... | Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed | https://proofwiki.org/wiki/Subset_of_Euclidean_Plane_whose_Product_of_Coordinates_are_Greater_Than_or_Equal_to_1_is_Closed | https://proofwiki.org/wiki/Subset_of_Euclidean_Plane_whose_Product_of_Coordinates_are_Greater_Than_or_Equal_to_1_is_Closed | [
"Examples of Closed Sets",
"Real Number Plane with Euclidean Topology"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Set",
"Definition:Closed Set/Topology"
] | [
"Definition:Topology Induced by Metric",
"Definition:Euclidean Metric/Real Number Plane",
"Definition:Relative Complement",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space"
] |
proofwiki-10535 | Projection on Real Euclidean Plane is Open Mapping | Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology.
Let $\rho: \R^2 \to \R$ be the first projection on $\R^2$ defined as:
:$\forall \tuple {x, y} \in \R^2: \map \rho {x, y} = x$
Then $\rho$ is an open mapping.
The same applies with the second projection on $\R^2$. | By definition, the real number plane with the usual (Euclidean) topology on $\R^2$ is the product space of $\struct {\R, d}$ with $\struct {\R, d}$, where $\struct {\R, d}$ is the real number line with the usual (Euclidean) topology
The result follows from Projection from Product Topology is Open.
{{qed}} | Let $\struct {\R^2, d}$ be the [[Definition:Real Number Plane with Euclidean Topology|real number plane with the usual (Euclidean) topology]].
Let $\rho: \R^2 \to \R$ be the [[Definition:First Projection|first projection]] on $\R^2$ defined as:
:$\forall \tuple {x, y} \in \R^2: \map \rho {x, y} = x$
Then $\rho$ is a... | By definition, the [[Definition:Real Number Plane with Euclidean Topology|real number plane with the usual (Euclidean) topology]] on $\R^2$ is the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $\struct {\R, d}$ with $\struct {\R, d}$, where $\struct {\R, d}$ is the [[Definition:Real Numb... | Projection on Real Euclidean Plane is Open Mapping | https://proofwiki.org/wiki/Projection_on_Real_Euclidean_Plane_is_Open_Mapping | https://proofwiki.org/wiki/Projection_on_Real_Euclidean_Plane_is_Open_Mapping | [
"Real Number Plane with Euclidean Topology",
"Examples of Open Mappings"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Projection (Mapping Theory)/First Projection",
"Definition:Open Mapping",
"Definition:Projection (Mapping Theory)/Second Projection"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Projection from Product Topology is Open"
] |
proofwiki-10536 | Projection on Real Euclidean Plane is not Closed Mapping | Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology.
Let $\rho: \R^2 \to \R$ be the first projection on $\R^2$ defined as:
:$\forall \tuple {x, y} \in \R^2: \map \rho {x, y} = x$
Then $\rho$ is not a closed mapping.
The same applies with the second projection on $\R^2$. | Consider the set $A \subseteq R^2$ of all points defined as:
:$A := \set {\tuple {x, y} \in \R^2: x y \ge 1}$
By Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed:
:$A$ is a closed set in $\struct {\R^2, d}$.
By inspection, it can be seen that the image of $A$ under $\rho$ ... | Let $\struct {\R^2, d}$ be the [[Definition:Real Number Plane with Euclidean Topology|real number plane with the usual (Euclidean) topology]].
Let $\rho: \R^2 \to \R$ be the [[Definition:First Projection|first projection]] on $\R^2$ defined as:
:$\forall \tuple {x, y} \in \R^2: \map \rho {x, y} = x$
Then $\rho$ is n... | Consider the [[Definition:Set|set]] $A \subseteq R^2$ of all points defined as:
:$A := \set {\tuple {x, y} \in \R^2: x y \ge 1}$
By [[Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed]]:
:$A$ is a [[Definition:Closed Set (Topology)|closed set]] in $\struct {\R^2, d}$.
By ... | Projection on Real Euclidean Plane is not Closed Mapping | https://proofwiki.org/wiki/Projection_on_Real_Euclidean_Plane_is_not_Closed_Mapping | https://proofwiki.org/wiki/Projection_on_Real_Euclidean_Plane_is_not_Closed_Mapping | [
"Real Number Plane with Euclidean Topology",
"Examples of Closed Mappings"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Projection (Mapping Theory)/First Projection",
"Definition:Closed Mapping",
"Definition:Projection (Mapping Theory)/Second Projection"
] | [
"Definition:Set",
"Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed",
"Definition:Closed Set/Topology",
"Definition:Image (Set Theory)/Mapping/Subset",
"Union of Open Sets of Metric Space is Open",
"Definition:Open Set/Topology",
"Definition:Closed Mapping"... |
proofwiki-10537 | Complement of Set of Rational Pairs in Real Euclidean Plane is Injectively Path-Connected | Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology.
Let $S \subseteq \R^2$ be the subset of $\R^2$ defined as:
:$\forall x, y \in \R^2: \tuple {x, y} \in S \iff x, y \in \Q$
Hence let $A := \R^2 \setminus S$:
:$\tuple {x, y} \in A$ {{iff}} either $x$ or $y$ or both is irrational.
Then ... | Let $\tuple {a, b} \in A$.
Consider any point $\tuple {x_1, y_1} \in A$ whose coordinates are both irrational.
By definition, either $a$ or $b$ is irrational.
{{WLOG}}, suppose $a$ is irrational.
Then the union of the straight lines $x = a, y = y_1$ is an injectively path-connected subset of $A$ connecting $\tuple {x_1... | Let $\struct {\R^2, d}$ be the [[Definition:Real Number Plane with Euclidean Topology|real number plane with the usual (Euclidean) topology]].
Let $S \subseteq \R^2$ be the [[Definition:Subset|subset]] of $\R^2$ defined as:
:$\forall x, y \in \R^2: \tuple {x, y} \in S \iff x, y \in \Q$
Hence let $A := \R^2 \setminus ... | Let $\tuple {a, b} \in A$.
Consider any point $\tuple {x_1, y_1} \in A$ whose [[Definition:Coordinate of Vector|coordinates]] are both [[Definition:Irrational Number|irrational]].
By definition, either $a$ or $b$ is [[Definition:Irrational Number|irrational]].
{{WLOG}}, suppose $a$ is [[Definition:Irrational Number|... | Complement of Set of Rational Pairs in Real Euclidean Plane is Injectively Path-Connected | https://proofwiki.org/wiki/Complement_of_Set_of_Rational_Pairs_in_Real_Euclidean_Plane_is_Injectively_Path-Connected | https://proofwiki.org/wiki/Complement_of_Set_of_Rational_Pairs_in_Real_Euclidean_Plane_is_Injectively_Path-Connected | [
"Real Number Plane with Euclidean Topology",
"Examples of Injectively Path-Connected Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Subset",
"Definition:Irrational Number",
"Definition:Injectively Path-Connected/Subset"
] | [
"Definition:Coordinate System/Coordinate",
"Definition:Irrational Number",
"Definition:Irrational Number",
"Definition:Irrational Number",
"Definition:Set Union",
"Definition:Line/Straight Line",
"Definition:Injectively Path-Connected/Subset",
"Definition:Element",
"Definition:Injectively Path-Conne... |
proofwiki-10538 | Empty Set is Compact | Let $T = \struct {S, \tau}$ be a topological space.
Then the empty set $\O$ is a compact subspace of $T$. | Recall the definition of compact subspace:
:$\struct {\O, \tau_\O}$ is '''compact in $T$''' {{iff}} every open cover $\CC \subseteq \tau_\O$ for $\O$ has a finite subcover.
The only open cover for $\O$ that is contained in $\O$ is $\set \O$ itself.
This has only one finite subcover, and that is $\set \O$.
This is a fin... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then the [[Definition:Empty Set|empty set]] $\O$ is a [[Definition:Compact Subspace|compact subspace]] of $T$. | Recall the definition of [[Definition:Compact Subspace|compact subspace]]:
:$\struct {\O, \tau_\O}$ is '''[[Definition:Compact Subspace|compact in $T$]]''' {{iff}} every [[Definition:Open Cover|open cover]] $\CC \subseteq \tau_\O$ for $\O$ has a [[Definition:Finite Subcover|finite subcover]].
The only [[Definition:Ope... | Empty Set is Compact | https://proofwiki.org/wiki/Empty_Set_is_Compact | https://proofwiki.org/wiki/Empty_Set_is_Compact | [
"Empty Set",
"Examples of Compact Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Empty Set",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Subcover/Finite",
"Definition:Compact Topological Space/Subspace",
"Category:Emp... |
proofwiki-10539 | Alexandroff Extension is Topology | Let $T = \struct {S, \tau}$ be a non-empty topological space.
Let $p$ be a new element not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$.
Then $\tau^*$ is a topology on $S^*$. | Recall the definition of the Alexandroff extension on $S$:
$U$ is open in $T^*$ {{iff}}:
:$U$ is an open set of $T$
or
:$U$ is the complement in $T^*$ of a closed and compact subset of $T$.
Each of the open set axioms is examined in turn: | Let $T = \struct {S, \tau}$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological space]].
Let $p$ be a [[Definition:New Element|new element]] not in $S$.
Let $S^* := S \cup \set p$.
Let $T^* = \struct {S^*, \tau^*}$ be the [[Definition:Alexandroff Extension|Alexandroff extension]] o... | Recall the definition of the [[Definition:Alexandroff Extension|Alexandroff extension]] on $S$:
$U$ is [[Definition:Open Set (Topology)|open]] in $T^*$ {{iff}}:
:$U$ is an [[Definition:Open Set (Topology)|open set]] of $T$
or
:$U$ is the [[Definition:Relative Complement|complement]] in $T^*$ of a [[Definition:Closed S... | Alexandroff Extension is Topology | https://proofwiki.org/wiki/Alexandroff_Extension_is_Topology | https://proofwiki.org/wiki/Alexandroff_Extension_is_Topology | [
"Alexandroff Extensions"
] | [
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:New Element",
"Definition:Alexandroff Extension",
"Definition:Topology"
] | [
"Definition:Alexandroff Extension",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Compact Topological Space/Subspace",
"Axiom:Open Set Axioms",
"Definition:Open Set/Topology",
"Definition:Open Set/Topol... |
proofwiki-10540 | Intersection of Closed Set with Compact Subspace is Compact | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be closed in $T$.
Let $K \subseteq S$ be compact in $T$.
Then $H \cap K$ is compact in $T$. | Let $\tau_K$ be the subspace topology on $K$.
Let $T_K = \left({K, \tau_K}\right)$ be the topological subspace determined by $K$.
By Closed Set in Topological Subspace, $H \cap K$ is closed in $T_K$.
By Closed Subspace of Compact Space is Compact, $H \cap K$ is compact in $T_K$.
By Compact in Subspace is Compact in Top... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be [[Definition:Closed Set (Topology)|closed]] in $T$.
Let $K \subseteq S$ be [[Definition:Compact Topological Subspace|compact]] in $T$.
Then $H \cap K$ is [[Definition:Compact Topological Subspace|compact]] in... | Let $\tau_K$ be the [[Definition:Subspace Topology|subspace topology]] on $K$.
Let $T_K = \left({K, \tau_K}\right)$ be the [[Definition:Topological Subspace|topological subspace]] determined by $K$.
By [[Closed Set in Topological Subspace]], $H \cap K$ is [[Definition:Closed Set (Topology)|closed]] in $T_K$.
By [[Cl... | Intersection of Closed Set with Compact Subspace is Compact/Proof 1 | https://proofwiki.org/wiki/Intersection_of_Closed_Set_with_Compact_Subspace_is_Compact | https://proofwiki.org/wiki/Intersection_of_Closed_Set_with_Compact_Subspace_is_Compact/Proof_1 | [
"Intersection of Closed Set with Compact Subspace is Compact",
"Compact Topological Spaces",
"Closed Sets"
] | [
"Definition:Topological Space",
"Definition:Closed Set/Topology",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Closed Set in Topological Subspace",
"Definition:Closed Set/Topology",
"Closed Subspace of Compact Space is Compact",
"Definition:Compact Topological Space/Subspace",
"Compact in Subspace is Compact in Topological Space",
"Definiti... |
proofwiki-10541 | Intersection of Closed Set with Compact Subspace is Compact | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be closed in $T$.
Let $K \subseteq S$ be compact in $T$.
Then $H \cap K$ is compact in $T$. | Let $\family {U_\alpha}$ be an open cover of $H \cap K$:
:$\ds H \cap K \subseteq \bigcup_\alpha U_\alpha$
Then:
:$\ds K \subseteq \bigcup_\alpha U_\alpha \cup \paren {S \setminus H}$
Since $H$ is closed in $T$, $\paren {S \setminus H}$ is open in $T$.
Hence $\family {U_\alpha} \cup S \setminus H$ is an open cover of $... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be [[Definition:Closed Set (Topology)|closed]] in $T$.
Let $K \subseteq S$ be [[Definition:Compact Topological Subspace|compact]] in $T$.
Then $H \cap K$ is [[Definition:Compact Topological Subspace|compact]] in... | Let $\family {U_\alpha}$ be an [[Definition:Open Cover|open cover]] of $H \cap K$:
:$\ds H \cap K \subseteq \bigcup_\alpha U_\alpha$
Then:
:$\ds K \subseteq \bigcup_\alpha U_\alpha \cup \paren {S \setminus H}$
Since $H$ is [[Definition:Closed Set (Topology)|closed]] in $T$, $\paren {S \setminus H}$ is [[Definition:Op... | Intersection of Closed Set with Compact Subspace is Compact/Proof 2 | https://proofwiki.org/wiki/Intersection_of_Closed_Set_with_Compact_Subspace_is_Compact | https://proofwiki.org/wiki/Intersection_of_Closed_Set_with_Compact_Subspace_is_Compact/Proof_2 | [
"Intersection of Closed Set with Compact Subspace is Compact",
"Compact Topological Spaces",
"Closed Sets"
] | [
"Definition:Topological Space",
"Definition:Closed Set/Topology",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Open Cover",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Definition:Compact Topological Space/Subspace",
"Definition:Subcover/Finite",
"Definition:Compact Topological Space/Subspace"
] |
proofwiki-10542 | Finite Union of Compact Sets is Compact | Let $T = \struct {S, \tau}$ be a topological space.
Let $n \in \N$ be a natural number.
Let $\sequence {U_i}_{1 \mathop \le i \mathop \le n}$ be a finite sequence of compact subsets of $T$.
Let $\UU_n := \ds \bigcup_{i \mathop = 1}^n U_i$ be the union of $\sequence {U_i}$.
Then $\UU_n$ is compact in $T$. | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\UU_n := \ds \bigcup_{i \mathop = 1}^n U_i$ is compact in $T$.
$\map P 0$ is the case:
:$\UU_0 := \ds \bigcup_{i \mathop = 1}^0 U_i$
From Union of Empty Set:
:$\ds \bigcup_{i \mathop = 1}^0 U_i = \O$
From Empty Set is Compact it follows ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $\sequence {U_i}_{1 \mathop \le i \mathop \le n}$ be a [[Definition:Finite Sequence|finite sequence]] of [[Definition:Compact Topological Subspace|compact subsets]... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\UU_n := \ds \bigcup_{i \mathop = 1}^n U_i$ is [[Definition:Compact Topological Subspace|compact]] in $T$.
$\map P 0$ is the case:
:$\UU_0 := \ds \bigcup_{i \mathop = ... | Finite Union of Compact Sets is Compact | https://proofwiki.org/wiki/Finite_Union_of_Compact_Sets_is_Compact | https://proofwiki.org/wiki/Finite_Union_of_Compact_Sets_is_Compact | [
"Compact Topological Spaces",
"Set Union"
] | [
"Definition:Topological Space",
"Definition:Natural Numbers",
"Definition:Finite Sequence",
"Definition:Compact Topological Space/Subspace",
"Definition:Set Union/Family of Sets",
"Definition:Compact Topological Space/Subspace"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Compact Topological Space/Subspace",
"Union of Empty Set",
"Empty Set is Compact",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspa... |
proofwiki-10543 | Principle of Mathematical Induction | Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is true
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$. | Let $S$ be the set defined as:
:$S := \set {n \in \N_{>0}: \map P n \text { is false} }$
{{AimForCont}} $S \ne \O$.
From the Well-Ordering Principle it follows that $S$ has a minimal element $m$.
From $(1)$ we have that $\map P 1$ holds.
Hence $1 \notin S$.
Therefore $m \ne 1$.
Therefore $m - 1 \in \N_{>0}$.
But $m$ is... | Let $\map P n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is [[Definition:True|true]]
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is [[Definition... | Let $S$ be the [[Definition:Set|set]] defined as:
:$S := \set {n \in \N_{>0}: \map P n \text { is false} }$
{{AimForCont}} $S \ne \O$.
From the [[Well-Ordering Principle]] it follows that $S$ has a [[Definition:Minimal Element|minimal element]] $m$.
From $(1)$ we have that $\map P 1$ holds.
Hence $1 \notin S$.
Th... | Principle of Mathematical Induction/One-Based/Proof 1 | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/One-Based/Proof_1 | [
"Principle of Mathematical Induction",
"Mathematical Induction",
"Number Theory",
"Named Theorems",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:True",
"Definition:True"
] | [
"Definition:Set",
"Well-Ordering Principle",
"Definition:Minimal/Element",
"Definition:Minimal/Element"
] |
proofwiki-10544 | Principle of Mathematical Induction | Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is true
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$. | Let $M$ be the set of all $n \in \N_{>0}$ for which $\map P n$ holds.
By $(1)$ we have that $1 \in M$.
By $(2)$ we have that if $k \in M$ then $k + 1 \in M$.
From the Axiomatization of $1$-Based Natural Numbers, Axiom $(\text F)$, it follows that $M = \N_{>0}$.
{{qed}} | Let $\map P n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is [[Definition:True|true]]
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is [[Definition... | Let $M$ be the [[Definition:Set|set]] of all $n \in \N_{>0}$ for which $\map P n$ holds.
By $(1)$ we have that $1 \in M$.
By $(2)$ we have that if $k \in M$ then $k + 1 \in M$.
From the [[Axiom:Axiomatization of 1-Based Natural Numbers|Axiomatization of $1$-Based Natural Numbers]], Axiom $(\text F)$, it follows that... | Principle of Mathematical Induction/One-Based/Proof 2 | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/One-Based/Proof_2 | [
"Principle of Mathematical Induction",
"Mathematical Induction",
"Number Theory",
"Named Theorems",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:True",
"Definition:True"
] | [
"Definition:Set",
"Axiom:Axiomatization of 1-Based Natural Numbers"
] |
proofwiki-10545 | Principle of Mathematical Induction | Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is true
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$. | We have that Natural Numbers are Non-Negative Integers.
Then we have that Integers form Well-Ordered Integral Domain.
The result follows from Induction on Well-Ordered Integral Domain.
{{qed}} | Let $\map P n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is [[Definition:True|true]]
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is [[Definition... | We have that [[Natural Numbers are Non-Negative Integers]].
Then we have that [[Integers form Well-Ordered Integral Domain]].
The result follows from [[Induction on Well-Ordered Integral Domain]].
{{qed}} | Principle of Mathematical Induction/One-Based/Proof 3 | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/One-Based/Proof_3 | [
"Principle of Mathematical Induction",
"Mathematical Induction",
"Number Theory",
"Named Theorems",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:True",
"Definition:True"
] | [
"Natural Numbers are Non-Negative Integers",
"Integers form Well-Ordered Integral Domain",
"Principle of Mathematical Induction/Well-Ordered Integral Domain"
] |
proofwiki-10546 | Principle of Mathematical Induction | Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is true
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$. | Let $\Z_{\ge n_0}$ denote the set:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S$ be the set of integers defined as:
:$S = \set {n \in \Z_{\ge n_0}: \map P n}$
That is, the set of all integers for which $n \ge n_0$ and for which $\map P n$ holds.
From Subset of Set with Propositional Function we have that:
:$S \s... | Let $\map P n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is [[Definition:True|true]]
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is [[Definition... | Let $\Z_{\ge n_0}$ denote the [[Definition:Set|set]]:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S$ be the [[Definition:Set|set]] of [[Definition:Integer|integers]] defined as:
:$S = \set {n \in \Z_{\ge n_0}: \map P n}$
That is, the set of all [[Definition:Integer|integers]] for which $n \ge n_0$ and for which... | Principle of Mathematical Induction/Proof | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/Proof | [
"Principle of Mathematical Induction",
"Mathematical Induction",
"Number Theory",
"Named Theorems",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:True",
"Definition:True"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Integer",
"Definition:Integer",
"Subset of Set with Propositional Function",
"Principle of Finite Induction"
] |
proofwiki-10547 | Principle of Mathematical Induction | Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is true
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$. | For each $n \in \N_{>0}$, let $\map {P'} n$ be defined as:
:$\map {P'} n := \map P 1 \land \dots \land \map P n$
It suffices to show that $\map {P'} n$ is true for all $n \in \N_{>0}$.
It is immediate from the assumption $\map P 1$ that $\map {P'} 1$ is true.
Now suppose that $\map {P'} n$ holds.
By $(2)$, this implies... | Let $\map P n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is [[Definition:True|true]]
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is [[Definition... | For each $n \in \N_{>0}$, let $\map {P'} n$ be defined as:
:$\map {P'} n := \map P 1 \land \dots \land \map P n$
It suffices to show that $\map {P'} n$ is true for all $n \in \N_{>0}$.
It is immediate from the assumption $\map P 1$ that $\map {P'} 1$ is [[Definition:True|true]].
Now suppose that $\map {P'} n$ hold... | Second Principle of Mathematical Induction/One-Based/Proof 1 | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction | https://proofwiki.org/wiki/Second_Principle_of_Mathematical_Induction/One-Based/Proof_1 | [
"Principle of Mathematical Induction",
"Mathematical Induction",
"Number Theory",
"Named Theorems",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:True",
"Definition:True"
] | [
"Definition:True",
"Principle of Mathematical Induction"
] |
proofwiki-10548 | Principle of Mathematical Induction | Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is true
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$. | Let $S \subseteq \N_{>0}$ containing those $n \in \N_{>0}$ for which $\map P n$ does not hold.
{{AimForCont}} $S \ne \O$.
Then by the Well-Ordering Principle $S$ contains a minimal element $s$.
We have that $s \ne 1$ because $\map P 1$ is true from $(1)$.
Thus there must exist some $k \in \N_{>0}$ such that $s = k + 1$... | Let $\map P n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is [[Definition:True|true]]
:$(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is [[Definition... | Let $S \subseteq \N_{>0}$ containing those $n \in \N_{>0}$ for which $\map P n$ does not hold.
{{AimForCont}} $S \ne \O$.
Then by the [[Well-Ordering Principle]] $S$ contains a [[Definition:Minimal Element|minimal element]] $s$.
We have that $s \ne 1$ because $\map P 1$ is true from $(1)$.
Thus there must exist som... | Second Principle of Mathematical Induction/One-Based/Proof 2 | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction | https://proofwiki.org/wiki/Second_Principle_of_Mathematical_Induction/One-Based/Proof_2 | [
"Principle of Mathematical Induction",
"Mathematical Induction",
"Number Theory",
"Named Theorems",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:True",
"Definition:True"
] | [
"Well-Ordering Principle",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Contradiction",
"Proof by Contradiction"
] |
proofwiki-10549 | Principle of Finite Induction | Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the set:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a subset of $\Z_{\ge n_0}$.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$
Then:
:$\forall n \ge n_0: n \in S$
That is:
:$S = ... | Consider $\N$ defined as a naturally ordered semigroup.
The result follows directly from Principle of Mathematical Induction for Naturally Ordered Semigroup: General Result.
{{qed}} | Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the [[Definition:Set|set]]:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a [[Definition:Subset|subset]] of $\Z_{\ge n_0}$.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$
Th... | Consider $\N$ defined as a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
The result follows directly from [[Principle of Mathematical Induction for Naturally Ordered Semigroup/General Result|Principle of Mathematical Induction for Naturally Ordered Semigroup: General Result]].
{{qed}} | Principle of Finite Induction/One-Based/Proof 1 | https://proofwiki.org/wiki/Principle_of_Finite_Induction | https://proofwiki.org/wiki/Principle_of_Finite_Induction/One-Based/Proof_1 | [
"Principle of Finite Induction",
"Mathematical Induction",
"Proof Techniques"
] | [
"Definition:Set",
"Definition:Subset"
] | [
"Definition:Naturally Ordered Semigroup",
"Principle of Mathematical Induction/Naturally Ordered Semigroup/General Result"
] |
proofwiki-10550 | Principle of Finite Induction | Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the set:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a subset of $\Z_{\ge n_0}$.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$
Then:
:$\forall n \ge n_0: n \in S$
That is:
:$S = ... | Let $T$ be the set of all $1$-based natural numbers not in $S$:
:$T = \N_{>0} \setminus S$
{{AimForCont}} $T$ is non-empty.
From the Well-Ordering Principle, $T$ has a smallest element.
Let this smallest element be denoted $a$.
We have been given that $1 \in S$.
So:
:$a > 1$
and so:
:$0 < a - 1 < a$
As $a$ is the small... | Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the [[Definition:Set|set]]:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a [[Definition:Subset|subset]] of $\Z_{\ge n_0}$.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$
Th... | Let $T$ be the [[Definition:Set|set]] of all [[Definition:1-Based Natural Numbers|$1$-based natural numbers]] not in $S$:
:$T = \N_{>0} \setminus S$
{{AimForCont}} $T$ is [[Definition:Non-Empty Set|non-empty]].
From the [[Well-Ordering Principle]], $T$ has a [[Definition:Smallest Element|smallest element]].
Let this... | Principle of Finite Induction/One-Based/Proof 2 | https://proofwiki.org/wiki/Principle_of_Finite_Induction | https://proofwiki.org/wiki/Principle_of_Finite_Induction/One-Based/Proof_2 | [
"Principle of Finite Induction",
"Mathematical Induction",
"Proof Techniques"
] | [
"Definition:Set",
"Definition:Subset"
] | [
"Definition:Set",
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Non-Empty Set",
"Well-Ordering Principle",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:By Hypothesis",
"Definition:Contradiction",
"Proof by Contradiction",... |
proofwiki-10551 | Principle of Finite Induction | Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the set:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a subset of $\Z_{\ge n_0}$.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$
Then:
:$\forall n \ge n_0: n \in S$
That is:
:$S = ... | Let $\Z_{\ge n_0} := \set {n \in \Z: n \ge n_0}$.
{{AimForCont}} $S \ne \Z_{\ge n_0}$.
Let $S' = \Z_{\ge n_0} \setminus S$.
Because $S \ne \Z_{\ge n_0}$ and $S \subseteq \Z_{\ge n_0}$, we have that $S' \ne \O$.
By definition, $\Z_{\ge n_0}$ is bounded below by $n_0$.
From Set of Integers Bounded Below by Integer has Sm... | Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the [[Definition:Set|set]]:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a [[Definition:Subset|subset]] of $\Z_{\ge n_0}$.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$
Th... | Let $\Z_{\ge n_0} := \set {n \in \Z: n \ge n_0}$.
{{AimForCont}} $S \ne \Z_{\ge n_0}$.
Let $S' = \Z_{\ge n_0} \setminus S$.
Because $S \ne \Z_{\ge n_0}$ and $S \subseteq \Z_{\ge n_0}$, we have that $S' \ne \O$.
By definition, $\Z_{\ge n_0}$ is [[Definition:Bounded Below Set|bounded below]] by $n_0$.
From [[Set of ... | Principle of Finite Induction/Proof 1 | https://proofwiki.org/wiki/Principle_of_Finite_Induction | https://proofwiki.org/wiki/Principle_of_Finite_Induction/Proof_1 | [
"Principle of Finite Induction",
"Mathematical Induction",
"Proof Techniques"
] | [
"Definition:Set",
"Definition:Subset"
] | [
"Definition:Bounded Below Set",
"Set of Integers Bounded Below by Integer has Smallest Element",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Proof by Contradiction"
] |
proofwiki-10552 | Principle of Finite Induction | Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the set:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a subset of $\Z_{\ge n_0}$.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$
Then:
:$\forall n \ge n_0: n \in S$
That is:
:$S = ... | {{questionable|This only takes on board a subset of $\N$, where we need a subset of $\Z$}}
Consider $\N$ defined as a naturally ordered semigroup.
The result follows directly from Principle of Mathematical Induction for Naturally Ordered Semigroup: General Result.
{{qed}} | Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the [[Definition:Set|set]]:
:$\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a [[Definition:Subset|subset]] of $\Z_{\ge n_0}$.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: n \in S \implies n + 1 \in S$
Th... | {{questionable|This only takes on board a subset of $\N$, where we need a subset of $\Z$}}
Consider $\N$ defined as a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
The result follows directly from [[Principle of Mathematical Induction for Naturally Ordered Semigroup/General Result|Principle ... | Principle of Finite Induction/Proof 2 | https://proofwiki.org/wiki/Principle_of_Finite_Induction | https://proofwiki.org/wiki/Principle_of_Finite_Induction/Proof_2 | [
"Principle of Finite Induction",
"Mathematical Induction",
"Proof Techniques"
] | [
"Definition:Set",
"Definition:Subset"
] | [
"Definition:Naturally Ordered Semigroup",
"Principle of Mathematical Induction/Naturally Ordered Semigroup/General Result"
] |
proofwiki-10553 | Principle of Mathematical Induction/Peano Structure | Let $\struct {P, s, 0}$ be a Peano structure.
Let $\map Q n$ be a propositional function depending on $n \in P$.
Suppose that:
:$(1): \quad \map Q 0$ is true
:$(2): \quad \forall n \in P: \map Q n \implies \map Q {\map s n}$
Then:
:$\forall n \in P: \map Q n$ | Let $A \subseteq P$ be defined by:
:$A := \set {n \in P: \map Q n}$
From $(1)$, $0 \in A$.
From $(2)$:
:$\forall n \in P: n \in A \implies \map s n \in A$
As this holds for all $n \in P$, it holds a fortiori for all $n \in A$.
Thus the condition:
:$n \in A \implies \map s n \in A$
is satisfied.
So by Axiom $(\text P 5)... | Let $\struct {P, s, 0}$ be a [[Definition:Peano Structure|Peano structure]].
Let $\map Q n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in P$.
Suppose that:
:$(1): \quad \map Q 0$ is [[Definition:True|true]]
:$(2): \quad \forall n \in P: \map Q n \implies \map Q {\map s n}$
... | Let $A \subseteq P$ be defined by:
:$A := \set {n \in P: \map Q n}$
From $(1)$, $0 \in A$.
From $(2)$:
:$\forall n \in P: n \in A \implies \map s n \in A$
As this holds for all $n \in P$, it holds [[Definition:A Fortiori|a fortiori]] for all $n \in A$.
Thus the condition:
:$n \in A \implies \map s n \in A$
is sati... | Principle of Mathematical Induction/Peano Structure | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/Peano_Structure | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/Peano_Structure | [
"Peano's Axioms",
"Principle of Mathematical Induction"
] | [
"Definition:Peano Structure",
"Definition:Propositional Function",
"Definition:True"
] | [
"Definition:A Fortiori",
"Axiom:Peano's Axioms"
] |
proofwiki-10554 | Principle of Finite Induction/Peano Structure | Let $\struct {P, s, 0}$ be a Peano structure.
Let $S \subseteq P$.
Suppose that:
:$(1): \quad 0 \in S$
:$(2): \quad \forall n: n \in S \implies \map s n \in S$
Then:
:$S = P$ | This is nothing but a reformulation of Axiom $(P5)$ of the Peano Axioms.
{{qed}}
Category:Peano's Axioms
Category:Principle of Finite Induction
m9ssfkhi9gsur18ot0230664iewgd4i | Let $\struct {P, s, 0}$ be a [[Definition:Peano Structure|Peano structure]].
Let $S \subseteq P$.
Suppose that:
:$(1): \quad 0 \in S$
:$(2): \quad \forall n: n \in S \implies \map s n \in S$
Then:
:$S = P$ | This is nothing but a reformulation of Axiom $(P5)$ of the [[Axiom:Peano's Axioms|Peano Axioms]].
{{qed}}
[[Category:Peano's Axioms]]
[[Category:Principle of Finite Induction]]
m9ssfkhi9gsur18ot0230664iewgd4i | Principle of Finite Induction/Peano Structure | https://proofwiki.org/wiki/Principle_of_Finite_Induction/Peano_Structure | https://proofwiki.org/wiki/Principle_of_Finite_Induction/Peano_Structure | [
"Peano's Axioms",
"Principle of Finite Induction"
] | [
"Definition:Peano Structure"
] | [
"Axiom:Peano's Axioms",
"Category:Peano's Axioms",
"Category:Principle of Finite Induction"
] |
proofwiki-10555 | Principle of Mathematical Induction/Naturally Ordered Semigroup | Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $T \subseteq S$ such that $0 \in T$ and $n \in T \implies n \circ 1 \in T$.
Then $T = S$. | {{AimForCont}} that $T \subsetneq S$.
That is, $T$ is a proper subset of $S$:
: $T \ne S$
Let $T' = S \setminus T$.
Then by Set Difference with Proper Subset:
:$T' \ne \O$
By {{NOSAxiom|1}}, $S$ is well-ordered.
By definition of well-ordered set, it follows that $T'$ has a smallest element $x$.
By definition of $T$:
:$... | Let $\struct {S, \circ, \preceq}$ be a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
Let $T \subseteq S$ such that $0 \in T$ and $n \in T \implies n \circ 1 \in T$.
Then $T = S$. | {{AimForCont}} that $T \subsetneq S$.
That is, $T$ is a [[Definition:Proper Subset|proper subset]] of $S$:
: $T \ne S$
Let $T' = S \setminus T$.
Then by [[Set Difference with Proper Subset]]:
:$T' \ne \O$
By {{NOSAxiom|1}}, $S$ is [[Definition:Well-Ordered Set|well-ordered]].
By definition of [[Definition:Well-Or... | Principle of Mathematical Induction/Naturally Ordered Semigroup | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/Naturally_Ordered_Semigroup | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/Naturally_Ordered_Semigroup | [
"Naturally Ordered Semigroup",
"Principle of Mathematical Induction"
] | [
"Definition:Naturally Ordered Semigroup"
] | [
"Definition:Proper Subset",
"Set Difference with Proper Subset",
"Definition:Well-Ordered Set",
"Definition:Well-Ordered Set",
"Definition:Smallest Element",
"Sum with One is Immediate Successor in Naturally Ordered Semigroup",
"Definition:Naturally Ordered Semigroup",
"Sum with One is Immediate Succe... |
proofwiki-10556 | Consecutive Fibonacci Numbers are Coprime | Let $F_k$ be the $k$th Fibonacci number.
Then:
:$\forall n \ge 2: \gcd \set {F_n, F_{n + 1} } = 1$
where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.
That is, a Fibonacci number and the one next to it are coprime. | From the definition of Fibonacci numbers:
:$F_1 = 1, F_2 = 1, F_3 = 2$
Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\gcd \set {F_n, F_{n + 1} } = 1$ | Let $F_k$ be the $k$th [[Definition:Fibonacci Numbers|Fibonacci number]].
Then:
:$\forall n \ge 2: \gcd \set {F_n, F_{n + 1} } = 1$
where $\gcd \set {a, b}$ denotes the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $a$ and $b$.
That is, a [[Definition:Fibonacci Numbers|Fibonacci num... | From the definition of [[Definition:Fibonacci Numbers|Fibonacci numbers]]:
:$F_1 = 1, F_2 = 1, F_3 = 2$
Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\gcd \set {F_n, F_{n + 1} } = 1$ | Consecutive Fibonacci Numbers are Coprime | https://proofwiki.org/wiki/Consecutive_Fibonacci_Numbers_are_Coprime | https://proofwiki.org/wiki/Consecutive_Fibonacci_Numbers_are_Coprime | [
"Fibonacci Numbers",
"Coprime Integers",
"Proofs by Induction"
] | [
"Definition:Fibonacci Number",
"Definition:Greatest Common Divisor/Integers",
"Definition:Fibonacci Number",
"Definition:Coprime/Integers"
] | [
"Definition:Fibonacci Number",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-10557 | Divisibility of Fibonacci Number | :$\forall m, n \in \Z_{> 2} : m \divides n \iff F_m \divides F_n$
where $\divides$ denotes divisibility. | From the initial definition of Fibonacci numbers:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
Let $n = k m - r$ where $0 \le r < m$
We have:
:$m \divides n \iff r = 0$
The proof proceeds by induction on $k$.
For all $k \in \N_{>0}$, let $\map P k$ be the proposition:
:$r = 0 \iff F_m \divides F_{k m - r}$ | :$\forall m, n \in \Z_{> 2} : m \divides n \iff F_m \divides F_n$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | From the initial definition of [[Definition:Fibonacci Number|Fibonacci numbers]]:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
Let $n = k m - r$ where $0 \le r < m$
We have:
:$m \divides n \iff r = 0$
The proof proceeds by [[Principle of Mathematical Induction|induction]] on $k$.
For all $k \in \N_{>0}$, let $\map P k$ ... | Divisibility of Fibonacci Number | https://proofwiki.org/wiki/Divisibility_of_Fibonacci_Number | https://proofwiki.org/wiki/Divisibility_of_Fibonacci_Number | [
"Fibonacci Numbers",
"Divisors",
"Proofs by Induction"
] | [
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Fibonacci Number",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-10558 | Honsberger's Identity | :$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$ | From the initial definition of Fibonacci numbers, we have:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
Proof by induction:
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\forall m \in \Z_{>0} : F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eq... | :$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$ | From the initial definition of [[Definition:Fibonacci Number|Fibonacci numbers]], we have:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\forall m \in \Z_{>0} : F_{m + n} = F_{m... | Honsberger's Identity/Proof 1 | https://proofwiki.org/wiki/Honsberger's_Identity | https://proofwiki.org/wiki/Honsberger's_Identity/Proof_1 | [
"Fibonacci Numbers",
"Honsberger's Identity"
] | [] | [
"Definition:Fibonacci Number",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Honsberger's Identity/Proof 1",
"Principle of Mathematical Induction"
] |
proofwiki-10559 | Honsberger's Identity | :$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$ | {{begin-eqn}}
{{eqn | r = F_{m - 1} F_n + F_m F_{n + 1}
| o =
}}
{{eqn | r = \dfrac {\phi^{m - 1} - \hat \phi^{m - 1} } {\sqrt 5} \dfrac {\phi^n - \hat \phi^n} {\sqrt 5} + \dfrac {\phi^m - \hat \phi^m} {\sqrt 5} \dfrac {\phi^{n + 1} - \hat \phi^{n + 1} } {\sqrt 5}
| c = Euler-Binet Formula
}}
{{eqn | r = \... | :$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$ | {{begin-eqn}}
{{eqn | r = F_{m - 1} F_n + F_m F_{n + 1}
| o =
}}
{{eqn | r = \dfrac {\phi^{m - 1} - \hat \phi^{m - 1} } {\sqrt 5} \dfrac {\phi^n - \hat \phi^n} {\sqrt 5} + \dfrac {\phi^m - \hat \phi^m} {\sqrt 5} \dfrac {\phi^{n + 1} - \hat \phi^{n + 1} } {\sqrt 5}
| c = [[Euler-Binet Formula]]
}}
{{eqn | r... | Honsberger's Identity/Proof 2 | https://proofwiki.org/wiki/Honsberger's_Identity | https://proofwiki.org/wiki/Honsberger's_Identity/Proof_2 | [
"Fibonacci Numbers",
"Honsberger's Identity"
] | [] | [
"Euler-Binet Formula",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Euler-Binet Formula"
] |
proofwiki-10560 | GCD of Fibonacci Numbers | :$\forall m, n \in \Z_{> 2}: \gcd \set {F_m, F_n} = F_{\gcd \set {m, n} }$
where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$. | From the initial definition of Fibonacci numbers, we have:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
{{WLOG}}, let $m \le n$.
Let $h$ be $\gcd \set {m, n}$.
Let $a$ and $b$ be integers such that $m = h a$ and $n = \map h {a + b}$.
$a$ and $a + b$ are coprime by Integers Divided by GCD are Coprime.
Therefore, $a$ and $b$ ar... | :$\forall m, n \in \Z_{> 2}: \gcd \set {F_m, F_n} = F_{\gcd \set {m, n} }$
where $\gcd \set {a, b}$ denotes the [[Definition:Greatest Common Divisor of Integers|greatest common divisor]] of $a$ and $b$. | From the initial definition of [[Definition:Fibonacci Number|Fibonacci numbers]], we have:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
{{WLOG}}, let $m \le n$.
Let $h$ be $\gcd \set {m, n}$.
Let $a$ and $b$ be [[Definition:Integer|integers]] such that $m = h a$ and $n = \map h {a + b}$.
$a$ and $a + b$ are [[Definition:C... | GCD of Fibonacci Numbers | https://proofwiki.org/wiki/GCD_of_Fibonacci_Numbers | https://proofwiki.org/wiki/GCD_of_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Greatest Common Divisor"
] | [
"Definition:Greatest Common Divisor/Integers"
] | [
"Definition:Fibonacci Number",
"Definition:Integer",
"Definition:Coprime/Integers",
"Integers Divided by GCD are Coprime",
"Integer Combination of Coprime Integers",
"Honsberger's Identity",
"GCD with Remainder",
"Divisibility of Fibonacci Number",
"Definition:Coprime/Integers",
"Consecutive Fibon... |
proofwiki-10561 | Catalan's Identity | :${F_n}^2 - F_{n - r} F_{n + r} = \left({-1}\right)^{n - r} {F_r}^2$ | From the definition of Fibonacci numbers:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
By Honsberger's Identity:
:$F_n = F_{n - r} F_{r - 1} + F_{n - r + 1} F_r$
Also:
{{begin-eqn}}
{{eqn | l = F_{n + r}
| r = F_{n - r} F_{2 r - 1} + F_{n - r + 1} F_{2 r}
| c = Honsberger's Identity
}}
{{eqn | r = F_{n - r} \paren... | :${F_n}^2 - F_{n - r} F_{n + r} = \left({-1}\right)^{n - r} {F_r}^2$ | From the definition of [[Definition:Fibonacci Number|Fibonacci numbers]]:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
By [[Honsberger's Identity]]:
:$F_n = F_{n - r} F_{r - 1} + F_{n - r + 1} F_r$
Also:
{{begin-eqn}}
{{eqn | l = F_{n + r}
| r = F_{n - r} F_{2 r - 1} + F_{n - r + 1} F_{2 r}
| c = [[Honsberger's ... | Catalan's Identity/Proof 1 | https://proofwiki.org/wiki/Catalan's_Identity | https://proofwiki.org/wiki/Catalan's_Identity/Proof_1 | [
"Fibonacci Numbers",
"Catalan's Identity"
] | [] | [
"Definition:Fibonacci Number",
"Honsberger's Identity",
"Honsberger's Identity",
"Honsberger's Identity",
"Cassini's Identity"
] |
proofwiki-10562 | Catalan's Identity | :${F_n}^2 - F_{n - r} F_{n + r} = \left({-1}\right)^{n - r} {F_r}^2$ | Proof by induction:
For all $n, r \in \N_{>0}$ where $n > r$, let $\map P {n, r}$ be the proposition:
:${F_n}^2 - F_{n - r} F_{n + r} = \paren {-1}^{n - r} {F_r}^2$
=== Basis for the Induction ===
$n = 1$ yields no suitable $r$, so we look at $n = 2$ instead, which only gives us $r = 1$.
$\map P {2, 1}$ is true:
:${F_2... | :${F_n}^2 - F_{n - r} F_{n + r} = \left({-1}\right)^{n - r} {F_r}^2$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n, r \in \N_{>0}$ where $n > r$, let $\map P {n, r}$ be the [[Definition:Proposition|proposition]]:
:${F_n}^2 - F_{n - r} F_{n + r} = \paren {-1}^{n - r} {F_r}^2$
=== Basis for the Induction ===
$n = 1$ yields no suitable $r$, so we look at $n = 2... | Catalan's Identity/Proof 2 | https://proofwiki.org/wiki/Catalan's_Identity | https://proofwiki.org/wiki/Catalan's_Identity/Proof_2 | [
"Fibonacci Numbers",
"Catalan's Identity"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Principle of Mathematical Induction",
"Catalan's Identity/Proof 2",
"Definition:Basis for the Induction",
"Definition:Induction Hypo... |
proofwiki-10563 | Fibonacci Number with Negative Index | :$\forall n \in \Z_{> 0} : F_{-n} = \paren {-1}^{n + 1} F_n$ | From the initial definition of Fibonacci numbers, we have:
:$F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
By definition of the extension of the Fibonacci numbers to negative integers:
:$F_n = F_{n + 2} - F_{n - 1}$
The proof proceeds by induction.
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$F_{-n} = \... | :$\forall n \in \Z_{> 0} : F_{-n} = \paren {-1}^{n + 1} F_n$ | From the initial definition of [[Definition:Fibonacci Number|Fibonacci numbers]], we have:
:$F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$
By definition of the [[Definition:Fibonacci Number for Negative Index|extension of the Fibonacci numbers to negative integers]]:
:$F_n = F_{n + 2} - F_{n - 1}$
The proof proceeds ... | Fibonacci Number with Negative Index | https://proofwiki.org/wiki/Fibonacci_Number_with_Negative_Index | https://proofwiki.org/wiki/Fibonacci_Number_with_Negative_Index | [
"Fibonacci Numbers",
"Proofs by Induction"
] | [] | [
"Definition:Fibonacci Number",
"Definition:Fibonacci Number/Negative",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-10564 | Fibonacci Number in terms of Larger Fibonacci Numbers | :$\forall m, n \in \Z_{>0} : \paren {-1}^n F_{m - n} = F_m F_{n - 1} - F_{m - 1} F_n$ | {{begin-eqn}}
{{eqn | l = F_{m - n}
| r = F_{m + \paren {-n} }
| c = {{Defof|Integer Subtraction}}
}}
{{eqn | r = F_{m - 1} F_{-n} + F_m F_{-n + 1}
| c = Honsberger's Identity
}}
{{eqn | r = \paren {-1}^{n + 1} F_{m - 1} F_n + \paren {-1}^n F_m F_{n - 1}
| c = Fibonacci Number with Negative Inde... | :$\forall m, n \in \Z_{>0} : \paren {-1}^n F_{m - n} = F_m F_{n - 1} - F_{m - 1} F_n$ | {{begin-eqn}}
{{eqn | l = F_{m - n}
| r = F_{m + \paren {-n} }
| c = {{Defof|Integer Subtraction}}
}}
{{eqn | r = F_{m - 1} F_{-n} + F_m F_{-n + 1}
| c = [[Honsberger's Identity]]
}}
{{eqn | r = \paren {-1}^{n + 1} F_{m - 1} F_n + \paren {-1}^n F_m F_{n - 1}
| c = [[Fibonacci Number with Negativ... | Fibonacci Number in terms of Larger Fibonacci Numbers | https://proofwiki.org/wiki/Fibonacci_Number_in_terms_of_Larger_Fibonacci_Numbers | https://proofwiki.org/wiki/Fibonacci_Number_in_terms_of_Larger_Fibonacci_Numbers | [
"Fibonacci Numbers"
] | [] | [
"Honsberger's Identity",
"Fibonacci Number with Negative Index",
"Category:Fibonacci Numbers"
] |
proofwiki-10565 | Fibonacci Number as Sum of Binomial Coefficients | {{begin-eqn}}
{{eqn | q = \forall n \in \Z_{>0}
| l = F_n
| r = \sum_{k \mathop = 0}^{\floor {\frac {n - 1} 2} } \dbinom {n - k - 1} k
| c =
}}
{{eqn | r = \binom {n - 1} 0 + \binom {n - 2} 1 + \binom {n - 3} 2 + \dotsb + \binom {n - j} {j - 1} + \binom {n - j - 1} j
| c = where $j = \floor {\d... | By definition of Fibonacci numbers:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, \ldots$
The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds F_n = \sum_{k \mathop = 0}^{\floor {\frac {n - 1} 2} } \dbinom {n - k - 1} k$ | {{begin-eqn}}
{{eqn | q = \forall n \in \Z_{>0}
| l = F_n
| r = \sum_{k \mathop = 0}^{\floor {\frac {n - 1} 2} } \dbinom {n - k - 1} k
| c =
}}
{{eqn | r = \binom {n - 1} 0 + \binom {n - 2} 1 + \binom {n - 3} 2 + \dotsb + \binom {n - j} {j - 1} + \binom {n - j - 1} j
| c = where $j = \floor {\d... | By definition of [[Definition:Fibonacci Numbers|Fibonacci numbers]]:
:$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, \ldots$
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds F_n = \sum_{k \mathop = 0}^{\floor ... | Fibonacci Number as Sum of Binomial Coefficients | https://proofwiki.org/wiki/Fibonacci_Number_as_Sum_of_Binomial_Coefficients | https://proofwiki.org/wiki/Fibonacci_Number_as_Sum_of_Binomial_Coefficients | [
"Fibonacci Numbers",
"Binomial Coefficients",
"Proofs by Induction",
"Fibonacci Number as Sum of Binomial Coefficients"
] | [] | [
"Definition:Fibonacci Number",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-10566 | Determinant of Matrix Product/General Case | Let $\mathbf A_1, \mathbf A_2, \cdots, \mathbf A_n$ be square matrices of order $n$, where $n > 1$.
Then:
:$\map \det {\mathbf A_1 \mathbf A_2 \cdots \mathbf A_n} = \map \det {\mathbf A_1} \map \det {\mathbf A_2} \cdots \map \det {\mathbf A_n}$ | Proof by induction: | Let $\mathbf A_1, \mathbf A_2, \cdots, \mathbf A_n$ be [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order $n$]], where $n > 1$.
Then:
:$\map \det {\mathbf A_1 \mathbf A_2 \cdots \mathbf A_n} = \map \det {\mathbf A_1} \map \det {\mathbf A_2} \cdots \map \det {\mathbf A_n}$ | Proof by [[Principle of Mathematical Induction|induction]]: | Determinant of Matrix Product/General Case | https://proofwiki.org/wiki/Determinant_of_Matrix_Product/General_Case | https://proofwiki.org/wiki/Determinant_of_Matrix_Product/General_Case | [
"Determinant of Matrix Product"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-10567 | Opposite Sides Equal implies Parallelogram | Let $ABCD$ be a convex quadrilateral with $AB = CD$ and $BC = AD$.
Then $ABCD$ is a parallelogram. | Join $AC$.
{{begin-eqn}}
{{eqn | n = 1
| l = AB
| r = CD
| c = {{hypothesis}}
}}
{{eqn | n = 2
| l = BC
| r = DA
| c = {{hypothesis}}
}}
{{eqn | n = 3
| l = AC
| r = CA
| c = Equality is Reflexive
}}
{{eqn | n = 4
| l = \Delta ABC
| r = \Delta CDA
... | Let $ABCD$ be a [[Definition:Convex Polygon|convex]] [[Definition:Quadrilateral|quadrilateral]] with $AB = CD$ and $BC = AD$.
Then $ABCD$ is a [[Definition:Parallelogram|parallelogram]]. | Join $AC$.
{{begin-eqn}}
{{eqn | n = 1
| l = AB
| r = CD
| c = {{hypothesis}}
}}
{{eqn | n = 2
| l = BC
| r = DA
| c = {{hypothesis}}
}}
{{eqn | n = 3
| l = AC
| r = CA
| c = [[Equality is Reflexive]]
}}
{{eqn | n = 4
| l = \Delta ABC
| r = \Delta CDA
... | Opposite Sides Equal implies Parallelogram | https://proofwiki.org/wiki/Opposite_Sides_Equal_implies_Parallelogram | https://proofwiki.org/wiki/Opposite_Sides_Equal_implies_Parallelogram | [
"Parallelograms"
] | [
"Definition:Convex Polygon",
"Definition:Quadrilateral",
"Definition:Quadrilateral/Parallelogram"
] | [
"Equality is Reflexive",
"Triangle Side-Side-Side Congruence",
"Equal Alternate Angles implies Parallel Lines",
"Equal Alternate Angles implies Parallel Lines",
"Definition:Quadrilateral/Parallelogram"
] |
proofwiki-10568 | Principle of Mathematical Induction/Naturally Ordered Semigroup/General Result | Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $p \in S$.
Let $T \subseteq S$ such that:
:$x \in T \implies p \preceq x \land \paren {x \in T \implies x \circ 1 \in T}$
Then:
:$S \setminus S_p \subseteq T$
where:
:$\setminus$ denotes set difference
:$S_p$ denotes the set of all elements of $S$ ... | Let $S_p$ be the set of all elements of $S$ preceding $p$:
:$S_p = \set {x \in S: x \prec p}$
Let $T' = T \cup S_p$.
Then the set $T'$ satisfies the conditions of the Principle of Mathematical Induction for a Naturally Ordered Semigroup.
From that result:
:$T' = S$
By Set Difference with Union is Set Difference:
:$S \s... | Let $\struct {S, \circ, \preceq}$ be a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
Let $p \in S$.
Let $T \subseteq S$ such that:
:$x \in T \implies p \preceq x \land \paren {x \in T \implies x \circ 1 \in T}$
Then:
:$S \setminus S_p \subseteq T$
where:
:$\setminus$ denotes [[Definition:... | Let $S_p$ be the [[Definition:Initial Segment|set of all elements of $S$ preceding $p$]]:
:$S_p = \set {x \in S: x \prec p}$
Let $T' = T \cup S_p$.
Then the set $T'$ satisfies the conditions of the [[Principle of Mathematical Induction for Naturally Ordered Semigroup|Principle of Mathematical Induction for a Naturall... | Principle of Mathematical Induction/Naturally Ordered Semigroup/General Result | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/Naturally_Ordered_Semigroup/General_Result | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/Naturally_Ordered_Semigroup/General_Result | [
"Naturally Ordered Semigroup",
"Principle of Mathematical Induction"
] | [
"Definition:Naturally Ordered Semigroup",
"Definition:Set Difference",
"Definition:Initial Segment"
] | [
"Definition:Initial Segment",
"Principle of Mathematical Induction/Naturally Ordered Semigroup",
"Set Difference with Union is Set Difference",
"Set Difference is Subset"
] |
proofwiki-10569 | Principle of Mathematical Induction for Minimally Inductive Set | Let $\omega$ be the minimally inductive set.
Let $S \subseteq \omega$.
Suppose that:
:$(1): \quad \O \in S$
:$(2): \quad \forall x: x \in S \implies x^+ \in S$
where $x^+$ is the successor set of $x$.
Then:
:$S = \omega$
</onlyinclude> | The hypotheses state precisely that $S$ is an inductive set.
Then the minimally inductive set $\omega$ being defined as the intersection of all inductive sets, we conclude that:
:$\omega \subseteq S$
by Intersection is Subset: General Result.
Thus, by definition of set equality:
:$S = \omega$
{{qed}} | Let $\omega$ be the [[Definition:Minimally Inductive Set|minimally inductive set]].
Let $S \subseteq \omega$.
Suppose that:
:$(1): \quad \O \in S$
:$(2): \quad \forall x: x \in S \implies x^+ \in S$
where $x^+$ is the [[Definition:Successor Set|successor set]] of $x$.
Then:
:$S = \omega$
</onlyinclude> | The hypotheses state precisely that $S$ is an [[Definition:Inductive Set|inductive set]].
Then the [[Definition:Minimally Inductive Set|minimally inductive set]] $\omega$ being defined as the [[Definition:Set Intersection|intersection]] of all [[Definition:Inductive Set|inductive sets]], we conclude that:
:$\omega \s... | Principle of Mathematical Induction for Minimally Inductive Set | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction_for_Minimally_Inductive_Set | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction_for_Minimally_Inductive_Set | [
"Minimally Inductive Set"
] | [
"Definition:Minimally Inductive Set",
"Definition:Successor Mapping/Successor Set"
] | [
"Definition:Inductive Set",
"Definition:Minimally Inductive Set",
"Definition:Set Intersection",
"Definition:Inductive Set",
"Intersection is Subset/General Result",
"Definition:Set Equality/Definition 2"
] |
proofwiki-10570 | Principle of Mathematical Induction for Natural Numbers in Real Numbers | Let $\struct {\R, +, \times, \le}$ be the field of real numbers.
Let $\N$ be the natural numbers in $\R$.
Suppose that $A \subseteq \N$ is an inductive set.
Then $A = \N$. | By definition of the natural numbers in $\R$:
:$\N = \ds \bigcap \II$
where $\II$ is the set of inductive sets in $\R$.
Since $A$ was supposed to be inductive, it follows that:
:$\N \subseteq A$
from Intersection is Subset: General Result.
Hence by definition of set equality:
:$A = \N$
{{qed}} | Let $\struct {\R, +, \times, \le}$ be the [[Definition:Field of Real Numbers|field of real numbers]].
Let $\N$ be the [[Definition:Natural Numbers in Real Numbers|natural numbers in $\R$]].
Suppose that $A \subseteq \N$ is an [[Definition:Inductive Set as Subset of Real Numbers|inductive set]].
Then $A = \N$. | By definition of the [[Definition:Natural Numbers in Real Numbers|natural numbers in $\R$]]:
:$\N = \ds \bigcap \II$
where $\II$ is the set of [[Definition:Inductive Set as Subset of Real Numbers|inductive sets]] in $\R$.
Since $A$ was supposed to be [[Definition:Inductive Set as Subset of Real Numbers|inductive]],... | Principle of Mathematical Induction for Natural Numbers in Real Numbers | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction_for_Natural_Numbers_in_Real_Numbers | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction_for_Natural_Numbers_in_Real_Numbers | [
"Natural Numbers in Real Numbers"
] | [
"Definition:Field of Real Numbers",
"Definition:Natural Numbers/Inductive Sets in Real Numbers",
"Definition:Inductive Set/Subset of Real Numbers"
] | [
"Definition:Natural Numbers/Inductive Sets in Real Numbers",
"Definition:Inductive Set/Subset of Real Numbers",
"Definition:Inductive Set/Subset of Real Numbers",
"Intersection is Subset/General Result",
"Definition:Set Equality/Definition 2"
] |
proofwiki-10571 | Chinese Remainder Theorem (Commutative Algebra) | Let $A$ be a commutative and unitary ring.
{{explain|Can this condition be weakened?}}
Let $I_1, \ldots, I_n$ for some $n \ge 1$ be ideals of $A$.
Then the ring homomorphism $\phi: A \to A / I_1 \times \cdots \times A / I_n$ defined as:
:$\map \phi x = \tuple {x + I_1, \ldots, x + I_n}$
has the kernel $\ds I := \bigcap... | The mapping $\phi$ is indeed a ring homomorphism, because each canonical projection $\phi_i: A \to A / I_i$ is a ring homomorphism.
The kernel of $\phi$ is given by:
:$\ds \ker \phi = \set {x \in A: \forall i, 1 \le i \le n : x \in I_i} = \bigcap_{1 \mathop \le i \mathop \le n} I_i =: I$
It remains then to be proved th... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative and unitary ring]].
{{explain|Can this condition be weakened?}}
Let $I_1, \ldots, I_n$ for some $n \ge 1$ be [[Definition:Ideal of Ring|ideals]] of $A$.
Then the [[Definition:Ring Homomorphism|ring homomorphism]] $\phi: A \to A / I_1 \times \cdots \tim... | The mapping $\phi$ is indeed a [[Definition:Ring Homomorphism|ring homomorphism]], because each [[Definition:Quotient Epimorphism|canonical projection]] $\phi_i: A \to A / I_i$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
The [[Definition:Kernel of Ring Homomorphism|kernel]] of $\phi$ is given by:
:$\ds \k... | Chinese Remainder Theorem (Commutative Algebra)/Proof 1 | https://proofwiki.org/wiki/Chinese_Remainder_Theorem_(Commutative_Algebra) | https://proofwiki.org/wiki/Chinese_Remainder_Theorem_(Commutative_Algebra)/Proof_1 | [
"Chinese Remainder Theorem (Commutative Algebra)",
"Chinese Remainder Theorem",
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Definition:Surjection",
"Definition:Ideal of Ring",
"Definition:Coprime Ideals",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"First Isom... | [
"Definition:Ring Homomorphism",
"Definition:Quotient Epimorphism",
"Definition:Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Definition:Surjection",
"Definition:Ideal of Ring",
"Definition:Coprime Ideals",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Cartesian Product/Coo... |
proofwiki-10572 | Chinese Remainder Theorem (Commutative Algebra) | Let $A$ be a commutative and unitary ring.
{{explain|Can this condition be weakened?}}
Let $I_1, \ldots, I_n$ for some $n \ge 1$ be ideals of $A$.
Then the ring homomorphism $\phi: A \to A / I_1 \times \cdots \times A / I_n$ defined as:
:$\map \phi x = \tuple {x + I_1, \ldots, x + I_n}$
has the kernel $\ds I := \bigcap... | Consider $\pi$ only as a homomorphism of groups.
Then Chinese Remainder Theorem (Groups) is applicable as Subgroup of Abelian Group is Normal.
It remains to demonstrate that the condition $I_i + I_j = R$ for all $i \ne j$ assumed here is equivalent to:
:$\ds \forall k \le n - 1: I_{k + 1} + \bigcap_{i \mathop = 1}^k I... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative and unitary ring]].
{{explain|Can this condition be weakened?}}
Let $I_1, \ldots, I_n$ for some $n \ge 1$ be [[Definition:Ideal of Ring|ideals]] of $A$.
Then the [[Definition:Ring Homomorphism|ring homomorphism]] $\phi: A \to A / I_1 \times \cdots \tim... | Consider $\pi$ only as a [[Definition:Group Homomorphism|homomorphism of groups]].
Then [[Chinese Remainder Theorem (Groups)]] is applicable as [[Subgroup of Abelian Group is Normal]].
It remains to demonstrate that the condition $I_i + I_j = R$ for all $i \ne j$ assumed here is equivalent to:
:$\ds \forall k \le n ... | Chinese Remainder Theorem (Commutative Algebra)/Proof 2 | https://proofwiki.org/wiki/Chinese_Remainder_Theorem_(Commutative_Algebra) | https://proofwiki.org/wiki/Chinese_Remainder_Theorem_(Commutative_Algebra)/Proof_2 | [
"Chinese Remainder Theorem (Commutative Algebra)",
"Chinese Remainder Theorem",
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Ring Homomorphism",
"Definition:Kernel of Ring Homomorphism",
"Definition:Surjection",
"Definition:Ideal of Ring",
"Definition:Coprime Ideals",
"Definition:Isomorphism (Abstract Algebra)/Ring Isomorphism",
"First Isom... | [
"Definition:Group Homomorphism",
"Chinese Remainder Theorem (Groups)",
"Subgroup of Abelian Group is Normal",
"Intersection of Ideals of Ring contains Product"
] |
proofwiki-10573 | Sine of i | :$\sin i = \paren {\dfrac e 2 - \dfrac 1 {2 e} } i$ | We have:
{{begin-eqn}}
{{eqn | n = 1
| l = \cos i + i \sin i
| r = e^{i \times i}
| c = Euler's Formula
}}
{{eqn | r = e^{-1}
| c = {{Defof|Imaginary Unit}}
}}
{{eqn | r = \frac 1 e
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | n = 2
| l = \cos i - i \sin i
| r = \map \cos {-i} + i \map... | :$\sin i = \paren {\dfrac e 2 - \dfrac 1 {2 e} } i$ | We have:
{{begin-eqn}}
{{eqn | n = 1
| l = \cos i + i \sin i
| r = e^{i \times i}
| c = [[Euler's Formula]]
}}
{{eqn | r = e^{-1}
| c = {{Defof|Imaginary Unit}}
}}
{{eqn | r = \frac 1 e
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | n = 2
| l = \cos i - i \sin i
| r = \map \cos {-i} + i... | Sine of i/Proof 1 | https://proofwiki.org/wiki/Sine_of_i | https://proofwiki.org/wiki/Sine_of_i/Proof_1 | [
"Sine of i",
"Sine Function",
"Imaginary Unit"
] | [] | [
"Euler's Formula",
"Cosine Function is Even",
"Sine Function is Odd",
"Euler's Formula"
] |
proofwiki-10574 | Sine of i | :$\sin i = \paren {\dfrac e 2 - \dfrac 1 {2 e} } i$ | {{begin-eqn}}
{{eqn | l = \sin i
| r = i \sinh 1
| c = Hyperbolic Sine in terms of Sine
}}
{{eqn | r = i \frac {e^1 - e^{-1} } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \paren {\frac e 2 - \frac 1 {2 e} } i
}}
{{end-eqn}}
{{qed}} | :$\sin i = \paren {\dfrac e 2 - \dfrac 1 {2 e} } i$ | {{begin-eqn}}
{{eqn | l = \sin i
| r = i \sinh 1
| c = [[Hyperbolic Sine in terms of Sine]]
}}
{{eqn | r = i \frac {e^1 - e^{-1} } 2
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \paren {\frac e 2 - \frac 1 {2 e} } i
}}
{{end-eqn}}
{{qed}} | Sine of i/Proof 2 | https://proofwiki.org/wiki/Sine_of_i | https://proofwiki.org/wiki/Sine_of_i/Proof_2 | [
"Sine of i",
"Sine Function",
"Imaginary Unit"
] | [] | [
"Hyperbolic Sine in terms of Sine"
] |
proofwiki-10575 | Cosine of i | :$\cos i = \dfrac e 2 + \dfrac 1 {2 e}$ | We have:
{{begin-eqn}}
{{eqn | n = 1
| l = \cos i + i \sin i
| r = e^{i \times i}
| c = Euler's Formula
}}
{{eqn | r = e^{-1}
| c = {{Defof|Imaginary Unit}}
}}
{{eqn | r = \frac 1 e
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | n = 2
| l = \cos i - i \sin i
| r = \map \cos {-i} + i \map... | :$\cos i = \dfrac e 2 + \dfrac 1 {2 e}$ | We have:
{{begin-eqn}}
{{eqn | n = 1
| l = \cos i + i \sin i
| r = e^{i \times i}
| c = [[Euler's Formula]]
}}
{{eqn | r = e^{-1}
| c = {{Defof|Imaginary Unit}}
}}
{{eqn | r = \frac 1 e
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | n = 2
| l = \cos i - i \sin i
| r = \map \cos {-i} + i... | Cosine of i/Proof 1 | https://proofwiki.org/wiki/Cosine_of_i | https://proofwiki.org/wiki/Cosine_of_i/Proof_1 | [
"Cosine of i",
"Cosine Function",
"Imaginary Unit"
] | [] | [
"Euler's Formula",
"Cosine Function is Even",
"Sine Function is Odd",
"Euler's Formula"
] |
proofwiki-10576 | Cosine of i | :$\cos i = \dfrac e 2 + \dfrac 1 {2 e}$ | {{begin-eqn}}
{{eqn | l = \cos i
| r = \cosh 1
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \frac {e^1 + e^{-1} } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac e 2 + \frac 1 {2 e}
}}
{{end-eqn}}
{{qed}} | :$\cos i = \dfrac e 2 + \dfrac 1 {2 e}$ | {{begin-eqn}}
{{eqn | l = \cos i
| r = \cosh 1
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \frac {e^1 + e^{-1} } 2
| c = {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \frac e 2 + \frac 1 {2 e}
}}
{{end-eqn}}
{{qed}} | Cosine of i/Proof 2 | https://proofwiki.org/wiki/Cosine_of_i | https://proofwiki.org/wiki/Cosine_of_i/Proof_2 | [
"Cosine of i",
"Cosine Function",
"Imaginary Unit"
] | [] | [
"Hyperbolic Cosine in terms of Cosine"
] |
proofwiki-10577 | Tangent of i | :$\tan i = \paren {\dfrac {e^2 - 1} {e^2 + 1} } i$ | {{begin-eqn}}
{{eqn | l = \tan i
| r = \frac {\sin i} {\cos i}
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {\paren {\frac e 2 - \frac 1 {2 e} } i} {\frac e 2 + \frac 1 {2 e} }
| c = Sine of $i$ and Cosine of $i$
}}
{{eqn | r = \paren {\frac {e - \frac 1 e} {e + \frac 1 e} } i
|... | :$\tan i = \paren {\dfrac {e^2 - 1} {e^2 + 1} } i$ | {{begin-eqn}}
{{eqn | l = \tan i
| r = \frac {\sin i} {\cos i}
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {\paren {\frac e 2 - \frac 1 {2 e} } i} {\frac e 2 + \frac 1 {2 e} }
| c = [[Sine of i|Sine of $i$]] and [[Cosine of i|Cosine of $i$]]
}}
{{eqn | r = \paren {\frac {e - \frac 1 ... | Tangent of i/Proof 1 | https://proofwiki.org/wiki/Tangent_of_i | https://proofwiki.org/wiki/Tangent_of_i/Proof_1 | [
"Tangent of i",
"Tangent Function",
"Imaginary Unit"
] | [] | [
"Sine of i",
"Cosine of i",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-10578 | Tangent of i | :$\tan i = \paren {\dfrac {e^2 - 1} {e^2 + 1} } i$ | {{begin-eqn}}
{{eqn | l = \tan i
| r = i \tanh 1
| c = Hyperbolic Tangent in terms of Tangent
}}
{{eqn | r = \paren {\frac {e^1 - e^{-1} } {e^1 + e^{-1} } } i
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \paren {\frac {e^2 - 1} {e^2 + 1} } i
| c = multiplying denominator and numerator by $e... | :$\tan i = \paren {\dfrac {e^2 - 1} {e^2 + 1} } i$ | {{begin-eqn}}
{{eqn | l = \tan i
| r = i \tanh 1
| c = [[Hyperbolic Tangent in terms of Tangent]]
}}
{{eqn | r = \paren {\frac {e^1 - e^{-1} } {e^1 + e^{-1} } } i
| c = {{Defof|Hyperbolic Tangent}}
}}
{{eqn | r = \paren {\frac {e^2 - 1} {e^2 + 1} } i
| c = multiplying [[Definition:Denominator|de... | Tangent of i/Proof 2 | https://proofwiki.org/wiki/Tangent_of_i | https://proofwiki.org/wiki/Tangent_of_i/Proof_2 | [
"Tangent of i",
"Tangent Function",
"Imaginary Unit"
] | [] | [
"Hyperbolic Tangent in terms of Tangent",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-10579 | Cosecant of i | :$\csc i = \paren {\dfrac {2 e} {1 - e^2} } i$ | {{begin-eqn}}
{{eqn | l = \csc i
| r = \frac 1 {\sin i}
| c = {{Defof|Complex Cosecant Function}}
}}
{{eqn | r = \frac 1 {\paren {\frac e 2 - \frac 1 {2 e} } i}
| c = Sine of $i$
}}
{{eqn | r = \paren {\frac 1 {\frac 1 {2 e} - \frac e 2} } i
| c = Reciprocal of $i$
}}
{{eqn | r = \paren {\frac {... | :$\csc i = \paren {\dfrac {2 e} {1 - e^2} } i$ | {{begin-eqn}}
{{eqn | l = \csc i
| r = \frac 1 {\sin i}
| c = {{Defof|Complex Cosecant Function}}
}}
{{eqn | r = \frac 1 {\paren {\frac e 2 - \frac 1 {2 e} } i}
| c = [[Sine of i|Sine of $i$]]
}}
{{eqn | r = \paren {\frac 1 {\frac 1 {2 e} - \frac e 2} } i
| c = [[Reciprocal of i|Reciprocal of $i... | Cosecant of i/Proof 1 | https://proofwiki.org/wiki/Cosecant_of_i | https://proofwiki.org/wiki/Cosecant_of_i/Proof_1 | [
"Cosecant of i",
"Cosecant Function",
"Imaginary Unit"
] | [] | [
"Sine of i",
"Reciprocal of i",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-10580 | Cosecant of i | :$\csc i = \paren {\dfrac {2 e} {1 - e^2} } i$ | {{begin-eqn}}
{{eqn | l = \csc i
| r = -i \csch 1
| c = Hyperbolic Cosecant in terms of Cosecant
}}
{{eqn | r = -\paren {\frac 2 {e^1 - e^{-1} } } i
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = -\paren {\frac {2 e} {e^2 - 1} } i
| c = multiplying denominator and numerator by $e$
}}
{{eqn |... | :$\csc i = \paren {\dfrac {2 e} {1 - e^2} } i$ | {{begin-eqn}}
{{eqn | l = \csc i
| r = -i \csch 1
| c = [[Hyperbolic Cosecant in terms of Cosecant]]
}}
{{eqn | r = -\paren {\frac 2 {e^1 - e^{-1} } } i
| c = {{Defof|Hyperbolic Cosecant}}
}}
{{eqn | r = -\paren {\frac {2 e} {e^2 - 1} } i
| c = multiplying [[Definition:Denominator|denominator]] ... | Cosecant of i/Proof 2 | https://proofwiki.org/wiki/Cosecant_of_i | https://proofwiki.org/wiki/Cosecant_of_i/Proof_2 | [
"Cosecant of i",
"Cosecant Function",
"Imaginary Unit"
] | [] | [
"Hyperbolic Cosecant in terms of Cosecant",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-10581 | Secant of i | :$\sec i = \dfrac {2 e} {e^2 + 1}$ | {{begin-eqn}}
{{eqn | l = \sec i
| r = \frac 1 {\cos i}
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \frac 1 {\frac e 2 + \frac 1 {2 e} }
| c = Cosine of $i$
}}
{{eqn | r = \frac {2 e} {e^2 + 1}
| c = multiplying denominator and numerator by $2 e$
}}
{{end-eqn}}
{{qed}} | :$\sec i = \dfrac {2 e} {e^2 + 1}$ | {{begin-eqn}}
{{eqn | l = \sec i
| r = \frac 1 {\cos i}
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = \frac 1 {\frac e 2 + \frac 1 {2 e} }
| c = [[Cosine of i|Cosine of $i$]]
}}
{{eqn | r = \frac {2 e} {e^2 + 1}
| c = multiplying [[Definition:Denominator|denominator]] and [[Definition:N... | Secant of i/Proof 1 | https://proofwiki.org/wiki/Secant_of_i | https://proofwiki.org/wiki/Secant_of_i/Proof_1 | [
"Secant of i",
"Secant Function",
"Imaginary Unit"
] | [] | [
"Cosine of i",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-10582 | Secant of i | :$\sec i = \dfrac {2 e} {e^2 + 1}$ | {{begin-eqn}}
{{eqn | l = \sec i
| r = \sech 1
| c = Hyperbolic Secant in terms of Secant
}}
{{eqn | r = \frac 2 {e^1 + e^{-1} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac {2 e} {e^2 + 1}
| c = multiplying denominator and numerator by $e$
}}
{{end-eqn}}
{{qed}} | :$\sec i = \dfrac {2 e} {e^2 + 1}$ | {{begin-eqn}}
{{eqn | l = \sec i
| r = \sech 1
| c = [[Hyperbolic Secant in terms of Secant]]
}}
{{eqn | r = \frac 2 {e^1 + e^{-1} }
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = \frac {2 e} {e^2 + 1}
| c = multiplying [[Definition:Denominator|denominator]] and [[Definition:Numerator|numerato... | Secant of i/Proof 2 | https://proofwiki.org/wiki/Secant_of_i | https://proofwiki.org/wiki/Secant_of_i/Proof_2 | [
"Secant of i",
"Secant Function",
"Imaginary Unit"
] | [] | [
"Hyperbolic Secant in terms of Secant",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-10583 | Cotangent of i | :$\cot i = \paren {\dfrac {1 + e^2} {1 - e^2} } i$ | {{begin-eqn}}
{{eqn | l = \cot i
| r = \frac {\cos i} {\sin i}
| c = {{Defof|Complex Cotangent Function}}
}}
{{eqn | r = \frac {\frac e 2 + \frac 1 {2 e} } {\left({\frac e 2 - \frac 1 {2 e} }\right) i}
| c = Cosine of $i$ and Sine of $i$
}}
{{eqn | r = \left({\frac {e + \frac 1 e} {e - \frac 1 e} }\ri... | :$\cot i = \paren {\dfrac {1 + e^2} {1 - e^2} } i$ | {{begin-eqn}}
{{eqn | l = \cot i
| r = \frac {\cos i} {\sin i}
| c = {{Defof|Complex Cotangent Function}}
}}
{{eqn | r = \frac {\frac e 2 + \frac 1 {2 e} } {\left({\frac e 2 - \frac 1 {2 e} }\right) i}
| c = [[Cosine of i|Cosine of $i$]] and [[Sine of i|Sine of $i$]]
}}
{{eqn | r = \left({\frac {e + \... | Cotangent of i/Proof 1 | https://proofwiki.org/wiki/Cotangent_of_i | https://proofwiki.org/wiki/Cotangent_of_i/Proof_1 | [
"Cotangent of i",
"Cotangent Function",
"Imaginary Unit"
] | [] | [
"Cosine of i",
"Sine of i",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Reciprocal of i"
] |
proofwiki-10584 | Cotangent of i | :$\cot i = \paren {\dfrac {1 + e^2} {1 - e^2} } i$ | {{begin-eqn}}
{{eqn | l = \cot i
| r = -i \coth 1
| c = Hyperbolic Cotangent in terms of Cotangent
}}
{{eqn | r = -\paren {\frac {e^1 + e^{-1} } {e^1 - e^{-1} } } i
| c = {{Defof|Hyperbolic Cotangent|index = 1}}
}}
{{eqn | r = -\paren {\frac {e^2 + 1} {e^2 - 1} } i
| c = multiplying denominator ... | :$\cot i = \paren {\dfrac {1 + e^2} {1 - e^2} } i$ | {{begin-eqn}}
{{eqn | l = \cot i
| r = -i \coth 1
| c = [[Hyperbolic Cotangent in terms of Cotangent]]
}}
{{eqn | r = -\paren {\frac {e^1 + e^{-1} } {e^1 - e^{-1} } } i
| c = {{Defof|Hyperbolic Cotangent|index = 1}}
}}
{{eqn | r = -\paren {\frac {e^2 + 1} {e^2 - 1} } i
| c = multiplying [[Defini... | Cotangent of i/Proof 2 | https://proofwiki.org/wiki/Cotangent_of_i | https://proofwiki.org/wiki/Cotangent_of_i/Proof_2 | [
"Cotangent of i",
"Cotangent Function",
"Imaginary Unit"
] | [] | [
"Hyperbolic Cotangent in terms of Cotangent",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-10585 | Sine in terms of Cosine | {{begin-eqn}}
{{eqn | l = \sin x
| r = +\sqrt {1 - \cos ^2 x}
| c = if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$
}}
{{eqn | l = \sin x
| r = -\sqrt {1 - \cos ^2 x}
| c = if there exists an integer $n$ such that $\paren {2 n + 1} \pi < x < \paren {2 n + 2} \pi$
}}... | {{begin-eqn}}
{{eqn | l = \cos^2 x + \sin^2 x
| r = 1
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | ll= \leadsto
| l = \sin^2 x
| r = 1 - \cos^2 x
}}
{{eqn | ll= \leadsto
| l = \sin x
| r = \pm \sqrt {1 - \cos^2 x}
}}
{{end-eqn}}
Then from Sign of Sine:
{{begin-eqn}}
{{eqn | l =... | {{begin-eqn}}
{{eqn | l = \sin x
| r = +\sqrt {1 - \cos ^2 x}
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$
}}
{{eqn | l = \sin x
| r = -\sqrt {1 - \cos ^2 x}
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $\pare... | {{begin-eqn}}
{{eqn | l = \cos^2 x + \sin^2 x
| r = 1
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | ll= \leadsto
| l = \sin^2 x
| r = 1 - \cos^2 x
}}
{{eqn | ll= \leadsto
| l = \sin x
| r = \pm \sqrt {1 - \cos^2 x}
}}
{{end-eqn}}
Then from [[Sign of Sine]]:
{{begin-eqn}}
... | Sine in terms of Cosine | https://proofwiki.org/wiki/Sine_in_terms_of_Cosine | https://proofwiki.org/wiki/Sine_in_terms_of_Cosine | [
"Sine Function",
"Cosine Function"
] | [
"Definition:Integer",
"Definition:Integer"
] | [
"Sum of Squares of Sine and Cosine",
"Sign of Sine",
"Definition:Integer",
"Definition:Integer"
] |
proofwiki-10586 | Secant in terms of Tangent | {{begin-eqn}}
{{eqn | l = \sec x
| r = +\sqrt {\tan ^2 x + 1}
| c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \sec x
| r = -\sqrt {\tan ^2 x + 1}
| c = if there exists an integer $n$ such that $\paren {2 n + \dfrac... | {{begin-eqn}}
{{eqn | l = \sec^2 x - \tan^2 x
| r = 1
| c = Difference of Squares of Secant and Tangent
}}
{{eqn | ll= \leadsto
| l = \sec^2 x
| r = \tan^2 x + 1
}}
{{eqn | ll= \leadsto
| l = \sec x
| r = \pm \sqrt {\tan ^2 x + 1}
}}
{{end-eqn}}
Also, from Sign of Secant:
:If there e... | {{begin-eqn}}
{{eqn | l = \sec x
| r = +\sqrt {\tan ^2 x + 1}
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \sec x
| r = -\sqrt {\tan ^2 x + 1}
| c = if there exists an [[Definition:Integer... | {{begin-eqn}}
{{eqn | l = \sec^2 x - \tan^2 x
| r = 1
| c = [[Difference of Squares of Secant and Tangent]]
}}
{{eqn | ll= \leadsto
| l = \sec^2 x
| r = \tan^2 x + 1
}}
{{eqn | ll= \leadsto
| l = \sec x
| r = \pm \sqrt {\tan ^2 x + 1}
}}
{{end-eqn}}
Also, from [[Sign of Secant]]:
:... | Secant in terms of Tangent | https://proofwiki.org/wiki/Secant_in_terms_of_Tangent | https://proofwiki.org/wiki/Secant_in_terms_of_Tangent | [
"Secant Function",
"Tangent Function"
] | [
"Definition:Integer",
"Definition:Integer"
] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Sign of Secant",
"Definition:Integer",
"Definition:Integer"
] |
proofwiki-10587 | Sine in terms of Tangent | {{begin-eqn}}
{{eqn | l = \sin x
| r = +\frac {\tan x} {\sqrt {1 + \tan^2 x} }
| c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \sin x
| r = -\frac {\tan x} {\sqrt {1 + \tan^2 x} }
| c = if there exists an integer $... | For the first part, if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$:
{{begin-eqn}}
{{eqn | l = \cos x
| r = +\frac 1 {\sqrt {1 + \tan^2 x} }
| c = Cosine in terms of Tangent
}}
{{eqn | ll= \leadsto
| l = \frac 1 {\paren {\frac 1 {\cos x} } }... | {{begin-eqn}}
{{eqn | l = \sin x
| r = +\frac {\tan x} {\sqrt {1 + \tan^2 x} }
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \sin x
| r = -\frac {\tan x} {\sqrt {1 + \tan^2 x} }
| c = if th... | For the first part, if there exists an [[Definition:Integer|integer]] $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$:
{{begin-eqn}}
{{eqn | l = \cos x
| r = +\frac 1 {\sqrt {1 + \tan^2 x} }
| c = [[Cosine in terms of Tangent]]
}}
{{eqn | ll= \leadsto
| l = \frac 1 {... | Sine in terms of Tangent | https://proofwiki.org/wiki/Sine_in_terms_of_Tangent | https://proofwiki.org/wiki/Sine_in_terms_of_Tangent | [
"Sine Function",
"Tangent Function"
] | [
"Definition:Integer",
"Definition:Integer"
] | [
"Definition:Integer",
"Cosine in terms of Tangent",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Tangent is Sine divided by Cosine",
"Definition:Integer",
"Cosine in terms of Tangent",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Tangent is Sine divi... |
proofwiki-10588 | Sine is Reciprocal of Cosecant | :$\sin \theta = \dfrac 1 {\csc \theta}$ | {{begin-eqn}}
{{eqn | l = \frac 1 {\sin \theta}
| r = \csc \theta
| c = Cosecant is Reciprocal of Sine
}}
{{eqn | ll= \leadsto
| l = \sin \theta
| r = \frac 1 {\csc \theta}
}}
{{end-eqn}}
{{qed}} | :$\sin \theta = \dfrac 1 {\csc \theta}$ | {{begin-eqn}}
{{eqn | l = \frac 1 {\sin \theta}
| r = \csc \theta
| c = [[Cosecant is Reciprocal of Sine]]
}}
{{eqn | ll= \leadsto
| l = \sin \theta
| r = \frac 1 {\csc \theta}
}}
{{end-eqn}}
{{qed}} | Sine is Reciprocal of Cosecant | https://proofwiki.org/wiki/Sine_is_Reciprocal_of_Cosecant | https://proofwiki.org/wiki/Sine_is_Reciprocal_of_Cosecant | [
"Sine Function",
"Cosecant Function"
] | [] | [
"Cosecant is Reciprocal of Sine"
] |
proofwiki-10589 | Tangent is Reciprocal of Cotangent | :$\tan \theta = \dfrac 1 {\cot \theta}$ | {{begin-eqn}}
{{eqn | l = \frac 1 {\tan \theta}
| r = \cot \theta
| c = Cotangent is Reciprocal of Tangent
}}
{{eqn | ll= \leadsto
| l = \tan \theta
| r = \frac 1 {\cot \theta}
}}
{{end-eqn}}
$\tan \theta$ is not defined when $\cos \theta = 0$, and $\cot \theta$ is not defined when $\sin \theta ... | :$\tan \theta = \dfrac 1 {\cot \theta}$ | {{begin-eqn}}
{{eqn | l = \frac 1 {\tan \theta}
| r = \cot \theta
| c = [[Cotangent is Reciprocal of Tangent]]
}}
{{eqn | ll= \leadsto
| l = \tan \theta
| r = \frac 1 {\cot \theta}
}}
{{end-eqn}}
$\tan \theta$ is not defined when $\cos \theta = 0$, and $\cot \theta$ is not defined when $\sin \... | Tangent is Reciprocal of Cotangent | https://proofwiki.org/wiki/Tangent_is_Reciprocal_of_Cotangent | https://proofwiki.org/wiki/Tangent_is_Reciprocal_of_Cotangent | [
"Tangent Function",
"Cotangent Function"
] | [] | [
"Cotangent is Reciprocal of Tangent"
] |
proofwiki-10590 | Cosine is Reciprocal of Secant | :$\cos \theta = \dfrac 1 {\sec \theta}$ | {{begin-eqn}}
{{eqn | l = \frac 1 {\cos \theta}
| r = \sec \theta
| c = Secant is Reciprocal of Cosine
}}
{{eqn | ll= \leadsto
| l = \cos \theta
| r = \frac 1 {\sec \theta}
}}
{{end-eqn}}
$\sec \theta$ and $\dfrac 1 {\cos \theta}$ are not defined when $\cos \theta = 0$.
{{qed}} | :$\cos \theta = \dfrac 1 {\sec \theta}$ | {{begin-eqn}}
{{eqn | l = \frac 1 {\cos \theta}
| r = \sec \theta
| c = [[Secant is Reciprocal of Cosine]]
}}
{{eqn | ll= \leadsto
| l = \cos \theta
| r = \frac 1 {\sec \theta}
}}
{{end-eqn}}
$\sec \theta$ and $\dfrac 1 {\cos \theta}$ are not defined when $\cos \theta = 0$.
{{qed}} | Cosine is Reciprocal of Secant | https://proofwiki.org/wiki/Cosine_is_Reciprocal_of_Secant | https://proofwiki.org/wiki/Cosine_is_Reciprocal_of_Secant | [
"Cosine Function",
"Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine"
] |
proofwiki-10591 | Sine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$
where:
:$\sin$ denotes the sine function (real and complex)
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine fun... | {{begin-eqn}}
{{eqn | l = \sinh \paren {a + b i}
| r = \sinh a \cosh \paren {b i} + \cosh a \sinh \paren {b i}
| c = Hyperbolic Sine of Sum
}}
{{eqn | r = \sinh a \cos b + \cosh a \sin \paren {b i}
| c = Cosine in terms of Hyperbolic Cosine
}}
{{eqn | r = \sinh a \cos b + i \cosh a \sin b
| c = ... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$
where:
:$\sin$ denotes the [[Definition:Sine Function|sine function]] ([[Definition:Real Sine Function|real]] and [[Definition:Com... | {{begin-eqn}}
{{eqn | l = \sinh \paren {a + b i}
| r = \sinh a \cosh \paren {b i} + \cosh a \sinh \paren {b i}
| c = [[Hyperbolic Sine of Sum]]
}}
{{eqn | r = \sinh a \cos b + \cosh a \sin \paren {b i}
| c = [[Cosine in terms of Hyperbolic Cosine]]
}}
{{eqn | r = \sinh a \cos b + i \cosh a \sin b
... | Hyperbolic Sine of Complex Number/Proof 1 | https://proofwiki.org/wiki/Sine_of_Complex_Number | https://proofwiki.org/wiki/Hyperbolic_Sine_of_Complex_Number/Proof_1 | [
"Sine Function",
"Complex Numbers",
"Sine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Sine",
"Definition:Sine/Real Function",
"Definition:Sine/Complex Function",
"Definition:Cosine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Hyperbolic Sine of Sum",
"Cosine in terms of Hyperbolic Cosine",
"Sine in terms of Hyperbolic Sine"
] |
proofwiki-10592 | Sine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$
where:
:$\sin$ denotes the sine function (real and complex)
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine fun... | {{begin-eqn}}
{{eqn | l = \sinh a \cos b + i \cosh a \sin b
| r = \frac {e^a - e^{-a} } 2 \frac {e^{i b} + e^{-i b} } 2 + i \frac {e^a + e^{-a} } 2 \frac {e^{i b} - e^{-i b} } {2 i}
| c = {{Defof|Hyperbolic Sine}}, Euler's Cosine Identity, {{Defof|Hyperbolic Cosine}}, Euler's Sine Identity
}}
{{eqn | r = \f... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$
where:
:$\sin$ denotes the [[Definition:Sine Function|sine function]] ([[Definition:Real Sine Function|real]] and [[Definition:Com... | {{begin-eqn}}
{{eqn | l = \sinh a \cos b + i \cosh a \sin b
| r = \frac {e^a - e^{-a} } 2 \frac {e^{i b} + e^{-i b} } 2 + i \frac {e^a + e^{-a} } 2 \frac {e^{i b} - e^{-i b} } {2 i}
| c = {{Defof|Hyperbolic Sine}}, [[Euler's Cosine Identity]], {{Defof|Hyperbolic Cosine}}, [[Euler's Sine Identity]]
}}
{{eqn ... | Hyperbolic Sine of Complex Number/Proof 2 | https://proofwiki.org/wiki/Sine_of_Complex_Number | https://proofwiki.org/wiki/Hyperbolic_Sine_of_Complex_Number/Proof_2 | [
"Sine Function",
"Complex Numbers",
"Sine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Sine",
"Definition:Sine/Real Function",
"Definition:Sine/Complex Function",
"Definition:Cosine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Euler's Cosine Identity",
"Euler's Sine Identity"
] |
proofwiki-10593 | Sine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$
where:
:$\sin$ denotes the sine function (real and complex)
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine fun... | {{begin-eqn}}
{{eqn | l = \sin \paren {a + b i}
| r = \sin a \cos \paren {b i} + \cos a \sin \paren {b i}
| c = Sine of Sum {{explain|Is Sine of Sum valid for complex numbers? Here is $ib$ complex.<br/>Of course it is, we just haven't proved it properly. Needs to be done. Sources quoted on the relevant page... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$
where:
:$\sin$ denotes the [[Definition:Sine Function|sine function]] ([[Definition:Real Sine Function|real]] and [[Definition:Com... | {{begin-eqn}}
{{eqn | l = \sin \paren {a + b i}
| r = \sin a \cos \paren {b i} + \cos a \sin \paren {b i}
| c = [[Sine of Sum]] {{explain|Is [[Sine of Sum]] valid for complex numbers? Here is $ib$ complex.<br/>Of course it is, we just haven't proved it properly. Needs to be done. Sources quoted on the relev... | Sine of Complex Number/Proof 1 | https://proofwiki.org/wiki/Sine_of_Complex_Number | https://proofwiki.org/wiki/Sine_of_Complex_Number/Proof_1 | [
"Sine Function",
"Complex Numbers",
"Sine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Sine",
"Definition:Sine/Real Function",
"Definition:Sine/Complex Function",
"Definition:Cosine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Sine of Sum",
"Sine of Sum",
"Hyperbolic Cosine in terms of Cosine",
"Hyperbolic Sine in terms of Sine"
] |
proofwiki-10594 | Sine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$
where:
:$\sin$ denotes the sine function (real and complex)
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine fun... | {{begin-eqn}}
{{eqn | l = \sin a \cosh b + i \cos a \sinh b
| r = \frac {e^{i a} - e^{-i a} } {2 i} \frac {e^b - e^{-b} } 2 + i \frac {e^{i a} + e^{-i a} } 2 \frac {e^b - e^{-b} } 2
| c = Euler's Sine Identity, Euler's Cosine Identity, {{Defof|Hyperbolic Sine}}, {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \f... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\sin \paren {a + b i} = \sin a \cosh b + i \cos a \sinh b$
where:
:$\sin$ denotes the [[Definition:Sine Function|sine function]] ([[Definition:Real Sine Function|real]] and [[Definition:Com... | {{begin-eqn}}
{{eqn | l = \sin a \cosh b + i \cos a \sinh b
| r = \frac {e^{i a} - e^{-i a} } {2 i} \frac {e^b - e^{-b} } 2 + i \frac {e^{i a} + e^{-i a} } 2 \frac {e^b - e^{-b} } 2
| c = [[Euler's Sine Identity]], [[Euler's Cosine Identity]], {{Defof|Hyperbolic Sine}}, {{Defof|Hyperbolic Cosine}}
}}
{{eqn ... | Sine of Complex Number/Proof 2 | https://proofwiki.org/wiki/Sine_of_Complex_Number | https://proofwiki.org/wiki/Sine_of_Complex_Number/Proof_2 | [
"Sine Function",
"Complex Numbers",
"Sine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Sine",
"Definition:Sine/Real Function",
"Definition:Sine/Complex Function",
"Definition:Cosine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Euler's Sine Identity",
"Euler's Cosine Identity",
"Euler's Sine Identity"
] |
proofwiki-10595 | Cosine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$
where:
:$\cos$ denotes the cosine function (real and complex)
:$\sin$ denotes the real sine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosi... | {{begin-eqn}}
{{eqn | l = \cos \paren {a + b i}
| r = \cos a \cos \paren {b i} - \sin a \sin \paren {b i}
| c = Cosine of Sum
}}
{{eqn | r = \cos a \cosh b - \sin a \sin \paren {b i}
| c = Hyperbolic Cosine in terms of Cosine
}}
{{eqn | r = \cos a \cosh b - i \sin a \sinh b
| c = Hyperbolic Sine... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$
where:
:$\cos$ denotes the [[Definition:Cosine Function|cosine function]] ([[Definition:Real Cosine Function|real]] and [[D... | {{begin-eqn}}
{{eqn | l = \cos \paren {a + b i}
| r = \cos a \cos \paren {b i} - \sin a \sin \paren {b i}
| c = [[Cosine of Sum]]
}}
{{eqn | r = \cos a \cosh b - \sin a \sin \paren {b i}
| c = [[Hyperbolic Cosine in terms of Cosine]]
}}
{{eqn | r = \cos a \cosh b - i \sin a \sinh b
| c = [[Hyper... | Cosine of Complex Number/Proof 1 | https://proofwiki.org/wiki/Cosine_of_Complex_Number | https://proofwiki.org/wiki/Cosine_of_Complex_Number/Proof_1 | [
"Cosine Function",
"Complex Numbers",
"Cosine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Cosine",
"Definition:Cosine/Real Function",
"Definition:Cosine/Complex Function",
"Definition:Sine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Cosine of Sum",
"Hyperbolic Cosine in terms of Cosine",
"Hyperbolic Sine in terms of Sine"
] |
proofwiki-10596 | Cosine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$
where:
:$\cos$ denotes the cosine function (real and complex)
:$\sin$ denotes the real sine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosi... | {{begin-eqn}}
{{eqn | l = \cos a \cosh b - i \sin a \sinh b
| r = \frac {e^{i a} + e^{-i a} } 2 \frac {e^b + e^{-b} } 2 - i \frac {e^{i a} - e^{-i a} } {2 i} \frac {e^b - e^{-b} } 2
| c = Euler's Cosine Identity, {{Defof|Hyperbolic Cosine}}, Euler's Sine Identity, {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \f... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$
where:
:$\cos$ denotes the [[Definition:Cosine Function|cosine function]] ([[Definition:Real Cosine Function|real]] and [[D... | {{begin-eqn}}
{{eqn | l = \cos a \cosh b - i \sin a \sinh b
| r = \frac {e^{i a} + e^{-i a} } 2 \frac {e^b + e^{-b} } 2 - i \frac {e^{i a} - e^{-i a} } {2 i} \frac {e^b - e^{-b} } 2
| c = [[Euler's Cosine Identity]], {{Defof|Hyperbolic Cosine}}, [[Euler's Sine Identity]], {{Defof|Hyperbolic Sine}}
}}
{{eqn ... | Cosine of Complex Number/Proof 2 | https://proofwiki.org/wiki/Cosine_of_Complex_Number | https://proofwiki.org/wiki/Cosine_of_Complex_Number/Proof_2 | [
"Cosine Function",
"Complex Numbers",
"Cosine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Cosine",
"Definition:Cosine/Real Function",
"Definition:Cosine/Complex Function",
"Definition:Sine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Euler's Cosine Identity",
"Euler's Sine Identity",
"Euler's Cosine Identity"
] |
proofwiki-10597 | Cosine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$
where:
:$\cos$ denotes the cosine function (real and complex)
:$\sin$ denotes the real sine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosi... | {{begin-eqn}}
{{eqn | l = \map \cosh {a + b i}
| r = \cosh a \map \cosh {b i} + \sinh a \map \sinh {b i}
| c = Hyperbolic Cosine of Sum
}}
{{eqn | r = \cosh a \cos b + \sinh a \map \sinh {b i}
| c = Cosine in terms of Hyperbolic Cosine
}}
{{eqn | r = \cosh a \cos b + i \sinh a \sin b
| c = Sine ... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$
where:
:$\cos$ denotes the [[Definition:Cosine Function|cosine function]] ([[Definition:Real Cosine Function|real]] and [[D... | {{begin-eqn}}
{{eqn | l = \map \cosh {a + b i}
| r = \cosh a \map \cosh {b i} + \sinh a \map \sinh {b i}
| c = [[Hyperbolic Cosine of Sum]]
}}
{{eqn | r = \cosh a \cos b + \sinh a \map \sinh {b i}
| c = [[Cosine in terms of Hyperbolic Cosine]]
}}
{{eqn | r = \cosh a \cos b + i \sinh a \sin b
| c... | Hyperbolic Cosine of Complex Number/Proof 1 | https://proofwiki.org/wiki/Cosine_of_Complex_Number | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Complex_Number/Proof_1 | [
"Cosine Function",
"Complex Numbers",
"Cosine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Cosine",
"Definition:Cosine/Real Function",
"Definition:Cosine/Complex Function",
"Definition:Sine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Hyperbolic Cosine of Sum",
"Cosine in terms of Hyperbolic Cosine",
"Sine in terms of Hyperbolic Sine"
] |
proofwiki-10598 | Cosine of Complex Number | Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$
where:
:$\cos$ denotes the cosine function (real and complex)
:$\sin$ denotes the real sine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosi... | {{begin-eqn}}
{{eqn | l = \cosh a \cos b - i \sinh a \sin b
| r = \frac {e^a + e^{-a} } 2 \frac {e^{i b} + e^{-i b} } 2 + i \frac {e^a - e^{-a} } {2 i} \frac {e^{i b} - e^{-i b} } 2
| c = {{Defof|Hyperbolic Cosine}}, Euler's Cosine Identity, {{Defof|Hyperbolic Sine}}, Euler's Sine Identity
}}
{{eqn | r = \f... | Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Let $i$ be the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\cos \left({a + b i}\right) = \cos a \cosh b - i \sin a \sinh b$
where:
:$\cos$ denotes the [[Definition:Cosine Function|cosine function]] ([[Definition:Real Cosine Function|real]] and [[D... | {{begin-eqn}}
{{eqn | l = \cosh a \cos b - i \sinh a \sin b
| r = \frac {e^a + e^{-a} } 2 \frac {e^{i b} + e^{-i b} } 2 + i \frac {e^a - e^{-a} } {2 i} \frac {e^{i b} - e^{-i b} } 2
| c = {{Defof|Hyperbolic Cosine}}, [[Euler's Cosine Identity]], {{Defof|Hyperbolic Sine}}, [[Euler's Sine Identity]]
}}
{{eqn ... | Hyperbolic Cosine of Complex Number/Proof 2 | https://proofwiki.org/wiki/Cosine_of_Complex_Number | https://proofwiki.org/wiki/Hyperbolic_Cosine_of_Complex_Number/Proof_2 | [
"Cosine Function",
"Complex Numbers",
"Cosine of Complex Number"
] | [
"Definition:Real Number",
"Definition:Complex Number/Imaginary Unit",
"Definition:Cosine",
"Definition:Cosine/Real Function",
"Definition:Cosine/Complex Function",
"Definition:Sine/Real Function",
"Definition:Hyperbolic Sine",
"Definition:Hyperbolic Cosine"
] | [
"Euler's Cosine Identity",
"Euler's Sine Identity"
] |
proofwiki-10599 | Cosine in terms of Sine | {{begin-eqn}}
{{eqn | l = \cos x
| r = +\sqrt {1 - \sin^2 x}
| c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \cos x
| r = -\sqrt {1 - \sin^2 x}
| c = if there exists an integer $n$ such that $\paren {2 n + \dfrac 1... | {{begin-eqn}}
{{eqn | l = \cos^2 x + \sin^2 x
| r = 1
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | ll= \leadsto
| l = \cos^2 x
| r = 1 - \sin^2 x
}}
{{eqn | ll= \leadsto
| l = \cos x
| r = \pm \sqrt {1 - \sin^2 x}
}}
{{end-eqn}}
Then from Sign of Cosine:
{{begin-eqn}}
{{eqn | l... | {{begin-eqn}}
{{eqn | l = \cos x
| r = +\sqrt {1 - \sin^2 x}
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \cos x
| r = -\sqrt {1 - \sin^2 x}
| c = if there exists an [[Definition:Integer|i... | {{begin-eqn}}
{{eqn | l = \cos^2 x + \sin^2 x
| r = 1
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | ll= \leadsto
| l = \cos^2 x
| r = 1 - \sin^2 x
}}
{{eqn | ll= \leadsto
| l = \cos x
| r = \pm \sqrt {1 - \sin^2 x}
}}
{{end-eqn}}
Then from [[Sign of Cosine]]:
{{begin-eqn}... | Cosine in terms of Sine | https://proofwiki.org/wiki/Cosine_in_terms_of_Sine | https://proofwiki.org/wiki/Cosine_in_terms_of_Sine | [
"Cosine Function",
"Sine Function"
] | [
"Definition:Integer",
"Definition:Integer"
] | [
"Sum of Squares of Sine and Cosine",
"Sign of Cosine",
"Definition:Integer",
"Definition:Integer"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.