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-20100 | Sides of Orthic Triangle of Acute Triangle | Let $\triangle ABC$ be an acute triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $\triangle DEF$ be the orthic triangle of $\triangle ABC$.
Then the sides of $\triangle DEF$ are $a \cos A$, $b \cos B$ and $c \cos C$. | :420px
Let $H$ be the orthocenter of $\triangle ABC$.
Let $R$ be the circumradius of $\triangle ABC$.
{{begin-eqn}}
{{eqn | l = \dfrac {EF} {\sin A}
| r = \dfrac {AE} {\sin \angle AFE}
| c = Law of Sines for $\triangle AFE$
}}
{{eqn | r = \dfrac {c \cos A} {\sin C}
| c =
}}
{{eqn | r = 2 R \cos A
... | Let $\triangle ABC$ be an [[Definition:Acute Triangle|acute triangle]] with [[Definition:Side of Polygon|sides]] $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $\triangle DEF$ be the [[Definition:Orthic Triangle|orthic triang... | :[[File:Orthic-Triangle.png|420px]]
Let $H$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
Let $R$ be the [[Definition:Circumradius|circumradius]] of $\triangle ABC$.
{{begin-eqn}}
{{eqn | l = \dfrac {EF} {\sin A}
| r = \dfrac {AE} {\sin \angle AFE}
| c = [[Law of Sines]] for $\triangl... | Sides of Orthic Triangle of Acute Triangle/Proof | https://proofwiki.org/wiki/Sides_of_Orthic_Triangle_of_Acute_Triangle | https://proofwiki.org/wiki/Sides_of_Orthic_Triangle_of_Acute_Triangle/Proof | [
"Sides of Orthic Triangle of Acute Triangle",
"Orthic Triangles",
"Acute Triangles"
] | [
"Definition:Triangle (Geometry)/Acute",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Orthic Triangle",
"Definition:Polygon/Side"
] | [
"File:Orthic-Triangle.png",
"Definition:Orthocenter",
"Definition:Circumradius",
"Law of Sines"
] |
proofwiki-20101 | Sides of Orthic Triangle of Obtuse Triangle | Let $\triangle ABC$ be an obtuse triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $\angle A$ be the obtuse angle of $\triangle ABC$.
Let $\triangle DEF$ be the orthic triangle of $\triangle ABC$.
Then the sides of $\triangle DEF$ are $-a \cos A$, $b \cos B$ and $c \cos C$. | :420px
Let $H$ be the orthocenter of $\triangle ABC$.
{{begin-eqn}}
{{eqn | l = \frac {EF} {\sin \angle EAF}
| r = \frac {AF} {\sin \angle AEF}
| c = Law of Sines for $\triangle AFE$
}}
{{eqn | ll= \leadsto
| l = \frac {EF} {\sin A}
| r = \frac {b \map \cos {180 \degrees - A} } {\sin B}
| ... | Let $\triangle ABC$ be an [[Definition:Obtuse Triangle|obtuse triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $\angle A$ be the [[Definition:Obtuse Angle|obtuse angle]]... | :[[File:Orthic-Triangle-Obtuse.png|420px]]
Let $H$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
{{begin-eqn}}
{{eqn | l = \frac {EF} {\sin \angle EAF}
| r = \frac {AF} {\sin \angle AEF}
| c = [[Law of Sines]] for $\triangle AFE$
}}
{{eqn | ll= \leadsto
| l = \frac {EF} {\sin A}... | Sides of Orthic Triangle of Obtuse Triangle/Proof | https://proofwiki.org/wiki/Sides_of_Orthic_Triangle_of_Obtuse_Triangle | https://proofwiki.org/wiki/Sides_of_Orthic_Triangle_of_Obtuse_Triangle/Proof | [
"Sides of Orthic Triangle of Obtuse Triangle",
"Orthic Triangles",
"Obtuse Triangles"
] | [
"Definition:Triangle (Geometry)/Obtuse",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Obtuse Angle",
"Definition:Orthic Triangle",
"Definition:Polygon/Side"
] | [
"File:Orthic-Triangle-Obtuse.png",
"Definition:Orthocenter",
"Law of Sines",
"Angles in Same Segment of Circle are Equal",
"Definition:Cyclic Quadrilateral",
"Law of Sines",
"Cosine of Supplementary Angle",
"Law of Sines",
"Angles in Same Segment of Circle are Equal",
"Definition:Cyclic Quadrilate... |
proofwiki-20102 | Inradius in Terms of Circumradius | Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $r$ denote the inradius of $\triangle ABC$.
Let $R$ denote the circumradius of $\triangle ABC$.
Then:
:$r = 4 R \sin \dfrac A 2 \sin \dfrac B 2 \sin \dfrac C 2$ | :400px
Let $D$, $E$ and $F$ be the points where the incircle is tangent to the sides $AC$, $AB$ and $CB$ respectively.
Let $s$ denote the semiperimeter of $\triangle ABC$.
From Tangent Points of Incircle in Terms of Semiperimeter:
{{begin-eqn}}
{{eqn | l = AD
| r = s - a
}}
{{eqn | l = BE
| r = s - b
}}
{{e... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $r$ denote the [[Definition:Inradius of Triangle|inradius]] of... | :[[File:IncenterLengthProof.png|400px]]
Let $D$, $E$ and $F$ be the [[Definition:Point|points]] where the [[Definition:Incircle of Triangle|incircle]] is [[Definition:Tangent to Circle|tangent]] to the [[Definition:Side of Polygon|sides]] $AC$, $AB$ and $CB$ respectively.
Let $s$ denote the [[Definition:Semiperimeter... | Inradius in Terms of Circumradius/Proof | https://proofwiki.org/wiki/Inradius_in_Terms_of_Circumradius | https://proofwiki.org/wiki/Inradius_in_Terms_of_Circumradius/Proof | [
"Inradius in Terms of Circumradius",
"Incircles of Triangles",
"Circumcircles of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Incircle of Triangle/Inradius",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"File:IncenterLengthProof.png",
"Definition:Point",
"Definition:Incircle of Triangle",
"Definition:Tangent Line/Circle",
"Definition:Polygon/Side",
"Definition:Semiperimeter",
"Tangent Points of Incircle in Terms of Semiperimeter",
"Law of Sines",
"Sum of Angles of Triangle equals Two Right Angles",... |
proofwiki-20103 | Exradius of Triangle in Terms of Circumradius | Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $\rho_a$ be the exradius of $\triangle ABC$ {{WRT}} $a$.
Let $R$ be the circumradius of $\triangle ABC$.
Then:
:$\rho_a = 4 R \sin \dfrac A 2 \cos \dfrac B 2 \cos \dfrac C 2$ | :500px
Let $r$ denote the inradius of $\triangle ABC$.
We have:
{{begin-eqn}}
{{eqn | l = r
| r = 4 R \sin \dfrac A 2 \sin \dfrac B 2 \sin \dfrac C 2
| c = Inradius in Terms of Circumradius
}}
{{eqn | ll= \leadsto
| l = \rho_a
| r = 4 R \sin \dfrac A 2 \map \sin {\dfrac {180 \degrees - B} 2} \ma... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $\rho_a$ be the [[Definition:Exradius of Triangle|exradius]] o... | :[[File:Area-of-Triangle-by-Exradius.png|500px]]
Let $r$ denote the [[Definition:Inradius of Triangle|inradius]] of $\triangle ABC$.
We have:
{{begin-eqn}}
{{eqn | l = r
| r = 4 R \sin \dfrac A 2 \sin \dfrac B 2 \sin \dfrac C 2
| c = [[Inradius in Terms of Circumradius]]
}}
{{eqn | ll= \leadsto
| l... | Exradius of Triangle in Terms of Circumradius/Proof 2 | https://proofwiki.org/wiki/Exradius_of_Triangle_in_Terms_of_Circumradius | https://proofwiki.org/wiki/Exradius_of_Triangle_in_Terms_of_Circumradius/Proof_2 | [
"Exradius of Triangle in Terms of Circumradius",
"Circumcircles of Triangles",
"Excircles of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Excircle of Triangle/Exradius",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"File:Area-of-Triangle-by-Exradius.png",
"Definition:Incircle of Triangle/Inradius",
"Inradius in Terms of Circumradius",
"Sine of Complement equals Cosine",
"Inradius in Terms of Circumradius"
] |
proofwiki-20104 | Area of Triangle in Terms of Exradius | Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $\rho_a$ be the exradius of $\triangle ABC$ {{WRT}} the excircle which is tangent to $a$.
Let $s$ be the semiperimeter of $\triangle ABC$.
Then the area $\AA$ of $\triangle ABC$ is given by:
:$\AA = \... | :500px
Let $C$ be the excircle of $\triangle ABC$ which is tangent to $a$.
By definition:
:$\rho_a$ is the radius of $C$
:$I_a$ is the center of $C$.
Then we have:
{{begin-eqn}}
{{eqn | l = \AA
| r = \map \Area {\triangle ABI_a} + \map \Area {\triangle ACI_a} - \map \Area {\triangle CBI_a}
| c = (see figure... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $\rho_a$ be the [[Definition:Exradius of Triangle|exradius]] o... | :[[File:Area-of-Triangle-by-Exradius.png|500px]]
Let $C$ be the [[Definition:Excircle of Triangle|excircle]] of $\triangle ABC$ which is [[Definition:Tangent to Circle|tangent]] to $a$.
By definition:
:$\rho_a$ is the [[Definition:Radius of Circle|radius]] of $C$
:$I_a$ is the [[Definition:Center of Circle|center]] ... | Area of Triangle in Terms of Exradius/Proof | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Exradius | https://proofwiki.org/wiki/Area_of_Triangle_in_Terms_of_Exradius/Proof | [
"Area of Triangle in Terms of Exradius",
"Excircles of Triangles",
"Areas of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Excircle of Triangle/Exradius",
"Definition:Excircle of Triangle",
"Definition:Tangent Line/Circle",
"Definition:Semiperimeter",
"Definition:Area"
] | [
"File:Area-of-Triangle-by-Exradius.png",
"Definition:Excircle of Triangle",
"Definition:Tangent Line/Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Area of Triangle in Terms of Side and Altitude"
] |
proofwiki-20105 | Complementary Idempotent of Complementary Idempotent is Idempotent | Let $\HH$ be a Hilbert space.
Let $I$ be an identity operator on $\HH$.
Let $A$ be an idempotent operator.
Let $B$ be the complementary idempotent of A.
Then the complementary idempotent of $B$ is $A$. | From Complementary Idempotent is Idempotent the complementary idempotent of $B$ is well-defined.
Let $C$ be the complementary idempotent of $B$.
We have:
{{begin-eqn}}
{{eqn | l = C
| r = I - B
| c = {{Defof|Complementary Idempotent}}
}}
{{eqn | r = I - \paren{I - A}
| c = {{Defof|Complementary Idempo... | Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $I$ be an [[Definition:Identity Operator|identity operator]] on $\HH$.
Let $A$ be an [[Definition:Idempotent Operator|idempotent operator]].
Let $B$ be the [[Definition:Complementary Idempotent|complementary idempotent]] of A.
Then the [[Definition:Co... | From [[Complementary Idempotent is Idempotent]] the [[Definition:Complementary Idempotent|complementary idempotent]] of $B$ is [[Definition:Well-Defined|well-defined]].
Let $C$ be the [[Definition:Complementary Idempotent|complementary idempotent]] of $B$.
We have:
{{begin-eqn}}
{{eqn | l = C
| r = I - B
... | Complementary Idempotent of Complementary Idempotent is Idempotent | https://proofwiki.org/wiki/Complementary_Idempotent_of_Complementary_Idempotent_is_Idempotent | https://proofwiki.org/wiki/Complementary_Idempotent_of_Complementary_Idempotent_is_Idempotent | [
"Linear Transformations on Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Identity Mapping",
"Definition:Idempotent Operator",
"Definition:Complementary Idempotent",
"Definition:Complementary Idempotent"
] | [
"Complementary Idempotent is Idempotent",
"Definition:Complementary Idempotent",
"Definition:Well-Defined",
"Definition:Complementary Idempotent",
"Category:Linear Transformations on Hilbert Spaces"
] |
proofwiki-20106 | Complementary Projection of Complementary Projection is Projection | Let $\HH$ be a Hilbert space.
Let $I$ be an identity operator on $\HH$.
Let $A$ be a projection.
Let $B$ be the complementary projection of A.
Then the complementary projection of $B$ is $A$. | From Complementary Projection is Projection the complementary projection of $B$ is well-defined.
Let $C$ be the complementary projection of $B$.
We have:
{{begin-eqn}}
{{eqn | l = C
| r = I - B
| c = {{Defof|Complementary Projection}}
}}
{{eqn | r = I - \paren{I - A}
| c = {{Defof|Complementary Projec... | Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $I$ be an [[Definition:Identity Operator|identity operator]] on $\HH$.
Let $A$ be a [[Definition:Projection (Hilbert Spaces)|projection]].
Let $B$ be the [[Definition:Complementary Projection|complementary projection]] of A.
Then the [[Definition:Comp... | From [[Complementary Projection is Projection]] the [[Definition:Complementary Projection|complementary projection]] of $B$ is [[Definition:Well-Defined|well-defined]].
Let $C$ be the [[Definition:Complementary Projection|complementary projection]] of $B$.
We have:
{{begin-eqn}}
{{eqn | l = C
| r = I - B
... | Complementary Projection of Complementary Projection is Projection | https://proofwiki.org/wiki/Complementary_Projection_of_Complementary_Projection_is_Projection | https://proofwiki.org/wiki/Complementary_Projection_of_Complementary_Projection_is_Projection | [
"Linear Transformations on Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Identity Mapping",
"Definition:Projection (Hilbert Spaces)",
"Definition:Complementary Projection",
"Definition:Complementary Projection"
] | [
"Complementary Projection is Projection",
"Definition:Complementary Projection",
"Definition:Well-Defined",
"Definition:Complementary Projection",
"Category:Linear Transformations on Hilbert Spaces"
] |
proofwiki-20107 | Complementary Projection is Complementary Idempotent | Let $\HH$ be a Hilbert space.
Let $A$ be a projection.
Let $B$ be the complementary projection of A.
Then $B$ is the complementary idempotent of $A$. | By the definition of projection, $A$ is an idempotent operator.
The result follows immediately from the definitions of:
* complementary projection
* complementary idempotent
where the constructions of the complementary projection and the complementary idempotent from $A$ are identical.
{{qed}}
Category:Linear Transform... | Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $A$ be a [[Definition:Projection (Hilbert Spaces)|projection]].
Let $B$ be the [[Definition:Complementary Projection|complementary projection]] of A.
Then $B$ is the [[Definition:Complementary Idempotent|complementary idempotent]] of $A$. | By the definition of [[Definition:Projection (Hilbert Spaces)|projection]], $A$ is an [[Definition:Idempotent Operator|idempotent operator]].
The result follows immediately from the definitions of:
* [[Definition:Complementary Projection|complementary projection]]
* [[Definition:Complementary Idempotent|complementary... | Complementary Projection is Complementary Idempotent | https://proofwiki.org/wiki/Complementary_Projection_is_Complementary_Idempotent | https://proofwiki.org/wiki/Complementary_Projection_is_Complementary_Idempotent | [
"Linear Transformations on Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Projection (Hilbert Spaces)",
"Definition:Complementary Projection",
"Definition:Complementary Idempotent"
] | [
"Definition:Projection (Hilbert Spaces)",
"Definition:Idempotent Operator",
"Definition:Complementary Projection",
"Definition:Complementary Idempotent",
"Definition:Complementary Projection",
"Definition:Complementary Idempotent",
"Category:Linear Transformations on Hilbert Spaces"
] |
proofwiki-20108 | Möbius Transformations form Group under Composition | Let $G$ be the set of Möbius transformations.
Let $\circ$ denote the composition of mappings.
Then $\struct {G, \circ}$ is a group. | Taking the group axioms in turn: | Let $G$ be the [[Definition:Set|set]] of [[Definition:Möbius Transformation|Möbius transformations]].
Let $\circ$ denote the [[Definition:Composition of Mappings|composition of mappings]].
Then $\struct {G, \circ}$ is a [[Definition:Group|group]]. | Taking the [[Axiom:Group Axioms|group axioms]] in turn: | Möbius Transformations form Group under Composition | https://proofwiki.org/wiki/Möbius_Transformations_form_Group_under_Composition | https://proofwiki.org/wiki/Möbius_Transformations_form_Group_under_Composition | [
"Möbius Transformations"
] | [
"Definition:Set",
"Definition:Möbius Transformation",
"Definition:Composition of Mappings",
"Definition:Group"
] | [
"Axiom:Group Axioms",
"Axiom:Group Axioms"
] |
proofwiki-20109 | Distance between Excenters of Triangle in Terms of Circumradius | Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $I_b$ and $I_c$ be the excenters of $\triangle ABC$ {{WRT}} $b$ and $c$ respectively.
Let $R$ be the circumradius of $\triangle ABC$.
Then:
:$I_b I_c = 4 R \cos \dfrac A 2$ | :560px
From Triangle is Orthic Triangle of Triangle formed from Excenters, we establish that $\triangle ABC$ is the orthic triangle of $\triangle I_a I_b I_c$.
Hence $I_b B$ is an altitude of $\triangle I_a I_b I_c$.
Thus $\angle I_b B I_a$ is a right angle.
From Altitudes of Triangle Bisect Angles of Orthic Triangle:
... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $I_b$ and $I_c$ be the [[Definition:Excenter of Triangle|excen... | :[[File:Orthic-Triangle-of-Excenters.png|560px]]
From [[Triangle is Orthic Triangle of Triangle formed from Excenters]], we establish that $\triangle ABC$ is the [[Definition:Orthic Triangle|orthic triangle]] of $\triangle I_a I_b I_c$.
Hence $I_b B$ is an [[Definition:Altitude of Triangle|altitude]] of $\triangle I_... | Distance between Excenters of Triangle in Terms of Circumradius/Proof | https://proofwiki.org/wiki/Distance_between_Excenters_of_Triangle_in_Terms_of_Circumradius | https://proofwiki.org/wiki/Distance_between_Excenters_of_Triangle_in_Terms_of_Circumradius/Proof | [
"Distance between Excenters of Triangle in Terms of Circumradius",
"Circumcircles of Triangles",
"Excenters of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Excircle of Triangle/Excenter",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"File:Orthic-Triangle-of-Excenters.png",
"Triangle is Orthic Triangle of Triangle formed from Excenters",
"Definition:Orthic Triangle",
"Definition:Altitude of Triangle",
"Definition:Right Angle",
"Altitudes of Triangle Bisect Angles of Orthic Triangle",
"Sum of Angles of Triangle equals Two Right Angle... |
proofwiki-20110 | Distance between Incenter and Excenter of Triangle in Terms of Circumradius | Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $I$ be the incenter of $\triangle ABC$.
Let $I_a$ be the excenter of $\triangle ABC$ {{WRT}} $a$.
Let $R$ be the circumradius of $\triangle ABC$.
Then:
:$I I_a = 4 R \sin \dfrac A 2$ | :560px
From Triangle is Orthic Triangle of Triangle formed from Excenters, we establish that $\triangle ABC$ is the orthic triangle of $\triangle I_a I_b I_c$.
By the Nine Point Circle Theorem, the Feuerbach circle of $\triangle I_a I_b I_c$ passes through each of $A$, $B$ and $C$.
Therefore the Feuerbach circle of $\t... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $I$ be the [[Definition:Incenter of Triangle|incenter]] of $\t... | :[[File:Orthic-Triangle-of-Excenters.png|560px]]
From [[Triangle is Orthic Triangle of Triangle formed from Excenters]], we establish that $\triangle ABC$ is the [[Definition:Orthic Triangle|orthic triangle]] of $\triangle I_a I_b I_c$.
By the [[Nine Point Circle Theorem]], the [[Definition:Feuerbach Circle|Feuerbach... | Distance between Incenter and Excenter of Triangle in Terms of Circumradius/Proof | https://proofwiki.org/wiki/Distance_between_Incenter_and_Excenter_of_Triangle_in_Terms_of_Circumradius | https://proofwiki.org/wiki/Distance_between_Incenter_and_Excenter_of_Triangle_in_Terms_of_Circumradius/Proof | [
"Distance between Incenter and Excenter of Triangle in Terms of Circumradius",
"Incenters of Triangles",
"Excenters of Triangles",
"Circumcircles of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Incircle of Triangle/Incenter",
"Definition:Excircle of Triangle/Excenter",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"File:Orthic-Triangle-of-Excenters.png",
"Triangle is Orthic Triangle of Triangle formed from Excenters",
"Definition:Orthic Triangle",
"Nine Point Circle Theorem",
"Definition:Feuerbach Circle",
"Definition:Feuerbach Circle",
"Definition:Circumcircle of Triangle",
"Definition:Circle/Radius",
"Defin... |
proofwiki-20111 | Condition for Linear Dependence of Linear Functionals in terms of Kernel | Let $V$ be a vector space over a field $\GF$.
Let $f, f_1, \ldots, f_n: V \to \GF$ be linear functionals.
Suppose that:
:$\ds \bigcap_{i \mathop = 1}^n \ker f_i \subseteq \ker f$
where $\ker f$ denotes the kernel of $f$.
Then there exist $\alpha_1, \ldots, \alpha_n \in \GF$ such that:
:$\ds \forall v \in V: \map f v = ... | For $i = 1, \ldots, n$, let $w_i$ be such that:
:$w_i \not\in \ker f_i$
:$w_i \in \ker f_j, j \ne i$
Suppose first that the $w_i$ all exist.
Let $v_i = \dfrac 1 {\map {f_i} {w_i} } w_i$.
Then since $f_i$ is linear:
:$\map {f_i} {v_i} = 1$
Furthermore for $j \ne i$, $\map {f_j} {v_i} = 0$.
Now let $v \in V$ be arbitrary... | Let $V$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Field (Abstract Algebra)|field]] $\GF$.
Let $f, f_1, \ldots, f_n: V \to \GF$ be [[Definition:Linear Functional|linear functionals]].
Suppose that:
:$\ds \bigcap_{i \mathop = 1}^n \ker f_i \subseteq \ker f$
where $\ker f$ denotes the [[Definit... | For $i = 1, \ldots, n$, let $w_i$ be such that:
:$w_i \not\in \ker f_i$
:$w_i \in \ker f_j, j \ne i$
Suppose first that the $w_i$ all exist.
Let $v_i = \dfrac 1 {\map {f_i} {w_i} } w_i$.
Then since $f_i$ is [[Definition:Linear Functional|linear]]:
:$\map {f_i} {v_i} = 1$
Furthermore for $j \ne i$, $\map {f_j} {v_... | Condition for Linear Dependence of Linear Functionals in terms of Kernel/Proof 1 | https://proofwiki.org/wiki/Condition_for_Linear_Dependence_of_Linear_Functionals_in_terms_of_Kernel | https://proofwiki.org/wiki/Condition_for_Linear_Dependence_of_Linear_Functionals_in_terms_of_Kernel/Proof_1 | [
"Condition for Linear Dependence of Linear Functionals in terms of Kernel",
"Linear Functionals"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Linear Functional",
"Definition:Kernel of Linear Transformation/Vector Space"
] | [
"Definition:Linear Functional"
] |
proofwiki-20112 | Condition for Linear Dependence of Linear Functionals in terms of Kernel | Let $V$ be a vector space over a field $\GF$.
Let $f, f_1, \ldots, f_n: V \to \GF$ be linear functionals.
Suppose that:
:$\ds \bigcap_{i \mathop = 1}^n \ker f_i \subseteq \ker f$
where $\ker f$ denotes the kernel of $f$.
Then there exist $\alpha_1, \ldots, \alpha_n \in \GF$ such that:
:$\ds \forall v \in V: \map f v = ... | Define $T : X \to {\GF}^n$ by:
:$\map T x = \paren {\map {g_1} x, \map {g_2} x, \ldots, \map {g_n} x}$
for each $x \in X$.
We show that $T$ is linear.
Let $x, y \in X$ and $\alpha, \beta \in \GF$.
We have:
{{begin-eqn}}
{{eqn | l = \map T {\alpha x + \beta y}
| r = \tuple {\map {g_1} {\alpha x + \beta y}, \map {g... | Let $V$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Field (Abstract Algebra)|field]] $\GF$.
Let $f, f_1, \ldots, f_n: V \to \GF$ be [[Definition:Linear Functional|linear functionals]].
Suppose that:
:$\ds \bigcap_{i \mathop = 1}^n \ker f_i \subseteq \ker f$
where $\ker f$ denotes the [[Definit... | Define $T : X \to {\GF}^n$ by:
:$\map T x = \paren {\map {g_1} x, \map {g_2} x, \ldots, \map {g_n} x}$
for each $x \in X$.
We show that $T$ is [[Definition:Linear Transformation|linear]].
Let $x, y \in X$ and $\alpha, \beta \in \GF$.
We have:
{{begin-eqn}}
{{eqn | l = \map T {\alpha x + \beta y}
| r = \tupl... | Condition for Linear Dependence of Linear Functionals in terms of Kernel/Proof 2 | https://proofwiki.org/wiki/Condition_for_Linear_Dependence_of_Linear_Functionals_in_terms_of_Kernel | https://proofwiki.org/wiki/Condition_for_Linear_Dependence_of_Linear_Functionals_in_terms_of_Kernel/Proof_2 | [
"Condition for Linear Dependence of Linear Functionals in terms of Kernel",
"Linear Functionals"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Linear Functional",
"Definition:Kernel of Linear Transformation/Vector Space"
] | [
"Definition:Linear Transformation",
"Definition:Linear Functional",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Image of Vector Subspace under Linear Transformation is Vector Subspace",
"Definition:Vector Subspace",
"Definition:Basis of Vector Space",
"Definition:Basis of ... |
proofwiki-20113 | Open Neighborhoods of Point form Directed Ordering | Let $\struct{ S, \tau }$ be a topological space.
Let $x \in S$.
Let $\NN \subseteq \tau$ be the set of open neighborhoods of $x$.
Then $\supseteq$, the ordering of $\NN$ by reverse inclusion, is a directed ordering on $\NN$. | By Subset Relation is Ordering and Dual Ordering is Ordering $\supseteq$ is an ordering on $\NN$.
To show that $\supseteq$ is directed, let $U, V \in \NN$.
Then $x \in U, V$, so that $x \in U \cap V$.
Hence $U \cap V \in \NN$ is an open neighborhood of $x$.
Moreover by Intersection is Subset:
:$U, V \supseteq U \cap V$... | Let $\struct{ S, \tau }$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Let $\NN \subseteq \tau$ be the set of [[Definition:Open Neighborhood of Point|open neighborhoods]] of $x$.
Then $\supseteq$, the [[Definition:Ordering by Reverse Inclusion|ordering of $\NN$ by reverse inclusion]], is a... | By [[Subset Relation is Ordering]] and [[Dual Ordering is Ordering]] $\supseteq$ is an [[Definition:Ordering|ordering]] on $\NN$.
To show that $\supseteq$ is [[Definition:Directed Ordering|directed]], let $U, V \in \NN$.
Then $x \in U, V$, so that $x \in U \cap V$.
Hence $U \cap V \in \NN$ is an [[Definition:Open Ne... | Open Neighborhoods of Point form Directed Ordering | https://proofwiki.org/wiki/Open_Neighborhoods_of_Point_form_Directed_Ordering | https://proofwiki.org/wiki/Open_Neighborhoods_of_Point_form_Directed_Ordering | [
"Topology",
"Directed Orderings",
"Subset Relation"
] | [
"Definition:Topological Space",
"Definition:Open Neighborhood/Point",
"Definition:Set Ordered by Subset Relation/Reverse Inclusion",
"Definition:Directed Ordering"
] | [
"Subset Relation is Ordering",
"Dual Ordering is Ordering",
"Definition:Ordering",
"Definition:Directed Ordering",
"Definition:Open Neighborhood/Point",
"Intersection is Subset",
"Definition:Directed Ordering"
] |
proofwiki-20114 | Finite Subsets form Directed Ordering | Let $I$ be a set.
Denote with $\FF$ the set of finite subsets of $I$.
Let $\subseteq$ be the subset relation on $\FF$.
Then $\subseteq$ is a directed ordering on $\FF$. | From Subset Relation is Ordering, we know that $\subseteq$ is an ordering.
Now let $F, G \in \FF$.
From Set Union Preserves Subsets, conclude that $F \cup G \subseteq I$ as $F, G \subseteq I$.
From Union of Finite Sets is Finite, $F \cup G$ is a finite set.
Hence $F \cup G \in \FF$.
Furthermore, $F \subseteq F \cup G$ ... | Let $I$ be a [[Definition:Set|set]].
Denote with $\FF$ the set of [[Definition:Finite Subset|finite subsets]] of $I$.
Let $\subseteq$ be the [[Definition:Subset Relation|subset relation]] on $\FF$.
Then $\subseteq$ is a [[Definition:Directed Ordering|directed ordering]] on $\FF$. | From [[Subset Relation is Ordering]], we know that $\subseteq$ is an [[Definition:Ordering|ordering]].
Now let $F, G \in \FF$.
From [[Set Union Preserves Subsets]], conclude that $F \cup G \subseteq I$ as $F, G \subseteq I$.
From [[Union of Finite Sets is Finite]], $F \cup G$ is a [[Definition:Finite Set|finite set]... | Finite Subsets form Directed Ordering | https://proofwiki.org/wiki/Finite_Subsets_form_Directed_Ordering | https://proofwiki.org/wiki/Finite_Subsets_form_Directed_Ordering | [
"Directed Orderings",
"Subset Relation"
] | [
"Definition:Set",
"Definition:Finite Subset",
"Definition:Subset Relation",
"Definition:Directed Ordering"
] | [
"Subset Relation is Ordering",
"Definition:Ordering",
"Set Union Preserves Subsets",
"Union of Finite Sets is Finite",
"Definition:Finite Set",
"Definition:Directed Ordering"
] |
proofwiki-20115 | Quotient Ring of Noetherian Ring is Noetherian | Let $A$ be a Noetherian ring.
Let $\mathfrak a \subseteq A$ be an ideal.
Let $A / \mathfrak a$ be the quotient ring of $A$ by $\mathfrak a$.
Then $A / \mathfrak a$ is a Noetherian ring. | Observe that:
:$0 \longrightarrow \mathfrak a \longrightarrow A \longrightarrow A / \mathfrak a \longrightarrow 0$
is a short exact sequence of $A$-modules.
By Short Exact Sequence Condition of Noetherian Modules, $A / \mathfrak a$ is a Noetherian $A$-module.
As $A / \mathfrak a$ is an $A / \mathfrak a$-module, $A / \m... | Let $A$ be a [[Definition:Noetherian Ring|Noetherian ring]].
Let $\mathfrak a \subseteq A$ be an [[Definition:Ideal of Ring|ideal]].
Let $A / \mathfrak a$ be the [[Definition:Quotient Ring|quotient ring]] of $A$ by $\mathfrak a$.
Then $A / \mathfrak a$ is a [[Definition:Noetherian Ring|Noetherian ring]]. | Observe that:
:$0 \longrightarrow \mathfrak a \longrightarrow A \longrightarrow A / \mathfrak a \longrightarrow 0$
is a [[Definition:Short Exact Sequence of Modules|short exact sequence]] of [[Definition:Module over Ring|$A$-modules]].
By [[Short Exact Sequence Condition of Noetherian Modules]], $A / \mathfrak a$ is a... | Quotient Ring of Noetherian Ring is Noetherian | https://proofwiki.org/wiki/Quotient_Ring_of_Noetherian_Ring_is_Noetherian | https://proofwiki.org/wiki/Quotient_Ring_of_Noetherian_Ring_is_Noetherian | [
"Noetherian Rings"
] | [
"Definition:Noetherian Ring",
"Definition:Ideal of Ring",
"Definition:Quotient Ring",
"Definition:Noetherian Ring"
] | [
"Definition:Short Exact Sequence of Modules",
"Definition:Module over Ring",
"Short Exact Sequence Condition of Noetherian Modules",
"Definition:Noetherian Module",
"Definition:Module over Ring",
"Definition:Module over Ring",
"Definition:Noetherian Module",
"Definition:Module over Ring",
"Definitio... |
proofwiki-20116 | Absolute Value of Trigonometric Function | Let $\theta$ be an angle embedded in a Cartesian plane.
Let $\theta$ be such that the vertex of $\theta$ is located at the origin while one arm is coincident with the $x$-axis.
Let $\phi$ be the acute angle made by the other arm with the $x$-axis.
Let $f: \R \to \R$ be a trigonometric function.
Then $\size {\map f \the... | One of the following applies:
:$\theta = \phi$
:$\theta = -\paren {\phi + \pi}$
:$\theta = \phi + \pi$
:$\theta = -\phi$
depending on the quadrant.
{{explain|prove the above}}
Substituting $\theta$ in $\size {\map f \theta}$ with any of the above expressions, we simplify by using
{{begin-eqn}}
{{eqn | l = \size {\map f... | Let $\theta$ be an [[Definition:Angle|angle]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]].
Let $\theta$ be such that the [[Definition:Vertex of Angle|vertex]] of $\theta$ is located at the [[Definition:Origin|origin]] while one [[Definition:Arm of Angle|arm]] is coincident with the [[Definition:X-Axis... | One of the following applies:
:$\theta = \phi$
:$\theta = -\paren {\phi + \pi}$
:$\theta = \phi + \pi$
:$\theta = -\phi$
depending on the [[Definition:Quadrant (Cartesian Coordinates)|quadrant]].
{{explain|prove the above}}
Substituting $\theta$ in $\size {\map f \theta}$ with any of the above expressions, we simplif... | Absolute Value of Trigonometric Function | https://proofwiki.org/wiki/Absolute_Value_of_Trigonometric_Function | https://proofwiki.org/wiki/Absolute_Value_of_Trigonometric_Function | [
"Trigonometric Functions"
] | [
"Definition:Angle",
"Definition:Cartesian Plane",
"Definition:Angle/Vertex",
"Definition:Coordinate System/Origin",
"Definition:Angle/Arm",
"Definition:Axis/X-Axis",
"Definition:Acute Angle",
"Definition:Angle/Arm",
"Definition:Axis/X-Axis",
"Definition:Trigonometric Function",
"Definition:Trigo... | [
"Definition:Cartesian Plane/Quadrants"
] |
proofwiki-20117 | Tangent of Angle minus Three Right Angles | :$\map \tan {x - \dfrac {3 \pi} 2} = \cot x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x - \dfrac {3 \pi} 2}
| r = -\map \tan {x - \dfrac {\pi} 2}
| c = as $\map \tan {x - \dfrac {\pi} 2}$ is in the opposite quadrant to $\map \tan {x - \dfrac {3 \pi} 2}$
}}
{{eqn | r = \cot \theta
| c = Tangent of Complement equals Cotangent
}}
{{end-eqn}}
{{qed}} | :$\map \tan {x - \dfrac {3 \pi} 2} = \cot x$ | {{begin-eqn}}
{{eqn | l = \map \tan {x - \dfrac {3 \pi} 2}
| r = -\map \tan {x - \dfrac {\pi} 2}
| c = as $\map \tan {x - \dfrac {\pi} 2}$ is in the opposite [[Definition:Quadrant (Cartesian Coordinates)|quadrant]] to $\map \tan {x - \dfrac {3 \pi} 2}$
}}
{{eqn | r = \cot \theta
| c = [[Tangent of Com... | Tangent of Angle minus Three Right Angles | https://proofwiki.org/wiki/Tangent_of_Angle_minus_Three_Right_Angles | https://proofwiki.org/wiki/Tangent_of_Angle_minus_Three_Right_Angles | [
"Tangent Function"
] | [] | [
"Definition:Cartesian Plane/Quadrants",
"Tangent of Complement equals Cotangent"
] |
proofwiki-20118 | Reciprocal of One Minus Cosine plus Reciprocal of One Plus Cosine | :$\dfrac 1 {1 - \cos x} + \dfrac 1 {1 + \cos x} = 2 \cosec^2 x$ | {{begin-eqn}}
{{eqn | l = \dfrac 1 {1 - \cos x} + \dfrac 1 {1 + \cos x}
| r = \dfrac {\paren {1 + \cos x} + \paren {1 - \cos x} } {\paren {1 - \cos x} \paren {1 + \cos x} }
| c = common denominator
}}
{{eqn | r = \dfrac 2 {1 - \cos^2 x}
| c = Difference of Two Squares and simplification
}}
{{eqn | r =... | :$\dfrac 1 {1 - \cos x} + \dfrac 1 {1 + \cos x} = 2 \cosec^2 x$ | {{begin-eqn}}
{{eqn | l = \dfrac 1 {1 - \cos x} + \dfrac 1 {1 + \cos x}
| r = \dfrac {\paren {1 + \cos x} + \paren {1 - \cos x} } {\paren {1 - \cos x} \paren {1 + \cos x} }
| c = [[Definition:Common Denominator|common denominator]]
}}
{{eqn | r = \dfrac 2 {1 - \cos^2 x}
| c = [[Difference of Two Squar... | Reciprocal of One Minus Cosine plus Reciprocal of One Plus Cosine | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Cosine_plus_Reciprocal_of_One_Plus_Cosine | https://proofwiki.org/wiki/Reciprocal_of_One_Minus_Cosine_plus_Reciprocal_of_One_Plus_Cosine | [
"Trigonometric Identities",
"Cosine Function"
] | [] | [
"Definition:Common Denominator",
"Difference of Two Squares",
"Sum of Squares of Sine and Cosine",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-20119 | Product of One Plus Cotangent with One Plus Tangent | :$\paren {1 + \cot x} \paren {1 + \tan x} = 2 + \csc x \sec x$ | {{begin-eqn}}
{{eqn | l = \paren {1 + \cot x} \paren {1 + \tan x}
| r = 1 + \cot x + \tan x + \cot x \tan x
| c = multiplying out
}}
{{eqn | r = 1 + \cot x + \tan x + \dfrac 1 {\tan x} \tan x
| c = Cotangent is Reciprocal of Tangent
}}
{{eqn | r = 2 + \cot x + \tan x
| c = simplifying
}}
{{eqn |... | :$\paren {1 + \cot x} \paren {1 + \tan x} = 2 + \csc x \sec x$ | {{begin-eqn}}
{{eqn | l = \paren {1 + \cot x} \paren {1 + \tan x}
| r = 1 + \cot x + \tan x + \cot x \tan x
| c = multiplying out
}}
{{eqn | r = 1 + \cot x + \tan x + \dfrac 1 {\tan x} \tan x
| c = [[Cotangent is Reciprocal of Tangent]]
}}
{{eqn | r = 2 + \cot x + \tan x
| c = simplifying
}}
{{e... | Product of One Plus Cotangent with One Plus Tangent | https://proofwiki.org/wiki/Product_of_One_Plus_Cotangent_with_One_Plus_Tangent | https://proofwiki.org/wiki/Product_of_One_Plus_Cotangent_with_One_Plus_Tangent | [
"Trigonometric Identities",
"Tangent Function",
"Cotangent Function"
] | [] | [
"Cotangent is Reciprocal of Tangent",
"Sum of Tangent and Cotangent"
] |
proofwiki-20120 | Cosine over Cosine of Complement plus Sine over Sine of Complement | :$\dfrac {\cos x} {\map \cos {90 \degrees - x} } + \dfrac {\sin x} {\map \sin {90 \degrees - x} } = \csc x \sec x$ | {{begin-eqn}}
{{eqn | l = \dfrac {\cos x} {\map \cos {90 \degrees - x} } + \dfrac {\sin x} {\map \sin {90 \degrees - x} }
| r = \dfrac {\cos x} {\sin x} + \dfrac {\sin x} {\cos x}
| c = Cosine of Complement equals Sine, Sine of Complement equals Cosine
}}
{{eqn | r = \cot x + \tan x
| c = Cotangent is... | :$\dfrac {\cos x} {\map \cos {90 \degrees - x} } + \dfrac {\sin x} {\map \sin {90 \degrees - x} } = \csc x \sec x$ | {{begin-eqn}}
{{eqn | l = \dfrac {\cos x} {\map \cos {90 \degrees - x} } + \dfrac {\sin x} {\map \sin {90 \degrees - x} }
| r = \dfrac {\cos x} {\sin x} + \dfrac {\sin x} {\cos x}
| c = [[Cosine of Complement equals Sine]], [[Sine of Complement equals Cosine]]
}}
{{eqn | r = \cot x + \tan x
| c = [[Co... | Cosine over Cosine of Complement plus Sine over Sine of Complement | https://proofwiki.org/wiki/Cosine_over_Cosine_of_Complement_plus_Sine_over_Sine_of_Complement | https://proofwiki.org/wiki/Cosine_over_Cosine_of_Complement_plus_Sine_over_Sine_of_Complement | [
"Trigonometric Identities",
"Sine Function",
"Cosine Function"
] | [] | [
"Cosine of Complement equals Sine",
"Sine of Complement equals Cosine",
"Cotangent is Cosine divided by Sine",
"Tangent is Sine divided by Cosine",
"Sum of Tangent and Cotangent"
] |
proofwiki-20121 | Sum of Fourth Powers of Sine and Cosine | :$\cos^4 x + \sin^4 x = 1 - 2 \cos^2 x \sin^2 x$ | {{begin-eqn}}
{{eqn | l = \cos^4 x + \sin^4 x
| r = \cos^2 x \paren {1 - \sin^2 x} + \sin^2 x \paren {1 - \cos^2 x}
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \cos^2 x + \sin^2 x - 2 \cos^2 x \sin^2 x
| c = simplification
}}
{{eqn | r = 1 - 2 \cos^2 x \sin^2 x
| c = Sum of Squares of... | :$\cos^4 x + \sin^4 x = 1 - 2 \cos^2 x \sin^2 x$ | {{begin-eqn}}
{{eqn | l = \cos^4 x + \sin^4 x
| r = \cos^2 x \paren {1 - \sin^2 x} + \sin^2 x \paren {1 - \cos^2 x}
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | r = \cos^2 x + \sin^2 x - 2 \cos^2 x \sin^2 x
| c = simplification
}}
{{eqn | r = 1 - 2 \cos^2 x \sin^2 x
| c = [[Sum of Squa... | Sum of Fourth Powers of Sine and Cosine/Proof 2 | https://proofwiki.org/wiki/Sum_of_Fourth_Powers_of_Sine_and_Cosine | https://proofwiki.org/wiki/Sum_of_Fourth_Powers_of_Sine_and_Cosine/Proof_2 | [
"Sum of Fourth Powers of Sine and Cosine",
"Trigonometric Identities",
"Sine Function",
"Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-20122 | Sum of Fourth Powers of Sine and Cosine | :$\cos^4 x + \sin^4 x = 1 - 2 \cos^2 x \sin^2 x$ | {{begin-eqn}}
{{eqn | l = \paren {\cos^2 x + \sin^2 x}^2
| r = 1
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | ll = \leadsto
| l = \cos^4 x + 2 \cos^2 x \sin^2 x + \sin^4 x
| r = 1
| c =
}}
{{eqn | ll = \leadsto
| l = \cos^4 x + \sin^4 x
| r = 1 - 2 \cos^2 x \sin^2 x
... | :$\cos^4 x + \sin^4 x = 1 - 2 \cos^2 x \sin^2 x$ | {{begin-eqn}}
{{eqn | l = \paren {\cos^2 x + \sin^2 x}^2
| r = 1
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | ll = \leadsto
| l = \cos^4 x + 2 \cos^2 x \sin^2 x + \sin^4 x
| r = 1
| c =
}}
{{eqn | ll = \leadsto
| l = \cos^4 x + \sin^4 x
| r = 1 - 2 \cos^2 x \sin^2 x
... | Sum of Fourth Powers of Sine and Cosine/Proof 3 | https://proofwiki.org/wiki/Sum_of_Fourth_Powers_of_Sine_and_Cosine | https://proofwiki.org/wiki/Sum_of_Fourth_Powers_of_Sine_and_Cosine/Proof_3 | [
"Sum of Fourth Powers of Sine and Cosine",
"Trigonometric Identities",
"Sine Function",
"Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine"
] |
proofwiki-20123 | Sum of Cubes of Sine and Cosine | :$\cos^3 x + \sin^3 x = \paren {\cos x + \sin x} \paren {1 - \cos x \sin x}$ | {{begin-eqn}}
{{eqn | l = \cos^3 x + \sin^3 x
| r = \cos x \paren {1 - \sin^2 x} + \sin x \paren {1 - \cos^2 x}
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | r = \cos x + \sin x - \cos x \sin x \paren {\cos x + \sin x}
| c = simplification
}}
{{eqn | r = \paren {\cos x + \sin x} \paren {1 - \cos ... | :$\cos^3 x + \sin^3 x = \paren {\cos x + \sin x} \paren {1 - \cos x \sin x}$ | {{begin-eqn}}
{{eqn | l = \cos^3 x + \sin^3 x
| r = \cos x \paren {1 - \sin^2 x} + \sin x \paren {1 - \cos^2 x}
| c = [[Sum of Squares of Sine and Cosine]]
}}
{{eqn | r = \cos x + \sin x - \cos x \sin x \paren {\cos x + \sin x}
| c = simplification
}}
{{eqn | r = \paren {\cos x + \sin x} \paren {1 - \... | Sum of Cubes of Sine and Cosine | https://proofwiki.org/wiki/Sum_of_Cubes_of_Sine_and_Cosine | https://proofwiki.org/wiki/Sum_of_Cubes_of_Sine_and_Cosine | [
"Trigonometric Identities",
"Sine Function",
"Cosine Function"
] | [] | [
"Sum of Squares of Sine and Cosine"
] |
proofwiki-20124 | Spectrum of Self-Adjoint Bounded Linear Operator is Real | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $T : \HH \to \HH$ be a bounded self-adjoint operator.
Let $\map \sigma T$ be the spectrum of $T$.
Then:
:$\map \sigma T \subseteq \R$ | This follows from:
:Spectrum of Self-Adjoint Densely-Defined Linear Operator is Real and Closed
:Spectrum of Bounded Linear Operator equal to Spectrum as Densely-Defined Linear Operator
{{qed}} | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $T : \HH \to \HH$ be a [[Definition:Self-Adjoint Operator|bounded self-adjoint operator]].
Let $\map \sigma T$ be the [[Definition:Spectrum of Bounded Linear Operator|spectrum]] of $T$.
Then:
:$\map \sigma T ... | This follows from:
:[[Spectrum of Self-Adjoint Densely-Defined Linear Operator is Real and Closed]]
:[[Spectrum of Bounded Linear Operator equal to Spectrum as Densely-Defined Linear Operator]]
{{qed}} | Spectrum of Self-Adjoint Bounded Linear Operator is Real/Proof 1 | https://proofwiki.org/wiki/Spectrum_of_Self-Adjoint_Bounded_Linear_Operator_is_Real | https://proofwiki.org/wiki/Spectrum_of_Self-Adjoint_Bounded_Linear_Operator_is_Real/Proof_1 | [
"Hermitian Operators",
"Spectra (Bounded Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Hermitian Operator",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator"
] | [
"Spectrum of Self-Adjoint Densely-Defined Linear Operator is Real and Closed",
"Spectrum of Bounded Linear Operator equal to Spectrum as Densely-Defined Linear Operator"
] |
proofwiki-20125 | Spectrum of Self-Adjoint Bounded Linear Operator is Real | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $T : \HH \to \HH$ be a bounded self-adjoint operator.
Let $\map \sigma T$ be the spectrum of $T$.
Then:
:$\map \sigma T \subseteq \R$ | Let $\lambda := a + i b \in \C \setminus \R$.
Note that $b \ne 0$.
For all $\phi \in \HH$:
{{begin-eqn}}
{{eqn | l = \norm {\paren {T - \lambda I} \phi}^2
| r = \norm {\paren {T - a I} \phi}^2 + b^2 \norm {\phi}^2
| c = as $\Re \innerprod {\paren {T - a I} \phi} {- i b \phi} = 0$
}}
{{eqn | o = \ge
| ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $T : \HH \to \HH$ be a [[Definition:Self-Adjoint Operator|bounded self-adjoint operator]].
Let $\map \sigma T$ be the [[Definition:Spectrum of Bounded Linear Operator|spectrum]] of $T$.
Then:
:$\map \sigma T ... | Let $\lambda := a + i b \in \C \setminus \R$.
Note that $b \ne 0$.
For all $\phi \in \HH$:
{{begin-eqn}}
{{eqn | l = \norm {\paren {T - \lambda I} \phi}^2
| r = \norm {\paren {T - a I} \phi}^2 + b^2 \norm {\phi}^2
| c = as $\Re \innerprod {\paren {T - a I} \phi} {- i b \phi} = 0$
}}
{{eqn | o = \ge
... | Spectrum of Self-Adjoint Bounded Linear Operator is Real/Proof 2 | https://proofwiki.org/wiki/Spectrum_of_Self-Adjoint_Bounded_Linear_Operator_is_Real | https://proofwiki.org/wiki/Spectrum_of_Self-Adjoint_Bounded_Linear_Operator_is_Real/Proof_2 | [
"Hermitian Operators",
"Spectra (Bounded Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Hermitian Operator",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator"
] | [
"Definition:Injective",
"Kernel of Linear Transformation is Orthocomplement of Image of Adjoint",
"Definition:Injective",
"Linear Subspace Dense iff Zero Orthocomplement",
"Definition:Everywhere Dense",
"Definition:Surjection",
"Definition:Invertible Bounded Linear Transformation"
] |
proofwiki-20126 | Convergent Sequences form Invariant Subspace of Bounded Sequences wrt Cesàro Summation Operator | Let $\ell^\infty$ be the space of bounded sequences.
Let $c$ be the space of convergent sequences.
Let $A : \ell^\infty \to \ell^\infty$ be the Cesàro summation operator.
Then $c$ is an invariant subspace of $\ell^\infty$ {{WRT}} $A$. | Let $\sequence {x_n}_{n \mathop \in \N} \in c$ be a sequence.
By definition, $\sequence {x_n}_{n \mathop \in \N}$ converges.
Let $\ds L = \lim_{n \mathop \to \infty} x_n$ be the limit of $\sequence {x_n}_{n \mathop \in \N}$.
Then:
:$\forall \epsilon \in \R_{> 0} : \exists N_1 \in \N : \forall n \in \N : n > N_1 \implie... | Let $\ell^\infty$ be the [[Definition:Space of Bounded Sequences|space of bounded sequences]].
Let $c$ be the [[Definition:Space of Convergent Sequences|space of convergent sequences]].
Let $A : \ell^\infty \to \ell^\infty$ be the [[Definition:Cesàro Summation Operator|Cesàro summation operator]].
Then $c$ is an [[... | Let $\sequence {x_n}_{n \mathop \in \N} \in c$ be a [[Definition:Sequence|sequence]].
By [[Definition:Space of Convergent Sequences|definition]], $\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence in Normed Vector Space|converges]].
Let $\ds L = \lim_{n \mathop \to \infty} x_n$ be the [[Definition:... | Convergent Sequences form Invariant Subspace of Bounded Sequences wrt Cesàro Summation Operator | https://proofwiki.org/wiki/Convergent_Sequences_form_Invariant_Subspace_of_Bounded_Sequences_wrt_Cesàro_Summation_Operator | https://proofwiki.org/wiki/Convergent_Sequences_form_Invariant_Subspace_of_Bounded_Sequences_wrt_Cesàro_Summation_Operator | [
"Convergent Real Sequences",
"Cesàro Summation Operator"
] | [
"Definition:Space of Bounded Sequences",
"Definition:Space of Convergent Sequences",
"Definition:Cesàro Summation Operator",
"Definition:Invariant Subspace/Normed Vector Space"
] | [
"Definition:Sequence",
"Definition:Space of Convergent Sequences",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Limit of Sequence/Normed Vector Space"
] |
proofwiki-20127 | First Fundamental Group of 1-Sphere | Let $\mathbb S^1$ be the $1$-sphere.
Let $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ be the first fundamental group of $\mathbb S^1$.
Let $\struct {\Z, +}$ be the additive group of integers.
Then $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ is isomorphic to $\struct {\Z, +}$. | We are given that $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ is the first fundamental group of $\mathbb S^1$.
Let $x_0 \in \mathbb S^1$.
Let $\struct{\mathbb S^1, x_0}$ be the pointed topological space for $\mathbb S^1$.
Since Fundamental Group is Independent of Base Point for Path-Connected Space, what to be proved... | Let $\mathbb S^1$ be the $1$-[[Definition:Sphere (Topology)|sphere]].
Let $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ be the first [[Definition:Fundamental Group|fundamental group]] of $\mathbb S^1$.
Let $\struct {\Z, +}$ be the [[Definition:Additive Group of Integers|additive group of integers]].
Then $\struct {... | We are [[Definition:Given|given]] that $\struct {\map {\pi _1} {\mathbb S^1}, \ast}$ is the first [[Definition:Fundamental Group|fundamental group]] of $\mathbb S^1$.
Let $x_0 \in \mathbb S^1$.
Let $\struct{\mathbb S^1, x_0}$ be the [[Definition:Pointed Topological Space|pointed topological space]] for $\mathbb S^1$.... | First Fundamental Group of 1-Sphere | https://proofwiki.org/wiki/First_Fundamental_Group_of_1-Sphere | https://proofwiki.org/wiki/First_Fundamental_Group_of_1-Sphere | [
"Homotopy Theory"
] | [
"Definition:Sphere/Topology",
"Definition:Fundamental Group",
"Definition:Additive Group of Integers",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Given",
"Definition:Fundamental Group",
"Definition:Pointed Topological Space",
"Fundamental Group is Independent of Base Point for Path-Connected Space",
"Definition:Mapping",
"Definition:Concatenation (Topology)",
"Definition:Homotopy Class",
"Definition:Isomorphism (Abstract Algebra)/Gr... |
proofwiki-20128 | Sum of Sides of Triangle in terms of Circumradius and Half Angle Cosines | Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$a + b + c = 8 R \cos \dfrac A 2 \cos \dfrac B 2 \cos \dfrac C 2$
where $R$ denotes the circumradius of $\triangle ABC$. | From Law of Sines:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
Hence:
{{begin-eqn}}
{{eqn | l = a
| r = 2 R \sin A
}}
{{eqn | l = b
| r = 2 R \sin B
}}
{{eqn | l = c
| r = 2 R \sin C
}}
{{eqn | ll= \leadsto
| l = a + b + c
| r = 2 R \paren {\sin A + \sin B + \sin C}
... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.
Then:
:$a + b + c = 8 R \cos \dfrac A 2 \cos \dfrac B 2 \cos \dfrac C 2$
where $R$ denotes the [[Definition:Circumradius of Triangle|circumradius]]... | From [[Law of Sines]]:
:$\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$
Hence:
{{begin-eqn}}
{{eqn | l = a
| r = 2 R \sin A
}}
{{eqn | l = b
| r = 2 R \sin B
}}
{{eqn | l = c
| r = 2 R \sin C
}}
{{eqn | ll= \leadsto
| l = a + b + c
| r = 2 R \paren {\sin A + \sin B + ... | Sum of Sides of Triangle in terms of Circumradius and Half Angle Cosines | https://proofwiki.org/wiki/Sum_of_Sides_of_Triangle_in_terms_of_Circumradius_and_Half_Angle_Cosines | https://proofwiki.org/wiki/Sum_of_Sides_of_Triangle_in_terms_of_Circumradius_and_Half_Angle_Cosines | [
"Triangles",
"Cosine Function"
] | [
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius"
] | [
"Law of Sines",
"Sum of Sines of Angles in Triangle"
] |
proofwiki-20129 | Sum of Cosines of Twice 2 Angles minus Cosine of Twice Third Angle of Triangle | Let $\triangle ABC$ be a triangle.
Then:
:$\cos 2 A + \cos 2 B - \cos 2 C = 1 - 4 \sin A \sin B \cos C$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = Sum of Angles of Triangle equals Two Right Angles
}}
{{eqn | n = 1
| ll= \leadsto
| l = A + B
| r = 180 \degrees - C
| c =
}}
{{end-eqn}}
That is, $C$ is the supplement of $A + B$.
Then:
{{begin-eqn}}
{... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\cos 2 A + \cos 2 B - \cos 2 C = 1 - 4 \sin A \sin B \cos C$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = [[Sum of Angles of Triangle equals Two Right Angles]]
}}
{{eqn | n = 1
| ll= \leadsto
| l = A + B
| r = 180 \degrees - C
| c =
}}
{{end-eqn}}
That is, $C$ is the [[Definition:Supplement of Angle|suppl... | Sum of Cosines of Twice 2 Angles minus Cosine of Twice Third Angle of Triangle | https://proofwiki.org/wiki/Sum_of_Cosines_of_Twice_2_Angles_minus_Cosine_of_Twice_Third_Angle_of_Triangle | https://proofwiki.org/wiki/Sum_of_Cosines_of_Twice_2_Angles_minus_Cosine_of_Twice_Third_Angle_of_Triangle | [
"Triangles",
"Cosine Function"
] | [
"Definition:Triangle (Geometry)"
] | [
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Supplementary Angles",
"Prosthaphaeresis Formulas/Cosine plus Cosine",
"Cosine of Supplementary Angle",
"Cosine of Difference",
"Cosine of Supplementary Angle",
"Cosine of Sum"
] |
proofwiki-20130 | Sum of Cosines of Twice Angles of Triangle | Let $\triangle ABC$ be a triangle.
Then:
:$\cos 2 A + \cos 2 B + \cos 2 C = -1 - 4 \cos A \cos B \cos C$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = Sum of Angles of Triangle equals Two Right Angles
}}
{{eqn | n = 1
| ll= \leadsto
| l = A + B
| r = 180 \degrees - C
| c =
}}
{{end-eqn}}
That is, $C$ is the supplement of $A + B$.
Then:
{{begin-eqn}}
{... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\cos 2 A + \cos 2 B + \cos 2 C = -1 - 4 \cos A \cos B \cos C$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = [[Sum of Angles of Triangle equals Two Right Angles]]
}}
{{eqn | n = 1
| ll= \leadsto
| l = A + B
| r = 180 \degrees - C
| c =
}}
{{end-eqn}}
That is, $C$ is the [[Definition:Supplement of Angle|suppl... | Sum of Cosines of Twice Angles of Triangle | https://proofwiki.org/wiki/Sum_of_Cosines_of_Twice_Angles_of_Triangle | https://proofwiki.org/wiki/Sum_of_Cosines_of_Twice_Angles_of_Triangle | [
"Triangles",
"Cosine Function"
] | [
"Definition:Triangle (Geometry)"
] | [
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Supplementary Angles",
"Prosthaphaeresis Formulas/Cosine plus Cosine",
"Cosine of Supplementary Angle",
"Cosine of Difference",
"Cosine of Supplementary Angle",
"Cosine of Sum"
] |
proofwiki-20131 | Sum of Sines of Twice Angles of Triangle | Let $\triangle ABC$ be a triangle.
Then:
:$\sin 2 A + \sin 2 B + \sin 2 C = 4 \sin A \sin B \sin C$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = Sum of Angles of Triangle equals Two Right Angles
}}
{{eqn | n = 1
| ll= \leadsto
| l = A + B
| r = 180 \degrees - C
| c =
}}
{{end-eqn}}
That is, $C$ is the supplement of $A + B$.
Then:
{{begin-eqn}}
{... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\sin 2 A + \sin 2 B + \sin 2 C = 4 \sin A \sin B \sin C$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = [[Sum of Angles of Triangle equals Two Right Angles]]
}}
{{eqn | n = 1
| ll= \leadsto
| l = A + B
| r = 180 \degrees - C
| c =
}}
{{end-eqn}}
That is, $C$ is the [[Definition:Supplement of Angle|suppl... | Sum of Sines of Twice Angles of Triangle | https://proofwiki.org/wiki/Sum_of_Sines_of_Twice_Angles_of_Triangle | https://proofwiki.org/wiki/Sum_of_Sines_of_Twice_Angles_of_Triangle | [
"Triangles",
"Sine Function"
] | [
"Definition:Triangle (Geometry)"
] | [
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Supplementary Angles",
"Prosthaphaeresis Formulas/Sine plus Sine",
"Sine of Supplementary Angle",
"Cosine of Difference",
"Double Angle Formulas/Sine",
"Cosine of Supplementary Angle",
"Cosine of Sum"
] |
proofwiki-20132 | Sine of Twice Angle minus Sum of Sines of Twice Other Two Angles of Triangle | Let $\triangle ABC$ be a triangle.
Then:
:$\sin 2 A - \sin 2 B - \sin 2 C = -4 \sin A \cos B \cos C$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = Sum of Angles of Triangle equals Two Right Angles
}}
{{eqn | n = 1
| ll= \leadsto
| l = A + B
| r = 180 \degrees - C
| c =
}}
{{end-eqn}}
That is, $C$ is the supplement of $A + B$.
Then:
{{begin-eqn}}
{... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\sin 2 A - \sin 2 B - \sin 2 C = -4 \sin A \cos B \cos C$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = [[Sum of Angles of Triangle equals Two Right Angles]]
}}
{{eqn | n = 1
| ll= \leadsto
| l = A + B
| r = 180 \degrees - C
| c =
}}
{{end-eqn}}
That is, $C$ is the [[Definition:Supplement of Angle|suppl... | Sine of Twice Angle minus Sum of Sines of Twice Other Two Angles of Triangle | https://proofwiki.org/wiki/Sine_of_Twice_Angle_minus_Sum_of_Sines_of_Twice_Other_Two_Angles_of_Triangle | https://proofwiki.org/wiki/Sine_of_Twice_Angle_minus_Sum_of_Sines_of_Twice_Other_Two_Angles_of_Triangle | [
"Triangles",
"Sine Function"
] | [
"Definition:Triangle (Geometry)"
] | [
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Supplementary Angles",
"Prosthaphaeresis Formulas/Sine minus Sine",
"Cosine of Supplementary Angle",
"Sine of Difference",
"Double Angle Formulas/Sine",
"Sine of Supplementary Angle",
"Cosine of Sum"
] |
proofwiki-20133 | Sum of Cosines of Four Times Angles of Triangle | Let $\triangle ABC$ be a triangle.
Then:
:$\cos 4 A + \cos 4 B + \cos 4 C = 4 \cos 2 A \cos 2 B \cos 2 C - 1$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = Sum of Angles of Triangle equals Two Right Angles
}}
{{eqn | ll= \leadsto
| l = 2 A + 2 B + 2 C
| r = 360 \degrees
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = 2 A + 2 B
| r = 360 \degrees - ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\cos 4 A + \cos 4 B + \cos 4 C = 4 \cos 2 A \cos 2 B \cos 2 C - 1$ | First we note that:
{{begin-eqn}}
{{eqn | l = A + B + C
| r = 180 \degrees
| c = [[Sum of Angles of Triangle equals Two Right Angles]]
}}
{{eqn | ll= \leadsto
| l = 2 A + 2 B + 2 C
| r = 360 \degrees
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = 2 A + 2 B
| r = 360 \degre... | Sum of Cosines of Four Times Angles of Triangle | https://proofwiki.org/wiki/Sum_of_Cosines_of_Four_Times_Angles_of_Triangle | https://proofwiki.org/wiki/Sum_of_Cosines_of_Four_Times_Angles_of_Triangle | [
"Triangles",
"Cosine Function"
] | [
"Definition:Triangle (Geometry)"
] | [
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Conjugate Angles",
"Prosthaphaeresis Formulas/Cosine plus Cosine",
"Cosine of Conjugate Angle",
"Cosine of Difference",
"Cosine of Conjugate Angle",
"Cosine of Sum"
] |
proofwiki-20134 | Spectrum of Bounded Linear Operator is Closed | Let $\struct {X, \norm \cdot_X}$ be a Banach space over $\C$..
Let $T$ be a bounded linear operator on $X$.
Then the spectrum $\map \sigma T$ of $T$ is a closed set in $\C$. | From Resolvent Set of Bounded Linear Operator is Open, the resolvent set $\map \rho T$ is open.
From the definition of spectrum, we have $\map \sigma T = \C \setminus \map \rho T$.
From the definition of a closed set, $\map \sigma T$ is closed set in $\C$.
{{qed}}
Category:Bounded Linear Operators
Category:Spectra (Bo... | Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Banach Space|Banach space]] over $\C$..
Let $T$ be a [[Definition:Bounded Linear Operator on Normed Vector Space|bounded linear operator]] on $X$.
Then the [[Definition:Spectrum of Bounded Linear Operator|spectrum]] $\map \sigma T$ of $T$ is a [[Definition:Closed Se... | From [[Resolvent Set of Bounded Linear Operator is Open]], the [[Definition:Resolvent Set of Bounded Linear Operator|resolvent set]] $\map \rho T$ is [[Definition:Open Set in Normed Vector Space|open]].
From the definition of [[Definition:Spectrum of Bounded Linear Operator|spectrum]], we have $\map \sigma T = \C \se... | Spectrum of Bounded Linear Operator is Closed | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_is_Closed | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_is_Closed | [
"Bounded Linear Operators",
"Spectra (Bounded Linear Operators)"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Operator/Normed Vector Space",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Definition:Closed Set/Complex Analysis"
] | [
"Resolvent Set of Bounded Linear Operator is Open",
"Definition:Resolvent Set/Bounded Linear Operator",
"Definition:Open Set/Normed Vector Space",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Definition:Closed Set/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space",
"C... |
proofwiki-20135 | Length of Angle Bisector in terms of Angle | Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $AD$ be the angle bisector of $\angle BAC$ that intersects $a$ at $D$.
:300px
Then:
:$AD = \dfrac {2 c b \cos \frac A 2} {b + c}$ | {{begin-eqn}}
{{eqn | l = \frac {BD} {DC}
| r = \frac c b
| c = Angle Bisector Theorem
}}
{{eqn | n = 1
| ll= \leadsto
| l = BD
| r = \frac {a c} {b + c}
| c = as $a = BD + DC$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \frac {AD} {\sin B}
| r = \frac {BD} {\sin \frac A 2}... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] with [[Definition:Side of Polygon|sides]] $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $AD$ be the [[Definition:Angle Bisector|angle bisector]] of $\angle... | {{begin-eqn}}
{{eqn | l = \frac {BD} {DC}
| r = \frac c b
| c = [[Angle Bisector Theorem]]
}}
{{eqn | n = 1
| ll= \leadsto
| l = BD
| r = \frac {a c} {b + c}
| c = as $a = BD + DC$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \frac {AD} {\sin B}
| r = \frac {BD} {\sin \fr... | Length of Angle Bisector in terms of Angle | https://proofwiki.org/wiki/Length_of_Angle_Bisector_in_terms_of_Angle | https://proofwiki.org/wiki/Length_of_Angle_Bisector_in_terms_of_Angle | [
"Length of Angle Bisector",
"Triangles",
"Angle Bisectors"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Angle Bisector",
"Definition:Intersection (Geometry)",
"File:LengthOfAngleBisector.png"
] | [
"Angle Bisector Theorem",
"Law of Sines",
"Law of Sines",
"Double Angle Formulas/Sine"
] |
proofwiki-20136 | Construction of Excircle to Triangle | It is possible to construct an excircle to a triangle tangent to any of its three sides. | We have that $\angle XBE = \angle CBE$ and $\angle BQE = \angle BRE$, a right angle, and $BE$ is common.
So from Triangle Angle-Side-Angle Congruence we have that $\triangle EQB = \triangle ERB$.
So $EQ = ER$.
For the same reason $ER = EP$.
So $EQ = ER = EP$.
So the circle drawn with radius $ER$ will pass through $P, Q... | It is possible to construct an [[Definition:Excircle of Triangle|excircle]] to a [[Definition:Triangle (Geometry)|triangle]] [[Definition:Tangent to Circle|tangent]] to any of its three [[Definition:Side of Polygon|sides]]. | We have that $\angle XBE = \angle CBE$ and $\angle BQE = \angle BRE$, a [[Definition:Right Angle|right angle]], and $BE$ is common.
So from [[Triangle Angle-Side-Angle Congruence]] we have that $\triangle EQB = \triangle ERB$.
So $EQ = ER$.
For the same reason $ER = EP$.
So $EQ = ER = EP$.
So the [[Definition:Circ... | Construction of Excircle to Triangle | https://proofwiki.org/wiki/Construction_of_Excircle_to_Triangle | https://proofwiki.org/wiki/Construction_of_Excircle_to_Triangle | [
"Excircles of Triangles"
] | [
"Definition:Excircle of Triangle",
"Definition:Triangle (Geometry)",
"Definition:Tangent Line/Circle",
"Definition:Polygon/Side"
] | [
"Definition:Right Angle",
"Triangle Angle-Side-Angle Congruence",
"Definition:Circle",
"Definition:Circle/Radius",
"Line at Right Angles to Diameter of Circle",
"Definition:Tangent Line/Circle",
"Category:Excircles of Triangles"
] |
proofwiki-20137 | Excenters and Incenter of Orthic Triangle/Acute Triangle | Let $\triangle ABC$ be an acute triangle.
Let $\triangle DEF$ be the orthic triangle of $\triangle ABC$ such that:
:$D$ is on $BC$
:$E$ is on $AC$
:$F$ is on $AB$
Then:
:the excenter of $\triangle DEF$ {{WRT}} $EF$ is $A$
:the excenter of $\triangle DEF$ {{WRT}} $DF$ is $B$
:the excenter of $\triangle DEF$ {{WRT}} $DE$... | :420px
From Altitudes of Triangle Bisect Angles of Orthic Triangle, $AD$ is the angle bisector of $\angle FDE$.
From Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular, the angle bisector of $\angle PDE$ is perpendicular to $AD$.
The line perpendicular to $AD$ is $BC$.
Similarly, fro... | Let $\triangle ABC$ be an [[Definition:Acute Triangle|acute triangle]].
Let $\triangle DEF$ be the [[Definition:Orthic Triangle|orthic triangle]] of $\triangle ABC$ such that:
:$D$ is on $BC$
:$E$ is on $AC$
:$F$ is on $AB$
Then:
:the [[Definition:Excenter of Triangle|excenter]] of $\triangle DEF$ {{WRT}} $EF$ is $A$... | :[[File:Excircle-of-Orthic-Triangle.png|420px]]
From [[Altitudes of Triangle Bisect Angles of Orthic Triangle]], $AD$ is the [[Definition:Angle Bisector|angle bisector]] of $\angle FDE$.
From [[Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular]], the [[Definition:Angle Bisector|an... | Excenters and Incenter of Orthic Triangle/Acute Triangle | https://proofwiki.org/wiki/Excenters_and_Incenter_of_Orthic_Triangle/Acute_Triangle | https://proofwiki.org/wiki/Excenters_and_Incenter_of_Orthic_Triangle/Acute_Triangle | [
"Excenters and Incenter of Orthic Triangle"
] | [
"Definition:Triangle (Geometry)/Acute",
"Definition:Orthic Triangle",
"Definition:Excircle of Triangle/Excenter",
"Definition:Excircle of Triangle/Excenter",
"Definition:Excircle of Triangle/Excenter",
"Definition:Incircle of Triangle/Incenter",
"Definition:Orthocenter"
] | [
"File:Excircle-of-Orthic-Triangle.png",
"Altitudes of Triangle Bisect Angles of Orthic Triangle",
"Definition:Angle Bisector",
"Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular",
"Definition:Angle Bisector",
"Definition:Right Angle/Perpendicular",
"Definition:Line/S... |
proofwiki-20138 | Orthic Triangle of Obtuse Triangle | Let $\triangle ABC$ be an obtuse triangle such that $A$ is the obtuse angle.
Let $H$ be the orthocenter of $\triangle ABC$.
Let $\triangle DEF$ be the orthic triangle of $\triangle ABC$.
Then $\triangle DEF$ is also the orthic triangle of $\triangle HBC$, which is an acute triangle. | :420px
By construction:
:$CE \perp BH$
:$BF \perp CH$
:$HD \perp BC$
Thus by definition $CE$, $BF$ and $HD$ are the altitudes of $\triangle HBC$.
Also by construction, $A$ lies on $CE$, $BF$ and $HD$.
Hence $\triangle DEF$ is the orthic triangle of $\triangle HBC$.
{{finish|Demonstrate that $\triangle HBC$ is acute}}
C... | Let $\triangle ABC$ be an [[Definition:Obtuse Triangle|obtuse triangle]] such that $A$ is the [[Definition:Obtuse Angle|obtuse angle]].
Let $H$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
Let $\triangle DEF$ be the [[Definition:Orthic Triangle|orthic triangle]] of $\triangle ABC$.
Then $\trian... | :[[File:Orthic-Triangle-Obtuse.png|420px]]
By construction:
:$CE \perp BH$
:$BF \perp CH$
:$HD \perp BC$
Thus by definition $CE$, $BF$ and $HD$ are the [[Definition:Altitude of Triangle|altitudes]] of $\triangle HBC$.
Also by construction, $A$ lies on $CE$, $BF$ and $HD$.
Hence $\triangle DEF$ is the [[Definition:O... | Orthic Triangle of Obtuse Triangle | https://proofwiki.org/wiki/Orthic_Triangle_of_Obtuse_Triangle | https://proofwiki.org/wiki/Orthic_Triangle_of_Obtuse_Triangle | [
"Orthic Triangles",
"Obtuse Triangles"
] | [
"Definition:Triangle (Geometry)/Obtuse",
"Definition:Obtuse Angle",
"Definition:Orthocenter",
"Definition:Orthic Triangle",
"Definition:Orthic Triangle",
"Definition:Triangle (Geometry)/Acute"
] | [
"File:Orthic-Triangle-Obtuse.png",
"Definition:Altitude of Triangle",
"Definition:Orthic Triangle",
"Definition:Triangle (Geometry)/Acute",
"Category:Orthic Triangles",
"Category:Obtuse Triangles"
] |
proofwiki-20139 | Orthic Triangle of Obtuse Triangle | Let $\triangle ABC$ be an obtuse triangle such that $A$ is the obtuse angle.
Let $H$ be the orthocenter of $\triangle ABC$.
Let $\triangle DEF$ be the orthic triangle of $\triangle ABC$.
Then $\triangle DEF$ is also the orthic triangle of $\triangle HBC$, which is an acute triangle. | :420px
Let $H$ be the orthocenter of $\triangle ABC$.
{{begin-eqn}}
{{eqn | l = \frac {EF} {\sin \angle EAF}
| r = \frac {AF} {\sin \angle AEF}
| c = Law of Sines for $\triangle AFE$
}}
{{eqn | ll= \leadsto
| l = \frac {EF} {\sin A}
| r = \frac {b \map \cos {180 \degrees - A} } {\sin B}
| ... | Let $\triangle ABC$ be an [[Definition:Obtuse Triangle|obtuse triangle]] such that $A$ is the [[Definition:Obtuse Angle|obtuse angle]].
Let $H$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
Let $\triangle DEF$ be the [[Definition:Orthic Triangle|orthic triangle]] of $\triangle ABC$.
Then $\trian... | :[[File:Orthic-Triangle-Obtuse.png|420px]]
Let $H$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
{{begin-eqn}}
{{eqn | l = \frac {EF} {\sin \angle EAF}
| r = \frac {AF} {\sin \angle AEF}
| c = [[Law of Sines]] for $\triangle AFE$
}}
{{eqn | ll= \leadsto
| l = \frac {EF} {\sin A}... | Sides of Orthic Triangle of Obtuse Triangle/Proof | https://proofwiki.org/wiki/Orthic_Triangle_of_Obtuse_Triangle | https://proofwiki.org/wiki/Sides_of_Orthic_Triangle_of_Obtuse_Triangle/Proof | [
"Orthic Triangles",
"Obtuse Triangles"
] | [
"Definition:Triangle (Geometry)/Obtuse",
"Definition:Obtuse Angle",
"Definition:Orthocenter",
"Definition:Orthic Triangle",
"Definition:Orthic Triangle",
"Definition:Triangle (Geometry)/Acute"
] | [
"File:Orthic-Triangle-Obtuse.png",
"Definition:Orthocenter",
"Law of Sines",
"Angles in Same Segment of Circle are Equal",
"Definition:Cyclic Quadrilateral",
"Law of Sines",
"Cosine of Supplementary Angle",
"Law of Sines",
"Angles in Same Segment of Circle are Equal",
"Definition:Cyclic Quadrilate... |
proofwiki-20140 | Pedal Triangle of Point on Circumcircle is Straight Line | Let $\triangle ABC$ be a triangle.
Let $P$ be an arbitrary point on the circumcircle of $\triangle ABC$.
The pedal triangle of $\triangle ABC$ {{WRT}} $P$ degenerates to a straight line segment. | {{ProofWanted|Z423x5c6: I think we can use Menelau's theorem.}} | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $P$ be an arbitrary [[Definition:Point|point]] on the [[Definition:Circumcircle|circumcircle]] of $\triangle ABC$.
The [[Definition:Pedal Triangle of Point|pedal triangle]] of $\triangle ABC$ {{WRT}} $P$ [[Definition:Degenerate Case|degenerate... | {{ProofWanted|Z423x5c6: I think we can use Menelau's theorem.}} | Pedal Triangle of Point on Circumcircle is Straight Line | https://proofwiki.org/wiki/Pedal_Triangle_of_Point_on_Circumcircle_is_Straight_Line | https://proofwiki.org/wiki/Pedal_Triangle_of_Point_on_Circumcircle_is_Straight_Line | [
"Pedal Triangles",
"Circumcircles of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Point",
"Definition:Circumcircle",
"Definition:Pedal Triangle/Point",
"Definition:Degenerate Case",
"Definition:Line/Straight Line Segment"
] | [] |
proofwiki-20141 | Line from Vertex of Triangle to Incenter is Angle Bisector | Let $\triangle ABC$ be a triangle.
Let $D$ be a point in the interior of $\triangle ABC$.
Then:
:$AD$ is the angle bisector of $A$
:$BD$ is the angle bisector of $B$
:$CD$ is the angle bisector of $C$
{{iff}}:
:$D$ is the incenter of $\triangle ABC$. | :420px | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $D$ be a [[Definition:Point|point]] in the [[Definition:Interior of Region|interior]] of $\triangle ABC$.
Then:
:$AD$ is the [[Definition:Angle Bisector|angle bisector]] of $A$
:$BD$ is the [[Definition:Angle Bisector|angle bisector]] of $B$
:$... | :[[File:Euclid-IV-4.png|420px]] | Line from Vertex of Triangle to Incenter is Angle Bisector | https://proofwiki.org/wiki/Line_from_Vertex_of_Triangle_to_Incenter_is_Angle_Bisector | https://proofwiki.org/wiki/Line_from_Vertex_of_Triangle_to_Incenter_is_Angle_Bisector | [
"Incenters of Triangles",
"Angle Bisectors"
] | [
"Definition:Triangle (Geometry)",
"Definition:Point",
"Definition:Region",
"Definition:Angle Bisector",
"Definition:Angle Bisector",
"Definition:Angle Bisector",
"Definition:Incircle of Triangle/Incenter"
] | [
"File:Euclid-IV-4.png"
] |
proofwiki-20142 | Hopf-Rinow Theorem | Let $\struct {M, g}$ be a connected Riemannian manifold.
Then $M$ is metrically complete {{iff}} it is geodesically complete. | {{ProofWanted}}
{{Namedfor|Heinz Hopf|name2 = Willi Ludwig August Rinow|cat = Hopf|cat2 = Rinow}} | Let $\struct {M, g}$ be a [[Definition:Connected Manifold|connected]] [[Definition:Riemannian Manifold|Riemannian manifold]].
Then $M$ is [[Definition:Metrically Complete Connected Riemannian Manifold|metrically complete]] {{iff}} it is [[Definition:Geodesically Complete Semi-Riemannian Manifold|geodesically complete... | {{ProofWanted}}
{{Namedfor|Heinz Hopf|name2 = Willi Ludwig August Rinow|cat = Hopf|cat2 = Rinow}} | Hopf-Rinow Theorem | https://proofwiki.org/wiki/Hopf-Rinow_Theorem | https://proofwiki.org/wiki/Hopf-Rinow_Theorem | [
"Riemannian Manifolds"
] | [
"Definition:Connected Manifold",
"Definition:Riemannian Manifold",
"Definition:Metrically Complete Connected Riemannian Manifold",
"Definition:Geodesically Complete Semi-Riemannian Manifold"
] | [] |
proofwiki-20143 | Triangle is Orthic Triangle of Triangle formed from Excenters | Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.
Let $I$ be the incenter of $\triangle ABC$.
Let $I_a$, $I_b$ and $I_c$ be the excenters of $\triangle ABC$ {{WRT}} $a$, $b$ and $c$ respectively.
Let $\triangle I_a I_b I_c$ be the triangle whose vertices... | :560px
From Construction of Excircle to Triangle, it is seen that:
:$A I_b$ is the angle bisector of $\angle PAC$
:$A I_c$ is the angle bisector of $\angle QAB$.
Hence $I_b A I_c$ is a straight line.
From the construction in Excenters and Incenter of Orthic Triangle, $A I I_a$ is a straight line which is perpendicular ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$ and $c$ [[Definition:Opposite (in Triangle)|opposite]] [[Definition:Vertex of Polygon|vertices]] $A$, $B$ and $C$ respectively.
Let $I$ be the [[Definition:Incenter of Triangle|incenter]] of $\t... | :[[File:Orthic-Triangle-of-Excenters.png|560px]]
From [[Construction of Excircle to Triangle]], it is seen that:
:$A I_b$ is the [[Definition:Angle Bisector|angle bisector]] of $\angle PAC$
:$A I_c$ is the [[Definition:Angle Bisector|angle bisector]] of $\angle QAB$.
Hence $I_b A I_c$ is a [[Definition:Straight Line|... | Triangle is Orthic Triangle of Triangle formed from Excenters | https://proofwiki.org/wiki/Triangle_is_Orthic_Triangle_of_Triangle_formed_from_Excenters | https://proofwiki.org/wiki/Triangle_is_Orthic_Triangle_of_Triangle_formed_from_Excenters | [
"Incenters of Triangles",
"Excenters of Triangles",
"Orthic Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Vertex",
"Definition:Incircle of Triangle/Incenter",
"Definition:Excircle of Triangle/Excenter",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Vertex",
"Definition:Exci... | [
"File:Orthic-Triangle-of-Excenters.png",
"Construction of Excircle to Triangle",
"Definition:Angle Bisector",
"Definition:Angle Bisector",
"Definition:Line/Straight Line",
"Excenters and Incenter of Orthic Triangle",
"Definition:Line/Straight Line",
"Definition:Right Angle/Perpendicular",
"Definitio... |
proofwiki-20144 | Angle Bisectors are Locus of Points Equidistant from Lines | Let $\LL_1$ and $\LL_2$ be straight lines in the plane.
The locus of points which are equidistant from $\LL_1$ and $\LL_2$ are the angle bisectors of $\LL_1$ and $\LL_2$. | Let $A'SA$ and $B'SB$ be the straight lines $\LL_1$ and $\LL_2$ respectively, intersecting at the point $S$.
Let $E$ denote the set of points equidistant from both $\LL_1$ and $\LL_2$.
Let $F$ denote the set of points on the angle bisectors of $\LL_1$ and $\LL_2$.
We are to show that $E = F$.
:500px
First we show that ... | Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] in [[Definition:The Plane|the plane]].
The [[Definition:Locus|locus]] of [[Definition:Point|points]] which are [[Definition:Equidistant|equidistant]] from $\LL_1$ and $\LL_2$ are the [[Definition:Angle Bisector|angle bisectors]] of $\LL_1$ and $\LL... | Let $A'SA$ and $B'SB$ be the [[Definition:Straight Line|straight lines]] $\LL_1$ and $\LL_2$ respectively, [[Definition:Intersection (Geometry)|intersecting]] at the [[Definition:Point|point]] $S$.
Let $E$ denote the [[Definition:Set|set]] of [[Definition:Point|points]] [[Definition:Equidistant|equidistant]] from both... | Angle Bisectors are Locus of Points Equidistant from Lines | https://proofwiki.org/wiki/Angle_Bisectors_are_Locus_of_Points_Equidistant_from_Lines | https://proofwiki.org/wiki/Angle_Bisectors_are_Locus_of_Points_Equidistant_from_Lines | [
"Angle Bisectors"
] | [
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Locus",
"Definition:Point",
"Definition:Equidistant",
"Definition:Angle Bisector"
] | [
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Definition:Point",
"Definition:Set",
"Definition:Point",
"Definition:Equidistant",
"Definition:Set",
"Definition:Point",
"Definition:Angle Bisector",
"File:Bisectors-of-angles.png",
"Definition:Angle Bisector",
"Definition... |
proofwiki-20145 | Perpendicular Bisector is Locus of Points Equidistant from Endpoints | Let $AB$ be a straight line segment.
The locus of points which are equidistant from $A$ and $B$ is the perpendicular bisector of $AB$. | Let $E$ denote the set of points equidistant from $A$ and $B$.
Let $F$ denote the set of points on the perpendicular bisector of $AB$.
We are to show that $E = F$.
:300px
First we show that $F \subseteq E$.
Let $P \in F$.
Hence by definition $P$ is on the perpendicular bisector of $AB$.
Let $C$ be the intersection of $... | Let $AB$ be a [[Definition:Straight Line Segment|straight line segment]].
The [[Definition:Locus|locus]] of [[Definition:Point|points]] which are [[Definition:Equidistant|equidistant]] from $A$ and $B$ is the [[Definition:Perpendicular Bisector|perpendicular bisector]] of $AB$. | Let $E$ denote the [[Definition:Set|set]] of [[Definition:Point|points]] [[Definition:Equidistant|equidistant]] from $A$ and $B$.
Let $F$ denote the [[Definition:Set|set]] of [[Definition:Point|points]] on the [[Definition:Perpendicular Bisector|perpendicular bisector]] of $AB$.
We are to show that $E = F$.
:[[File... | Perpendicular Bisector is Locus of Points Equidistant from Endpoints | https://proofwiki.org/wiki/Perpendicular_Bisector_is_Locus_of_Points_Equidistant_from_Endpoints | https://proofwiki.org/wiki/Perpendicular_Bisector_is_Locus_of_Points_Equidistant_from_Endpoints | [
"Equidistant",
"Perpendicular Bisectors"
] | [
"Definition:Line/Straight Line Segment",
"Definition:Locus",
"Definition:Point",
"Definition:Equidistant",
"Definition:Perpendicular Bisector"
] | [
"Definition:Set",
"Definition:Point",
"Definition:Equidistant",
"Definition:Set",
"Definition:Point",
"Definition:Perpendicular Bisector",
"File:Perpendicular-Bisector-Equidistance.png",
"Definition:Perpendicular Bisector",
"Definition:Intersection (Geometry)",
"Definition:Perpendicular Bisector",... |
proofwiki-20146 | Polygon has Salient Angle | Let $P$ be a polygon.
Then $P$ has at least one salient angle. | Recall the definition of salient angle:
:A salient angle is an internal angle which is less than $180 \degrees$.
An internal angle which is not a salient angle is a re-entrant angle.
Let $C$ be a circle such that all vertices of $P$ lie within the interior of $C$.
Let $A$ be the vertex of $P$ that has the minimal dista... | Let $P$ be a [[Definition:Polygon|polygon]].
Then $P$ has at least one [[Definition:Salient Angle|salient angle]]. | Recall the definition of [[Definition:Salient Angle|salient angle]]:
:A [[Definition:Salient Angle|salient angle]] is an [[Definition:Internal Angle|internal angle]] which is less than $180 \degrees$.
An [[Definition:Internal Angle|internal angle]] which is not a [[Definition:Salient Angle|salient angle]] is a [[Defin... | Polygon has Salient Angle | https://proofwiki.org/wiki/Polygon_has_Salient_Angle | https://proofwiki.org/wiki/Polygon_has_Salient_Angle | [
"Polygons",
"Internal Angles"
] | [
"Definition:Polygon",
"Definition:Salient Angle"
] | [
"Definition:Salient Angle",
"Definition:Salient Angle",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Internal Angle",
"Definition:Salient Angle",
"Definition:Re-entrant Angle",
"Definition:Circle",
"Definition:Polygon/Vertex",
"Definition:Region",
"Definition:Polygon/Vertex",
"Defini... |
proofwiki-20147 | Mean Ergodic Theorem | Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.
Let $\map {L^2_\C} \mu$ be the complex-valued $L^2$ space of $\mu$.
Let $U_T : \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ of $T$ be the Koopman operator.
Let $I := \set {f \in \map {L^2_\C} \mu : \map {U_T} f = f}$.
Then for each $f \in \map {L^2_\C... | Recall $L^2$ space forms Hilbert space.
That is, $\map {L^2_\C} \mu$ is a Hilbert space with the $L^2$ inner product:
:$\ds \innerprod f g := \int f \; \overline g \rd \mu$
Let $\norm \cdot$ be the norm of $\map {L^2_\C} \mu$.
By Koopman Operator is Isometry:
:$\forall f \in \map {L^2_\C} \mu : \norm {\map U f} = \norm... | Let $\struct {X, \BB, \mu, T}$ be a [[Definition:Measure-Preserving Dynamical System|measure-preserving dynamical system]].
Let $\map {L^2_\C} \mu$ be the [[Definition:Complex-Valued Function|complex-valued]] [[Definition:Lp Space|$L^2$ space]] of $\mu$.
Let $U_T : \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ of $T$ be t... | Recall [[L-2 Space forms Hilbert Space|$L^2$ space forms Hilbert space]].
That is, $\map {L^2_\C} \mu$ is a [[Definition:Hilbert Space|Hilbert space]] with the [[Definition:L-2 Inner Product|$L^2$ inner product]]:
:$\ds \innerprod f g := \int f \; \overline g \rd \mu$
Let $\norm \cdot$ be the [[Definition:Inner Prod... | Mean Ergodic Theorem | https://proofwiki.org/wiki/Mean_Ergodic_Theorem | https://proofwiki.org/wiki/Mean_Ergodic_Theorem | [
"Mean Ergodic Theorem",
"Ergodic Theory",
"Operator Theory"
] | [
"Definition:Measure-Preserving Dynamical System",
"Definition:Complex-Valued Function",
"Definition:Lp Space",
"Definition:Koopman Operator on Complex L-2 Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Lp Norm",
"Definition:Composition of Mappings",
"Definition:Orthogonal Pr... | [
"L-2 Space forms Hilbert Space",
"Definition:Hilbert Space",
"Definition:L-2 Inner Product",
"Definition:Inner Product Norm",
"Koopman Operator is Isometry",
"Mean Ergodic Theorem (Hilbert Space)"
] |
proofwiki-20148 | Medians of Triangle Meet at Centroid/Corollary | Let $AA'$ be produced beyond $BC$ to $X$, where $A'X = A'G$.
Then the straight lines $BX$ and $CX$ are parallel to $CC'$ and $BB'$ respectively, and $\dfrac 2 3$ of their length. | {{improve|modify the diagram to add $C'$ to it}}
:360px
Consider the quadrilateral $\Box BGCX$.
Its diagonals are $GX$ and $BC$.
By construction, they bisect each other.
From Quadrilateral with Bisecting Diagonals is Parallelogram, $\Box BGCX$ is a parallelogram.
From Position of Centroid of Triangle on Median:
:$BG$ i... | Let $AA'$ be [[Definition:Production|produced]] beyond $BC$ to $X$, where $A'X = A'G$.
Then the [[Definition:Straight Line|straight lines]] $BX$ and $CX$ are [[Definition:Parallel Lines|parallel]] to $CC'$ and $BB'$ respectively, and $\dfrac 2 3$ of their [[Definition:Length of Line|length]]. | {{improve|modify the diagram to add $C'$ to it}}
:[[File:Medians-meet-at-Centroid.png|360px]]
Consider the [[Definition:Quadrilateral|quadrilateral]] $\Box BGCX$.
Its [[Definition:Diagonal of Parallelogram|diagonals]] are $GX$ and $BC$.
By construction, they [[Definition:Bisection|bisect]] each other.
From [[Quadri... | Medians of Triangle Meet at Centroid/Corollary | https://proofwiki.org/wiki/Medians_of_Triangle_Meet_at_Centroid/Corollary | https://proofwiki.org/wiki/Medians_of_Triangle_Meet_at_Centroid/Corollary | [
"Medians of Triangle Meet at Centroid"
] | [
"Definition:Production",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Linear Measure/Length"
] | [
"File:Medians-meet-at-Centroid.png",
"Definition:Quadrilateral",
"Definition:Diameter of Parallelogram",
"Definition:Bisection",
"Quadrilateral with Bisecting Diagonals is Parallelogram",
"Definition:Quadrilateral/Parallelogram",
"Position of Centroid of Triangle on Median",
"Definition:Linear Measure... |
proofwiki-20149 | External Center of Similitude of Circles with respect to Radii | Let $A$ and $B$ be the centers of two circles $\bigcirc Ar$ and $\bigcirc BR$ whose radii are $r$ and $R$ respectively, $r \ne R$.
Let $\bigcirc Ar$ and $\bigcirc BR$ be such that neither is completely enclosed inside the other.
Let $T$ be the external center of similitude of $\bigcirc Ar$ and $\bigcirc BR$.
Let $P$ an... | :500px | Let $A$ and $B$ be the [[Definition:Center of Circle|centers]] of two [[Definition:Circle|circles]] $\bigcirc Ar$ and $\bigcirc BR$ whose [[Definition:Radius of Circle|radii]] are $r$ and $R$ respectively, $r \ne R$.
Let $\bigcirc Ar$ and $\bigcirc BR$ be such that neither is completely enclosed inside the other.
Let... | :[[File:External-Center-of-Similitude.png|500px]] | External Center of Similitude of Circles with respect to Radii | https://proofwiki.org/wiki/External_Center_of_Similitude_of_Circles_with_respect_to_Radii | https://proofwiki.org/wiki/External_Center_of_Similitude_of_Circles_with_respect_to_Radii | [
"Centers of Similitude"
] | [
"Definition:Circle/Center",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:External Center of Similitude",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle/Radius",
"Definition:Parallel (Geometry)/Lines",
"Definition:Collinear"
] | [
"File:External-Center-of-Similitude.png"
] |
proofwiki-20150 | Area of Regular Hexagon | Let $H$ be a regular hexagon.
Let the length of one side of $H$ be $s$.
Let $\AA$ be the area of $H$.
Then:
:$\AA = \dfrac {3 \sqrt 3} 2 s^2$ | From Regular Hexagon is composed of Equilateral Triangles, it follows that a regular hexagon can be dissected into six congruent equilateral triangles:
:300px
Let $\AA_T$ be the area of the bottom triangle.
Then by Area of Equilateral Triangle:
: $ \AA_T = \dfrac{\sqrt 3} 4 s^2 $
As $H$ consists of six congruent trian... | Let $H$ be a [[Definition:Regular Hexagon|regular hexagon]].
Let the [[Definition:Length (Linear Measure)|length]] of one [[Definition:Side of Polygon|side]] of $H$ be $s$.
Let $\AA$ be the [[Definition:Area|area]] of $H$.
Then:
:$\AA = \dfrac {3 \sqrt 3} 2 s^2$ | From [[Regular Hexagon is composed of Equilateral Triangles]], it follows that a [[Definition:Regular Hexagon|regular hexagon]] can be [[Definition:Dissection|dissected]] into six [[Definition:Congruence (Geometry)|congruent]] [[Definition:Equilateral Triangle|equilateral triangles]]:
:[[File:Regular Hexagon.svg|300px... | Area of Regular Hexagon/Proof 1 | https://proofwiki.org/wiki/Area_of_Regular_Hexagon | https://proofwiki.org/wiki/Area_of_Regular_Hexagon/Proof_1 | [
"Area of Regular Hexagon",
"Hexagons"
] | [
"Definition:Hexagon/Regular",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Area"
] | [
"Regular Hexagon is composed of Equilateral Triangles",
"Definition:Hexagon/Regular",
"Definition:Dissection",
"Definition:Congruence (Geometry)",
"Definition:Triangle (Geometry)/Equilateral",
"File:Regular Hexagon.svg",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Area of Equilateral Trian... |
proofwiki-20151 | Area of Regular Hexagon | Let $H$ be a regular hexagon.
Let the length of one side of $H$ be $s$.
Let $\AA$ be the area of $H$.
Then:
:$\AA = \dfrac {3 \sqrt 3} 2 s^2$ | A regular hexagon is a regular 6-sided polygon.
Therefore:
{{begin-eqn}}
{{eqn | l = \AA
| r = \dfrac 1 4 \times 6 \times s^2 \times \cot \dfrac \pi 6
| c = Area of Regular Polygon
}}
{{eqn | r = \dfrac 3 2 \times s^2 \times \sqrt 3
| c = Cotangent of $30 \degrees$
}}
{{eqn | r = \dfrac {3 \sqrt 3} 2 ... | Let $H$ be a [[Definition:Regular Hexagon|regular hexagon]].
Let the [[Definition:Length (Linear Measure)|length]] of one [[Definition:Side of Polygon|side]] of $H$ be $s$.
Let $\AA$ be the [[Definition:Area|area]] of $H$.
Then:
:$\AA = \dfrac {3 \sqrt 3} 2 s^2$ | A [[Definition:Regular Hexagon|regular hexagon]] is a [[Definition:Regular Polygon|regular]] [[Definition:Multilateral Polygon|6-sided polygon]].
Therefore:
{{begin-eqn}}
{{eqn | l = \AA
| r = \dfrac 1 4 \times 6 \times s^2 \times \cot \dfrac \pi 6
| c = [[Area of Regular Polygon]]
}}
{{eqn | r = \dfrac 3... | Area of Regular Hexagon/Proof 2 | https://proofwiki.org/wiki/Area_of_Regular_Hexagon | https://proofwiki.org/wiki/Area_of_Regular_Hexagon/Proof_2 | [
"Area of Regular Hexagon",
"Hexagons"
] | [
"Definition:Hexagon/Regular",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Area"
] | [
"Definition:Hexagon/Regular",
"Definition:Polygon/Regular",
"Definition:Polygon/Multilateral",
"Area of Regular Polygon",
"Cotangent of 30 Degrees"
] |
proofwiki-20152 | Product Rule for Derivatives/General Result/3 Factors | Let $\map u x$, $\map v x$ and $\map w x$ be real functions differentiable on the open interval $I$.
Then:
:$\forall x \in I: \map {\dfrac \d {\d x} } {u v w} = u v \dfrac {\d w} {\d x} + u w \dfrac {\d v} {\d x} + v w \dfrac {\d u} {\d x}$ | Let $y = u v w$.
Then:
{{begin-eqn}}
{{eqn | q =
| l = y
| r = u \paren {v w}
| c =
}}
{{eqn | r = \dfrac {\d u} {\d x} \paren {v w} + u \map {\dfrac \d {\d x} } {v w}
| c = Product Rule for Derivatives
}}
{{eqn | r = v w \dfrac {\d u} {\d x} + u \paren {w \dfrac {\d v} {\d x} + v \dfrac {\d w... | Let $\map u x$, $\map v x$ and $\map w x$ be [[Definition:Real Function|real functions]] [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $I$.
Then:
:$\forall x \in I: \map {\dfrac \d {\d x} } {u v w} = u v \dfrac {\d w} {\d x} + u w \dfrac {\d v} {\d x} +... | Let $y = u v w$.
Then:
{{begin-eqn}}
{{eqn | q =
| l = y
| r = u \paren {v w}
| c =
}}
{{eqn | r = \dfrac {\d u} {\d x} \paren {v w} + u \map {\dfrac \d {\d x} } {v w}
| c = [[Product Rule for Derivatives]]
}}
{{eqn | r = v w \dfrac {\d u} {\d x} + u \paren {w \dfrac {\d v} {\d x} + v \dfrac... | Product Rule for Derivatives/General Result/3 Factors | https://proofwiki.org/wiki/Product_Rule_for_Derivatives/General_Result/3_Factors | https://proofwiki.org/wiki/Product_Rule_for_Derivatives/General_Result/3_Factors | [
"Product Rule for Derivatives"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Product Rule for Derivatives",
"Product Rule for Derivatives"
] |
proofwiki-20153 | Median of Trapezium is Parallel to Bases/Sufficient Condition | Let $\Box ABCD$ be a trapezium such that $AB$ and $DC$ are the parallel sides.
:300px
Let $E$ be the midpoint of $AD$.
Let $F$ lie on $BC$.
Let $EF$ be parallel to both $AB$ and $DC$.
Then $F$ is the midpoint of $BC$. | :300px
Let $DH$ be constructed parallel to $BC$ to cut $AB$ at $H$.
From the Parallel Transversal Theorem:
:$DG : GH = DE : EA$
and so $G$ is the midpoint of $AH$.
That is:
:$(1): \quad DG = GH$
Then we have that:
:$DC$ is parallel to $GF$
and:
:$DG$ is parallel to $CF$
so, by definition, $\Box GFCD$ is a parallelogram... | Let $\Box ABCD$ be a [[Definition:Trapezium|trapezium]] such that $AB$ and $DC$ are the [[Definition:Parallel Lines|parallel]] [[Definition:Side of Polygon|sides]].
:[[File:Median-of-Trapezoid.png|300px]]
Let $E$ be the [[Definition:Midpoint of Line|midpoint]] of $AD$.
Let $F$ lie on $BC$.
Let $EF$ be [[Definition... | :[[File:Median-of-Trapezoid-Proof.png|300px]]
Let $DH$ be constructed [[Definition:Parallel Lines|parallel]] to $BC$ to cut $AB$ at $H$.
From the [[Parallel Transversal Theorem]]:
:$DG : GH = DE : EA$
and so $G$ is the [[Definition:Midpoint of Line|midpoint]] of $AH$.
That is:
:$(1): \quad DG = GH$
Then we have t... | Median of Trapezium is Parallel to Bases/Sufficient Condition | https://proofwiki.org/wiki/Median_of_Trapezium_is_Parallel_to_Bases/Sufficient_Condition | https://proofwiki.org/wiki/Median_of_Trapezium_is_Parallel_to_Bases/Sufficient_Condition | [
"Median of Trapezium is Parallel to Bases"
] | [
"Definition:Quadrilateral/Trapezium",
"Definition:Parallel (Geometry)/Lines",
"Definition:Polygon/Side",
"File:Median-of-Trapezoid.png",
"Definition:Line/Midpoint",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Midpoint"
] | [
"File:Median-of-Trapezoid-Proof.png",
"Definition:Parallel (Geometry)/Lines",
"Parallel Transversal Theorem",
"Definition:Line/Midpoint",
"Definition:Parallel (Geometry)/Lines",
"Definition:Parallel (Geometry)/Lines",
"Definition:Quadrilateral/Parallelogram",
"Definition:Parallel (Geometry)/Lines",
... |
proofwiki-20154 | Median of Trapezium is Parallel to Bases | Let $\Box ABCD$ be a trapezium such that $AB$ and $DC$ are the bases.
:300px
Let $E$ be the midpoint of $AD$.
Let $F$ lie on $BC$.
Then:
:$EF$ is parallel to both $AB$ and $DC$
{{iff}}:
:$F$ is the midpoint of $BC$.
That is, the median of $\Box ABCD$ is parallel to the bases of $\Box ABCD$. | === Sufficient Condition ===
{{:Median of Trapezium is Parallel to Bases/Sufficient Condition}}{{qed|lemma}} | Let $\Box ABCD$ be a [[Definition:Trapezium|trapezium]] such that $AB$ and $DC$ are the [[Definition:Base of Trapezium|bases]].
:[[File:Median-of-Trapezoid.png|300px]]
Let $E$ be the [[Definition:Midpoint of Line|midpoint]] of $AD$.
Let $F$ lie on $BC$.
Then:
:$EF$ is [[Definition:Parallel Lines|parallel]] to both... | === [[Median of Trapezium is Parallel to Bases/Sufficient Condition|Sufficient Condition]] ===
{{:Median of Trapezium is Parallel to Bases/Sufficient Condition}}{{qed|lemma}} | Median of Trapezium is Parallel to Bases | https://proofwiki.org/wiki/Median_of_Trapezium_is_Parallel_to_Bases | https://proofwiki.org/wiki/Median_of_Trapezium_is_Parallel_to_Bases | [
"Median of Trapezium is Parallel to Bases",
"Medians of Trapezia"
] | [
"Definition:Quadrilateral/Trapezium",
"Definition:Quadrilateral/Trapezium/Base",
"File:Median-of-Trapezoid.png",
"Definition:Line/Midpoint",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Midpoint",
"Definition:Median of Trapezium",
"Definition:Parallel (Geometry)/Lines",
"Definition:Quadr... | [
"Median of Trapezium is Parallel to Bases/Sufficient Condition"
] |
proofwiki-20155 | Median of Trapezium is Parallel to Bases/Necessary Condition | Let $\Box ABCD$ be a trapezium such that $AB$ and $DC$ are the parallel sides.
:300px
Let $E$ be the midpoint of $AD$.
Let $F$ be the midpoint of $BC$.
Then $EF$ is parallel to both $AB$ and $DC$. | :300px
{{AimForCont}} $EF$ is not parallel to $DC$.
By Playfair's axiom, there exists a unique straight line through $E$ which ''is'' parallel to $DC$.
Let $EF'$ be this line.
From Median of Trapezium is Parallel to Bases: Sufficient Condition, $F'$ is the midpoint of $BC$.
But {{hypothesis}} $F$ is also the midpoint o... | Let $\Box ABCD$ be a [[Definition:Trapezium|trapezium]] such that $AB$ and $DC$ are the [[Definition:Parallel Lines|parallel]] [[Definition:Side of Polygon|sides]].
:[[File:Median-of-Trapezoid.png|300px]]
Let $E$ be the [[Definition:Midpoint of Line|midpoint]] of $AD$.
Let $F$ be the [[Definition:Midpoint of Line|mi... | :[[File:Median-of-Trapezoid-Proof-2.png|300px]]
{{AimForCont}} $EF$ is not [[Definition:Parallel Lines|parallel]] to $DC$.
By [[Axiom:Playfair's Axiom|Playfair's axiom]], there exists a [[Definition:Unique|unique]] [[Definition:Straight Line|straight line]] through $E$ which ''is'' [[Definition:Parallel Lines|paralle... | Median of Trapezium is Parallel to Bases/Necessary Condition | https://proofwiki.org/wiki/Median_of_Trapezium_is_Parallel_to_Bases/Necessary_Condition | https://proofwiki.org/wiki/Median_of_Trapezium_is_Parallel_to_Bases/Necessary_Condition | [
"Median of Trapezium is Parallel to Bases"
] | [
"Definition:Quadrilateral/Trapezium",
"Definition:Parallel (Geometry)/Lines",
"Definition:Polygon/Side",
"File:Median-of-Trapezoid.png",
"Definition:Line/Midpoint",
"Definition:Line/Midpoint",
"Definition:Parallel (Geometry)/Lines"
] | [
"File:Median-of-Trapezoid-Proof-2.png",
"Definition:Parallel (Geometry)/Lines",
"Axiom:Playfair's Axiom",
"Definition:Unique",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Median of Trapezium is Parallel to Bases/Sufficient Condition",
"D... |
proofwiki-20156 | Probability Density Function of Exponential Distribution | Let $X$ be a continuous random variable with the exponential distribution with parameter $\beta$.
Then the probability density function of $X$ is given by:
:$\map {f_X} x = \begin{cases} \dfrac 1 \beta e^{-\frac x \beta} & : x \ge 0 \\ 0 & : \text{otherwise} \end{cases}$ | By definition of exponential distribution:
:$\map {F_X} \Omega = \R_{\ge 0}$
:$\map \Pr {X < x} = 1 - e^{-\frac x \beta}$
where $0 < \beta$.
By definition of probability density function:
:$\forall x \in \R: \map {f_X} x = \begin {cases} \map {F_X'} x & : x \in \Sigma \\ 0 & : x \notin \Sigma \end {cases}$
where $\map ... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Exponential Distribution|exponential distribution with parameter $\beta$]].
Then the [[Definition:Probability Density Function|probability density function]] of $X$ is given by:
:$\map {f_X} x = \begin{cases} \dfra... | By definition of [[Definition:Exponential Distribution|exponential distribution]]:
:$\map {F_X} \Omega = \R_{\ge 0}$
:$\map \Pr {X < x} = 1 - e^{-\frac x \beta}$
where $0 < \beta$.
By definition of [[Definition:Probability Density Function|probability density function]]:
:$\forall x \in \R: \map {f_X} x = \begin {c... | Probability Density Function of Exponential Distribution | https://proofwiki.org/wiki/Probability_Density_Function_of_Exponential_Distribution | https://proofwiki.org/wiki/Probability_Density_Function_of_Exponential_Distribution | [
"Probability Density Function of Exponential Distribution",
"Exponential Distribution",
"Examples of Probability Density Functions"
] | [
"Definition:Random Variable/Continuous",
"Definition:Exponential Distribution",
"Definition:Probability Density Function"
] | [
"Definition:Exponential Distribution",
"Definition:Probability Density Function",
"Definition:Derivative/Real Function"
] |
proofwiki-20157 | Primitive of Reciprocal of Root of a minus x by Cube of Root of x minus b | :$\ds \int \dfrac {\d x} {\paren {a - x}^{1/2} \paren {x - b}^{3/2} } = \dfrac 2 {b - a} \sqrt {\dfrac {a - x} {x - b} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cos^2 \theta + b \sin^2 \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d \theta}
| r = 2 a \cos \theta \paren {-\sin \theta} + 2 b \sin \theta \cos \theta
| c = Chain Rule for Derivatives, Derivative of Cosine Function, Derivative of S... | :$\ds \int \dfrac {\d x} {\paren {a - x}^{1/2} \paren {x - b}^{3/2} } = \dfrac 2 {b - a} \sqrt {\dfrac {a - x} {x - b} } + C$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cos^2 \theta + b \sin^2 \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d \theta}
| r = 2 a \cos \theta \paren {-\sin \theta} + 2 b \sin \theta \cos \theta
| c = [[Chain Rule for Derivatives]], [[Derivative of Cosine Function]], [[Deri... | Primitive of Reciprocal of Root of a minus x by Cube of Root of x minus b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_minus_x_by_Cube_of_Root_of_x_minus_b | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_minus_x_by_Cube_of_Root_of_x_minus_b/Proof_1 | [
"Primitive of Reciprocal of Root of a minus x by Cube of Root of x minus b",
"Primitives involving Root of a x + b and Root of p x + q"
] | [] | [
"Derivative of Composite Function",
"Derivative of Cosine Function",
"Derivative of Sine Function",
"Sum of Squares of Sine and Cosine",
"Sum of Squares of Sine and Cosine",
"Primitive of Square of Secant Function"
] |
proofwiki-20158 | Equivalence of Definitions of Simple Connectedness | Let $T = \struct{S, \tau}$ be a path-connected topological space.
{{TFAE| def = Simply Connected}} | === Definition by fundamental group implies Definition by path-homotopy of loops ===
Let $x \in S$.
From Fundamental Group is Independent of Base Point for Path-Connected Space, it follows that all fundamental groups $\map {\pi_1}{T, x}$ are isomorphic to one group denoted $\map {\pi_1}{ T }$.
By assumption, it follows... | Let $T = \struct{S, \tau}$ be a [[Definition:Path-Connected Space|path-connected]] [[Definition:Topological Space|topological space]].
{{TFAE| def = Simply Connected}} | === [[Definition:Simply Connected/Definition 1|Definition by fundamental group]] implies [[Definition:Simply Connected/Definition 2|Definition by path-homotopy of loops]] ===
Let $x \in S$.
From [[Fundamental Group is Independent of Base Point for Path-Connected Space]], it follows that all [[Definition:Fundamental G... | Equivalence of Definitions of Simple Connectedness | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Simple_Connectedness | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Simple_Connectedness | [
"Simply Connected Spaces"
] | [
"Definition:Path-Connected/Topological Space",
"Definition:Topological Space"
] | [
"Definition:Simply Connected/Definition 1",
"Definition:Simply Connected/Definition 2",
"Fundamental Group is Independent of Base Point for Path-Connected Space",
"Definition:Fundamental Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Group",
"Definition:Trivial Group"... |
proofwiki-20159 | Primitive of x squared by Exponential of x | :$\ds \int x^2 e^x \rd x = e^x \paren {x^2 - 2 x + 2} + C$ | From Primitive of $x^2 e^{a x}$:
{{:Primitive of x squared by Exponential of a x}}
The result follows by setting $a = 1$.
{{qed}} | :$\ds \int x^2 e^x \rd x = e^x \paren {x^2 - 2 x + 2} + C$ | From [[Primitive of x squared by Exponential of a x|Primitive of $x^2 e^{a x}$]]:
{{:Primitive of x squared by Exponential of a x}}
The result follows by setting $a = 1$.
{{qed}} | Primitive of x squared by Exponential of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Exponential_of_x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Exponential_of_x/Proof_1 | [
"Primitive of x squared by Exponential of x",
"Primitives involving Exponential Function",
"Primitive of x squared by Exponential of x",
"Primitives involving Exponential Function"
] | [] | [
"Primitive of x squared by Exponential of a x"
] |
proofwiki-20160 | Primitive of x squared by Exponential of x | :$\ds \int x^2 e^x \rd x = e^x \paren {x^2 - 2 x + 2} + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = Derivative of Power
}}
{{end-eqn}}
and l... | :$\ds \int x^2 e^x \rd x = e^x \paren {x^2 - 2 x + 2} + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = x^2
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = 2 x
| c = [[Derivative o... | Primitive of x squared by Exponential of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Exponential_of_x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Exponential_of_x/Proof_2 | [
"Primitive of x squared by Exponential of x",
"Primitives involving Exponential Function",
"Primitive of x squared by Exponential of x",
"Primitives involving Exponential Function"
] | [] | [
"Definition:Primitive",
"Power Rule for Derivatives",
"Primitive of Exponential Function",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x by Exponential of a x"
] |
proofwiki-20161 | Primitive of x by Arctangent of x | :$\ds \int x \arctan x \rd x = \frac {x^2 + 1} 2 \arctan x - \frac x 2 + C$ | From Primitive of $x \arctan \dfrac x a$:
{{:Primitive of x by Arctangent of x over a}}
The result follows on setting $a = 1$.
{{qed}} | :$\ds \int x \arctan x \rd x = \frac {x^2 + 1} 2 \arctan x - \frac x 2 + C$ | From [[Primitive of x by Arctangent of x over a|Primitive of $x \arctan \dfrac x a$]]:
{{:Primitive of x by Arctangent of x over a}}
The result follows on setting $a = 1$.
{{qed}} | Primitive of x by Arctangent of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_x_by_Arctangent_of_x | https://proofwiki.org/wiki/Primitive_of_x_by_Arctangent_of_x/Proof_1 | [
"Primitive of x by Arctangent of x",
"Primitives involving Inverse Tangent Function"
] | [] | [
"Primitive of x by Arctangent of x over a"
] |
proofwiki-20162 | Primitive of x by Arctangent of x | :$\ds \int x \arctan x \rd x = \frac {x^2 + 1} 2 \arctan x - \frac x 2 + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {x^2 + 1}
| c = Derivative of $\arct... | :$\ds \int x \arctan x \rd x = \frac {x^2 + 1} 2 \arctan x - \frac x 2 + C$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arctan x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac 1 {x^2 + 1}
... | Primitive of x by Arctangent of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_x_by_Arctangent_of_x | https://proofwiki.org/wiki/Primitive_of_x_by_Arctangent_of_x/Proof_2 | [
"Primitive of x by Arctangent of x",
"Primitives involving Inverse Tangent Function"
] | [] | [
"Definition:Primitive",
"Derivative of Arctangent Function",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of x squared over x squared plus a squared"
] |
proofwiki-20163 | Primitive of Cube of Secant Function | :$\ds \int \sec^3 x \rd x = \frac 1 2 \paren {\sec x \tan x + \ln \size {\sec x + \tan x} } + C$ | From Primitive of $\sec^3 a x$:
{{:Primitive of Cube of Secant of a x}}
The result follows on setting $a = 1$.
{{qed}} | :$\ds \int \sec^3 x \rd x = \frac 1 2 \paren {\sec x \tan x + \ln \size {\sec x + \tan x} } + C$ | From [[Primitive of Cube of Secant of a x|Primitive of $\sec^3 a x$]]:
{{:Primitive of Cube of Secant of a x}}
The result follows on setting $a = 1$.
{{qed}} | Primitive of Cube of Secant Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Secant_Function | https://proofwiki.org/wiki/Primitive_of_Cube_of_Secant_Function/Proof_1 | [
"Primitive of Cube of Secant Function",
"Primitives involving Secant Function",
"Primitive of Cube of Secant Function",
"Primitives involving Secant Function"
] | [] | [
"Primitive of Cube of Secant of a x"
] |
proofwiki-20164 | Primitive of Cube of Secant Function | :$\ds \int \sec^3 x \rd x = \frac 1 2 \paren {\sec x \tan x + \ln \size {\sec x + \tan x} } + C$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sec a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sec a x \tan a x
| c = Derivative of Funct... | :$\ds \int \sec^3 x \rd x = \frac 1 2 \paren {\sec x \tan x + \ln \size {\sec x + \tan x} } + C$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sec a x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = a \sec a x \ta... | Primitive of Cube of Secant Function/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Secant_Function | https://proofwiki.org/wiki/Primitive_of_Cube_of_Secant_Function/Proof_2 | [
"Primitive of Cube of Secant Function",
"Primitives involving Secant Function",
"Primitive of Cube of Secant Function",
"Primitives involving Secant Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Derivative of Function of Constant Multiple",
"Derivative of Secant Function",
"Primitive of Square of Secant Function",
"Integration by Parts",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Linear Combination of Integrals/Indefinite",
"Primitive of Secant Funct... |
proofwiki-20165 | Primitive of Composite Function/Corollary | Let $f$ and $g$ be a real functions which are integrable.
Let the composite function $g \circ f$ also be integrable.
Then:
{{begin-eqn}}
{{eqn | l = \int \map {\paren {g \circ f} } x \map {f'} x \rd x
| r = \int \map g u \rd u
| c =
}}
{{end-eqn}}
where $u = \map f x$. | {{begin-eqn}}
{{eqn | l = \map F x
| r = \int \map {\paren {g \circ f} } x \map {f'} x \rd x
}}
{{eqn | r = \int \map g {\map f x} \map {f'} x \rd x
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \int \map g u \map {f'} x \rd x
| c = where $u = \map f x$
}}
{{eqn | ll= \leadsto
| l = \fr... | Let $f$ and $g$ be a [[Definition:Real Function|real functions]] which are [[Definition:Integrable Function|integrable]].
Let the [[Definition:Composition of Mappings|composite function]] $g \circ f$ also be [[Definition:Integrable Function|integrable]].
Then:
{{begin-eqn}}
{{eqn | l = \int \map {\paren {g \circ f} ... | {{begin-eqn}}
{{eqn | l = \map F x
| r = \int \map {\paren {g \circ f} } x \map {f'} x \rd x
}}
{{eqn | r = \int \map g {\map f x} \map {f'} x \rd x
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \int \map g u \map {f'} x \rd x
| c = where $u = \map f x$
}}
{{eqn | ll= \leadsto
| l = \fr... | Primitive of Composite Function/Corollary | https://proofwiki.org/wiki/Primitive_of_Composite_Function/Corollary | https://proofwiki.org/wiki/Primitive_of_Composite_Function/Corollary | [
"Primitive of Composite Function"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Composition of Mappings",
"Definition:Integrable Function"
] | [
"Derivative of Composite Function",
"Category:Primitive of Composite Function"
] |
proofwiki-20166 | Primitive of Cube of Sine Function | :$\ds \int \sin^3 x \rd x = \frac {\cos^3 x} 3 - \cos x + C$ | From Primitive of $\sin^3 a x$:
{{:Primitive of Cube of Sine of a x}}
The result follows by setting $a = 1$.
{{qed}} | :$\ds \int \sin^3 x \rd x = \frac {\cos^3 x} 3 - \cos x + C$ | From [[Primitive of Cube of Sine of a x|Primitive of $\sin^3 a x$]]:
{{:Primitive of Cube of Sine of a x}}
The result follows by setting $a = 1$.
{{qed}} | Primitive of Cube of Sine Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Cube_of_Sine_Function | https://proofwiki.org/wiki/Primitive_of_Cube_of_Sine_Function/Proof_1 | [
"Primitive of Cube of Sine Function",
"Primitives involving Sine Function"
] | [] | [
"Primitive of Cube of Sine of a x"
] |
proofwiki-20167 | Primitive of Fourth Power of Cosine Function | :$\ds \int \cos^4 x \rd x = \frac {3 x} 8 + \frac {\sin 2 x} 4 + \frac {\sin 4 x} {32} + C$ | From Primitive of $\cos^4 a x$:
{{:Primitive of Fourth Power of Cosine of a x}}
The result follows by setting $a = 1$.
{{qed}} | :$\ds \int \cos^4 x \rd x = \frac {3 x} 8 + \frac {\sin 2 x} 4 + \frac {\sin 4 x} {32} + C$ | From [[Primitive of Fourth Power of Cosine of a x|Primitive of $\cos^4 a x$]]:
{{:Primitive of Fourth Power of Cosine of a x}}
The result follows by setting $a = 1$.
{{qed}} | Primitive of Fourth Power of Cosine Function | https://proofwiki.org/wiki/Primitive_of_Fourth_Power_of_Cosine_Function | https://proofwiki.org/wiki/Primitive_of_Fourth_Power_of_Cosine_Function | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Fourth Power of Cosine of a x"
] |
proofwiki-20168 | Sine of 22.5 Degrees | :$\sin 22.5 \degrees = \sin \dfrac \pi 8 = \dfrac 1 2 \sqrt {2 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sin 22.5 \degrees
| r = \sin \frac {45 \degrees} 2
| c =
}}
{{eqn | r = +\sqrt {\frac {1 - \cos 45 \degrees} 2}
| c = Half Angle Formula for Sine
}}
{{eqn | r = \sqrt {\frac {1 - \frac {\sqrt 2} 2} 2}
| c = {{cos|45}}
}}
{{eqn | r = \sqrt {\frac {2 - \sqrt 2} 4}
... | :$\sin 22.5 \degrees = \sin \dfrac \pi 8 = \dfrac 1 2 \sqrt {2 - \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \sin 22.5 \degrees
| r = \sin \frac {45 \degrees} 2
| c =
}}
{{eqn | r = +\sqrt {\frac {1 - \cos 45 \degrees} 2}
| c = [[Half Angle Formula for Sine]]
}}
{{eqn | r = \sqrt {\frac {1 - \frac {\sqrt 2} 2} 2}
| c = {{cos|45}}
}}
{{eqn | r = \sqrt {\frac {2 - \sqrt 2} 4}
... | Sine of 22.5 Degrees | https://proofwiki.org/wiki/Sine_of_22.5_Degrees | https://proofwiki.org/wiki/Sine_of_22.5_Degrees | [
"Sine Function"
] | [] | [
"Half Angle Formulas/Sine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Square Root"
] |
proofwiki-20169 | Cosine of 22.5 Degrees | :$\cos 22.5 \degrees = \cos \dfrac \pi 8 = \dfrac 1 2 \sqrt {2 + \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \cos 22.5 \degrees
| r = \cos \frac {45 \degrees} 2
| c =
}}
{{eqn | r = +\sqrt {\frac {1 + \cos 45 \degrees} 2}
| c = Half Angle Formula for Cosine
}}
{{eqn | r = \sqrt {\frac {1 + \frac {\sqrt 2} 2} 2}
| c = {{cos|45}}
}}
{{eqn | r = \sqrt {\frac {2 + \sqrt 2} 4}
... | :$\cos 22.5 \degrees = \cos \dfrac \pi 8 = \dfrac 1 2 \sqrt {2 + \sqrt 2}$ | {{begin-eqn}}
{{eqn | l = \cos 22.5 \degrees
| r = \cos \frac {45 \degrees} 2
| c =
}}
{{eqn | r = +\sqrt {\frac {1 + \cos 45 \degrees} 2}
| c = [[Half Angle Formula for Cosine]]
}}
{{eqn | r = \sqrt {\frac {1 + \frac {\sqrt 2} 2} 2}
| c = {{cos|45}}
}}
{{eqn | r = \sqrt {\frac {2 + \sqrt 2} 4}... | Cosine of 22.5 Degrees | https://proofwiki.org/wiki/Cosine_of_22.5_Degrees | https://proofwiki.org/wiki/Cosine_of_22.5_Degrees | [
"Cosine Function"
] | [] | [
"Half Angle Formulas/Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-20170 | Definite Integral from 0 to 1 of Even Powers of Logarithm of 1 - x over x | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
:$\ds \int_0^1 \map {\ln^{2n} } {\dfrac {1 - x} x} \rd x = \paren {-1}^{n + 1} B_{2 n} \paren {2^{2 n} - 2} \pi^{2 n}$
:where $B_{2 n}$ is the $2 n$th Bernoulli number. | let:
{{begin-eqn}}
{{eqn | l = \map \ln {\dfrac {1 - x} x}
| r = -u
| c = Integration by Substitution
}}
{{eqn | l = \dfrac {1 - x} x
| r = e^{-u}
| c =
}}
{{eqn | l = \dfrac 1 x - 1
| r = e^{-u}
| c =
}}
{{eqn | l = \dfrac 1 x
| r = 1 + e^{-u}
| c =
}}
{{eqn | l = x
... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
:$\ds \int_0^1 \map {\ln^{2n} } {\dfrac {1 - x} x} \rd x = \paren {-1}^{n + 1} B_{2 n} \paren {2^{2 n} - 2} \pi^{2 n}$
:where $B_{2 n}$ is the $2 n$th [[Definition:Bernoulli Numbers|Bernoulli number]]. | let:
{{begin-eqn}}
{{eqn | l = \map \ln {\dfrac {1 - x} x}
| r = -u
| c = [[Integration by Substitution]]
}}
{{eqn | l = \dfrac {1 - x} x
| r = e^{-u}
| c =
}}
{{eqn | l = \dfrac 1 x - 1
| r = e^{-u}
| c =
}}
{{eqn | l = \dfrac 1 x
| r = 1 + e^{-u}
| c =
}}
{{eqn | l = x
... | Definite Integral from 0 to 1 of Even Powers of Logarithm of 1 - x over x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Even_Powers_of_Logarithm_of_1_-_x_over_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Even_Powers_of_Logarithm_of_1_-_x_over_x | [
"Definite Integrals involving Logarithm Function",
"Bernoulli Numbers",
"Riemann Zeta Function at Even Integers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Bernoulli Numbers"
] | [
"Integration by Substitution",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Exponential Function/Corollary 1",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Infinite Geo... |
proofwiki-20171 | Primitive of Square Root | :$\ds \int \sqrt x \rd x = \dfrac {2 x^{3 / 2} } 3 = \dfrac {2 \sqrt x^3} 3$ | From Primitive of Power:
{{:Primitive of Power}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \sqrt x \rd x
| r = \int x^{1/2} \rd x
| c = {{Defof|Square Root}}
}}
{{eqn | r = \dfrac {x^{1/2 + 1} } {1/2 + 1} + C
| c = Primitive of Power
}}
{{eqn | r = \dfrac {x^{3/2} } {3/2} + C
| c = simplification
}}... | :$\ds \int \sqrt x \rd x = \dfrac {2 x^{3 / 2} } 3 = \dfrac {2 \sqrt x^3} 3$ | From [[Primitive of Power]]:
{{:Primitive of Power}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \sqrt x \rd x
| r = \int x^{1/2} \rd x
| c = {{Defof|Square Root}}
}}
{{eqn | r = \dfrac {x^{1/2 + 1} } {1/2 + 1} + C
| c = [[Primitive of Power]]
}}
{{eqn | r = \dfrac {x^{3/2} } {3/2} + C
| c = simplif... | Primitive of Square Root | https://proofwiki.org/wiki/Primitive_of_Square_Root | https://proofwiki.org/wiki/Primitive_of_Square_Root | [
"Examples of Use of Primitive of Power",
"Primitives",
"Square Roots"
] | [] | [
"Primitive of Power",
"Primitive of Power",
"Category:Examples of Use of Primitive of Power",
"Category:Primitives",
"Category:Square Roots"
] |
proofwiki-20172 | Condition for Conditional Expectation to be Almost Surely Non-Negative | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra of $\Sigma$.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Then we have:
:$\expect {... | === Sufficient Condition ===
Suppose that:
:$\expect {X \chi_A} \ge 0$ for each $A \in \GG$.
Then we have, by the definition of the conditional expectation of $X$ given $\GG$:
:$\expect {\expect {X \mid \GG} \chi_A} \ge 0$ for each $A \in \GG$.
Set:
:$A = \set {\omega \in \Omega : \map {\paren {\expect {X \mid \GG} } ... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]] of $\Sigma$.
Let $\... | === Sufficient Condition ===
Suppose that:
:$\expect {X \chi_A} \ge 0$ for each $A \in \GG$.
Then we have, by the definition of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation of $X$ given $\GG$]]:
:$\expect {\expect {X \mid \GG} \chi_A} \ge 0$ for each $A \in \GG$.
Set:
:$A = \... | Condition for Conditional Expectation to be Almost Surely Non-Negative | https://proofwiki.org/wiki/Condition_for_Conditional_Expectation_to_be_Almost_Surely_Non-Negative | https://proofwiki.org/wiki/Condition_for_Conditional_Expectation_to_be_Almost_Surely_Non-Negative | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable",
"Definition:Sub-Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere",
"Definition:Characteristic Function"
] | [
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Measurable Function",
"Characterization of Measurable Functions",
"Expectation is Monotone",
"Measurable Function Zero A.E. iff Absolute Value has Zero Integral",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
... |
proofwiki-20173 | Conditional Expectation is Monotone | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be integrable random variables on $\struct {\Omega, \Sigma, \Pr}$ such that:
:$X \le Y$ almost everywhere.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra of $\Sigma$.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation... | We have:
:$Y - X \ge 0$ almost everywhere.
So, for each $A \in \GG$ we have:
:$\paren {Y - X} \cdot 1_A \ge 0$ almost everywhere.
So, from Expectation is Monotone:
:$\expect {\paren {Y - X} \cdot 1_A} \ge 0$
for each $A \in \GG$.
So, from Condition for Conditional Expectation to be Almost Surely Non-Negative, we have:
... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Integrable Random Variable|integrable random variables]] on $\struct {\Omega, \Sigma, \Pr}$ such that:
:$X \le Y$ [[Definition:Almost Everywhere|almost everywhere]].
Let $\GG \subseteq \Sigma... | We have:
:$Y - X \ge 0$ [[Definition:Almost Everywhere|almost everywhere]].
So, for each $A \in \GG$ we have:
:$\paren {Y - X} \cdot 1_A \ge 0$ [[Definition:Almost Everywhere|almost everywhere]].
So, from [[Expectation is Monotone]]:
:$\expect {\paren {Y - X} \cdot 1_A} \ge 0$
for each $A \in \GG$.
So, from [[Co... | Conditional Expectation is Monotone | https://proofwiki.org/wiki/Conditional_Expectation_is_Monotone | https://proofwiki.org/wiki/Conditional_Expectation_is_Monotone | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable",
"Definition:Almost Everywhere",
"Definition:Sub-Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere"
] | [
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Expectation is Monotone",
"Condition for Conditional Expectation to be Almost Surely Non-Negative",
"Definition:Almost Everywhere",
"Conditional Expectation is Linear",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"C... |
proofwiki-20174 | Conditional Expectation Conditioned on Trivial Sigma-Algebra | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable random variable.
Let $\expect {X \mid \set {\O, \Omega} }$ be a version of the conditional expectation of $X$ given $\set {\O, \Omega}$.
Then:
:$\expect {X \mid \set {\O, \Omega} } = \expect X$ almost everywhere. | We check that $\expect X$ is a version of the conditional expectation of $X$ given $\set {\O, \Omega}$, so that we get:
:$\expect {X \mid \set {\O, \Omega} } = \expect X$ almost everywhere.
from Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra.
Note that $\expect X$ is $\GG$-me... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]].
Let $\expect {X \mid \set {\O, \Omega} }$ be a version of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation o... | We check that $\expect X$ is a version of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation of $X$ given $\set {\O, \Omega}$]], so that we get:
:$\expect {X \mid \set {\O, \Omega} } = \expect X$ [[Definition:Almost Everywhere|almost everywhere]].
from [[Existence and Essential Uniquene... | Conditional Expectation Conditioned on Trivial Sigma-Algebra | https://proofwiki.org/wiki/Conditional_Expectation_Conditioned_on_Trivial_Sigma-Algebra | https://proofwiki.org/wiki/Conditional_Expectation_Conditioned_on_Trivial_Sigma-Algebra | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere"
] | [
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere",
"Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra",
"Definition:Measurable Function",
"Constant Function is Measurable",
"Integral of Integrable Function over Null Set... |
proofwiki-20175 | Conditional Expectation Conditioned on Event of Non-Zero Probability | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $B \in \Sigma$ be an event with:
:$\map \Pr B > 0$
Let:
:$\GG = \map \sigma B = \set {\O, B, B^c, \Omega}$
where $\map \sigma B$ is the $\sigma$-algebra generated by $B$.
Let:
:$... | We show that:
:$\ds Z = \alpha \cdot 1_B + \beta \cdot 1_{B^c}$ is a version of the conditional expectation of $X$ given $\GG$.
We will then be done by Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra.
From Characteristic Function Measurable iff Set Measurable, $1_B$ and $1_{B^... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $B \in \Sigma$ be an [[Definition:Event|event]] with:
:$\map \Pr B > 0$
Let:
:$\GG = \map \sigma B ... | We show that:
:$\ds Z = \alpha \cdot 1_B + \beta \cdot 1_{B^c}$ is a version of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation of $X$ given $\GG$]].
We will then be done by [[Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra]].
From [[Charac... | Conditional Expectation Conditioned on Event of Non-Zero Probability | https://proofwiki.org/wiki/Conditional_Expectation_Conditioned_on_Event_of_Non-Zero_Probability | https://proofwiki.org/wiki/Conditional_Expectation_Conditioned_on_Event_of_Non-Zero_Probability | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable",
"Definition:Event",
"Definition:Sigma-Algebra Generated by Collection of Subsets",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere"
] | [
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra",
"Characteristic Function Measurable iff Set Measurable",
"Definition:Measurable Function",
"Pointwise Product of Measurable Functions is Measurable",... |
proofwiki-20176 | Rule for Extracting Random Variable from Conditional Expectation of Product | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ and $Y$ be integrable random variables such that:
:$X Y$ is integrable
and:
:$Y$ is $\GG$-measurable.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Let... | Let $X$ and $Y$ be non-negative random variables.
We show first that the statement holds if $Y = \chi_A$ for some $A \in \GG$.
We show that $Y \expect {X \mid \GG}$ is a version of the conditional expectation of $X Y$ given $\GG$.
We will then obtain the demand from Existence and Essential Uniqueness of Conditional E... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]].
Let $X$ and $Y$ be [[Definition:Integrable Random Variable|integrable random variables]] such that:
:$X Y$ is [[Definition:Integrable Ra... | Let $X$ and $Y$ be non-negative [[Definition:Real-Valued Random Variable|random variables]].
We show first that the statement holds if $Y = \chi_A$ for some $A \in \GG$.
We show that $Y \expect {X \mid \GG}$ is a version of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation of $X Y$ g... | Rule for Extracting Random Variable from Conditional Expectation of Product | https://proofwiki.org/wiki/Rule_for_Extracting_Random_Variable_from_Conditional_Expectation_of_Product | https://proofwiki.org/wiki/Rule_for_Extracting_Random_Variable_from_Conditional_Expectation_of_Product | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Integrable Random Variable",
"Definition:Measurable Function",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Alge... | [
"Definition:Random Variable/Real-Valued",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra",
"Characteristic Function of Intersection",
"Characteristic Function of Intersection",
"Definition:Almost ... |
proofwiki-20177 | Triangle Inequality for Conditional Expectation | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ be an integrable random variable.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Let $\expect {\size X \mid \GG}$ be a version of the conditional expectat... | From Conditional Expectation is Monotone, we have:
:$\expect {X^+ \mid \GG} \ge 0$ almost everywhere
and:
:$\expect {X^- \mid \GG} \ge 0$ almost everywhere
where $X^+$ and $X^-$ are the positive and negative parts respectively.
Now, almost everywhere we have:
{{begin-eqn}}
{{eqn | l = \size {\expect {X \mid \GG} }
... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]].
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]].
Let $\expect {X \mid \GG}$ be a version of the [[Defi... | From [[Conditional Expectation is Monotone]], we have:
:$\expect {X^+ \mid \GG} \ge 0$ [[Definition:Almost Everywhere|almost everywhere]]
and:
:$\expect {X^- \mid \GG} \ge 0$ [[Definition:Almost Everywhere|almost everywhere]]
where $X^+$ and $X^-$ are the [[Definition:Positive Part|positive]] and [[Definition:Nega... | Triangle Inequality for Conditional Expectation | https://proofwiki.org/wiki/Triangle_Inequality_for_Conditional_Expectation | https://proofwiki.org/wiki/Triangle_Inequality_for_Conditional_Expectation | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere"
] | [
"Conditional Expectation is Monotone",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Positive Part",
"Definition:Negative Part",
"Definition:Almost Everywhere",
"Conditional Expectation is Linear",
"Triangle Inequality/Real Numbers",
"Conditional Expectation is Linear",... |
proofwiki-20178 | Conditional Expectation of Constant | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $c \in \R$.
Define $X : \Omega \to \R$ by $\map X \omega = c$ for each $\omega \in \Omega$.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Then:
:$\expect {X ... | From Constant Function is Measurable, $X$ is a real-valued random variable.
The result then follows immediately from Conditional Expectation of Measurable Random Variable.
{{qed}}
Category:Conditional Expectation
7t34wd6gqauyohn7r6a2sr8ztllvgxz | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]].
Let $c \in \R$.
Define $X : \Omega \to \R$ by $\map X \omega = c$ for each $\omega \in \Omega$.
Let $\expect {X \mid \GG}$ be a version ... | From [[Constant Function is Measurable]], $X$ is a [[Definition:Real-Valued Random Variable|real-valued random variable]].
The result then follows immediately from [[Conditional Expectation of Measurable Random Variable]].
{{qed}}
[[Category:Conditional Expectation]]
7t34wd6gqauyohn7r6a2sr8ztllvgxz | Conditional Expectation of Constant | https://proofwiki.org/wiki/Conditional_Expectation_of_Constant | https://proofwiki.org/wiki/Conditional_Expectation_of_Constant | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere"
] | [
"Constant Function is Measurable",
"Definition:Random Variable/Real-Valued",
"Conditional Expectation of Measurable Random Variable",
"Category:Conditional Expectation"
] |
proofwiki-20179 | Conditional Expectation of Non-Negative Random Variable is Non-Negative | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ be an integrable random variable such that:
:$X \ge 0$ almost everywhere.
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Then:
:$\expect {X \mid \GG} \ge ... | From Conditional Expectation is Monotone, we have:
:$\expect {X \mid \GG} \ge \expect {0 \mid \GG}$ almost everywhere.
From Conditional Expectation of Constant, we have:
:$\expect {0 \mid \GG} = 0$ almost everywhere.
So:
:$\expect {X \mid \GG} \ge 0$ almost everywhere.
{{qed}} | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]].
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]] such that:
:$X \ge 0$ [[Definition:Almost Everywhere|a... | From [[Conditional Expectation is Monotone]], we have:
:$\expect {X \mid \GG} \ge \expect {0 \mid \GG}$ [[Definition:Almost Everywhere|almost everywhere]].
From [[Conditional Expectation of Constant]], we have:
:$\expect {0 \mid \GG} = 0$ [[Definition:Almost Everywhere|almost everywhere]].
So:
:$\expect {X \mid \G... | Conditional Expectation of Non-Negative Random Variable is Non-Negative | https://proofwiki.org/wiki/Conditional_Expectation_of_Non-Negative_Random_Variable_is_Non-Negative | https://proofwiki.org/wiki/Conditional_Expectation_of_Non-Negative_Random_Variable_is_Non-Negative | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Almost Everywhere",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere"
] | [
"Conditional Expectation is Monotone",
"Definition:Almost Everywhere",
"Conditional Expectation of Constant",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere"
] |
proofwiki-20180 | Conditional Fatou's Lemma | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an non-negative integrable random variable.
Let $\sequence {X_n}_{n \in \N}$ be an sequence of non-negative integrable random variables.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
For each $n \in \N$, let $\expect {X_n \mid \GG}$ be a ver... | From Conditional Expectation of Measurable Random Variable, we have:
:$\expect {X_n \mid \GG} = \expect {\expect {X_n \mid \GG} \mid \GG}$
It therefore suffices to show, from Conditional Expectation is Linear:
:$\ds \expect {\liminf_{n \mathop \to \infty} \expect {X_n \mid \GG} - \liminf_{n \mathop \to \infty} X_n \mid... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Positive Real Function|non-negative]] [[Definition:Integrable Random Variable|integrable random variable]].
Let $\sequence {X_n}_{n \in \N}$ be an [[Definition:Sequence|sequence]] of [[Definition:Po... | From [[Conditional Expectation of Measurable Random Variable]], we have:
:$\expect {X_n \mid \GG} = \expect {\expect {X_n \mid \GG} \mid \GG}$
It therefore suffices to show, from [[Conditional Expectation is Linear]]:
:$\ds \expect {\liminf_{n \mathop \to \infty} \expect {X_n \mid \GG} - \liminf_{n \mathop \to \inft... | Conditional Fatou's Lemma | https://proofwiki.org/wiki/Conditional_Fatou's_Lemma | https://proofwiki.org/wiki/Conditional_Fatou's_Lemma | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Positive Real Function",
"Definition:Integrable Random Variable",
"Definition:Sequence",
"Definition:Positive Real Function",
"Definition:Integrable Random Variable",
"Definition:Sub-Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Alge... | [
"Conditional Expectation of Measurable Random Variable",
"Conditional Expectation is Linear",
"Definition:Almost Everywhere",
"Condition for Conditional Expectation to be Almost Surely Non-Negative",
"Integral of Integrable Function is Additive/Corollary 2",
"Fatou's Lemma for Integrals",
"Fatou's Lemma... |
proofwiki-20181 | Conditional Reverse Fatou's Lemma | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable random variable.
Let $\sequence {X_n}_{n \mathop \in \N}$ be an sequence of non-negative integrable random variables such that:
:there exists an integrable random variable $Y$ such that:
::$\size {X_n} \le Y$ almost surely.
Let $\GG \... | We should first verify that a version of the conditional expectation of $\ds \limsup_{n \mathop \to \infty} X_n$ conditioned on $\GG$ exists.
We have:
:$-Y \le X_n \le Y$
and so:
:$\ds -Y \le \limsup_{n \mathop \to \infty} X_n \le Y$ almost surely
so that:
:$\ds -\infty < -\expect Y \le \expect {\limsup_{n \mathop \to... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]].
Let $\sequence {X_n}_{n \mathop \in \N}$ be an [[Definition:Sequence|sequence]] of [[Definition:Positive Real Function|non-negative]] [[Defin... | We should first verify that a version of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation of $\ds \limsup_{n \mathop \to \infty} X_n$ conditioned on $\GG$]] exists.
We have:
:$-Y \le X_n \le Y$
and so:
:$\ds -Y \le \limsup_{n \mathop \to \infty} X_n \le Y$ [[Definition:Almost Everyw... | Conditional Reverse Fatou's Lemma | https://proofwiki.org/wiki/Conditional_Reverse_Fatou's_Lemma | https://proofwiki.org/wiki/Conditional_Reverse_Fatou's_Lemma | [
"Conditional Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable",
"Definition:Sequence",
"Definition:Positive Real Function",
"Definition:Integrable Random Variable",
"Definition:Integrable Random Variable",
"Definition:Almost Everywhere",
"Definition:Sub-Sigma-Algebra",
"Definition:Condition... | [
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere",
"Expectation is Monotone",
"Definition:Integrable Random Variable",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Conditional Fatou's Lemma",
"Definition:Almost Everywhere",
"Conditiona... |
proofwiki-20182 | Conditional Dominated Convergence Theorem | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable random variable.
Let $\sequence {X_n}_{n \in \N}$ be an sequence of non-negative integrable random variables converging almost surely to $X$, such that:
:there exists an integrable random variable $Y$ such that:
::$\size {X_n} \le Y$ ... | Let $\expect {Y \mid \GG}$ be a version of the conditional expectation of $Y$ conditioned on $\GG$.
For each $n \in \N$, let $\expect {X_n \mid \GG}$ be a version of the conditional expectation of $X_n$ conditioned on $\GG$.
Since we have:
:$\size {X_n} \le Y$ almost surely.
and $Y$ is integrable, we have:
:$\ds \lim... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Integrable Random Variable|integrable random variable]].
Let $\sequence {X_n}_{n \in \N}$ be an [[Definition:Sequence|sequence]] of [[Definition:Positive Real Function|non-negative]] [[Definition:In... | Let $\expect {Y \mid \GG}$ be a version of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation of $Y$ conditioned on $\GG$]].
For each $n \in \N$, let $\expect {X_n \mid \GG}$ be a version of the [[Definition:Conditional Expectation on Sigma-Algebra|conditional expectation of $X_n$ condit... | Conditional Dominated Convergence Theorem | https://proofwiki.org/wiki/Conditional_Dominated_Convergence_Theorem | https://proofwiki.org/wiki/Conditional_Dominated_Convergence_Theorem | [
"Conditional Expectation",
"Lebesgue's Dominated Convergence Theorem"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable",
"Definition:Sequence",
"Definition:Positive Real Function",
"Definition:Integrable Random Variable",
"Definition:Almost Sure Convergence",
"Definition:Integrable Random Variable",
"Definition:Almost Everywhere",
"Definition:Sub... | [
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere",
"Definition:Integrable Random Variable",
"Definition:Almost Everywhere",
"Conditional Reverse Fatou's Lemma",
"Conditional Fatou's Lemma",
"... |
proofwiki-20183 | Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ be a integrable random variable.
Let $\map \sigma X$ be the $\sigma$-algebra generated by $X$.
Let $\HH \subseteq \Sigma$ be a sub-$\sigma$-algebra that is independent of $\map \sigma {\map \sigma X... | First take $X$ to be a non-negative random variable.
Let:
:$\SS = \set {G \cap H : G \in \GG, \, H \in \HH}$
We aim to apply Uniqueness of Measures to a suitable measure with $\SS$.
We start by showing that $\Omega \in \SS$, $\SS$ is a $\pi$-system, and that $\SS$ generates $\map \sigma {\GG, \HH}$.
First note that:
:... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\GG \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]].
Let $X$ be a [[Definition:Integrable Random Variable|integrable random variable]].
Let $\map \sigma X$ be the [[Definition:Sigma-Algebra ... | First take $X$ to be a non-negative [[Definition:Real-Valued Random Variable|random variable]].
Let:
:$\SS = \set {G \cap H : G \in \GG, \, H \in \HH}$
We aim to apply [[Uniqueness of Measures]] to a suitable measure with $\SS$.
We start by showing that $\Omega \in \SS$, $\SS$ is a [[Definition:Pi-System|$\pi$-sys... | Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra | https://proofwiki.org/wiki/Conditional_Expectation_Unchanged_on_Conditioning_on_Independent_Sigma-Algebra | https://proofwiki.org/wiki/Conditional_Expectation_Unchanged_on_Conditioning_on_Independent_Sigma-Algebra | [
"Conditional Expectation",
"Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra"
] | [
"Definition:Probability Space",
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Sigma-Algebra Generated by Collection of Mappings",
"Definition:Sub-Sigma-Algebra",
"Definition:Independent Sigma-Algebras",
"Definition:Sigma-Algebra Generated by Collection of Subsets",... | [
"Definition:Random Variable/Real-Valued",
"Uniqueness of Measures",
"Definition:Pi-System",
"Definition:Pi-System",
"Intersection is Associative",
"Definition:Sigma-Algebra",
"Definition:Sigma-Algebra Generated by Collection of Subsets",
"Definition:Sigma-Algebra",
"Definition:Measure with Density",... |
proofwiki-20184 | Expectation of Product of Independent Random Variables is Product of Expectations | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be non-negative real-valued random variables that are independent.
Then:
:$\expect {X Y} = \expect X \expect Y$ | We first prove the claim in the case that $X = \chi_A$ for $A \in \Sigma$ and $Y = \chi_B$ for $B \in \Sigma$.
In particular, we have $A \in \map \sigma X$ and $B \in \map \sigma Y$ where $\map \sigma X$ and $\map \sigma Y$ are the $\sigma$-algebras generated by $A$ and $B$ respectively.
Then, we have:
{{begin-eqn}}
{... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Non-Negative Real Number|non-negative]] [[Definition:Real-Valued Random Variable|real-valued random variables]] that are [[Definition:Independent Random Variables/General Definition|independent... | We first prove the claim in the case that $X = \chi_A$ for $A \in \Sigma$ and $Y = \chi_B$ for $B \in \Sigma$.
In particular, we have $A \in \map \sigma X$ and $B \in \map \sigma Y$ where $\map \sigma X$ and $\map \sigma Y$ are the [[Definition:Sigma-Algebra Generated by Collection of Mappings|$\sigma$-algebras genera... | Expectation of Product of Independent Random Variables is Product of Expectations | https://proofwiki.org/wiki/Expectation_of_Product_of_Independent_Random_Variables_is_Product_of_Expectations | https://proofwiki.org/wiki/Expectation_of_Product_of_Independent_Random_Variables_is_Product_of_Expectations | [
"Expectation of Product of Independent Random Variables is Product of Expectations",
"Expectation"
] | [
"Definition:Probability Space",
"Definition:Positive/Real Number",
"Definition:Random Variable/Real-Valued",
"Definition:Independent Random Variables/General Definition"
] | [
"Definition:Sigma-Algebra Generated by Collection of Mappings",
"Characteristic Function of Intersection",
"Integral of Characteristic Function",
"Integral of Characteristic Function",
"Definition:Random Variable/Real-Valued",
"Definition:Simple Function",
"Definition:Random Variable/Real-Valued",
"De... |
proofwiki-20185 | Set Difference is Right Distributive over Set Intersection/General Case | Let $U$ be a collection of sets.
Let $T$ be a set.
Then:
:$\ds \bigcap_{X \mathop \in U} \paren {X \setminus T} = \paren {\bigcap_{X \mathop \in U} X} \setminus T$
That is, the difference with an intersection equals the intersection of the differences. | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcap_{X \mathop \in U} \paren {X \setminus T}
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \forall X \in U
| l = x
| o = \in
| r = X \setminus T
| c = {{Defof|Set Intersection}}
}}
{{eqn | ll= \leadstoandfrom
| q = \fora... | Let $U$ be a [[Definition:Collection|collection]] of [[Definition:Set|sets]].
Let $T$ be a [[Definition:Set|set]].
Then:
:$\ds \bigcap_{X \mathop \in U} \paren {X \setminus T} = \paren {\bigcap_{X \mathop \in U} X} \setminus T$
That is, the [[Definition:Set Difference|difference]] with an [[Definition:Set Intersect... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcap_{X \mathop \in U} \paren {X \setminus T}
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \forall X \in U
| l = x
| o = \in
| r = X \setminus T
| c = {{Defof|Set Intersection}}
}}
{{eqn | ll= \leadstoandfrom
| q = \fora... | Set Difference is Right Distributive over Set Intersection/General Case/Proof | https://proofwiki.org/wiki/Set_Difference_is_Right_Distributive_over_Set_Intersection/General_Case | https://proofwiki.org/wiki/Set_Difference_is_Right_Distributive_over_Set_Intersection/General_Case/Proof | [
"Set Difference is Right Distributive over Set Intersection"
] | [
"Definition:Collection",
"Definition:Set",
"Definition:Set",
"Definition:Set Difference",
"Definition:Set Intersection",
"Definition:Set Intersection",
"Definition:Set Difference"
] | [] |
proofwiki-20186 | Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra/Corollary | Let $\HH \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ be a integrable random variable such that:
:$\map \sigma X$ is independent of $\HH$
where $\map \sigma X$ is the $\sigma$-algebra generated by $X$.
Let $\expect {X \mid \HH}$ be a version of the conditional expectation of $X$ given $\GG$.
Then:
:$\expect ... | Note that:
:$\map \sigma X = \map \sigma {\set {\O, \Omega}, \map \sigma X}$
So by Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra, we have:
:$\expect {X \mid \map \sigma {\set {\O, \Omega}, \HH} } = \expect {X \mid \set {\O, \Omega} }$
From Conditional Expectation Conditioned on Trivia... | Let $\HH \subseteq \Sigma$ be a [[Definition:Sub-Sigma-Algebra|sub-$\sigma$-algebra]].
Let $X$ be a [[Definition:Integrable Random Variable|integrable random variable]] such that:
:$\map \sigma X$ is [[Definition:Independent Sigma-Algebras|independent]] of $\HH$
where $\map \sigma X$ is the [[Definition:Sigma-Algeb... | Note that:
:$\map \sigma X = \map \sigma {\set {\O, \Omega}, \map \sigma X}$
So by [[Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra]], we have:
:$\expect {X \mid \map \sigma {\set {\O, \Omega}, \HH} } = \expect {X \mid \set {\O, \Omega} }$
From [[Conditional Expectation Conditioned... | Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra/Corollary | https://proofwiki.org/wiki/Conditional_Expectation_Unchanged_on_Conditioning_on_Independent_Sigma-Algebra/Corollary | https://proofwiki.org/wiki/Conditional_Expectation_Unchanged_on_Conditioning_on_Independent_Sigma-Algebra/Corollary | [
"Conditional Expectation",
"Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra"
] | [
"Definition:Sub-Sigma-Algebra",
"Definition:Integrable Random Variable",
"Definition:Independent Sigma-Algebras",
"Definition:Sigma-Algebra Generated by Collection of Mappings",
"Definition:Conditional Expectation/General Case/Sigma-Algebra",
"Definition:Almost Everywhere"
] | [
"Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra",
"Conditional Expectation Conditioned on Trivial Sigma-Algebra",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere"
] |
proofwiki-20187 | Sum of Independent Random Variables with Mean Zero is Martingale | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a sequence of integrable independent random variables with:
:$\expect {X_n} = 0$ for each $n \in \N$
and:
:$X_0 = 0$
For $n \ge 0$ define:
:$\ds S_n = \sum_{i \mathop = 0}^n X_i$
Let $\sequence {\FF_n^X}_{n \mathop \... | We first show that $\sequence {S_n}_{n \mathop \ge 0}$ is $\sequence {\FF_n^X}_{n \mathop \ge 0}$-adapted.
From the definition of the $\sigma$-algebra generated by a collection of mappings, we have:
:$X_i$ is $\map \sigma {X_0, \ldots, X_n}$-measurable for $0 \le i \le n$.
So from Pointwise Sum of Measurable Functions ... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a [[Definition:Sequence|sequence]] of [[Definition:Integrable Random Variable|integrable]] [[Definition:Independent Random Variables/General Definition|independent random variables]] ... | We first show that $\sequence {S_n}_{n \mathop \ge 0}$ is [[Definition:Adapted Stochastic Process|$\sequence {\FF_n^X}_{n \mathop \ge 0}$-adapted]].
From the definition of the [[Definition:Sigma-Algebra Generated by Collection of Mappings|$\sigma$-algebra generated by a collection of mappings]], we have:
:$X_i$ is [[... | Sum of Independent Random Variables with Mean Zero is Martingale | https://proofwiki.org/wiki/Sum_of_Independent_Random_Variables_with_Mean_Zero_is_Martingale | https://proofwiki.org/wiki/Sum_of_Independent_Random_Variables_with_Mean_Zero_is_Martingale | [
"Martingales"
] | [
"Definition:Probability Space",
"Definition:Sequence",
"Definition:Integrable Random Variable",
"Definition:Independent Random Variables/General Definition",
"Definition:Natural Filtration/Discrete Time",
"Definition:Martingale/Discrete Time"
] | [
"Definition:Adapted Stochastic Process",
"Definition:Sigma-Algebra Generated by Collection of Mappings",
"Definition:Measurable Function",
"Pointwise Sum of Measurable Functions is Measurable/General Result",
"Definition:Measurable Function",
"Definition:Natural Filtration",
"Definition:Adapted Stochast... |
proofwiki-20188 | Equivalence of Definitions of Stopping Time in Discrete Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $T : \Omega \to \Z_{\ge 0} \cup \set {\infty}$ be a function.
{{TFAE|def = Stopping Time|stopping time}} | === Definition 1 implies Definition 2 ===
Suppose that:
:$\set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
for all $t \in \Z_{\ge 0}$.
Setting $t = 0$ this certainly implies:
:$\set {\omega \in \Omega : \map T \omega = t} \in \FF_t$
Now take $t \ge 1$ a positive integer.
We have:
:$\set {\omega \in \Omega : \... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $T : \Omega \to \Z_{\ge 0} \cup \set {\infty}$ be a [[Definition:Function|function]].
{{TFAE|def = Stopping Time|stopping time}} | === Definition 1 implies Definition 2 ===
Suppose that:
:$\set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
for all $t \in \Z_{\ge 0}$.
Setting $t = 0$ this certainly implies:
:$\set {\omega \in \Omega : \map T \omega = t} \in \FF_t$
Now take $t \ge 1$ a [[Definition:Positive Integer|positive integer]].
... | Equivalence of Definitions of Stopping Time in Discrete Time | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Stopping_Time_in_Discrete_Time | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Stopping_Time_in_Discrete_Time | [
"Stopping Times"
] | [
"Definition:Filtered Probability Space",
"Definition:Function"
] | [
"Definition:Positive/Integer",
"Definition:Filtration of Sigma-Algebra",
"Sigma-Algebra Closed under Set Difference",
"Definition:Filtration of Sigma-Algebra"
] |
proofwiki-20189 | Pointwise Minimum of Stopping Times is Stopping Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $T$ and $S$ be stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $S \wedge T$ be the pointwise minimum of $S$ and $T$.
Then $S \wedge T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$. | We have, for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$:
:$\map {\paren {S \wedge T} } \omega \le t$ {{iff}} $\map S \omega \le t$ or $\map T \omega \le t$
That is:
:$\set {\omega \in \Omega : \map {\paren {S \wedge T} } \omega \le t} = \set {\omega \in \Omega : \map S \omega \le t} \cup \set {\omega \in \Omega : \ma... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $T$ and $S$ be [[Definition:Stopping Time/Discrete Time|stopping times]] with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $S \wedge T$ be the [[Definition:Pointwise Minimum... | We have, for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$:
:$\map {\paren {S \wedge T} } \omega \le t$ {{iff}} $\map S \omega \le t$ or $\map T \omega \le t$
That is:
:$\set {\omega \in \Omega : \map {\paren {S \wedge T} } \omega \le t} = \set {\omega \in \Omega : \map S \omega \le t} \cup \set {\omega \in \Omega : ... | Pointwise Minimum of Stopping Times is Stopping Time | https://proofwiki.org/wiki/Pointwise_Minimum_of_Stopping_Times_is_Stopping_Time | https://proofwiki.org/wiki/Pointwise_Minimum_of_Stopping_Times_is_Stopping_Time | [
"Stopping Times"
] | [
"Definition:Filtered Probability Space",
"Definition:Stopping Time/Discrete Time",
"Definition:Pointwise Minimum of Mappings/Extended Real-Valued Functions",
"Definition:Stopping Time/Discrete Time"
] | [
"Definition:Stopping Time/Discrete Time",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union/Finite Union",
"Category:Stopping Times"
] |
proofwiki-20190 | Constant Function is Stopping Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $T_\ast$ be an extended natural number.
Define $T : \Omega \to \Z_{\ge 0} \cup \set \infty$ by:
:$\map T \omega = T_\ast$
for each $\omega \in \Omega$.
Then $T$ is a stopping time with respect to $\sequence {\FF_n}_{... | First, if $T_\ast = \infty$, we have:
:$\set {\omega \in \Omega : \map T \omega \le t} = \O$
for all $t \in \Z_{\ge 0}$.
Since each $\FF_t$ is a $\sigma$-algebra we therefore have:
:$\set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
for all $t \in \Z_{\ge 0}$ in the case $T_\ast = \infty$.
Now let $T_\ast < ... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $T_\ast$ be an [[Definition:Extended Natural Number|extended natural number]].
Define $T : \Omega \to \Z_{\ge 0} \cup \set \infty$ by:
:$\map T \omega = T_\ast$
for each ... | First, if $T_\ast = \infty$, we have:
:$\set {\omega \in \Omega : \map T \omega \le t} = \O$
for all $t \in \Z_{\ge 0}$.
Since each $\FF_t$ is a [[Definition:Sigma-Algebra|$\sigma$-algebra]] we therefore have:
:$\set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
for all $t \in \Z_{\ge 0}$ in the case $T_\... | Constant Function is Stopping Time | https://proofwiki.org/wiki/Constant_Function_is_Stopping_Time | https://proofwiki.org/wiki/Constant_Function_is_Stopping_Time | [
"Stopping Times"
] | [
"Definition:Filtered Probability Space",
"Definition:Extended Natural Numbers",
"Definition:Stopping Time/Discrete Time"
] | [
"Definition:Sigma-Algebra",
"Definition:Sigma-Algebra",
"Definition:Stopping Time/Discrete Time",
"Category:Stopping Times"
] |
proofwiki-20191 | Pointwise Infimum of Stopping Times is Stopping Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {T_n}_{n \in \N}$ be a sequence of stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let:
:$\ds T = \inf_{n \in \N} T_n$
be the pointwise infimum of the $\sequence {T_n}_{n \in \N}$.
Then $T$ is ... | We have, for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$:
:$\map T \omega \le t$ {{iff}} $\map {T_n} \omega \le t$ for some $n \in \N$.
That is:
:$\ds \set {\omega \in \Omega : \map T \omega \le t} = \bigcup_{n \in \N} \set {\omega \in \Omega : \map {T_n} \omega \le t}$
for each $t \in \Z_{\ge 0}$.
Now fix $t \in \Z_{... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {T_n}_{n \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Stopping Time/Discrete Time|stopping times]] with respect to $\sequence {\FF_n}_{n \ge 0}$.
... | We have, for $t \in \Z_{\ge 0}$ and $\omega \in \Omega$:
:$\map T \omega \le t$ {{iff}} $\map {T_n} \omega \le t$ for some $n \in \N$.
That is:
:$\ds \set {\omega \in \Omega : \map T \omega \le t} = \bigcup_{n \in \N} \set {\omega \in \Omega : \map {T_n} \omega \le t}$
for each $t \in \Z_{\ge 0}$.
Now fix $t \in... | Pointwise Infimum of Stopping Times is Stopping Time | https://proofwiki.org/wiki/Pointwise_Infimum_of_Stopping_Times_is_Stopping_Time | https://proofwiki.org/wiki/Pointwise_Infimum_of_Stopping_Times_is_Stopping_Time | [
"Stopping Times"
] | [
"Definition:Filtered Probability Space",
"Definition:Sequence",
"Definition:Stopping Time/Discrete Time",
"Definition:Pointwise Infimum of Extended Real-Valued Functions",
"Definition:Stopping Time/Discrete Time"
] | [
"Definition:Stopping Time/Discrete Time",
"Definition:Sigma-Algebra",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union/Countable Union",
"Definition:Stopping Time/Discrete Time",
"Category:Stopping Times"
] |
proofwiki-20192 | Sum of Stopping Times is Stopping Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $T$ and $S$ be stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$.
Then the pointwise sum $T + S$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$. | Let $t \in \Z_{\ge 0}$.
Then, if for $\omega \in \Omega$ we have $\map T \omega + \map S \omega = t$, we have:
:$\map T \omega \le t$
and:
:$\map S \omega \le t$
If we have:
:$\map S \omega = s \le t$
and:
:$\map S \omega + \map T \omega = t$
we have:
:$\map T \omega = t - s$
So, we have:
:$\ds \set {\omega \in... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $T$ and $S$ be [[Definition:Stopping Time/Discrete Time|stopping times]] with respect to $\sequence {\FF_n}_{n \ge 0}$.
Then the [[Definition:Pointwise Addition of Extended ... | Let $t \in \Z_{\ge 0}$.
Then, if for $\omega \in \Omega$ we have $\map T \omega + \map S \omega = t$, we have:
:$\map T \omega \le t$
and:
:$\map S \omega \le t$
If we have:
:$\map S \omega = s \le t$
and:
:$\map S \omega + \map T \omega = t$
we have:
:$\map T \omega = t - s$
So, we have:
:$\ds \set... | Sum of Stopping Times is Stopping Time | https://proofwiki.org/wiki/Sum_of_Stopping_Times_is_Stopping_Time | https://proofwiki.org/wiki/Sum_of_Stopping_Times_is_Stopping_Time | [
"Stopping Times"
] | [
"Definition:Filtered Probability Space",
"Definition:Stopping Time/Discrete Time",
"Definition:Pointwise Addition of Extended Real-Valued Functions",
"Definition:Stopping Time/Discrete Time"
] | [
"Definition:Stopping Time/Discrete Time",
"Definition:Filtration of Sigma-Algebra",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union/Finite Union",
"Definition:Stopping Time/Discrete Time",
"Category:Stopping Times"
] |
proofwiki-20193 | Shift of Stopping Time is Stopping Time | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a filtered probability space.
Let $T$ be a stopping time {{WRT}} $\sequence {\FF_n}_{n \mathop \ge 0}$.
Let $t$ be a extended natural number.
Then $T + t$ is a stopping time {{WRT}} $\sequence {\FF_n}_{n \mathop \ge 0}$. | By Constant Function is Stopping Time, $t$ is a stopping time {{WRT}} $\sequence {\FF_n}_{n \mathop \ge 0}$.
By Sum of Stopping Times is Stopping Time, $T + t$ is a stopping time {{WRT}} $\sequence {\FF_n}_{n \mathop \ge 0}$.
{{qed}}
Category:Stopping Times
1z3amfn8zilqrdlwhf6uhp4xj1pap34 | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $T$ be a [[Definition:Stopping Time/Discrete Time|stopping time]] {{WRT}} $\sequence {\FF_n}_{n \mathop \ge 0}$.
Let $t$ be a [[Definition:Extended Natural Number|ext... | By [[Constant Function is Stopping Time]], $t$ is a [[Definition:Stopping Time/Discrete Time|stopping time]] {{WRT}} $\sequence {\FF_n}_{n \mathop \ge 0}$.
By [[Sum of Stopping Times is Stopping Time]], $T + t$ is a [[Definition:Stopping Time/Discrete Time|stopping time]] {{WRT}} $\sequence {\FF_n}_{n \mathop \ge 0}$.... | Shift of Stopping Time is Stopping Time | https://proofwiki.org/wiki/Shift_of_Stopping_Time_is_Stopping_Time | https://proofwiki.org/wiki/Shift_of_Stopping_Time_is_Stopping_Time | [
"Stopping Times"
] | [
"Definition:Filtered Probability Space",
"Definition:Stopping Time/Discrete Time",
"Definition:Extended Natural Numbers",
"Definition:Stopping Time/Discrete Time"
] | [
"Constant Function is Stopping Time",
"Definition:Stopping Time/Discrete Time",
"Sum of Stopping Times is Stopping Time",
"Definition:Stopping Time/Discrete Time",
"Category:Stopping Times"
] |
proofwiki-20194 | Stopped Sigma-Algebra is Sigma-Algebra | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $\FF_T$ be the stopped $\sigma$-algebra associated with $T$.
Then $\FF_T$ is a $\sigma$-algebra. | We show that $\Omega \in \FF_T$, that $\FF_T$ is closed under countable intersection, and relative complement.
For each $t \in \Z_{\ge 0}$ we have:
:$\Omega \cap \set {\omega \in \Omega : \map T \omega \le t} = \set {\omega \in \Omega : \map T \omega \le t}$
Since $T$ is a stopping time with respect to $\sequence {\F... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $T$ be a [[Definition:Stopping Time|stopping time]] with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $\FF_T$ be the [[Definition:Stopped Sigma-Algebra|stopped $\sigma$-alg... | We show that $\Omega \in \FF_T$, that $\FF_T$ is [[Definition:Closed under Operation|closed]] under [[Definition:Countable Intersection|countable intersection]], and [[Definition:Relative Complement|relative complement]].
For each $t \in \Z_{\ge 0}$ we have:
:$\Omega \cap \set {\omega \in \Omega : \map T \omega \le... | Stopped Sigma-Algebra is Sigma-Algebra | https://proofwiki.org/wiki/Stopped_Sigma-Algebra_is_Sigma-Algebra | https://proofwiki.org/wiki/Stopped_Sigma-Algebra_is_Sigma-Algebra | [
"Stopped Sigma-Algebras",
"Sigma-Algebras",
"Examples of Sigma-Algebras"
] | [
"Definition:Filtered Probability Space",
"Definition:Stopping Time",
"Definition:Stopped Sigma-Algebra",
"Definition:Sigma-Algebra"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Intersection/Countable Intersection",
"Definition:Relative Complement",
"Definition:Stopping Time",
"Definition:Sequence",
"Intersection is Associative",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definitio... |
proofwiki-20195 | Stopped Sigma-Algebra preserves Inequality between Stopping Times | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $T$ and $S$ be stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$ such that:
:$\map S \omega \le \map T \omega$
for each $\omega \in \Omega$.
Let $\FF_S$ and $\FF_T$ be the stopped $\sigma$-algebras associat... | Let $A \in \FF_S$ and $t \in \Z_{\ge 0}$.
If $\omega \in \Omega$ is such that:
:$\map T \omega \le t$
we have:
:$\map S \omega \le t$
So:
:$\set {\omega \in \Omega : \map T \omega \le t} \subseteq \set {\omega \in \Omega : \map S \omega \le t}$
for each $t \in \Z_{\ge 0}$.
So, from Intersection with Subset is Subset... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $T$ and $S$ be [[Definition:Stopping Time|stopping times]] with respect to $\sequence {\FF_n}_{n \ge 0}$ such that:
:$\map S \omega \le \map T \omega$
for each $\omega \in ... | Let $A \in \FF_S$ and $t \in \Z_{\ge 0}$.
If $\omega \in \Omega$ is such that:
:$\map T \omega \le t$
we have:
:$\map S \omega \le t$
So:
:$\set {\omega \in \Omega : \map T \omega \le t} \subseteq \set {\omega \in \Omega : \map S \omega \le t}$
for each $t \in \Z_{\ge 0}$.
So, from [[Intersection with Subset... | Stopped Sigma-Algebra preserves Inequality between Stopping Times | https://proofwiki.org/wiki/Stopped_Sigma-Algebra_preserves_Inequality_between_Stopping_Times | https://proofwiki.org/wiki/Stopped_Sigma-Algebra_preserves_Inequality_between_Stopping_Times | [
"Stopped Sigma-Algebras"
] | [
"Definition:Filtered Probability Space",
"Definition:Stopping Time",
"Definition:Stopped Sigma-Algebra"
] | [
"Intersection with Subset is Subset",
"Intersection is Associative",
"Definition:Stopping Time",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Intersection/Finite Intersection"
] |
proofwiki-20196 | Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $X_T$ be the adapted stochastic process $\sequen... | We have that if $\map T \omega = t$ for $\omega \in \Omega$ and $t \in \Z_{\ge 0}$ then:
:$\map {X_T} \omega = \map {X_t} \omega$
We aim to show that for each Borel set $A \subseteq \R$:
:$\ds \set {\omega \in \Omega : \map {X_T} \omega \in A} \in \FF_T$
That is, we want to show that:
:$\set {\omega \in \Omega : \map ... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Adapted Stochastic Process|$\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process]].
Let $T$ be a [[Definition:Stopping Time|... | We have that if $\map T \omega = t$ for $\omega \in \Omega$ and $t \in \Z_{\ge 0}$ then:
:$\map {X_T} \omega = \map {X_t} \omega$
We aim to show that for each [[Definition:Borel Set|Borel set]] $A \subseteq \R$:
:$\ds \set {\omega \in \Omega : \map {X_T} \omega \in A} \in \FF_T$
That is, we want to show that:
:$\... | Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Algebra | https://proofwiki.org/wiki/Adapted_Stochastic_Process_at_Stopping_Time_is_Measurable_with_respect_to_Stopped_Sigma-Algebra | https://proofwiki.org/wiki/Adapted_Stochastic_Process_at_Stopping_Time_is_Measurable_with_respect_to_Stopped_Sigma-Algebra | [
"Adapted Stochastic Processes",
"Stopped Sigma-Algebras",
"Stopping Times",
"Adapted Stochastic Processes"
] | [
"Definition:Filtered Probability Space",
"Definition:Adapted Stochastic Process",
"Definition:Stopping Time",
"Definition:Adapted Stochastic Process",
"Definition:Adapted Stochastic Process at Stopping Time",
"Definition:Stopped Sigma-Algebra",
"Definition:Measurable Function"
] | [
"Definition:Borel Sigma-Algebra/Borel Set",
"Definition:Stopping Time",
"Definition:Filtration of Sigma-Algebra",
"Definition:Adapted Stochastic Process",
"Definition:Measurable Function",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Intersection/Finite Intersection",
"... |
proofwiki-20197 | Stopped Process is Adapted Stochastic Process | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $\sequence {X_n}_{n \ge 0}$ be a $\sequence {\F... | Let $n \in \Z_{\ge 0}$.
From Constant Function is Stopping Time, $n$ is a stopping time.
From Pointwise Minimum of Stopping Times is Stopping Time, $n \wedge T$ is a stopping time, where $\wedge$ is the pointwise minimum.
From Adapted Stochastic Process at Stopping Time is Measurable with respect to Stopped Sigma-Alge... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Adapted Stochastic Process|$\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process]].
Let $T$ be a [[Definition:Stopping Time|... | Let $n \in \Z_{\ge 0}$.
From [[Constant Function is Stopping Time]], $n$ is a [[Definition:Stopping Time/Discrete Time|stopping time]].
From [[Pointwise Minimum of Stopping Times is Stopping Time]], $n \wedge T$ is a [[Definition:Stopping Time/Discrete Time|stopping time]], where $\wedge$ is the [[Definition:Pointwi... | Stopped Process is Adapted Stochastic Process | https://proofwiki.org/wiki/Stopped_Process_is_Adapted_Stochastic_Process | https://proofwiki.org/wiki/Stopped_Process_is_Adapted_Stochastic_Process | [
"Stopped Processes",
"Adapted Stochastic Processes"
] | [
"Definition:Filtered Probability Space",
"Definition:Adapted Stochastic Process",
"Definition:Stopping Time",
"Definition:Adapted Stochastic Process",
"Definition:Stopped Process",
"Definition:Adapted Stochastic Process"
] | [
"Constant Function is Stopping Time",
"Definition:Stopping Time/Discrete Time",
"Pointwise Minimum of Stopping Times is Stopping Time",
"Definition:Stopping Time/Discrete Time",
"Definition:Pointwise Minimum of Mappings/Extended Real-Valued Functions",
"Adapted Stochastic Process at Stopping Time is Measu... |
proofwiki-20198 | Stopped Supermartingale is Supermartingale | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-supermartingale.
Let $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let $\sequence {X_n^T}_{n \ge 0}$ be the stopped process.
Then ... | By Stopped Process is Adapted Stochastic Process, $\sequence {X_n^T}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process.
From Integrable Adapted Stochastic Process at Stopping Time is Integrable:
:$X_n^T$ is integrable for each $n \in \Z_{\ge 0}$.
Note that by definition we have for $\omega \in \O... | Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a [[Definition:Filtered Probability Space|filtered probability space]].
Let $\sequence {X_n}_{n \ge 0}$ be an [[Definition:Supermartingale|$\sequence {\FF_n}_{n \ge 0}$-supermartingale]].
Let $T$ be a [[Definition:Stopping Time|stopping time]] with r... | By [[Stopped Process is Adapted Stochastic Process]], $\sequence {X_n^T}_{n \ge 0}$ is a [[Definition:Adapted Stochastic Process|$\sequence {\FF_n}_{n \ge 0}$-adapted stochastic process]].
From [[Integrable Adapted Stochastic Process at Stopping Time is Integrable]]:
:$X_n^T$ is [[Definition:Integrable Random Variabl... | Stopped Supermartingale is Supermartingale | https://proofwiki.org/wiki/Stopped_Supermartingale_is_Supermartingale | https://proofwiki.org/wiki/Stopped_Supermartingale_is_Supermartingale | [
"Supermartingales",
"Stopped Processes",
"Stopped Supermartingale is Supermartingale"
] | [
"Definition:Filtered Probability Space",
"Definition:Supermartingale",
"Definition:Stopping Time",
"Definition:Stopped Process",
"Definition:Supermartingale"
] | [
"Stopped Process is Adapted Stochastic Process",
"Definition:Adapted Stochastic Process",
"Integrable Adapted Stochastic Process at Stopping Time is Integrable",
"Definition:Integrable Random Variable",
"Definition:Stopping Time",
"Definition:Filtration of Sigma-Algebra",
"Characteristic Function Measur... |
proofwiki-20199 | Primitive of x squared by Cosine of x | :$\ds \int x^2 \cos x \rd x = x^2 \sin x + 2 x \cos x + 2 \sin x + C$ | From Primitive of $x^2 \cos a x$:
{{:Primitive of x squared by Cosine of a x}}
The result follows on setting $a = 1$.
{{qed}} | :$\ds \int x^2 \cos x \rd x = x^2 \sin x + 2 x \cos x + 2 \sin x + C$ | From [[Primitive of x squared by Cosine of a x|Primitive of $x^2 \cos a x$]]:
{{:Primitive of x squared by Cosine of a x}}
The result follows on setting $a = 1$.
{{qed}} | Primitive of x squared by Cosine of x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Cosine_of_x | https://proofwiki.org/wiki/Primitive_of_x_squared_by_Cosine_of_x | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of x squared by Cosine of a x"
] |
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