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-21200 | Associator of Associative Algebra is Zero | Let $\struct {A_R, \oplus}$ be an associative algebra.
Let $\sqbrk {a, b, c}$ denote the associator of $a, b, c \in A_R$.
Then:
:$\forall a, b, c \in A_R: \sqbrk {a, b, c} = \mathbf 0_R$ | {{begin-eqn}}
{{eqn | q = \forall a, b, c \in A_R
| l = a \oplus \paren {b \oplus c}
| r = \paren {a \oplus b} \oplus c
| c = {{Defof|Associative Algebra}}
}}
{{eqn | ll= \leadsto
| l = a \oplus \paren {b \oplus c} - \paren {a \oplus b} \oplus c
| r = \mathbf 0_R
| c =
}}
{{end-eqn}... | Let $\struct {A_R, \oplus}$ be an [[Definition:Associative Algebra|associative algebra]].
Let $\sqbrk {a, b, c}$ denote the [[Definition:Associator|associator]] of $a, b, c \in A_R$.
Then:
:$\forall a, b, c \in A_R: \sqbrk {a, b, c} = \mathbf 0_R$ | {{begin-eqn}}
{{eqn | q = \forall a, b, c \in A_R
| l = a \oplus \paren {b \oplus c}
| r = \paren {a \oplus b} \oplus c
| c = {{Defof|Associative Algebra}}
}}
{{eqn | ll= \leadsto
| l = a \oplus \paren {b \oplus c} - \paren {a \oplus b} \oplus c
| r = \mathbf 0_R
| c =
}}
{{end-eqn}... | Associator of Associative Algebra is Zero | https://proofwiki.org/wiki/Associator_of_Associative_Algebra_is_Zero | https://proofwiki.org/wiki/Associator_of_Associative_Algebra_is_Zero | [
"Associators",
"Associative Algebras"
] | [
"Definition:Associative Algebra",
"Definition:Associator"
] | [] |
proofwiki-21201 | Summation over Union of Disjoint Finite Index Sets | Let $\struct{G, +}$ be a commutative monoid.
Let $I$ and $J$ be disjoint finite indexing sets.
Let $K = I \cup J$.
Let $\family{g_k}_{k \mathop \in K}$ be an indexed family of elements of $G$.
Then:
:$\ds \sum_{k \mathop \in K} g_k = \paren{\sum_{i \mathop \in I} g_i} + \paren{\sum_{j \mathop \in J} g_j}$
where:
:$\ds ... | Let $\set{i_1, i_2, \ldots, i_n}$ be an enumeration of $I$.
Let $\set{j_1, j_2, \ldots, j_m}$ be an enumeration of $J$.
Let $k: \closedint 1 {n+m}$ be the mapping defined by:
:$k_l = \begin{cases}
i_l & : \text{ if } 1 \le l \le n \\
j_{l-n} & : \text{ if } l > n \\
\end{cases}$
From Union of Bijections with Disjoint ... | Let $\struct{G, +}$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Monoid|monoid]].
Let $I$ and $J$ be [[Definition:Disjoint Sets|disjoint]] [[Definition:Finite Set|finite]] [[Definition:Indexing Set|indexing sets]].
Let $K = I \cup J$.
Let $\family{g_k}_{k \mathop \in K}$ be an [[Definition:Ind... | Let $\set{i_1, i_2, \ldots, i_n}$ be an [[Definition:Enumeration|enumeration]] of $I$.
Let $\set{j_1, j_2, \ldots, j_m}$ be an [[Definition:Enumeration|enumeration]] of $J$.
Let $k: \closedint 1 {n+m}$ be the [[Definition:Mapping|mapping]] defined by:
:$k_l = \begin{cases}
i_l & : \text{ if } 1 \le l \le n \\
j_{l-n... | Summation over Union of Disjoint Finite Index Sets | https://proofwiki.org/wiki/Summation_over_Union_of_Disjoint_Finite_Index_Sets | https://proofwiki.org/wiki/Summation_over_Union_of_Disjoint_Finite_Index_Sets | [
"Summations"
] | [
"Definition:Commutative Semigroup",
"Definition:Monoid",
"Definition:Disjoint Sets",
"Definition:Finite Set",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Summation over Finite Index",
"Definition:Summation over Finite Index",
"Definition:Restricti... | [
"Definition:Enumeration",
"Definition:Enumeration",
"Definition:Mapping",
"Union of Bijections with Disjoint Domains and Codomains is Bijection",
"Definition:Enumeration",
"Definition:Summation",
"Category:Summations"
] |
proofwiki-21202 | Unitization of Commutative Algebra over Field is Commutative | Let $K$ be a field.
Let $A$ be a commutative algebra over $K$.
Let $A_+$ be the unitization of $A$.
Then $A_+$ is commutative. | Let $\tuple {x, \lambda}, \tuple {y, \mu} \in A_+$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \tuple {x, \lambda} \tuple {y, \mu}
| r = \tuple {x y + \lambda y + \mu x, \lambda \mu}
}}
{{eqn | r = \tuple {y x + \lambda y + \mu x, \mu \lambda}
| c = $A$ is commutative, $K$ is a field
}}
{{eqn | r = \tuple {y, \mu} \... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative algebra]] over $K$.
Let $A_+$ be the [[Definition:Unitization of Algebra over Field|unitization]] of $A$.
Then $A_+$ is [[Definition:Commutative Algebra (Abstract Algebra)|commuta... | Let $\tuple {x, \lambda}, \tuple {y, \mu} \in A_+$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \tuple {x, \lambda} \tuple {y, \mu}
| r = \tuple {x y + \lambda y + \mu x, \lambda \mu}
}}
{{eqn | r = \tuple {y x + \lambda y + \mu x, \mu \lambda}
| c = $A$ is [[Definition:Commutative Algebra (Abstract Algebra)|commuta... | Unitization of Commutative Algebra over Field is Commutative | https://proofwiki.org/wiki/Unitization_of_Commutative_Algebra_over_Field_is_Commutative | https://proofwiki.org/wiki/Unitization_of_Commutative_Algebra_over_Field_is_Commutative | [
"Unitizations of Algebras over Fields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Unitization of Algebra over Field",
"Definition:Commutative Algebra (Abstract Algebra)"
] | [
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Field (Abstract Algebra)",
"Definition:Commutative Algebra (Abstract Algebra)",
"Category:Unitizations of Algebras over Fields"
] |
proofwiki-21203 | Existence of Choice Function for Family of Non-Empty Sets of Natural Numbers | Let $\FF$ be a family of non-empty sets of natural numbers.
Then there exists a choice function for $\FF$. | {{ProofWanted|Not clear what axiom we are expected to use to start this.}} | Let $\FF$ be a [[Definition:Indexed Family of Sets|family]] of [[Definition:Non-Empty Set|non-empty sets]] of [[Definition:Natural Number|natural numbers]].
Then there exists a [[Definition:Choice Function|choice function]] for $\FF$. | {{ProofWanted|Not clear what axiom we are expected to use to start this.}} | Existence of Choice Function for Family of Non-Empty Sets of Natural Numbers | https://proofwiki.org/wiki/Existence_of_Choice_Function_for_Family_of_Non-Empty_Sets_of_Natural_Numbers | https://proofwiki.org/wiki/Existence_of_Choice_Function_for_Family_of_Non-Empty_Sets_of_Natural_Numbers | [
"Choice Functions"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Non-Empty Set",
"Definition:Natural Numbers",
"Definition:Choice Function"
] | [] |
proofwiki-21204 | Existence of Choice Function for Family of Finite Non-Empty Sets of Real Numbers | Let $\FF$ be a family of finite non-empty sets of real numbers.
Then there exists a choice function for $\FF$. | {{ProofWanted|Not clear what axiom we are expected to use to start this.}} | Let $\FF$ be a [[Definition:Indexed Family of Sets|family]] of [[Definition:Finite Set|finite]] [[Definition:Non-Empty Set|non-empty sets]] of [[Definition:Real Number|real numbers]].
Then there exists a [[Definition:Choice Function|choice function]] for $\FF$. | {{ProofWanted|Not clear what axiom we are expected to use to start this.}} | Existence of Choice Function for Family of Finite Non-Empty Sets of Real Numbers | https://proofwiki.org/wiki/Existence_of_Choice_Function_for_Family_of_Finite_Non-Empty_Sets_of_Real_Numbers | https://proofwiki.org/wiki/Existence_of_Choice_Function_for_Family_of_Finite_Non-Empty_Sets_of_Real_Numbers | [
"Choice Functions"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Real Number",
"Definition:Choice Function"
] | [] |
proofwiki-21205 | Unitization of Algebra over Field preserves Subalgebra Relation | Let $K$ be a field.
Let $A$ be a non-unital commutative algebra over $K$.
Let $B$ be a subalgebra of $A$.
Let $A_+$ and $B_+$ be the unitizations of $A$ and $B$ respectively.
Then $B_+$ is a unital subalgebra of $A_+$. | Let $\struct {x, s}, \tuple {y, t} \in B_+$.
Let $\lambda \in K$.
Then, we have:
:$\tuple {x, s} + \lambda \tuple {y, t} = \tuple {x + \lambda y, s + \lambda t}$
Since $\tuple {x, s}, \tuple {y, t} \in B_+$, we have $x, y \in B$.
Since $B$ is a subalgebra of $A$, we have that:
:$B$ is a vector subspace of $A$
and henc... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be a [[Definition:Unital Algebra|non-unital]] [[Definition:Commutative Algebra (Abstract Algebra)|commutative algebra]] over $K$.
Let $B$ be a [[Definition:Subalgebra|subalgebra]] of $A$.
Let $A_+$ and $B_+$ be the [[Definition:Unitization of Algeb... | Let $\struct {x, s}, \tuple {y, t} \in B_+$.
Let $\lambda \in K$.
Then, we have:
:$\tuple {x, s} + \lambda \tuple {y, t} = \tuple {x + \lambda y, s + \lambda t}$
Since $\tuple {x, s}, \tuple {y, t} \in B_+$, we have $x, y \in B$.
Since $B$ is a [[Definition:Subalgebra|subalgebra]] of $A$, we have that:
:$B$ is a [... | Unitization of Algebra over Field preserves Subalgebra Relation | https://proofwiki.org/wiki/Unitization_of_Algebra_over_Field_preserves_Subalgebra_Relation | https://proofwiki.org/wiki/Unitization_of_Algebra_over_Field_preserves_Subalgebra_Relation | [
"Unitizations of Algebras over Fields"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Unital Algebra",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Subalgebra",
"Definition:Unitization of Algebra over Field",
"Definition:Unital Subalgebra"
] | [
"Definition:Subalgebra",
"Definition:Vector Subspace",
"One-Step Vector Subspace Test",
"Definition:Vector Subspace",
"Definition:Subalgebra",
"Definition:Unitization of Algebra over Field",
"Definition:Subalgebra",
"Definition:Vector Subspace",
"Definition:Subalgebra",
"Definition:Unital Algebra"... |
proofwiki-21206 | Spectrum of Element in Unital Subalgebra/Corollary | Let $A$ be a non-unital algebra over $\C$.
Let $B$ be a subalgebra of $A$.
Let $x \in B$.
Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the spectra of $x$ in $A$ and $B$ respectively.
Then:
:$\map {\sigma_A} x \subseteq \map {\sigma_B} x$ | Let $A_+$ and $B_+$ be the unitizations of $A$ and $B$ respectively.
From Unitization of Algebra over Field preserves Subalgebra Relation, we have:
:$B_+$ is a unital subalgebra of $A_+$.
From Spectrum of Element in Unital Subalgebra, we have that:
:$\map {\sigma_{A_+} } {\tuple {x, 0} } \subseteq \map {\sigma_{B_+} } ... | Let $A$ be a [[Definition:Unital Algebra|non-unital]] [[Definition:Algebra over Field|algebra]] over $\C$.
Let $B$ be a [[Definition:Subalgebra|subalgebra]] of $A$.
Let $x \in B$.
Let $\map {\sigma_A} x$ and $\map {\sigma_B} x$ be the [[Definition:Spectrum (Spectral Theory)|spectra]] of $x$ in $A$ and $B$ respectiv... | Let $A_+$ and $B_+$ be the [[Definition:Unitization of Algebra over Field|unitizations]] of $A$ and $B$ respectively.
From [[Unitization of Algebra over Field preserves Subalgebra Relation]], we have:
:$B_+$ is a [[Definition:Unital Subalgebra|unital subalgebra]] of $A_+$.
From [[Spectrum of Element in Unital Subalge... | Spectrum of Element in Unital Subalgebra/Corollary | https://proofwiki.org/wiki/Spectrum_of_Element_in_Unital_Subalgebra/Corollary | https://proofwiki.org/wiki/Spectrum_of_Element_in_Unital_Subalgebra/Corollary | [
"Spectrum of Element in Unital Subalgebra"
] | [
"Definition:Unital Algebra",
"Definition:Algebra over Field",
"Definition:Subalgebra",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Unitization of Algebra over Field",
"Unitization of Algebra over Field preserves Subalgebra Relation",
"Definition:Unital Subalgebra",
"Spectrum of Element in Unital Subalgebra",
"Definition:Spectrum (Spectral Theory)/Non-Unital Algebra",
"Category:Spectrum of Element in Unital Subalgebra"
] |
proofwiki-21207 | Closure of Subalgebra in Normed Algebra is Subalgebra | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$.
Let $B$ be a subalgebra of $A$.
Then $B^-$ is a subalgebra of $A$. | From Closure of Subspace of Normed Vector Space is Subspace, $B^-$ is a vector subspace of $A$.
Now let $x, y \in B^-$.
From the definition of a closed set in a normed vector space, there exists sequences $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ valued in $B$ such that:
:$x_n \to x... | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $\GF$.
Let $B$ be a [[Definition:Subalgebra|subalgebra]] of $A$.
Then $B^-$ is a [[Definition:Subalgebra|subalgebra]] of $A$. | From [[Closure of Subspace of Normed Vector Space is Subspace]], $B^-$ is a [[Definition:Vector Subspace|vector subspace]] of $A$.
Now let $x, y \in B^-$.
From the definition of a [[Definition:Closed Set of Normed Vector Space|closed set in a normed vector space]], there exists [[Definition:Sequence|sequences]] $\se... | Closure of Subalgebra in Normed Algebra is Subalgebra | https://proofwiki.org/wiki/Closure_of_Subalgebra_in_Normed_Algebra_is_Subalgebra | https://proofwiki.org/wiki/Closure_of_Subalgebra_in_Normed_Algebra_is_Subalgebra | [
"Normed Algebras"
] | [
"Definition:Normed Algebra",
"Definition:Subalgebra",
"Definition:Subalgebra"
] | [
"Closure of Subspace of Normed Vector Space is Subspace",
"Definition:Vector Subspace",
"Definition:Closed Set/Normed Vector Space",
"Definition:Sequence",
"Product Rule for Sequence in Normed Algebra",
"Definition:Closure (Topology)",
"Definition:Vector Subspace",
"Definition:Subalgebra",
"Category... |
proofwiki-21208 | Maximal Subalgebra in Normed Algebra is Closed | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$.
Let $B$ be a subalgebra of $A$ that is maximal with respect to set inclusion.
Then $B$ is closed. | From Closure of Subalgebra in Normed Algebra is Subalgebra, the closure $B^-$ of $B$ is a subalgebra with $B \subseteq B^-$.
Since $B$ is maximal with respect to set inclusion, we have that $B = B^-$.
From Set is Closed iff Equals Topological Closure, we conclude that $B$ is closed.
{{qed}}
Category:Normed Algebras
kzk... | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $\GF$.
Let $B$ be a [[Definition:Subalgebra|subalgebra]] of $A$ that is [[Definition:Maximal Element|maximal]] with respect to [[Definition:Set Inclusion|set inclusion]].
Then $B$ is [[Definit... | From [[Closure of Subalgebra in Normed Algebra is Subalgebra]], the [[Definition:Topological Closure|closure]] $B^-$ of $B$ is a [[Definition:Subalgebra|subalgebra]] with $B \subseteq B^-$.
Since $B$ is [[Definition:Maximal Element|maximal]] with respect to [[Definition:Set Inclusion|set inclusion]], we have that $B =... | Maximal Subalgebra in Normed Algebra is Closed | https://proofwiki.org/wiki/Maximal_Subalgebra_in_Normed_Algebra_is_Closed | https://proofwiki.org/wiki/Maximal_Subalgebra_in_Normed_Algebra_is_Closed | [
"Normed Algebras"
] | [
"Definition:Normed Algebra",
"Definition:Subalgebra",
"Definition:Maximal/Element",
"Definition:Subset",
"Definition:Closed Set"
] | [
"Closure of Subalgebra in Normed Algebra is Subalgebra",
"Definition:Closure (Topology)",
"Definition:Subalgebra",
"Definition:Maximal/Element",
"Definition:Subset",
"Set is Closed iff Equals Topological Closure",
"Definition:Closed Set",
"Category:Normed Algebras"
] |
proofwiki-21209 | Existence of Vitali Set implies Subset of Unit Interval with Inner Measure Zero and Outer Measure 1 | Let us posit the existence of a Vitali set.
Then there exists a subset of the closed unit interval which has an inner measure of $0$ and an outer measure of $1$. | {{ProofWanted|But first we need to know what an inner measure is.}} | Let us posit the existence of a [[Definition:Vitali Set|Vitali set]].
Then there exists a [[Definition:Subset|subset]] of the [[Definition:Closed Unit Interval|closed unit interval]] which has an [[Definition:Inner Measure|inner measure]] of $0$ and an [[Definition:Outer Measure|outer measure]] of $1$. | {{ProofWanted|But first we need to know what an inner measure is.}} | Existence of Vitali Set implies Subset of Unit Interval with Inner Measure Zero and Outer Measure 1 | https://proofwiki.org/wiki/Existence_of_Vitali_Set_implies_Subset_of_Unit_Interval_with_Inner_Measure_Zero_and_Outer_Measure_1 | https://proofwiki.org/wiki/Existence_of_Vitali_Set_implies_Subset_of_Unit_Interval_with_Inner_Measure_Zero_and_Outer_Measure_1 | [
"Vitali Sets"
] | [
"Definition:Vitali Set",
"Definition:Subset",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Inner Measure",
"Definition:Outer Measure"
] | [] |
proofwiki-21210 | Maximal Commutative Subalgebra of Unital Algebra is Unital | Let $K$ be a field.
Let $A$ be a unital algebra over $K$.
Let $B$ be a commutative subalgebra of $A$ that is maximal with respect to set inclusion.
Then $B$ is unital. | We show that $B + K {\mathbf 1}_A$ is a subalgebra of $A$.
First we show that $B + K {\mathbf 1}_A$ is a vector subspace of $A$.
Let $x + t {\mathbf 1}_A, y + s {\mathbf 1}_A \in B + K {\mathbf 1}_A$ for $x, y \in B$ and $s, t \in K$.
Let $\lambda \in K$.
We have:
:$\paren {x + t {\mathbf 1}_A} + \lambda \paren {y + s ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ be a [[Definition:Unital Algebra|unital algebra]] over $K$.
Let $B$ be a [[Definition:Commutative Algebra (Abstract Algebra)|commutative]] [[Definition:Subalgebra|subalgebra]] of $A$ that is [[Definition:Maximal Element|maximal]] with respect to [[De... | We show that $B + K {\mathbf 1}_A$ is a [[Definition:Subalgebra|subalgebra]] of $A$.
First we show that $B + K {\mathbf 1}_A$ is a [[Definition:Vector Subspace|vector subspace]] of $A$.
Let $x + t {\mathbf 1}_A, y + s {\mathbf 1}_A \in B + K {\mathbf 1}_A$ for $x, y \in B$ and $s, t \in K$.
Let $\lambda \in K$.
We ... | Maximal Commutative Subalgebra of Unital Algebra is Unital | https://proofwiki.org/wiki/Maximal_Commutative_Subalgebra_of_Unital_Algebra_is_Unital | https://proofwiki.org/wiki/Maximal_Commutative_Subalgebra_of_Unital_Algebra_is_Unital | [
"Algebras over Fields",
"Unital Algebras",
"Commutative Algebras",
"Subalgebras"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Unital Algebra",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Subalgebra",
"Definition:Maximal/Element",
"Definition:Subset",
"Definition:Unital Algebra"
] | [
"Definition:Subalgebra",
"Definition:Vector Subspace",
"Definition:Subalgebra",
"Definition:Vector Subspace",
"Definition:Vector Subspace",
"Definition:Subalgebra",
"Definition:Vector Subspace",
"Definition:Subalgebra",
"Definition:Maximal/Element",
"Definition:Subset",
"Category:Algebras over F... |
proofwiki-21211 | Absolute Net Convergence Equivalent to Absolute Convergence | Let $V$ be a Banach space.
Let $\sequence {v_n}_{n \mathop \in \N}$ be a sequence of elements in $V$.
Let $r \in \R_{\mathop \ge 0}$
Then the following two statements are equivalent:
:$(1): \quad$ the generalized sum $\ds \sum \set {v_n: n \in \N}$ is absolutely net convergent to $r$
:$(2): \quad$ the series $\ds \sum_... | === Statement $(1)$ implies Statement $(2)$ ===
Let the generalized sum $\ds \sum \set {v_n: n \in \N}$ be absolutely net convergent to $r$.
{{:Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Net Convergence implies Absolute Convergence}}{{qed|lemma}} | Let $V$ be a [[Definition:Banach Space|Banach space]].
Let $\sequence {v_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of elements in $V$.
Let $r \in \R_{\mathop \ge 0}$
Then the following two statements are equivalent:
:$(1): \quad$ the [[Definition:Generalized Sum|generalized sum]] $\ds \sum \set ... | === [[Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Net Convergence implies Absolute Convergence|Statement $(1)$ implies Statement $(2)$]] ===
Let the [[Definition:Generalized Sum|generalized sum]] $\ds \sum \set {v_n: n \in \N}$ be [[Definition:Absolute Net Convergence|absolutely net convergent... | Absolute Net Convergence Equivalent to Absolute Convergence | https://proofwiki.org/wiki/Absolute_Net_Convergence_Equivalent_to_Absolute_Convergence | https://proofwiki.org/wiki/Absolute_Net_Convergence_Equivalent_to_Absolute_Convergence | [
"Banach Spaces",
"Nets (Set Theory)",
"Absolute Net Convergence Equivalent to Absolute Convergence"
] | [
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Series",
"Definition:Absolutely Convergent Series"
] | [
"Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Net Convergence implies Absolute Convergence",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence"
] |
proofwiki-21212 | Gelfand's Spectral Radius Formula/Banach Algebra | Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $x \in A$.
Let $\map {r_A} x$ be the spectral radius of $x$ in $A$.
Then, we have:
:$\ds \map {r_A} x = \inf_{n \mathop \in \N_{> 0} } \norm {x^n}^{1/n} = \lim_{n \mathop \to \infty} \norm {x^n}^{1/n}$ | First suppose that $A$ is unital.
Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$.
Let $\map {\rho_A} x$ be the resolvent set of $x$ in $A$.
Let $\lambda \in \map {\sigma_A} x$.
Let $n \in \N_{> 0}$.
From Spectral Mapping Theorem for Polynomials, we have $\lambda^n \in \map {\sigma_A} {x^n}$.
From Spectrum of ... | Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Banach Algebra|Banach algebra]] over $\C$.
Let $x \in A$.
Let $\map {r_A} x$ be the [[Definition:Spectral Radius/Banach Algebra|spectral radius]] of $x$ in $A$.
Then, we have:
:$\ds \map {r_A} x = \inf_{n \mathop \in \N_{> 0} } \norm {x^n}^{1/n} = \lim_{n \... | First suppose that $A$ is [[Definition:Unital Banach Algebra|unital]].
Let $\map {\sigma_A} x$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $x$ in $A$.
Let $\map {\rho_A} x$ be the [[Definition:Resolvent Set|resolvent set]] of $x$ in $A$.
Let $\lambda \in \map {\sigma_A} x$.
Let $n \in \N_{> 0}$. ... | Gelfand's Spectral Radius Formula/Banach Algebra | https://proofwiki.org/wiki/Gelfand's_Spectral_Radius_Formula/Banach_Algebra | https://proofwiki.org/wiki/Gelfand's_Spectral_Radius_Formula/Banach_Algebra | [
"Gelfand's Spectral Radius Formula",
"Spectral Theory of Banach Algebras"
] | [
"Definition:Banach Algebra",
"Definition:Spectral Radius/Banach Algebra"
] | [
"Definition:Unital Banach Algebra",
"Definition:Spectrum (Spectral Theory)",
"Definition:Resolvent Set",
"Spectral Mapping Theorem for Polynomials",
"Spectrum of Element of Banach Algebra is Bounded",
"Definition:Supremum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Spectrum of Ele... |
proofwiki-21213 | Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Net Convergence implies Absolute Convergence | Let $V$ be a Banach space.
Let $\sequence {v_n}_{n \mathop \in \N}$ be a sequence of elements in $V$.
Let $r \in \R_{\mathop \ge 0}$
Let the generalized sum $\ds \sum \set {v_n: n \in \N}$ be absolutely net convergent to $r$.
Then:
:the series $\ds \sum_{n \mathop = 1}^\infty v_n$ is absolutely convergent to $r$. | Let $\epsilon \in \R_{\mathop > 0}$.
From Characterization of Convergent Net in Metric Space:
:$(1) \quad \exists F \subset \N: F $ is finite $: \forall E \subseteq \N : E \supseteq F: E$ is finite $\implies \size{\ds \sum_{n \mathop \in E} \norm{v_n} - r} < \epsilon$
Let $N = \max \set{n : v_n \in F}$.
We have:
:$F \s... | Let $V$ be a [[Definition:Banach Space|Banach space]].
Let $\sequence {v_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of elements in $V$.
Let $r \in \R_{\mathop \ge 0}$
Let the [[Definition:Generalized Sum|generalized sum]] $\ds \sum \set {v_n: n \in \N}$ be [[Definition:Absolute Net Convergence|abso... | Let $\epsilon \in \R_{\mathop > 0}$.
From [[Characterization of Convergent Net in Metric Space]]:
:$(1) \quad \exists F \subset \N: F $ is [[Definition:Finite Set|finite]] $: \forall E \subseteq \N : E \supseteq F: E$ is [[Definition:Finite Set|finite]] $\implies \size{\ds \sum_{n \mathop \in E} \norm{v_n} - r} < \ep... | Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Net Convergence implies Absolute Convergence | https://proofwiki.org/wiki/Absolute_Net_Convergence_Equivalent_to_Absolute_Convergence/Absolute_Net_Convergence_implies_Absolute_Convergence | https://proofwiki.org/wiki/Absolute_Net_Convergence_Equivalent_to_Absolute_Convergence/Absolute_Net_Convergence_implies_Absolute_Convergence | [
"Absolute Net Convergence Equivalent to Absolute Convergence"
] | [
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Series",
"Definition:Absolutely Convergent Series"
] | [
"Characterization of Convergent Net in Metric Space",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Summation over Finite Index",
"Definition:Series",
"Definition:Absolutely Convergent Series"
] |
proofwiki-21214 | Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Convergence implies Absolute Net Convergence | Let $V$ be a Banach space.
Let $\sequence {v_n}_{n \mathop \in \N}$ be a sequence of elements in $V$.
Let $r \in \R_{\mathop \ge 0}$
Let the series $\ds \sum_{n \mathop = 1}^\infty v_n$ be absolutely convergent to $r$.
Then:
:the generalized sum $\ds \sum \set {v_n: n \in \N}$ is absolutely net convergent to $r$. | Let $\epsilon \in \R_{\mathop \ge 0}$.
By definition of absolutely convergent:
:$(2) \quad \exists N \in \N : \forall m \ge N : \size{\ds \sum_{n \mathop = 0}^m \norm{v_m} - r} < \dfrac \epsilon 3$
Let:
:$F = \closedint 0 N$
Let:
:$E \subseteq \N : E \supseteq F : E$ is finite.
Let:
:$m = \max \set{n : n \in E}$
Let:
:... | Let $V$ be a [[Definition:Banach Space|Banach space]].
Let $\sequence {v_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of elements in $V$.
Let $r \in \R_{\mathop \ge 0}$
Let the [[Definition:Series|series]] $\ds \sum_{n \mathop = 1}^\infty v_n$ be [[Definition:Absolutely Convergent Series|absolutely c... | Let $\epsilon \in \R_{\mathop \ge 0}$.
By definition of [[Definition:Absolutely Convergent Series|absolutely convergent]]:
:$(2) \quad \exists N \in \N : \forall m \ge N : \size{\ds \sum_{n \mathop = 0}^m \norm{v_m} - r} < \dfrac \epsilon 3$
Let:
:$F = \closedint 0 N$
Let:
:$E \subseteq \N : E \supseteq F : E$ is... | Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Convergence implies Absolute Net Convergence | https://proofwiki.org/wiki/Absolute_Net_Convergence_Equivalent_to_Absolute_Convergence/Absolute_Convergence_implies_Absolute_Net_Convergence | https://proofwiki.org/wiki/Absolute_Net_Convergence_Equivalent_to_Absolute_Convergence/Absolute_Convergence_implies_Absolute_Net_Convergence | [
"Absolute Net Convergence Equivalent to Absolute Convergence"
] | [
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Series",
"Definition:Absolutely Convergent Series",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence"
] | [
"Definition:Absolutely Convergent Series",
"Definition:Finite Set",
"Set Difference and Intersection form Partition",
"Set Difference Intersection with Second Set is Empty Set",
"Set Difference over Subset",
"Summation over Union of Disjoint Finite Index Sets",
"Triangle Inequality/Real Numbers",
"Sum... |
proofwiki-21215 | Generalized Hilbert Sequence Space is Metric Space/Well-Defined | Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
:$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
:$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
Let $d_2: A ... | From Characterization of Hausdorff Property in terms of Nets:
:a convergent net in $\R$ has a unique limit.
To show that $d_2$ is well-defined, it is sufficient to show:
:$\ds \forall x = \family {x_i}, y = \family {y_i} \in A:$ the generalized sum $\ds \sum_{i \mathop \in I} \paren {x_i - y_i}^2$ converges
Let $x = \... | Let $\alpha$ be an [[Definition:Infinite Cardinal|infinite cardinal number]].
Let $I$ be an [[Definition:Indexed Set|indexed set]] of [[Definition:Cardinality|cardinality]] $\alpha$.
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real-Valued Function|real-valued functions]] $x : I \to \R$ such that:
:$(1)... | From [[Characterization of Hausdorff Property in terms of Nets]]:
:a [[Definition:Convergent Net|convergent net]] in $\R$ has a unique [[Definition:Limit of Net|limit]].
To show that $d_2$ is [[Definition:Well-Defined|well-defined]], it is sufficient to show:
:$\ds \forall x = \family {x_i}, y = \family {y_i} \in A:$... | Generalized Hilbert Sequence Space is Metric Space/Well-Defined | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Well-Defined | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Well-Defined | [
"Generalized Hilbert Sequence Space is Metric Space"
] | [
"Definition:Infinite Cardinal",
"Definition:Indexing Set/Indexed Set",
"Definition:Cardinality",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Countable Set",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Real-Valued Function",
"Definition:Well-Defined"
... | [
"Characterization of Hausdorff Property in terms of Nets",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Well-Defined",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Convergent Net",
"Characterization of Generalized Hilbert Sequence Space",
"Definition:... |
proofwiki-21216 | Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M1 | Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
:$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
:$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
Let $d_2: A ... | Let $x \in A$.
From Lemma:
:$\exists y \in \ell^2 :$
::$\map {d_2} {x, x} = \map {d_{\ell^2}} {y, y}$
We have:
{{begin-eqn}}
{{eqn | l = \map {d_2} {x, x}
| r = \map {d_{\ell^2} } {y, y}
| c = Lemma
}}
{{eqn | r = 0
| c = {{Metric-space-axiom|4}} applied to $d_{\ell^2}$
}}
{{end-eqn}}
The result follo... | Let $\alpha$ be an [[Definition:Infinite Cardinal|infinite cardinal number]].
Let $I$ be an [[Definition:Indexed Set|indexed set]] of [[Definition:Cardinality|cardinality]] $\alpha$.
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real-Valued Function|real-valued functions]] $x : I \to \R$ such that:
:$(1)... | Let $x \in A$.
From [[Generalized Hilbert Sequence Space is Metric Space/Lemma 1|Lemma]]:
:$\exists y \in \ell^2 :$
::$\map {d_2} {x, x} = \map {d_{\ell^2}} {y, y}$
We have:
{{begin-eqn}}
{{eqn | l = \map {d_2} {x, x}
| r = \map {d_{\ell^2} } {y, y}
| c = [[Generalized Hilbert Sequence Space is Metric Sp... | Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M1 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Metric_Space_Axiom_M1 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Metric_Space_Axiom_M1 | [
"Generalized Hilbert Sequence Space is Metric Space"
] | [
"Definition:Infinite Cardinal",
"Definition:Indexing Set/Indexed Set",
"Definition:Cardinality",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Countable Set",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Real-Valued Function"
] | [
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1",
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1"
] |
proofwiki-21217 | Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M2 | Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
:$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
:$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
Let $d_2: A ... | Let $x_1, x_2, x_3 \in A$.
From Lemma:
:$\exists y_1, y_2, y_3 \in \ell^2 : $
::$\forall i, j \in \set{1, 2, 3} : \map {d_2} {x_i, x_j} = \map {d_{\ell^2}} {y_i, y_j}$
We have:
{{begin-eqn}}
{{eqn | l = \map {d_2} {x_1, x_3}
| r = \map {d_{\ell^2} } {y_1, y_3}
| c = Lemma
}}
{{eqn | o = \le
| r = \map... | Let $\alpha$ be an [[Definition:Infinite Cardinal|infinite cardinal number]].
Let $I$ be an [[Definition:Indexed Set|indexed set]] of [[Definition:Cardinality|cardinality]] $\alpha$.
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real-Valued Function|real-valued functions]] $x : I \to \R$ such that:
:$(1)... | Let $x_1, x_2, x_3 \in A$.
From [[Generalized Hilbert Sequence Space is Metric Space/Lemma 1|Lemma]]:
:$\exists y_1, y_2, y_3 \in \ell^2 : $
::$\forall i, j \in \set{1, 2, 3} : \map {d_2} {x_i, x_j} = \map {d_{\ell^2}} {y_i, y_j}$
We have:
{{begin-eqn}}
{{eqn | l = \map {d_2} {x_1, x_3}
| r = \map {d_{\ell^2} ... | Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M2 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Metric_Space_Axiom_M2 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Metric_Space_Axiom_M2 | [
"Generalized Hilbert Sequence Space is Metric Space"
] | [
"Definition:Infinite Cardinal",
"Definition:Indexing Set/Indexed Set",
"Definition:Cardinality",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Countable Set",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Real-Valued Function"
] | [
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1",
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1",
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1"
] |
proofwiki-21218 | Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M3 | Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
:$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
:$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
Let $d_2: A ... | Let $x_1, x_2 \in A$.
From Lemma:
:$\exists y_1, y_2 \in \ell^2 : $
::$\map {d_2} {x_1, x_2} = \map {d_{\ell^2}} {y_1, y_2}$ and $\map {d_2} {x_2, x_1} = \map {d_{\ell^2}} {y_2, y_1}$
We have:
{{begin-eqn}}
{{eqn | l = \map {d_2} {x_1, x_2}
| r = \map {d_{\ell^2} } {y_1, y_2}
| c = Lemma
}}
{{eqn | r = \map... | Let $\alpha$ be an [[Definition:Infinite Cardinal|infinite cardinal number]].
Let $I$ be an [[Definition:Indexed Set|indexed set]] of [[Definition:Cardinality|cardinality]] $\alpha$.
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real-Valued Function|real-valued functions]] $x : I \to \R$ such that:
:$(1)... | Let $x_1, x_2 \in A$.
From [[Generalized Hilbert Sequence Space is Metric Space/Lemma 1|Lemma]]:
:$\exists y_1, y_2 \in \ell^2 : $
::$\map {d_2} {x_1, x_2} = \map {d_{\ell^2}} {y_1, y_2}$ and $\map {d_2} {x_2, x_1} = \map {d_{\ell^2}} {y_2, y_1}$
We have:
{{begin-eqn}}
{{eqn | l = \map {d_2} {x_1, x_2}
| r = \... | Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M3 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Metric_Space_Axiom_M3 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Metric_Space_Axiom_M3 | [
"Generalized Hilbert Sequence Space is Metric Space"
] | [
"Definition:Infinite Cardinal",
"Definition:Indexing Set/Indexed Set",
"Definition:Cardinality",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Countable Set",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Real-Valued Function"
] | [
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1",
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1",
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1"
] |
proofwiki-21219 | Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M4 | Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
:$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
:$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
Let $d_2: A ... | Let $x_1, x_2 \in A : x_1 \ne x_2$.
From Lemma:
:$\exists y_1, y_2 \in \ell^2 :$
::$y_1 \ne y_2$
::$\map {d_2} {x_1, x_2} = \map {d_{\ell^2}} {y_1, y_2}$
We have:
{{begin-eqn}}
{{eqn | l = \map {d_2} {x_1, x_2}
| r = \map {d_{\ell^2} } {y_1, y_2}
| c = Lemma
}}
{{eqn | o = >
| r = 0
| c = {{Metr... | Let $\alpha$ be an [[Definition:Infinite Cardinal|infinite cardinal number]].
Let $I$ be an [[Definition:Indexed Set|indexed set]] of [[Definition:Cardinality|cardinality]] $\alpha$.
Let $A$ be the [[Definition:Set|set]] of all [[Definition:Real-Valued Function|real-valued functions]] $x : I \to \R$ such that:
:$(1)... | Let $x_1, x_2 \in A : x_1 \ne x_2$.
From [[Generalized Hilbert Sequence Space is Metric Space/Lemma 1|Lemma]]:
:$\exists y_1, y_2 \in \ell^2 :$
::$y_1 \ne y_2$
::$\map {d_2} {x_1, x_2} = \map {d_{\ell^2}} {y_1, y_2}$
We have:
{{begin-eqn}}
{{eqn | l = \map {d_2} {x_1, x_2}
| r = \map {d_{\ell^2} } {y_1, y_2}
... | Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M4 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Metric_Space_Axiom_M4 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Metric_Space_Axiom_M4 | [
"Generalized Hilbert Sequence Space is Metric Space"
] | [
"Definition:Infinite Cardinal",
"Definition:Indexing Set/Indexed Set",
"Definition:Cardinality",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Countable Set",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Real-Valued Function"
] | [
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1",
"Generalized Hilbert Sequence Space is Metric Space/Lemma 1"
] |
proofwiki-21220 | Indices of Complete Bipartite Graph Commute | :$K_{m, n}$ is isomorphic to $K_{n, m}$
for all $m, n \in \N$. | Let $K_{m, n}$ be represented by the graph $G = \struct {A \mid B, E}$, where $\card A = m$ and $\card B = n$.
Let $K_{n, m}$ be represented by the graph $G' = \struct {A' \mid B', E'}$, where $\card {A'} = n$ and $\card {B'} = m$
Let:
{{begin-eqn}}
{{eqn | l = A
| r = \set {a_1, a_2, \ldots, a_m}
}}
{{eqn | l = ... | :$K_{m, n}$ is [[Definition:Graph Isomorphism|isomorphic]] to $K_{n, m}$
for all $m, n \in \N$. | Let $K_{m, n}$ be represented by the [[Definition:Graph (Graph Theory)|graph]] $G = \struct {A \mid B, E}$, where $\card A = m$ and $\card B = n$.
Let $K_{n, m}$ be represented by the [[Definition:Graph (Graph Theory)|graph]] $G' = \struct {A' \mid B', E'}$, where $\card {A'} = n$ and $\card {B'} = m$
Let:
{{begin-eq... | Indices of Complete Bipartite Graph Commute | https://proofwiki.org/wiki/Indices_of_Complete_Bipartite_Graph_Commute | https://proofwiki.org/wiki/Indices_of_Complete_Bipartite_Graph_Commute | [
"Complete Bipartite Graphs"
] | [
"Definition:Isomorphism (Graph Theory)"
] | [
"Definition:Graph (Graph Theory)",
"Definition:Graph (Graph Theory)",
"Definition:Isomorphism (Graph Theory)",
"Definition:Isomorphism (Graph Theory)",
"Definition:Surjection",
"Cardinality of Codomain of Surjection",
"Definition:Bijection",
"Definition:Isomorphism (Graph Theory)",
"Category:Complet... |
proofwiki-21221 | Complete Bipartite Graphs which are Trees | :$K_{0, 0}$ is a tree
:$K_{1, n}$ and $K_{n, 1}$ is a tree for all $n$
and no other complete bipartite graphs are trees. | We note from Null Graph is Complete Bipartite Graph and Null Graph is Tree that $K_{0, 0}$ is a complete bipartite graph that is also a tree.
Next we note that the order of $K_{1, n}$ is $n + 1$.
Indeed, there is $1$ vertex in $A$ and $n$ vertices in $B$, for a total of $n + 1$.
Then we note that the size of $K_{1, n}$... | :$K_{0, 0}$ is a [[Definition:Tree (Graph Theory)|tree]]
:$K_{1, n}$ and $K_{n, 1}$ is a [[Definition:Tree (Graph Theory)|tree]] for all $n$
and no other [[Definition:Complete Bipartite Graph|complete bipartite graphs]] are [[Definition:Tree (Graph Theory)|trees]]. | We note from [[Null Graph is Complete Bipartite Graph]] and [[Null Graph is Tree]] that $K_{0, 0}$ is a [[Definition:Complete Bipartite Graph|complete bipartite graph]] that is also a [[Definition:Tree (Graph Theory)|tree]].
Next we note that the [[Definition:Order of Graph|order]] of $K_{1, n}$ is $n + 1$.
Indeed, ... | Complete Bipartite Graphs which are Trees | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Trees | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Trees | [
"Complete Bipartite Graphs",
"Tree Theory"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)",
"Definition:Complete Bipartite Graph",
"Definition:Tree (Graph Theory)"
] | [
"Null Graph is Complete Bipartite Graph",
"Null Graph is Tree",
"Definition:Complete Bipartite Graph",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Size",
... |
proofwiki-21222 | Condition for Complete Bipartite Graph to be Edgeless | :$K_{m, n}$ is edgeless {{iff}} either $m = 0$ or $n = 0$. | Consider $K_{m, n}$ where both $m > 0$ and $n > 0$.
Then:
:$\exists u \in A, v \in V: \set {u, v} \in E$
and it follows by definition that $K_{m, n}$ is not edgeless.
Consider $K_{m, n}$ where either $m = 0$ or $n = 0$.
Hence one of the partite sets of $G$ is empty.
Hence there are no vertices for the vertices in the o... | :$K_{m, n}$ is [[Definition:Edgeless Graph|edgeless]] {{iff}} either $m = 0$ or $n = 0$. | Consider $K_{m, n}$ where both $m > 0$ and $n > 0$.
Then:
:$\exists u \in A, v \in V: \set {u, v} \in E$
and it follows by definition that $K_{m, n}$ is not [[Definition:Edgeless Graph|edgeless]].
Consider $K_{m, n}$ where either $m = 0$ or $n = 0$.
Hence one of the [[Definition:Partite Set|partite sets]] of $G$ i... | Condition for Complete Bipartite Graph to be Edgeless | https://proofwiki.org/wiki/Condition_for_Complete_Bipartite_Graph_to_be_Edgeless | https://proofwiki.org/wiki/Condition_for_Complete_Bipartite_Graph_to_be_Edgeless | [
"Complete Bipartite Graphs",
"Edgeless Graphs"
] | [
"Definition:Edgeless Graph"
] | [
"Definition:Edgeless Graph",
"Definition:Bipartite Graph/Partite Set",
"Definition:Empty Set",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Bipartite Graph/Partite Set",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Edgel... |
proofwiki-21223 | Complete Bipartite Graphs which are Regular | :$K_{n, n}$ is $n$-regular for all $n$
:$K_{n, 0}$ and $K_{0, n}$ are $0$-regular for all $n$
and no other complete bipartite graphs are regular. | First consider $K_{n, 0}$ and $K_{0, n}$ for $n \in \N$.
From Condition for Complete Bipartite Graph to be Edgeless, every vertex in $K_{n, 0}$ and $K_{0, n}$ has degree $0$.
From Graph is 0-Regular iff Edgeless, $K_{n, 0}$ and $K_{0, n}$ are $0$-regular.
Now consider $K_{m, n}$ where $m, n > 0$.
By definition of compl... | :$K_{n, n}$ is [[Definition:Regular Graph|$n$-regular]] for all $n$
:$K_{n, 0}$ and $K_{0, n}$ are [[Definition:Regular Graph|$0$-regular]] for all $n$
and no other [[Definition:Complete Bipartite Graph|complete bipartite graphs]] are [[Definition:Regular Graph|regular]]. | First consider $K_{n, 0}$ and $K_{0, n}$ for $n \in \N$.
From [[Condition for Complete Bipartite Graph to be Edgeless]], every [[Definition:Vertex of Graph|vertex]] in $K_{n, 0}$ and $K_{0, n}$ has [[Definition:Degree of Vertex|degree]] $0$.
From [[Graph is 0-Regular iff Edgeless]], $K_{n, 0}$ and $K_{0, n}$ are [[De... | Complete Bipartite Graphs which are Regular | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Regular | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Regular | [
"Complete Bipartite Graphs",
"Regular Graphs"
] | [
"Definition:Regular Graph",
"Definition:Regular Graph",
"Definition:Complete Bipartite Graph",
"Definition:Regular Graph"
] | [
"Condition for Complete Bipartite Graph to be Edgeless",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Graph is 0-Regular iff Edgeless",
"Definition:Regular Graph",
"Definition:Complete Bipartite Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph ... |
proofwiki-21224 | Null Graph is Tree | The null graph $N_0$ is a tree. | By definition, a tree is a connected graph with no cycles.
From Null Graph is Connected, $N_0$ is a connected graph.
The null graph has no edges.
Hence the null graph has no cycles vacuously.
Hence the result.
{{qed}}
Category:Null Graph
Category:Tree Theory
8jigtkxiab6o3m5vr1clnpw9vhrb49t | The [[Definition:Null Graph|null graph]] $N_0$ is a [[Definition:Tree (Graph Theory)|tree]]. | By definition, a [[Definition:Tree (Graph Theory)|tree]] is a [[Definition:Connected Graph|connected graph]] with no [[Definition:Cycle (Graph Theory)|cycles]].
From [[Null Graph is Connected]], $N_0$ is a [[Definition:Connected Graph|connected graph]].
The [[Definition:Null Graph|null graph]] has no [[Definition:Edg... | Null Graph is Tree | https://proofwiki.org/wiki/Null_Graph_is_Tree | https://proofwiki.org/wiki/Null_Graph_is_Tree | [
"Null Graph",
"Tree Theory"
] | [
"Definition:Null Graph",
"Definition:Tree (Graph Theory)"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Cycle (Graph Theory)",
"Null Graph is Connected",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Null Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Null Graph",
"Definition:Cycle (Graph... |
proofwiki-21225 | Null Graph is Complete Bipartite Graph | The null graph $N_0$ is the complete bipartite graph $K_{0, 0}$. | By definition, the complete bipartite graph $K_{0, 0}$ has no vertices.
Hence the result by definition of null graph.
{{qed}}
Category:Null Graph
Category:Complete Bipartite Graphs
f5awimvrq2u1ymz8as26brg0mpjd8ou | The [[Definition:Null Graph|null graph]] $N_0$ is the [[Definition:Complete Bipartite Graph|complete bipartite graph]] $K_{0, 0}$. | By definition, the [[Definition:Complete Bipartite Graph|complete bipartite graph]] $K_{0, 0}$ has no [[Definition:Vertex of Graph|vertices]].
Hence the result by definition of [[Definition:Null Graph|null graph]].
{{qed}}
[[Category:Null Graph]]
[[Category:Complete Bipartite Graphs]]
f5awimvrq2u1ymz8as26brg0mpjd8ou | Null Graph is Complete Bipartite Graph | https://proofwiki.org/wiki/Null_Graph_is_Complete_Bipartite_Graph | https://proofwiki.org/wiki/Null_Graph_is_Complete_Bipartite_Graph | [
"Null Graph",
"Complete Bipartite Graphs"
] | [
"Definition:Null Graph",
"Definition:Complete Bipartite Graph"
] | [
"Definition:Complete Bipartite Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Null Graph",
"Category:Null Graph",
"Category:Complete Bipartite Graphs"
] |
proofwiki-21226 | Null Graph is Connected | The null graph $N_0$ is a connected graph. | By definition, a graph $G$ is '''connected''' {{iff}} every pair of vertices in $G$ is connected.
There are no vertices in $N_0$.
Hence the result holds vacuously.
{{qed}}
Category:Null Graph
Category:Connectedness (Graph Theory)
aol2vr7k45vxn5ugdxzz4ahq5yqz681 | The [[Definition:Null Graph|null graph]] $N_0$ is a [[Definition:Connected Graph|connected graph]]. | By definition, a [[Definition:Graph (Graph Theory)|graph]] $G$ is '''connected''' {{iff}} every pair of [[Definition:Vertex of Graph|vertices]] in $G$ is [[Definition:Connected Vertices|connected]].
There are no [[Definition:Vertex of Graph|vertices]] in $N_0$.
Hence the result holds [[Definition:Vacuous Truth|vacuou... | Null Graph is Connected | https://proofwiki.org/wiki/Null_Graph_is_Connected | https://proofwiki.org/wiki/Null_Graph_is_Connected | [
"Null Graph",
"Connectedness (Graph Theory)"
] | [
"Definition:Null Graph",
"Definition:Connected (Graph Theory)/Graph"
] | [
"Definition:Graph (Graph Theory)",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Connected (Graph Theory)/Vertices",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Vacuous Truth",
"Category:Null Graph",
"Category:Connectedness (Graph Theory)"
] |
proofwiki-21227 | Complete Bipartite Graphs which are Complete Graphs | :$K_{0, 0}$ is the complete graph $K_0$
:$K_{0, 1}$ and $K_{1, 0}$ are the complete graph $K_1$
:$K_{1, 1}$ is the complete graph $K_2$
and no other complete bipartite graphs are complete. | $K_{0, 0}$ is the null graph by Null Graph is Complete Bipartite Graph.
Then by Null Graph is Complete Graph, $K_{0, 0}$ is the complete graph $K_0$.
$K_{0, 1}$ and $K_{1, 0}$ consist of one vertex and no edges.
Then by Complete Graph of Order 1 is Edgeless, $K_{0, 1}$ and $K_{1, 0}$ are both the complete graph $K_1$.
... | :$K_{0, 0}$ is the [[Definition:Complete Graph|complete graph]] $K_0$
:$K_{0, 1}$ and $K_{1, 0}$ are the [[Definition:Complete Graph|complete graph]] $K_1$
:$K_{1, 1}$ is the [[Definition:Complete Graph|complete graph]] $K_2$
and no other [[Definition:Complete Bipartite Graph|complete bipartite graphs]] are [[Definiti... | $K_{0, 0}$ is the [[Definition:Null Graph|null graph]] by [[Null Graph is Complete Bipartite Graph]].
Then by [[Null Graph is Complete Graph]], $K_{0, 0}$ is the [[Definition:Complete Graph|complete graph]] $K_0$.
$K_{0, 1}$ and $K_{1, 0}$ consist of one [[Definition:Vertex of Graph|vertex]] and no [[Definition:Edge... | Complete Bipartite Graphs which are Complete Graphs | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Complete_Graphs | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Complete_Graphs | [
"Complete Bipartite Graphs",
"Complete Graphs"
] | [
"Definition:Complete Graph",
"Definition:Complete Graph",
"Definition:Complete Graph",
"Definition:Complete Bipartite Graph",
"Definition:Complete Graph"
] | [
"Definition:Null Graph",
"Null Graph is Complete Bipartite Graph",
"Null Graph is Complete Graph",
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Complete Graph of Order 1 is Edgeless",
"Definition:Complete Graph",
"Definition:Complete... |
proofwiki-21228 | Null Graph is Complete Graph | The null graph $N_0$ is the complete graph $K_0$. | By definition, the complete graph $K_0$ has no vertices.
Hence the result by definition of null graph.
{{qed}}
Category:Null Graph
Category:Complete Graphs
1qjrnfj3536tq3zcradqlwnzftwbjap | The [[Definition:Null Graph|null graph]] $N_0$ is the [[Definition:Complete Graph|complete graph]] $K_0$. | By definition, the [[Definition:Complete Graph|complete graph]] $K_0$ has no [[Definition:Vertex of Graph|vertices]].
Hence the result by definition of [[Definition:Null Graph|null graph]].
{{qed}}
[[Category:Null Graph]]
[[Category:Complete Graphs]]
1qjrnfj3536tq3zcradqlwnzftwbjap | Null Graph is Complete Graph | https://proofwiki.org/wiki/Null_Graph_is_Complete_Graph | https://proofwiki.org/wiki/Null_Graph_is_Complete_Graph | [
"Null Graph",
"Complete Graphs",
"Null Graph",
"Complete Graphs"
] | [
"Definition:Null Graph",
"Definition:Complete Graph"
] | [
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Null Graph",
"Category:Null Graph",
"Category:Complete Graphs"
] |
proofwiki-21229 | Complete Bipartite Graphs which are Path Graphs | :$K_{0, 0}$ is the path graph $P_0$
:$K_{0, 1}$ and $K_{1, 0}$ are the path graph $P_1$
:$K_{1, 1}$ is the path graph $P_2$
:$K_{1, 2}$ and $K_{2, 1}$ are the path graphs $P_3$
and no other complete bipartite graphs are path graphs. | $K_{0, 0}$, $K_{0, 1}$ and $K_{1, 0}$ have no edges.
Hence they are path graphs vacuously.
{{qed|lemma}}
That $K_{1, 1}$ is the path graph $P_2$ can be determined by inspection:
:100px
{{qed|lemma}}
Similarly, that $K_{1, 2}$ and $K_{2, 1}$ are the path graph $P_3$ can be determined by inspection:
:200px
{{qed|lemma}}
... | :$K_{0, 0}$ is the [[Definition:Path Graph|path graph]] $P_0$
:$K_{0, 1}$ and $K_{1, 0}$ are the [[Definition:Path Graph|path graph]] $P_1$
:$K_{1, 1}$ is the [[Definition:Path Graph|path graph]] $P_2$
:$K_{1, 2}$ and $K_{2, 1}$ are the [[Definition:Path Graph|path graphs]] $P_3$
and no other [[Definition:Complete Bipa... | $K_{0, 0}$, $K_{0, 1}$ and $K_{1, 0}$ have no [[Definition:Edge of Graph|edges]].
Hence they are [[Definition:Path Graph|path graphs]] [[Definition:Vacuous Truth|vacuously]].
{{qed|lemma}}
That $K_{1, 1}$ is the [[Definition:Path Graph|path graph]] $P_2$ can be determined by inspection:
:[[File:K1-1.png|100px]]
{{q... | Complete Bipartite Graphs which are Path Graphs | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Path_Graphs | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Path_Graphs | [
"Complete Bipartite Graphs",
"Path Graphs"
] | [
"Definition:Path Graph",
"Definition:Path Graph",
"Definition:Path Graph",
"Definition:Path Graph",
"Definition:Complete Bipartite Graph",
"Definition:Path Graph"
] | [
"Definition:Graph (Graph Theory)/Edge",
"Definition:Path Graph",
"Definition:Vacuous Truth",
"Definition:Path Graph",
"File:K1-1.png",
"Definition:Path Graph",
"File:K1-2.png",
"Path Graph is Tree",
"Complete Bipartite Graphs which are Trees",
"Definition:Path Graph",
"Definition:Graph (Graph Th... |
proofwiki-21230 | Complete Bipartite Graphs which are Cycle Graphs | :$K_{2, 2}$ is the cycle graph $C_4$
and no other complete bipartite graphs are cycle graphs. | From Cycle Graph is 2-Regular, a cycle graph is $2$-regular.
From Complete Bipartite Graphs which are Regular, the only $2$-regular complete bipartite graph is $K_{2, 2}$.
Hence the result.
{{qed}}
Category:Complete Bipartite Graphs
Category:Cycle Graphs
21rembf08qx1bdc9wo92rj7333nd9hi | :$K_{2, 2}$ is the [[Definition:Cycle Graph|cycle graph]] $C_4$
and no other [[Definition:Complete Bipartite Graph|complete bipartite graphs]] are [[Definition:Cycle Graph|cycle graphs]]. | From [[Cycle Graph is 2-Regular]], a [[Definition:Cycle Graph|cycle graph]] is [[Definition:Regular Graph|$2$-regular]].
From [[Complete Bipartite Graphs which are Regular]], the only [[Definition:Regular Graph|$2$-regular]] [[Definition:Complete Bipartite Graph|complete bipartite graph]] is $K_{2, 2}$.
Hence the res... | Complete Bipartite Graphs which are Cycle Graphs | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Cycle_Graphs | https://proofwiki.org/wiki/Complete_Bipartite_Graphs_which_are_Cycle_Graphs | [
"Complete Bipartite Graphs",
"Cycle Graphs"
] | [
"Definition:Cycle Graph",
"Definition:Complete Bipartite Graph",
"Definition:Cycle Graph"
] | [
"Cycle Graph is 2-Regular",
"Definition:Cycle Graph",
"Definition:Regular Graph",
"Complete Bipartite Graphs which are Regular",
"Definition:Regular Graph",
"Definition:Complete Bipartite Graph",
"Category:Complete Bipartite Graphs",
"Category:Cycle Graphs"
] |
proofwiki-21231 | Singleton Graph is Unique | The singleton graph $N_1$ is unique (up to isomorphism). | $N_1$ can be expressed as:
:$N_1 := \struct {\set v, \O}$
where:
:$\set v$ is the set of vertices
:$\O$ is the set of edges, which is empty by definition.
Suppose there exists another singleton graph $N_1' = \struct {\set v', \O}$.
Let $\phi: \set v \to \set {v'}$ be the mapping from $N_1$ to $N_1'$ defined as:
:$\map... | The [[Definition:Singleton Graph|singleton graph]] $N_1$ is [[Definition:Unique|unique]] (up to [[Definition:Graph Isomorphism|isomorphism]]). | $N_1$ can be expressed as:
:$N_1 := \struct {\set v, \O}$
where:
:$\set v$ is the [[Definition:Set|set]] of [[Definition:Vertex of Graph|vertices]]
:$\O$ is the [[Definition:Set|set]] of [[Definition:Edge of Graph|edges]], which is [[Definition:Empty Set|empty]] by definition.
Suppose there exists another [[Definiti... | Singleton Graph is Unique | https://proofwiki.org/wiki/Singleton_Graph_is_Unique | https://proofwiki.org/wiki/Singleton_Graph_is_Unique | [
"Singleton Graph"
] | [
"Definition:Singleton Graph",
"Definition:Unique",
"Definition:Isomorphism (Graph Theory)"
] | [
"Definition:Set",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Set",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Empty Set",
"Definition:Singleton Graph",
"Definition:Mapping",
"Mapping from Singleton is Injection",
"Definition:Injection",
"Mapping to Singleton is Surjection",
"... |
proofwiki-21232 | Singleton Graph is Edgeless | The singleton graph $N_1$ is edgeless. | Recall the definition of $N_1$:
{{:Definition:Singleton Graph}}
As $N_1$ is simple, the only edges are between distinct vertices.
As $N_1$ has only one vertex, $N_1$ can have no edges.
{{qed}} | The [[Definition:Singleton Graph|singleton graph]] $N_1$ is [[Definition:Edgeless Graph|edgeless]]. | Recall the definition of $N_1$:
{{:Definition:Singleton Graph}}
As $N_1$ is [[Definition:Simple Graph|simple]], the only [[Definition:Edge of Graph|edges]] are between [[Definition:Distinct Elements|distinct]] [[Definition:Vertex of Graph|vertices]].
As $N_1$ has only one [[Definition:Vertex of Graph|vertex]], $N_1$ ... | Singleton Graph is Edgeless | https://proofwiki.org/wiki/Singleton_Graph_is_Edgeless | https://proofwiki.org/wiki/Singleton_Graph_is_Edgeless | [
"Singleton Graph",
"Edgeless Graphs"
] | [
"Definition:Singleton Graph",
"Definition:Edgeless Graph"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Distinct/Plural",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge"
] |
proofwiki-21233 | Singleton Graph is Complete | The singleton graph $N_1$ is complete. | Recall the definition of $N_1$:
{{:Definition:Singleton Graph}}
Recall the definition of complete graph:
{{:Definition:Complete Graph}}
As $N_1$ has only one vertex, this follows vacuously.
{{qed}} | The [[Definition:Singleton Graph|singleton graph]] $N_1$ is [[Definition:Complete Graph|complete]]. | Recall the definition of $N_1$:
{{:Definition:Singleton Graph}}
Recall the definition of [[Definition:Complete Graph|complete graph]]:
{{:Definition:Complete Graph}}
As $N_1$ has only one [[Definition:Vertex of Graph|vertex]], this follows [[Definition:Vacuous Truth|vacuously]].
{{qed}} | Singleton Graph is Complete | https://proofwiki.org/wiki/Singleton_Graph_is_Complete | https://proofwiki.org/wiki/Singleton_Graph_is_Complete | [
"Singleton Graph",
"Complete Graphs"
] | [
"Definition:Singleton Graph",
"Definition:Complete Graph"
] | [
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Vacuous Truth"
] |
proofwiki-21234 | Singleton Graph is Regular | The singleton graph $N_1$ is regular of degree $0$. | Recall the definition of $N_1$:
{{:Definition:Singleton Graph}}
Recall the definition of regular graph:
{{:Definition:Regular Graph}}
As $N_1$ has only one vertex, this follows vacuously.
{{qed}}
Category:Singleton Graph
Category:Regular Graphs
16yimaydpdlp9kinzrqoek1gwe9ud7y | The [[Definition:Singleton Graph|singleton graph]] $N_1$ is [[Definition:Regular Graph|regular of degree $0$]]. | Recall the definition of $N_1$:
{{:Definition:Singleton Graph}}
Recall the definition of [[Definition:Regular Graph|regular graph]]:
{{:Definition:Regular Graph}}
As $N_1$ has only one [[Definition:Vertex of Graph|vertex]], this follows [[Definition:Vacuous Truth|vacuously]].
{{qed}}
[[Category:Singleton Graph]]
[[... | Singleton Graph is Regular | https://proofwiki.org/wiki/Singleton_Graph_is_Regular | https://proofwiki.org/wiki/Singleton_Graph_is_Regular | [
"Singleton Graph",
"Singleton Graph",
"Regular Graphs"
] | [
"Definition:Singleton Graph",
"Definition:Regular Graph"
] | [
"Definition:Regular Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Vacuous Truth",
"Category:Singleton Graph",
"Category:Regular Graphs"
] |
proofwiki-21235 | Image of Fredholm Operator of Banach Spaces is Closed | Let $\Bbb F \in \set {\R, \C}$.
Let $\tuple {X, \norm \cdot_X}, \tuple {Y, \norm \cdot_Y}$ be Banach spaces over $\Bbb F$.
Let $T: X \to Y$ be a Fredholm operator.
Let $\Img T$ be the image of $T$ .
Then $\Img T$ is closed. | By Kernel of Bounded Linear Transformation is Closed Linear Subspace, $\map \ker T \subseteq X$ is closed.
Thus, by Characterization of Complete Normed Quotient Vector Spaces, $X / \map \ker T$ is a Banach space.
Define the injective linear transformation $\tilde T : X / \map \ker T \to Y$ by:
:$\map {\tilde T} {\eqcla... | Let $\Bbb F \in \set {\R, \C}$.
Let $\tuple {X, \norm \cdot_X}, \tuple {Y, \norm \cdot_Y}$ be [[Definition:Banach Space|Banach spaces]] over $\Bbb F$.
Let $T: X \to Y$ be a [[Definition:Fredholm Operator|Fredholm operator]].
Let $\Img T$ be the [[Definition:Image of Mapping|image]] of $T$ .
Then $\Img T$ is [[Def... | By [[Kernel of Bounded Linear Transformation is Closed Linear Subspace]], $\map \ker T \subseteq X$ is [[Definition:Closed Set (Topology)|closed]].
Thus, by [[Characterization of Complete Normed Quotient Vector Spaces]], $X / \map \ker T$ is a [[Definition:Banach Space|Banach space]].
Define the [[Definition:Injectio... | Image of Fredholm Operator of Banach Spaces is Closed | https://proofwiki.org/wiki/Image_of_Fredholm_Operator_of_Banach_Spaces_is_Closed | https://proofwiki.org/wiki/Image_of_Fredholm_Operator_of_Banach_Spaces_is_Closed | [
"Fredholm Operators",
"Functional Analysis"
] | [
"Definition:Banach Space",
"Definition:Fredholm Operator",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Closed Set/Topology"
] | [
"Kernel of Bounded Linear Transformation is Closed Linear Subspace",
"Definition:Closed Set/Topology",
"Characterization of Complete Normed Quotient Vector Spaces",
"Definition:Banach Space",
"Definition:Injection",
"Definition:Linear Transformation",
"Definition:Injection",
"Definition:Dimension of V... |
proofwiki-21236 | Generalized Hilbert Sequence Space is Metric Space/Lemma 1 | Let $x_1, x_2, \ldots, x_m \in A$.
Then there exists $y_1, y_2, \ldots, y_m \in \ell^2$:
:$\forall a,b \in \closedint 1 m : y_a \ne y_b \iff x_a \ne x_b$
:$\forall a,b \in \closedint 1 m : \map {d_{\ell^2} } {y_a, y_b} = \map {d_2} {x_a, x_b}$ | For each $k \in \closedint 1 m$, let:
:$\ds \sum_{i \mathop \in I} \paren{x_k}_i^2$ converge to $r_k \in \R$.
From Characterization of Generalized Hilbert Sequence Space, there exists enumeration $J = \set{j_0, j_1, j_2, \ldots}$ of a countable set of $I$:
:$\forall k \in \closedint 1 m : \set{i \in I : \paren{x_k}_i ... | Let $x_1, x_2, \ldots, x_m \in A$.
Then there exists $y_1, y_2, \ldots, y_m \in \ell^2$:
:$\forall a,b \in \closedint 1 m : y_a \ne y_b \iff x_a \ne x_b$
:$\forall a,b \in \closedint 1 m : \map {d_{\ell^2} } {y_a, y_b} = \map {d_2} {x_a, x_b}$ | For each $k \in \closedint 1 m$, let:
:$\ds \sum_{i \mathop \in I} \paren{x_k}_i^2$ [[Definition:Convergent Net|converge]] to $r_k \in \R$.
From [[Characterization of Generalized Hilbert Sequence Space]], there exists [[Definition:Enumeration|enumeration]] $J = \set{j_0, j_1, j_2, \ldots}$ of a [[Definition:Countabl... | Generalized Hilbert Sequence Space is Metric Space/Lemma 1 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Lemma_1 | https://proofwiki.org/wiki/Generalized_Hilbert_Sequence_Space_is_Metric_Space/Lemma_1 | [
"Generalized Hilbert Sequence Space is Metric Space"
] | [] | [
"Definition:Convergent Net",
"Characterization of Generalized Hilbert Sequence Space",
"Definition:Enumeration",
"Definition:Countable Set",
"Definition:Subset",
"P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space",
"Characterization of G... |
proofwiki-21237 | Acceleration of Point in Straight Line | Let $\mathbf a$ be the acceleration of a particle $P$ in space.
Let $P$ be moving along a straight line $\LL$ whose positive direction has been established.
Then the motion of $P$ can be defined by:
:$\mathbf a = \dfrac {\d^2 s} {\d t^2} \mathbf i$
where $\mathbf i$ denotes the unit vector in the positive direction of ... | We have {{hypothesis}} that $P$ moves along a straight line $\LL$.
Then the rate of change of displacement perpendicular to $\LL$ is zero.
Let the $\LL$ be embedded in a Cartesian space $\CC$.
From the components of acceleration vector, we have:
:$\mathbf a = \dfrac {\d \mathbf v} {\d t} = \dfrac {\d^2 \mathbf r} {\d t... | Let $\mathbf a$ be the [[Definition:Acceleration|acceleration]] of a [[Definition:Particle|particle]] $P$ in [[Definition:Ordinary Space|space]].
Let $P$ be [[Definition:Motion|moving]] along a [[Definition:Straight Line|straight line]] $\LL$ whose [[Definition:Positive Direction|positive direction]] has been establis... | We have {{hypothesis}} that $P$ [[Definition:Motion|moves]] along a [[Definition:Straight Line|straight line]] $\LL$.
Then the [[Definition:Rate of Change|rate of change]] of [[Definition:Displacement|displacement]] [[Definition:Perpendicular|perpendicular]] to $\LL$ is [[Definition:Zero Vector|zero]].
Let the $\LL$... | Acceleration of Point in Straight Line | https://proofwiki.org/wiki/Acceleration_of_Point_in_Straight_Line | https://proofwiki.org/wiki/Acceleration_of_Point_in_Straight_Line | [
"Acceleration"
] | [
"Definition:Acceleration",
"Definition:Particle",
"Definition:Ordinary Space",
"Definition:Motion",
"Definition:Line/Straight Line",
"Definition:Axis/Positive Direction",
"Definition:Motion",
"Definition:Unit Vector",
"Definition:Axis/Positive Direction"
] | [
"Definition:Motion",
"Definition:Line/Straight Line",
"Definition:Rate of Change",
"Definition:Displacement",
"Definition:Right Angle/Perpendicular",
"Definition:Zero Vector",
"Definition:Cartesian Product/Cartesian Space",
"Component of Vector/Examples/Acceleration",
"Definition:Velocity",
"Defin... |
proofwiki-21238 | Acceleration of Point in Plane in Intrinsic Coordinates | Let $\mathbf a$ be the acceleration of a particle $P$ in space.
Let $P$ be moving in a plane $\PP$.
Then the motion of $P$ can be expressed in intrinsic coordinates as:
:$\mathbf a = \dfrac {\d \map v t} {\d t} \mathbf s + \dfrac {\paren {\map v t}^2} \rho \bspsi$
where:
:$\mathbf s$ denotes the unit vector along the t... | {{ProofWanted|Needs some more background work on intrinsic coordinates for this to make sense<br/> See https://dspace.mit.edu/bitstream/handle/1721.1/60691/16-07-fall-2004/contents/lecture-notes/d4.pdf}} | Let $\mathbf a$ be the [[Definition:Acceleration|acceleration]] of a [[Definition:Particle|particle]] $P$ in [[Definition:Ordinary Space|space]].
Let $P$ be [[Definition:Motion|moving]] in a [[Definition:Plane|plane]] $\PP$.
Then the [[Definition:Motion|motion]] of $P$ can be expressed in [[Definition:Whewell Equati... | {{ProofWanted|Needs some more background work on intrinsic coordinates for this to make sense<br/> See https://dspace.mit.edu/bitstream/handle/1721.1/60691/16-07-fall-2004/contents/lecture-notes/d4.pdf}} | Acceleration of Point in Plane in Intrinsic Coordinates | https://proofwiki.org/wiki/Acceleration_of_Point_in_Plane_in_Intrinsic_Coordinates | https://proofwiki.org/wiki/Acceleration_of_Point_in_Plane_in_Intrinsic_Coordinates | [
"Acceleration"
] | [
"Definition:Acceleration",
"Definition:Particle",
"Definition:Ordinary Space",
"Definition:Motion",
"Definition:Plane Surface",
"Definition:Motion",
"Definition:Intrinsic Equation/Whewell Equation",
"Definition:Unit Vector",
"Definition:Tangential Direction",
"Definition:Unit Vector",
"Definitio... | [] |
proofwiki-21239 | Tangential Component of Acceleration of Uniform Circular Motion is Zero | Let $P$ be a particle moving in space in uniform circular motion.
Let $\mathbf a$ denote the acceleration of $P$.
Then the tangential component of $\mathbf a$ is zero. | From Acceleration of Point in Plane in Intrinsic Coordinates, the motion of $P$ can be expressed in intrinsic coordinates as:
:$\mathbf a = \dfrac {\d \map v t} {\d t} \mathbf s + \dfrac {\paren {\map v t}^2} \rho \bspsi$
where:
:$\mathbf s$ denotes the unit vector along the tangential direction of $P$
:$\bspsi$ denote... | Let $P$ be a [[Definition:Particle|particle]] moving in [[Definition:Ordinary Space|space]] in [[Definition:Uniform Circular Motion|uniform circular motion]].
Let $\mathbf a$ denote the [[Definition:Acceleration|acceleration]] of $P$.
Then the [[Definition:Tangential Component of Acceleration|tangential component]] o... | From [[Acceleration of Point in Plane in Intrinsic Coordinates]], the [[Definition:Motion|motion]] of $P$ can be expressed in [[Definition:Whewell Equation|intrinsic coordinates]] as:
:$\mathbf a = \dfrac {\d \map v t} {\d t} \mathbf s + \dfrac {\paren {\map v t}^2} \rho \bspsi$
where:
:$\mathbf s$ denotes the [[Defin... | Tangential Component of Acceleration of Uniform Circular Motion is Zero | https://proofwiki.org/wiki/Tangential_Component_of_Acceleration_of_Uniform_Circular_Motion_is_Zero | https://proofwiki.org/wiki/Tangential_Component_of_Acceleration_of_Uniform_Circular_Motion_is_Zero | [
"Acceleration"
] | [
"Definition:Particle",
"Definition:Ordinary Space",
"Definition:Uniform Circular Motion",
"Definition:Acceleration",
"Definition:Tangential Component of Acceleration",
"Definition:Zero (Number)"
] | [
"Acceleration of Point in Plane in Intrinsic Coordinates",
"Definition:Motion",
"Definition:Intrinsic Equation/Whewell Equation",
"Definition:Unit Vector",
"Definition:Tangential Direction",
"Definition:Unit Vector",
"Definition:Center of Curvature",
"Definition:Motion",
"Definition:Speed",
"Defin... |
proofwiki-21240 | Countably Infinite Set has Enumeration | Let $S$ be a countably inifnite set.
Then there exists a countably infinite enumeration $\set{s_1, s_2, s_3, \ldots}$ of $S$. | By definition of countably inifnite set:
:there exists a bijection $f:S \to \N$
From Inverse of Bijection is Bijection:
:$f^{-1} : \N \to S$ is a bijection
Let $s = f^{-1}$.
It follows that $s : \N \to S$ is a countably infinite enumeration $\set{s_1, s_2, s_3, \ldots}$ by definition.
{{qed}}
s8g9y1tnv04add54jfjs9o548e... | Let $S$ be a [[Definition:Countably Infinite Set|countably inifnite set]].
Then there exists a [[Definition:Countably Infinite Enumeration|countably infinite enumeration]] $\set{s_1, s_2, s_3, \ldots}$ of $S$. | By definition of [[Definition:Countably Infinite Set|countably inifnite set]]:
:there exists a [[Definition:Bijection|bijection]] $f:S \to \N$
From [[Inverse of Bijection is Bijection]]:
:$f^{-1} : \N \to S$ is a [[Definition:Bijection|bijection]]
Let $s = f^{-1}$.
It follows that $s : \N \to S$ is a [[Definition:Co... | Countably Infinite Set has Enumeration | https://proofwiki.org/wiki/Countably_Infinite_Set_has_Enumeration | https://proofwiki.org/wiki/Countably_Infinite_Set_has_Enumeration | [] | [
"Definition:Countably Infinite/Set",
"Definition:Enumeration/Countably Infinite"
] | [
"Definition:Countably Infinite/Set",
"Definition:Bijection",
"Inverse of Bijection is Bijection",
"Definition:Bijection",
"Definition:Enumeration/Countably Infinite"
] |
proofwiki-21241 | Generalized Sum of Constant Zero Converges to Zero | Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\family{g_i}_{i \in I}$ be the indexed family of $G$ defined by:
:$\forall i \in I : g_i = 0_G$
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $0_G$ | Let $\FF$ denote the set of finite subsets of $I$.
From Power of Identity is Identity:
:$\forall F \in \FF : \ds \map \phi F = \sum_{i \mathop \in F} g_i = 0_G$
Hence the net $\ds \sum \set {g_i: i \in I}$ is a constant mapping.
From Constant Net is Convergent:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$ con... | Let $G$ be a [[Definition:Commutative Semigroup|commutative]] [[Definition:Topological Semigroup|topological semigroup]] with [[Definition:Identity Element|identity]] $0_G$.
Let $\family{g_i}_{i \in I}$ be the [[Definition:Indexed Family|indexed family]] of $G$ defined by:
:$\forall i \in I : g_i = 0_G$
Then:
:the ... | Let $\FF$ denote the [[Definition:Set|set]] of [[Definition:Finite Set|finite]] [[Definition:Subset|subsets]] of $I$.
From [[Power of Identity is Identity]]:
:$\forall F \in \FF : \ds \map \phi F = \sum_{i \mathop \in F} g_i = 0_G$
Hence the [[Definition:Net (Preordered Set)|net]] $\ds \sum \set {g_i: i \in I}$ is ... | Generalized Sum of Constant Zero Converges to Zero | https://proofwiki.org/wiki/Generalized_Sum_of_Constant_Zero_Converges_to_Zero | https://proofwiki.org/wiki/Generalized_Sum_of_Constant_Zero_Converges_to_Zero | [
"Generalized Sums"
] | [
"Definition:Commutative Semigroup",
"Definition:Topological Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Indexing Set/Family",
"Definition:Generalized Sum",
"Definition:Convergent Net"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Subset",
"Power of Identity is Identity",
"Definition:Net (Preordered Set)",
"Definition:Constant Mapping",
"Constant Net is Convergent",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Category:Generalized Sums"
] |
proofwiki-21242 | Generalized Sum Restricted to Non-zero Summands/Corollary | Let $K \subseteq I : \set{i \in I : g_i \ne 0_G} \subseteq K$
Let $h \in G$.
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$
{{iff}}:
:the generalized sum $\ds \sum_{k \mathop \in K} g_k$ converges to $h$ | Let $J = \set{i \in I : g_i \ne 0_G}$.
We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop \in I} g_i
| o = \to
| r = h
}}
{{eqn | ll = \leadstoandfrom
| l = \sum_{j \mathop \in J} g_j
| o = \to
| r = h
| c = Generalized Sum Restricted to Non-zero Summands
}}
{{eqn | ll = \leadstoandf... | Let $K \subseteq I : \set{i \in I : g_i \ne 0_G} \subseteq K$
Let $h \in G$.
Then:
:the [[Definition:Generalized Sum|generalized sum]] $\ds \sum_{i \mathop \in I} g_i$ [[Definition:Convergent Net|converges]] to $h$
{{iff}}:
:the [[Definition:Generalized Sum|generalized sum]] $\ds \sum_{k \mathop \in K} g_k$ [[Defini... | Let $J = \set{i \in I : g_i \ne 0_G}$.
We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop \in I} g_i
| o = \to
| r = h
}}
{{eqn | ll = \leadstoandfrom
| l = \sum_{j \mathop \in J} g_j
| o = \to
| r = h
| c = [[Generalized Sum Restricted to Non-zero Summands]]
}}
{{eqn | ll = \leads... | Generalized Sum Restricted to Non-zero Summands/Corollary | https://proofwiki.org/wiki/Generalized_Sum_Restricted_to_Non-zero_Summands/Corollary | https://proofwiki.org/wiki/Generalized_Sum_Restricted_to_Non-zero_Summands/Corollary | [
"Generalized Sum Restricted to Non-zero Summands"
] | [
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:Generalized Sum",
"Definition:Convergent Net"
] | [
"Generalized Sum Restricted to Non-zero Summands",
"Generalized Sum Restricted to Non-zero Summands",
"Category:Generalized Sum Restricted to Non-zero Summands"
] |
proofwiki-21243 | Characterization of Generalized Hilbert Sequence Space | Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $H^\alpha = \struct{A, d_2}$ be the generalized Hilbert sequence space of weight $\alpha$ where:
:$A$ denotes the set of all real-valued functions $x : I \to \R$ such that:
::$(1)\quad \set{i \in I: x_i \ne 0}$ is counta... | === Necessary Condition ===
Let $x_1, x_2, \ldots, x_m \in A$.
{{:Characterization of Generalized Hilbert Sequence Space/Necessary Condition}}{{qed|lemma}} | Let $\alpha$ be an [[Definition:Infinite Cardinal|infinite cardinal number]].
Let $I$ be an [[Definition:Indexed Set|indexed set]] of [[Definition:Cardinality|cardinality]] $\alpha$.
Let $H^\alpha = \struct{A, d_2}$ be the [[Definition:Generalized Hilbert Sequence Space|generalized Hilbert sequence space of weight $... | === [[Characterization of Generalized Hilbert Sequence Space/Necessary Condition|Necessary Condition]] ===
Let $x_1, x_2, \ldots, x_m \in A$.
{{:Characterization of Generalized Hilbert Sequence Space/Necessary Condition}}{{qed|lemma}} | Characterization of Generalized Hilbert Sequence Space | https://proofwiki.org/wiki/Characterization_of_Generalized_Hilbert_Sequence_Space | https://proofwiki.org/wiki/Characterization_of_Generalized_Hilbert_Sequence_Space | [
"Generalized Hilbert Sequence Spaces",
"Characterization of Generalized Hilbert Sequence Space"
] | [
"Definition:Infinite Cardinal",
"Definition:Indexing Set/Indexed Set",
"Definition:Cardinality",
"Definition:Generalized Hilbert Sequence Space",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Countable Set",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:... | [
"Characterization of Generalized Hilbert Sequence Space/Necessary Condition"
] |
proofwiki-21244 | Angle is not Invariant under Affine Transformation | The angle between two (straight) lines does not necessarily stay the same under an affine transformation. | {{tidy}}
{{Proofread}}
In order to prove this, we can supply a counter example. Consider the transformation:
\[ \LL : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \\ \LL : \paren {x, y} \mapsto \paren{x, x + y} \]
We can check that $\LL$ is an affine transformation. Indeed, if we denote $A = \paren{x_A, y_A}$ and $B = \paren{... | The [[Definition:Angle|angle]] between two [[Definition:Straight Line|(straight) lines]] does not necessarily stay the same under an [[Definition:Affine Transformation|affine transformation]]. | {{tidy}}
{{Proofread}}
In order to prove this, we can supply a counter example. Consider the transformation:
\[ \LL : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \\ \LL : \paren {x, y} \mapsto \paren{x, x + y} \]
We can check that $\LL$ is an affine transformation. Indeed, if we denote $A = \paren{x_A, y_A}$ and $B = \pare... | Angle is not Invariant under Affine Transformation | https://proofwiki.org/wiki/Angle_is_not_Invariant_under_Affine_Transformation | https://proofwiki.org/wiki/Angle_is_not_Invariant_under_Affine_Transformation | [
"Angles",
"Affine Transformations"
] | [
"Definition:Angle",
"Definition:Line/Straight Line",
"Definition:Affine Transformation"
] | [
"Definition:Euclidean Plane",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis"
] |
proofwiki-21245 | Term of Computable Real Sequence is Computable | Let $\sequence {x_i}$ be a computable real sequence.
Then, for each $i \in \N$:
:$x_i$ is a computable real number. | By definition of computable real sequence, there exists a total recursive function $f : \N^2 \to \N$ such that:
:For every $m,n \in \N$, $\map f {m, n}$ codes an integer $k$ such that:
::$\dfrac {k - 1} {n + 1} < x_m < \dfrac {k + 1} {n + 1}$
Let $g : \N \to \N$ be defined as:
:$\map g n = \map f {i, n}$
By:
* Constant... | Let $\sequence {x_i}$ be a [[Definition:Computable Real Sequence|computable real sequence]].
Then, for each $i \in \N$:
:$x_i$ is a [[Definition:Computable Real Number|computable real number]]. | By definition of [[Definition:Computable Real Sequence|computable real sequence]], there exists a [[Definition:Total Recursive Function|total recursive function]] $f : \N^2 \to \N$ such that:
:For every $m,n \in \N$, $\map f {m, n}$ [[Definition:Code Number for Integer|codes an integer]] $k$ such that:
::$\dfrac {k - 1... | Term of Computable Real Sequence is Computable | https://proofwiki.org/wiki/Term_of_Computable_Real_Sequence_is_Computable | https://proofwiki.org/wiki/Term_of_Computable_Real_Sequence_is_Computable | [
"Computability Theory"
] | [
"Definition:Computable Real Sequence",
"Definition:Computable Real Number"
] | [
"Definition:Computable Real Sequence",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Constant Function is Primitive Recursive",
"Primitive Recursive Function is Total Recursive Function",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Defi... |
proofwiki-21246 | Composition of Sequentially Computable Real Functions is Sequentially Computable | Let $f,g : \R \to \R$ be sequentially computable real functions.
Let $h : \R \to \R$ be defined as:
:$\map h x = \map f {\map g x}$
Then $h$ is sequentially computable. | Let $\sequence {x_n}$ be a computable real sequence.
As $g$ is sequentially computable:
:$\sequence {\map g {x_n}}$
is computable.
As $f$ is sequentially computable:
:$\sequence {\map f {\map g {x_n}}}$
is computable.
But:
:$\map f {\map g {x_n}} = \map h {x_n}$
Therefore:
:$\sequence {\map h {x_n}}$
is computable.
As ... | Let $f,g : \R \to \R$ be [[Definition:Sequentially Computable Real Function|sequentially computable real functions]].
Let $h : \R \to \R$ be defined as:
:$\map h x = \map f {\map g x}$
Then $h$ is [[Definition:Sequentially Computable Real Function|sequentially computable]]. | Let $\sequence {x_n}$ be a [[Definition:Computable Real Sequence|computable real sequence]].
As $g$ is [[Definition:Sequentially Computable Real Function|sequentially computable]]:
:$\sequence {\map g {x_n}}$
is [[Definition:Computable Real Sequence|computable]].
As $f$ is [[Definition:Sequentially Computable Real Fu... | Composition of Sequentially Computable Real Functions is Sequentially Computable | https://proofwiki.org/wiki/Composition_of_Sequentially_Computable_Real_Functions_is_Sequentially_Computable | https://proofwiki.org/wiki/Composition_of_Sequentially_Computable_Real_Functions_is_Sequentially_Computable | [
"Computability Theory"
] | [
"Definition:Sequentially Computable Real Function",
"Definition:Sequentially Computable Real Function"
] | [
"Definition:Computable Real Sequence",
"Definition:Sequentially Computable Real Function",
"Definition:Computable Real Sequence",
"Definition:Sequentially Computable Real Function",
"Definition:Computable Real Sequence",
"Definition:Computable Real Sequence",
"Definition:Sequentially Computable Real Fun... |
proofwiki-21247 | Composition of Computably Uniformly Continuous Real Functions is Computably Uniformly Continuous | Let $f,g : \R \to \R$ be computably uniformly continuous real functions.
Let $h : \R \to \R$ be defined as:
:$\map h x = \map f {\map g x}$
Then $h$ is computably uniformly continuous. | By definition of computably uniformly continuous, there exist total recursive functions $d_f, d_g : \N \to \N$ such that:
:For all $n \in \N$ and $x,y \in \R$ such that:
::$\size {x - y} < \dfrac 1 {\map {d_f} n + 1}$
:it holds that:
::$\size {\map f x - \map f y} < \dfrac 1 {n + 1}$
and:
:For all $n \in \N$ and $x,y \... | Let $f,g : \R \to \R$ be [[Definition:Computably Uniformly Continuous Real Function|computably uniformly continuous real functions]].
Let $h : \R \to \R$ be defined as:
:$\map h x = \map f {\map g x}$
Then $h$ is [[Definition:Computably Uniformly Continuous Real Function|computably uniformly continuous]]. | By definition of [[Definition:Computably Uniformly Continuous Real Function|computably uniformly continuous]], there exist [[Definition:Total Recursive Function|total recursive functions]] $d_f, d_g : \N \to \N$ such that:
:For all $n \in \N$ and $x,y \in \R$ such that:
::$\size {x - y} < \dfrac 1 {\map {d_f} n + 1}$
:... | Composition of Computably Uniformly Continuous Real Functions is Computably Uniformly Continuous | https://proofwiki.org/wiki/Composition_of_Computably_Uniformly_Continuous_Real_Functions_is_Computably_Uniformly_Continuous | https://proofwiki.org/wiki/Composition_of_Computably_Uniformly_Continuous_Real_Functions_is_Computably_Uniformly_Continuous | [
"Computability Theory"
] | [
"Definition:Computably Uniformly Continuous Real Function",
"Definition:Computably Uniformly Continuous Real Function"
] | [
"Definition:Computably Uniformly Continuous Real Function",
"Definition:Total Recursive Function",
"Definition:Substitution (Mathematical Logic)",
"Definition:Recursive/Function",
"Definition:Recursive",
"Definition:Total Recursive Function",
"Definition:Total Recursive Function",
"Definition:Computab... |
proofwiki-21248 | Composition of Computable Real Functions is Computable | Let $f,g : \R \to \R$ be computable real functions.
Let $h : \R \to \R$ be defined as:
:$\map h x = \map f {\map g x}$
Then $h$ is computable. | Follows immediately from:
* Composition of Sequentially Computable Real Functions is Sequentially Computable
* Composition of Computably Uniformly Continuous Real Functions is Computably Uniformly Continuous
{{qed}}
Category:Computability Theory
ofoolc2elb33ytmx1gwfgj53r7qe4hk | Let $f,g : \R \to \R$ be [[Definition:Computable Real Function|computable real functions]].
Let $h : \R \to \R$ be defined as:
:$\map h x = \map f {\map g x}$
Then $h$ is [[Definition:Computable Real Function|computable]]. | Follows immediately from:
* [[Composition of Sequentially Computable Real Functions is Sequentially Computable]]
* [[Composition of Computably Uniformly Continuous Real Functions is Computably Uniformly Continuous]]
{{qed}}
[[Category:Computability Theory]]
ofoolc2elb33ytmx1gwfgj53r7qe4hk | Composition of Computable Real Functions is Computable | https://proofwiki.org/wiki/Composition_of_Computable_Real_Functions_is_Computable | https://proofwiki.org/wiki/Composition_of_Computable_Real_Functions_is_Computable | [
"Computability Theory"
] | [
"Definition:Computable Real Function",
"Definition:Computable Real Function"
] | [
"Composition of Sequentially Computable Real Functions is Sequentially Computable",
"Composition of Computably Uniformly Continuous Real Functions is Computably Uniformly Continuous",
"Category:Computability Theory"
] |
proofwiki-21249 | Constant Sequence of Computable Real Number is Computable | Let $a \in \R$ be a computable real number.
Let $\sequence {x_n}$ be defined as:
:$x_n = a$
Then, $\sequence {x_n}$ is a computable real sequence. | By definition of computable real number, there exists a total recursive function $f : \N \to \N$ such that:
:For every $n \in \N$, $\map f n$ codes an integer $k$ such that:
::$\dfrac {k - 1} {n + 1} < a < \dfrac {k + 1} {n + 1}$
Let $g : \N^2 \to \N$ be defined as:
:$\map g {m, n} = \map f n$
As $f$ is total recursive... | Let $a \in \R$ be a [[Definition:Computable Real Number|computable real number]].
Let $\sequence {x_n}$ be defined as:
:$x_n = a$
Then, $\sequence {x_n}$ is a [[Definition:Computable Real Sequence|computable real sequence]]. | By definition of [[Definition:Computable Real Number|computable real number]], there exists a [[Definition:Total Recursive Function|total recursive function]] $f : \N \to \N$ such that:
:For every $n \in \N$, $\map f n$ [[Definition:Code Number for Integer|codes an integer]] $k$ such that:
::$\dfrac {k - 1} {n + 1} < a... | Constant Sequence of Computable Real Number is Computable | https://proofwiki.org/wiki/Constant_Sequence_of_Computable_Real_Number_is_Computable | https://proofwiki.org/wiki/Constant_Sequence_of_Computable_Real_Number_is_Computable | [
"Computability Theory"
] | [
"Definition:Computable Real Number",
"Definition:Computable Real Sequence"
] | [
"Definition:Computable Real Number",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Definition:Total Recursive Function",
"Definition:Total Recursive Function",
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Computable Real Sequen... |
proofwiki-21250 | Image of Computable Real Number over Sequentially Computable Real Function is Computable | Let $f : \R \to \R$ be a sequentially computable real function.
Let $a \in \R$ be a computable real number.
Then, $\map f a$ is a computable real number. | Let $\sequence {x_n}$ be defined as:
:$x_n = a$
By Constant Sequence of Computable Real Number is Computable:
:$\sequence {x_n}$ is a computable real sequence.
As $f$ is sequentially computable:
:$\sequence {\map f {x_n}}$ is computable.
By Term of Computable Real Sequence is Computable:
:$\map f {x_0}$ is computable.
... | Let $f : \R \to \R$ be a [[Definition:Sequentially Computable Real Function|sequentially computable real function]].
Let $a \in \R$ be a [[Definition:Computable Real Number|computable real number]].
Then, $\map f a$ is a [[Definition:Computable Real Number|computable real number]]. | Let $\sequence {x_n}$ be defined as:
:$x_n = a$
By [[Constant Sequence of Computable Real Number is Computable]]:
:$\sequence {x_n}$ is a [[Definition:Computable Real Sequence|computable real sequence]].
As $f$ is [[Definition:Sequentially Computable Real Function|sequentially computable]]:
:$\sequence {\map f {x_n}}... | Image of Computable Real Number over Sequentially Computable Real Function is Computable | https://proofwiki.org/wiki/Image_of_Computable_Real_Number_over_Sequentially_Computable_Real_Function_is_Computable | https://proofwiki.org/wiki/Image_of_Computable_Real_Number_over_Sequentially_Computable_Real_Function_is_Computable | [
"Computability Theory"
] | [
"Definition:Sequentially Computable Real Function",
"Definition:Computable Real Number",
"Definition:Computable Real Number"
] | [
"Constant Sequence of Computable Real Number is Computable",
"Definition:Computable Real Sequence",
"Definition:Sequentially Computable Real Function",
"Definition:Computable Real Sequence",
"Term of Computable Real Sequence is Computable",
"Definition:Computable Real Number",
"Definition:Computable Rea... |
proofwiki-21251 | Identity Function is Sequentially Computable Real Function | Let $I_\R : \R \to \R$ denote the identity function on $\R$.
Then $I_\R$ is a sequentially computable real function. | Let $\sequence {x_n}$ be a computable real sequence.
By definition of identity function:
:$\map {I_\R} x = x$
Therefore:
:$\sequence {\map {I_\R} {x_n}} = \sequence {x_n}$
and is thus computable.
As $\sequence {x_n}$ was arbitrary, it follows that $I_\R$ is sequentially computable by definition.
{{qed}}
Category:Comput... | Let $I_\R : \R \to \R$ denote the [[Definition:Identity Function|identity function]] on $\R$.
Then $I_\R$ is a [[Definition:Sequentially Computable Real Function|sequentially computable real function]]. | Let $\sequence {x_n}$ be a [[Definition:Computable Real Sequence|computable real sequence]].
By definition of [[Definition:Identity Function|identity function]]:
:$\map {I_\R} x = x$
Therefore:
:$\sequence {\map {I_\R} {x_n}} = \sequence {x_n}$
and is thus [[Definition:Computable Real Sequence|computable]].
As $\se... | Identity Function is Sequentially Computable Real Function | https://proofwiki.org/wiki/Identity_Function_is_Sequentially_Computable_Real_Function | https://proofwiki.org/wiki/Identity_Function_is_Sequentially_Computable_Real_Function | [
"Computability Theory"
] | [
"Definition:Identity Mapping",
"Definition:Sequentially Computable Real Function"
] | [
"Definition:Computable Real Sequence",
"Definition:Identity Mapping",
"Definition:Computable Real Sequence",
"Definition:Sequentially Computable Real Function",
"Category:Computability Theory"
] |
proofwiki-21252 | Identity Function is Computably Uniformly Continuous Real Function | Let $I_\R : \R \to \R$ denote the identity function on $\R$.
Then, $I_\R$ is a computably uniformly continuous real function. | Let $d : \N \to \N$ be defined as:
:$\map d n = n$
As $d$ is precisely the projection function $\pr_1^1$, it is primitive recursive.
By Primitive Recursive Function is Total Recursive Function, $d$ is a total recursive function.
Let $n \in \N$ and $x, y \in \R$ satisfy:
:$\size {x - y} < \dfrac 1 {\map d n + 1}$
By def... | Let $I_\R : \R \to \R$ denote the [[Definition:Identity Function|identity function]] on $\R$.
Then, $I_\R$ is a [[Definition:Computably Uniformly Continuous Real Function|computably uniformly continuous real function]]. | Let $d : \N \to \N$ be defined as:
:$\map d n = n$
As $d$ is precisely the [[Definition:Projection Function|projection function]] $\pr_1^1$, it is [[Definition:Primitive Recursive Function|primitive recursive]].
By [[Primitive Recursive Function is Total Recursive Function]], $d$ is a [[Definition:Total Recursive Fun... | Identity Function is Computably Uniformly Continuous Real Function | https://proofwiki.org/wiki/Identity_Function_is_Computably_Uniformly_Continuous_Real_Function | https://proofwiki.org/wiki/Identity_Function_is_Computably_Uniformly_Continuous_Real_Function | [
"Computability Theory"
] | [
"Definition:Identity Mapping",
"Definition:Computably Uniformly Continuous Real Function"
] | [
"Definition:Basic Primitive Recursive Function/Projection Function",
"Definition:Primitive Recursive/Function",
"Primitive Recursive Function is Total Recursive Function",
"Definition:Total Recursive Function",
"Definition:Identity Mapping",
"Definition:Computably Uniformly Continuous Real Function",
"C... |
proofwiki-21253 | Identity Function is Computable Real Function | Let $I_\R : \R \to \R$ denote the identity function on $\R$.
Then $I_\R$ is a computable real function. | Follows immediately from:
* Identity Function is Sequentially Computable Real Function
* Identity Function is Computably Uniformly Continuous Real Function
{{qed}}
Category:Computability Theory
fqylmaqzzlnf5gfo39vyysdlhkjki9y | Let $I_\R : \R \to \R$ denote the [[Definition:Identity Function|identity function]] on $\R$.
Then $I_\R$ is a [[Definition:Computable Real Function|computable real function]]. | Follows immediately from:
* [[Identity Function is Sequentially Computable Real Function]]
* [[Identity Function is Computably Uniformly Continuous Real Function]]
{{qed}}
[[Category:Computability Theory]]
fqylmaqzzlnf5gfo39vyysdlhkjki9y | Identity Function is Computable Real Function | https://proofwiki.org/wiki/Identity_Function_is_Computable_Real_Function | https://proofwiki.org/wiki/Identity_Function_is_Computable_Real_Function | [
"Computability Theory",
"Computability Theory"
] | [
"Definition:Identity Mapping",
"Definition:Computable Real Function"
] | [
"Identity Function is Sequentially Computable Real Function",
"Identity Function is Computably Uniformly Continuous Real Function",
"Category:Computability Theory"
] |
proofwiki-21254 | Absolute Value of Integer is Primitive Recursive | For every $n \in \N$, let $n$ code the integer $k_n$.
Let $a : \N \to \N$ be defined as:
:$\map a n = \size {k_n}$
Then, $a$ is primitive recursive. | Let $a : \N \to \N$ be defined as:
:$\map a n = \map {\operatorname{quot}} {n + 1, 2}$
By:
* Constant Function is Primitive Recursive
* Quotient is Primitive Recursive
* Successor Function is Primitive Recursive
it follows that $a$ is primitive recursive.
By definition of code number for integer, either:
:$n = 2 k_n - ... | For every $n \in \N$, let $n$ [[Definition:Code Number for Integer|code the integer]] $k_n$.
Let $a : \N \to \N$ be defined as:
:$\map a n = \size {k_n}$
Then, $a$ is [[Definition:Primitive Recursive Function|primitive recursive]]. | Let $a : \N \to \N$ be defined as:
:$\map a n = \map {\operatorname{quot}} {n + 1, 2}$
By:
* [[Constant Function is Primitive Recursive]]
* [[Quotient is Primitive Recursive]]
* [[Definition:Basic Primitive Recursive Function/Successor Function|Successor Function is Primitive Recursive]]
it follows that $a$ is [[Defin... | Absolute Value of Integer is Primitive Recursive | https://proofwiki.org/wiki/Absolute_Value_of_Integer_is_Primitive_Recursive | https://proofwiki.org/wiki/Absolute_Value_of_Integer_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Function"
] | [
"Constant Function is Primitive Recursive",
"Quotient is Primitive Recursive",
"Definition:Basic Primitive Recursive Function/Successor Function",
"Definition:Primitive Recursive/Function",
"Definition:Code Number for Integer",
"Category:Primitive Recursive Functions"
] |
proofwiki-21255 | Set of Strictly Positive Integers is Primitive Recursive | Let $P \subseteq \N$ be the set of all $n \in \N$ such that:
:$n$ codes an integer $k$ such that $k > 0$.
Then $P$ is a primitive recursive set. | Let $p : \N \to \N$ be defined as:
:$\map p n = \map {\operatorname{rem}} {n, 2}$
By:
* Constant Function is Primitive Recursive
* Remainder is Primitive Recursive
Suppose $n$ codes an integer $k$ such that $k > 0$.
Then:
:$n = 2 k - 1$
Therefore:
{{begin-eqn}}
{{eqn | l = \map p n
| r = \map {\operatorname{rem} ... | Let $P \subseteq \N$ be the [[Definition:Set|set]] of all $n \in \N$ such that:
:$n$ [[Definition:Code Number for Integer|codes an integer]] $k$ such that $k > 0$.
Then $P$ is a [[Definition:Primitive Recursive Set|primitive recursive set]]. | Let $p : \N \to \N$ be defined as:
:$\map p n = \map {\operatorname{rem}} {n, 2}$
By:
* [[Constant Function is Primitive Recursive]]
* [[Remainder is Primitive Recursive]]
Suppose $n$ [[Definition:Code Number for Integer|codes an integer]] $k$ such that $k > 0$.
Then:
:$n = 2 k - 1$
Therefore:
{{begin-eqn}}
{{eqn |... | Set of Strictly Positive Integers is Primitive Recursive | https://proofwiki.org/wiki/Set_of_Strictly_Positive_Integers_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_of_Strictly_Positive_Integers_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Set",
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Set"
] | [
"Constant Function is Primitive Recursive",
"Remainder is Primitive Recursive",
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Primitive Recursive/Set",
"Category:Primitive Recursive Functions"
] |
proofwiki-21256 | Code Number for Non-Negative Integer is Primitive Recursive | Let $c : \N \to \N$ be defined as:
:$\map c n = m$
where $m$ is the code number for the integer $n : \Z$.
Then $c$ is a primitive recursive function. | Let $c : \N \to \N$ be defined as:
:$\map c n = \begin{cases}
\map {\operatorname{pred}} {n + n} & n > 0 \\
0 & n = 0
\end{cases}$
That $c$ is primitive recursive follows from:
* Definition by Cases is Primitive Recursive
* Predecessor Function is Primitive Recursive
* Addition is Primitive Recursive
* Ordering Relatio... | Let $c : \N \to \N$ be defined as:
:$\map c n = m$
where $m$ is the [[Definition:Code Number for Integer|code number for the integer]] $n : \Z$.
Then $c$ is a [[Definition:Primitive Recursive Function|primitive recursive function]]. | Let $c : \N \to \N$ be defined as:
:$\map c n = \begin{cases}
\map {\operatorname{pred}} {n + n} & n > 0 \\
0 & n = 0
\end{cases}$
That $c$ is [[Definition:Primitive Recursive Function|primitive recursive]] follows from:
* [[Definition by Cases is Primitive Recursive]]
* [[Predecessor Function is Primitive Recursive]]... | Code Number for Non-Negative Integer is Primitive Recursive | https://proofwiki.org/wiki/Code_Number_for_Non-Negative_Integer_is_Primitive_Recursive | https://proofwiki.org/wiki/Code_Number_for_Non-Negative_Integer_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursive/Function",
"Definition by Cases is Primitive Recursive",
"Predecessor Function is Primitive Recursive",
"Addition is Primitive Recursive",
"Ordering Relations are Primitive Recursive",
"Equality Relation is Primitive Recursive",
"Category:Primitive Recursive Functions"
] |
proofwiki-21257 | Code Number for Non-Positive Integer is Primitive Recursive | Let $c : \N \to \N$ be defined as:
:$\map c n = m$
where $m$ is the code number for the integer $-n : \Z$.
Then $c$ is a primitive recursive function. | Let $c : \N \to \N$ be defined as:
:$\map c n = n + n$
which is primitive recursive by:
* Addition is Primitive Recursive
For every $n \in \N$, we have:
:$-n \le 0$
Thus:
:$m = -2 \paren {-n} = 2 n$
Therefore:
:$\map c n = m$
{{qed}}
Category:Primitive Recursive Functions
osaywp6nxnog01v2qyhmzy9ro60wlie | Let $c : \N \to \N$ be defined as:
:$\map c n = m$
where $m$ is the [[Definition:Code Number for Integer|code number for the integer]] $-n : \Z$.
Then $c$ is a [[Definition:Primitive Recursive Function|primitive recursive function]]. | Let $c : \N \to \N$ be defined as:
:$\map c n = n + n$
which is [[Definition:Primitive Recursive Function|primitive recursive]] by:
* [[Addition is Primitive Recursive]]
For every $n \in \N$, we have:
:$-n \le 0$
Thus:
:$m = -2 \paren {-n} = 2 n$
Therefore:
:$\map c n = m$
{{qed}}
[[Category:Primitive Recursive Fu... | Code Number for Non-Positive Integer is Primitive Recursive | https://proofwiki.org/wiki/Code_Number_for_Non-Positive_Integer_is_Primitive_Recursive | https://proofwiki.org/wiki/Code_Number_for_Non-Positive_Integer_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursive/Function",
"Addition is Primitive Recursive",
"Category:Primitive Recursive Functions"
] |
proofwiki-21258 | Set of Strictly Negative Integers is Primitive Recursive | Let $N \subseteq \N$ be the set of all $n \in \N$ such that:
:$n$ codes an integer $k$ such that $k < 0$.
Then $N$ is a primitive recursive set. | By Set of Strictly Positive Integers is Primitive Recursive:
:$P = \set {n \in \N : k > 0}$
is primitive recursive.
By Complement of Primitive Recursive Set:
:$P^c = \set {n \in \N : k \le 0}$
is primitive recursive.
It is clear that:
:$N = P^c \setminus \set {n \in \N : k = 0}$
By Set Difference as Intersection with R... | Let $N \subseteq \N$ be the [[Definition:Set|set]] of all $n \in \N$ such that:
:$n$ [[Definition:Code Number for Integer|codes an integer]] $k$ such that $k < 0$.
Then $N$ is a [[Definition:Primitive Recursive Set|primitive recursive set]]. | By [[Set of Strictly Positive Integers is Primitive Recursive]]:
:$P = \set {n \in \N : k > 0}$
is [[Definition:Primitive Recursive Set|primitive recursive]].
By [[Complement of Primitive Recursive Set]]:
:$P^c = \set {n \in \N : k \le 0}$
is [[Definition:Primitive Recursive Set|primitive recursive]].
It is clear tha... | Set of Strictly Negative Integers is Primitive Recursive | https://proofwiki.org/wiki/Set_of_Strictly_Negative_Integers_is_Primitive_Recursive | https://proofwiki.org/wiki/Set_of_Strictly_Negative_Integers_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Set",
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Set"
] | [
"Set of Strictly Positive Integers is Primitive Recursive",
"Definition:Primitive Recursive/Set",
"Complement of Primitive Recursive Set",
"Definition:Primitive Recursive/Set",
"Set Difference as Intersection with Relative Complement",
"Intersection of Primitive Recursive Sets",
"Complement of Primitive... |
proofwiki-21259 | Multiplication of Integers is Primitive Recursive | Let $t : \N^2 \to \N$ be defined as:
:$\map t {m, n} = p$
where:
:$m$ is the code number for the integer $k$
:$n$ is the code number for the integer $\ell$
:$p$ is the code number for the integer $k \times \ell$
Then $t$ is a primitive recursive function. | We have:
:$\size {k \times \ell} = \size k \times \size \ell$
By:
* Multiplication is Primitive Recursive
* Absolute Value of Integer is Primitive Recursive
we have that:
:$\map a {m, n} = \size {k \times \ell}$
is primitive recursive.
Additionally, we have that:
:$k \times \ell > 0$
{{iff}} either:
:$k > 0 \land \ell ... | Let $t : \N^2 \to \N$ be defined as:
:$\map t {m, n} = p$
where:
:$m$ is the [[Definition:Code Number for Integer|code number for the integer]] $k$
:$n$ is the [[Definition:Code Number for Integer|code number for the integer]] $\ell$
:$p$ is the [[Definition:Code Number for Integer|code number for the integer]] $k \tim... | We have:
:$\size {k \times \ell} = \size k \times \size \ell$
By:
* [[Multiplication is Primitive Recursive]]
* [[Absolute Value of Integer is Primitive Recursive]]
we have that:
:$\map a {m, n} = \size {k \times \ell}$
is [[Definition:Primitive Recursive Function|primitive recursive]].
Additionally, we have that:
:... | Multiplication of Integers is Primitive Recursive | https://proofwiki.org/wiki/Multiplication_of_Integers_is_Primitive_Recursive | https://proofwiki.org/wiki/Multiplication_of_Integers_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Function"
] | [
"Multiplication is Primitive Recursive",
"Absolute Value of Integer is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Set of Strictly Positive Integers is Primitive Recursive",
"Set of Strictly Negative Integers is Primitive Recursive",
"Set Operations on Primitive Recursive Relations",... |
proofwiki-21260 | Signum Function on Integers is Primitive Recursive | Let $\sgn_\Z : \Z \to \set {-1, 0, 1}$ denote the signum function on the integers.
Let $s : \N \to \N$ be defined as:
:$\map s n = m$
where:
:$n$ codes the integer $k$
:$m$ codes the integer $\map {\sgn_\Z} k$
Then $s$ is a primitive recursive function. | Let $s : \N \to \N$ be defined as:
:$\map s n = \begin{cases}
1 & : k > 0 \\
2 & : k < 0 \\
0 & : \text{otherwise}
\end{cases}$
By:
* Set of Strictly Positive Integers is Primitive Recursive
* Set of Strictly Negative Integers is Primitive Recursive
we have that:
:$k > 0 \iff n \in P$
:$k < 0 \iff n \in N$
are primitiv... | Let $\sgn_\Z : \Z \to \set {-1, 0, 1}$ denote the [[Definition:Signum Function|signum function]] on the [[Definition:Integer|integers]].
Let $s : \N \to \N$ be defined as:
:$\map s n = m$
where:
:$n$ [[Definition:Code Number for Integer|codes the integer]] $k$
:$m$ [[Definition:Code Number for Integer|codes the intege... | Let $s : \N \to \N$ be defined as:
:$\map s n = \begin{cases}
1 & : k > 0 \\
2 & : k < 0 \\
0 & : \text{otherwise}
\end{cases}$
By:
* [[Set of Strictly Positive Integers is Primitive Recursive]]
* [[Set of Strictly Negative Integers is Primitive Recursive]]
we have that:
:$k > 0 \iff n \in P$
:$k < 0 \iff n \in N$
are... | Signum Function on Integers is Primitive Recursive | https://proofwiki.org/wiki/Signum_Function_on_Integers_is_Primitive_Recursive | https://proofwiki.org/wiki/Signum_Function_on_Integers_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Signum Function",
"Definition:Integer",
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Function"
] | [
"Set of Strictly Positive Integers is Primitive Recursive",
"Set of Strictly Negative Integers is Primitive Recursive",
"Definition:Primitive Recursive/Relation",
"Constant Function is Primitive Recursive",
"Definition by Cases is Primitive Recursive/Corollary",
"Definition:Primitive Recursive/Function",
... |
proofwiki-21261 | Addition of Integers is Primitive Recursive | Let $a : \N^2 \to \N$ be defined as:
:$\map a {m, n} = p$
where:
:$m$ codes an integer $k$
:$n$ codes an integer $\ell$
:$p$ codes the integer $k + \ell$
Then $a$ is a primitive recursive function. | {{Proofread}}
Define:
:$<nowiki>\map a {m, n} = \begin{cases}
\map {\sgn_\Z} k \times_\Z \paren {\size k + \size \ell}_\Z & : \map {\sgn_\Z} k = \map {\sgn_\Z} \ell \\
\map {\sgn_\Z} k \times_\Z \paren {\size k {\dot -} \size \ell}_\Z & : \map {\sgn_\Z} k \ne \map {\sgn_\Z} \ell \land \size k \ge \size \ell \\
\map {\s... | Let $a : \N^2 \to \N$ be defined as:
:$\map a {m, n} = p$
where:
:$m$ [[Definition:Code Number for Integer|codes an integer]] $k$
:$n$ [[Definition:Code Number for Integer|codes an integer]] $\ell$
:$p$ [[Definition:Code Number for Integer|codes the integer]] $k + \ell$
Then $a$ is a [[Definition:Primitive Recursive F... | {{Proofread}}
Define:
:$<nowiki>\map a {m, n} = \begin{cases}
\map {\sgn_\Z} k \times_\Z \paren {\size k + \size \ell}_\Z & : \map {\sgn_\Z} k = \map {\sgn_\Z} \ell \\
\map {\sgn_\Z} k \times_\Z \paren {\size k {\dot -} \size \ell}_\Z & : \map {\sgn_\Z} k \ne \map {\sgn_\Z} \ell \land \size k \ge \size \ell \\
\map {\s... | Addition of Integers is Primitive Recursive | https://proofwiki.org/wiki/Addition_of_Integers_is_Primitive_Recursive | https://proofwiki.org/wiki/Addition_of_Integers_is_Primitive_Recursive | [
"Integer Addition",
"Primitive Recursive Functions"
] | [
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Multiplication/Integers",
"Definition:Code Number for Integer",
"Definition:Signum Function",
"Definition:Partial Subtraction",
"Ordering Relations are Primitive Recursive",
"Equality Relation is Primitive Recursive",
"Set Operations on Primitive Recursive Relations",
"Signum Function on I... |
proofwiki-21262 | Negation of Integer is Primitive Recursive | Let $f^- : \N \to \N$ be defined as:
:$\map {f^-} {x_\Z} = \paren {-x}_\Z$
where:
:$k_\Z$ denotes the code number for the integer $k$
Then $f^-$ is a primitive recursive function. | Define $f^-$ as:
$\quad\map {f^-} {x_\Z} = \begin{cases}
\map {c^-} {\size x} & : x > 0 \\
\map {c^+} {\size x} & : \text{otherwise}
\end{cases}$
where:
:$\map {c^-} x = \paren {-x}_\Z$
:$\map {c^+} x = x_\Z$
The function is primitive recursive by:
* Definition by Cases is Primitive Recursive/Corollary
* Set of Strictl... | Let $f^- : \N \to \N$ be defined as:
:$\map {f^-} {x_\Z} = \paren {-x}_\Z$
where:
:$k_\Z$ denotes the [[Definition:Code Number for Integer|code number for the integer]] $k$
Then $f^-$ is a [[Definition:Primitive Recursive Function|primitive recursive function]]. | Define $f^-$ as:
$\quad\map {f^-} {x_\Z} = \begin{cases}
\map {c^-} {\size x} & : x > 0 \\
\map {c^+} {\size x} & : \text{otherwise}
\end{cases}$
where:
:$\map {c^-} x = \paren {-x}_\Z$
:$\map {c^+} x = x_\Z$
The function is [[Definition:Primitive Recursive Function|primitive recursive]] by:
* [[Definition by Cases ... | Negation of Integer is Primitive Recursive | https://proofwiki.org/wiki/Negation_of_Integer_is_Primitive_Recursive | https://proofwiki.org/wiki/Negation_of_Integer_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Code Number for Integer",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Primitive Recursive/Function",
"Definition by Cases is Primitive Recursive/Corollary",
"Set of Strictly Positive Integers is Primitive Recursive",
"Code Number for Non-Negative Integer is Primitive Recursive",
"Code Number for Non-Positive Integer is Primitive Recursive",
"Category:Primitive R... |
proofwiki-21263 | Entropy of Measure-Preserving Transformation with respect to Finite Sub-Sigma Algebra is Well-Defined | Let $\struct {X, \BB, \mu}$ be a probability space.
Let $T: X \to X$ be a $\mu$-preserving transformation.
Let $\AA \subseteq \BB$ be a finite sub-$\sigma$-algebra.
Then the entropy of $T$ with respect to $\AA$:
:$\ds \map h {T, \AA} := \lim_{n \mathop \to \infty} \frac 1 n \map H {\bigvee_{k \mathop = 0}^{n - 1} T^{-k... | Let:
:$\ds a_n := \map H {\bigvee_{k \mathop = 0}^{n - 1} T^{-k} \AA}$
We need to show that the limit:
:$\ds \lim_{n \mathop \to \infty} \frac {a_n} n$
exists.
In view of Fekete's Subadditive Lemma, it suffices to show the subadditivity of $\sequence {a_n}$.
To this end, let $m, n \ge 1$.
Then:
{{begin-eqn}}
{{eqn | l ... | Let $\struct {X, \BB, \mu}$ be a [[Definition:Probability Space|probability space]].
Let $T: X \to X$ be a $\mu$-[[Definition:Measure-Preserving Transformation|preserving transformation]].
Let $\AA \subseteq \BB$ be a [[Definition:Finite Sub-Sigma-Algebra|finite sub-$\sigma$-algebra]].
Then the [[Definition:Entropy... | Let:
:$\ds a_n := \map H {\bigvee_{k \mathop = 0}^{n - 1} T^{-k} \AA}$
We need to show that the [[Definition:Limit Point|limit]]:
:$\ds \lim_{n \mathop \to \infty} \frac {a_n} n$
exists.
In view of [[Fekete's Subadditive Lemma]], it suffices to show the [[Definition:Subadditive Sequence|subadditivity]] of $\sequence ... | Entropy of Measure-Preserving Transformation with respect to Finite Sub-Sigma Algebra is Well-Defined | https://proofwiki.org/wiki/Entropy_of_Measure-Preserving_Transformation_with_respect_to_Finite_Sub-Sigma_Algebra_is_Well-Defined | https://proofwiki.org/wiki/Entropy_of_Measure-Preserving_Transformation_with_respect_to_Finite_Sub-Sigma_Algebra_is_Well-Defined | [
"Definitions/Ergodic Theory"
] | [
"Definition:Probability Space",
"Definition:Measure-Preserving Transformation",
"Definition:Finite Sub-Sigma-Algebra",
"Definition:Entropy of Measure-Preserving Transformation with respect to Finite Sub-Sigma Algebra",
"Definition:Well-Defined"
] | [
"Definition:Limit Point",
"Fekete's Subadditive Lemma",
"Definition:Subadditive Sequence",
"Category:Definitions/Ergodic Theory"
] |
proofwiki-21264 | Quotient of Integers is Primitive Recursive | Let $\operatorname{quot}_\Z : \N^2 \to \N$ be defined as:
:$\map {\operatorname{quot}_\Z} {a_\Z, b_\Z} = \begin{cases}
q_\Z & : b \ne 0 \\
0 & : b = 0
\end{cases}$
where:
:$k_\Z$ denotes the code number for the integer $k$
:$q$ is the quotient of $a$ on division by $b$
Then $\operatorname{quot}_\Z$ is primitive recursi... | By definition:
:$a = q b + r$
where $0 \le r < \size b$.
We will first show that $\operatorname{quot}_{\Z,\N} : \N^2 \to \N$ defined as:
:$\map {\operatorname{quot}_{\Z,\N}} {a_\Z, b} = \begin{cases}
q_\Z & : b \ne 0 \\
0 & : b = 0
\end{cases}$
with the difference being that $b \in \N$ instead of $\Z$.
If $a \ge 0$, we... | Let $\operatorname{quot}_\Z : \N^2 \to \N$ be defined as:
:$\map {\operatorname{quot}_\Z} {a_\Z, b_\Z} = \begin{cases}
q_\Z & : b \ne 0 \\
0 & : b = 0
\end{cases}$
where:
:$k_\Z$ denotes the [[Definition:Code Number for Integer|code number for the integer]] $k$
:$q$ is the [[Definition:Quotient (Integer Division)|quoti... | By definition:
:$a = q b + r$
where $0 \le r < \size b$.
We will first show that $\operatorname{quot}_{\Z,\N} : \N^2 \to \N$ defined as:
:$\map {\operatorname{quot}_{\Z,\N}} {a_\Z, b} = \begin{cases}
q_\Z & : b \ne 0 \\
0 & : b = 0
\end{cases}$
with the difference being that $b \in \N$ instead of $\Z$.
If $a \ge 0$,... | Quotient of Integers is Primitive Recursive | https://proofwiki.org/wiki/Quotient_of_Integers_is_Primitive_Recursive | https://proofwiki.org/wiki/Quotient_of_Integers_is_Primitive_Recursive | [
"Primitive Recursive Functions"
] | [
"Definition:Code Number for Integer",
"Definition:Quotient (Integer Division)",
"Definition:Primitive Recursive/Function"
] | [
"Quotient is Primitive Recursive",
"Definition:Quotient (Integer Division)",
"Quotient is Primitive Recursive",
"Definition:Primitive Recursive/Function",
"Definition by Cases is Primitive Recursive",
"Quotient is Primitive Recursive",
"Predecessor Function is Primitive Recursive",
"Addition of Intege... |
proofwiki-21265 | Restriction of Restriction is Restriction | Let $\RR$ be a relation on $S \times T$.
Let $X \subseteq S$, $Y \subseteq T$.
Let $W \subseteq X$, $V \subseteq Y$.
Then:
:$\paren{\RR {\restriction_{X \times Y} } } {\restriction_{W \times V} } = \RR {\restriction_{W \times V} }$
That is, the restriction of $\RR {\restriction_{X \times Y} }$ to $W \times V$ is the re... | From Cartesian Product of Subsets:
:$W \times V \subseteq X \times Y$
We have:
{{begin-eqn}}
{{eqn | l = \paren{\RR {\restriction_{X \times Y} } } {\restriction_{W \times V} }
| r = \RR {\restriction_{X \times Y} } \cap W \times V
| c = {{Defof|Restriction of Relation}}
}}
{{eqn | r = \paren {\RR \cap X \ti... | Let $\RR$ be a [[Definition:Relation|relation]] on $S \times T$.
Let $X \subseteq S$, $Y \subseteq T$.
Let $W \subseteq X$, $V \subseteq Y$.
Then:
:$\paren{\RR {\restriction_{X \times Y} } } {\restriction_{W \times V} } = \RR {\restriction_{W \times V} }$
That is, the [[Definition:Restriction of Relation|restrict... | From [[Cartesian Product of Subsets]]:
:$W \times V \subseteq X \times Y$
We have:
{{begin-eqn}}
{{eqn | l = \paren{\RR {\restriction_{X \times Y} } } {\restriction_{W \times V} }
| r = \RR {\restriction_{X \times Y} } \cap W \times V
| c = {{Defof|Restriction of Relation}}
}}
{{eqn | r = \paren {\RR \cap ... | Restriction of Restriction is Restriction | https://proofwiki.org/wiki/Restriction_of_Restriction_is_Restriction | https://proofwiki.org/wiki/Restriction_of_Restriction_is_Restriction | [
"Restrictions"
] | [
"Definition:Relation",
"Definition:Restriction/Relation",
"Definition:Restriction/Relation"
] | [
"Cartesian Product of Subsets",
"Intersection is Associative",
"Intersection with Subset is Subset",
"Category:Restrictions"
] |
proofwiki-21266 | Characterization of Generalized Hilbert Sequence Space/Necessary Condition | Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $H^\alpha = \struct{A, d_2}$ be the generalized Hilbert sequence space of weight $\alpha$ where:
:$A$ denotes the set of all real-valued functions $x : I \to \R$ such that:
::$(1)\quad \set{i \in I: x_i \ne 0}$ is counta... | By definition of $A$:
:$\forall k \in \closedint 1 m : \set{i \in I : \paren{x_k}_i \ne 0}$ is countable
From Infinite Set has Countably Infinite Subset, let:
:$I' \subseteq I$ be countably infinite
Let:
:$J = I' \cup \ds \bigcup_{k \mathop = 1}^m \set{i \in I : \paren{x_k}_i \ne 0}$
From Countable Union of Countable S... | Let $\alpha$ be an [[Definition:Infinite Cardinal|infinite cardinal number]].
Let $I$ be an [[Definition:Indexed Set|indexed set]] of [[Definition:Cardinality|cardinality]] $\alpha$.
Let $H^\alpha = \struct{A, d_2}$ be the [[Definition:Generalized Hilbert Sequence Space|generalized Hilbert sequence space of weight $... | By definition of $A$:
:$\forall k \in \closedint 1 m : \set{i \in I : \paren{x_k}_i \ne 0}$ is [[Definition:Countable Set|countable]]
From [[Infinite Set has Countably Infinite Subset]], let:
:$I' \subseteq I$ be [[Definition:Countably Infinite|countably infinite]]
Let:
:$J = I' \cup \ds \bigcup_{k \mathop = 1}^m \... | Characterization of Generalized Hilbert Sequence Space/Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Generalized_Hilbert_Sequence_Space/Necessary_Condition | https://proofwiki.org/wiki/Characterization_of_Generalized_Hilbert_Sequence_Space/Necessary_Condition | [
"Characterization of Generalized Hilbert Sequence Space"
] | [
"Definition:Infinite Cardinal",
"Definition:Indexing Set/Indexed Set",
"Definition:Cardinality",
"Definition:Generalized Hilbert Sequence Space",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Countable Set",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:... | [
"Definition:Countable Set",
"Infinite Set has Countably Infinite Subset",
"Definition:Countably Infinite",
"Countable Union of Countable Sets is Countable",
"Definition:Countable Set",
"Definition:Contrapositive Statement",
"Subset of Finite Set is Finite",
"Definition:Countably Infinite",
"Countabl... |
proofwiki-21267 | Characterization of Generalized Hilbert Sequence Space/Sufficient Condition | Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $H^\alpha = \struct{A, d_2}$ be the generalized Hilbert sequence space of weight $\alpha$ where:
:$A$ denotes the set of all real-valued functions $x : I \to \R$ such that:
::$(1)\quad \set{i \in I: x_i \ne 0}$ is counta... | By definition of $\ell^2$:
:$\forall k \in \closedint 1 m : \ds \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n}}^2 < \infty$
We have:
{{begin-eqn}}
{{eqn | q = \forall k \in \closedint 1 m
| l = \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n} }^2
| r = \sum_{n \mathop = 0}^\infty \size{\paren{\par... | Let $\alpha$ be an [[Definition:Infinite Cardinal|infinite cardinal number]].
Let $I$ be an [[Definition:Indexed Set|indexed set]] of [[Definition:Cardinality|cardinality]] $\alpha$.
Let $H^\alpha = \struct{A, d_2}$ be the [[Definition:Generalized Hilbert Sequence Space|generalized Hilbert sequence space of weight $... | By definition of $\ell^2$:
:$\forall k \in \closedint 1 m : \ds \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n}}^2 < \infty$
We have:
{{begin-eqn}}
{{eqn | q = \forall k \in \closedint 1 m
| l = \sum_{n \mathop = 0}^\infty \paren{\paren{x_k}_{j_n} }^2
| r = \sum_{n \mathop = 0}^\infty \size{\paren{\p... | Characterization of Generalized Hilbert Sequence Space/Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Generalized_Hilbert_Sequence_Space/Sufficient_Condition | https://proofwiki.org/wiki/Characterization_of_Generalized_Hilbert_Sequence_Space/Sufficient_Condition | [
"Characterization of Generalized Hilbert Sequence Space"
] | [
"Definition:Infinite Cardinal",
"Definition:Indexing Set/Indexed Set",
"Definition:Cardinality",
"Definition:Generalized Hilbert Sequence Space",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Countable Set",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Definition:... | [
"Square of Real Number is Non-Negative",
"Generalized Sum with Countable Non-zero Summands",
"Generalized Sum with Countable Non-zero Summands"
] |
proofwiki-21268 | Angle Bisector Theorem/Exterior Angle | Let $\triangle ABC$ be a triangle.
Let $AB$ be produced past $A$ to $D$.
Let the external angle $CAD$ be bisected by $AE$ where $BE$ is $BC$ produced.
:400px
Then:
:$BE : EC = AB : AC$ | 400px
Construct $CF$ parallel to the angle bisector $AE$.
By Parallelism implies Equal Corresponding Angles:
:$\angle AFC = \angle DAE$
By Parallelism implies Equal Alternate Angles:
:$\angle ACF = \angle CAE$
Given that $\angle DAC$ is bisected:
:$\angle DAE = \angle CAE$
By {{EuclidCommonNotionLink|1}}:
:$\angle AFC ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $AB$ be [[Definition:Production|produced]] past $A$ to $D$.
Let the [[Definition:External Angle|external angle]] $CAD$ be [[Definition:Bisection|bisected]] by $AE$ where $BE$ is $BC$ [[Definition:Production|produced]].
:[[File:Euclid-VI-3a.png... | [[File:Euclid-VI-3b.png|400px]]
Construct $CF$ [[Definition:Parallel Lines|parallel]] to the [[Definition:Angle Bisector|angle bisector]] $AE$.
By [[Parallelism implies Equal Corresponding Angles]]:
:$\angle AFC = \angle DAE$
By [[Parallelism implies Equal Alternate Angles]]:
:$\angle ACF = \angle CAE$
Given that $... | Angle Bisector Theorem/Exterior Angle | https://proofwiki.org/wiki/Angle_Bisector_Theorem/Exterior_Angle | https://proofwiki.org/wiki/Angle_Bisector_Theorem/Exterior_Angle | [
"Angle Bisector Theorem"
] | [
"Definition:Triangle (Geometry)",
"Definition:Production",
"Definition:Polygon/External Angle",
"Definition:Bisection",
"Definition:Production",
"File:Euclid-VI-3a.png"
] | [
"File:Euclid-VI-3b.png",
"Definition:Parallel (Geometry)/Lines",
"Definition:Angle Bisector",
"Parallelism implies Equal Corresponding Angles",
"Parallelism implies Equal Alternate Angles",
"Definition:Angle Bisector",
"Triangle with Two Equal Angles is Isosceles",
"Definition:Triangle (Geometry)/Isos... |
proofwiki-21269 | Angular Momentum of Particle moving with Constant Circular Motion | Let $P$ be a particle of mass $m$ moving:
:with constant speed $v$
:in a circle of radius $r$ about a point $O$.
Then the angular momentum has magnitude $m v r$. | The angular momentum $\mathbf L$ of a $P$ with respect to a point is given by the formula:
:$\mathbf L = \mathbf r \times \mathbf p$
where
:$\mathbf r$ is the position of $P$
:$\mathbf p$ is the linear momentum of $P$
We have that $P$ is moving in a circle of radius $r = \size {\mathbf r}$.
Define a coordinate system ... | Let $P$ be a [[Definition:Particle|particle]] of [[Definition:Mass|mass]] $m$ [[Definition:Motion|moving]]:
:with [[Definition:Constant|constant]] [[Definition:Speed|speed]] $v$
:in a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ about a [[Definition:Point|point]] $O$.
Then the [[Definiti... | The [[Definition:Angular Momentum|angular momentum]] $\mathbf L$ of a $P$ with respect to a [[Definition:Point|point]] is given by the formula:
:$\mathbf L = \mathbf r \times \mathbf p$
where
:$\mathbf r$ is the [[Definition:Position Vector|position]] of $P$
:$\mathbf p$ is the [[Definition:Linear Momentum|linear mome... | Angular Momentum of Particle moving with Constant Circular Motion | https://proofwiki.org/wiki/Angular_Momentum_of_Particle_moving_with_Constant_Circular_Motion | https://proofwiki.org/wiki/Angular_Momentum_of_Particle_moving_with_Constant_Circular_Motion | [
"Angular Momentum",
"Kinematics"
] | [
"Definition:Particle",
"Definition:Mass",
"Definition:Motion",
"Definition:Constant",
"Definition:Speed",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Point",
"Definition:Angular Momentum",
"Definition:Magnitude"
] | [
"Definition:Angular Momentum",
"Definition:Point",
"Definition:Position Vector",
"Definition:Linear Momentum",
"Definition:Motion",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Coordinate System",
"Definition:Coordinate System/Origin",
"Definition:Circle/Center",
"Definition:Cart... |
proofwiki-21270 | Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable | Let $D \subseteq \R^n$ and $E \subseteq \R^m$ be subsets of real cartesian space.
Let $f : D \to \R$ be sequentially computable.
Let $g_1, g_2, \dotsc, g_n : E \to \R$ be sequentially computable.
Suppose that, for every $\bsx \in E$:
:$\tuple {\map {g_1} \bsx, \map {g_2} \bsx, \dotsc, \map {g_n} \bsx} \in D$
Then $h : ... | Let $\tuple {\sequence {x_{1,k}}_k, \dotsc, \sequence {x_{m,k}}_k}$ be an $m$-tuple of computable real sequences in $E$.
By definition of sequentially computable:
:$\sequence {\map {g_i} {x_{1,k}, \dotsc, x_{m,k}}}_k$ is sequentially computable
for every $1 \le i \le n$
Additionally, for every $k$:
:$\tuple {\map {g_1}... | Let $D \subseteq \R^n$ and $E \subseteq \R^m$ be [[Definition:Subset|subsets]] of [[Definition:Real Cartesian Space|real cartesian space]].
Let $f : D \to \R$ be [[Definition:Sequentially Computable Real-Valued Function|sequentially computable]].
Let $g_1, g_2, \dotsc, g_n : E \to \R$ be [[Definition:Sequentially Com... | Let $\tuple {\sequence {x_{1,k}}_k, \dotsc, \sequence {x_{m,k}}_k}$ be an $m$-[[Definition:Ordered Tuple|tuple]] of [[Definition:Computable Real Sequence|computable real sequences]] in $E$.
By definition of [[Definition:Sequentially Computable Real-Valued Function|sequentially computable]]:
:$\sequence {\map {g_i} {x_... | Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable | https://proofwiki.org/wiki/Composition_of_Sequentially_Computable_Real-Valued_Functions_is_Sequentially_Computable | https://proofwiki.org/wiki/Composition_of_Sequentially_Computable_Real-Valued_Functions_is_Sequentially_Computable | [
"Computability Theory"
] | [
"Definition:Subset",
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Sequentially Computable Real-Valued Function",
"Definition:Sequentially Computable Real-Valued Function",
"Definition:Sequentially Computable Real-Valued Function"
] | [
"Definition:Ordered Tuple",
"Definition:Computable Real Sequence",
"Definition:Sequentially Computable Real-Valued Function",
"Definition:Sequentially Computable Real-Valued Function",
"Definition:Ordered Tuple",
"Definition:Computable Real Sequence",
"Definition:Sequentially Computable Real-Valued Func... |
proofwiki-21271 | Composition of Computably Uniformly Continuous Real-Valued Functions is Computably Uniformly Continuous | Let $D \subseteq \R^n$ and $E \subseteq \R^m$ be subsets of real cartesian space.
Let $f : D \to \R$ be computably uniformly continuous.
Let $g_1, g_2, \dotsc, g_n : E \to \R$ be computably uniformly continuous.
Suppose that, for every $\bsx \in E$:
:$\tuple {\map {g_1} \bsx, \map {g_2} \bsx, \dotsc, \map {g_n} \bsx} \... | By definition of computably uniformly continuous, for each of:
:$f, g_1, \dotsc, g_n$
there exists a total recursive function $d_f : \N \to \N$ such that, for all $k \in \N$ and $\bsx, \bsy \in \Dom f$:
:$\norm {\bsx - \bsy} < \dfrac 1 {\map {d_f} k + 1} \implies \size {\map f \bsx - \map f \bsy} < \dfrac 1 {k + 1}$
De... | Let $D \subseteq \R^n$ and $E \subseteq \R^m$ be [[Definition:Subset|subsets]] of [[Definition:Real Cartesian Space|real cartesian space]].
Let $f : D \to \R$ be [[Definition:Computably Uniformly Continuous Real-Valued Function|computably uniformly continuous]].
Let $g_1, g_2, \dotsc, g_n : E \to \R$ be [[Definition:... | By definition of [[Definition:Computably Uniformly Continuous Real-Valued Function|computably uniformly continuous]], for each of:
:$f, g_1, \dotsc, g_n$
there exists a [[Definition:Total Recursive Function|total recursive function]] $d_f : \N \to \N$ such that, for all $k \in \N$ and $\bsx, \bsy \in \Dom f$:
:$\norm {... | Composition of Computably Uniformly Continuous Real-Valued Functions is Computably Uniformly Continuous | https://proofwiki.org/wiki/Composition_of_Computably_Uniformly_Continuous_Real-Valued_Functions_is_Computably_Uniformly_Continuous | https://proofwiki.org/wiki/Composition_of_Computably_Uniformly_Continuous_Real-Valued_Functions_is_Computably_Uniformly_Continuous | [
"Computability Theory"
] | [
"Definition:Subset",
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Computably Uniformly Continuous Real-Valued Function",
"Definition:Computably Uniformly Continuous Real-Valued Function",
"Definition:Computably Uniformly Continuous Real-Valued Function"
] | [
"Definition:Computably Uniformly Continuous Real-Valued Function",
"Definition:Total Recursive Function",
"Definition:Max Operation/General Definition",
"Definition:Substitution (Mathematical Logic)",
"Definition:Max Operation",
"Maximum Function is Primitive Recursive",
"Multiplication is Primitive Rec... |
proofwiki-21272 | Composition of Computable Real-Valued Functions is Computable | Let $D \subseteq \R^n$ and $E \subseteq \R^m$ be subsets of real cartesian space.
Let $f : D \to \R$ be computable.
Let $g_1, g_2, \dotsc, g_n : E \to \R$ be computable.
Suppose that, for every $\bsx \in E$:
:$\tuple {\map {g_1} \bsx, \map {g_2} \bsx, \dotsc, \map {g_n} \bsx} \in D$
Then $h : E \to \R$ defined as:
:$\m... | Follows immediately from:
* Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable
* Composition of Computably Uniformly Continuous Real-Valued Functions is Computably Uniformly Continuous
{{qed}}
Category:Computability Theory
bhipvqt9fyp3vmilaala6eqa0y7ifej | Let $D \subseteq \R^n$ and $E \subseteq \R^m$ be [[Definition:Subset|subsets]] of [[Definition:Real Cartesian Space|real cartesian space]].
Let $f : D \to \R$ be [[Definition:Computable Real-Valued Function|computable]].
Let $g_1, g_2, \dotsc, g_n : E \to \R$ be [[Definition:Computable Real-Valued Function|computable... | Follows immediately from:
* [[Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable]]
* [[Composition of Computably Uniformly Continuous Real-Valued Functions is Computably Uniformly Continuous]]
{{qed}}
[[Category:Computability Theory]]
bhipvqt9fyp3vmilaala6eqa0y7ifej | Composition of Computable Real-Valued Functions is Computable | https://proofwiki.org/wiki/Composition_of_Computable_Real-Valued_Functions_is_Computable | https://proofwiki.org/wiki/Composition_of_Computable_Real-Valued_Functions_is_Computable | [
"Computability Theory"
] | [
"Definition:Subset",
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Computable Real-Valued Function",
"Definition:Computable Real-Valued Function",
"Definition:Computable Real-Valued Function"
] | [
"Composition of Sequentially Computable Real-Valued Functions is Sequentially Computable",
"Composition of Computably Uniformly Continuous Real-Valued Functions is Computably Uniformly Continuous",
"Category:Computability Theory"
] |
proofwiki-21273 | Bounded Generalized Sum is Absolutely Convergent | Let $V$ be a Banach space.
Let $\family {v_i}_{i \mathop \in I}$ be an indexed family of elements of $V$.
Let $\FF$ denote the set of finite subsets of $I$.
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} v_i$ is absolutely net convergent
{{iff}}
:there exists $M \in \R_{\mathop \ge 0}$ such that for all $F \in ... | === Necessary Condition ===
Let the generalized sum $\ds \sum_{i \mathop \in I} v_i$ be absolutely net convergent.
Let:
:$M = \ds \sum_{i \mathop \in I} \norm{v_i}$
{{AimForCont}}
:$\exists F_0 \in \FF : \sum_{i \mathop \in F_0} \norm{v_i} > M$
Let:
:$(1) \quad \epsilon \in \R_{\mathop > 0} : \epsilon < \paren{\ds \su... | Let $V$ be a [[Definition:Banach Space|Banach space]].
Let $\family {v_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] of $V$.
Let $\FF$ denote the [[Definition:Set|set]] of [[Definition:Finite Set|finite]] [[Definition:Subset|subsets]] of $I$.
Then:
:the... | === Necessary Condition ===
Let the [[Definition:Generalized Sum|generalized sum]] $\ds \sum_{i \mathop \in I} v_i$ be [[Definition:Absolute Net Convergence|absolutely net convergent]].
Let:
:$M = \ds \sum_{i \mathop \in I} \norm{v_i}$
{{AimForCont}}
:$\exists F_0 \in \FF : \sum_{i \mathop \in F_0} \norm{v_i} > M$... | Bounded Generalized Sum is Absolutely Convergent | https://proofwiki.org/wiki/Bounded_Generalized_Sum_is_Absolutely_Convergent | https://proofwiki.org/wiki/Bounded_Generalized_Sum_is_Absolutely_Convergent | [
"Banach Spaces",
"Generalized Sums"
] | [
"Definition:Banach Space",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Summation over Finite Index"
] | [
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Generalized Sum/Absolute Net Convergence",
"Set is Subset of Union",
"Summation over Union of Disjoint Finite Index Sets",
"Definition:Contradiction",
"Summation over Union of Disjoint Finite Index Sets",
... |
proofwiki-21274 | Velocity of Point on Rotating Body | Let $B$ be a body undergoing rotational motion around an axis $A$.
Let the angular velocity of $B$ about $A$ be $\bsomega$.
Let $O$ be an arbitrary point on $A$.
Let $P$ be a point in $B$ whose position vector {{WRT}} $O$ is $\mathbf r$.
Then the velocity $\mathbf v$ of $P$ is given by:
:$\mathbf v = \bsomega \times \m... | {{begin-eqn}}
{{eqn | l = \bsomega \times \mathbf r
| r = \paren {\frac {\mathbf r \times \mathbf v} {\size {\mathbf r}^2} } \times \mathbf r
| c = {{Defof|Angular Velocity}}
}}
{{eqn | r = \frac 1 {\size {\mathbf r}^2} \paren {\paren {\mathbf r \cdot \mathbf r} \mathbf v - \paren {\mathbf v \cdot \mathbf r... | Let $B$ be a [[Definition:Body|body]] undergoing [[Definition:Rotation (Geometry)|rotational]] [[Definition:Motion|motion]] around an [[Definition:Axis of Rotation|axis]] $A$.
Let the [[Definition:Angular Velocity|angular velocity]] of $B$ about $A$ be $\bsomega$.
Let $O$ be an arbitrary [[Definition:Point|point]] on... | {{begin-eqn}}
{{eqn | l = \bsomega \times \mathbf r
| r = \paren {\frac {\mathbf r \times \mathbf v} {\size {\mathbf r}^2} } \times \mathbf r
| c = {{Defof|Angular Velocity}}
}}
{{eqn | r = \frac 1 {\size {\mathbf r}^2} \paren {\paren {\mathbf r \cdot \mathbf r} \mathbf v - \paren {\mathbf v \cdot \mathbf r... | Velocity of Point on Rotating Body | https://proofwiki.org/wiki/Velocity_of_Point_on_Rotating_Body | https://proofwiki.org/wiki/Velocity_of_Point_on_Rotating_Body | [
"Velocity",
"Angular Velocity",
"Kinematics"
] | [
"Definition:Body",
"Definition:Rotation (Geometry)",
"Definition:Motion",
"Definition:Rotation (Geometry)/Axis",
"Definition:Angular Velocity",
"Definition:Point",
"Definition:Point",
"Definition:Position Vector",
"Definition:Velocity"
] | [
"Lagrange's Formula/Corollary",
"Dot Product of Vector with Itself",
"Definition:Rotation (Geometry)",
"Definition:Motion",
"Dot Product of Perpendicular Vectors"
] |
proofwiki-21275 | Nagata-Smirnov Metrization Theorem/Lemma 1 | :$\forall x \in S$ and $n \in \N$:
::the generalized sum $\ds \sum_{B \mathop \in \BB_n} \map {f_{\tuple {B, n} }^2} x$ converges | Let $s \in S$ and $m \in \N$.
By definition of locally finite set of subsets:
:$\exists U \in \tau : s \in U : \set {B \in \BB_m : B \cap U \ne \O}$ is finite
Hence:
:$\set {B \in \BB_m : s \in B}$ is finite
It follows that:
:$\set {\tuple {B, m} \in I : \map {f_{\tuple {B, m} } } s \ne 0}$ is finite
From Generalized S... | :$\forall x \in S$ and $n \in \N$:
::the [[Definition:Generalized Sum|generalized sum]] $\ds \sum_{B \mathop \in \BB_n} \map {f_{\tuple {B, n} }^2} x$ [[Definition:Convergent Net|converges]] | Let $s \in S$ and $m \in \N$.
By definition of [[Definition:Locally Finite Set of Subsets|locally finite set of subsets]]:
:$\exists U \in \tau : s \in U : \set {B \in \BB_m : B \cap U \ne \O}$ is [[Definition:Finite Set|finite]]
Hence:
:$\set {B \in \BB_m : s \in B}$ is [[Definition:Finite Set|finite]]
It follows ... | Nagata-Smirnov Metrization Theorem/Lemma 1 | https://proofwiki.org/wiki/Nagata-Smirnov_Metrization_Theorem/Lemma_1 | https://proofwiki.org/wiki/Nagata-Smirnov_Metrization_Theorem/Lemma_1 | [
"Nagata-Smirnov Metrization Theorem"
] | [
"Definition:Generalized Sum",
"Definition:Convergent Net"
] | [
"Definition:Locally Finite Set of Subsets",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Generalized Sum with Finite Non-zero Summands",
"Definition:Generalized Sum",
"Definition:Convergent Net",
"Category:Nagata-Smirnov Metrization Theorem"
] |
proofwiki-21276 | Nagata-Smirnov Metrization Theorem/Lemma 2 | :for all $n \in \N$ and $x \in S$:
::the generalized sum $\ds \sum_{\tuple{B, k} \mathop \in I_n} \sqbrk{\dfrac 1 {\paren{\sqrt 2}^k} \dfrac {\map {f_{\tuple{B, k}}} x} {\sqrt {1 + \map {g_k} x}}}^2$ converges
and:
:$\ds \sum_{\tuple{B, k} \mathop \in I_n} \sqbrk{\dfrac 1 {\paren{\sqrt 2}^k} \dfrac {\map {f_{\tuple{B,... | Let $n \in \N$.
Let $x \in S$.
Let $\FF$ denote the set of finite subsets of $I_n$.
Let $F \in \FF$.
Hence:
:$\set{k \in \N : \exists B \in \BB_k : \tuple{B, k} \in F}$ is finite.
Let $\set{n_1, n_2, \ldots, n_m} = \set{k \in \N : \exists B \in \BB_k : \tuple{B, k} \in F}$.
We have:
{{begin-eqn}}
{{eqn | l = \sum_{\tup... | :for all $n \in \N$ and $x \in S$:
::the [[Definition:Generalized Sum|generalized sum]] $\ds \sum_{\tuple{B, k} \mathop \in I_n} \sqbrk{\dfrac 1 {\paren{\sqrt 2}^k} \dfrac {\map {f_{\tuple{B, k}}} x} {\sqrt {1 + \map {g_k} x}}}^2$ [[Definition:Convergent Net|converges]]
and:
:$\ds \sum_{\tuple{B, k} \mathop \in I_n}... | Let $n \in \N$.
Let $x \in S$.
Let $\FF$ denote the [[Definition:Set|set]] of [[Definition:Finite Set|finite subsets]] of $I_n$.
Let $F \in \FF$.
Hence:
:$\set{k \in \N : \exists B \in \BB_k : \tuple{B, k} \in F}$ is [[Definition:Finite Set|finite]].
Let $\set{n_1, n_2, \ldots, n_m} = \set{k \in \N : \exists B ... | Nagata-Smirnov Metrization Theorem/Lemma 2 | https://proofwiki.org/wiki/Nagata-Smirnov_Metrization_Theorem/Lemma_2 | https://proofwiki.org/wiki/Nagata-Smirnov_Metrization_Theorem/Lemma_2 | [
"Nagata-Smirnov Metrization Theorem"
] | [
"Definition:Generalized Sum",
"Definition:Convergent Net"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Square/Function",
"Definition:Square Root",
"Summation over Union of Disjoint Finite Index Sets",
"Absolutely Convergent Generalized Sum Converges to Supremum",
"Sum of Infinite Geometric Sequence",
"Bounded Generalized ... |
proofwiki-21277 | Continuous Image of Everywhere Dense Set is Everywhere Dense | Let $A_T = \struct {A, \tau_A}$ and $B_T = \struct {B, \tau_B}$ be topological spaces.
Let $f : A \to B$ be an everywhere continuous surjection.
Let $S \subseteq A$ be everywhere dense in $A_T$.
Then, $f \sqbrk S$ is everywhere dense in $B_T$. | {{begin-eqn}}
{{eqn | l = \paren {f \sqbrk S}^-
| o = \supseteq
| r = f \sqbrk {S^-}
| c = Continuity Defined by Closure
}}
{{eqn | r = f \sqbrk A
| c = {{Defof|Everywhere Dense|index = 1}}
}}
{{eqn | r = B
| c = {{Defof|Surjection|index = 2}}
}}
{{end-eqn}}
As $\paren {f \sqbrk S}^- \subs... | Let $A_T = \struct {A, \tau_A}$ and $B_T = \struct {B, \tau_B}$ be [[Definition:Topological Space|topological spaces]].
Let $f : A \to B$ be an [[Definition:Everywhere Continuous Mapping (Topology)|everywhere continuous]] [[Definition:Surjection|surjection]].
Let $S \subseteq A$ be [[Definition:Everywhere Dense|every... | {{begin-eqn}}
{{eqn | l = \paren {f \sqbrk S}^-
| o = \supseteq
| r = f \sqbrk {S^-}
| c = [[Continuity Defined by Closure]]
}}
{{eqn | r = f \sqbrk A
| c = {{Defof|Everywhere Dense|index = 1}}
}}
{{eqn | r = B
| c = {{Defof|Surjection|index = 2}}
}}
{{end-eqn}}
As $\paren {f \sqbrk S}^- ... | Continuous Image of Everywhere Dense Set is Everywhere Dense | https://proofwiki.org/wiki/Continuous_Image_of_Everywhere_Dense_Set_is_Everywhere_Dense | https://proofwiki.org/wiki/Continuous_Image_of_Everywhere_Dense_Set_is_Everywhere_Dense | [
"Denseness"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Surjection",
"Definition:Everywhere Dense",
"Definition:Everywhere Dense"
] | [
"Continuity Defined by Closure",
"Definition:Set Equality",
"Definition:Everywhere Dense/Definition 1",
"Category:Denseness"
] |
proofwiki-21278 | Cross Product of Antiparallel Vectors is Zero | Let $\mathbf u$ and $\mathbf v$ be antiparallel vectors.
Then:
:$\mathbf u \times \mathbf v = 0$
where $\times$ denotes vector cross product. | We have {{hypothesis}} that $\mathbf u$ and $\mathbf v$ are antiparallel.
Hence we can define $\mathbf u$ and $\mathbf v$ as:
{{begin-eqn}}
{{eqn | l = \mathbf u
| r = x \mathbf i + y \mathbf j + z \mathbf k
}}
{{eqn | l = \mathbf v
| r = \paren {-x} \mathbf i + \paren {-y} \mathbf j + \paren {-z} \mathbf k... | Let $\mathbf u$ and $\mathbf v$ be [[Definition:Antiparallel Vectors|antiparallel vectors]].
Then:
:$\mathbf u \times \mathbf v = 0$
where $\times$ denotes [[Definition:Vector Cross Product|vector cross product]]. | We have {{hypothesis}} that $\mathbf u$ and $\mathbf v$ are [[Definition:Antiparallel Vectors|antiparallel]].
Hence we can define $\mathbf u$ and $\mathbf v$ as:
{{begin-eqn}}
{{eqn | l = \mathbf u
| r = x \mathbf i + y \mathbf j + z \mathbf k
}}
{{eqn | l = \mathbf v
| r = \paren {-x} \mathbf i + \paren ... | Cross Product of Antiparallel Vectors is Zero | https://proofwiki.org/wiki/Cross_Product_of_Antiparallel_Vectors_is_Zero | https://proofwiki.org/wiki/Cross_Product_of_Antiparallel_Vectors_is_Zero | [
"Antiparallel Vectors",
"Vector Cross Product"
] | [
"Definition:Antiparallel Vectors",
"Definition:Vector Cross Product"
] | [
"Definition:Antiparallel Vectors",
"Definition:Real Number",
"Definition:Vector Cross Product",
"Definition:Array/Row",
"Definition:Array/Row",
"Determinant with Row Multiplied by Constant"
] |
proofwiki-21279 | Dot Product of Antiparallel Vectors is Negative | Let $\mathbf u$ and $\mathbf v$ be non-zero antiparallel vectors.
Then:
:$\mathbf u \cdot \mathbf v < 0$
where $\cdot$ denotes dot product. | We have {{hypothesis}} that $\mathbf u$ and $\mathbf v$ are antiparallel.
Hence we can define $\mathbf u$ and $\mathbf v$ as:
{{begin-eqn}}
{{eqn | l = \mathbf u
| r = x \mathbf i + y \mathbf j + z \mathbf k
}}
{{eqn | l = \mathbf v
| r = \paren {-x} \mathbf i + \paren {-y} \mathbf j + \paren {-z} \mathbf k... | Let $\mathbf u$ and $\mathbf v$ be non-[[Definition:Zero Vector|zero]] [[Definition:Antiparallel Vectors|antiparallel vectors]].
Then:
:$\mathbf u \cdot \mathbf v < 0$
where $\cdot$ denotes [[Definition:Dot Product|dot product]]. | We have {{hypothesis}} that $\mathbf u$ and $\mathbf v$ are [[Definition:Antiparallel Vectors|antiparallel]].
Hence we can define $\mathbf u$ and $\mathbf v$ as:
{{begin-eqn}}
{{eqn | l = \mathbf u
| r = x \mathbf i + y \mathbf j + z \mathbf k
}}
{{eqn | l = \mathbf v
| r = \paren {-x} \mathbf i + \paren ... | Dot Product of Antiparallel Vectors is Negative | https://proofwiki.org/wiki/Dot_Product_of_Antiparallel_Vectors_is_Negative | https://proofwiki.org/wiki/Dot_Product_of_Antiparallel_Vectors_is_Negative | [
"Antiparallel Vectors",
"Dot Product"
] | [
"Definition:Zero Vector",
"Definition:Antiparallel Vectors",
"Definition:Dot Product"
] | [
"Definition:Antiparallel Vectors",
"Definition:Real Number",
"Definition:Zero (Number)",
"Definition:Dot Product"
] |
proofwiki-21280 | Approximation/Examples/Sine of x | For small values of $x$ measured in radians:
:$\sin x \approx x$ | From Limit of Sine of X over X at Zero:
:$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$
Hence the result.
{{qed}} | For small values of $x$ measured in [[Definition:Radian|radians]]:
:$\sin x \approx x$ | From [[Limit of Sine of X over X at Zero]]:
:$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$
Hence the result.
{{qed}} | Approximation/Examples/Sine of x | https://proofwiki.org/wiki/Approximation/Examples/Sine_of_x | https://proofwiki.org/wiki/Approximation/Examples/Sine_of_x | [
"Sine Function",
"Examples of Approximations"
] | [
"Definition:Angular Measure/Radian"
] | [
"Limit of Sinc Function at Zero"
] |
proofwiki-21281 | Absolutely Convergent Generalized Sum Converges to Supremum | Let $V$ be a Banach space.
Let $\family {v_i}_{i \mathop \in I}$ be an indexed family of elements of $V$.
Let $\FF$ denote the set of finite subsets of $I$.
Let the generalized sum $\ds \sum_{i \mathop \in I} v_i$ converge absolutely to $c \in \R$.
Then:
:$c = \sup \set{\ds \sum_{i \mathop \in F} \norm{v_i} : F \in \FF... | {{AimForCont}}:
:$\exists E \in \FF : \ds \sum_{i \mathop \in E} \norm{v_i} > c$
Let:
:$0 < \epsilon < \ds \sum_{i \mathop \in F} \norm{v_i} - c$
Let $F \in \FF$.
Let $E' = F \cup E$.
We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop \in E'} \norm{v_i}
| r = \sum_{i \mathop \in F} \norm{v_i} + \sum_{i \mathop \i... | Let $V$ be a [[Definition:Banach Space|Banach space]].
Let $\family {v_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Element|elements]] of $V$.
Let $\FF$ denote the [[Definition:Set|set]] of [[Definition:Finite Set|finite]] [[Definition:Subset|subsets]] of $I$.
Let the [[... | {{AimForCont}}:
:$\exists E \in \FF : \ds \sum_{i \mathop \in E} \norm{v_i} > c$
Let:
:$0 < \epsilon < \ds \sum_{i \mathop \in F} \norm{v_i} - c$
Let $F \in \FF$.
Let $E' = F \cup E$.
We have:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop \in E'} \norm{v_i}
| r = \sum_{i \mathop \in F} \norm{v_i} + \sum_{i \mat... | Absolutely Convergent Generalized Sum Converges to Supremum | https://proofwiki.org/wiki/Absolutely_Convergent_Generalized_Sum_Converges_to_Supremum | https://proofwiki.org/wiki/Absolutely_Convergent_Generalized_Sum_Converges_to_Supremum | [
"Banach Spaces",
"Generalized Sums"
] | [
"Definition:Banach Space",
"Definition:Indexing Set/Family",
"Definition:Element",
"Definition:Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Absolute Net Convergence"
] | [
"Summation over Union of Disjoint Finite Index Sets",
"Summation over Union of Disjoint Finite Index Sets",
"Definition:Contradiction",
"Definition:Hypothesis",
"Definition:Generalized Sum/Absolute Net Convergence",
"Definition:Generalized Sum/Absolute Net Convergence",
"Characterizing Property of Supre... |
proofwiki-21282 | Characterization of T1 Space using Neighborhood Basis | Let $T = \struct {S, \tau}$ be a topological space.
For each $x \in S$, let $\NN_x$ be a neighborhood basis at $x$.
Then:
:$T$ is a $T_1$ Space
{{iff}}
:$\forall x, y \in S : x \ne y$, both:
::$\exists N \in \NN_x : y \notin N$
:and:
::$\exists M \in \NN_y : x \notin M$ | === Necessary Condition ===
Let $T$ be a $T_1$ Space.
Let $x, y \in S$ such that $x \ne y$.
By definition of $T_1$ Space:
:$\exists U \in \tau : x \in U, y \notin U$
From Set is Open iff Neighborhood of all its Points:
:$U$ is a neighborhood of $x$
By definition of neighborhood basis:
:$\exists N \in \NN_x : N \subsete... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
For each $x \in S$, let $\NN_x$ be a [[Definition:Neighborhood Basis|neighborhood basis]] at $x$.
Then:
:$T$ is a [[Definition:T1 Space|$T_1$ Space]]
{{iff}}
:$\forall x, y \in S : x \ne y$, both:
::$\exists N \in \NN_x : y \notin... | === Necessary Condition ===
Let $T$ be a [[Definition:T1 Space|$T_1$ Space]].
Let $x, y \in S$ such that $x \ne y$.
By definition of [[Definition:T1 Space|$T_1$ Space]]:
:$\exists U \in \tau : x \in U, y \notin U$
From [[Set is Open iff Neighborhood of all its Points]]:
:$U$ is a [[Definition:Neighborhood (Topolog... | Characterization of T1 Space using Neighborhood Basis | https://proofwiki.org/wiki/Characterization_of_T1_Space_using_Neighborhood_Basis | https://proofwiki.org/wiki/Characterization_of_T1_Space_using_Neighborhood_Basis | [
"T1 Spaces",
"Local Bases"
] | [
"Definition:Topological Space",
"Definition:Neighborhood Basis",
"Definition:T1 Space"
] | [
"Definition:T1 Space",
"Definition:T1 Space",
"Set is Open iff Neighborhood of all its Points",
"Definition:Neighborhood (Topology)",
"Definition:Neighborhood Basis",
"Definition:Neighborhood",
"Definition:T1 Space"
] |
proofwiki-21283 | Control Path in Flow Chart is Unique | Let $C = \struct {F, P, V, E}$ be a flow chart.
Let $\struct {X, \set {f_g}, \set {p_q}}$ be an interpretation for $C$.
Let $b \in V$ and $x \in X$, and $N \in \N_{> 0}$.
Then, there is at most one length $N$ control path in $\struct {C, X}$, starting from $\tuple {b, x}$. | We will proceed by induction on $N$. | Let $C = \struct {F, P, V, E}$ be a [[Definition:Flow Chart|flow chart]].
Let $\struct {X, \set {f_g}, \set {p_q}}$ be an [[Definition:Flow Chart/Interpretation|interpretation]] for $C$.
Let $b \in V$ and $x \in X$, and $N \in \N_{> 0}$.
Then, there is at most one length $N$ [[Definition:Flow Chart/Control Path|con... | We will proceed by [[Definition:Mathematical Induction|induction]] on $N$. | Control Path in Flow Chart is Unique | https://proofwiki.org/wiki/Control_Path_in_Flow_Chart_is_Unique | https://proofwiki.org/wiki/Control_Path_in_Flow_Chart_is_Unique | [
"Algorithms"
] | [
"Definition:Flow Chart",
"Definition:Flow Chart/Interpretation",
"Definition:Flow Chart/Control Path"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-21284 | Characterization of T1 Space using Basis | Let $T = \struct {S, \tau}$ be a topological space.
Let $\BB$ be a basis for $T$.
Then:
:$T$ is a $T_1$ Space
{{iff}}
:$\forall x, y \in S : x \ne y$, both:
::$\exists B_x \in \BB : x \in B_x, y \notin B_x$
:and:
::$\exists B_y \in \BB : y \in B_y, x \notin B_y$ | === Necessary Condition ===
From Basis induces Local Basis:
:$\forall x \in S : \BB_x = \set{B \in \BB : x \in B}$ is a local basis of $x$
By definition of local basis:
:$\forall x \in S : \BB_x$ is a neighborhood basis of open sets
From Characterization of $T_1$ Space using Neighborhood Basis:
:$\forall x, y \in S : x... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\BB$ be a [[Definition:Analytic Basis|basis]] for $T$.
Then:
:$T$ is a [[Definition:T1 Space|$T_1$ Space]]
{{iff}}
:$\forall x, y \in S : x \ne y$, both:
::$\exists B_x \in \BB : x \in B_x, y \notin B_x$
:and:
::$\exists B_y \... | === Necessary Condition ===
From [[Basis induces Local Basis]]:
:$\forall x \in S : \BB_x = \set{B \in \BB : x \in B}$ is a [[Definition:Local Basis|local basis]] of $x$
By definition of [[Definition:Local Basis|local basis]]:
:$\forall x \in S : \BB_x$ is a [[Definition:Neighborhood Basis|neighborhood basis]] of [[D... | Characterization of T1 Space using Basis | https://proofwiki.org/wiki/Characterization_of_T1_Space_using_Basis | https://proofwiki.org/wiki/Characterization_of_T1_Space_using_Basis | [
"T1 Spaces",
"Topological Bases"
] | [
"Definition:Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:T1 Space"
] | [
"Basis induces Local Basis",
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Neighborhood Basis",
"Definition:Open Set/Topology",
"Characterization of T1 Space using Neighborhood Basis"
] |
proofwiki-21285 | Choquet's Theorem | Let $X$ be a locally convex vector space over $\R$.
Let $K$ be a non-empty metrizable compact convex subspace of $X$.
Let $K_e$ be the set of extreme points of $K$.
Then each $x \in K$ is a barycenter of a Borel probability measure $m_x$ on $K$ such that:
:$\map {m_x} {K_e} = 1$
That is:
:$\ds \forall \ell \in X^\ast :... | {{ProofWanted}}
{{Namedfor|Gustave Alfred Arthur Choquet|cat = Choquet}} | Let $X$ be a [[Definition:Locally Convex Topological Vector Space|locally convex vector space]] over $\R$.
Let $K$ be a [[Definition:Non-Empty|non-empty]] [[Definition:Metrizable Topology|metrizable]] [[Definition:Compact Topological Space|compact]] [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Topologi... | {{ProofWanted}}
{{Namedfor|Gustave Alfred Arthur Choquet|cat = Choquet}} | Choquet's Theorem | https://proofwiki.org/wiki/Choquet's_Theorem | https://proofwiki.org/wiki/Choquet's_Theorem | [
"Topological Vector Spaces"
] | [
"Definition:Locally Convex Topological Vector Space",
"Definition:Non-Empty",
"Definition:Metrizable Space",
"Definition:Compact Topological Space",
"Definition:Convex Set (Vector Space)",
"Definition:Topological Subspace",
"Definition:Set",
"Definition:Extreme Point of Convex Set",
"Definition:Bary... | [] |
proofwiki-21286 | Ackermann-Péter Function at (1,y) | For every $y \in \N$:
:$\map A {1, y} = y + 2$
where $A$ is the Ackermann-Péter function. | Proceed by induction on $y$. | For every $y \in \N$:
:$\map A {1, y} = y + 2$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | Proceed by [[Definition:Mathematical Induction|induction]] on $y$. | Ackermann-Péter Function at (1,y) | https://proofwiki.org/wiki/Ackermann-Péter_Function_at_(1,y) | https://proofwiki.org/wiki/Ackermann-Péter_Function_at_(1,y) | [
"Ackermann-Péter Function"
] | [
"Definition:Ackermann-Péter Function"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-21287 | Ackermann-Péter Function at (2,y) | For every $y \in \N$:
:$\map A {2, y} = 2 y + 3$
where $A$ is the Ackermann-Péter function. | Proceed by induction on $y$. | For every $y \in \N$:
:$\map A {2, y} = 2 y + 3$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | Proceed by [[Definition:Mathematical Induction|induction]] on $y$. | Ackermann-Péter Function at (2,y) | https://proofwiki.org/wiki/Ackermann-Péter_Function_at_(2,y) | https://proofwiki.org/wiki/Ackermann-Péter_Function_at_(2,y) | [
"Ackermann-Péter Function"
] | [
"Definition:Ackermann-Péter Function"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-21288 | Ackermann-Péter Function is Greater than Second Argument | For all $x, y \in \N$:
:$\map A {x, y} > y$
where $A$ is the Ackermann-Péter function. | First, perform induction on $x$. | For all $x, y \in \N$:
:$\map A {x, y} > y$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | First, perform [[Definition:Mathematical Induction|induction]] on $x$. | Ackermann-Péter Function is Greater than Second Argument | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Greater_than_Second_Argument | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Greater_than_Second_Argument | [] | [
"Definition:Ackermann-Péter Function"
] | [
"Definition:Mathematical Induction",
"Definition:Mathematical Induction"
] |
proofwiki-21289 | Ackermann-Péter Function is Strictly Increasing on Second Argument | For all $x, y \in \N$:
:$\map A {x, y + 1} > \map A {x, y}$
where $A$ is the Ackermann-Péter function. | Proceed by induction on $x$. | For all $x, y \in \N$:
:$\map A {x, y + 1} > \map A {x, y}$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | Proceed by [[Definition:Mathematical Induction|induction]] on $x$. | Ackermann-Péter Function is Strictly Increasing on Second Argument | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Strictly_Increasing_on_Second_Argument | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Strictly_Increasing_on_Second_Argument | [
"Ackermann-Péter Function is Strictly Increasing on Second Argument",
"Ackermann-Péter Function"
] | [
"Definition:Ackermann-Péter Function"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-21290 | Ackermann-Péter Function is Strictly Increasing on Second Argument/General Result | For all $x, y, z \in \N$ such that:
:$y < z$
we have:
:$\map A {x, y} < \map A {x, z}$
where $A$ is the Ackermann-Péter function. | Let $z$ be expressed as:
:$z = y + k$
for some $k \in \N_{>0}$.
Proceed by induction on $k$. | For all $x, y, z \in \N$ such that:
:$y < z$
we have:
:$\map A {x, y} < \map A {x, z}$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | Let $z$ be expressed as:
:$z = y + k$
for some $k \in \N_{>0}$.
Proceed by [[Definition:Mathematical Induction|induction]] on $k$. | Ackermann-Péter Function is Strictly Increasing on Second Argument/General Result | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Strictly_Increasing_on_Second_Argument/General_Result | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Strictly_Increasing_on_Second_Argument/General_Result | [
"Ackermann-Péter Function is Strictly Increasing on Second Argument"
] | [
"Definition:Ackermann-Péter Function"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-21291 | Euler Phi Function is not Completely Multiplicative | The Euler $\phi$ function is not a completely multiplicative function.
That is, it is not always the case that:
:$\map \phi {m n} = \map \phi m \map \phi n$
where $m, n \in \Z_{>0}$. | ;Proof by Counterexample
{{begin-eqn}}
{{eqn | l = \map \phi 6
| r = 2
| c = {{EulerPhiLink|6}}
}}
{{eqn | l = \map \phi {10}
| r = 4
| c = {{EulerPhiLink|10}}
}}
{{eqn | l = \map \phi {60}
| r = 16
| c = {{EulerPhiLink|60}}
}}
{{end-eqn}}
Hence we see:
:$6 \times 10 = 60$
but:
:$\ma... | The [[Definition:Euler Phi Function|Euler $\phi$ function]] is not a [[Definition:Completely Multiplicative Function|completely multiplicative function]].
That is, it is not always the case that:
:$\map \phi {m n} = \map \phi m \map \phi n$
where $m, n \in \Z_{>0}$. | ;[[Proof by Counterexample]]
{{begin-eqn}}
{{eqn | l = \map \phi 6
| r = 2
| c = {{EulerPhiLink|6}}
}}
{{eqn | l = \map \phi {10}
| r = 4
| c = {{EulerPhiLink|10}}
}}
{{eqn | l = \map \phi {60}
| r = 16
| c = {{EulerPhiLink|60}}
}}
{{end-eqn}}
Hence we see:
:$6 \times 10 = 60$
but:... | Euler Phi Function is not Completely Multiplicative | https://proofwiki.org/wiki/Euler_Phi_Function_is_not_Completely_Multiplicative | https://proofwiki.org/wiki/Euler_Phi_Function_is_not_Completely_Multiplicative | [
"Completely Multiplicative Functions",
"Euler Phi Function"
] | [
"Definition:Euler Phi Function",
"Definition:Completely Multiplicative Function"
] | [
"Proof by Counterexample"
] |
proofwiki-21292 | Increasing Second Argument of Ackermann-Péter Function is Not Greater than Increasing First Argument | For all $x, y \in \N$:
:$\map A {x + 1, y} \ge \map A {x, y + 1}$
where $A$ is the Ackermann-Péter function. | Proceed by induction on $y$. | For all $x, y \in \N$:
:$\map A {x + 1, y} \ge \map A {x, y + 1}$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | Proceed by [[Definition:Mathematical Induction|induction]] on $y$. | Increasing Second Argument of Ackermann-Péter Function is Not Greater than Increasing First Argument | https://proofwiki.org/wiki/Increasing_Second_Argument_of_Ackermann-Péter_Function_is_Not_Greater_than_Increasing_First_Argument | https://proofwiki.org/wiki/Increasing_Second_Argument_of_Ackermann-Péter_Function_is_Not_Greater_than_Increasing_First_Argument | [
"Ackermann-Péter Function"
] | [
"Definition:Ackermann-Péter Function"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-21293 | Ackermann-Péter Function is Strictly Increasing on First Argument | For all $x, y \in \N$:
:$\map A {x + 1, y} > \map A {x, y}$
where $A$ is the Ackermann-Péter function. | {{begin-eqn}}
{{eqn | l = \map A {x + 1, y}
| o = \ge
| r = \map A {x, y + 1}
| c = Increasing Second Argument of Ackermann-Péter Function is Not Greater than Increasing First Argument
}}
{{eqn | o = >
| r = \map A {x, y}
| c = Ackermann-Péter Function is Strictly Increasing on Second Argu... | For all $x, y \in \N$:
:$\map A {x + 1, y} > \map A {x, y}$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | {{begin-eqn}}
{{eqn | l = \map A {x + 1, y}
| o = \ge
| r = \map A {x, y + 1}
| c = [[Increasing Second Argument of Ackermann-Péter Function is Not Greater than Increasing First Argument]]
}}
{{eqn | o = >
| r = \map A {x, y}
| c = [[Ackermann-Péter Function is Strictly Increasing on Secon... | Ackermann-Péter Function is Strictly Increasing on First Argument | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Strictly_Increasing_on_First_Argument | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Strictly_Increasing_on_First_Argument | [
"Ackermann-Péter Function is Strictly Increasing on First Argument",
"Ackermann-Péter Function"
] | [
"Definition:Ackermann-Péter Function"
] | [
"Increasing Second Argument of Ackermann-Péter Function is Not Greater than Increasing First Argument",
"Ackermann-Péter Function is Strictly Increasing on Second Argument",
"Category:Ackermann-Péter Function is Strictly Increasing on First Argument",
"Category:Ackermann-Péter Function"
] |
proofwiki-21294 | Ackermann-Péter Function is Strictly Increasing on First Argument/General Result | Forall $x, y, z \in \N$ such that:
:$x < y$
we have:
:$\map A {x, z} < \map A {y, z}$
where $A$ is the Ackermann-Péter function. | Let $y$ be expressed as:
:$y = x + k$
for some $k \in \N_{>0}$.
Proceed by induction on $k$. | Forall $x, y, z \in \N$ such that:
:$x < y$
we have:
:$\map A {x, z} < \map A {y, z}$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | Let $y$ be expressed as:
:$y = x + k$
for some $k \in \N_{>0}$.
Proceed by [[Definition:Mathematical Induction|induction]] on $k$. | Ackermann-Péter Function is Strictly Increasing on First Argument/General Result | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Strictly_Increasing_on_First_Argument/General_Result | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_Strictly_Increasing_on_First_Argument/General_Result | [
"Ackermann-Péter Function is Strictly Increasing on First Argument"
] | [
"Definition:Ackermann-Péter Function"
] | [
"Definition:Mathematical Induction"
] |
proofwiki-21295 | Strict Upper Bound for Composition of Ackermann-Péter Functions | For all $x, y, z \in \N$:
:$\map A {x + y + 2, z} > \map A {x, \map A {y, z}}$
where $A$ is the Ackermann-Péter function. | We have:
:$x + y + 1 > y$
giving us:
:$\paren 1 \quad \map A {x + y + 1, z} > \map A {y, z}$
by Ackermann-Péter Function is Strictly Increasing on First Argument.
Therefore:
{{begin-eqn}}
{{eqn | l = \map A {x + y + 2, z}
| o = \ge
| r = \map A {x + y + 1, z + 1}
| c = Increasing Second Argument of Ac... | For all $x, y, z \in \N$:
:$\map A {x + y + 2, z} > \map A {x, \map A {y, z}}$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | We have:
:$x + y + 1 > y$
giving us:
:$\paren 1 \quad \map A {x + y + 1, z} > \map A {y, z}$
by [[Ackermann-Péter Function is Strictly Increasing on First Argument/General Result|Ackermann-Péter Function is Strictly Increasing on First Argument]].
Therefore:
{{begin-eqn}}
{{eqn | l = \map A {x + y + 2, z}
| o =... | Strict Upper Bound for Composition of Ackermann-Péter Functions | https://proofwiki.org/wiki/Strict_Upper_Bound_for_Composition_of_Ackermann-Péter_Functions | https://proofwiki.org/wiki/Strict_Upper_Bound_for_Composition_of_Ackermann-Péter_Functions | [
"Ackermann-Péter Function"
] | [
"Definition:Ackermann-Péter Function"
] | [
"Ackermann-Péter Function is Strictly Increasing on First Argument/General Result",
"Increasing Second Argument of Ackermann-Péter Function is Not Greater than Increasing First Argument",
"Ackermann-Péter Function is Strictly Increasing on Second Argument/General Result",
"Ackermann-Péter Function is Strictly... |
proofwiki-21296 | Strict Upper Bound for Sum of Ackermann-Péter Functions | For all $x, y, z \in \N$:
:$\map A {\map \max {x, y} + 4, z} > \map A {x, z} + \map A {y, z}$
where $A$ is the Ackermann-Péter function. | {{begin-eqn}}
{{eqn | l = \map A {\map \max {x, y} + 4, z}
| o = >
| r = \map A {2, \map A {\map \max {x, y}, z} }
| c = Strict Upper Bound for Composition of Ackermann-Péter Functions
}}
{{eqn | r = 2 \map A {\map \max {x, y}, z} + 3
| c = Ackermann-Péter Function at (2,y)
}}
{{eqn | o = >
... | For all $x, y, z \in \N$:
:$\map A {\map \max {x, y} + 4, z} > \map A {x, z} + \map A {y, z}$
where $A$ is the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]]. | {{begin-eqn}}
{{eqn | l = \map A {\map \max {x, y} + 4, z}
| o = >
| r = \map A {2, \map A {\map \max {x, y}, z} }
| c = [[Strict Upper Bound for Composition of Ackermann-Péter Functions]]
}}
{{eqn | r = 2 \map A {\map \max {x, y}, z} + 3
| c = [[Ackermann-Péter Function at (2,y)]]
}}
{{eqn | o ... | Strict Upper Bound for Sum of Ackermann-Péter Functions | https://proofwiki.org/wiki/Strict_Upper_Bound_for_Sum_of_Ackermann-Péter_Functions | https://proofwiki.org/wiki/Strict_Upper_Bound_for_Sum_of_Ackermann-Péter_Functions | [
"Ackermann-Péter Function"
] | [
"Definition:Ackermann-Péter Function"
] | [
"Strict Upper Bound for Composition of Ackermann-Péter Functions",
"Ackermann-Péter Function at (2,y)",
"Ackermann-Péter Function is Strictly Increasing on First Argument/General Result",
"Category:Ackermann-Péter Function"
] |
proofwiki-21297 | Ackermann-Péter Function is not Primitive Recursive | The Ackermann-Péter function is not primitive recursive. | Let $A : \N^2 \to \N$ denote the Ackermann-Péter function.
It suffices to show that, for every primitive recursive $f : \N^k \to \N$, there exists some $t_f \in \N$ such that:
:$\forall x_1, \dotsc, x_k \in \N: \map f {x_1, \dotsc, x_k} < \map A {t_f, \map \max {x_1, \dotsc, x_k}}$
For, if $A$ were primitive recursive,... | The [[Definition:Ackermann-Péter Function|Ackermann-Péter function]] is not [[Definition:Primitive Recursive Function|primitive recursive]]. | Let $A : \N^2 \to \N$ denote the [[Definition:Ackermann-Péter Function|Ackermann-Péter function]].
It suffices to show that, for every [[Definition:Primitive Recursive Function|primitive recursive]] $f : \N^k \to \N$, there exists some $t_f \in \N$ such that:
:$\forall x_1, \dotsc, x_k \in \N: \map f {x_1, \dotsc, x_k... | Ackermann-Péter Function is not Primitive Recursive | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_not_Primitive_Recursive | https://proofwiki.org/wiki/Ackermann-Péter_Function_is_not_Primitive_Recursive | [
"Ackermann-Péter Function",
"Primitive Recursive Functions"
] | [
"Definition:Ackermann-Péter Function",
"Definition:Primitive Recursive/Function"
] | [
"Definition:Ackermann-Péter Function",
"Definition:Primitive Recursive/Function",
"Definition:Primitive Recursive/Function",
"Definition:Contradiction",
"Definition:Primitive Recursive/Function",
"Definition:Ackermann-Péter Function",
"Definition:Primitive Recursive/Function"
] |
proofwiki-21298 | Division on Numbers is Not Associative | The operation of division on the numbers is not associative.
That is, in general:
:$a \div \paren {b \div c} \ne \paren {a \div b} \div c$ | By definition of division:
{{begin-eqn}}
{{eqn | l = a \div \paren {b \div c}
| r = a \times \paren {\dfrac 1 {b \times \dfrac 1 c} }
| c =
}}
{{eqn | r = a \times \paren {\dfrac c b}
| c =
}}
{{eqn | r = \dfrac {a c} b
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \paren {a \div b} \div c
... | The operation of [[Definition:Division|division]] on the [[Definition:Number|numbers]] is not [[Definition:Associative Operation|associative]].
That is, in general:
:$a \div \paren {b \div c} \ne \paren {a \div b} \div c$ | By definition of [[Definition:Division|division]]:
{{begin-eqn}}
{{eqn | l = a \div \paren {b \div c}
| r = a \times \paren {\dfrac 1 {b \times \dfrac 1 c} }
| c =
}}
{{eqn | r = a \times \paren {\dfrac c b}
| c =
}}
{{eqn | r = \dfrac {a c} b
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l =... | Division on Numbers is Not Associative | https://proofwiki.org/wiki/Division_on_Numbers_is_Not_Associative | https://proofwiki.org/wiki/Division_on_Numbers_is_Not_Associative | [
"Numbers",
"Division",
"Examples of Associative Operations"
] | [
"Definition:Division",
"Definition:Number",
"Definition:Associative Operation"
] | [
"Definition:Division"
] |
proofwiki-21299 | Real Addition is Sequentially Computable | Let $f : \R^2 \to \R$ be defined as:
:$\map f {x, y} = x + y$
Then $f$ is sequentially computable. | Follows immediately from Sum of Computable Real Sequences is Computable.
{{qed}}
Category:Computability Theory
ggmlqw7st6j6105q4n1v41mushddqum | Let $f : \R^2 \to \R$ be defined as:
:$\map f {x, y} = x + y$
Then $f$ is [[Definition:Sequentially Computable Real-Valued Function|sequentially computable]]. | Follows immediately from [[Sum of Computable Real Sequences is Computable]].
{{qed}}
[[Category:Computability Theory]]
ggmlqw7st6j6105q4n1v41mushddqum | Real Addition is Sequentially Computable | https://proofwiki.org/wiki/Real_Addition_is_Sequentially_Computable | https://proofwiki.org/wiki/Real_Addition_is_Sequentially_Computable | [
"Computability Theory"
] | [
"Definition:Sequentially Computable Real-Valued Function"
] | [
"Sum of Computable Real Sequences is Computable",
"Category:Computability Theory"
] |
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