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-18800 | Elements of Subgroup of Dipper Semigroup are not Invertible in Dipper | Let $m, n \in \Z$ be integers such that $m, n > 0$.
Let $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ denote the dipper semigroup.
Let $\struct {H, +_{m, n} }$ be the subgroup of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ where $H = \set {k \in \N: m \le k < m + n}$
Then the elements of $\struct {H, +_... | From Identity of Subgroup of Dipper Semigroup is not Identity of Dipper, the identity of $\struct {H, +_{m, n} }$ is not an identity of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.
Hence (indirectly) from Identity of Submagma containing Identity of Magma is Same Identity, $\struct {N_{< \paren {m \mathop + n} ... | Let $m, n \in \Z$ be [[Definition:Integer|integers]] such that $m, n > 0$.
Let $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$ denote the [[Definition:Dipper Semigroup|dipper semigroup]].
Let $\struct {H, +_{m, n} }$ be the [[Definition:Subgroup|subgroup]] of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$... | From [[Identity of Subgroup of Dipper Semigroup is not Identity of Dipper]], the [[Definition:Identity Element|identity]] of $\struct {H, +_{m, n} }$ is not an [[Definition:Identity Element|identity]] of $\struct {N_{< \paren {m \mathop + n} }, +_{m, n} }$.
Hence (indirectly) from [[Identity of Submagma containing Ide... | Elements of Subgroup of Dipper Semigroup are not Invertible in Dipper | https://proofwiki.org/wiki/Elements_of_Subgroup_of_Dipper_Semigroup_are_not_Invertible_in_Dipper | https://proofwiki.org/wiki/Elements_of_Subgroup_of_Dipper_Semigroup_are_not_Invertible_in_Dipper | [
"Dipper Semigroups",
"Examples of Inverse Elements",
"Elements of Subgroup of Dipper Semigroup are not Invertible in Dipper"
] | [
"Definition:Integer",
"Definition:Dipper Semigroup",
"Definition:Subgroup",
"Definition:Element",
"Definition:Invertible Element"
] | [
"Identity of Subgroup of Dipper Semigroup is not Identity of Dipper",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Identity of Submagma containing Identity of Magma is Same Identity",
"Definition:Identity (Abstract Algebra)/Two-Si... |
proofwiki-18801 | De Rham Cohomology of Sphere | Let $S^n$ denote the $n$-Sphere.
Then the de Rham Cohomology of $S^n$ are:
:$\map {H^k} {S^n} = \begin {cases} \Z^2 & : k = 0, n = 0 \\ \Z & : k = 0 \text { or } k = n, n > 0 \\ 0 & : 0 < k < n \end {cases}$
and Higher de Rham Cohomology Vanishes.
{{explain|The condition for the middle case is not uniquely readable, an... | {{ProofWanted|Mayer–Vietoris sequence and Induction}}
Category:de Rham Cohomology
7qa3jz0yfip0lvrb06zkmt8d50t2hit | Let $S^n$ denote the [[Definition:Sphere (Topology)|$n$-Sphere]].
Then the [[Definition:De Rham Cohomology|de Rham Cohomology]] of $S^n$ are:
:$\map {H^k} {S^n} = \begin {cases} \Z^2 & : k = 0, n = 0 \\ \Z & : k = 0 \text { or } k = n, n > 0 \\ 0 & : 0 < k < n \end {cases}$
and [[Higher de Rham Cohomology Vanishes]]... | {{ProofWanted|Mayer–Vietoris sequence and Induction}}
[[Category:de Rham Cohomology]]
7qa3jz0yfip0lvrb06zkmt8d50t2hit | De Rham Cohomology of Sphere | https://proofwiki.org/wiki/De_Rham_Cohomology_of_Sphere | https://proofwiki.org/wiki/De_Rham_Cohomology_of_Sphere | [
"de Rham Cohomology"
] | [
"Definition:Sphere/Topology",
"Definition:De Rham Cohomology",
"Higher de Rham Cohomology Vanishes"
] | [
"Category:de Rham Cohomology"
] |
proofwiki-18802 | Isomorphism between Symmetry Group of Regular Pentagon and Subgroup of Symmetric Group | Let $\PP = ABCDE$ denote a regular pentagon.
Let $\struct {\PP, \circ}$ be the symmetry group of $\PP$, where the various symmetries are identified as:
:the identity mapping $e$
:the rotations $r, r^2, r^3, r^4$ of $72^\circ, 144^\circ, 216^\circ, 288^\circ$ around the center of $\PP$ anticlockwise respectively
:the re... | Let the $A$, $B$, $C$, $D$ and $E$ of $\PP$ be identified with the integers $1$, $2$, $3$, $4$ and $5$ of $S_5$.
Let each of the symmetries of $\PP$ be identified with permutations of $S_n$ according to where the symmetry moves the vertices of $\PP$.
We express these in two-row notation, which we construct by inspectio... | Let $\PP = ABCDE$ denote a [[Definition:Regular Pentagon|regular pentagon]].
Let $\struct {\PP, \circ}$ be the [[Definition:Symmetry Group of Regular Pentagon|symmetry group]] of $\PP$, where the various [[Definition:Symmetry (Geometry)|symmetries]] are identified as:
:the [[Definition:Identity Mapping|identity mappin... | Let the $A$, $B$, $C$, $D$ and $E$ of $\PP$ be identified with the [[Definition:Integer|integers]] $1$, $2$, $3$, $4$ and $5$ of $S_5$.
Let each of the [[Definition:Symmetry (Geometry)|symmetries]] of $\PP$ be identified with [[Definition:Permutation|permutations]] of $S_n$ according to where the [[Definition:Symmetry... | Isomorphism between Symmetry Group of Regular Pentagon and Subgroup of Symmetric Group | https://proofwiki.org/wiki/Isomorphism_between_Symmetry_Group_of_Regular_Pentagon_and_Subgroup_of_Symmetric_Group | https://proofwiki.org/wiki/Isomorphism_between_Symmetry_Group_of_Regular_Pentagon_and_Subgroup_of_Symmetric_Group | [
"Symmetry Group of Regular Pentagon",
"Symmetric Group on 5 Letters"
] | [
"Definition:Pentagon/Regular",
"Definition:Symmetry Group of Regular Pentagon",
"Definition:Symmetry (Geometry)",
"Definition:Identity Mapping",
"Definition:Rotation (Geometry)/Plane",
"Definition:Polygon/Regular/Center",
"Definition:Anticlockwise",
"Definition:Reflection (Geometry)/Plane",
"Definit... | [
"Definition:Integer",
"Definition:Symmetry (Geometry)",
"Definition:Permutation",
"Definition:Symmetry (Geometry)",
"Definition:Vertex",
"Definition:Permutation on n Letters/Two-Row Notation",
"Definition:Permutation",
"Definition:Element",
"Definition:Permutation on n Letters/Cycle Notation",
"De... |
proofwiki-18803 | Isomorphisms between Symmetry Groups of Isosceles Triangle and Equilateral Triangle | Let $\TT = ABC$ be an isosceles triangle whose apex is $A$.
Let $\struct {\TT, \circ}$ be the symmetry group of $\TT$, where the symmetries are identified as:
:the identity mapping $e$
:the reflection $d$ in the line through $A$ and the midpoint of $BC$.
:240px
Let $\SS = A'B'C'$ be an equilateral triangle.
We define i... | We have that $\struct {\TT, \circ}$ is of order $2$.
We also have that:
{{begin-eqn}}
{{eqn | o =
| r = \set {e, \tuple {12} }
}}
{{eqn | o =
| r = \set {e, \tuple {13} }
}}
{{eqn | o =
| r = \set {e, \tuple {23} }
}}
{{end-eqn}}
are also groups of order $2$.
From Parity Group is Only Group with 2 ... | Let $\TT = ABC$ be an [[Definition:Isosceles Triangle|isosceles triangle]] whose [[Definition:Apex of Isosceles Triangle|apex]] is $A$.
Let $\struct {\TT, \circ}$ be the [[Definition:Symmetry Group of Isosceles Triangle|symmetry group]] of $\TT$, where the [[Definition:Symmetry (Geometry)|symmetries]] are identified a... | We have that $\struct {\TT, \circ}$ is of [[Definition:Order of Group|order $2$]].
We also have that:
{{begin-eqn}}
{{eqn | o =
| r = \set {e, \tuple {12} }
}}
{{eqn | o =
| r = \set {e, \tuple {13} }
}}
{{eqn | o =
| r = \set {e, \tuple {23} }
}}
{{end-eqn}}
are also [[Definition:Group|groups]]... | Isomorphisms between Symmetry Groups of Isosceles Triangle and Equilateral Triangle | https://proofwiki.org/wiki/Isomorphisms_between_Symmetry_Groups_of_Isosceles_Triangle_and_Equilateral_Triangle | https://proofwiki.org/wiki/Isomorphisms_between_Symmetry_Groups_of_Isosceles_Triangle_and_Equilateral_Triangle | [
"Symmetry Group of Isosceles Triangle",
"Symmetry Group of Equilateral Triangle"
] | [
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles/Apex",
"Definition:Symmetry Group of Isosceles Triangle",
"Definition:Symmetry (Geometry)",
"Definition:Identity Mapping",
"Definition:Reflection (Geometry)/Plane",
"Definition:Line/Straight Line",
"Definition:Line/M... | [
"Definition:Order of Structure",
"Definition:Group",
"Definition:Order of Structure",
"Parity Group is Only Group with 2 Elements",
"Definition:Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] |
proofwiki-18804 | Set of Associating Elements forms Subsemigroup of Magma | Let $S$ be a set.
Let $\oplus$ be an operation on $S$ such that $\struct {S, \oplus}$ is a magma.
Let $T \subseteq S$ be the subset of $S$ defined as:
:$T = \set {x \in S: \paren {x \oplus y} \oplus z = x \oplus \paren {y \oplus z} }$
Suppose $T \ne \O$.
Then $\struct {T, \oplus {\restriction_T} }$ is a subsemigroup of... | Taking the semigroup axioms in turn: | Let $S$ be a [[Definition:Set|set]].
Let $\oplus$ be an [[Definition:Binary Operation|operation]] on $S$ such that $\struct {S, \oplus}$ is a [[Definition:Magma|magma]].
Let $T \subseteq S$ be the [[Definition:Subset|subset]] of $S$ defined as:
:$T = \set {x \in S: \paren {x \oplus y} \oplus z = x \oplus \paren {y ... | Taking the [[Axiom:Semigroup Axioms|semigroup axioms]] in turn: | Set of Associating Elements forms Subsemigroup of Magma | https://proofwiki.org/wiki/Set_of_Associating_Elements_forms_Subsemigroup_of_Magma | https://proofwiki.org/wiki/Set_of_Associating_Elements_forms_Subsemigroup_of_Magma | [
"Subsemigroups",
"Associativity"
] | [
"Definition:Set",
"Definition:Operation/Binary Operation",
"Definition:Magma",
"Definition:Subset",
"Definition:Subsemigroup"
] | [
"Axiom:Semigroup Axioms",
"Axiom:Semigroup Axioms"
] |
proofwiki-18805 | Subsemigroup of Cancellable Mappings is Subgroup of Invertible Mappings | Let $S$ be a set.
Let $S^S$ denote the set of mappings from $S$ to itself.
Let $\CC \subseteq S^S$ denote the set of cancellable mappings on $S$.
Let $\MM \subseteq S^S$ denote the set of invertible mappings on $S$.
Then:
:the subsemigroup $\struct {\CC, \circ}$ of $\struct {S^S, \circ}$ coincides with the subgroup $\s... | From Set of Invertible Mappings forms Symmetric Group, we have that $\struct {\MM, \circ}$ is a group.
Hence, by definition, $\struct {\MM, \circ}$ is a subgroup of $\struct {S^S, \circ}$.
Recall from Bijection iff Left and Right Inverse that a mapping is invertible {{iff}} it is a bijection.
By definition, a cancellab... | Let $S$ be a [[Definition:Set|set]].
Let $S^S$ denote the [[Definition:Set|set]] of [[Definition:Mapping|mappings]] from $S$ to itself.
Let $\CC \subseteq S^S$ denote the [[Definition:Set|set]] of [[Definition:Cancellable Mapping|cancellable mappings]] on $S$.
Let $\MM \subseteq S^S$ denote the [[Definition:Set|set... | From [[Set of Invertible Mappings forms Symmetric Group]], we have that $\struct {\MM, \circ}$ is a [[Definition:Group|group]].
Hence, by definition, $\struct {\MM, \circ}$ is a [[Definition:Subgroup|subgroup]] of $\struct {S^S, \circ}$.
Recall from [[Bijection iff Left and Right Inverse]] that a [[Definition:Mapping... | Subsemigroup of Cancellable Mappings is Subgroup of Invertible Mappings | https://proofwiki.org/wiki/Subsemigroup_of_Cancellable_Mappings_is_Subgroup_of_Invertible_Mappings | https://proofwiki.org/wiki/Subsemigroup_of_Cancellable_Mappings_is_Subgroup_of_Invertible_Mappings | [
"Cancellability",
"Inverse Mappings",
"Examples of Subgroups",
"Examples of Subsemigroups"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Set",
"Definition:Cancellable Mapping",
"Definition:Set",
"Definition:Inverse Mapping",
"Definition:Subsemigroup",
"Definition:Subgroup",
"Definition:Composition of Mappings"
] | [
"Set of Invertible Mappings forms Symmetric Group",
"Definition:Group",
"Definition:Subgroup",
"Bijection iff Left and Right Inverse",
"Definition:Mapping",
"Definition:Inverse Mapping",
"Definition:Bijection",
"Definition:Cancellable Mapping",
"Definition:Mapping",
"Definition:Left Cancellable Ma... |
proofwiki-18806 | Mapping is Idempotent iff Restriction to Image is Identity Mapping | Let $S$ be a set.
Let $S^S$ denote the set of mappings from $S$ to itself.
Let $f \in S^S$ be a mapping on $S$.
Then:
:$f$ is idempotent
{{iff}}:
:the restriction of $f$ to $\Img f$ is the identity mapping. | Recall the definitions:
:$\Img f$ denotes the image set of $f$
:The identity mapping $I_S$ is defined as:
::$\forall x \in f: \map {I_S} x = x$
:An idempotent mapping is a mapping with the property:
::$f \circ f = f$
:where $\circ$ denotes composition of mappings. | Let $S$ be a [[Definition:Set|set]].
Let $S^S$ denote the [[Definition:Set|set]] of [[Definition:Mapping|mappings]] from $S$ to itself.
Let $f \in S^S$ be a [[Definition:Mapping|mapping]] on $S$.
Then:
:$f$ is [[Definition:Idempotent Mapping|idempotent]]
{{iff}}:
:the [[Definition:Restriction of Mapping|restriction... | Recall the definitions:
:$\Img f$ denotes the [[Definition:Image of Mapping|image set of $f$]]
:The [[Definition:Identity Mapping|identity mapping]] $I_S$ is defined as:
::$\forall x \in f: \map {I_S} x = x$
:An [[Definition:Idempotent Mapping|idempotent mapping]] is a [[Definition:Mapping|mapping]] with the propert... | Mapping is Idempotent iff Restriction to Image is Identity Mapping | https://proofwiki.org/wiki/Mapping_is_Idempotent_iff_Restriction_to_Image_is_Identity_Mapping | https://proofwiki.org/wiki/Mapping_is_Idempotent_iff_Restriction_to_Image_is_Identity_Mapping | [
"Identity Mappings",
"Idempotence",
"Restrictions"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Idempotence/Mapping",
"Definition:Restriction/Mapping",
"Definition:Identity Mapping"
] | [
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Identity Mapping",
"Definition:Idempotence/Mapping",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Identity Mapping",
"Definition:Idempotence/Mapping",
"Definition:Idempotence/Mapping",
"Definition:Identity Mappin... |
proofwiki-18807 | Non-Cancellable Elements of Semigroup form Subsemigroup | Let $\struct {S, \circ}$ be a semigroup.
Let $T \subseteq S$ be the subset of $S$ containing the elements of $S$ which are specifically not cancellable in $\struct {S, \circ}$.
Then $\struct {T, \circ}$ forms a subsemigroup of $S$. | Recall the definition of cancellable element:
An element $x \in \struct {S, \circ}$ is '''cancellable''' {{iff}}:
:$\forall a, b \in S: x \circ a = x \circ b \implies a = b$
:$\forall a, b \in S: a \circ x = b \circ x \implies a = b$
From the Subsemigroup Closure Test it is sufficient to demonstrate that:
:$\forall x, ... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $T \subseteq S$ be the [[Definition:Subset|subset]] of $S$ containing the [[Definition:Element|elements]] of $S$ which are specifically not [[Definition:Cancellable Element|cancellable]] in $\struct {S, \circ}$.
Then $\struct {T, \circ}$ forms a [... | Recall the definition of [[Definition:Cancellable Element|cancellable element]]:
An [[Definition:Element|element]] $x \in \struct {S, \circ}$ is '''[[Definition:Cancellable Element|cancellable]]''' {{iff}}:
:$\forall a, b \in S: x \circ a = x \circ b \implies a = b$
:$\forall a, b \in S: a \circ x = b \circ x \implies... | Non-Cancellable Elements of Semigroup form Subsemigroup | https://proofwiki.org/wiki/Non-Cancellable_Elements_of_Semigroup_form_Subsemigroup | https://proofwiki.org/wiki/Non-Cancellable_Elements_of_Semigroup_form_Subsemigroup | [
"Cancellability",
"Subsemigroups"
] | [
"Definition:Semigroup",
"Definition:Subset",
"Definition:Element",
"Definition:Cancellable Element",
"Definition:Subsemigroup"
] | [
"Definition:Cancellable Element",
"Definition:Element",
"Definition:Cancellable Element",
"Subsemigroup Closure Test",
"Definition:Cancellable Element"
] |
proofwiki-18808 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6 | :$\exists r \in \Q, n \in \Z, y \in \Z_p$:
::$(1) \quad x = r + p^n y$
::$(2) \quad$ the canonical expansion of $y$ is periodic. | Let $\ldots d_i \ldots d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_{-m}$ be the canonical expansion of $x$.
By definition of eventually periodic there exists a finite sequence of $k$ digits of $x$:
:$\tuple {d_{n + k - 1} \ldots d_{n + 1} d_n }$
such that $n \ge 0$ and for all $s \in \Z_{\ge 0}$ and for all $j \in \set {0, 2,... | :$\exists r \in \Q, n \in \Z, y \in \Z_p$:
::$(1) \quad x = r + p^n y$
::$(2) \quad$ the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $y$ is [[Definition:Periodic P-adic Expansion|periodic]]. | Let $\ldots d_i \ldots d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_{-m}$ be the [[Definition:Canonical P-adic Expansion|canonical expansion]] of $x$.
By definition of [[Definition:Eventually Periodic P-adic Expansion|eventually periodic]] there exists a [[Definition:Finite Sequence|finite sequence]] of $k$ [[Definition:Digi... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_6 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_6 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [
"Definition:Canonical P-adic Expansion",
"Definition:Periodic P-adic Expansion"
] | [
"Definition:Canonical P-adic Expansion",
"Definition:Eventually Periodic P-adic Expansion",
"Definition:Finite Sequence",
"Definition:Digit",
"Definition:Common Divisor/Integers",
"Definition:Series",
"Definition:P-adic Integer",
"Definition:Canonical P-adic Expansion",
"Definition:Canonical P-adic ... |
proofwiki-18809 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 7 | :$1 + p^k + p^{2 k} + p^{3 k} + \cdots = \dfrac 1 {1 - p^k}$ | Let $S_n$ be the partial sum:
:$\ds S_n = \sum_{j \mathop = 0}^n p^{j k}$
We have:
{{begin-eqn}}
{{eqn | l = \paren {1 - p^k} S_n
| r = \paren {1 - p^k} \sum_{j \mathop = 0}^n p^{j k}
}}
{{eqn | r = \paren {\sum_{j \mathop = 0}^n p^{j k} } - p^k \paren {\sum_{j \mathop = 0}^n p^{j k} }
| c = distributing th... | :$1 + p^k + p^{2 k} + p^{3 k} + \cdots = \dfrac 1 {1 - p^k}$ | Let $S_n$ be the [[Definition:Partial Sum|partial sum]]:
:$\ds S_n = \sum_{j \mathop = 0}^n p^{j k}$
We have:
{{begin-eqn}}
{{eqn | l = \paren {1 - p^k} S_n
| r = \paren {1 - p^k} \sum_{j \mathop = 0}^n p^{j k}
}}
{{eqn | r = \paren {\sum_{j \mathop = 0}^n p^{j k} } - p^k \paren {\sum_{j \mathop = 0}^n p^{j k} ... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 7 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_7 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_7 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Definition:Series/Sequence of Partial Sums",
"Definition:Series/Sequence of Partial Sums",
"Definition:Series/Sequence of Partial Sums",
"Definition:Series",
"Definition:Telescoping Series",
"Properties of Norm on Division Ring/Norm of Negative",
"Combination Theorem for Sequences/Normed Division Ring/... |
proofwiki-18810 | Locally Integrable f(x+ct) is Weak Solution to Transport Equation | Consider the transport equation:
:$\dfrac {\partial u} {\partial t} - c \dfrac {\partial u} {\partial x} = 0$
with the initial condition:
:$\map u {x, 0} = \map f x$
where $c \in \R$.
Then it has a weak solution of the form:
:$\map u {x, t} := \map f {x + ct}$
where $f \in \map {L^1_{loc} } \R$ is a locally integrable... | Let $\map u {x, t} = \map f {x + ct}$ be a locally integrable function.
We have that a locally integrable function defines a distribution.
Let $T_u \in \map {\DD'} {\R^2}$ be a Schwartz distribution associated with $u$.
Let $\phi \in \map \DD {\R^2}$ be a test function.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {\partial}... | Consider the [[Definition:Transport Equation|transport equation]]:
:$\dfrac {\partial u} {\partial t} - c \dfrac {\partial u} {\partial x} = 0$
with the [[Definition:Initial Condition|initial condition]]:
:$\map u {x, 0} = \map f x$
where $c \in \R$.
Then it has a [[Definition:Weak Solution|weak solution]] of the... | Let $\map u {x, t} = \map f {x + ct}$ be a [[Definition:Locally Integrable Function|locally integrable function]].
We have that a [[Locally Integrable Function defines Distribution|locally integrable function defines a distribution]].
Let $T_u \in \map {\DD'} {\R^2}$ be a [[Definition:Schwartz Distribution|Schwartz d... | Locally Integrable f(x+ct) is Weak Solution to Transport Equation | https://proofwiki.org/wiki/Locally_Integrable_f(x+ct)_is_Weak_Solution_to_Transport_Equation | https://proofwiki.org/wiki/Locally_Integrable_f(x+ct)_is_Weak_Solution_to_Transport_Equation | [
"Examples of Weak Solutions"
] | [
"Definition:Transport Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Weak Solution",
"Definition:Integrable Function/Locally Integrable Function"
] | [
"Definition:Integrable Function/Locally Integrable Function",
"Locally Integrable Function defines Distribution",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Variable/Real",
"Definition:Mapping",
"Definition:Variable/Real",
"Definition:Real Function",
"Change of Varia... |
proofwiki-18811 | Smooth Vector Field as Sum of Smooth Horizontal and Vertical Vector Fields | Let $\tilde M, M$ be smooth manifolds.
Let $\pi : \tilde M \to M$ be a smooth submersion.
Let $\tilde g$ be a Riemannian metric on $\tilde M$.
Let $W$ be a smooth vector field on $\tilde M$.
Then $W$ can be uniquely decomposed as a sum:
:$W = W^H + W^V$
where $W^H$ and $W^V$ are smooth horizontal and vertical vector fi... | {{ProofWanted|some of the assumptions may be unnecessary}} | Let $\tilde M, M$ be [[Definition:Smooth Manifold|smooth manifolds]].
Let $\pi : \tilde M \to M$ be a [[Definition:Submersion|smooth submersion]].
Let $\tilde g$ be a [[Definition:Riemannian Metric|Riemannian metric]] on $\tilde M$.
Let $W$ be a [[Definition:Smooth Vector Field|smooth vector field]] on $\tilde M$.
... | {{ProofWanted|some of the assumptions may be unnecessary}} | Smooth Vector Field as Sum of Smooth Horizontal and Vertical Vector Fields | https://proofwiki.org/wiki/Smooth_Vector_Field_as_Sum_of_Smooth_Horizontal_and_Vertical_Vector_Fields | https://proofwiki.org/wiki/Smooth_Vector_Field_as_Sum_of_Smooth_Horizontal_and_Vertical_Vector_Fields | [
"Riemannian Geometry"
] | [
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Submersion",
"Definition:Riemannian Metric",
"Definition:Smooth Vector Field",
"Definition:Unique",
"Definition:Smooth Vector Field",
"Definition:Horizontal Vector Field",
"Definition:Vertical Vector Field",
"Definition:Vector Field"
] | [] |
proofwiki-18812 | Sets of Operations on Set of 3 Elements/Automorphism Group of A | :$\AA$ has $3$ elements. | Recall the definition of (group) automorphism:
:$\phi$ is an automorphism on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a permutation of $S$
::$\phi$ is a homomorphism on $\struct {S, \circ}$: $\forall a, b \in S: \map \phi {a \circ b} = \map \phi a \circ \map \phi b$
Hence $\AA$ can be defined as the set of operations ... | :$\AA$ has $3$ [[Definition:Element|elements]]. | Recall the definition of [[Definition:Group Automorphism|(group) automorphism]]:
:$\phi$ is an [[Definition:Group Automorphism|automorphism]] on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a [[Definition:Permutation|permutation]] of $S$
::$\phi$ is a [[Definition:Group Homomorphism|homomorphism]] on $\struct {S, \circ}$... | Sets of Operations on Set of 3 Elements/Automorphism Group of A | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_A | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_A | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Element"
] | [
"Definition:Group Automorphism",
"Definition:Group Automorphism",
"Definition:Permutation",
"Definition:Group Homomorphism",
"Definition:Set",
"Definition:Operation/Binary Operation",
"Definition:Permutation",
"Definition:Group Automorphism",
"Definition:Set",
"Definition:Permutation",
"Definiti... |
proofwiki-18813 | Sets of Operations on Set of 3 Elements/Automorphism Group of B | :$\BB$ has $3^3 - 3$ elements. | Recall the definition of (group) automorphism:
:$\phi$ is an automorphism on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a permutation of $S$
::$\phi$ is a homomorphism on $\struct {S, \circ}$: $\forall a, b \in S: \map \phi {a \circ b} = \map \phi a \circ \map \phi b$
From Identity Mapping is Group Automorphism, $I_S$ i... | :$\BB$ has $3^3 - 3$ [[Definition:Element|elements]]. | Recall the definition of [[Definition:Group Automorphism|(group) automorphism]]:
:$\phi$ is an [[Definition:Group Automorphism|automorphism]] on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a [[Definition:Permutation|permutation]] of $S$
::$\phi$ is a [[Definition:Group Homomorphism|homomorphism]] on $\struct {S, \circ}$... | Sets of Operations on Set of 3 Elements/Automorphism Group of B | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_B | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_B | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Element"
] | [
"Definition:Group Automorphism",
"Definition:Group Automorphism",
"Definition:Permutation",
"Definition:Group Homomorphism",
"Identity Mapping is Automorphism/Groups",
"Definition:Group Automorphism",
"Definition:Element",
"Definition:Group Product/Product Element",
"Definition:Group Product/Product... |
proofwiki-18814 | Sets of Operations on Set of 3 Elements/Automorphism Group of C n | :Each of $\CC_1$, $\CC_2$ and $\CC_3$ has $3^4 - 3$ elements. | Recall the definition of (group) automorphism:
:$\phi$ is an automorphism on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a permutation of $S$
::$\phi$ is a homomorphism on $\struct {S, \circ}$: $\forall a, b \in S: \map \phi {a \circ b} = \map \phi a \circ \map \phi b$
From Identity Mapping is Group Automorphism, $I_S$ i... | :Each of $\CC_1$, $\CC_2$ and $\CC_3$ has $3^4 - 3$ [[Definition:Element|elements]]. | Recall the definition of [[Definition:Group Automorphism|(group) automorphism]]:
:$\phi$ is an [[Definition:Group Automorphism|automorphism]] on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a [[Definition:Permutation|permutation]] of $S$
::$\phi$ is a [[Definition:Group Homomorphism|homomorphism]] on $\struct {S, \circ}$... | Sets of Operations on Set of 3 Elements/Automorphism Group of C n | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Element"
] | [
"Definition:Group Automorphism",
"Definition:Group Automorphism",
"Definition:Permutation",
"Definition:Group Homomorphism",
"Identity Mapping is Automorphism/Groups",
"Definition:Group Automorphism",
"Definition:Operation/Binary Operation",
"Definition:Group Automorphism",
"Definition:Permutation",... |
proofwiki-18815 | Sets of Operations on Set of 3 Elements/Automorphism Group of D | :$\DD$ has $19 \, 422$ elements. | Let $n$ denote the cardinality of $\DD$.
Equivalently, $n$ equals the number of operations $\circ$ on $S$ on which the only automorphism is $I_S$.
Recall these definitions:
Let $\AA$, $\BB$, $\CC_1$, $\CC_2$ and $\CC_3$ be respectively the set of all operations $\circ$ on $S$ such that the groups of automorphisms of $\... | :$\DD$ has $19 \, 422$ [[Definition:Element|elements]]. | Let $n$ denote the [[Definition:Cardinality|cardinality]] of $\DD$.
Equivalently, $n$ equals the number of [[Definition:Binary Operation|operations]] $\circ$ on $S$ on which the only [[Definition:Group Automorphism|automorphism]] is $I_S$.
Recall these definitions:
Let $\AA$, $\BB$, $\CC_1$, $\CC_2$ and $\CC_3$ be ... | Sets of Operations on Set of 3 Elements/Automorphism Group of D | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_D | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_D | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Element"
] | [
"Definition:Cardinality",
"Definition:Operation/Binary Operation",
"Definition:Group Automorphism",
"Definition:Set",
"Definition:Operation/Binary Operation",
"Definition:Group",
"Definition:Group Automorphism",
"Definition:Symmetric Group",
"Definition:Identity Mapping",
"Definition:Operation/Bin... |
proofwiki-18816 | Equivalence of Definitions of Quaternion Modulus | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion, where $a, b, c, d \in \R$.
{{TFAE|def = Quaternion Modulus}} | Let $\mathbf x = \begin{bmatrix} a + b i & c + d i \\ -c + d i & a - b i \end{bmatrix}$ be the matrix form of quaternion $\mathbf x$.
{{begin-eqn}}
{{eqn | l = \size {\mathbf x}
| r = \sqrt {\map \det {\mathbf x} }
| c =
}}
{{eqn | r = \sqrt {\map \det {\begin{bmatrix} a + b i & c + d i \\ -c + d i & a - b ... | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a [[Definition:Quaternion|quaternion]], where $a, b, c, d \in \R$.
{{TFAE|def = Quaternion Modulus}} | Let $\mathbf x = \begin{bmatrix} a + b i & c + d i \\ -c + d i & a - b i \end{bmatrix}$ be the [[Matrix Form of Quaternion|matrix form of quaternion $\mathbf x$]].
{{begin-eqn}}
{{eqn | l = \size {\mathbf x}
| r = \sqrt {\map \det {\mathbf x} }
| c =
}}
{{eqn | r = \sqrt {\map \det {\begin{bmatrix} a + b i... | Equivalence of Definitions of Quaternion Modulus | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Quaternion_Modulus | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Quaternion_Modulus | [
"Quaternion Modulus"
] | [
"Definition:Quaternion"
] | [
"Matrix Form of Quaternion",
"Category:Quaternion Modulus"
] |
proofwiki-18817 | Smooth Vector Field has Unique Smooth Horizontal Lift | Let $\tilde M, M$ be smooth manifolds.
Let $\pi : \tilde M \to M$ be a smooth submersion.
Let $\tilde g$ be a Riemannian metric on $\tilde M$.
Let $W$ be a smooth vector field on $M$.
Then $W$ has the unique smooth horizontal lift to $\tilde M$. | {{ProofWanted|some of the assumptions may be unnecessary}} | Let $\tilde M, M$ be [[Definition:Smooth Manifold|smooth manifolds]].
Let $\pi : \tilde M \to M$ be a [[Definition:Submersion|smooth submersion]].
Let $\tilde g$ be a [[Definition:Riemannian Metric|Riemannian metric]] on $\tilde M$.
Let $W$ be a [[Definition:Smooth Vector Field|smooth vector field]] on $M$.
Then $... | {{ProofWanted|some of the assumptions may be unnecessary}} | Smooth Vector Field has Unique Smooth Horizontal Lift | https://proofwiki.org/wiki/Smooth_Vector_Field_has_Unique_Smooth_Horizontal_Lift | https://proofwiki.org/wiki/Smooth_Vector_Field_has_Unique_Smooth_Horizontal_Lift | [
"Riemannian Geometry"
] | [
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Submersion",
"Definition:Riemannian Metric",
"Definition:Smooth Vector Field",
"Definition:Unique",
"Definition:Smooth Vector Field",
"Definition:Horizontal Lift"
] | [] |
proofwiki-18818 | Sets of Operations on Set of 3 Elements/Automorphism Group of C n/Lemma 1 | $c$ is an idempotent element under $\circ$, that is:
:$c \circ c = c$ | Recall the definition of (group) automorphism:
:$\phi$ is an automorphism on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a permutation of $S$
::$\phi$ is a homomorphism on $\struct {S, \circ}$: $\forall a, b \in S: \map \phi {a \circ b} = \map \phi a \circ \map \phi b$
Let us denote $\tuple {a, b}$ as the mapping $r: S \... | $c$ is an [[Definition:Idempotent Element|idempotent element]] under $\circ$, that is:
:$c \circ c = c$ | Recall the definition of [[Definition:Group Automorphism|(group) automorphism]]:
:$\phi$ is an [[Definition:Group Automorphism|automorphism]] on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a [[Definition:Permutation|permutation]] of $S$
::$\phi$ is a [[Definition:Group Homomorphism|homomorphism]] on $\struct {S, \circ}$... | Sets of Operations on Set of 3 Elements/Automorphism Group of C n/Lemma 1 | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n/Lemma_1 | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n/Lemma_1 | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Idempotence/Element"
] | [
"Definition:Group Automorphism",
"Definition:Group Automorphism",
"Definition:Permutation",
"Definition:Group Homomorphism",
"Definition:Mapping",
"Definition:Idempotence/Element",
"Definition:Idempotence/Element",
"Category:Sets of Operations on Set of 3 Elements"
] |
proofwiki-18819 | Sets of Operations on Set of 3 Elements/Automorphism Group of C n/Lemma 2 | {{begin-eqn}}
{{eqn | l = a \circ a = a
| o = \iff
| r = b \circ b = b
}}
{{eqn | l = a \circ a = b
| o = \iff
| r = b \circ b = a
}}
{{eqn | l = a \circ a = c
| o = \iff
| r = b \circ b = c
}}
{{eqn | l = a \circ b = a
| o = \iff
| r = b \circ a = b
}}
{{eqn | l = a \cir... | Recall the definition of (group) automorphism:
:$\phi$ is an automorphism on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a permutation of $S$
::$\phi$ is a homomorphism on $\struct {S, \circ}$: $\forall a, b \in S: \map \phi {a \circ b} = \map \phi a \circ \map \phi b$
Let us denote $\tuple {a, b}$ as the mapping $r: S \... | {{begin-eqn}}
{{eqn | l = a \circ a = a
| o = \iff
| r = b \circ b = b
}}
{{eqn | l = a \circ a = b
| o = \iff
| r = b \circ b = a
}}
{{eqn | l = a \circ a = c
| o = \iff
| r = b \circ b = c
}}
{{eqn | l = a \circ b = a
| o = \iff
| r = b \circ a = b
}}
{{eqn | l = a \cir... | Recall the definition of [[Definition:Group Automorphism|(group) automorphism]]:
:$\phi$ is an [[Definition:Group Automorphism|automorphism]] on $\struct {S, \circ}$ {{iff}}:
::$\phi$ is a [[Definition:Permutation|permutation]] of $S$
::$\phi$ is a [[Definition:Group Homomorphism|homomorphism]] on $\struct {S, \circ}$... | Sets of Operations on Set of 3 Elements/Automorphism Group of C n/Lemma 2 | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n/Lemma_2 | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n/Lemma_2 | [
"Sets of Operations on Set of 3 Elements"
] | [] | [
"Definition:Group Automorphism",
"Definition:Group Automorphism",
"Definition:Permutation",
"Definition:Group Homomorphism",
"Definition:Mapping",
"Sets of Operations on Set of 3 Elements/Automorphism Group of C n/Lemma 1",
"Definition:Group Product/Product Element",
"Category:Sets of Operations on Se... |
proofwiki-18820 | Sets of Operations on Set of 3 Elements/Automorphism Group of A/Isomorphism Classes | :The elements of $\AA$ are each in its own isomorphism class. | Recall from Automorphism Group of $\AA$ the elements of $\AA$, expressed in Cayley table form:
:$\begin {array} {c|ccc}
\to & a & b & c \\
\hline
a & a & b & c \\
b & a & b & c \\
c & a & b & c \\
\end {array}
\qquad
\begin {array} {c|ccc}
\gets & a & b & c \\
\hline
a & a & a & a \\
b & b & b & b \\
c & c & c & c \\
\... | :The [[Definition:Element|elements]] of $\AA$ are each in its own [[Definition:Isomorphism Class (Algebraic Structures)|isomorphism class]]. | Recall from [[Sets of Operations on Set of 3 Elements/Automorphism Group of A|Automorphism Group of $\AA$]] the [[Definition:Element|elements]] of $\AA$, expressed in [[Definition:Cayley Table|Cayley table]] form:
:$\begin {array} {c|ccc}
\to & a & b & c \\
\hline
a & a & b & c \\
b & a & b & c \\
c & a & b & c \\
\en... | Sets of Operations on Set of 3 Elements/Automorphism Group of A/Isomorphism Classes | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_A/Isomorphism_Classes | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_A/Isomorphism_Classes | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Element",
"Definition:Isomorphism Class (Algebraic Structures)"
] | [
"Sets of Operations on Set of 3 Elements/Automorphism Group of A",
"Definition:Element",
"Definition:Cayley Table",
"Algebraic Structures formed by Left and Right Operations are not Isomorphic for Cardinality Greater than 1",
"Definition:Isomorphism (Abstract Algebra)",
"Sets of Operations on Set of 3 Ele... |
proofwiki-18821 | Sets of Operations on Set of 3 Elements/Isomorphism Classes | Let $\MM$ be the set of all operations $\circ$ on $S$.
Then the elements of $\MM$ are divided in $3330$ isomorphism classes.
That is, up to isomorphism, there are $3330$ operations on $S$. | From Automorphism Group of $\AA$: Isomorphism Classes:
:each element of $\AA$ is in its own isomorphism class.
Hence $\AA$ contributes $3$ isomorphism classes.
From Automorphism Group of $\BB$: Isomorphism Classes:
:the $24$ elements of $\BB$ form $12$ isomorphism classes in pairs.
From Automorphism Group of $\CC_n$: I... | Let $\MM$ be the [[Definition:Set|set]] of all [[Definition:Binary Operation|operations]] $\circ$ on $S$.
Then the [[Definition:Element|elements]] of $\MM$ are divided in $3330$ [[Definition:Isomorphism Class (Algebraic Structures)|isomorphism classes]].
That is, up to [[Definition:Isomorphism (Abstract Algebra)|isom... | From [[Sets of Operations on Set of 3 Elements/Automorphism Group of A/Isomorphism Classes|Automorphism Group of $\AA$: Isomorphism Classes]]:
:each [[Definition:Element|element]] of $\AA$ is in its own [[Definition:Isomorphism Class (Algebraic Structures)|isomorphism class]].
Hence $\AA$ contributes $3$ [[Definition:... | Sets of Operations on Set of 3 Elements/Isomorphism Classes | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Isomorphism_Classes | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Isomorphism_Classes | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Set",
"Definition:Operation/Binary Operation",
"Definition:Element",
"Definition:Isomorphism Class (Algebraic Structures)",
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Operation/Binary Operation"
] | [
"Sets of Operations on Set of 3 Elements/Automorphism Group of A/Isomorphism Classes",
"Definition:Element",
"Definition:Isomorphism Class (Algebraic Structures)",
"Definition:Isomorphism Class (Algebraic Structures)",
"Sets of Operations on Set of 3 Elements/Automorphism Group of B/Isomorphism Classes",
... |
proofwiki-18822 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 8 | :$\ds \lim_{n \mathop \to \infty} \dfrac a {p^{n+1}} = 0$ | From Sequence of Powers of Rational Number less than One:
:$\ds \lim_{n \mathop \to \infty} \dfrac 1 {p^n} = 0$
From Multiple Rule for Sequences:
:$\ds \lim_{n \mathop \to \infty} \dfrac a p \cdot \paren{\dfrac 1 {p^n} } = \dfrac a p \cdot 0 = 0$
The result follows.
{{qed}}
Category:Canonical P-adic Expansion of Ration... | :$\ds \lim_{n \mathop \to \infty} \dfrac a {p^{n+1}} = 0$ | From [[Sequence of Powers of Rational Number less than One]]:
:$\ds \lim_{n \mathop \to \infty} \dfrac 1 {p^n} = 0$
From [[Multiple Rule for Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \dfrac a p \cdot \paren{\dfrac 1 {p^n} } = \dfrac a p \cdot 0 = 0$
The result follows.
{{qed}}
[[Category:Canonical P-adic Expans... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 8 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_8 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_8 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Sequence of Powers of Number less than One/Rational Numbers",
"Combination Theorem for Sequences/Multiple Rule",
"Category:Canonical P-adic Expansion of Rational is Eventually Periodic"
] |
proofwiki-18823 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 9 | :$\ds \lim_{n \mathop \to \infty} \dfrac {a - \paren{p^{n+1} - 1} b } {p^{n+1}} = -b$ | From Sequence of Reciprocals is Null Sequence:
:$\ds \lim_{n \mathop \to \infty} \dfrac 1 n = 0$
From Combined Sum Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \dfrac {n - 1} n = \lim_{n \mathop \to \infty} 1 - \dfrac 1 n = 1$
From Limit of Subsequence equals Limit of Real Sequence:
:$\ds \lim_{n \matho... | :$\ds \lim_{n \mathop \to \infty} \dfrac {a - \paren{p^{n+1} - 1} b } {p^{n+1}} = -b$ | From [[Sequence of Reciprocals is Null Sequence]]:
:$\ds \lim_{n \mathop \to \infty} \dfrac 1 n = 0$
From [[Combined Sum Rule for Real Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \dfrac {n - 1} n = \lim_{n \mathop \to \infty} 1 - \dfrac 1 n = 1$
From [[Limit of Subsequence equals Limit of Real Sequence]]:
:$\ds ... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 9 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_9 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_9 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Sequence of Powers of Reciprocals is Null Sequence/Corollary",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Limit of Subsequence equals Limit of Sequence/Real Numbers",
"Combination Theorem for Sequences/Real/Combined Sum Rule"
] |
proofwiki-18824 | Sets of Operations on Set of 3 Elements/Automorphism Group of A/Operations with Identity | :None of the operations of $\AA$ has an identity element. | Recall from Automorphism Group of $\AA$ the elements of $\AA$, expressed in Cayley table form:
:$\begin{array}{c|ccc}
\to & a & b & c \\
\hline
a & a & b & c \\
b & a & b & c \\
c & a & b & c \\
\end{array}
\qquad
\begin{array}{c|ccc}
\gets & a & b & c \\
\hline
a & a & a & a \\
b & b & b & b \\
c & c & c & c \\
\end{a... | :None of the [[Definition:Binary Operation|operations]] of $\AA$ has an [[Definition:Identity Element|identity element]]. | Recall from [[Sets of Operations on Set of 3 Elements/Automorphism Group of A|Automorphism Group of $\AA$]] the [[Definition:Element|elements]] of $\AA$, expressed in [[Definition:Cayley Table|Cayley table]] form:
:$\begin{array}{c|ccc}
\to & a & b & c \\
\hline
a & a & b & c \\
b & a & b & c \\
c & a & b & c \\
\end{... | Sets of Operations on Set of 3 Elements/Automorphism Group of A/Operations with Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_A/Operations_with_Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_A/Operations_with_Identity | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Sets of Operations on Set of 3 Elements/Automorphism Group of A",
"Definition:Element",
"Definition:Cayley Table"
] |
proofwiki-18825 | Sets of Operations on Set of 3 Elements/Automorphism Group of B/Operations with Identity | :None of the operations of $\BB$ has an identity element. | Recall Automorphism Group of $\BB$.
Consider each of the categories of $\BB$ induced by each of $a \circ a$, $a \circ b$ and $a \circ c$, illustrated by the partially-filled Cayley tables to which they give rise:
;$(1): \quad a \circ a$
:$\begin {array} {c|ccc}
\circ & a & b & c \\
\hline
a & a & & \\
b & & b & ... | :None of the [[Definition:Binary Operation|operations]] of $\BB$ has an [[Definition:Identity Element|identity element]]. | Recall [[Sets of Operations on Set of 3 Elements/Automorphism Group of B|Automorphism Group of $\BB$]].
Consider each of the categories of $\BB$ induced by each of $a \circ a$, $a \circ b$ and $a \circ c$, illustrated by the partially-filled [[Definition:Cayley Table|Cayley tables]] to which they give rise:
;$(1): ... | Sets of Operations on Set of 3 Elements/Automorphism Group of B/Operations with Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_B/Operations_with_Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_B/Operations_with_Identity | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Sets of Operations on Set of 3 Elements/Automorphism Group of B",
"Definition:Cayley Table",
"Definition:Cayley Table",
"Definition:Operation/Binary Operation",
"Definition:Cayley Table",
"Definition:Cayley Table",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-S... |
proofwiki-18826 | Sets of Operations on Set of 3 Elements/Automorphism Group of C n/Operations with Identity | :$9$ of the operations of each of $\CC_1$, $\CC_2$ and $\CC_3$ has an identity element. | {{WLOG}}, we will analyse the nature of $\CC_1$.
Recall this lemma: | :$9$ of the [[Definition:Binary Operation|operations]] of each of $\CC_1$, $\CC_2$ and $\CC_3$ has an [[Definition:Identity Element|identity element]]. | {{WLOG}}, we will analyse the nature of $\CC_1$.
Recall this [[Definition:Lemma|lemma]]: | Sets of Operations on Set of 3 Elements/Automorphism Group of C n/Operations with Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n/Operations_with_Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n/Operations_with_Identity | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Lemma"
] |
proofwiki-18827 | Element of Horizontal Space as Horizontal Lift of Vector Field | Let $\tilde M, M$ be smooth manifolds.
Let $\pi : \tilde M \to M$ be a smooth submersion.
Let $\tilde g$ be a Riemannian metric on $\tilde M$.
Let $H_x$ be a horizontal tangent space of $\tilde M$ at $x$.
Let $\map {\mathfrak{X}} M$ be the space of smooth vector fields of $M$.
Then for every $x \in \tilde M$ and every ... | {{ProofWanted|some of the assumptions may be unnecessary}} | Let $\tilde M, M$ be [[Definition:Smooth Manifold|smooth manifolds]].
Let $\pi : \tilde M \to M$ be a [[Definition:Submersion|smooth submersion]].
Let $\tilde g$ be a [[Definition:Riemannian Metric|Riemannian metric]] on $\tilde M$.
Let $H_x$ be a [[Definition:Horizontal Tangent Space|horizontal tangent space]] of $... | {{ProofWanted|some of the assumptions may be unnecessary}} | Element of Horizontal Space as Horizontal Lift of Vector Field | https://proofwiki.org/wiki/Element_of_Horizontal_Space_as_Horizontal_Lift_of_Vector_Field | https://proofwiki.org/wiki/Element_of_Horizontal_Space_as_Horizontal_Lift_of_Vector_Field | [
"Riemannian Geometry"
] | [
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Submersion",
"Definition:Riemannian Metric",
"Definition:Horizontal Tangent Space",
"Definition:Space of Smooth Vector Fields on Riemannian Manifold",
"Definition:Vector Field",
"Definition:Horizontal Lift"
] | [] |
proofwiki-18828 | Sets of Operations on Set of 3 Elements/Automorphism Group of D/Operations with Identity | :$216$ of the operations of $\DD$ has an identity element. | Let $n$ denote the number of operations of $\DD$ which have an identity element.
Recall these definitions:
Let $\AA$, $\BB$, $\CC_1$, $\CC_2$ and $\CC_3$ be respectively the set of all operations $\circ$ on $S$ such that the groups of automorphisms of $\struct {S, \circ}$ are as follows:
{{begin-eqn}}
{{eqn | l = \AA
... | :$216$ of the [[Definition:Binary Operation|operations]] of $\DD$ has an [[Definition:Identity Element|identity element]]. | Let $n$ denote the number of [[Definition:Binary Operation|operations]] of $\DD$ which have an [[Definition:Identity Element|identity element]].
Recall these definitions:
Let $\AA$, $\BB$, $\CC_1$, $\CC_2$ and $\CC_3$ be respectively the [[Definition:Set|set]] of all [[Definition:Binary Operation|operations]] $\circ... | Sets of Operations on Set of 3 Elements/Automorphism Group of D/Operations with Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_D/Operations_with_Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_D/Operations_with_Identity | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Set",
"Definition:Operation/Binary Operation",
"Definition:Group",
"Definition:Group Automorphism",
"Definition:Symmetric Group",
"Definition:Identity Mapping",
"Definition:Operation/Bin... |
proofwiki-18829 | Sets of Operations on Set of 3 Elements/Operations with Identity | Let $\NN$ be the set of all operations $\circ$ on $S$ which have an identity element.
Then the elements of $\NN$ are divided in $45$ isomorphism classes.
That is, up to isomorphism, there are $45$ operations on $S$ which have an identity element. | From Automorphism Group of $\AA$: Operations with Identity:
:there are no elements of $\AA$ which have an identity element.
From Automorphism Group of $\BB$: Operations with Identity:
:there are no elements of $\BB$ which have an identity element.
From Automorphism Group of $\CC_n$: Operations with Identity:
:there are... | Let $\NN$ be the [[Definition:Set|set]] of all [[Definition:Binary Operation|operations]] $\circ$ on $S$ which have an [[Definition:Identity Element|identity element]].
Then the [[Definition:Element|elements]] of $\NN$ are divided in $45$ [[Definition:Isomorphism Class (Algebraic Structures)|isomorphism classes]].
Th... | From [[Sets of Operations on Set of 3 Elements/Automorphism Group of A/Operations with Identity|Automorphism Group of $\AA$: Operations with Identity]]:
:there are no [[Definition:Element|elements]] of $\AA$ which have an [[Definition:Identity Element|identity element]].
From [[Sets of Operations on Set of 3 Elements/... | Sets of Operations on Set of 3 Elements/Operations with Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Operations_with_Identity | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Operations_with_Identity | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Set",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Element",
"Definition:Isomorphism Class (Algebraic Structures)",
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Operation/Binary Operation",
"Definition:Identit... | [
"Sets of Operations on Set of 3 Elements/Automorphism Group of A/Operations with Identity",
"Definition:Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Sets of Operations on Set of 3 Elements/Automorphism Group of B/Operations with Identity",
"Definition:Element",
"Definition:Ident... |
proofwiki-18830 | Sets of Operations on Set of 3 Elements/Automorphism Group of A/Commutative Operations | :Exactly $1$ of the operations of $\AA$ is commutative. | Recall from Automorphism Group of $\AA$ the elements of $\AA$, expressed in Cayley table form:
:<nowiki>$\begin {array} {c|ccc}
\to & a & b & c \\
\hline
a & a & b & c \\
b & a & b & c \\
c & a & b & c \\
\end {array}
\qquad
\begin {array} {c|ccc}
\gets & a & b & c \\
\hline
a & a & a & a \\
b & b & b & b \\
c & c & c ... | :Exactly $1$ of the [[Definition:Binary Operation|operations]] of $\AA$ is [[Definition:Commutative Operation|commutative]]. | Recall from [[Sets of Operations on Set of 3 Elements/Automorphism Group of A|Automorphism Group of $\AA$]] the [[Definition:Element|elements]] of $\AA$, expressed in [[Definition:Cayley Table|Cayley table]] form:
:<nowiki>$\begin {array} {c|ccc}
\to & a & b & c \\
\hline
a & a & b & c \\
b & a & b & c \\
c & a & b & ... | Sets of Operations on Set of 3 Elements/Automorphism Group of A/Commutative Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_A/Commutative_Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_A/Commutative_Operations | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation"
] | [
"Sets of Operations on Set of 3 Elements/Automorphism Group of A",
"Definition:Element",
"Definition:Cayley Table",
"Cayley Table for Commutative Operation is Symmetrical about Main Diagonal",
"Definition:Cayley Table",
"Definition:Commutative/Operation"
] |
proofwiki-18831 | Sets of Operations on Set of 3 Elements/Automorphism Group of B/Commutative Operations | :Exactly $8$ of the operations of $\BB$ is commutative. | Recall Automorphism Group of $\BB$.
Consider each of the categories of $\BB$ induced by each of $a \circ a$, $a \circ b$ and $a \circ c$, illustrated by the partially-filled Cayley tables to which they give rise:
;$(1): \quad a \circ a$
:$\begin {array} {c|ccc}
\circ & a & b & c \\
\hline
a & a & & \\
b & & b & ... | :Exactly $8$ of the [[Definition:Binary Operation|operations]] of $\BB$ is [[Definition:Commutative Operation|commutative]]. | Recall [[Sets of Operations on Set of 3 Elements/Automorphism Group of B|Automorphism Group of $\BB$]].
Consider each of the categories of $\BB$ induced by each of $a \circ a$, $a \circ b$ and $a \circ c$, illustrated by the partially-filled [[Definition:Cayley Table|Cayley tables]] to which they give rise:
;$(1): ... | Sets of Operations on Set of 3 Elements/Automorphism Group of B/Commutative Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_B/Commutative_Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_B/Commutative_Operations | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation"
] | [
"Sets of Operations on Set of 3 Elements/Automorphism Group of B",
"Definition:Cayley Table",
"Cayley Table for Commutative Operation is Symmetrical about Main Diagonal",
"Definition:Cayley Table",
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation",
"Definition:Operation/Binary ... |
proofwiki-18832 | Sets of Operations on Set of 3 Elements/Automorphism Group of C n/Commutative Operations | $8$ of the operations of each of $\CC_1$, $\CC_2$ and $\CC_3$ is commutative. | {{WLOG}}, we will analyse the nature of $\CC_1$.
Recall this lemma: | $8$ of the [[Definition:Binary Operation|operations]] of each of $\CC_1$, $\CC_2$ and $\CC_3$ is [[Definition:Commutative Operation|commutative]]. | {{WLOG}}, we will analyse the nature of $\CC_1$.
Recall this [[Definition:Lemma|lemma]]: | Sets of Operations on Set of 3 Elements/Automorphism Group of C n/Commutative Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n/Commutative_Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_C_n/Commutative_Operations | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation"
] | [
"Definition:Lemma"
] |
proofwiki-18833 | Sets of Operations on Set of 3 Elements/Automorphism Group of D/Commutative Operations | :$696$ of the operations of $\DD$ is commutative. | Let $n$ denote the number of commutative operations of $\DD$.
Recall these definitions:
Let $\AA$, $\BB$, $\CC_1$, $\CC_2$ and $\CC_3$ be respectively the set of all operations $\circ$ on $S$ such that the groups of automorphisms of $\struct {S, \circ}$ are as follows:
{{begin-eqn}}
{{eqn | l = \AA
| o = :
... | :$696$ of the [[Definition:Binary Operation|operations]] of $\DD$ is [[Definition:Commutative Operation|commutative]]. | Let $n$ denote the number of [[Definition:Commutative Operation|commutative operations]] of $\DD$.
Recall these definitions:
Let $\AA$, $\BB$, $\CC_1$, $\CC_2$ and $\CC_3$ be respectively the [[Definition:Set|set]] of all [[Definition:Binary Operation|operations]] $\circ$ on $S$ such that the [[Definition:Group|grou... | Sets of Operations on Set of 3 Elements/Automorphism Group of D/Commutative Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_D/Commutative_Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Automorphism_Group_of_D/Commutative_Operations | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Operation/Binary Operation",
"Definition:Commutative/Operation"
] | [
"Definition:Commutative/Operation",
"Definition:Set",
"Definition:Operation/Binary Operation",
"Definition:Group",
"Definition:Group Automorphism",
"Definition:Symmetric Group",
"Definition:Identity Mapping",
"Definition:Commutative/Operation",
"Definition:Commutative/Operation",
"Definition:Commu... |
proofwiki-18834 | Sets of Operations on Set of 3 Elements/Commutative Operations | Let $\PP$ be the set of all commutative operations $\circ$ on $S$.
Then the elements of $\PP$ are divided in $129$ isomorphism classes.
That is, up to isomorphism, there are $129$ commutative operations on $S$ which have an identity element. | From Automorphism Group of $\AA$: Commutative Operations:
:there is exactly $1$ commutative operation in $\AA$.
From Automorphism Group of $\BB$: Commutative Operations:
:there are $8$ commutative operations in $\BB$.
From Automorphism Group of $\CC_n$: Commutative Operations:
:there are $3 \times 8$ commutative operat... | Let $\PP$ be the [[Definition:Set|set]] of all [[Definition:Commutative Operation|commutative operations]] $\circ$ on $S$.
Then the [[Definition:Element|elements]] of $\PP$ are divided in $129$ [[Definition:Isomorphism Class (Algebraic Structures)|isomorphism classes]].
That is, up to [[Definition:Isomorphism (Abstra... | From [[Sets of Operations on Set of 3 Elements/Automorphism Group of A/Commutative Operations|Automorphism Group of $\AA$: Commutative Operations]]:
:there is exactly $1$ [[Definition:Commutative Operation|commutative operation]] in $\AA$.
From [[Sets of Operations on Set of 3 Elements/Automorphism Group of B/Commutat... | Sets of Operations on Set of 3 Elements/Commutative Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Commutative_Operations | https://proofwiki.org/wiki/Sets_of_Operations_on_Set_of_3_Elements/Commutative_Operations | [
"Sets of Operations on Set of 3 Elements"
] | [
"Definition:Set",
"Definition:Commutative/Operation",
"Definition:Element",
"Definition:Isomorphism Class (Algebraic Structures)",
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Commutative/Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Sets of Operations on Set of 3 Elements/Automorphism Group of A/Commutative Operations",
"Definition:Commutative/Operation",
"Sets of Operations on Set of 3 Elements/Automorphism Group of B/Commutative Operations",
"Definition:Commutative/Operation",
"Sets of Operations on Set of 3 Elements/Automorphism Gr... |
proofwiki-18835 | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 10 | Let:
:$n, k \in \N : k > 0 : r_n = r_{n + k}$
Then:
:$d_{n + 1} = d_{n + k + 1}$
:$r_{n + 1} = r_{n + k + 1}$ | We have:
{{begin-eqn}}
{{eqn | l = d_{n + 1} b + p r_{n + 1}
| r = r_n
| c = {{hypothesis}}
}}
{{eqn | r = r_{n + k}
| c = {{hypothesis}}
}}
{{eqn | r = d_{n + k + 1} b + p r_{n + k + 1}
| c = {{hypothesis}}
}}
{{eqn | ll = \leadsto
| l = p \paren{ r_{n + k + 1} - r_{n + 1} }
| r = \... | Let:
:$n, k \in \N : k > 0 : r_n = r_{n + k}$
Then:
:$d_{n + 1} = d_{n + k + 1}$
:$r_{n + 1} = r_{n + k + 1}$ | We have:
{{begin-eqn}}
{{eqn | l = d_{n + 1} b + p r_{n + 1}
| r = r_n
| c = {{hypothesis}}
}}
{{eqn | r = r_{n + k}
| c = {{hypothesis}}
}}
{{eqn | r = d_{n + k + 1} b + p r_{n + k + 1}
| c = {{hypothesis}}
}}
{{eqn | ll = \leadsto
| l = p \paren{ r_{n + k + 1} - r_{n + 1} }
| r = \... | Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 10 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_10 | https://proofwiki.org/wiki/Canonical_P-adic_Expansion_of_Rational_is_Eventually_Periodic/Lemma_10 | [
"Canonical P-adic Expansion of Rational is Eventually Periodic"
] | [] | [
"Definition:Coprime/Integers",
"Euclid's Lemma",
"Definition:P-adic Digit",
"Category:Canonical P-adic Expansion of Rational is Eventually Periodic"
] |
proofwiki-18836 | Locally Integrable (f(x+ct) + f(x-ct))/2 is Weak Solution to Wave Equation | Consider the wave equation:
:$\dfrac {\partial^2 u} {\partial t^2} - c^2 \dfrac {\partial^2 u} {\partial x^2} = 0$
with the initial conditions:
:$\map u {x, 0} = \map f x$
:$\map {\dfrac {\partial u} {\partial t}} {x, 0} = 0$
and $c \in \R$.
Then it has a weak solution of the form:
:$\map u {x, t} := \dfrac {\map f {x ... | Let $\map u {x, t} = \map f {x + ct}$ be a locally integrable function.
We have that a locally integrable function defines a distribution.
Let $T_u \in \map {\DD'} {\R^2}$ be a Schwartz distribution associated with $u$.
Let $\phi \in \map \DD {\R^2}$ be a test function.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {\df... | Consider the [[Definition:Wave Equation|wave equation]]:
:$\dfrac {\partial^2 u} {\partial t^2} - c^2 \dfrac {\partial^2 u} {\partial x^2} = 0$
with the [[Definition:Initial Condition|initial conditions]]:
:$\map u {x, 0} = \map f x$
:$\map {\dfrac {\partial u} {\partial t}} {x, 0} = 0$
and $c \in \R$.
Then it h... | Let $\map u {x, t} = \map f {x + ct}$ be a [[Definition:Locally Integrable Function|locally integrable function]].
We have that a [[Locally Integrable Function defines Distribution|locally integrable function defines a distribution]].
Let $T_u \in \map {\DD'} {\R^2}$ be a [[Definition:Schwartz Distribution|Schwartz d... | Locally Integrable (f(x+ct) + f(x-ct))/2 is Weak Solution to Wave Equation | https://proofwiki.org/wiki/Locally_Integrable_(f(x+ct)_+_f(x-ct))/2_is_Weak_Solution_to_Wave_Equation | https://proofwiki.org/wiki/Locally_Integrable_(f(x+ct)_+_f(x-ct))/2_is_Weak_Solution_to_Wave_Equation | [
"Examples of Weak Solutions",
"Wave Equation"
] | [
"Definition:Wave Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Weak Solution",
"Definition:Integrable Function/Locally Integrable Function"
] | [
"Definition:Integrable Function/Locally Integrable Function",
"Locally Integrable Function defines Distribution",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Distributional Partial Derivatives Commute",
"Locally Integrable f(x+ct) is Weak Solution to Transport Equation",
"Definition... |
proofwiki-18837 | Gilmer-Parker Theorem | Let $\struct {R, +, *}$ be a GCD Domain.
Let $R \sqbrk x$ be a polynomial ring over $R$.
Then $R \sqbrk x$ is also a GCD Domain. | {{tidy|Under way, this will take a long time as this page is very far from following the house rules.
Use <code><nowiki>{{eqn}}</nowiki></code> template.}}
{{MissingLinks}}
Let $K$ be the field of quotients of $R$.
Let $R \xrightarrow \varphi R \sqbrk x \xrightarrow \psi K \sqbrk x$
where $\varphi, \psi$ - embedding ho... | Let $\struct {R, +, *}$ be a [[Definition:GCD Domain|GCD Domain]].
Let $R \sqbrk x$ be a [[Definition:Polynomial Ring|polynomial ring]] over $R$.
Then $R \sqbrk x$ is also a [[Definition:GCD Domain|GCD Domain]]. | {{tidy|Under way, this will take a long time as this page is very far from following the house rules.
Use <code><nowiki>{{eqn}}</nowiki></code> template.}}
{{MissingLinks}}
Let $K$ be the [[Definition:Field of Quotients|field of quotients of $R$]].
Let $R \xrightarrow \varphi R \sqbrk x \xrightarrow \psi K \sqbrk x$
... | Gilmer-Parker Theorem | https://proofwiki.org/wiki/Gilmer-Parker_Theorem | https://proofwiki.org/wiki/Gilmer-Parker_Theorem | [
"GCD Domains",
"Polynomial Rings"
] | [
"Definition:GCD Domain",
"Definition:Polynomial Ring",
"Definition:GCD Domain"
] | [
"Definition:Field of Quotients",
"Definition:Ring Monomorphism",
"Definition:Primitive Polynomial (Ring Theory)",
"Definition:Content of Polynomial/GCD Domain",
"Definition:Euclidean Domain",
"Definition:GCD Domain",
"Definition:Associate",
"Euclid's Lemma"
] |
proofwiki-18838 | Weak Solution to Dx u = Heaviside Step Function | Let $H: \R \to \closedint 0 1$ be the Heaviside step function.
Let $u : \R \to \R$ be such that:
:<nowiki>$\map u x = \begin{cases}
c & : x < 0 \\
x + c & : x > 0
\end{cases}$</nowiki>
where $c \in \R$.
Let $T_u$ be the Schwartz distribution associated with $u$.
Then $u$ is a weak solution of:
:$u' = H$
That is, in t... | $u$ is continuous on $\R$ and continously differentiable on $\R \setminus \set 0$.
For $x < 0$ we have $\map {u'} x = 0$.
For $x > 0$ we have $\map {u'} x = 1$.
That is:
:$\map {u'} x = \map H x$
Furthermore:
:$\ds \lim_{x \mathop \to 0^-} = 0$
:$\ds \lim_{x \mathop \to 0^+} = 1$
By the jump rule:
{{begin-eqn}}
{{eqn |... | Let $H: \R \to \closedint 0 1$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $u : \R \to \R$ be such that:
:<nowiki>$\map u x = \begin{cases}
c & : x < 0 \\
x + c & : x > 0
\end{cases}$</nowiki>
where $c \in \R$.
Let $T_u$ be the [[Definition:Schwartz Distribution|Schwartz distributio... | $u$ is [[Definition:Everywhere Continuous Real Function|continuous]] on $\R$ and [[Definition:Continuously Differentiable Real Function|continously differentiable]] on $\R \setminus \set 0$.
For $x < 0$ we have $\map {u'} x = 0$.
For $x > 0$ we have $\map {u'} x = 1$.
That is:
:$\map {u'} x = \map H x$
Furthermore... | Weak Solution to Dx u = Heaviside Step Function | https://proofwiki.org/wiki/Weak_Solution_to_Dx_u_=_Heaviside_Step_Function | https://proofwiki.org/wiki/Weak_Solution_to_Dx_u_=_Heaviside_Step_Function | [
"Examples of Weak Solutions"
] | [
"Definition:Heaviside Step Function",
"Definition:Schwartz Distribution",
"Definition:Differential Equation/Solution/Weak Solution",
"Definition:Distributional Derivative"
] | [
"Definition:Continuous Real Function/Everywhere",
"Definition:Continuously Differentiable/Real Function",
"Jump Rule"
] |
proofwiki-18839 | Laplace Transform of Derivative with Finite Discontinuities | Let $f$ have a finite number of jump discontinuities at $t = a_i$ for $i = 1, 2, \ldots, n$.
Then:
:$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0 - \ds \sum_{i \mathop = 1}^n e^{-a_i s} \paren {\map f {a_i^+} - \map f {a_i^-} }$ | {{tidy|Rewrite the following in house style}}
The proof is found similarly to the proof in Laplace Transform of Derivative but requires breaking the integral into $n$ integrals of finite range between the discontinuities and one improper integral from $a_n^+$ to $+\infty$. Integration by parts yields the summation when... | Let $f$ have a finite number of [[Definition:Jump Discontinuity|jump discontinuities]] at $t = a_i$ for $i = 1, 2, \ldots, n$.
Then:
:$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0 - \ds \sum_{i \mathop = 1}^n e^{-a_i s} \paren {\map f {a_i^+} - \map f {a_i^-} }$ | {{tidy|Rewrite the following in house style}}
The proof is found similarly to the proof in [[Laplace Transform of Derivative]] but requires breaking the integral into $n$ integrals of finite range between the discontinuities and one improper integral from $a_n^+$ to $+\infty$. Integration by parts yields the summation... | Laplace Transform of Derivative with Finite Discontinuities | https://proofwiki.org/wiki/Laplace_Transform_of_Derivative_with_Finite_Discontinuities | https://proofwiki.org/wiki/Laplace_Transform_of_Derivative_with_Finite_Discontinuities | [
"Laplace Transforms of Derivatives"
] | [
"Definition:Discontinuity (Real Analysis)/Jump"
] | [
"Laplace Transform of Derivative",
"Laplace Transform of Derivative",
"Laplace Transform of Derivative"
] |
proofwiki-18840 | Power Set is Nonempty | Let $S$ be a set.
Then:
:$\powerset S \ne \O$ | By Empty Set is Element of Power Set:
:$\O \in \powerset S$
Thus we conclude that $\powerset S$ is non-empty.
{{qed}}
Category:Empty Set
Category:Power Set
j0xb6oxhss17kiz835w4sxlq42kj82i | Let $S$ be a [[Definition:Set|set]].
Then:
:$\powerset S \ne \O$ | By [[Empty Set is Element of Power Set]]:
:$\O \in \powerset S$
Thus we conclude that $\powerset S$ is [[Definition:Non-Empty Set|non-empty]].
{{qed}}
[[Category:Empty Set]]
[[Category:Power Set]]
j0xb6oxhss17kiz835w4sxlq42kj82i | Power Set is Nonempty | https://proofwiki.org/wiki/Power_Set_is_Nonempty | https://proofwiki.org/wiki/Power_Set_is_Nonempty | [
"Empty Set",
"Power Set"
] | [
"Definition:Set"
] | [
"Empty Set is Element of Power Set",
"Definition:Non-Empty Set",
"Category:Empty Set",
"Category:Power Set"
] |
proofwiki-18841 | Multiplication of Distribution induced by Locally Integrable Function by Smooth Function | Let $f \in \map {L^1_{loc} } {\R^d}$ be a locally integrable function.
Let $\alpha \in \map {C^\infty} {\R^d}$ be a smooth function.
Let $T_f \in \map {\DD'} {\R^d}$ be a Schwartz distribution induced by $f$.
Then in the distributional sense it holds that:
:$\alpha T_f = T_{\alpha f}$ | Let $\Omega \subseteq \R^d$ be a compact subset.
Then for all $\mathbf x \in \Omega$ we have that $\map \alpha {\mathbf x}$ is bounded.
Hence, $\alpha f$ is locally integrable.
Let $\phi \in \map \DD {\R^d}$ be a test function.
Then:
{{begin-eqn}}
{{eqn | l = \alpha \map {T_f} \phi
| r = \map {T_f} {\alpha \phi}
... | Let $f \in \map {L^1_{loc} } {\R^d}$ be a [[Definition:Locally Integrable Function|locally integrable function]].
Let $\alpha \in \map {C^\infty} {\R^d}$ be a [[Definition:Smooth Function|smooth function]].
Let $T_f \in \map {\DD'} {\R^d}$ be a [[Definition:Schwartz Distribution|Schwartz distribution]] induced by $f$... | Let $\Omega \subseteq \R^d$ be a [[Definition:Compact Subset of Real Euclidean Space|compact subset]].
Then for all $\mathbf x \in \Omega$ we have that $\map \alpha {\mathbf x}$ is [[Definition:Bounded Mapping|bounded]].
Hence, $\alpha f$ is [[Definition:Locally Integrable Function|locally integrable]].
Let $\phi \i... | Multiplication of Distribution induced by Locally Integrable Function by Smooth Function | https://proofwiki.org/wiki/Multiplication_of_Distribution_induced_by_Locally_Integrable_Function_by_Smooth_Function | https://proofwiki.org/wiki/Multiplication_of_Distribution_induced_by_Locally_Integrable_Function_by_Smooth_Function | [
"Schwartz Distributions"
] | [
"Definition:Integrable Function/Locally Integrable Function",
"Definition:Smooth Real Function",
"Definition:Schwartz Distribution",
"Definition:Schwartz Distribution"
] | [
"Definition:Compact Space/Euclidean Space",
"Definition:Bounded Mapping",
"Definition:Integrable Function/Locally Integrable Function",
"Definition:Test Function",
"Locally Integrable Function defines Distribution",
"Locally Integrable Function defines Distribution"
] |
proofwiki-18842 | Cartesian Product is Unique | Let $A$ and $B$ be classes.
If there exists a '''cartesian product''' of $A$ and $B$, then it is unique. | Let $C_1$ and $C_2$ be cartesian products of $A$ and $B$.
Then by the cartesian product definition, for an arbitrary $a$:
:$a \in C_1 \iff \exists x \in A: \exists y \in B: a = \tuple {x, y}$
:$a \in C_2 \iff \exists x \in A: \exists y \in B: a = \tuple {x, y}$
By Biconditional is Transitive:
:$a \in C_1 \iff a \in C_2... | Let $A$ and $B$ be [[Definition:Class (Class Theory)|classes]].
If there exists a '''[[Definition:Cartesian Product (Class Theory)|cartesian product]]''' of $A$ and $B$, then it is [[Definition:Unique|unique]]. | Let $C_1$ and $C_2$ be [[Definition:Cartesian Product (Class Theory)|cartesian products]] of $A$ and $B$.
Then by the [[Definition:Cartesian Product (Class Theory)|cartesian product]] definition, for an arbitrary $a$:
:$a \in C_1 \iff \exists x \in A: \exists y \in B: a = \tuple {x, y}$
:$a \in C_2 \iff \exists x \in ... | Cartesian Product is Unique | https://proofwiki.org/wiki/Cartesian_Product_is_Unique | https://proofwiki.org/wiki/Cartesian_Product_is_Unique | [
"Cartesian Product"
] | [
"Definition:Class (Class Theory)",
"Definition:Cartesian Product/Class Theory",
"Definition:Unique"
] | [
"Definition:Cartesian Product/Class Theory",
"Definition:Cartesian Product/Class Theory",
"Biconditional is Transitive",
"Axiom:Axiom of Extension/Class Theory",
"Definition:Cartesian Product/Class Theory",
"Definition:Unique"
] |
proofwiki-18843 | Biconditional of Proposition and its Negation | :$\vdash \neg (p \iff \neg p)$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
$\begin{array}{|c|cccc|} \hline
\neg & (p & \iff & \neg & p)\\
\hline
\T & \F & \F & \T & \F \\
\T & \T & \F & \F & \T \\
\hline
\end{arr... | :$\vdash \neg (p \iff \neg p)$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] in the appropriate columns match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|c|cccc|} \hline
\neg & (p & \iff & \neg & p)\\
\hl... | Biconditional of Proposition and its Negation/Proof by Truth Table | https://proofwiki.org/wiki/Biconditional_of_Proposition_and_its_Negation | https://proofwiki.org/wiki/Biconditional_of_Proposition_and_its_Negation/Proof_by_Truth_Table | [
"Biconditional",
"Biconditional of Proposition and its Negation"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-18844 | Axiom of Specification from Replacement and Empty Set | The {{axiom-link|Specification|Sets}} is a consequence of:
:the {{axiom-link|Replacement}}
and
:the {{axiom-link|the Empty Set|Set Theory}}. | {{Proofread|proof is tedious}}
Let $A$ be an arbitrary set.
Let $\map P x$ be an arbitrary propositional function.
It is to be shown that there exists a set $B$ consisting of exactly the $y \in A$ such that $\map P y$.
That is:
:$\forall A: \exists B: \forall y: \paren {y \in B \iff \paren {y \in A \land \map P y} }$
B... | The {{axiom-link|Specification|Sets}} is a consequence of:
:the {{axiom-link|Replacement}}
and
:the {{axiom-link|the Empty Set|Set Theory}}. | {{Proofread|proof is tedious}}
Let $A$ be an arbitrary [[Definition:Set|set]].
Let $\map P x$ be an arbitrary [[Definition:Propositional Function|propositional function]].
It is to be shown that there exists a [[Definition:Set|set]] $B$ consisting of exactly the $y \in A$ such that $\map P y$.
That is:
:$\forall A... | Axiom of Specification from Replacement and Empty Set | https://proofwiki.org/wiki/Axiom_of_Specification_from_Replacement_and_Empty_Set | https://proofwiki.org/wiki/Axiom_of_Specification_from_Replacement_and_Empty_Set | [
"Zermelo-Fraenkel Axioms",
"Empty Set"
] | [] | [
"Definition:Set",
"Definition:Propositional Function",
"Definition:Set",
"Law of Excluded Middle",
"Definition:Empty Set",
"Rule of Explosion",
"Rule of Explosion",
"Universal Generalisation",
"Existential Generalisation",
"Definition:Propositional Function",
"Definition:Mapping",
"Definition:... |
proofwiki-18845 | Cycle Graph is Eulerian | Let $G$ be a cycle graph.
Then $G$ is Eulerian. | From Cycle Graph is Connected, $G$ is a connected graph.
From Cycle Graph is $2$-Regular, $G$ is $2$-regular.
It follows directly from Characteristics of Eulerian Graph that $G$ is Eulerian.
{{Qed}}
Category:Cycle Graphs
Category:Eulerian Graphs
jt94p8yky7cp9lr3s8g8bw2rngonfnd | Let $G$ be a [[Definition:Cycle Graph|cycle graph]].
Then $G$ is [[Definition:Eulerian Graph|Eulerian]]. | From [[Cycle Graph is Connected]], $G$ is a [[Definition:Connected Graph|connected graph]].
From [[Cycle Graph is 2-Regular|Cycle Graph is $2$-Regular]], $G$ is [[Definition:Regular Graph|$2$-regular]].
It follows directly from [[Characteristics of Eulerian Graph]] that $G$ is [[Definition:Eulerian Graph|Eulerian]].
... | Cycle Graph is Eulerian | https://proofwiki.org/wiki/Cycle_Graph_is_Eulerian | https://proofwiki.org/wiki/Cycle_Graph_is_Eulerian | [
"Cycle Graphs",
"Eulerian Graphs"
] | [
"Definition:Cycle Graph",
"Definition:Eulerian Graph"
] | [
"Cycle Graph is Connected",
"Definition:Connected (Graph Theory)/Graph",
"Cycle Graph is 2-Regular",
"Definition:Regular Graph",
"Characteristics of Eulerian Graph",
"Definition:Eulerian Graph",
"Category:Cycle Graphs",
"Category:Eulerian Graphs"
] |
proofwiki-18846 | Cycle Graph is 2-Regular | Let $G$ be a cycle graph.
Then $G$ is regular. | Let $G$ be a cycle graph.
By definition, a '''cycle graph''' is a graph which consists of a single cycle $C$.
By definition, a '''cycle''' is a circuit in which no vertex except the first (which is also the last) appears more than once.
By definition, a '''circuit''' is a closed trail with at least one edge.
By definit... | Let $G$ be a [[Definition:Cycle Graph|cycle graph]].
Then $G$ is [[Definition:Regular Graph|regular]]. | Let $G$ be a [[Definition:Cycle Graph|cycle graph]].
By definition, a '''[[Definition:Cycle Graph|cycle graph]]''' is a [[Definition:Graph (Graph Theory)|graph]] which consists of a single [[Definition:Cycle (Graph Theory)|cycle]] $C$.
By definition, a '''[[Definition:Cycle (Graph Theory)|cycle]]''' is a [[Definition... | Cycle Graph is 2-Regular | https://proofwiki.org/wiki/Cycle_Graph_is_2-Regular | https://proofwiki.org/wiki/Cycle_Graph_is_2-Regular | [
"Cycle Graphs",
"Regular Graphs"
] | [
"Definition:Cycle Graph",
"Definition:Regular Graph"
] | [
"Definition:Cycle Graph",
"Definition:Cycle Graph",
"Definition:Graph (Graph Theory)",
"Definition:Cycle (Graph Theory)",
"Definition:Cycle (Graph Theory)",
"Definition:Circuit (Graph Theory)",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Circuit (Graph Theory)",
"Definition:Walk (Graph The... |
proofwiki-18847 | Cycle Graph is Connected | Let $G = \struct {V, E}$ be a cycle graph.
Then $G$ is connected. | A cycle graph is defined as a (simple) graph which consists of a single cycle.
So a cycle graph consists of just one component, and hence is connected.
{{qed}}
Category:Cycle Graphs
Category:Connectedness (Graph Theory)
0enbhaoaa4eus769o1qcyjxp639ol87 | Let $G = \struct {V, E}$ be a [[Definition:Cycle Graph|cycle graph]].
Then $G$ is [[Definition:Connected Graph|connected]]. | A [[Definition:Cycle Graph|cycle graph]] is defined as a [[Definition:Simple Graph|(simple) graph]] which consists of a single [[Definition:Cycle (Graph Theory)|cycle]].
So a [[Definition:Cycle Graph|cycle graph]] consists of just one [[Definition:Component of Graph|component]], and hence is [[Definition:Connected Gra... | Cycle Graph is Connected | https://proofwiki.org/wiki/Cycle_Graph_is_Connected | https://proofwiki.org/wiki/Cycle_Graph_is_Connected | [
"Cycle Graphs",
"Connectedness (Graph Theory)"
] | [
"Definition:Cycle Graph",
"Definition:Connected (Graph Theory)/Graph"
] | [
"Definition:Cycle Graph",
"Definition:Simple Graph",
"Definition:Cycle (Graph Theory)",
"Definition:Cycle Graph",
"Definition:Component of Graph",
"Definition:Connected (Graph Theory)/Graph",
"Category:Cycle Graphs",
"Category:Connectedness (Graph Theory)"
] |
proofwiki-18848 | Cycle Graph is Bipartite iff Order is Even | Let $n \in \N$ be a natural number.
Let $C_n$ be the cycle graph of order $n$.
Then $C_n$ is a bipartite graph {{iff}} $n$ is even. | Let $V$ be the set of vertices of $C_n$.
Let the elements of $V$ be denoted $v_1, v_2, \ldots, v_n$.
Then by definition of cycle graph, by appropriate selection of subscripts, $C_n$ consists of one cycle $C$ that can be expressed as:
:$C := \tuple {v_1 v_2 \ldots, v_n v_1}$
Let $v_k \in C_n$.
Then for $1 < k < n$, $v_k... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $C_n$ be the [[Definition:Cycle Graph|cycle graph]] of [[Definition:Order of Graph|order]] $n$.
Then $C_n$ is a [[Definition:Bipartite Graph|bipartite graph]] {{iff}} $n$ is [[Definition:Even Integer|even]]. | Let $V$ be the [[Definition:Set|set]] of [[Definition:Vertex of Graph|vertices]] of $C_n$.
Let the [[Definition:Element|elements]] of $V$ be denoted $v_1, v_2, \ldots, v_n$.
Then by definition of [[Definition:Cycle Graph|cycle graph]], by appropriate selection of subscripts, $C_n$ consists of one [[Definition:Cycle (... | Cycle Graph is Bipartite iff Order is Even | https://proofwiki.org/wiki/Cycle_Graph_is_Bipartite_iff_Order_is_Even | https://proofwiki.org/wiki/Cycle_Graph_is_Bipartite_iff_Order_is_Even | [
"Cycle Graphs",
"Bipartite Graphs",
"Cycle Graph is Bipartite iff Order is Even"
] | [
"Definition:Natural Numbers",
"Definition:Cycle Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Bipartite Graph",
"Definition:Even Integer"
] | [
"Definition:Set",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Element",
"Definition:Cycle Graph",
"Definition:Cycle (Graph Theory)",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Adjacent ... |
proofwiki-18849 | Cycle Graph of Order 1 is Loop-Graph | Let $C_1$ denote the cycle graph of order $1$.
Then $C_1$ is a loop-graph. | By definition, the vertex set of $C_1$ is singleton, $\set v$, say.
The only vertex of $C_1$ that an edge can be incident to is $v$.
Hence there exists an edge which is incident to $v$ at both ends.
That is, $C_1$ has a loop.
Hence the result by definition of loop-graph.
{{qed}}
Category:Examples of Cycle Graphs
Catego... | Let $C_1$ denote the [[Definition:Cycle Graph|cycle graph]] of [[Definition:Order of Graph|order $1$]].
Then $C_1$ is a [[Definition:Loop-Graph|loop-graph]]. | By definition, the [[Definition:Vertex Set|vertex set]] of $C_1$ is [[Definition:Singleton|singleton]], $\set v$, say.
The only [[Definition:Vertex of Graph|vertex]] of $C_1$ that an [[Definition:Edge of Graph|edge]] can be [[Definition:Incident (Undirected Graph)|incident]] to is $v$.
Hence there exists an [[Definit... | Cycle Graph of Order 1 is Loop-Graph | https://proofwiki.org/wiki/Cycle_Graph_of_Order_1_is_Loop-Graph | https://proofwiki.org/wiki/Cycle_Graph_of_Order_1_is_Loop-Graph | [
"Examples of Cycle Graphs",
"Examples of Loop-Graphs"
] | [
"Definition:Cycle Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Loop-Graph"
] | [
"Definition:Vertex Set",
"Definition:Singleton",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Incident (Graph Theory)/Undirected Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Incident (Graph Theory)/Undirected Graph",
"Definition:Loop (Gra... |
proofwiki-18850 | Cycle Graph of Order 2 is Multigraph | Let $C_2$ denote the cycle graph of order $2$.
Then $C_2$ is a multigraph. | By definition, the vertex set of $C_2$ is doubleton, $\set {v_1, v_2}$, say.
By definition of cycle graph, there is a circuit $v_1 v_2 v_1$.
That is:
:there exists an edge which is incident to $v_1$ and $v_2$
:there exists an edge which is incident to $v_2$ and $v_1$
That is, there are $2$ edges which are both incident... | Let $C_2$ denote the [[Definition:Cycle Graph|cycle graph]] of [[Definition:Order of Graph|order $2$]].
Then $C_2$ is a [[Definition:Multigraph|multigraph]]. | By definition, the [[Definition:Vertex Set|vertex set]] of $C_2$ is [[Definition:Doubleton|doubleton]], $\set {v_1, v_2}$, say.
By definition of [[Definition:Cycle Graph|cycle graph]], there is a [[Definition:Circuit (Graph Theory)|circuit]] $v_1 v_2 v_1$.
That is:
:there exists an [[Definition:Edge of Graph|edge]] w... | Cycle Graph of Order 2 is Multigraph | https://proofwiki.org/wiki/Cycle_Graph_of_Order_2_is_Multigraph | https://proofwiki.org/wiki/Cycle_Graph_of_Order_2_is_Multigraph | [
"Examples of Cycle Graphs",
"Examples of Multigraphs"
] | [
"Definition:Cycle Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Multigraph"
] | [
"Definition:Vertex Set",
"Definition:Doubleton",
"Definition:Cycle Graph",
"Definition:Circuit (Graph Theory)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Incident (Graph Theory)/Undirected Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Incident (Graph Theory)/Undirected Graph",
... |
proofwiki-18851 | Cycle Graph of Order 3 is Complete Graph | Let $C_3$ denote the cycle graph of order $2$.
Then $C_3$ is the complete graph of of order $3$. | Let the vertex set of $C_3$ is $\set {v_1, v_2, v_3}$.
By definition of cycle graph, $C_3$ consists of the cycle $v_1 v_2 v_3 v_1$.
It is seen by inspection that:
:$v_1$ is adjacent to $v_2$ and $v_3$
:$v_2$ is adjacent to $v_1$ and $v_3$
:$v_3$ is adjacent to $v_1$ and $v_2$.
Hence the result by definition of complete... | Let $C_3$ denote the [[Definition:Cycle Graph|cycle graph]] of [[Definition:Order of Graph|order $2$]].
Then $C_3$ is the [[Definition:Complete Graph|complete graph]] of of [[Definition:Order of Graph|order $3$]]. | Let the [[Definition:Vertex Set|vertex set]] of $C_3$ is $\set {v_1, v_2, v_3}$.
By definition of [[Definition:Cycle Graph|cycle graph]], $C_3$ consists of the [[Definition:Cycle (Graph Theory)|cycle]] $v_1 v_2 v_3 v_1$.
It is seen by inspection that:
:$v_1$ is [[Definition:Adjacent Vertices of Graph|adjacent]] to $v... | Cycle Graph of Order 3 is Complete Graph | https://proofwiki.org/wiki/Cycle_Graph_of_Order_3_is_Complete_Graph | https://proofwiki.org/wiki/Cycle_Graph_of_Order_3_is_Complete_Graph | [
"Examples of Cycle Graphs",
"Examples of Complete Graphs"
] | [
"Definition:Cycle Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Order"
] | [
"Definition:Vertex Set",
"Definition:Cycle Graph",
"Definition:Cycle (Graph Theory)",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Complete Graph",
"Category:Examples of Cycle Graphs",
"Catego... |
proofwiki-18852 | Negated Restricted Existential Quantifier | Let $x$ and $A$ be sets.
Let $\map P x$ be a propositional function.
:$\neg \exists x \in A : \map P x \iff \forall x \in A : \neg \map P x $ | From left to right:
{{begin-eqn}}
{{eqn | q = \neg \exists x \in A
| o = \map P x
}}
{{eqn | ll= \leadsto
| q = \neg \exists x
| o = x \in A \land \map P x
| c = {{defof|Restricted Existential Quantifier}}
}}
{{eqn | ll= \leadsto
| q = \forall x
| o = \neg \paren{x \in A \land \map ... | Let $x$ and $A$ be [[Definition:Set|sets]].
Let $\map P x$ be a [[Definition:Propositional Function|propositional function]].
:$\neg \exists x \in A : \map P x \iff \forall x \in A : \neg \map P x $ | From left to right:
{{begin-eqn}}
{{eqn | q = \neg \exists x \in A
| o = \map P x
}}
{{eqn | ll= \leadsto
| q = \neg \exists x
| o = x \in A \land \map P x
| c = {{defof|Restricted Existential Quantifier}}
}}
{{eqn | ll= \leadsto
| q = \forall x
| o = \neg \paren{x \in A \land \map... | Negated Restricted Existential Quantifier | https://proofwiki.org/wiki/Negated_Restricted_Existential_Quantifier | https://proofwiki.org/wiki/Negated_Restricted_Existential_Quantifier | [] | [
"Definition:Set",
"Definition:Propositional Function"
] | [
"De Morgan's Laws (Predicate Logic)/Denial of Existence",
"Modus Ponendo Tollens/Variant",
"Modus Ponendo Tollens/Variant",
"De Morgan's Laws (Predicate Logic)/Denial of Existence"
] |
proofwiki-18853 | Negated Restricted Universal Quantifier | Let $x$ and $A$ be sets.
Let $\map P x$ be a propositional function.
:$\neg \forall x \in A : \map P x \iff \exists x \in A : \neg \map P x $ | === Sufficient Condition ===
{{begin-eqn}}
{{eqn | q = \neg \forall x \in A
| o = \map P x
}}
{{eqn | ll= \leadsto
| q = \neg \forall x
| o = x \in A \implies \map P x
| c = {{defof|Restricted Universal Quantifier}}
}}
{{eqn | ll= \leadsto
| q = \exists x
| o = \neg \paren{x \in A \... | Let $x$ and $A$ be [[Definition:Set|sets]].
Let $\map P x$ be a [[Definition:Propositional Function|propositional function]].
:$\neg \forall x \in A : \map P x \iff \exists x \in A : \neg \map P x $ | === Sufficient Condition ===
{{begin-eqn}}
{{eqn | q = \neg \forall x \in A
| o = \map P x
}}
{{eqn | ll= \leadsto
| q = \neg \forall x
| o = x \in A \implies \map P x
| c = {{defof|Restricted Universal Quantifier}}
}}
{{eqn | ll= \leadsto
| q = \exists x
| o = \neg \paren{x \in A ... | Negated Restricted Universal Quantifier | https://proofwiki.org/wiki/Negated_Restricted_Universal_Quantifier | https://proofwiki.org/wiki/Negated_Restricted_Universal_Quantifier | [
"Universal Quantifier"
] | [
"Definition:Set",
"Definition:Propositional Function"
] | [
"De Morgan's Laws (Predicate Logic)/Denial of Universality",
"Conjunction with Negative is Equivalent to Negation of Conditional",
"Conjunction with Negative is Equivalent to Negation of Conditional",
"De Morgan's Laws (Predicate Logic)/Denial of Universality"
] |
proofwiki-18854 | Mediant is Dependent upon Representation | Let $r, s \in \Q$ be rational numbers.
Let $r$ and $s$ be expressed as:
{{begin-eqn}}
{{eqn | l = r
| r = \dfrac a b
}}
{{eqn | l = s
| r = \dfrac c d
}}
{{end-eqn}}
where $a, b, c, d$ are integers such that $b > 0, d > 0$.
Then the mediant of $r$ and $s$ is dependent upon the specific integers chosen for $... | ;Proof by Counterexample
Let $r = \dfrac 1 2$ and $s = 1$.
We have:
:$r = \dfrac 1 2 = \dfrac 2 4 = \dfrac 3 6$
Then the mediant of $r = \dfrac 2 4$ and $s = \dfrac 1 1$ gives:
:$\dfrac {2 + 1} {4 + 1} = \dfrac 3 5$
but the mediant of $r = \dfrac 1 2$ and $s = \dfrac 1 1$ gives:
:$\dfrac {1 + 1} {2 + 1} = \dfrac 2 3$... | Let $r, s \in \Q$ be [[Definition:Rational Number|rational numbers]].
Let $r$ and $s$ be expressed as:
{{begin-eqn}}
{{eqn | l = r
| r = \dfrac a b
}}
{{eqn | l = s
| r = \dfrac c d
}}
{{end-eqn}}
where $a, b, c, d$ are [[Definition:Integer|integers]] such that $b > 0, d > 0$.
Then the [[Definition:Med... | ;[[Proof by Counterexample]]
Let $r = \dfrac 1 2$ and $s = 1$.
We have:
:$r = \dfrac 1 2 = \dfrac 2 4 = \dfrac 3 6$
Then the [[Definition:Mediant|mediant]] of $r = \dfrac 2 4$ and $s = \dfrac 1 1$ gives:
:$\dfrac {2 + 1} {4 + 1} = \dfrac 3 5$
but the [[Definition:Mediant|mediant]] of $r = \dfrac 1 2$ and $s = \d... | Mediant is Dependent upon Representation | https://proofwiki.org/wiki/Mediant_is_Dependent_upon_Representation | https://proofwiki.org/wiki/Mediant_is_Dependent_upon_Representation | [
"Mediants"
] | [
"Definition:Rational Number",
"Definition:Integer",
"Definition:Mediant",
"Definition:Integer"
] | [
"Proof by Counterexample",
"Definition:Mediant",
"Definition:Mediant"
] |
proofwiki-18855 | All Infima Preserving Mapping is Upper Adjoint of Galois Connection/Lemma 1 | :$g$ is an increasing mapping. | Define a mapping $d: T \to S$:
:$\forall t \in T: \map d t := \map \inf {g^{-1} \sqbrk {t^\succsim} }$
Let $x, y \in S$ such that
:$x \preceq y$
By Upper Closure is Decreasing:
:$y^\succeq \subseteq x^\succeq$
By Infimum of Upper Closure of Element:
:$\map \inf {x^\succeq} = x$ and $\map \inf {y^\succeq} = y$
By defini... | :$g$ is an [[Definition:Increasing Mapping|increasing mapping]]. | Define a [[Definition:Mapping|mapping]] $d: T \to S$:
:$\forall t \in T: \map d t := \map \inf {g^{-1} \sqbrk {t^\succsim} }$
Let $x, y \in S$ such that
:$x \preceq y$
By [[Upper Closure is Decreasing]]:
:$y^\succeq \subseteq x^\succeq$
By [[Infimum of Upper Closure of Element]]:
:$\map \inf {x^\succeq} = x$ and $\m... | All Infima Preserving Mapping is Upper Adjoint of Galois Connection/Lemma 1 | https://proofwiki.org/wiki/All_Infima_Preserving_Mapping_is_Upper_Adjoint_of_Galois_Connection/Lemma_1 | https://proofwiki.org/wiki/All_Infima_Preserving_Mapping_is_Upper_Adjoint_of_Galois_Connection/Lemma_1 | [
"All Infima Preserving Mapping is Upper Adjoint of Galois Connection"
] | [
"Definition:Increasing/Mapping"
] | [
"Definition:Mapping",
"Upper Closure is Decreasing",
"Infimum of Upper Closure of Element",
"Definition:Mapping Preserves Infimum/All",
"Definition:Mapping Preserves Infimum/Subset",
"Definition:Mapping Preserves Infimum/Subset",
"Definition:Mapping Preserves Infimum/Subset",
"Image of Subset under Ma... |
proofwiki-18856 | Category Axioms are Self-Dual/Object Category Theory | Let $\mathrm {CT}$ be the collection of seven axioms on Characterization of Metacategory via Equations.
Then:
:$\mathrm {CT} = \mathrm {CT}^*$
where $\mathrm {CT}^*$ consists of the dual statements of those in $\mathrm{CT}$. | The seven axioms are:
{{begin-eqn}}
{{eqn | l = \operatorname {dom} \operatorname {id}_A = A
| o = \qquad
| r = \operatorname {cod} \operatorname {id}_A = A
}}
{{eqn | l = f \circ 1_{\operatorname {dom} f} = f
| o =
| r = 1_{\operatorname {cod} f} \circ f = f
}}
{{eqn | l = \map {\operatorname ... | Let $\mathrm {CT}$ be the collection of seven axioms on [[Characterization of Metacategory via Equations]].
Then:
:$\mathrm {CT} = \mathrm {CT}^*$
where $\mathrm {CT}^*$ consists of the [[Definition:Dual Statement (Category Theory)|dual statements]] of those in $\mathrm{CT}$. | The seven axioms are:
{{begin-eqn}}
{{eqn | l = \operatorname {dom} \operatorname {id}_A = A
| o = \qquad
| r = \operatorname {cod} \operatorname {id}_A = A
}}
{{eqn | l = f \circ 1_{\operatorname {dom} f} = f
| o =
| r = 1_{\operatorname {cod} f} \circ f = f
}}
{{eqn | l = \map {\operatorname... | Category Axioms are Self-Dual/Object Category Theory | https://proofwiki.org/wiki/Category_Axioms_are_Self-Dual/Object_Category_Theory | https://proofwiki.org/wiki/Category_Axioms_are_Self-Dual/Object_Category_Theory | [
"Category Axioms are Self-Dual"
] | [
"Characterization of Metacategory via Equations",
"Definition:Dual Statement (Category Theory)"
] | [
"Definition:Dual Statement (Category Theory)",
"Definition:Bound Variable"
] |
proofwiki-18857 | Distribution Space over Smooth Functions is Unitary Module | The distribution space over smooth functions is a unitary module. | Let $\phi \in \map \DD {\R^d}$ be a test function. | The [[Definition:Distribution Space|distribution space]] over [[Definition:Smooth Function|smooth functions]] is a [[Definition:Unitary Module|unitary module]]. | Let $\phi \in \map \DD {\R^d}$ be a [[Definition:Test Function|test function]]. | Distribution Space over Smooth Functions is Unitary Module | https://proofwiki.org/wiki/Distribution_Space_over_Smooth_Functions_is_Unitary_Module | https://proofwiki.org/wiki/Distribution_Space_over_Smooth_Functions_is_Unitary_Module | [
"Schwartz Distributions",
"Examples of Unitary Modules"
] | [
"Definition:Distribution Space",
"Definition:Smooth Real Function",
"Definition:Unitary Module over Ring"
] | [
"Definition:Test Function"
] |
proofwiki-18858 | Smooth Real Function times Derivative of Dirac Delta Distribution/Corollary | Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Then in the distributional sense it holds that:
:$x \delta' = -\delta$ | From Smooth Real Function times Derivative of Dirac Delta Distribution:
:$\alpha \cdot \delta' = \map \alpha 0 \delta' - \map {\alpha'} 0 \delta$
where $\alpha$ is a smooth function.
If $\map \alpha x = x$, then:
:$x \delta' = -\delta$
{{qed}} | Let $\delta \in \map {\DD'} \R$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Then in the [[Definition:Schwartz Distribution|distributional sense]] it holds that:
:$x \delta' = -\delta$ | From [[Smooth Real Function times Derivative of Dirac Delta Distribution]]:
:$\alpha \cdot \delta' = \map \alpha 0 \delta' - \map {\alpha'} 0 \delta$
where $\alpha$ is a [[Definition:Smooth Function|smooth function]].
If $\map \alpha x = x$, then:
:$x \delta' = -\delta$
{{qed}} | Smooth Real Function times Derivative of Dirac Delta Distribution/Corollary | https://proofwiki.org/wiki/Smooth_Real_Function_times_Derivative_of_Dirac_Delta_Distribution/Corollary | https://proofwiki.org/wiki/Smooth_Real_Function_times_Derivative_of_Dirac_Delta_Distribution/Corollary | [
"Smooth Real Function times Derivative of Dirac Delta Distribution",
"Dirac Delta Function"
] | [
"Definition:Dirac Delta Distribution",
"Definition:Schwartz Distribution"
] | [
"Smooth Real Function times Derivative of Dirac Delta Distribution",
"Definition:Smooth Real Function"
] |
proofwiki-18859 | Hamiltonian Graph is not necessarily Ore Graph | Let $G = \struct {V, E}$ be a simple graph of order $n \ge 3$.
Let $G$ be a Hamiltonian graph.
Then $G$ is not necessarily an Ore graph. | Proof by Counterexample:
Recall the definition of an Ore graph:
:For each pair of non-adjacent vertices $u, v \in V$:
::$\deg u + \deg v \ge n$
Let $n \in \N$ such that $n \ge 5$.
Consider the cycle graph $C_n$.
We have from Cycle Graph is Hamiltonian that $C_n$ is a Hamiltonian graph.
We also have from Cycle Graph is ... | Let $G = \struct {V, E}$ be a [[Definition:Simple Graph|simple graph]] of [[Definition:Order of Graph|order $n \ge 3$]].
Let $G$ be a [[Definition:Hamiltonian Graph|Hamiltonian graph]].
Then $G$ is not necessarily an [[Definition:Ore Graph|Ore graph]]. | [[Proof by Counterexample]]:
Recall the definition of an [[Definition:Ore Graph|Ore graph]]:
:For each pair of [[Definition:Adjacent Vertices (Undirected Graph)|non-adjacent]] [[Definition:Vertex of Graph|vertices]] $u, v \in V$:
::$\deg u + \deg v \ge n$
Let $n \in \N$ such that $n \ge 5$.
Consider the [[Definiti... | Hamiltonian Graph is not necessarily Ore Graph | https://proofwiki.org/wiki/Hamiltonian_Graph_is_not_necessarily_Ore_Graph | https://proofwiki.org/wiki/Hamiltonian_Graph_is_not_necessarily_Ore_Graph | [
"Hamiltonian Graphs",
"Ore Graphs"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Hamiltonian Graph",
"Definition:Ore Graph"
] | [
"Proof by Counterexample",
"Definition:Ore Graph",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Cycle Graph",
"Cycle Graph is Hamiltonian",
"Definition:Hamiltonian Graph",
"Cycle Graph is 2-Regular",
"Definition:Graph (Graph Th... |
proofwiki-18860 | Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma | Let $\closedint a b$ be a closed real interval.
Let $c$ be a real number.
Let $a < c < b$.
Let $f$ be a real function defined on $\closedint a b$.
Let $\map L S$ be the lower Darboux sum of $f$ on $\closedint a b$ where $S$ is a subdivision of $\closedint a b$.
Let $P$ and $Q$ be finite subdivisions of $\closedint a b$... | This is an instance of Lower Sum of Refinement.
{{qed}} | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $c$ be a [[Definition:Real Number|real number]].
Let $a < c < b$.
Let $f$ be a [[Definition:Real Function|real function]] defined on $\closedint a b$.
Let $\map L S$ be the [[Definition:Lower Darboux Sum|lower Darboux sum]] of ... | This is an instance of [[Lower Sum of Refinement]].
{{qed}} | Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma/Proof 1 | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions/Lemma | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions/Lemma/Proof_1 | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Number",
"Definition:Real Function",
"Definition:Lower Darboux Sum",
"Definition:Subdivision of Interval",
"Definition:Subdivision of Interval/Finite"
] | [
"Lower Sum of Refinement"
] |
proofwiki-18861 | Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma | Let $\closedint a b$ be a closed real interval.
Let $c$ be a real number.
Let $a < c < b$.
Let $f$ be a real function defined on $\closedint a b$.
Let $\map L S$ be the lower Darboux sum of $f$ on $\closedint a b$ where $S$ is a subdivision of $\closedint a b$.
Let $P$ and $Q$ be finite subdivisions of $\closedint a b$... | Let $P = \set {x_0, x_1, \ldots, x_n}$.
Suppose that:
:$c \in P$
Then:
:$Q = P$
We have:
:$\map L P \ge \map L P$
:$\leadsto \map L Q \ge \map L P$ as $Q = P$
This finishes the proof for this case.
The only other possibility for $c$ is:
:$x_{j-1} < c < x_j$
where $1 \le j \le n$.
Let $m_i$ be the infimum of $f$ on the ... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $c$ be a [[Definition:Real Number|real number]].
Let $a < c < b$.
Let $f$ be a [[Definition:Real Function|real function]] defined on $\closedint a b$.
Let $\map L S$ be the [[Definition:Lower Darboux Sum|lower Darboux sum]] of ... | Let $P = \set {x_0, x_1, \ldots, x_n}$.
Suppose that:
:$c \in P$
Then:
:$Q = P$
We have:
:$\map L P \ge \map L P$
:$\leadsto \map L Q \ge \map L P$ as $Q = P$
This finishes the proof for this case.
The only other possibility for $c$ is:
:$x_{j-1} < c < x_j$
where $1 \le j \le n$.
Let $m_i$ be the [[Definition:... | Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma/Proof 2 | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions/Lemma | https://proofwiki.org/wiki/Sum_of_Integrals_on_Adjacent_Intervals_for_Integrable_Functions/Lemma/Proof_2 | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Number",
"Definition:Real Function",
"Definition:Lower Darboux Sum",
"Definition:Subdivision of Interval",
"Definition:Subdivision of Interval/Finite"
] | [
"Definition:Infimum of Set/Real Numbers",
"Definition:Real Interval/Closed",
"Definition:Infimum of Set/Real Numbers",
"Definition:Real Interval/Closed",
"Definition:Infimum of Set/Real Numbers",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Closed",
"Definition:Subset",
"Definition:L... |
proofwiki-18862 | Intersection of Class and Set is Set | Let $C$ be the class:
:$C = \set { u : \map \phi {u, p_1, \ldots, p_n} }$
Then for all sets $X$, $C \cap X$ is a set. | {{NotZFC}}
By the definition of class intersection:
:$a \in C \cap X \implies a \in C \land a \in X$
Thus:
:$a \in C \cap X \implies a \in X$
The subclass definition gives:
:$C \cap X \subseteq X$
By Subclass of Set is Set, $C \cap X$ is a set.
{{qed}} | Let $C$ be the [[Definition:Class (Class Theory)|class]]:
:$C = \set { u : \map \phi {u, p_1, \ldots, p_n} }$
Then for all [[Definition:Set|sets]] $X$, $C \cap X$ is a [[Definition:Set|set]]. | {{NotZFC}}
By the definition of [[Definition:Class Intersection|class intersection]]:
:$a \in C \cap X \implies a \in C \land a \in X$
Thus:
:$a \in C \cap X \implies a \in X$
The [[Definition:Subclass|subclass]] definition gives:
:$C \cap X \subseteq X$
By [[Subclass of Set is Set]], $C \cap X$ is a [[Definition:... | Intersection of Class and Set is Set | https://proofwiki.org/wiki/Intersection_of_Class_and_Set_is_Set | https://proofwiki.org/wiki/Intersection_of_Class_and_Set_is_Set | [
"Gödel-Bernays Class Theory",
"Class Intersection",
"Subclasses"
] | [
"Definition:Class (Class Theory)",
"Definition:Set",
"Definition:Set"
] | [
"Definition:Class Intersection",
"Definition:Subclass",
"Subclass of Set is Set",
"Definition:Set"
] |
proofwiki-18863 | If Set Exists then Empty Set Exists | If at least one set exists, then there exists an empty set. | {{NotZFC}}
Let $S$ be a set.
By the axiom of class comprehension, there is an empty class:
:$\O = \set { x : x \ne x }$
Since $x \in \O$ is never true, it follows vacuously that:
:$x \in \O \implies x \in S$
By the subclass definition:
:$\O \subseteq S$
By Subclass of Set is Set, $\O$ is a set.
{{qed}} | If at least one [[Definition:Set|set]] exists, then there exists an [[Definition:Empty Set|empty set]]. | {{NotZFC}}
Let $S$ be a [[Definition:Set|set]].
By the [[Axiom:Class Comprehension Schema|axiom of class comprehension]], there is an [[Definition:Empty Set|empty class]]:
:$\O = \set { x : x \ne x }$
Since $x \in \O$ is never true, it follows [[Definition:Vacuous Truth|vacuously]] that:
:$x \in \O \implies x \in S$... | If Set Exists then Empty Set Exists | https://proofwiki.org/wiki/If_Set_Exists_then_Empty_Set_Exists | https://proofwiki.org/wiki/If_Set_Exists_then_Empty_Set_Exists | [
"Gödel-Bernays Class Theory",
"Empty Set"
] | [
"Definition:Set",
"Definition:Empty Set"
] | [
"Definition:Set",
"Axiom:Class Comprehension Schema",
"Definition:Empty Set",
"Definition:Vacuous Truth",
"Definition:Subclass",
"Subclass of Set is Set",
"Definition:Set"
] |
proofwiki-18864 | Powerset is not Subset of its Set | Let $A$ be a set.
Then:
:$\powerset A \not \subseteq A$ | {{AimForCont}} that $\powerset A \subseteq A$, and define:
:$C = \set {x \in \powerset A : x \notin x}$
We have that $C \subseteq \powerset A$, as it contains only the $x \in \powerset A$ meeting the condition $x \notin x$.
Since $\powerset A \subseteq A$, we have:
:$C \subseteq A$
and thus
:$C \in \powerset A$
We can ... | Let $A$ be a [[Definition:Set|set]].
Then:
:$\powerset A \not \subseteq A$ | {{AimForCont}} that $\powerset A \subseteq A$, and define:
:$C = \set {x \in \powerset A : x \notin x}$
We have that $C \subseteq \powerset A$, as it contains only the $x \in \powerset A$ meeting the condition $x \notin x$.
Since $\powerset A \subseteq A$, we have:
:$C \subseteq A$
and thus
:$C \in \powerset A$
We ... | Powerset is not Subset of its Set/Proof 1 | https://proofwiki.org/wiki/Powerset_is_not_Subset_of_its_Set | https://proofwiki.org/wiki/Powerset_is_not_Subset_of_its_Set/Proof_1 | [
"Powerset is not Subset of its Set",
"Power Set",
"Subsets"
] | [
"Definition:Set"
] | [
"Definition:Contradiction",
"Russell's Paradox"
] |
proofwiki-18865 | Powerset is not Subset of its Set | Let $A$ be a set.
Then:
:$\powerset A \not \subseteq A$ | {{AimForCont}} that $\powerset A \subseteq A$.
Let $I: \powerset A \to A$ be the identity mapping.
$I$ is an injection by Identity Mapping is Injection.
But by No Injection from Power Set to Set, this is a contradiction.
{{qed}} | Let $A$ be a [[Definition:Set|set]].
Then:
:$\powerset A \not \subseteq A$ | {{AimForCont}} that $\powerset A \subseteq A$.
Let $I: \powerset A \to A$ be the [[Definition:Identity Mapping|identity mapping]].
$I$ is an [[Definition:Injection|injection]] by [[Identity Mapping is Injection]].
But by [[No Injection from Power Set to Set]], this is a [[Definition:Contradiction|contradiction]].
{{... | Powerset is not Subset of its Set/Proof 2 | https://proofwiki.org/wiki/Powerset_is_not_Subset_of_its_Set | https://proofwiki.org/wiki/Powerset_is_not_Subset_of_its_Set/Proof_2 | [
"Powerset is not Subset of its Set",
"Power Set",
"Subsets"
] | [
"Definition:Set"
] | [
"Definition:Identity Mapping",
"Definition:Injection",
"Identity Mapping is Injection",
"No Injection from Power Set to Set",
"Definition:Contradiction"
] |
proofwiki-18866 | Powerset is not Subset of its Set | Let $A$ be a set.
Then:
:$\powerset A \not \subseteq A$ | {{AimForCont}} that $\powerset A \subseteq A$.
Since $A \in \powerset A$, this implies:
:$A \in A$
But this contradicts Set is Not Element of Itself.
{{qed}} | Let $A$ be a [[Definition:Set|set]].
Then:
:$\powerset A \not \subseteq A$ | {{AimForCont}} that $\powerset A \subseteq A$.
Since $A \in \powerset A$, this implies:
:$A \in A$
But this [[Definition:Contradiction|contradicts]] [[Set is Not Element of Itself]].
{{qed}} | Powerset is not Subset of its Set/Proof 3 | https://proofwiki.org/wiki/Powerset_is_not_Subset_of_its_Set | https://proofwiki.org/wiki/Powerset_is_not_Subset_of_its_Set/Proof_3 | [
"Powerset is not Subset of its Set",
"Power Set",
"Subsets"
] | [
"Definition:Set"
] | [
"Definition:Contradiction",
"Set is Not Element of Itself"
] |
proofwiki-18867 | Integer which is Multiplied by Last Digit when moving Last Digit to First | Let $N$ be a positive integer expressed in decimal notation in the form:
:$N = \sqbrk {a_k a_{k - 1} a_{k - 2} \ldots a_2 a_1}_{10}$
Let $N$ be such that when you multiply it by $a_1$, you get:
:$a_1 N = \sqbrk {a_1 a_k a_{k - 1} \ldots a_3 a_2}_{10}$
Then at least one such $N$ is equal to the recurring part of the fra... | Let us consider:
:$q = 0 \cdotp \dot a_k a_{k - 1} a_{k - 2} \ldots a_2 \dot a_1$
Let:
:$a_1 q = 0 \cdotp \dot a_1 a_k a_{k - 1} \ldots a_3 \dot a_2$
Then:
{{begin-eqn}}
{{eqn | l = 10 a_1 q
| r = a_1 \cdotp \dot a_k a_{k - 1} a_{k - 2} \ldots a_2 \dot a_1
| c =
}}
{{eqn | ll= \leadsto
| l = 10 a_1 q... | Let $N$ be a [[Definition:Positive Integer|positive integer]] expressed in [[Definition:Decimal Notation|decimal notation]] in the form:
:$N = \sqbrk {a_k a_{k - 1} a_{k - 2} \ldots a_2 a_1}_{10}$
Let $N$ be such that when you [[Definition:Integer Multiplication|multiply]] it by $a_1$, you get:
:$a_1 N = \sqbrk {a_1... | Let us consider:
:$q = 0 \cdotp \dot a_k a_{k - 1} a_{k - 2} \ldots a_2 \dot a_1$
Let:
:$a_1 q = 0 \cdotp \dot a_1 a_k a_{k - 1} \ldots a_3 \dot a_2$
Then:
{{begin-eqn}}
{{eqn | l = 10 a_1 q
| r = a_1 \cdotp \dot a_k a_{k - 1} a_{k - 2} \ldots a_2 \dot a_1
| c =
}}
{{eqn | ll= \leadsto
| l = 10 a_... | Integer which is Multiplied by Last Digit when moving Last Digit to First | https://proofwiki.org/wiki/Integer_which_is_Multiplied_by_Last_Digit_when_moving_Last_Digit_to_First | https://proofwiki.org/wiki/Integer_which_is_Multiplied_by_Last_Digit_when_moving_Last_Digit_to_First | [
"Recreational Mathematics"
] | [
"Definition:Positive/Integer",
"Definition:Decimal Notation",
"Definition:Multiplication/Integers",
"Definition:Basis Expansion/Recurrence/Recurring Part",
"Definition:Fraction"
] | [
"Category:Recreational Mathematics"
] |
proofwiki-18868 | Weak Solution to Dx u + 3yu = 0 with Heaviside Step Function Boundary Condition | Consider the boundary value problem:
:<nowiki>$\begin{cases}
\dfrac {\partial u} {\partial x} + 3 y u = 0 & : x \in \R_{>0},~ y \in \R \\
& \\
\map u {0, y} = \map H y & : y \in \R \\
\end{cases}$</nowiki>
Then it has a weak solution of the form:
:$u = e^{-3 y x} \map H y$ | Let $u = e^{-3 y x} \map H y$
We have that:
:Heaviside Step Function is Locally Integrable
:Locally Integrable Function defines Distribution
:Multiplication of Distribution induced by Locally Integrable Function by Smooth Function
Hence, we can define a distribution $T_u \in \map {\DD'} {\R^2}$ associated with $u$.
The... | Consider the [[Definition:Boundary Value Problem|boundary value problem]]:
:<nowiki>$\begin{cases}
\dfrac {\partial u} {\partial x} + 3 y u = 0 & : x \in \R_{>0},~ y \in \R \\
& \\
\map u {0, y} = \map H y & : y \in \R \\
\end{cases}$</nowiki>
Then it has a [[Definition:Weak Solution|weak solution]] of the form:
:$... | Let $u = e^{-3 y x} \map H y$
We have that:
:[[Heaviside Step Function is Locally Integrable]]
:[[Locally Integrable Function defines Distribution]]
:[[Multiplication of Distribution induced by Locally Integrable Function by Smooth Function]]
Hence, we can define a [[Definition:Schwartz Distribution|distribution]] $... | Weak Solution to Dx u + 3yu = 0 with Heaviside Step Function Boundary Condition | https://proofwiki.org/wiki/Weak_Solution_to_Dx_u_+_3yu_=_0_with_Heaviside_Step_Function_Boundary_Condition | https://proofwiki.org/wiki/Weak_Solution_to_Dx_u_+_3yu_=_0_with_Heaviside_Step_Function_Boundary_Condition | [
"Examples of Weak Solutions"
] | [
"Definition:Boundary Value Problem",
"Definition:Differential Equation/Solution/Weak Solution"
] | [
"Heaviside Step Function is Locally Integrable",
"Locally Integrable Function defines Distribution",
"Multiplication of Distribution induced by Locally Integrable Function by Smooth Function",
"Definition:Schwartz Distribution",
"Definition:Schwartz Distribution",
"Multiplication of Distribution induced b... |
proofwiki-18869 | Divisors of One More than Power of 10/Number of Zero Digits Congruent to 2 Modulo 3 | Let $N$ be a natural number of the form:
:$N = 1000 \ldots 01$
where the number of zero digits between the two $1$ digits is of the form $3 k - 1$.
Then $N$ has divisors:
::$1 \underbrace {00 \ldots 0}_{\text {$k - 1$ $0$'s} } 1$
:where the number of zero digits between the two $1$ digits is $k - 1$
::$\underbrace {99 ... | By definition, $N$ can be expressed as:
:$N = 10^{3 k} + 1$
Let $a := 10^k$.
Then we have:
{{begin-eqn}}
{{eqn | l = N
| r = a^3 + 1
| c =
}}
{{eqn | r = \paren {a + 1} \paren {a^2 - a + 1}
| c = Sum of Two Cubes
}}
{{end-eqn}}
where it is noted that:
{{begin-eqn}}
{{eqn | l = \underbrace {99 \ldots ... | Let $N$ be a [[Definition:Natural Number|natural number]] of the form:
:$N = 1000 \ldots 01$
where the number of [[Definition:Zero Digit|zero digits]] between the two $1$ [[Definition:Digit|digits]] is of the form $3 k - 1$.
Then $N$ has [[Definition:Divisor of Integer|divisors]]:
::$1 \underbrace {00 \ldots 0}_{\text... | By definition, $N$ can be expressed as:
:$N = 10^{3 k} + 1$
Let $a := 10^k$.
Then we have:
{{begin-eqn}}
{{eqn | l = N
| r = a^3 + 1
| c =
}}
{{eqn | r = \paren {a + 1} \paren {a^2 - a + 1}
| c = [[Sum of Two Cubes]]
}}
{{end-eqn}}
where it is noted that:
{{begin-eqn}}
{{eqn | l = \underbrace {99 ... | Divisors of One More than Power of 10/Number of Zero Digits Congruent to 2 Modulo 3 | https://proofwiki.org/wiki/Divisors_of_One_More_than_Power_of_10/Number_of_Zero_Digits_Congruent_to_2_Modulo_3 | https://proofwiki.org/wiki/Divisors_of_One_More_than_Power_of_10/Number_of_Zero_Digits_Congruent_to_2_Modulo_3 | [
"Divisors of One More than Power of 10"
] | [
"Definition:Natural Numbers",
"Definition:Zero Digit",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer",
"Definition:Zero Digit",
"Definition:Digit"
] | [
"Sum of Two Odd Powers/Examples/Sum of Two Cubes"
] |
proofwiki-18870 | Divisors of One More than Power of 10/Number of Zero Digits Even | Let $N$ be a natural number of the form:
:$N = 1 \underbrace {000 \ldots 0}_{\text {$2 k$ $0$'s} } 1$
that is, where the number of zero digits between the two $1$ digits is even.
Then $N$ can be expressed as:
:$N = 11 \times \underbrace {9090 \ldots 90}_{\text {$k - 1$ $90$'s} } 91$ | By definition, $N$ can be expressed as:
:$N = 10^{2 k + 1} + 1$
Then we have:
{{begin-eqn}}
{{eqn | l = N
| r = 10^{2 k + 1} + 1
| c =
}}
{{eqn | r = \paren {10 + 1} \sum_{j \mathop = 0}^{2 k} \paren {-1}^j 10^{2 k - j}
| c = Sum of Two Odd Powers
}}
{{eqn | ll= \leadsto
| l = \dfrac N {11}
... | Let $N$ be a [[Definition:Natural Number|natural number]] of the form:
:$N = 1 \underbrace {000 \ldots 0}_{\text {$2 k$ $0$'s} } 1$
that is, where the number of [[Definition:Zero Digit|zero digits]] between the two $1$ [[Definition:Digit|digits]] is [[Definition:Even Integer|even]].
Then $N$ can be expressed as:
:$N ... | By definition, $N$ can be expressed as:
:$N = 10^{2 k + 1} + 1$
Then we have:
{{begin-eqn}}
{{eqn | l = N
| r = 10^{2 k + 1} + 1
| c =
}}
{{eqn | r = \paren {10 + 1} \sum_{j \mathop = 0}^{2 k} \paren {-1}^j 10^{2 k - j}
| c = [[Sum of Two Odd Powers]]
}}
{{eqn | ll= \leadsto
| l = \dfrac N {1... | Divisors of One More than Power of 10/Number of Zero Digits Even | https://proofwiki.org/wiki/Divisors_of_One_More_than_Power_of_10/Number_of_Zero_Digits_Even | https://proofwiki.org/wiki/Divisors_of_One_More_than_Power_of_10/Number_of_Zero_Digits_Even | [
"Divisors of One More than Power of 10"
] | [
"Definition:Natural Numbers",
"Definition:Zero Digit",
"Definition:Digit",
"Definition:Even Integer"
] | [
"Sum of Two Odd Powers"
] |
proofwiki-18871 | Measurable Set with Negative Measure has Negative Subset | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $A \in \Sigma$ have:
:$-\infty < \map \mu A < 0$
Then:
:there exists a $\mu$-negative set $B$ such that $B \subseteq A$ and $\map \mu B \le \map \mu A$. | Let:
:$\delta_1 = \sup \set {\map \mu E : E \in \Sigma \text { and } E \subseteq A}$
possibly infinite.
Since $\O \in \Sigma$ and $\O \subseteq A$, we have:
:$\map \mu \O = 0 \in \set {\map \mu E : E \in \Sigma \text { and } E \subseteq A}$
and so, from the definition of supremum, we have:
:$\delta_1 \ge 0$.
Again ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $A \in \Sigma$ have:
:$-\infty < \map \mu A < 0$
Then:
:there exists a [[Definition:Negative Set|$\mu$-negative set]] $B$ such that $B \subse... | Let:
:$\delta_1 = \sup \set {\map \mu E : E \in \Sigma \text { and } E \subseteq A}$
possibly [[Definition:Infinite Set|infinite]].
Since $\O \in \Sigma$ and $\O \subseteq A$, we have:
:$\map \mu \O = 0 \in \set {\map \mu E : E \in \Sigma \text { and } E \subseteq A}$
and so, from the definition of [[Definition... | Measurable Set with Negative Measure has Negative Subset | https://proofwiki.org/wiki/Measurable_Set_with_Negative_Measure_has_Negative_Subset | https://proofwiki.org/wiki/Measurable_Set_with_Negative_Measure_has_Negative_Subset | [
"Signed Measures",
"Negative Sets"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Negative Set"
] | [
"Definition:Infinite Set",
"Definition:Supremum of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Disjoint Sets",
"Definition:Disjoint Sets",
"Definition:Signed Measure",
"Definition:Countably Additive Function",
"Definition:Count... |
proofwiki-18872 | Complement of Horizontal Section of Set is Horizontal Section of Complement | Let $X$ and $Y$ be sets.
Let $E \subseteq X \times Y$.
Let $y \in Y$.
Then:
:$\paren {\paren {X \times Y} \setminus E}^y = X \setminus E^y$
where:
:$\paren {\paren {X \times Y} \setminus E}^y$ is the $y$-horizontal section of the set difference $\paren {X \times Y} \setminus E$
:$E^y$ is the $y$-horizontal section o... | Note that from the definition of set difference, we have that:
:$x \in X \setminus E^y$
{{iff}}:
:$x \in X$ and $x \not \in E^y$.
That is, from the definition of the $y$-horizontal section:
:$x \in X$ and $\tuple {x, y} \not \in E$.
This is equivalent to:
:$\tuple {x, y} \in \paren {X \times Y} \setminus E$
From th... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $E \subseteq X \times Y$.
Let $y \in Y$.
Then:
:$\paren {\paren {X \times Y} \setminus E}^y = X \setminus E^y$
where:
:$\paren {\paren {X \times Y} \setminus E}^y$ is the [[Definition:Horizontal Section of Set|$y$-horizontal section]] of the [[Definition:Set Dif... | Note that from the definition of [[Definition:Set Difference|set difference]], we have that:
:$x \in X \setminus E^y$
{{iff}}:
:$x \in X$ and $x \not \in E^y$.
That is, from the definition of the [[Definition:Horizontal Section of Set|$y$-horizontal section]]:
:$x \in X$ and $\tuple {x, y} \not \in E$.
This is... | Complement of Horizontal Section of Set is Horizontal Section of Complement | https://proofwiki.org/wiki/Complement_of_Horizontal_Section_of_Set_is_Horizontal_Section_of_Complement | https://proofwiki.org/wiki/Complement_of_Horizontal_Section_of_Set_is_Horizontal_Section_of_Complement | [
"Horizontal Section of Sets"
] | [
"Definition:Set",
"Definition:Horizontal Section of Set",
"Definition:Set Difference",
"Definition:Horizontal Section of Set"
] | [
"Definition:Set Difference",
"Definition:Horizontal Section of Set",
"Definition:Horizontal Section of Set",
"Category:Horizontal Section of Sets"
] |
proofwiki-18873 | Complement of Vertical Section of Set is Vertical Section of Complement | Let $X$ and $Y$ be sets.
Let $E \subseteq X \times Y$.
Let $x \in X$.
Then:
:$\paren {\paren {X \times Y} \setminus E}_x = Y \setminus E_x$
where:
:$\paren {\paren {X \times Y} \setminus E}_x$ is the $x$-vertical section of the set difference $\paren {X \times Y} \setminus E$
:$E_x$ is the $x$-vertical section of $E... | Note that from the definition of set difference, we have that:
:$y \in Y \setminus E_x$
{{iff}}:
:$y \in Y$ and $y \not \in E_x$.
That is, from the definition of the $x$-vertical section:
:$y \in Y$ and $\tuple {x, y} \not \in E$.
This is equivalent to:
:$\tuple {x, y} \in \paren {X \times Y} \setminus E$
From the ... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $E \subseteq X \times Y$.
Let $x \in X$.
Then:
:$\paren {\paren {X \times Y} \setminus E}_x = Y \setminus E_x$
where:
:$\paren {\paren {X \times Y} \setminus E}_x$ is the [[Definition:Vertical Section of Set|$x$-vertical section]] of the [[Definition:Set Differe... | Note that from the definition of [[Definition:Set Difference|set difference]], we have that:
:$y \in Y \setminus E_x$
{{iff}}:
:$y \in Y$ and $y \not \in E_x$.
That is, from the definition of the [[Definition:Vertical Section of Set|$x$-vertical section]]:
:$y \in Y$ and $\tuple {x, y} \not \in E$.
This is equ... | Complement of Vertical Section of Set is Vertical Section of Complement | https://proofwiki.org/wiki/Complement_of_Vertical_Section_of_Set_is_Vertical_Section_of_Complement | https://proofwiki.org/wiki/Complement_of_Vertical_Section_of_Set_is_Vertical_Section_of_Complement | [
"Vertical Section of Sets"
] | [
"Definition:Set",
"Definition:Vertical Section of Set",
"Definition:Set Difference",
"Definition:Vertical Section of Set"
] | [
"Definition:Set Difference",
"Definition:Vertical Section of Set",
"Definition:Vertical Section of Set",
"Category:Vertical Section of Sets"
] |
proofwiki-18874 | Union of Horizontal Sections is Horizontal Section of Union | Let $X$, $Y$ and $A$ be sets.
Let $\set {E_\alpha : \alpha \in A}$ be a set of subsets of $X \times Y$.
Let $y \in Y$.
Then:
:$\ds \paren {\bigcup_{\alpha \mathop \in A} E_\alpha}^y = \bigcup_{\alpha \mathop \in A} \paren {E_\alpha}^y$
where:
:$\ds \paren {\bigcup_{\alpha \mathop \in A} E_\alpha}^y$ is the $y$-horiz... | Note that:
:$\ds x \in \bigcup_{\alpha \mathop \in A} \paren {E_\alpha}^y$
{{iff}}:
:$x \in \paren {E_\alpha}^y$ for some $\alpha \in A$.
From the definition of the $y$-horizontal section, this is equivalent to:
:$\tuple {x, y} \in E_\alpha$ for some $\alpha \in A$.
This in turn is equivalent to:
:$\ds \tuple {x, y}... | Let $X$, $Y$ and $A$ be [[Definition:Set|sets]].
Let $\set {E_\alpha : \alpha \in A}$ be a [[Definition:Set|set]] of subsets of $X \times Y$.
Let $y \in Y$.
Then:
:$\ds \paren {\bigcup_{\alpha \mathop \in A} E_\alpha}^y = \bigcup_{\alpha \mathop \in A} \paren {E_\alpha}^y$
where:
:$\ds \paren {\bigcup_{\alpha ... | Note that:
:$\ds x \in \bigcup_{\alpha \mathop \in A} \paren {E_\alpha}^y$
{{iff}}:
:$x \in \paren {E_\alpha}^y$ for some $\alpha \in A$.
From the definition of the [[Definition:Horizontal Section of Set|$y$-horizontal section]], this is equivalent to:
:$\tuple {x, y} \in E_\alpha$ for some $\alpha \in A$.
This... | Union of Horizontal Sections is Horizontal Section of Union | https://proofwiki.org/wiki/Union_of_Horizontal_Sections_is_Horizontal_Section_of_Union | https://proofwiki.org/wiki/Union_of_Horizontal_Sections_is_Horizontal_Section_of_Union | [
"Set Union",
"Horizontal Section of Sets"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Horizontal Section of Set",
"Definition:Horizontal Section of Set"
] | [
"Definition:Horizontal Section of Set",
"Definition:Horizontal Section of Set",
"Category:Set Union",
"Category:Horizontal Section of Sets"
] |
proofwiki-18875 | Horizontal Section of Cartesian Product | Let $X$ and $Y$ be sets.
Let $A \subseteq X$ and $B \subseteq Y$, so that $A \times B \subseteq X \times Y$.
Let $y \in Y$.
Then:
:$\paren {A \times B}^y = \begin{cases}A & y \in B \\ \O & y \not \in B\end{cases}$
where $\paren {A \times B}^y$ is a horizontal section of $A \times B$. | Let $y \in B$.
From the definition of the horizontal section, we have:
:$x \in \paren {A \times B}^y$
{{iff}}:
:$\tuple {x, y} \in A \times B$
Since $y \in B$, this equivalent to:
:$x \in A$
So:
:$x \in \paren {A \times B}^y$ {{iff}} $x \in A$
giving:
:$\paren {A \times B}^y = A$ if $y \in B$.
Now let $y \in Y \set... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $A \subseteq X$ and $B \subseteq Y$, so that $A \times B \subseteq X \times Y$.
Let $y \in Y$.
Then:
:$\paren {A \times B}^y = \begin{cases}A & y \in B \\ \O & y \not \in B\end{cases}$
where $\paren {A \times B}^y$ is a [[Definition:Horizontal Section of Set|hori... | Let $y \in B$.
From the definition of the [[Definition:Horizontal Section of Set|horizontal section]], we have:
:$x \in \paren {A \times B}^y$
{{iff}}:
:$\tuple {x, y} \in A \times B$
Since $y \in B$, this equivalent to:
:$x \in A$
So:
:$x \in \paren {A \times B}^y$ {{iff}} $x \in A$
giving:
:$\paren {A \... | Horizontal Section of Cartesian Product | https://proofwiki.org/wiki/Horizontal_Section_of_Cartesian_Product | https://proofwiki.org/wiki/Horizontal_Section_of_Cartesian_Product | [
"Horizontal Section of Sets",
"Cartesian Product"
] | [
"Definition:Set",
"Definition:Horizontal Section of Set"
] | [
"Definition:Horizontal Section of Set",
"Definition:Set Difference",
"Category:Horizontal Section of Sets",
"Category:Cartesian Product"
] |
proofwiki-18876 | Horizontal Section of Measurable Set is Measurable | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be measurable spaces.
Let $E \in \Sigma_X \otimes \Sigma_Y$ where $\Sigma_X \otimes \Sigma_Y$ is the product $\sigma$-algebra of $\Sigma_X$ and $\Sigma_Y$.
Let $y \in Y$.
Then:
:$E^y \in \Sigma_X$
where $E^y$ is the $y$-horizontal section of $E$. | Define $f_y : X \to X \times Y$ by:
:$\map {f_y} x = \tuple {x, y}$
for each $x \in X$.
Note that we have $\tuple {x, y} \in E$ {{iff}} $x \in E^y$ from the definition of the horizontal section.
In other words:
:$\paren {f_y}^{-1} \sqbrk E = E^y$
We now show that $f_y$ is $\Sigma_X/\paren {\Sigma_X \otimes \Sigma_Y}... | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be [[Definition:Measurable Space|measurable spaces]].
Let $E \in \Sigma_X \otimes \Sigma_Y$ where $\Sigma_X \otimes \Sigma_Y$ is the [[Definition:Product Sigma-Algebra|product $\sigma$-algebra]] of $\Sigma_X$ and $\Sigma_Y$.
Let $y \in Y$.
Then:
:$E^y \in ... | Define $f_y : X \to X \times Y$ by:
:$\map {f_y} x = \tuple {x, y}$
for each $x \in X$.
Note that we have $\tuple {x, y} \in E$ {{iff}} $x \in E^y$ from the definition of the [[Definition:Horizontal Section of Set|horizontal section]].
In other words:
:$\paren {f_y}^{-1} \sqbrk E = E^y$
We now show that $f_y$ ... | Horizontal Section of Measurable Set is Measurable/Proof 2 | https://proofwiki.org/wiki/Horizontal_Section_of_Measurable_Set_is_Measurable | https://proofwiki.org/wiki/Horizontal_Section_of_Measurable_Set_is_Measurable/Proof_2 | [
"Horizontal Section of Sets",
"Horizontal Section of Measurable Set is Measurable"
] | [
"Definition:Measurable Space",
"Definition:Product Sigma-Algebra",
"Definition:Horizontal Section of Set"
] | [
"Definition:Horizontal Section of Set",
"Definition:Measurable Mapping",
"Mapping Measurable iff Measurable on Generator",
"Definition:Product Sigma-Algebra",
"Definition:Generated Sigma-Algebra",
"Horizontal Section of Cartesian Product",
"Definition:Sigma-Algebra",
"Mapping Measurable iff Measurable... |
proofwiki-18877 | Union of Vertical Sections is Vertical Section of Union | Let $X$ and $Y$ be sets.
Let $\set {E_\alpha : \alpha \in A}$ be a set of subsets of $X \times Y$.
Let $x \in X$.
Then:
:$\ds \paren {\bigcup_{\alpha \in A} E_\alpha}_x = \bigcup_{\alpha \in A} \paren {E_\alpha}_x$
where:
:$\ds \paren {\bigcup_{\alpha \in A} E_\alpha}_x$ is the $x$-vertical section of $\ds \bigcup_{... | Note that:
:$\ds y \in \bigcup_{\alpha \in A} \paren {E_\alpha}_x$
{{iff}}:
:$y \in \paren {E_\alpha}_x$ for some $\alpha \in A$.
From the definition of the $x$-vertical section, this is equivalent to:
:$\tuple {x, y} \in E_\alpha$ for some $\alpha \in A$.
This in turn is equivalent to:
:$\ds \tuple {x, y} \in \bigc... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $\set {E_\alpha : \alpha \in A}$ be a [[Definition:Set|set]] of subsets of $X \times Y$.
Let $x \in X$.
Then:
:$\ds \paren {\bigcup_{\alpha \in A} E_\alpha}_x = \bigcup_{\alpha \in A} \paren {E_\alpha}_x$
where:
:$\ds \paren {\bigcup_{\alpha \in A} E_\alpha}_x$ i... | Note that:
:$\ds y \in \bigcup_{\alpha \in A} \paren {E_\alpha}_x$
{{iff}}:
:$y \in \paren {E_\alpha}_x$ for some $\alpha \in A$.
From the definition of the [[Definition:Vertical Section of Set|$x$-vertical section]], this is equivalent to:
:$\tuple {x, y} \in E_\alpha$ for some $\alpha \in A$.
This in turn is ... | Union of Vertical Sections is Vertical Section of Union | https://proofwiki.org/wiki/Union_of_Vertical_Sections_is_Vertical_Section_of_Union | https://proofwiki.org/wiki/Union_of_Vertical_Sections_is_Vertical_Section_of_Union | [
"Set Union",
"Vertical Section of Sets"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Vertical Section of Set",
"Definition:Vertical Section of Set"
] | [
"Definition:Vertical Section of Set",
"Definition:Vertical Section of Set",
"Category:Set Union",
"Category:Vertical Section of Sets"
] |
proofwiki-18878 | Tempered Distribution Space is Proper Subset of Distribution Space | Let $\map {\DD'} \R$ be the distribution space.
Let $\map {\SS'} \R$ be the tempered distribution space.
Then $\map {\SS'} \R$ is a proper subset of $\map {\DD'} \R$:
:$\map {\SS'} \R \subsetneqq \map {\DD'} \R$ | By Convergence of Sequence of Test Functions in Test Function Space implies Convergence in Schwartz Space we have that $\map {\SS'} \R \subseteq \map {\DD'} \R$.
{{Research|how?}}
Consider the real function $\map f x = e^{x^2}$.
We have that:
:Real Power Function for Positive Integer Power is Continuous
:Exponential Fu... | Let $\map {\DD'} \R$ be the [[Definition:Distribution Space|distribution space]].
Let $\map {\SS'} \R$ be the [[Definition:Tempered Distribution Space|tempered distribution space]].
Then $\map {\SS'} \R$ is a [[Definition:Proper Subset|proper subset]] of $\map {\DD'} \R$:
:$\map {\SS'} \R \subsetneqq \map {\DD'} \R... | By [[Convergence of Sequence of Test Functions in Test Function Space implies Convergence in Schwartz Space]] we have that $\map {\SS'} \R \subseteq \map {\DD'} \R$.
{{Research|how?}}
Consider the [[Definition:Real Function|real function]] $\map f x = e^{x^2}$.
We have that:
:[[Real Power Function for Positive Integ... | Tempered Distribution Space is Proper Subset of Distribution Space | https://proofwiki.org/wiki/Tempered_Distribution_Space_is_Proper_Subset_of_Distribution_Space | https://proofwiki.org/wiki/Tempered_Distribution_Space_is_Proper_Subset_of_Distribution_Space | [
"Tempered Distributions"
] | [
"Definition:Distribution Space",
"Definition:Tempered Distribution Space",
"Definition:Proper Subset"
] | [
"Convergence of Sequence of Test Functions in Test Function Space implies Convergence in Schwartz Space",
"Definition:Real Function",
"Real Power Function for Positive Integer Power is Continuous",
"Exponential Function is Continuous/Real Numbers",
"Composite of Continuous Mappings is Continuous",
"Defini... |
proofwiki-18879 | Measure of Vertical Section of Measurable Set gives Measurable Function | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.
For each $E \in \Sigma_X \otimes \Sigma_Y$, define the function $f_E : X \to \overline \R$ by:
:$\map {f_E} x = \map {\nu} {E_x}$
for each $x \in X$ where:
:$\Sigma_X \otimes \Sigma_Y$ is the product $\sigma$-algebra ... | From Vertical Section of Measurable Set is Measurable, the function $f_E$ is certainly well-defined for each $E \in \Sigma_X \otimes \Sigma_Y$.
First suppose that $\nu$ is a finite measure.
Let:
:$\mathcal F = \set {E \in \Sigma_X \otimes \Sigma_Y : f_E \text { is } \Sigma_X\text{-measurable} }$
We aim to show that: ... | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be [[Definition:Sigma-Finite Measure Space|$\sigma$-finite]] [[Definition:Measure Space|measure spaces]].
For each $E \in \Sigma_X \otimes \Sigma_Y$, define the [[Definition:Extended Real-Valued Function|function]] $f_E : X \to \overline \R$ by:
:$\ma... | From [[Vertical Section of Measurable Set is Measurable]], the [[Definition:Extended Real-Valued Function|function]] $f_E$ is certainly well-defined for each $E \in \Sigma_X \otimes \Sigma_Y$.
First suppose that $\nu$ is a [[Definition:Finite Measure|finite measure]].
Let:
:$\mathcal F = \set {E \in \Sigma_X \oti... | Measure of Vertical Section of Measurable Set gives Measurable Function | https://proofwiki.org/wiki/Measure_of_Vertical_Section_of_Measurable_Set_gives_Measurable_Function | https://proofwiki.org/wiki/Measure_of_Vertical_Section_of_Measurable_Set_gives_Measurable_Function | [
"Vertical Section of Sets"
] | [
"Definition:Sigma-Finite Measure Space",
"Definition:Measure Space",
"Definition:Extended Real-Valued Function",
"Definition:Product Sigma-Algebra",
"Definition:Vertical Section of Set",
"Definition:Measurable Function"
] | [
"Vertical Section of Measurable Set is Measurable",
"Definition:Extended Real-Valued Function",
"Definition:Finite Measure",
"Definition:Measurable Function",
"Measure of Vertical Section of Cartesian Product",
"Definition:Measurable Set",
"Definition:Measurable Function",
"Pointwise Scalar Multiple o... |
proofwiki-18880 | Hahn Decomposition Theorem | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Then there exists disjoint sets $P$ and $N$ such that:
:$(1): \quad$ $P$ is a $\mu$-positive set and $N$ is a $\mu$-negative set
:$(2): \quad$ $X = P \cup N$
:$(3): \quad$ for any other $\mu$-positive set $P'$ and ... | Note that $\mu$ can attain at most one of $+\infty$ and $-\infty$.
Suppose first that $\mu$ does not attain the value $-\infty$.
Set:
:$s_1 = \inf \set {\map \mu D : D \in \Sigma \text { and } D \subseteq X}$
Since $\O \subseteq X$ and $\map \mu \O = 0$ we have:
:$0 \in \set {\map \mu D : D \in \Sigma \text { and } D ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Then there exists [[Definition:Disjoint Sets|disjoint sets]] $P$ and $N$ such that:
:$(1): \quad$ $P$ is a [[Definition:Positive Set|$\mu$-positive ... | Note that $\mu$ can attain at most one of $+\infty$ and $-\infty$.
Suppose first that $\mu$ does not attain the value $-\infty$.
Set:
:$s_1 = \inf \set {\map \mu D : D \in \Sigma \text { and } D \subseteq X}$
Since $\O \subseteq X$ and $\map \mu \O = 0$ we have:
:$0 \in \set {\map \mu D : D \in \Sigma \text { an... | Hahn Decomposition Theorem | https://proofwiki.org/wiki/Hahn_Decomposition_Theorem | https://proofwiki.org/wiki/Hahn_Decomposition_Theorem | [
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Disjoint Sets",
"Definition:Positive Set",
"Definition:Negative Set",
"Definition:Positive Set",
"Definition:Negative Set",
"Definition:Symmetric Difference",
"Definition:Null Set"
] | [
"Definition:Infimum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Measurable Set with Negative Measure has Negative Subset",
"Definition:Negative Set",
"Definition:Pairwise Disjoint",
"Definition:Infimum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Definition:Mea... |
proofwiki-18881 | Jordan Decomposition Theorem | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\tuple {P, N}$ be a Hahn decomposition of $\mu$.
Then there exists measures $\mu^+$ and $\mu^-$ on $\struct {X, \Sigma}$ such that:
:$\mu = \mu^+ - \mu^-$
Further, at least one of $\mu^+$ and $\mu^-$ is finit... | From the definition of a Hahn decomposition, the set $P$ is $\mu$-positive, the set $N$ is $\mu$-negative and:
:$X = P \cup N$
with $P$ and $N$ disjoint.
From Sigma-Algebra Closed under Countable Intersection, we have:
:$A \cap P \in \Sigma$
and:
:$A \cap N \in \Sigma$
for each $A \in \Sigma$.
Now, for each $A \in \... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $\tuple {P, N}$ be a [[Definition:Hahn Decomposition|Hahn decomposition]] of $\mu$.
Then there exists [[Definition:Measure (Measure Theory)|meas... | From the definition of a [[Definition:Hahn Decomposition|Hahn decomposition]], the set $P$ is [[Definition:Positive Set|$\mu$-positive]], the set $N$ is [[Definition:Negative Set|$\mu$-negative]] and:
:$X = P \cup N$
with $P$ and $N$ [[Definition:Disjoint Sets|disjoint]].
From [[Sigma-Algebra Closed under Countabl... | Jordan Decomposition Theorem | https://proofwiki.org/wiki/Jordan_Decomposition_Theorem | https://proofwiki.org/wiki/Jordan_Decomposition_Theorem | [
"Jordan Decomposition Theorem",
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Hahn Decomposition",
"Definition:Measure (Measure Theory)",
"Definition:Finite Measure"
] | [
"Definition:Hahn Decomposition",
"Definition:Positive Set",
"Definition:Negative Set",
"Definition:Disjoint Sets",
"Sigma-Algebra Closed under Countable Intersection",
"Definition:Measure (Measure Theory)",
"Definition:Signed Measure",
"Definition:Measure (Measure Theory)",
"Definition:Positive Set"... |
proofwiki-18882 | Intersection of Positive Set and Negative Set is Null Set | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $P$ be a $\mu$-positive set.
Let $N$ be a $\mu$-negative set.
Then:
:$P \cap N$ is a $\mu$-null set. | Note that, from Sigma-Algebra Closed under Countable Intersection:
:$P \cap N \in \Sigma$
We aim to show that:
:for each $E \in \Sigma$ with $E \subseteq P \cap N$ we have $\map \mu E = 0$.
Note first that from Intersection is Subset, we have:
:$P \cap N \subseteq P$
so that:
:$E \subseteq P$
So, since $P$ is $\mu$-p... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $P$ be a [[Definition:Positive Set|$\mu$-positive set]].
Let $N$ be a [[Definition:Negative Set|$\mu$-negative set]].
Then:
:$P \cap N$ is a... | Note that, from [[Sigma-Algebra Closed under Countable Intersection]]:
:$P \cap N \in \Sigma$
We aim to show that:
:for each $E \in \Sigma$ with $E \subseteq P \cap N$ we have $\map \mu E = 0$.
Note first that from [[Intersection is Subset]], we have:
:$P \cap N \subseteq P$
so that:
:$E \subseteq P$
So, sinc... | Intersection of Positive Set and Negative Set is Null Set | https://proofwiki.org/wiki/Intersection_of_Positive_Set_and_Negative_Set_is_Null_Set | https://proofwiki.org/wiki/Intersection_of_Positive_Set_and_Negative_Set_is_Null_Set | [
"Positive Sets",
"Negative Sets"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Positive Set",
"Definition:Negative Set",
"Definition:Null Set/Signed Measure"
] | [
"Sigma-Algebra Closed under Countable Intersection",
"Intersection is Subset",
"Definition:Positive Set",
"Intersection is Subset",
"Definition:Negative Set",
"Definition:Null Set/Signed Measure",
"Category:Positive Sets",
"Category:Negative Sets"
] |
proofwiki-18883 | Non-Negative Signed Measure is Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$ such that:
:$\map \mu A \ge 0$
for each $A \in \Sigma$.
Then $\mu$ is a measure on $\struct {X, \Sigma}$. | We verify each of the conditions given in the definition of a measure.
From the definition of a signed measure, $\mu$ is a function $\Sigma \to \overline \R$. | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$ such that:
:$\map \mu A \ge 0$
for each $A \in \Sigma$.
Then $\mu$ is a [[Definition:Measure (Measure Theory)|measure]] on $\struct {X, \Sigma}$. | We verify each of the conditions given in the definition of a [[Definition:Measure (Measure Theory)|measure]].
From the definition of a [[Definition:Signed Measure|signed measure]], $\mu$ is a [[Definition:Extended Real-Valued Function|function]] $\Sigma \to \overline \R$. | Non-Negative Signed Measure is Measure | https://proofwiki.org/wiki/Non-Negative_Signed_Measure_is_Measure | https://proofwiki.org/wiki/Non-Negative_Signed_Measure_is_Measure | [
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Measure (Measure Theory)"
] | [
"Definition:Measure (Measure Theory)",
"Definition:Signed Measure",
"Definition:Extended Real-Valued Function",
"Definition:Signed Measure",
"Definition:Signed Measure",
"Definition:Measure (Measure Theory)"
] |
proofwiki-18884 | Null Sets Closed under Countable Union/Signed Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\sequence {N_i}_{i \mathop \in \N}$ be a sequence of $\mu$-null sets.
Then:
:$\ds N = \bigcup_{i \mathop = 1}^\infty N_i$
is a $\mu$-null set. | From Countable Union of Measurable Sets as Disjoint Union of Measurable Sets, there exists a sequence of pairwise disjoint $\Sigma$-measurable sets $\sequence {A_i}_{i \in \N}$ such that:
:$\ds N = \bigcup_{i \mathop = 1}^\infty A_i$
We now show that if $E \in \Sigma$ has $E \subseteq N$, then $\map \mu E = 0$.
Write... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $\sequence {N_i}_{i \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Null Set/Signed Measure|$\mu$-null sets]].
Then:
:$... | From [[Countable Union of Measurable Sets as Disjoint Union of Measurable Sets]], there exists a [[Definition:Sequence|sequence]] of [[Definition:Pairwise Disjoint|pairwise disjoint]] [[Definition:Measurable Set|$\Sigma$-measurable sets]] $\sequence {A_i}_{i \in \N}$ such that:
:$\ds N = \bigcup_{i \mathop = 1}^\inft... | Null Sets Closed under Countable Union/Signed Measure | https://proofwiki.org/wiki/Null_Sets_Closed_under_Countable_Union/Signed_Measure | https://proofwiki.org/wiki/Null_Sets_Closed_under_Countable_Union/Signed_Measure | [
"Null Sets Closed under Countable Union",
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Sequence",
"Definition:Null Set/Signed Measure",
"Definition:Null Set/Signed Measure"
] | [
"Countable Union of Measurable Sets as Disjoint Union of Measurable Sets",
"Definition:Sequence",
"Definition:Pairwise Disjoint",
"Definition:Measurable Set",
"Intersection with Subset is Subset",
"Intersection Distributes over Union/Family of Sets",
"Definition:Countably Additive Function",
"Intersec... |
proofwiki-18885 | Uniqueness of Jordan Decomposition | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\tuple {P_1, N_1}$ and $\tuple {P_2, N_2}$ be Hahn decompositions of $\mu$.
Let $\tuple {\mu^+_1, \mu^-_1}$ be the Jordan decomposition of $\mu$ corresponding to $\tuple {P_1, N_1}$.
Let $\tuple {\mu^+_2, \mu... | We first prove two useful identities. | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $\tuple {P_1, N_1}$ and $\tuple {P_2, N_2}$ be [[Definition:Hahn Decomposition|Hahn decompositions]] of $\mu$.
Let $\tuple {\mu^+_1, \mu^-_1}$ be... | We first prove two useful identities. | Uniqueness of Jordan Decomposition | https://proofwiki.org/wiki/Uniqueness_of_Jordan_Decomposition | https://proofwiki.org/wiki/Uniqueness_of_Jordan_Decomposition | [
"Signed Measures",
"Uniqueness of Jordan Decomposition"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Hahn Decomposition",
"Definition:Jordan Decomposition",
"Definition:Jordan Decomposition",
"Definition:Jordan Decomposition",
"Definition:Signed Measure"
] | [] |
proofwiki-18886 | Uniqueness of Jordan Decomposition/Lemma | Let $\tuple {P, N}$ be a Hahn decomposition of $\mu$.
Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$ corresponding to $\tuple {P, N}$.
Then, for each $A \in \Sigma$, we have:
:$\map {\mu^+} A = \sup \set {\map \mu B : B \in \Sigma \text { and } B \subseteq A}$
and:
:$\map {\mu^-} A = \sup \set {-\ma... | Since $\tuple {\mu^+, \mu^-}$ is a Jordan decomposition of $\mu$, we have:
:$\mu = \mu^+ - \mu^-$
with $\mu^+$ and $\mu^-$ measures.
Let $A \in \Sigma$.
We first show:
:$\map {\mu^+} A = \sup \set {\map \mu B : B \in \Sigma \text { and } B \subseteq A}$
Let $B \in \Sigma$ have $B \subseteq A$.
We have:
{{begin-eqn}}... | Let $\tuple {P, N}$ be a [[Definition:Hahn Decomposition|Hahn decomposition]] of $\mu$.
Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$ corresponding to $\tuple {P, N}$.
Then, for each $A \in \Sigma$, we have:
:$\map {\mu^+} A = \sup \set {\map \mu B : B \in \S... | Since $\tuple {\mu^+, \mu^-}$ is a [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$, we have:
:$\mu = \mu^+ - \mu^-$
with $\mu^+$ and $\mu^-$ [[Definition:Measure (Measure Theory)|measures]].
Let $A \in \Sigma$.
We first show:
:$\map {\mu^+} A = \sup \set {\map \mu B : B \in \Sigma \text { and... | Uniqueness of Jordan Decomposition/Lemma | https://proofwiki.org/wiki/Uniqueness_of_Jordan_Decomposition/Lemma | https://proofwiki.org/wiki/Uniqueness_of_Jordan_Decomposition/Lemma | [
"Uniqueness of Jordan Decomposition"
] | [
"Definition:Hahn Decomposition",
"Definition:Jordan Decomposition"
] | [
"Definition:Jordan Decomposition",
"Definition:Measure (Measure Theory)",
"Definition:Measure (Measure Theory)",
"Measure is Monotone",
"Definition:Upper Bound of Set/Real Numbers",
"Intersection is Subset",
"Sigma-Algebra Closed under Countable Intersection",
"Definition:Jordan Decomposition",
"Def... |
proofwiki-18887 | Characterization of Absolutely Continuous Measures | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $\nu$ be a finite measure on $\struct {X, \Sigma}$.
Then $\nu$ is absolutely continuous with respect to $\mu$ {{iff}}:
:for each $\epsilon > 0$ there exists $\delta > 0$ such that for each $A \in \Sigma$ with $\map \m... | === Necessary Condition ===
We prove the contrapositive, then the result follows from Rule of Transposition.
Suppose that:
:for some $\epsilon > 0$, there exists no $\delta > 0$ such that for each $A \in \Sigma$ with $\map \mu A < \delta$, we have $\map \nu A < \epsilon$.
That is:
:for some $\epsilon > 0$, for all $\... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Measure (Measure Theory)|measure]] on $\struct {X, \Sigma}$.
Let $\nu$ be a [[Definition:Finite Measure|finite measure]] on $\struct {X, \Sigma}$.
Then $\nu$ is [[Definition:Absolutely Continuous Measure|ab... | === Necessary Condition ===
We prove the [[Definition:Contrapositive|contrapositive]], then the result follows from [[Rule of Transposition]].
Suppose that:
:for some $\epsilon > 0$, there exists no $\delta > 0$ such that for each $A \in \Sigma$ with $\map \mu A < \delta$, we have $\map \nu A < \epsilon$.
That is:... | Characterization of Absolutely Continuous Measures | https://proofwiki.org/wiki/Characterization_of_Absolutely_Continuous_Measures | https://proofwiki.org/wiki/Characterization_of_Absolutely_Continuous_Measures | [
"Absolutely Continuous Measures"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Finite Measure",
"Definition:Absolute Continuity/Measure"
] | [
"Definition:Contrapositive Statement",
"Rule of Transposition",
"Definition:Absolute Continuity/Measure",
"Sum of Infinite Geometric Sequence",
"Borel-Cantelli Lemma",
"Definition:Finite Measure",
"Definition:Finite Extended Real Number",
"Set is Subset of Union",
"Definition:Sequence",
"Definitio... |
proofwiki-18888 | Radon-Nikodym Theorem | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ and $\nu$ be $\sigma$-finite measures on $\struct {X, \Sigma}$ such that:
:$\nu$ is absolutely continuous with respect to $\mu$.
Then there exists a $\Sigma$-measurable function $g : X \to \hointr 0 \infty$ such that:
:$\ds \map \nu A = \int_A g \rd \mu$
for ... | === Existence ===
We first prove the case of $\mu$ and $\nu$ finite.
Define $\FF$ to be the set of $\Sigma$-measurable functions $f : X \to \overline \R_{\ge 0}$ with:
:$\ds \int_A f \rd \mu \le \map \nu A$
for each $A \in \Sigma$.
We show that $\FF$ is non-empty.
From Measurable Function Zero A.E. iff Absolute Valu... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ and $\nu$ be [[Definition:Sigma-Finite Measure|$\sigma$-finite measures]] on $\struct {X, \Sigma}$ such that:
:$\nu$ is [[Definition:Absolutely Continuous Measure|absolutely continuous]] with respect to $\mu$.
Then there exi... | === Existence ===
We first prove the case of $\mu$ and $\nu$ [[Definition:Finite Measure|finite]].
Define $\FF$ to be the [[Definition:Set|set]] of [[Definition:Measurable Function|$\Sigma$-measurable functions]] $f : X \to \overline \R_{\ge 0}$ with:
:$\ds \int_A f \rd \mu \le \map \nu A$
for each $A \in \Sigma$... | Radon-Nikodym Theorem | https://proofwiki.org/wiki/Radon-Nikodym_Theorem | https://proofwiki.org/wiki/Radon-Nikodym_Theorem | [
"Radon-Nikodym Theorem",
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Sigma-Finite Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Measurable Function",
"Definition:Almost Everywhere"
] | [
"Definition:Finite Measure",
"Definition:Set",
"Definition:Measurable Function",
"Definition:Non-Empty Set",
"Measurable Function Zero A.E. iff Absolute Value has Zero Integral",
"Definition:Constant Mapping",
"Definition:Real-Valued Function",
"Definition:Finite Measure",
"Definition:Increasing Seq... |
proofwiki-18889 | Vertical Section of Measurable Set is Measurable | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be measurable spaces.
Let $E \in \Sigma_X \otimes \Sigma_Y$ where $\Sigma_X \otimes \Sigma_Y$ is the product $\sigma$-algebra of $\Sigma_X$ and $\Sigma_Y$.
Let $x \in X$.
Then:
:$E_x \in \Sigma_Y$
where $E_x$ is the $x$-vertical section of $E$. | Let:
:$\FF = \set {E \subseteq X \times Y : E_x \in \Sigma_Y}$
We will show that $\FF$ contains each $S_1 \times S_2$ with $S_1 \in \Sigma_X$ and $S_2 \in \Sigma_Y$.
We will then show that $\FF$ is a $\sigma$-algebra, at which point we will have:
:$\map \sigma {\set {S_1 \times S_2 : S_1 \in \Sigma_X, \, S_2 \in \Sig... | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be [[Definition:Measurable Space|measurable spaces]].
Let $E \in \Sigma_X \otimes \Sigma_Y$ where $\Sigma_X \otimes \Sigma_Y$ is the [[Definition:Product Sigma-Algebra|product $\sigma$-algebra]] of $\Sigma_X$ and $\Sigma_Y$.
Let $x \in X$.
Then:
:$E_x \in ... | Let:
:$\FF = \set {E \subseteq X \times Y : E_x \in \Sigma_Y}$
We will show that $\FF$ contains each $S_1 \times S_2$ with $S_1 \in \Sigma_X$ and $S_2 \in \Sigma_Y$.
We will then show that $\FF$ is a [[Definition:Sigma-Algebra|$\sigma$-algebra]], at which point we will have:
:$\map \sigma {\set {S_1 \times S_2 : ... | Vertical Section of Measurable Set is Measurable | https://proofwiki.org/wiki/Vertical_Section_of_Measurable_Set_is_Measurable | https://proofwiki.org/wiki/Vertical_Section_of_Measurable_Set_is_Measurable | [
"Vertical Section of Sets"
] | [
"Definition:Measurable Space",
"Definition:Product Sigma-Algebra",
"Definition:Vertical Section of Set"
] | [
"Definition:Sigma-Algebra",
"Sigma-Algebra Contains Generated Sigma-Algebra of Subset",
"Definition:Product Sigma-Algebra",
"Vertical Section of Cartesian Product",
"Definition:Sigma-Algebra",
"Definition:Sigma-Algebra",
"Definition:Closed under Mapping",
"Definition:Set Union/Countable Union",
"Def... |
proofwiki-18890 | Measure of Limit of Increasing Sequence of Measurable Sets | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $E \in \Sigma$.
Let $\sequence {E_n}_{n \mathop \in \N}$ be an increasing sequence of $\Sigma$-measurable sets such that:
:$E_n \uparrow E$
where $E_n \uparrow E$ denotes limit of increasing sequence of sets.
Then:
:$\ds \map \mu E = \lim_{n \mathop \to \infty}... | We define a sequence $\sequence {F_n}_{n \mathop \in \N}$ inductively.
Set $F_1 = E_1$.
For $n > 1$, define:
:$F_n = E_n \setminus E_{n - 1}$
From the definition of set difference we have:
:$E_n \setminus E_{n - 1} = E_n \cap \paren {X \setminus E_{n - 1} }$
Since $\Sigma$ is closed under complementation, we have:
... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $E \in \Sigma$.
Let $\sequence {E_n}_{n \mathop \in \N}$ be an [[Definition:Increasing Sequence of Sets|increasing sequence]] of [[Definition:Measurable Set|$\Sigma$-measurable sets]] such that:
:$E_n \uparrow E$
where $E_n \upar... | We define a [[Definition:Sequence|sequence]] $\sequence {F_n}_{n \mathop \in \N}$ inductively.
Set $F_1 = E_1$.
For $n > 1$, define:
:$F_n = E_n \setminus E_{n - 1}$
From the definition of [[Definition:Set Difference|set difference]] we have:
:$E_n \setminus E_{n - 1} = E_n \cap \paren {X \setminus E_{n - 1} }$... | Measure of Limit of Increasing Sequence of Measurable Sets | https://proofwiki.org/wiki/Measure_of_Limit_of_Increasing_Sequence_of_Measurable_Sets | https://proofwiki.org/wiki/Measure_of_Limit_of_Increasing_Sequence_of_Measurable_Sets | [
"Increasing Sequences of Sets",
"Measure Theory",
"Measures",
"Measures",
"Increasing Sequences of Sets"
] | [
"Definition:Measure Space",
"Definition:Increasing Sequence of Sets",
"Definition:Measurable Set",
"Definition:Limit of Increasing Sequence of Sets"
] | [
"Definition:Sequence",
"Definition:Set Difference",
"Definition:Closed under Mapping",
"Definition:Relative Complement",
"Sigma-Algebra Closed under Countable Intersection",
"Definition:Increasing Sequence of Sets",
"Definition:Disjoint Sets",
"Definition:Disjoint Sets",
"Definition:Increasing Seque... |
proofwiki-18891 | Signed Measure of Limit of Increasing Sequence of Measurable Sets | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $E \in \Sigma$.
Let $\sequence {E_n}_{n \mathop \in \N}$ be an increasing sequence of $\Sigma$-measurable sets such that:
:$E_n \uparrow E$
where $E_n \uparrow E$ denotes the limit of an increasing sequence ... | Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then $\mu^+$ and $\mu^-$ are measures with:
:$\mu = \mu^+ - \mu^-$
where at least one of $\mu^+$ and $\mu^-$ is finite.
Then we have:
:$\map \mu E = \map {\mu^+} E - \map {\mu^-} E$
From Measure of Limit of Increasing Sequence of Measurable Sets, we ha... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $E \in \Sigma$.
Let $\sequence {E_n}_{n \mathop \in \N}$ be an [[Definition:Increasing Sequence of Sets|increasing sequence]] of [[Definition:Me... | Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
Then $\mu^+$ and $\mu^-$ are [[Definition:Measure (Measure Theory)|measures]] with:
:$\mu = \mu^+ - \mu^-$
where at least one of $\mu^+$ and $\mu^-$ is [[Definition:Finite Measure|finite]].
Then we have:
:$\map ... | Signed Measure of Limit of Increasing Sequence of Measurable Sets | https://proofwiki.org/wiki/Signed_Measure_of_Limit_of_Increasing_Sequence_of_Measurable_Sets | https://proofwiki.org/wiki/Signed_Measure_of_Limit_of_Increasing_Sequence_of_Measurable_Sets | [
"Signed Measures",
"Increasing Sequences of Sets"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Increasing Sequence of Sets",
"Definition:Measurable Set",
"Definition:Limit of Increasing Sequence of Sets"
] | [
"Definition:Jordan Decomposition",
"Definition:Measure (Measure Theory)",
"Definition:Finite Measure",
"Measure of Limit of Increasing Sequence of Measurable Sets",
"Combination Theorem for Sequences/Real/Difference Rule",
"Category:Signed Measures",
"Category:Increasing Sequences of Sets"
] |
proofwiki-18892 | Signed Measure of Limit of Decreasing Sequence of Measurable Sets | Let $\struct {X, \Sigma}$ be a measurable space.
Let $E \in \Sigma$.
Let $\sequence {E_n}_{n \mathop \in \N}$ be an decreasing sequence of $\Sigma$-measurable sets such that:
:$E_n \downarrow E$
where $E_n \downarrow E$ denotes limit of decreasing sequence of sets, and:
:there exists $m \in \N$ such that $\map \mu ... | Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then $\mu^+$ and $\mu^-$ are measures with:
:$\mu = \mu^+ - \mu^-$
where at least one of $\mu^+$ and $\mu^-$ is finite.
Then we have:
:$\map \mu E = \map {\mu^+} E - \map {\mu^-} E$
and:
:$\map \mu {E_m} = \map {\mu^+} {E_m} - \map {\mu^-} {E_m}$
Sinc... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $E \in \Sigma$.
Let $\sequence {E_n}_{n \mathop \in \N}$ be an [[Definition:Decreasing Sequence of Sets|decreasing sequence]] of [[Definition:Measurable Set|$\Sigma$-measurable sets]] such that:
:$E_n \downarrow E$
where $E_n \... | Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
Then $\mu^+$ and $\mu^-$ are [[Definition:Measure (Measure Theory)|measures]] with:
:$\mu = \mu^+ - \mu^-$
where at least one of $\mu^+$ and $\mu^-$ is [[Definition:Finite Measure|finite]].
Then we have:
:$\map ... | Signed Measure of Limit of Decreasing Sequence of Measurable Sets | https://proofwiki.org/wiki/Signed_Measure_of_Limit_of_Decreasing_Sequence_of_Measurable_Sets | https://proofwiki.org/wiki/Signed_Measure_of_Limit_of_Decreasing_Sequence_of_Measurable_Sets | [
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Decreasing Sequence of Sets",
"Definition:Measurable Set",
"Definition:Limit of Decreasing Sequence of Sets",
"Definition:Finite Extended Real Number"
] | [
"Definition:Jordan Decomposition",
"Definition:Measure (Measure Theory)",
"Definition:Finite Measure",
"Definition:Finite Extended Real Number",
"Definition:Finite Extended Real Number",
"Combination Theorem for Sequences/Real/Difference Rule"
] |
proofwiki-18893 | Union is Increasing Sequence of Sets | Let $\sequence {D_n}_{n \mathop \in \N}$ be a sequence of sets.
Then:
:the sequence $\ds \sequence {\bigcup_{k \mathop = 1}^n D_k}_{n \mathop \in \N}$ is increasing. | We have:
:$\ds \bigcup_{k \mathop = 1}^{n + 1} D_k = D_{n + 1} \cup \bigcup_{k \mathop = 1}^n D_k$
From Set is Subset of Union, we have:
:$\ds \bigcup_{k \mathop = 1}^n D_k \subseteq D_{n + 1} \cup \bigcup_{k \mathop = 1}^n D_k$
so:
:$\ds \bigcup_{k \mathop = 1}^n D_k \subseteq \bigcup_{k \mathop = 1}^{n + 1} D_k$
So... | Let $\sequence {D_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]].
Then:
:the [[Definition:Sequence|sequence]] $\ds \sequence {\bigcup_{k \mathop = 1}^n D_k}_{n \mathop \in \N}$ is [[Definition:Increasing Sequence of Sets|increasing]]. | We have:
:$\ds \bigcup_{k \mathop = 1}^{n + 1} D_k = D_{n + 1} \cup \bigcup_{k \mathop = 1}^n D_k$
From [[Set is Subset of Union]], we have:
:$\ds \bigcup_{k \mathop = 1}^n D_k \subseteq D_{n + 1} \cup \bigcup_{k \mathop = 1}^n D_k$
so:
:$\ds \bigcup_{k \mathop = 1}^n D_k \subseteq \bigcup_{k \mathop = 1}^{n + 1... | Union is Increasing Sequence of Sets | https://proofwiki.org/wiki/Union_is_Increasing_Sequence_of_Sets | https://proofwiki.org/wiki/Union_is_Increasing_Sequence_of_Sets | [
"Set Union",
"Increasing Sequences of Sets"
] | [
"Definition:Sequence",
"Definition:Set",
"Definition:Sequence",
"Definition:Increasing Sequence of Sets"
] | [
"Set is Subset of Union",
"Definition:Increasing Sequence of Sets",
"Category:Set Union",
"Category:Increasing Sequences of Sets"
] |
proofwiki-18894 | Vertical Section preserves Subsets | Let $X$ and $Y$ be sets.
Let $A \subseteq B$ be subsets of $X \times Y$.
Let $x \in X$.
Then:
:$A_x \subseteq B_x$
where $A_x$ is the $x$-vertical section of $A$ and $B_x$ is the $x$-vertical section of $B$. | Note that if:
:$y \in A_x$
from the definition of $x$-vertical section, we have:
:$\tuple {x, y} \in A$
so:
:$\tuple {x, y} \in B$
So, from the definition of $x$-vertical section, we have:
:$y \in B_x$
So:
:if $y \in A_x$ then $y \in B_x$.
That is:
:$A_x \subseteq B_x$
{{qed}}
Category:Vertical Section of Sets
0z9c... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $A \subseteq B$ be [[Definition:Subset|subsets]] of $X \times Y$.
Let $x \in X$.
Then:
:$A_x \subseteq B_x$
where $A_x$ is the [[Definition:Vertical Section of Set|$x$-vertical section]] of $A$ and $B_x$ is the [[Definition:Vertical Section of Set|$x$-vertical se... | Note that if:
:$y \in A_x$
from the definition of [[Definition:Vertical Section of Set|$x$-vertical section]], we have:
:$\tuple {x, y} \in A$
so:
:$\tuple {x, y} \in B$
So, from the definition of [[Definition:Vertical Section of Set|$x$-vertical section]], we have:
:$y \in B_x$
So:
:if $y \in A_x$ then $y ... | Vertical Section preserves Subsets | https://proofwiki.org/wiki/Vertical_Section_preserves_Subsets | https://proofwiki.org/wiki/Vertical_Section_preserves_Subsets | [
"Vertical Section of Sets"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Vertical Section of Set",
"Definition:Vertical Section of Set"
] | [
"Definition:Vertical Section of Set",
"Definition:Vertical Section of Set",
"Category:Vertical Section of Sets"
] |
proofwiki-18895 | Vertical Section preserves Increasing Sequences of Sets | Let $X$ and $Y$ be sets.
Let $\sequence {A_n}_{n \mathop \in \N}$ be an increasing sequence in $X \times Y$.
Let $x \in X$.
Then:
:$\sequence {\paren {A_n}_x}_{n \mathop \in \N}$ is an increasing sequence. | Since $\sequence {A_n}_{n \mathop \in \N}$ is increasing, we have:
:$A_n \subseteq A_{n + 1}$
for each $n$.
From Vertical Section preserves Subsets, we have:
:$\paren {A_n}_x \subseteq \paren {A_{n + 1} }_x$
for each $n$.
So:
:$\sequence {\paren {A_n}_x}_{n \mathop \in \N}$ is an increasing sequence.
{{qed}}
Categor... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $\sequence {A_n}_{n \mathop \in \N}$ be an [[Definition:Increasing Sequence of Sets|increasing sequence]] in $X \times Y$.
Let $x \in X$.
Then:
:$\sequence {\paren {A_n}_x}_{n \mathop \in \N}$ is an [[Definition:Increasing Sequence of Sets|increasing sequence]]. | Since $\sequence {A_n}_{n \mathop \in \N}$ is [[Definition:Increasing Sequence of Sets|increasing]], we have:
:$A_n \subseteq A_{n + 1}$
for each $n$.
From [[Vertical Section preserves Subsets]], we have:
:$\paren {A_n}_x \subseteq \paren {A_{n + 1} }_x$
for each $n$.
So:
:$\sequence {\paren {A_n}_x}_{n \math... | Vertical Section preserves Increasing Sequences of Sets | https://proofwiki.org/wiki/Vertical_Section_preserves_Increasing_Sequences_of_Sets | https://proofwiki.org/wiki/Vertical_Section_preserves_Increasing_Sequences_of_Sets | [
"Increasing Sequences of Sets",
"Vertical Section of Sets",
"Increasing Sequences of Sets"
] | [
"Definition:Set",
"Definition:Increasing Sequence of Sets",
"Definition:Increasing Sequence of Sets"
] | [
"Definition:Increasing Sequence of Sets",
"Vertical Section preserves Subsets",
"Definition:Increasing Sequence of Sets",
"Category:Vertical Section of Sets",
"Category:Increasing Sequences of Sets"
] |
proofwiki-18896 | Horizontal Section preserves Increasing Sequences of Sets | Let $X$ and $Y$ be sets.
Let $\sequence {A_n}_{n \mathop \in \N}$ be an increasing sequence in $X \times Y$.
Let $y \in Y$.
Then:
:$\sequence {\paren {A_n}^y}_{n \mathop \in \N}$ is an increasing sequence. | Since $\sequence {A_n}_{n \mathop \in \N}$ is increasing, we have:
:$A_n \subseteq A_{n + 1}$
for each $n$.
From Horizontal Section preserves Subsets, we have:
:$\paren {A_n}^y \subseteq \paren {A_{n + 1} }^y$
for each $n$.
So:
:$\sequence {\paren {A_n}^y}_{n \mathop \in \N}$ is an increasing sequence.
{{qed}}
Categ... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $\sequence {A_n}_{n \mathop \in \N}$ be an [[Definition:Increasing Sequence of Sets|increasing sequence]] in $X \times Y$.
Let $y \in Y$.
Then:
:$\sequence {\paren {A_n}^y}_{n \mathop \in \N}$ is an [[Definition:Increasing Sequence of Sets|increasing sequence]]. | Since $\sequence {A_n}_{n \mathop \in \N}$ is [[Definition:Increasing Sequence of Sets|increasing]], we have:
:$A_n \subseteq A_{n + 1}$
for each $n$.
From [[Horizontal Section preserves Subsets]], we have:
:$\paren {A_n}^y \subseteq \paren {A_{n + 1} }^y$
for each $n$.
So:
:$\sequence {\paren {A_n}^y}_{n \ma... | Horizontal Section preserves Increasing Sequences of Sets | https://proofwiki.org/wiki/Horizontal_Section_preserves_Increasing_Sequences_of_Sets | https://proofwiki.org/wiki/Horizontal_Section_preserves_Increasing_Sequences_of_Sets | [
"Increasing Sequences of Sets",
"Horizontal Section of Sets",
"Increasing Sequences of Sets"
] | [
"Definition:Set",
"Definition:Increasing Sequence of Sets",
"Definition:Increasing Sequence of Sets"
] | [
"Definition:Increasing Sequence of Sets",
"Horizontal Section preserves Subsets",
"Definition:Increasing Sequence of Sets",
"Category:Horizontal Section of Sets",
"Category:Increasing Sequences of Sets"
] |
proofwiki-18897 | Horizontal Section preserves Subsets | Let $X$ and $Y$ be sets.
Let $A \subseteq B$ be subsets of $X \times Y$.
Let $y \in Y$.
Then:
:$A^y \subseteq B^y$
where $A^y$ is the $y$-horizontal section of $A$ and $B^y$ is the $y$-horizontal section of $B$. | Note that if:
:$x \in A^y$
from the definition of $x$-vertical section, we have:
:$\tuple {x, y} \in A$
so:
:$\tuple {x, y} \in B$
So, from the definition of $x$-vertical section, we have:
:$x \in B^y$
So:
:if $x \in A^y$ then $x \in B^y$.
That is:
:$A^y \subseteq B^y$
{{qed}}
Category:Horizontal Section of Sets
bj... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $A \subseteq B$ be [[Definition:Subset|subsets]] of $X \times Y$.
Let $y \in Y$.
Then:
:$A^y \subseteq B^y$
where $A^y$ is the [[Definition:Horizontal Section of Set|$y$-horizontal section]] of $A$ and $B^y$ is the [[Definition:Horizontal Section of Set|$y$-horiz... | Note that if:
:$x \in A^y$
from the definition of [[Definition:Vertical Section of Set|$x$-vertical section]], we have:
:$\tuple {x, y} \in A$
so:
:$\tuple {x, y} \in B$
So, from the definition of [[Definition:Vertical Section of Set|$x$-vertical section]], we have:
:$x \in B^y$
So:
:if $x \in A^y$ then $x ... | Horizontal Section preserves Subsets | https://proofwiki.org/wiki/Horizontal_Section_preserves_Subsets | https://proofwiki.org/wiki/Horizontal_Section_preserves_Subsets | [
"Horizontal Section of Sets"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Horizontal Section of Set",
"Definition:Horizontal Section of Set"
] | [
"Definition:Vertical Section of Set",
"Definition:Vertical Section of Set",
"Category:Horizontal Section of Sets"
] |
proofwiki-18898 | Vertical Section of Cartesian Product | Let $X$ and $Y$ be sets.
Let $A \subseteq X$ and $B \subseteq Y$, so that $A \times B \subseteq X \times Y$.
Let $x \in X$.
Then:
:$\paren {A \times B}_x = \begin{cases}B & x \in A \\ \O & x \not \in A\end{cases}$
where $\paren {A \times B}_x$ is the $x$-vertical section of $A \times B$. | Let $x \in A$.
From the definition of the horizontal section, we have:
:$y \in \paren {A \times B}_x$
{{iff}}:
:$\tuple {x, y} \in A \times B$
Since $x \in A$, this equivalent to:
:$y \in B$
So:
:$y \in \paren {A \times B}_x$ {{iff}} $y \in B$
giving:
:$\paren {A \times B}_x = B$ if $x \in A$.
Now let $x \in X \set... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $A \subseteq X$ and $B \subseteq Y$, so that $A \times B \subseteq X \times Y$.
Let $x \in X$.
Then:
:$\paren {A \times B}_x = \begin{cases}B & x \in A \\ \O & x \not \in A\end{cases}$
where $\paren {A \times B}_x$ is the [[Definition:Vertical Section of Set|$x$-... | Let $x \in A$.
From the definition of the [[Definition:Horizontal Section of Set|horizontal section]], we have:
:$y \in \paren {A \times B}_x$
{{iff}}:
:$\tuple {x, y} \in A \times B$
Since $x \in A$, this equivalent to:
:$y \in B$
So:
:$y \in \paren {A \times B}_x$ {{iff}} $y \in B$
giving:
:$\paren {A \... | Vertical Section of Cartesian Product | https://proofwiki.org/wiki/Vertical_Section_of_Cartesian_Product | https://proofwiki.org/wiki/Vertical_Section_of_Cartesian_Product | [
"Vertical Section of Sets",
"Cartesian Product"
] | [
"Definition:Set",
"Definition:Vertical Section of Set"
] | [
"Definition:Horizontal Section of Set",
"Definition:Set Difference",
"Category:Vertical Section of Sets",
"Category:Cartesian Product"
] |
proofwiki-18899 | Integral of Vertical Section of Measurable Function gives Measurable Function | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.
Let $f : X \times Y \to \overline \R_{\ge 0}$ be a positive $\Sigma_X \otimes \Sigma_Y$-measurable function, where $\Sigma_X \otimes \Sigma_Y$ is the product $\sigma$-algebra of $\Sigma_X$ and $\Sigma_Y$.
Define the fun... | First we prove the case of:
:$f = \chi_E$
where $E$ is a $\Sigma_X \otimes \Sigma_Y$-measurable set.
From Vertical Section of Characteristic Function is Characteristic Function of Vertical Section, we have:
:$f_x = \chi_{E_x}$
From Vertical Section of Measurable Function is Measurable, we also have:
:$f_x$ is $\Sigma... | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be [[Definition:Sigma-Finite Measure Space|$\sigma$-finite]] [[Definition:Measure Space|measure spaces]].
Let $f : X \times Y \to \overline \R_{\ge 0}$ be a [[Definition:Positive Measurable Function|positive $\Sigma_X \otimes \Sigma_Y$-measurable functi... | First we prove the case of:
:$f = \chi_E$
where $E$ is a [[Definition:Measurable Set|$\Sigma_X \otimes \Sigma_Y$-measurable set]].
From [[Vertical Section of Characteristic Function is Characteristic Function of Vertical Section]], we have:
:$f_x = \chi_{E_x}$
From [[Vertical Section of Measurable Function is Mea... | Integral of Vertical Section of Measurable Function gives Measurable Function | https://proofwiki.org/wiki/Integral_of_Vertical_Section_of_Measurable_Function_gives_Measurable_Function | https://proofwiki.org/wiki/Integral_of_Vertical_Section_of_Measurable_Function_gives_Measurable_Function | [
"Measurable Functions",
"Vertical Section of Functions"
] | [
"Definition:Sigma-Finite Measure Space",
"Definition:Measure Space",
"Definition:Measurable Function/Positive",
"Definition:Product Sigma-Algebra",
"Definition:Extended Real-Valued Function",
"Definition:Vertical Section of Function",
"Definition:Measurable Function"
] | [
"Definition:Measurable Set",
"Vertical Section of Characteristic Function is Characteristic Function of Vertical Section",
"Vertical Section of Measurable Function is Measurable",
"Definition:Measurable Function",
"Definition:Integral of Positive Measurable Function",
"Integral of Characteristic Function/... |
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