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-8500 | Irrational Number Space is not Locally Compact Hausdorff Space | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is not a locally compact Hausdorff Space. | For $\struct {\R \setminus \Q, \tau_d}$ to be a locally compact Hausdorff Space, it is required that every point of $\R \setminus \Q$ has a compact neighborhood.
Let $x \in \R \setminus \Q$.
Let $N \subseteq \R \setminus \Q$ be a neighborhood of $x$.
Then:
:$\exists U \in \tau: x \in U \subseteq N \subseteq \R \setminu... | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is not a [[Definition:Locally Compact Hausdorff Space|locally compact Hausd... | For $\struct {\R \setminus \Q, \tau_d}$ to be a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff Space]], it is required that every point of $\R \setminus \Q$ has a [[Definition:Compact Topological Subspace|compact]] [[Definition:Neighborhood of Point|neighborhood]].
Let $x \in \R \setminus \Q$.... | Irrational Number Space is not Locally Compact Hausdorff Space | https://proofwiki.org/wiki/Irrational_Number_Space_is_not_Locally_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Irrational_Number_Space_is_not_Locally_Compact_Hausdorff_Space | [
"Irrational Number Space",
"Examples of Locally Compact Hausdorff Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Locally Compact Hausdorff Space"
] | [
"Definition:Locally Compact Hausdorff Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Compact Topological Space/Subspace",
"Compact Set of Irrational Numbers is Nowhere Dense",
"Definition:Nowher... |
proofwiki-8501 | Rational Number Space is not Weakly Sigma-Locally Compact | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is not weakly $\sigma$-locally compact. | From Rational Number Space is not Locally Compact Hausdorff Space, $\struct {\Q, \tau_d}$ is not weakly locally compact.
Hence the result from definition of weakly $\sigma$-locally compact.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is not [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact]]. | From [[Rational Number Space is not Locally Compact Hausdorff Space]], $\struct {\Q, \tau_d}$ is not [[Definition:Weakly Locally Compact Space|weakly locally compact]].
Hence the result from definition of [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact]].
{{qed}} | Rational Number Space is not Weakly Sigma-Locally Compact | https://proofwiki.org/wiki/Rational_Number_Space_is_not_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Rational_Number_Space_is_not_Weakly_Sigma-Locally_Compact | [
"Rational Number Space",
"Examples of Weakly Sigma-Locally Compact Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Rational Number Space is not Locally Compact Hausdorff Space",
"Definition:Weakly Locally Compact Space",
"Definition:Weakly Sigma-Locally Compact Space"
] |
proofwiki-8502 | Irrational Number Space is not Weakly Sigma-Locally Compact | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is not weakly $\sigma$-locally compact. | From Irrational Number Space is not Locally Compact Hausdorff Space, $\struct {\R \setminus \Q, \tau_d}$ is not weakly locally compact.
Hence the result from definition of weakly $\sigma$-locally compact.
{{qed}} | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is not [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-loca... | From [[Irrational Number Space is not Locally Compact Hausdorff Space]], $\struct {\R \setminus \Q, \tau_d}$ is not [[Definition:Weakly Locally Compact Space|weakly locally compact]].
Hence the result from definition of [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact]].
{{qed}} | Irrational Number Space is not Weakly Sigma-Locally Compact | https://proofwiki.org/wiki/Irrational_Number_Space_is_not_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Irrational_Number_Space_is_not_Weakly_Sigma-Locally_Compact | [
"Irrational Number Space",
"Examples of Weakly Sigma-Locally Compact Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Irrational Number Space is not Locally Compact Hausdorff Space",
"Definition:Weakly Locally Compact Space",
"Definition:Weakly Sigma-Locally Compact Space"
] |
proofwiki-8503 | Rational Number Space is Sigma-Compact | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is $\sigma$-compact. | From Rational Numbers are Countably Infinite, $\Q$ is countable.
Hence the result from definition of Countable Space is $\sigma$-Compact.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is [[Definition:Sigma-Compact Space|$\sigma$-compact]]. | From [[Rational Numbers are Countably Infinite]], $\Q$ is [[Definition:Countably Infinite Set|countable]].
Hence the result from definition of [[Countable Space is Sigma-Compact|Countable Space is $\sigma$-Compact]].
{{qed}} | Rational Number Space is Sigma-Compact | https://proofwiki.org/wiki/Rational_Number_Space_is_Sigma-Compact | https://proofwiki.org/wiki/Rational_Number_Space_is_Sigma-Compact | [
"Rational Number Space",
"Examples of Sigma-Compact Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Sigma-Compact Space"
] | [
"Rational Numbers are Countably Infinite",
"Definition:Countably Infinite/Set",
"Countable Space is Sigma-Compact"
] |
proofwiki-8504 | Rationals plus Irrational are Everywhere Dense in Irrationals | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Let $x \in \R \setminus \Q$ be an arbitrary irrational number.
Let $S_x$ be the set defined as:
:$S_x := \set {x + q: q \in \Q}$
Then $S_x$ is everywhere dense in $\struct {\R \setminus \Q, \tau_d}$. | Let $y \in \R \setminus \Q$.
Let $U \subseteq \R \setminus \Q$ be an open set of $\struct {\R \setminus \Q, \tau_d}$ such that $x \in U$.
From Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\R$ form a basis for $\struct {\R, \tau_d}$.
By Basis for Topological Subspace, the set ... | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Let $x \in \R \setminus \Q$ be an arbitrary [[Definition:Irrational Number|irrational number]].
Let $S_x$ be the [[... | Let $y \in \R \setminus \Q$.
Let $U \subseteq \R \setminus \Q$ be an [[Definition:Open Set (Topology)|open set]] of $\struct {\R \setminus \Q, \tau_d}$ such that $x \in U$.
From [[Basis for Euclidean Topology on Real Number Line]], the set of all [[Definition:Open Real Interval|open real intervals]] of $\R$ form a b... | Rationals plus Irrational are Everywhere Dense in Irrationals | https://proofwiki.org/wiki/Rationals_plus_Irrational_are_Everywhere_Dense_in_Irrationals | https://proofwiki.org/wiki/Rationals_plus_Irrational_are_Everywhere_Dense_in_Irrationals | [
"Irrational Number Space",
"Examples of Everywhere Dense"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Irrational Number",
"Definition:Set",
"Definition:Everywhere Dense"
] | [
"Definition:Open Set/Topology",
"Basis for Euclidean Topology on Real Number Line",
"Definition:Real Interval/Open",
"Basis for Topological Subspace",
"Definition:Real Interval/Open",
"Between two Real Numbers exists Rational Number",
"Definition:Everywhere Dense"
] |
proofwiki-8505 | Rational Number Space is Totally Separated | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is totally separated. | Let $x, y \in \Q$.
From Between two Rational Numbers exists Irrational Number:
:$\exists \alpha \in \R \setminus \Q: x < \alpha < y$
Consider the unbounded open real intervals:
:$A := \openint \gets \alpha$, $B := \openint \alpha \to$
Let:
:$U := A \cap \Q, V := B \cap \Q$
Let $\beta \in \Q$.
Then either:
:$(1): \quad ... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is [[Definition:Totally Separated Space|totally separated]]. | Let $x, y \in \Q$.
From [[Between two Rational Numbers exists Irrational Number]]:
:$\exists \alpha \in \R \setminus \Q: x < \alpha < y$
Consider the [[Definition:Unbounded Open Real Interval|unbounded open real intervals]]:
:$A := \openint \gets \alpha$, $B := \openint \alpha \to$
Let:
:$U := A \cap \Q, V := B \cap... | Rational Number Space is Totally Separated | https://proofwiki.org/wiki/Rational_Number_Space_is_Totally_Separated | https://proofwiki.org/wiki/Rational_Number_Space_is_Totally_Separated | [
"Rational Number Space",
"Examples of Totally Separated Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Totally Separated Space"
] | [
"Between two Rational Numbers exists Irrational Number",
"Definition:Real Interval/Unbounded Open",
"Definition:Separation (Topology)",
"Definition:Totally Separated Space"
] |
proofwiki-8506 | Irrational Number Space is Totally Separated | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is totally separated. | Let $x, y \in \R \setminus \Q$.
From Between two Real Numbers exists Rational Number:
:$\exists \alpha \in \Q: x < \alpha < y$
Consider the unbounded open real intervals:
:$A := \openint \gets \alpha$, $B := \openint \alpha \to$
Let:
:$U := A \cap \paren {\R \setminus \Q}$, $V := B \cap \paren {\R \setminus \Q}$
Let $\... | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is [[Definition:Totally Separated Space|totally separated]]. | Let $x, y \in \R \setminus \Q$.
From [[Between two Real Numbers exists Rational Number]]:
:$\exists \alpha \in \Q: x < \alpha < y$
Consider the [[Definition:Unbounded Open Real Interval|unbounded open real intervals]]:
:$A := \openint \gets \alpha$, $B := \openint \alpha \to$
Let:
:$U := A \cap \paren {\R \setminus ... | Irrational Number Space is Totally Separated | https://proofwiki.org/wiki/Irrational_Number_Space_is_Totally_Separated | https://proofwiki.org/wiki/Irrational_Number_Space_is_Totally_Separated | [
"Irrational Number Space",
"Examples of Totally Separated Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Totally Separated Space"
] | [
"Between two Real Numbers exists Rational Number",
"Definition:Real Interval/Unbounded Open",
"Definition:Separation (Topology)",
"Definition:Totally Separated Space"
] |
proofwiki-8507 | Rational Number Space is not Scattered | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is not scattered. | For a space to be scattered, it needs by definition to have no subset which is dense-in-itself.
From Rational Number Space is Dense-in-itself, $\struct {\Q, \tau_d}$ is dense-in-itself.
Hence the result.
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is not [[Definition:Scattered Space|scattered]]. | For a [[Definition:Topological Space|space]] to be [[Definition:Scattered Space|scattered]], it needs by definition to have no [[Definition:Subset|subset]] which is [[Definition:Dense-in-itself|dense-in-itself]].
From [[Rational Number Space is Dense-in-itself]], $\struct {\Q, \tau_d}$ is [[Definition:Dense-in-itself|... | Rational Number Space is not Scattered | https://proofwiki.org/wiki/Rational_Number_Space_is_not_Scattered | https://proofwiki.org/wiki/Rational_Number_Space_is_not_Scattered | [
"Rational Number Space",
"Examples of Scattered Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Scattered Space"
] | [
"Definition:Topological Space",
"Definition:Scattered Space",
"Definition:Subset",
"Definition:Dense-in-itself",
"Rational Number Space is Dense-in-itself",
"Definition:Dense-in-itself"
] |
proofwiki-8508 | Rational Number Space is Dense-in-itself | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is dense-in-itself. | Let $x \in \Q$.
Let $U \subseteq \R$ be an open set of $\struct {\Q, \tau_d}$ such that $x \in U$.
From Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\R$ form a basis for $\struct {\R, \tau_d}$.
By Basis for Topological Subspace, the set of all intersections of $\Q$ and open r... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is [[Definition:Dense-in-itself|dense-in-itself]]. | Let $x \in \Q$.
Let $U \subseteq \R$ be an [[Definition:Open Set (Topology)|open set]] of $\struct {\Q, \tau_d}$ such that $x \in U$.
From [[Basis for Euclidean Topology on Real Number Line]], the set of all [[Definition:Open Real Interval|open real intervals]] of $\R$ form a basis for $\struct {\R, \tau_d}$.
By [[... | Rational Number Space is Dense-in-itself | https://proofwiki.org/wiki/Rational_Number_Space_is_Dense-in-itself | https://proofwiki.org/wiki/Rational_Number_Space_is_Dense-in-itself | [
"Rational Number Space",
"Examples of Dense-in-itself"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Dense-in-itself"
] | [
"Definition:Open Set/Topology",
"Basis for Euclidean Topology on Real Number Line",
"Definition:Real Interval/Open",
"Basis for Topological Subspace",
"Definition:Real Interval/Open",
"Between two Real Numbers exists Rational Number",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Dense-in... |
proofwiki-8509 | Irrational Number Space is Dense-in-itself | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is dense-in-itself. | Let $x \in \R \setminus \Q$.
Let $U \subseteq \R$ be an open set of $\struct {\R \setminus \Q, \tau_d}$ such that $x \in U$.
From Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\R$ form a basis for $\struct {\R, \tau_d}$.
By Basis for Topological Subspace, the set of all inters... | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is [[Definition:Dense-in-itself|dense-in-itself]]. | Let $x \in \R \setminus \Q$.
Let $U \subseteq \R$ be an [[Definition:Open Set (Topology)|open set]] of $\struct {\R \setminus \Q, \tau_d}$ such that $x \in U$.
From [[Basis for Euclidean Topology on Real Number Line]], the set of all [[Definition:Open Real Interval|open real intervals]] of $\R$ form a basis for $\st... | Irrational Number Space is Dense-in-itself | https://proofwiki.org/wiki/Irrational_Number_Space_is_Dense-in-itself | https://proofwiki.org/wiki/Irrational_Number_Space_is_Dense-in-itself | [
"Irrational Number Space",
"Examples of Dense-in-itself"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Dense-in-itself"
] | [
"Definition:Open Set/Topology",
"Basis for Euclidean Topology on Real Number Line",
"Definition:Real Interval/Open",
"Basis for Topological Subspace",
"Definition:Real Interval/Open",
"Between two Real Numbers exists Irrational Number",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Dense-... |
proofwiki-8510 | Irrational Number Space is not Scattered | Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is not scattered. | For a space to be scattered, it needs by definition to have no subset which is dense-in-itself.
From Irrational Number Space is Dense-in-itself, $\struct {\R \setminus \Q, \tau_d}$ is dense-in-itself.
Hence the result.
{{qed}} | Let $\struct {\R \setminus \Q, \tau_d}$ be the [[Definition:Irrational Number Space|irrational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is not [[Definition:Scattered Space|scattered]]. | For a [[Definition:Topological Space|space]] to be [[Definition:Scattered Space|scattered]], it needs by definition to have no [[Definition:Subset|subset]] which is [[Definition:Dense-in-itself|dense-in-itself]].
From [[Irrational Number Space is Dense-in-itself]], $\struct {\R \setminus \Q, \tau_d}$ is [[Definition:D... | Irrational Number Space is not Scattered | https://proofwiki.org/wiki/Irrational_Number_Space_is_not_Scattered | https://proofwiki.org/wiki/Irrational_Number_Space_is_not_Scattered | [
"Irrational Number Space",
"Examples of Scattered Spaces"
] | [
"Definition:Irrational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Scattered Space"
] | [
"Definition:Topological Space",
"Definition:Scattered Space",
"Definition:Subset",
"Definition:Dense-in-itself",
"Irrational Number Space is Dense-in-itself",
"Definition:Dense-in-itself"
] |
proofwiki-8511 | Open Unit Interval on Rational Number Space is Totally Bounded | Let $T = \struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then:
:$\openint 0 1 \cap \Q$ is totally bounded in $T$
where $\openint 0 1$ is the open unit interval. | Let $\epsilon \in \R_{>0}$.
By the Archimedean Property of $\R$:
:$\exists n \in \N: \dfrac 1 n < \epsilon$
We pick the numbers $\dfrac i n \in \openint 0 1 \cap \Q$, where $i \in \N$ and $0 < i < n$.
Then for all $x \in \openint 0 1 \cap \Q$ and $x \ge \dfrac 1 n$:
{{begin-eqn}}
{{eqn | l = \inf_{0 \mathop < i \mathop... | Let $T = \struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then:
:$\openint 0 1 \cap \Q$ is [[Definition:Totally Bounded Metric Space|totally bounded]] in $T$
where $\openint 0 1$ is the [... | Let $\epsilon \in \R_{>0}$.
By the [[Definition:Archimedean Property|Archimedean Property of $\R$]]:
:$\exists n \in \N: \dfrac 1 n < \epsilon$
We pick the numbers $\dfrac i n \in \openint 0 1 \cap \Q$, where $i \in \N$ and $0 < i < n$.
Then for all $x \in \openint 0 1 \cap \Q$ and $x \ge \dfrac 1 n$:
{{begin-eqn}}
... | Open Unit Interval on Rational Number Space is Totally Bounded | https://proofwiki.org/wiki/Open_Unit_Interval_on_Rational_Number_Space_is_Totally_Bounded | https://proofwiki.org/wiki/Open_Unit_Interval_on_Rational_Number_Space_is_Totally_Bounded | [
"Rational Number Space",
"Examples of Totally Bounded Metric Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Totally Bounded Metric Space",
"Definition:Real Interval/Unit Interval/Open"
] | [
"Definition:Archimedean Property",
"Definition:Integer",
"Real Number minus Floor",
"Definition:Totally Bounded Metric Space"
] |
proofwiki-8512 | Integer Reciprocal Space is Topological Space | Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$.
Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n: n \in \Z_{>0} }$
Then the integer reciprocal space $\struct {A, \tau_d}$ is a topological space. | We have that $A \subseteq \R$.
By definition, $\struct {A, \tau_d}$ is a subspace of $\struct {\R, \tau_d}$.
Hence the result from Topological Subspace is Topological Space.
{{qed}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number line]] $\R$ under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Let $A \subseteq \R$ be the [[Definition:Set|set]] of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n: n \in \Z_{>0} }$
T... | We have that $A \subseteq \R$.
By definition, $\struct {A, \tau_d}$ is a [[Definition:Topological Subspace|subspace]] of $\struct {\R, \tau_d}$.
Hence the result from [[Topological Subspace is Topological Space]].
{{qed}} | Integer Reciprocal Space is Topological Space | https://proofwiki.org/wiki/Integer_Reciprocal_Space_is_Topological_Space | https://proofwiki.org/wiki/Integer_Reciprocal_Space_is_Topological_Space | [
"Integer Reciprocal Space"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Set",
"Definition:Integer Reciprocal Space",
"Definition:Topological Space"
] | [
"Definition:Topological Subspace",
"Topological Subspace is Topological Space"
] |
proofwiki-8513 | Closure of Integer Reciprocal Space | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n: n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.
Then:
:$A^- = A \cup \set 0$
where $A^-$ denotes the closure of $A$ in $\R$. | By definition, the closure of $A$ is:
:$A \cup A'$
where $A'$ is the derived set of $A$.
By definition of derived set, $A'$ consists of all the limit points of $A$ in $\R$.
From Zero is Limit Point of Integer Reciprocal Space, the only limit point of $A$ is $0$.
Hence the result:
:$A^- = A \cup \set 0$
{{qed}} | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n: n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
... | By definition, the [[Definition:Closure (Topology)|closure]] of $A$ is:
:$A \cup A'$
where $A'$ is the [[Definition:Derived Set|derived set]] of $A$.
By definition of [[Definition:Derived Set|derived set]], $A'$ consists of all the [[Definition:Limit Point of Set|limit points]] of $A$ in $\R$.
From [[Zero is Limit Po... | Closure of Integer Reciprocal Space | https://proofwiki.org/wiki/Closure_of_Integer_Reciprocal_Space | https://proofwiki.org/wiki/Closure_of_Integer_Reciprocal_Space | [
"Integer Reciprocal Space",
"Examples of Set Closures"
] | [
"Definition:Set",
"Definition:Integer Reciprocal Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Closure (Topology)"
] | [
"Definition:Closure (Topology)",
"Definition:Derived Set",
"Definition:Derived Set",
"Definition:Limit Point/Topology/Set",
"Zero is Limit Point of Integer Reciprocal Space",
"Definition:Limit Point/Topology/Set"
] |
proofwiki-8514 | Zero is Limit Point of Integer Reciprocal Space | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.
Then $0$ is the only limit point of $A$ in $\R$. | There are three cases to consider: | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all [[Definition:Point of Set|points]] on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] under the [[Definition:Euclidean Topology on Real Number Line|usu... | There are three cases to consider: | Zero is Limit Point of Integer Reciprocal Space | https://proofwiki.org/wiki/Zero_is_Limit_Point_of_Integer_Reciprocal_Space | https://proofwiki.org/wiki/Zero_is_Limit_Point_of_Integer_Reciprocal_Space | [
"Integer Reciprocal Space",
"Examples of Limit Points"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Integer Reciprocal Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Limit Point/Topology/Set"
] | [] |
proofwiki-8515 | Zero is Limit Point of Integer Reciprocal Space Union with Closed Interval | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $T = \struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.
Let $B$ be the uncountable set:
:$B := A \cup \closedint 2 3$
where $\closedint 2 3$ is a closed interval of $\... | Let $U$ be an open set of $\R$ which contains $0$.
From Open Sets in Real Number Line, there exists an open interval $I$ of the form:
:$I := \openint {-a} b \subseteq U$
By the Axiom of Archimedes:
:$\exists n \in \N: n > \dfrac 1 b$
and so:
:$\exists n \in \N: \dfrac 1 n < b$
But $\dfrac 1 n \in B$.
Thus an open set $... | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $T = \struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]... | Let $U$ be an [[Definition:Open Set (Topology)|open set]] of $\R$ which contains $0$.
From [[Open Sets in Real Number Line]], there exists an [[Definition:Open Real Interval|open interval]] $I$ of the form:
:$I := \openint {-a} b \subseteq U$
By the [[Axiom of Archimedes]]:
:$\exists n \in \N: n > \dfrac 1 b$
and so... | Zero is Limit Point of Integer Reciprocal Space Union with Closed Interval | https://proofwiki.org/wiki/Zero_is_Limit_Point_of_Integer_Reciprocal_Space_Union_with_Closed_Interval | https://proofwiki.org/wiki/Zero_is_Limit_Point_of_Integer_Reciprocal_Space_Union_with_Closed_Interval | [
"Integer Reciprocal Space",
"Examples of Limit Points"
] | [
"Definition:Set",
"Definition:Integer Reciprocal Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Uncountable/Set",
"Definition:Real Interval/Closed",
"Definition:Limit Point/Topology/Set"
] | [
"Definition:Open Set/Topology",
"Open Sets in Real Number Line",
"Definition:Real Interval/Open",
"Axiom of Archimedes",
"Definition:Open Set/Topology",
"Definition:Limit Point/Topology/Set"
] |
proofwiki-8516 | Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.
Let $B$ be the uncountable set:
:$B := A \cup \closedint 2 3$
where $\closedint 2 3$ is a closed interval of $\R$.
... | Let $U$ be an open set of $\R$ which contains $0$.
From Open Sets in Real Number Line, there exists an open interval $I$ of the form:
:$I := \openint {-a} b \subseteq U$
By the Axiom of Archimedes:
:$\exists n \in \N: n > \dfrac 1 b$
and so:
:$\exists n \in \N: \dfrac 1 n < b$
Let:
:$M := \set {m \in \N: m \ge n}$
Then... | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
... | Let $U$ be an [[Definition:Open Set (Topology)|open set]] of $\R$ which contains $0$.
From [[Open Sets in Real Number Line]], there exists an [[Definition:Open Real Interval|open interval]] $I$ of the form:
:$I := \openint {-a} b \subseteq U$
By the [[Axiom of Archimedes]]:
:$\exists n \in \N: n > \dfrac 1 b$
and so... | Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval | https://proofwiki.org/wiki/Zero_is_Omega-Accumulation_Point_of_Integer_Reciprocal_Space_Union_with_Closed_Interval | https://proofwiki.org/wiki/Zero_is_Omega-Accumulation_Point_of_Integer_Reciprocal_Space_Union_with_Closed_Interval | [
"Integer Reciprocal Space",
"Examples of Omega-Accumulation Points"
] | [
"Definition:Set",
"Definition:Integer Reciprocal Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Uncountable/Set",
"Definition:Real Interval/Closed",
"Definition:Omega-Accumulation Point"
] | [
"Definition:Open Set/Topology",
"Open Sets in Real Number Line",
"Definition:Real Interval/Open",
"Axiom of Archimedes",
"Definition:Open Set/Topology",
"Definition:Countably Infinite/Set",
"Definition:Element",
"Definition:Omega-Accumulation Point"
] |
proofwiki-8517 | Zero is not Condensation Point of Integer Reciprocal Space Union with Closed Interval | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.
Let $B$ be the uncountable set:
:$B := A \cup \closedint 2 3$
where $\closedint 2 3$ is a closed interval of $\R$.
... | Let $U$ be an open set of $\R$ which contains $0$.
From Open Sets in Real Number Line, there exists an open interval $I$ of the form:
:$I := \openint {-a} b \subseteq U$
From Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval, there is a countably infinite number of points of $B$ in... | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
... | Let $U$ be an [[Definition:Open Set (Topology)|open set]] of $\R$ which contains $0$.
From [[Open Sets in Real Number Line]], there exists an [[Definition:Open Real Interval|open interval]] $I$ of the form:
:$I := \openint {-a} b \subseteq U$
From [[Zero is Omega-Accumulation Point of Integer Reciprocal Space Union w... | Zero is not Condensation Point of Integer Reciprocal Space Union with Closed Interval | https://proofwiki.org/wiki/Zero_is_not_Condensation_Point_of_Integer_Reciprocal_Space_Union_with_Closed_Interval | https://proofwiki.org/wiki/Zero_is_not_Condensation_Point_of_Integer_Reciprocal_Space_Union_with_Closed_Interval | [
"Integer Reciprocal Space",
"Examples of Condensation Points"
] | [
"Definition:Set",
"Definition:Integer Reciprocal Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Uncountable/Set",
"Definition:Real Interval/Closed",
"Definition:Condensation Point"
] | [
"Definition:Open Set/Topology",
"Open Sets in Real Number Line",
"Definition:Real Interval/Open",
"Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval",
"Definition:Countably Infinite/Set",
"Definition:Uncountable/Set"
] |
proofwiki-8518 | Integer Reciprocal Space contains Cauchy Sequence with no Limit | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.
Then $A$ has a Cauchy sequence which has no limit in $A$. | Let $\sequence {x_n}$ be the sequence $1, \dfrac 1 2, \dfrac 1 3, \ldots$
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.
By the Axiom of Archimedes:
:$\exists \N \in n: n > \dfrac 1 \epsilon$
and so:
:$\exists \N \in n: \dfrac 1 n < \epsilon$
As:
:$0 < \dfrac 1 {n + 1} < \dfrac 1 n$
it follows that:
:... | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all points on $\R$ defined as:
:$A := \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
... | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence]] $1, \dfrac 1 2, \dfrac 1 3, \ldots$
Let $\epsilon \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
By the [[Axiom of Archimedes]]:
:$\exists \N \in n: n > \dfrac 1 \epsilon$
and so:
:$\exists \N \in n: \d... | Integer Reciprocal Space contains Cauchy Sequence with no Limit | https://proofwiki.org/wiki/Integer_Reciprocal_Space_contains_Cauchy_Sequence_with_no_Limit | https://proofwiki.org/wiki/Integer_Reciprocal_Space_contains_Cauchy_Sequence_with_no_Limit | [
"Integer Reciprocal Space",
"Examples of Cauchy Sequences"
] | [
"Definition:Set",
"Definition:Integer Reciprocal Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Cauchy Sequence",
"Definition:Limit of Sequence/Topological Space"
] | [
"Definition:Real Sequence",
"Definition:Strictly Positive/Real Number",
"Axiom of Archimedes",
"Definition:Cauchy Sequence",
"Sequence of Powers of Reciprocals is Null Sequence",
"Definition:Basic Null Sequence"
] |
proofwiki-8519 | Integer Reciprocal Space with Zero is not Locally Connected | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology.
Then $A$ is not locally connected. | We have:
:Integer Reciprocal Space with Zero is Totally Separated
:Totally Separated Space is Totally Disconnected
:Totally Disconnected and Locally Connected Space is Discrete
We also have:
:Topological Space is Discrete iff All Points are Isolated
:Zero is Limit Point of Integer Reciprocal Space
From definition of li... | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all points on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] with [[Definition:Zero (Number)|zero]] under the [[Definition:Euclidean Topology ... | We have:
:[[Integer Reciprocal Space with Zero is Totally Separated]]
:[[Totally Separated Space is Totally Disconnected]]
:[[Totally Disconnected and Locally Connected Space is Discrete]]
We also have:
:[[Topological Space is Discrete iff All Points are Isolated]]
:[[Zero is Limit Point of Integer Reciprocal Space]]
... | Integer Reciprocal Space with Zero is not Locally Connected | https://proofwiki.org/wiki/Integer_Reciprocal_Space_with_Zero_is_not_Locally_Connected | https://proofwiki.org/wiki/Integer_Reciprocal_Space_with_Zero_is_not_Locally_Connected | [
"Integer Reciprocal Space",
"Examples of Locally Connected Spaces"
] | [
"Definition:Set",
"Definition:Integer Reciprocal Space",
"Definition:Zero (Number)",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Locally Connected Space"
] | [
"Integer Reciprocal Space with Zero is Totally Separated",
"Totally Separated Space is Totally Disconnected",
"Totally Disconnected and Locally Connected Space is Discrete",
"Topological Space is Discrete iff All Points are Isolated",
"Zero is Limit Point of Integer Reciprocal Space",
"Definition:Limit Po... |
proofwiki-8520 | Discrete Space is Locally Connected | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is locally connected. | Let $T = \struct {S, \tau}$ be a discrete space.
From Discrete Space is Locally Path-Connected, $T$ is locally path-connected.
From Locally Path-Connected Space is Locally Connected, it follows that $T$ is locally connected.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is [[Definition:Locally Connected Space|locally connected]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Space|discrete space]].
From [[Discrete Space is Locally Path-Connected]], $T$ is [[Definition:Locally Path-Connected Space|locally path-connected]].
From [[Locally Path-Connected Space is Locally Connected]], it follows that $T$ is [[Definition:Locally Connected... | Discrete Space is Locally Connected | https://proofwiki.org/wiki/Discrete_Space_is_Locally_Connected | https://proofwiki.org/wiki/Discrete_Space_is_Locally_Connected | [
"Discrete Topologies",
"Examples of Locally Connected Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Locally Connected Space"
] | [
"Definition:Discrete Topology",
"Discrete Space is Locally Path-Connected",
"Definition:Locally Path-Connected Space",
"Locally Path-Connected Space is Locally Connected",
"Definition:Locally Connected Space"
] |
proofwiki-8521 | Local Connectedness is not Preserved under Continuous Mapping | Let $\struct {A, \tau_1}$ and $\struct {B, \tau_2}$ be topological spaces.
Let $f: A \to B$ be a continuous mapping.
Let $\struct {\Img f, \tau_3}$ be the image of $f$ with the subspace topology of $B$.
Let $\struct {A, \tau_1}$ be locally connected.
Then it is not necessarily the case that $\struct {\Img f, \tau_3}$ i... | ;Proof by Counterexample
Let $\struct {A, \tau_1}$ be an arbitrary countable discrete space.
Let $B \subseteq \R$ be the set of all points of $\R$ defined as:
:$B := \set 0 \cup \set {\dfrac 1 n: n \in \Z_{>0} }$
Let $\struct {B, \tau_2}$ be the integer reciprocal space with zero under the usual (Euclidean) topology.
F... | Let $\struct {A, \tau_1}$ and $\struct {B, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: A \to B$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]].
Let $\struct {\Img f, \tau_3}$ be the [[Definition:Image of Mapping|image of $f$]] with [[Definition:Topological... | ;[[Proof by Counterexample]]
Let $\struct {A, \tau_1}$ be an [[Definition:Arbitrary|arbitrary]] [[Definition:Countable Discrete Space|countable discrete space]].
Let $B \subseteq \R$ be the [[Definition:Set|set]] of all [[Definition:Point of Set|points]] of $\R$ defined as:
:$B := \set 0 \cup \set {\dfrac 1 n: n \in ... | Local Connectedness is not Preserved under Continuous Mapping | https://proofwiki.org/wiki/Local_Connectedness_is_not_Preserved_under_Continuous_Mapping | https://proofwiki.org/wiki/Local_Connectedness_is_not_Preserved_under_Continuous_Mapping | [
"Locally Connected Spaces",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Topological Subspace",
"Definition:Locally Connected Space",
"Definition:Locally Connected Space"
] | [
"Proof by Counterexample",
"Definition:Arbitrary",
"Definition:Discrete Topology/Countable",
"Definition:Set",
"Definition:Element",
"Definition:Integer Reciprocal Space",
"Definition:Zero (Number)",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Discrete Space is Locally Connect... |
proofwiki-8522 | Integer Reciprocal Space with Zero is Totally Separated | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology.
Then $A$ is totally separated. | Let $a, b \in A$ such that $a < b$.
From Between two Rational Numbers exists Irrational Number:
:$\exists \alpha \in \R \setminus \Q: a < \alpha < b$
Because $\forall x \in A: x \in \Q$ it follows that $\alpha \notin A$.
Consider the half-open intervals $S = \hointr 0 \alpha$ and $T = \hointl \alpha 1$
Let:
:$U := S \c... | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all points on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] with [[Definition:Zero (Number)|zero]] under the [[Definition:Euclidean Topology ... | Let $a, b \in A$ such that $a < b$.
From [[Between two Rational Numbers exists Irrational Number]]:
:$\exists \alpha \in \R \setminus \Q: a < \alpha < b$
Because $\forall x \in A: x \in \Q$ it follows that $\alpha \notin A$.
Consider the [[Definition:Half-Open Real Interval|half-open intervals]] $S = \hointr 0 \alp... | Integer Reciprocal Space with Zero is Totally Separated | https://proofwiki.org/wiki/Integer_Reciprocal_Space_with_Zero_is_Totally_Separated | https://proofwiki.org/wiki/Integer_Reciprocal_Space_with_Zero_is_Totally_Separated | [
"Integer Reciprocal Space",
"Examples of Totally Separated Spaces"
] | [
"Definition:Set",
"Definition:Integer Reciprocal Space",
"Definition:Zero (Number)",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Totally Separated Space"
] | [
"Between two Rational Numbers exists Irrational Number",
"Definition:Real Interval/Half-Open",
"Definition:Separation (Topology)",
"Definition:Totally Separated Space"
] |
proofwiki-8523 | Components of Integer Reciprocal Space with Zero are Single Points | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology.
Then the components of $A$ are singletons. | From Integer Reciprocal Space with Zero is Totally Separated, $A$ is totally separated.
From Totally Separated Space is Totally Disconnected, $A$ is totally disconnected.
The result follows by definition of totally disconnected.
{{qed}} | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all [[Definition:Point of Set|points]] on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] with [[Definition:Zero (Number)|zero]] under the [[De... | From [[Integer Reciprocal Space with Zero is Totally Separated]], $A$ is [[Definition:Totally Separated Space|totally separated]].
From [[Totally Separated Space is Totally Disconnected]], $A$ is [[Definition:Totally Disconnected Space|totally disconnected]].
The result follows by definition of [[Definition:Totally ... | Components of Integer Reciprocal Space with Zero are Single Points | https://proofwiki.org/wiki/Components_of_Integer_Reciprocal_Space_with_Zero_are_Single_Points | https://proofwiki.org/wiki/Components_of_Integer_Reciprocal_Space_with_Zero_are_Single_Points | [
"Integer Reciprocal Space",
"Examples of Topological Components"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Integer Reciprocal Space",
"Definition:Zero (Number)",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Component (Topology)",
"Definition:Singleton"
] | [
"Integer Reciprocal Space with Zero is Totally Separated",
"Definition:Totally Separated Space",
"Totally Separated Space is Totally Disconnected",
"Definition:Totally Disconnected Space",
"Definition:Totally Disconnected Space"
] |
proofwiki-8524 | Quasicomponents of Integer Reciprocal Space with Zero are Single Points | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology.
Then the quasicomponents of $A$ are singletons. | From Integer Reciprocal Space with Zero is Totally Separated, $A$ is totally separated.
The result follows from Totally Separated iff Quasicomponents are Singletons.
{{qed}} | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all [[Definition:Point of Set|points]] on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] with [[Definition:Zero (Number)|zero]] under the [[De... | From [[Integer Reciprocal Space with Zero is Totally Separated]], $A$ is [[Definition:Totally Separated Space|totally separated]].
The result follows from [[Totally Separated iff Quasicomponents are Singletons]].
{{qed}} | Quasicomponents of Integer Reciprocal Space with Zero are Single Points | https://proofwiki.org/wiki/Quasicomponents_of_Integer_Reciprocal_Space_with_Zero_are_Single_Points | https://proofwiki.org/wiki/Quasicomponents_of_Integer_Reciprocal_Space_with_Zero_are_Single_Points | [
"Integer Reciprocal Space",
"Examples of Quasicomponents"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Integer Reciprocal Space",
"Definition:Zero (Number)",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Quasicomponent",
"Definition:Singleton"
] | [
"Integer Reciprocal Space with Zero is Totally Separated",
"Definition:Totally Separated Space",
"Equivalence of Definitions of Totally Separated Space"
] |
proofwiki-8525 | Integer Reciprocal Space with Zero is not Extremally Disconnected | Let $A \subseteq \R$ be the set of all points on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology.
Then $A$ is not extremally disconnected. | $\struct {A, \tau_d}$ is a metric space.
We have:
:Extremally Disconnected Metric Space is Discrete
We also have:
:Topological Space is Discrete iff All Points are Isolated
:Zero is Limit Point of Integer Reciprocal Space
From definition of limit point:
:$0$ is not an isolated point of $A$
Hence integer reciprocal spac... | Let $A \subseteq \R$ be the [[Definition:Set|set]] of all points on $\R$ defined as:
:$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$
Let $\struct {A, \tau_d}$ be the [[Definition:Integer Reciprocal Space|integer reciprocal space]] with [[Definition:Zero (Number)|zero]] under the [[Definition:Euclidean Topology ... | $\struct {A, \tau_d}$ is a [[Definition:Metric Space|metric space]].
We have:
:[[Extremally Disconnected Metric Space is Discrete]]
We also have:
:[[Topological Space is Discrete iff All Points are Isolated]]
:[[Zero is Limit Point of Integer Reciprocal Space]]
From definition of [[Definition:Limit Point of Set|limi... | Integer Reciprocal Space with Zero is not Extremally Disconnected | https://proofwiki.org/wiki/Integer_Reciprocal_Space_with_Zero_is_not_Extremally_Disconnected | https://proofwiki.org/wiki/Integer_Reciprocal_Space_with_Zero_is_not_Extremally_Disconnected | [
"Integer Reciprocal Space",
"Examples of Extremally Disconnected Spaces"
] | [
"Definition:Set",
"Definition:Integer Reciprocal Space",
"Definition:Zero (Number)",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Extremally Disconnected Space"
] | [
"Definition:Metric Space",
"Extremally Disconnected Metric Space is Discrete",
"Topological Space is Discrete iff All Points are Isolated",
"Zero is Limit Point of Integer Reciprocal Space",
"Definition:Limit Point/Topology/Set",
"Definition:Isolated Point (Topology)",
"Definition:Integer Reciprocal Spa... |
proofwiki-8526 | Interior of Intersection may not equal Intersection of Interiors | Let $T$ be a topological space.
Let $\mathbb H$ be a set of subsets of $T$.
That is, let $\mathbb H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$.
Then it is not necessarily the case that:
:$\ds \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ = \bigcap_{H \mathop \in \mathbb H} H^\circ$
where $H... | From Intersection of Interiors contains Interior of Intersection it is seen that it is always the case that:
:$\ds \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ \subseteq \bigcap_{H \mathop \in \mathbb H} H^\circ$
From Interior of Finite Intersection equals Intersection of Interiors it is seen that if $\mathbb H$ ... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $\mathbb H$ be a [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $T$.
That is, let $\mathbb H \subseteq \powerset T$ where $\powerset T$ is the [[Definition:Power Set|power set]] of $T$.
Then it is not necessarily the case that:
:$\ds ... | From [[Intersection of Interiors contains Interior of Intersection]] it is seen that it is always the case that:
:$\ds \paren {\bigcap_{H \mathop \in \mathbb H} H}^\circ \subseteq \bigcap_{H \mathop \in \mathbb H} H^\circ$
From [[Interior of Finite Intersection equals Intersection of Interiors]] it is seen that if $\m... | Interior of Intersection may not equal Intersection of Interiors | https://proofwiki.org/wiki/Interior_of_Intersection_may_not_equal_Intersection_of_Interiors | https://proofwiki.org/wiki/Interior_of_Intersection_may_not_equal_Intersection_of_Interiors | [
"Set Interiors",
"Set Intersection"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Interior (Topology)"
] | [
"Intersection of Interiors contains Interior of Intersection",
"Interior of Finite Intersection equals Intersection of Interiors",
"Definition:Finite Set",
"Definition:Infinite Set",
"Proof by Counterexample",
"Definition:Real Interval/Open",
"Open Sets in Real Number Line",
"Definition:Open Set/Topol... |
proofwiki-8527 | Interior of Union of Adjacent Open Intervals | Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the union of the adjacent open intervals:
:$A := \openint a b \cup \openint b c$
Then:
:$A = A^\circ$
where $A^\circ$ is the interior of $A$. | From Open Sets in Real Number Line, $A$ is open in $\R$.
The result follows from Interior of Open Set.
{{qed}} | Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the [[Definition:Union of Adjacent Open Intervals|union of the adjacent open intervals]]:
:$A := \openint a b \cup \openint b c$
Then:
:$A = A^\circ$
where $A^\circ$ is the [[Definition:Interior (Topology)|interior]] of $A$. | From [[Open Sets in Real Number Line]], $A$ is [[Definition:Open Set (Topology)|open]] in $\R$.
The result follows from [[Interior of Open Set]].
{{qed}} | Interior of Union of Adjacent Open Intervals | https://proofwiki.org/wiki/Interior_of_Union_of_Adjacent_Open_Intervals | https://proofwiki.org/wiki/Interior_of_Union_of_Adjacent_Open_Intervals | [
"Union of Adjacent Open Intervals",
"Examples of Set Interiors"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Interior (Topology)"
] | [
"Open Sets in Real Number Line",
"Definition:Open Set/Topology",
"Interior of Open Set"
] |
proofwiki-8528 | Interior of Closure of Interior of Union of Adjacent Open Intervals | Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the union of the adjacent open intervals:
:$A := \openint a b \cup \openint b c$
Then:
:$A^{\circ - \circ} = A^{- \circ} = \openint a c$
where:
:$A^\circ$ is the interior of $A$
:$A^-$ is the closure of $A$. | From Interior of Union of Adjacent Open Intervals:
:$A^\circ = A$
From Closure of Union of Adjacent Open Intervals:
:$A^- = \closedint a c$
From Interior of Closed Real Interval is Open Real Interval:
:$\closedint a c^\circ = \openint a c$
whence the result.
{{qed}} | Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the [[Definition:Union of Adjacent Open Intervals|union of the adjacent open intervals]]:
:$A := \openint a b \cup \openint b c$
Then:
:$A^{\circ - \circ} = A^{- \circ} = \openint a c$
where:
:$A^\circ$ is the [[Definition:Interior (Topology)|interior]] of $A$
:$A^-$ ... | From [[Interior of Union of Adjacent Open Intervals]]:
:$A^\circ = A$
From [[Closure of Union of Adjacent Open Intervals]]:
:$A^- = \closedint a c$
From [[Interior of Closed Real Interval is Open Real Interval]]:
:$\closedint a c^\circ = \openint a c$
whence the result.
{{qed}} | Interior of Closure of Interior of Union of Adjacent Open Intervals | https://proofwiki.org/wiki/Interior_of_Closure_of_Interior_of_Union_of_Adjacent_Open_Intervals | https://proofwiki.org/wiki/Interior_of_Closure_of_Interior_of_Union_of_Adjacent_Open_Intervals | [
"Union of Adjacent Open Intervals",
"Examples of Set Interiors",
"Examples of Set Closures"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)"
] | [
"Interior of Union of Adjacent Open Intervals",
"Closure of Union of Adjacent Open Intervals",
"Interior of Closed Real Interval is Open Real Interval"
] |
proofwiki-8529 | Closure of Interior of Closure of Union of Adjacent Open Intervals | Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the union of the adjacent open intervals:
:$A := \openint a b \cup \openint b c$
Then:
:$A^{- \circ -} = A^{\circ -} = A^- = \closedint a c$
where:
:$A^\circ$ is the interior of $A$
:$A^-$ is the closure of $A$. | {{begin-eqn}}
{{eqn | l = A^{\circ -}
| r = A^-
| c = Interior of Union of Adjacent Open Intervals: $A^\circ = A$
}}
{{eqn | r = \closedint a c
| c = Closure of Union of Adjacent Open Intervals
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = A^{- \circ -}
| r = \openint a c^-
| c = Interior... | Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the [[Definition:Union of Adjacent Open Intervals|union of the adjacent open intervals]]:
:$A := \openint a b \cup \openint b c$
Then:
:$A^{- \circ -} = A^{\circ -} = A^- = \closedint a c$
where:
:$A^\circ$ is the [[Definition:Interior (Topology)|interior]] of $A$
:$A... | {{begin-eqn}}
{{eqn | l = A^{\circ -}
| r = A^-
| c = [[Interior of Union of Adjacent Open Intervals]]: $A^\circ = A$
}}
{{eqn | r = \closedint a c
| c = [[Closure of Union of Adjacent Open Intervals]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = A^{- \circ -}
| r = \openint a c^-
| c =... | Closure of Interior of Closure of Union of Adjacent Open Intervals | https://proofwiki.org/wiki/Closure_of_Interior_of_Closure_of_Union_of_Adjacent_Open_Intervals | https://proofwiki.org/wiki/Closure_of_Interior_of_Closure_of_Union_of_Adjacent_Open_Intervals | [
"Union of Adjacent Open Intervals",
"Examples of Set Interiors",
"Examples of Set Closures"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)"
] | [
"Interior of Union of Adjacent Open Intervals",
"Closure of Union of Adjacent Open Intervals",
"Interior of Closure of Interior of Union of Adjacent Open Intervals",
"Closure of Open Ball in Metric Space"
] |
proofwiki-8530 | Exterior of Exterior of Union of Adjacent Open Intervals | Let $A$ be the union of the adjacent open intervals:
:$A := \openint a b \cup \openint b c$
Then:
:$A^{ee} = \openint a c$
where $A^e$ is the exterior of $A$. | By definition of exterior, $A^e$ is the complement relative to $\R$ of the closure of $A$ in $\R$.
Thus:
{{begin-eqn}}
{{eqn | l = A^{ee}
| r = \paren {\relcomp \R {A^-} }^e
| c = {{Defof|Exterior (Topology)|Exterior}}
}}
{{eqn | r = \paren {\relcomp \R {\closedint a c} }^e
| c = Closure of Interior o... | Let $A$ be the [[Definition:Union of Adjacent Open Intervals|union of the adjacent open intervals]]:
:$A := \openint a b \cup \openint b c$
Then:
:$A^{ee} = \openint a c$
where $A^e$ is the [[Definition:Exterior (Topology)|exterior]] of $A$. | By definition of [[Definition:Exterior (Topology)|exterior]], $A^e$ is the [[Definition:Relative Complement|complement relative to $\R$]] of the [[Definition:Closure (Topology)|closure]] of $A$ in $\R$.
Thus:
{{begin-eqn}}
{{eqn | l = A^{ee}
| r = \paren {\relcomp \R {A^-} }^e
| c = {{Defof|Exterior (Topo... | Exterior of Exterior of Union of Adjacent Open Intervals | https://proofwiki.org/wiki/Exterior_of_Exterior_of_Union_of_Adjacent_Open_Intervals | https://proofwiki.org/wiki/Exterior_of_Exterior_of_Union_of_Adjacent_Open_Intervals | [
"Union of Adjacent Open Intervals",
"Examples of Set Exteriors"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Exterior (Topology)"
] | [
"Definition:Exterior (Topology)",
"Definition:Relative Complement",
"Definition:Closure (Topology)",
"Closure of Interior of Closure of Union of Adjacent Open Intervals",
"Closure of Finite Union equals Union of Closures",
"Closure of Open Ball in Metric Space"
] |
proofwiki-8531 | Interior may not equal Exterior of Exterior | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$ be a subset of the underlying set $S$ of $T$.
Let $A^e$ be the exterior of $A$.
Let $A^\circ$ be the interior of $A$.
Then it is not necessarily the case that:
:$A^{ee} = A^\circ$ | We have from Interior is Subset of Exterior of Exterior:
:$A^\circ \subseteq A^{ee}$
It remains to be shown that there exist $A \subseteq S$ such that:
:$A^\circ \ne A^{ee}$
Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the union of the adjacent open intervals:
:$A := \openint a b \cup \openint b c$
From Exterior o... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq S$ be a [[Definition:Subset|subset]] of the [[Definition:Underlying Set of Topological Space|underlying set]] $S$ of $T$.
Let $A^e$ be the [[Definition:Exterior (Topology)|exterior]] of $A$.
Let $A^\circ$ be the [... | We have from [[Interior is Subset of Exterior of Exterior]]:
:$A^\circ \subseteq A^{ee}$
It remains to be shown that there exist $A \subseteq S$ such that:
:$A^\circ \ne A^{ee}$
Let $a, b, c \in R$ where $a < b < c$.
Let $A$ be the [[Definition:Union of Adjacent Open Intervals|union of the adjacent open intervals]]... | Interior may not equal Exterior of Exterior/Proof 1 | https://proofwiki.org/wiki/Interior_may_not_equal_Exterior_of_Exterior | https://proofwiki.org/wiki/Interior_may_not_equal_Exterior_of_Exterior/Proof_1 | [
"Interior may not equal Exterior of Exterior",
"Set Exteriors",
"Set Interiors"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Underlying Set/Topological Space",
"Definition:Exterior (Topology)",
"Definition:Interior (Topology)"
] | [
"Interior is Subset of Exterior of Exterior",
"Definition:Union of Adjacent Open Intervals",
"Exterior of Exterior of Union of Adjacent Open Intervals",
"Interior of Union of Adjacent Open Intervals"
] |
proofwiki-8532 | Interior may not equal Exterior of Exterior | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$ be a subset of the underlying set $S$ of $T$.
Let $A^e$ be the exterior of $A$.
Let $A^\circ$ be the interior of $A$.
Then it is not necessarily the case that:
:$A^{ee} = A^\circ$ | Proof by Counterexample:
Let $\struct {S, \preccurlyeq}$ be a totally ordered set.
Let $T = \struct {S, \tau}$ be the Sorgenfrey line on $\struct {S, \preccurlyeq}$.
Let $A \subseteq S$ denote the subset of $S$ defined as:
:$A = \openint a {+\infty}$
By Exterior of Exterior in Sorgenfrey Line is not necessarily Interio... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq S$ be a [[Definition:Subset|subset]] of the [[Definition:Underlying Set of Topological Space|underlying set]] $S$ of $T$.
Let $A^e$ be the [[Definition:Exterior (Topology)|exterior]] of $A$.
Let $A^\circ$ be the [... | [[Proof by Counterexample]]:
Let $\struct {S, \preccurlyeq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $T = \struct {S, \tau}$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]] on $\struct {S, \preccurlyeq}$.
Let $A \subseteq S$ denote the [[Definition:Subset|subset]] of $S$ defined as:
:$... | Interior may not equal Exterior of Exterior/Proof 2 | https://proofwiki.org/wiki/Interior_may_not_equal_Exterior_of_Exterior | https://proofwiki.org/wiki/Interior_may_not_equal_Exterior_of_Exterior/Proof_2 | [
"Interior may not equal Exterior of Exterior",
"Set Exteriors",
"Set Interiors"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Underlying Set/Topological Space",
"Definition:Exterior (Topology)",
"Definition:Interior (Topology)"
] | [
"Proof by Counterexample",
"Definition:Totally Ordered Set",
"Definition:Sorgenfrey Line",
"Definition:Subset",
"Exterior of Exterior in Sorgenfrey Line is not necessarily Interior"
] |
proofwiki-8533 | Open Real Interval is Regular Open | Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.
Let $\openint a b$ be an open interval of $\R$.
Then $\openint a b$ is regular open in $\struct {\R, \tau_d}$. | From Open Sets in Real Number Line, $\openint a b$ is open in $\struct {\R, \tau_d}$.
From Closure of Open Real Interval is Closed Real Interval:
:$\openint a b^- = \closedint a b$
where $\openint a b^-$ denotes the closure of $\openint a b$.
From Interior of Closed Real Interval is Open Real Interval:
:$\closedint a b... | Let $\struct {\R, \tau}$ denote the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\openint a b$ be an [[Definition:Open Real Interval|open interval]] of $\R$.
Then $\openint a b$ is [[Definition:Regular Open Set|regular open]] in $\struct {\R, \tau... | From [[Open Sets in Real Number Line]], $\openint a b$ is [[Definition:Open Set (Topology)|open]] in $\struct {\R, \tau_d}$.
From [[Closure of Open Real Interval is Closed Real Interval]]:
:$\openint a b^- = \closedint a b$
where $\openint a b^-$ denotes the [[Definition:Closure (Topology)|closure]] of $\openint a b$.... | Open Real Interval is Regular Open | https://proofwiki.org/wiki/Open_Real_Interval_is_Regular_Open | https://proofwiki.org/wiki/Open_Real_Interval_is_Regular_Open | [
"Real Number Line with Euclidean Topology",
"Real Intervals",
"Examples of Regular Open Sets"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval/Open",
"Definition:Regular Open Set"
] | [
"Open Sets in Real Number Line",
"Definition:Open Set/Topology",
"Closure of Open Real Interval is Closed Real Interval",
"Definition:Closure (Topology)",
"Interior of Closed Real Interval is Open Real Interval",
"Definition:Interior (Topology)",
"Definition:Regular Open Set"
] |
proofwiki-8534 | Closure of Open Real Interval is Closed Real Interval | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\openint a b$ be an open interval of $\R$.
Then the closure of $\openint a b$ is the closed interval $\closedint a b$. | From Limit Points of Open Real Interval, the limit points of $\openint a b$ consist of:
:the points $\openint a b$ itself
and
:the points $a$ and $b$.
By definition, the closure of $\openint a b$ is the union of $\openint a b$ and its limit points.
Hence the result, by definition of the closed interval $\closedint a b$... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\openint a b$ be an [[Definition:Open Real Interval|open interval]] of $\R$.
Then the [[Definition:Closure (Topology)|closure]] of $\openint a b$ is the [[Definition:C... | From [[Limit Points of Open Real Interval]], the [[Definition:Limit Point of Set|limit points]] of $\openint a b$ consist of:
:the points $\openint a b$ itself
and
:the points $a$ and $b$.
By definition, the [[Definition:Closure (Topology)|closure]] of $\openint a b$ is the [[Definition:Set Union|union]] of $\openint ... | Closure of Open Real Interval is Closed Real Interval | https://proofwiki.org/wiki/Closure_of_Open_Real_Interval_is_Closed_Real_Interval | https://proofwiki.org/wiki/Closure_of_Open_Real_Interval_is_Closed_Real_Interval | [
"Open Sets",
"Set Closures",
"Real Number Line with Euclidean Topology"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval/Open",
"Definition:Closure (Topology)",
"Definition:Real Interval/Closed"
] | [
"Limit Points of Open Real Interval",
"Definition:Limit Point/Topology/Set",
"Definition:Closure (Topology)",
"Definition:Set Union",
"Definition:Limit Point/Topology/Set",
"Definition:Real Interval/Closed",
"Category:Open Sets",
"Category:Set Closures",
"Category:Real Number Line with Euclidean Top... |
proofwiki-8535 | Limit Points of Open Real Interval | Let $\struct {\R, \tau_d}$ be the real number line under the usual (Euclidean) topology.
Let $\openint a b$ be an open interval of $\R$.
Then the limit points of $\openint a b$ are:
:all the points in $\openint a b$
as well as:
:the points $a$ and $b$. | Let $x \in \openint a b$.
Then by definition $a < x < b$.
Let $N_x$ be an open neighborhood of $x$ in $\R$.
So by Open Superset is Open Neighborhood, $N_x$ is an open set of $\R$ containing $x$.
By Open Sets in Real Number Line, $N_x$ is a countable union of pairwise disjoint open intervals.
As $x \in N_x$, it follows ... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number line]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $\openint a b$ be an [[Definition:Open Real Interval|open interval]] of $\R$.
Then the [[Definition:Limit Point of Set|limit points]] of ... | Let $x \in \openint a b$.
Then by definition $a < x < b$.
Let $N_x$ be an [[Definition:Open Neighborhood of Point|open neighborhood of $x$]] in $\R$.
So by [[Open Superset is Open Neighborhood]], $N_x$ is an [[Definition:Open Set (Topology)|open set]] of $\R$ containing $x$.
By [[Open Sets in Real Number Line]], $N... | Limit Points of Open Real Interval | https://proofwiki.org/wiki/Limit_Points_of_Open_Real_Interval | https://proofwiki.org/wiki/Limit_Points_of_Open_Real_Interval | [
"Examples of Limit Points",
"Real Intervals"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval/Open",
"Definition:Limit Point/Topology/Set"
] | [
"Definition:Open Neighborhood/Point",
"Open Superset is Open Neighborhood",
"Definition:Open Set/Topology",
"Open Sets in Real Number Line",
"Definition:Set Union/Countable Union",
"Definition:Pairwise Disjoint",
"Definition:Real Interval/Open",
"Real Numbers are Densely Ordered",
"Definition:Open N... |
proofwiki-8536 | Interior of Closed Real Interval is Open Real Interval | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\closedint a b$ be a closed interval of $\R$.
Then the interior of $\closedint a b$ is the open interval $\openint a b$. | By definition, the interior of $\closedint a b$ is the largest open set contained in $\closedint a b$.
From Open Sets in Real Number Line it follows that $\openint a b$ is an open set of $\struct {\R, \tau_d}$.
By definition of open interval, $\openint a b$ is contained in $\closedint a b$.
Suppose $U$ is an open set o... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed interval]] of $\R$.
Then the [[Definition:Interior (Topology)|interior]] of $\closedint a b$ is the [[Def... | By definition, the [[Definition:Interior (Topology)|interior]] of $\closedint a b$ is the largest [[Definition:Open Set (Topology)|open set]] contained in $\closedint a b$.
From [[Open Sets in Real Number Line]] it follows that $\openint a b$ is an [[Definition:Open Set (Topology)|open set]] of $\struct {\R, \tau_d}$.... | Interior of Closed Real Interval is Open Real Interval | https://proofwiki.org/wiki/Interior_of_Closed_Real_Interval_is_Open_Real_Interval | https://proofwiki.org/wiki/Interior_of_Closed_Real_Interval_is_Open_Real_Interval | [
"Closed Sets",
"Examples of Set Interiors",
"Real Number Line with Euclidean Topology"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval/Closed",
"Definition:Interior (Topology)",
"Definition:Real Interval/Open"
] | [
"Definition:Interior (Topology)",
"Definition:Open Set/Topology",
"Open Sets in Real Number Line",
"Definition:Open Set/Topology",
"Definition:Real Interval/Open",
"Definition:Open Set/Topology",
"Open Sets in Real Number Line",
"Real Numbers are Densely Ordered",
"Definition:Open Set/Topology",
"... |
proofwiki-8537 | Union of Regular Open Sets is not necessarily Regular Open | Let $T = \struct {S, \tau}$ be a topological space.
Let $U$ and $V$ be regular open sets of $T$.
Then $U \cup V$ is not also necessarily a regular open set of $T$. | {{Recall|Regular Open Set|regular open set}}
{{:Definition:Regular Open Set}}
;Proof by Counterexample
By Open Real Interval is Regular Open, the open real intervals:
:$\openint 0 {\dfrac 1 2}, \openint {\dfrac 1 2} 1$
are both regular open sets of $\R$.
Consider $A$, the union of the adjacent open intervals:
:$A := \o... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U$ and $V$ be [[Definition:Regular Open Set|regular open sets]] of $T$.
Then $U \cup V$ is not also necessarily a [[Definition:Regular Open Set|regular open set]] of $T$. | {{Recall|Regular Open Set|regular open set}}
{{:Definition:Regular Open Set}}
;[[Proof by Counterexample]]
By [[Open Real Interval is Regular Open]], the [[Definition:Open Real Interval|open real intervals]]:
:$\openint 0 {\dfrac 1 2}, \openint {\dfrac 1 2} 1$
are both [[Definition:Regular Open Set|regular open sets... | Union of Regular Open Sets is not necessarily Regular Open | https://proofwiki.org/wiki/Union_of_Regular_Open_Sets_is_not_necessarily_Regular_Open | https://proofwiki.org/wiki/Union_of_Regular_Open_Sets_is_not_necessarily_Regular_Open | [
"Regular Open Sets",
"Open Sets",
"Set Union"
] | [
"Definition:Topological Space",
"Definition:Regular Open Set",
"Definition:Regular Open Set"
] | [
"Proof by Counterexample",
"Open Real Interval is Regular Open",
"Definition:Real Interval/Open",
"Definition:Regular Open Set",
"Definition:Union of Adjacent Open Intervals",
"Interior of Closure of Interior of Union of Adjacent Open Intervals",
"Definition:Regular Open Set"
] |
proofwiki-8538 | Closure of Intersection may not equal Intersection of Closures | Let $T = \struct {S, \tau}$ be a topological space.
Let $H_1$ and $H_2$ be subsets of $S$.
Let ${H_1}^-$ and ${H_2}^-$ denote the closures of $H_1$ and $H_2$ respectively.
Then it is not necessarily the case that:
:$\paren {H_1 \cap H_2}^- = {H_1}^- \cap {H_2}^-$ | Let $\struct {\R, \tau}$ be the real number line under the usual (Euclidean) topology.
Let $\Q$ denote the set of rational numbers.
Let $\R \setminus \Q$ denote the set of irrational numbers.
From Closure of Intersection of Rationals and Irrationals is Empty Set:
:$\paren {\Q \cap \paren {\R \setminus \Q} }^- = \O$
Fro... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H_1$ and $H_2$ be [[Definition:Subset|subsets]] of $S$.
Let ${H_1}^-$ and ${H_2}^-$ denote the [[Definition:Closure (Topology)|closures]] of $H_1$ and $H_2$ respectively.
Then it is not necessarily the case that:
:$\paren {H_1... | Let $\struct {\R, \tau}$ be the [[Definition:Real Number Line|real number line]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $\Q$ denote the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]].
Let $\R \setminus \Q$ denote the [[Definition:Set|... | Closure of Intersection may not equal Intersection of Closures/Proof 1 | https://proofwiki.org/wiki/Closure_of_Intersection_may_not_equal_Intersection_of_Closures | https://proofwiki.org/wiki/Closure_of_Intersection_may_not_equal_Intersection_of_Closures/Proof_1 | [
"Closure of Intersection may not equal Intersection of Closures",
"Set Closures",
"Set Intersection"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closure (Topology)"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Set",
"Definition:Rational Number",
"Definition:Set",
"Definition:Irrational Number",
"Closure of Intersection of Rationals and Irrationals is Empty Set",
"Intersection of Closures ... |
proofwiki-8539 | Closure of Intersection may not equal Intersection of Closures | Let $T = \struct {S, \tau}$ be a topological space.
Let $H_1$ and $H_2$ be subsets of $S$.
Let ${H_1}^-$ and ${H_2}^-$ denote the closures of $H_1$ and $H_2$ respectively.
Then it is not necessarily the case that:
:$\paren {H_1 \cap H_2}^- = {H_1}^- \cap {H_2}^-$ | Let $\struct {\R, \tau_d}$ be the real number line under the usual (Euclidean) topology.
Let $H_1 = \openint 0 {\dfrac 1 2}$ and $H_2 = \openint {\dfrac 1 2} 1$.
By inspection it can be seen that:
:$H_1 \cap H_2 = \O$
Thus from Closure of Empty Set is Empty Set:
:$\paren {H_1 \cap H_2}^- = \O$
From Closure of Open Real... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H_1$ and $H_2$ be [[Definition:Subset|subsets]] of $S$.
Let ${H_1}^-$ and ${H_2}^-$ denote the [[Definition:Closure (Topology)|closures]] of $H_1$ and $H_2$ respectively.
Then it is not necessarily the case that:
:$\paren {H_1... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number line]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $H_1 = \openint 0 {\dfrac 1 2}$ and $H_2 = \openint {\dfrac 1 2} 1$.
By inspection it can be seen that:
:$H_1 \cap H_2 = \O$
Thus from [[C... | Closure of Intersection may not equal Intersection of Closures/Proof 2 | https://proofwiki.org/wiki/Closure_of_Intersection_may_not_equal_Intersection_of_Closures | https://proofwiki.org/wiki/Closure_of_Intersection_may_not_equal_Intersection_of_Closures/Proof_2 | [
"Closure of Intersection may not equal Intersection of Closures",
"Set Closures",
"Set Intersection"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closure (Topology)"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Closure of Empty Set is Empty Set",
"Closure of Open Real Interval is Closed Real Interval"
] |
proofwiki-8540 | Closure of Empty Set is Empty Set | Let $T = \struct {S, \tau}$ be a topological space.
Then the closure of the empty set $\O$ in $T$ is $\O$. | From Empty Set is Closed in Topological Space, $\O$ is closed in $T$.
The result follows from Closed Set equals its Closure.
{{qed}}
Category:Set Closures
Category:Empty Set
ev0qv3muvwdrhiss3hjy1kk3syrue4h | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then the [[Definition:Closure (Topology)|closure]] of the [[Definition:Empty Set|empty set]] $\O$ in $T$ is $\O$. | From [[Empty Set is Closed in Topological Space]], $\O$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
The result follows from [[Closed Set equals its Closure]].
{{qed}}
[[Category:Set Closures]]
[[Category:Empty Set]]
ev0qv3muvwdrhiss3hjy1kk3syrue4h | Closure of Empty Set is Empty Set | https://proofwiki.org/wiki/Closure_of_Empty_Set_is_Empty_Set | https://proofwiki.org/wiki/Closure_of_Empty_Set_is_Empty_Set | [
"Set Closures",
"Empty Set"
] | [
"Definition:Topological Space",
"Definition:Closure (Topology)",
"Definition:Empty Set"
] | [
"Empty Set is Closed/Topological Space",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Category:Set Closures",
"Category:Empty Set"
] |
proofwiki-8541 | Closed Real Interval is Regular Closed | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\closedint a b$ be a closed interval of $\R$.
Then $\closedint a b$ is a regular closed set in $\struct {\R, \tau_d}$. | {{Recall|Regular Closed Set|regular closed set}}
{{:Definition:Regular Closed Set}}
From Closed Real Interval is Closed in Real Number Line, $\closedint a b$ is closed in $\struct {\R, \tau_d}$.
From Interior of Closed Real Interval is Open Real Interval:
:$\closedint a b^\circ = \openint a b$
where $\closedint a b^\ci... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed interval]] of $\R$.
Then $\closedint a b$ is a [[Definition:Regular Closed Set|regular closed set]] in $\s... | {{Recall|Regular Closed Set|regular closed set}}
{{:Definition:Regular Closed Set}}
From [[Closed Real Interval is Closed in Real Number Line]], $\closedint a b$ is [[Definition:Closed Set (Topology)|closed]] in $\struct {\R, \tau_d}$.
From [[Interior of Closed Real Interval is Open Real Interval]]:
:$\closedint a b^... | Closed Real Interval is Regular Closed | https://proofwiki.org/wiki/Closed_Real_Interval_is_Regular_Closed | https://proofwiki.org/wiki/Closed_Real_Interval_is_Regular_Closed | [
"Real Number Line with Euclidean Topology",
"Real Intervals",
"Examples of Regular Closed Sets"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval/Closed",
"Definition:Regular Closed Set"
] | [
"Closed Real Interval is Closed in Real Number Line",
"Definition:Closed Set/Topology",
"Interior of Closed Real Interval is Open Real Interval",
"Definition:Interior (Topology)",
"Closure of Open Real Interval is Closed Real Interval",
"Definition:Closure (Topology)",
"Definition:Regular Closed Set"
] |
proofwiki-8542 | Closed Real Interval is Closed in Real Number Line | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\closedint a b$ be a closed interval of $\R$.
Then $\closedint a b$ is closed (in the topological sense) in $\struct {\R, \tau_d}$. | From Open Sets in Real Number Line:
:$U := \openint \gets a \cup \openint b \to$
is an open set in $\struct {\R, \tau_d}$.
Consider the complement relative to $\R$:
:$V := \relcomp \R U = \closedint a b$
By definition, $\relcomp \R U$ is closed (in the topological sense) in $\struct {\R, \tau_d}$.
But by construction:
... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed interval]] of $\R$.
Then $\closedint a b$ is [[Definition:Closed Set (Topology)|closed (in the topological... | From [[Open Sets in Real Number Line]]:
:$U := \openint \gets a \cup \openint b \to$
is an [[Definition:Open Set (Topology)|open set]] in $\struct {\R, \tau_d}$.
Consider the [[Definition:Relative Complement|complement relative to $\R$]]:
:$V := \relcomp \R U = \closedint a b$
By definition, $\relcomp \R U$ is [[Defi... | Closed Real Interval is Closed in Real Number Line | https://proofwiki.org/wiki/Closed_Real_Interval_is_Closed_in_Real_Number_Line | https://proofwiki.org/wiki/Closed_Real_Interval_is_Closed_in_Real_Number_Line | [
"Closed Sets",
"Real Number Line with Euclidean Topology"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval/Closed",
"Definition:Closed Set/Topology"
] | [
"Open Sets in Real Number Line",
"Definition:Open Set/Topology",
"Definition:Relative Complement",
"Definition:Closed Set/Topology"
] |
proofwiki-8543 | Intersection of Regular Closed Sets is not necessarily Regular Closed | Let $T = \struct {S, \tau}$ be a topological space.
Let $U$ and $V$ be regular closed sets of $T$.
Then $U \cap V$ is not also necessarily a regular closed set of $T$. | ;Proof by Counterexample
By Closed Real Interval is Regular Closed, the closed real intervals:
:$\closedint 0 {\dfrac 1 2}, \closedint {\dfrac 1 2} 1$
are both regular closed sets of $\R$.
Consider $A$, the intersection of the two half-unit closed intervals:
:$A := \closedint 0 {\dfrac 1 2} \cap \closedint {\dfrac 1 2}... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U$ and $V$ be [[Definition:Regular Closed Set|regular closed]] sets of $T$.
Then $U \cap V$ is not also necessarily a [[Definition:Regular Closed Set|regular closed]] set of $T$. | ;[[Proof by Counterexample]]
By [[Closed Real Interval is Regular Closed]], the [[Definition:Closed Real Interval|closed real intervals]]:
:$\closedint 0 {\dfrac 1 2}, \closedint {\dfrac 1 2} 1$
are both [[Definition:Regular Closed Set|regular closed]] sets of $\R$.
Consider $A$, the [[Definition:Set Intersection|i... | Intersection of Regular Closed Sets is not necessarily Regular Closed | https://proofwiki.org/wiki/Intersection_of_Regular_Closed_Sets_is_not_necessarily_Regular_Closed | https://proofwiki.org/wiki/Intersection_of_Regular_Closed_Sets_is_not_necessarily_Regular_Closed | [
"Regular Closed Sets",
"Closed Sets",
"Set Intersection"
] | [
"Definition:Topological Space",
"Definition:Regular Closed Set",
"Definition:Regular Closed Set"
] | [
"Proof by Counterexample",
"Closed Real Interval is Regular Closed",
"Definition:Real Interval/Closed",
"Definition:Regular Closed Set",
"Definition:Set Intersection",
"Definition:Real Interval/Closed",
"Interior of Closed Real Interval is Open Real Interval",
"Closure of Empty Set is Empty Set",
"D... |
proofwiki-8544 | Interior of Union is not necessarily Union of Interiors | Let $T = \struct {S, \tau}$ be a topological space.
Let $H_1$ and $H_2$ be subsets of $S$.
Let ${H_1}^\circ$ and ${H_2}^\circ$ denote the interiors of $H_1$ and $H_2$ respectively.
Then it is not necessarily the case that:
:$\paren {H_1 \cup H_2}^\circ = {H_1}^\circ \cup {H_2}^\circ$ | From Union of Interiors is Subset of Interior of Union:
:$\paren {H_1 \cup H_2}^\circ \supseteq {H_1}^\circ \cup {H_2}^\circ$
It remains to be shown that it is not necessarily the case that:
:$\paren {H_1 \cup H_2}^\circ = {H_1}^\circ \cup {H_2}^\circ$
;Proof by Counterexample
Let $\struct {\R, \tau_d}$ be the real num... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H_1$ and $H_2$ be [[Definition:Subset|subsets]] of $S$.
Let ${H_1}^\circ$ and ${H_2}^\circ$ denote the [[Definition:Interior (Topology)|interiors]] of $H_1$ and $H_2$ respectively.
Then it is not necessarily the case that:
:$\... | From [[Union of Interiors is Subset of Interior of Union]]:
:$\paren {H_1 \cup H_2}^\circ \supseteq {H_1}^\circ \cup {H_2}^\circ$
It remains to be shown that it is not necessarily the case that:
:$\paren {H_1 \cup H_2}^\circ = {H_1}^\circ \cup {H_2}^\circ$
;[[Proof by Counterexample]]
Let $\struct {\R, \tau_d}$ be ... | Interior of Union is not necessarily Union of Interiors | https://proofwiki.org/wiki/Interior_of_Union_is_not_necessarily_Union_of_Interiors | https://proofwiki.org/wiki/Interior_of_Union_is_not_necessarily_Union_of_Interiors | [
"Set Interiors",
"Set Union"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Interior (Topology)"
] | [
"Union of Interiors is Subset of Interior of Union",
"Proof by Counterexample",
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Interior of Closed Real Interval is Open Real Interval",
"Interior of Closed Real Interval is Open Real Interval"
] |
proofwiki-8545 | Non-Homeomorphic Sets may be Homeomorphic to Subsets of Each Other | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $H_1 \subseteq S_1$ and $H_2 \subseteq S_2$.
Then it is possible for:
{{begin-itemize}}
{{item|(1):|$T_1$ to be homeomorphic to $H_2$}}
{{item|(2):|$T_2$ to be homeomorphic to $H_1$}}
{{end-itemize}}
but:
{{begin-itemize}}
{{... | Let $\struct {\R, \tau_d}$ be the real number line under the usual (Euclidean) topology.
Let $S_1 := \closedint 0 1$ be the closed unit interval.
Let $S_2 := \openint 0 1$ be the open unit interval.
Let $H_1 := \openint 0 1$ and $H_2 := \closedint {\dfrac 1 4} {\dfrac 3 4}$.
Then:
{{begin-itemize}}
{{item|(1):|$\struct... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $H_1 \subseteq S_1$ and $H_2 \subseteq S_2$.
Then it is possible for:
{{begin-itemize}}
{{item|(1):|$T_1$ to be [[Definition:Homeomorphic Topological Spaces|homeomorphic]] to $H_2$}}
{{item... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number line]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $S_1 := \closedint 0 1$ be the [[Definition:Closed Unit Interval|closed unit interval]].
Let $S_2 := \openint 0 1$ be the [[Definition:Open... | Non-Homeomorphic Sets may be Homeomorphic to Subsets of Each Other | https://proofwiki.org/wiki/Non-Homeomorphic_Sets_may_be_Homeomorphic_to_Subsets_of_Each_Other | https://proofwiki.org/wiki/Non-Homeomorphic_Sets_may_be_Homeomorphic_to_Subsets_of_Each_Other | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Real Interval/Unit Interval/Open",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Mapping",
"Definition:Homeomorphism/Topo... |
proofwiki-8546 | Superspace of Homeomorphic Subspaces may not have Homeomorphism to Itself containing Subspace Homeomorphism | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $H_1 \subseteq S_1$ and $H_2 \subseteq S_2$.
Let $H_1$ and $H_2$ be a homeomorphic.
Then it may be the case that there does not exist a homeomorphism $g: T_1 \to T_2$ such that:
:$g \restriction_{H_1} = f$
where:
:$g \restric... | ;Proof by Counterexample
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $H_1 := \set 0 \cup \closedint 1 2 \cup \set 3$, where $\closedint 1 2$ denotes the closed interval from $1$ to $2$.
Let $H_2 := \closedint 0 1 \cup \set 2 \cup \set 3$.
$H_1$ and $H_2$ are homeomorphic,... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $H_1 \subseteq S_1$ and $H_2 \subseteq S_2$.
Let $H_1$ and $H_2$ be a [[Definition:Homeomorphic Topological Spaces|homeomorphic]].
Then it may be the case that there does not exist a [[Def... | ;[[Proof by Counterexample]]
Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $H_1 := \set 0 \cup \closedint 1 2 \cup \set 3$, where $\closedint 1 2$ denotes the [[Definition:Closed Real Interval|closed interval]] from ... | Superspace of Homeomorphic Subspaces may not have Homeomorphism to Itself containing Subspace Homeomorphism | https://proofwiki.org/wiki/Superspace_of_Homeomorphic_Subspaces_may_not_have_Homeomorphism_to_Itself_containing_Subspace_Homeomorphism | https://proofwiki.org/wiki/Superspace_of_Homeomorphic_Subspaces_may_not_have_Homeomorphism_to_Itself_containing_Subspace_Homeomorphism | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Restriction/Mapping",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Proof by Counterexample",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Real Interval/Closed",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Mapping",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces",
"Definition... |
proofwiki-8547 | Expectation of Exponential Distribution | Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$
Then the expectation of $X$ is given by:
:$\expect X = \beta$ | The expectation of a continuous random variable $X$ with sample space $\Omega_X$ is given by:
:$\ds \expect X := \int_{x \mathop \in \Omega_X} x \map {f_X} x \rd x$
where $f_X$ is the probability density function of $X$.
For the exponential distribution:
:$\Omega_X = \hointr 0 \infty$
From Probability Density Function ... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] of the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$ for some $\beta \in \R_{> 0}$
Then the [[Definition:Expectation of Continuous Random Variable|expectation]] of $X$ is given by:
:$\expect X ... | The [[Definition:Expectation of Continuous Random Variable|expectation]] of a [[Definition:Continuous Random Variable|continuous random variable]] $X$ with [[Definition:Sample Space|sample space]] $\Omega_X$ is given by:
:$\ds \expect X := \int_{x \mathop \in \Omega_X} x \map {f_X} x \rd x$
where $f_X$ is the [[Defin... | Expectation of Exponential Distribution/Proof 1 | https://proofwiki.org/wiki/Expectation_of_Exponential_Distribution | https://proofwiki.org/wiki/Expectation_of_Exponential_Distribution/Proof_1 | [
"Expectation of Exponential Distribution",
"Expectation",
"Exponential Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Exponential Distribution",
"Definition:Expectation/Continuous"
] | [
"Definition:Expectation/Continuous",
"Definition:Random Variable/Continuous",
"Definition:Sample Space",
"Definition:Probability Density Function",
"Definition:Exponential Distribution",
"Probability Density Function of Exponential Distribution",
"Definition:Exponential Function/Real",
"Integration by... |
proofwiki-8548 | Expectation of Exponential Distribution | Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$
Then the expectation of $X$ is given by:
:$\expect X = \beta$ | From Moment Generating Function of Exponential Distribution, the moment generating function $M_X$ of $X$, is given by:
:$\map {M_X} t = \dfrac 1 {1 - \beta t}$
By Moment in terms of Moment Generating Function:
:$\expect X = \map {M_X'} 0$
We have:
{{begin-eqn}}
{{eqn | l = \map {M_X'} t
| r = \map {\frac \d {\d... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] of the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$ for some $\beta \in \R_{> 0}$
Then the [[Definition:Expectation of Continuous Random Variable|expectation]] of $X$ is given by:
:$\expect X ... | From [[Moment Generating Function of Exponential Distribution]], the [[Definition:Moment Generating Function|moment generating function]] $M_X$ of $X$, is given by:
:$\map {M_X} t = \dfrac 1 {1 - \beta t}$
By [[Moment in terms of Moment Generating Function]]:
:$\expect X = \map {M_X'} 0$
We have:
{{begin-eqn}}
{{... | Expectation of Exponential Distribution/Proof 2 | https://proofwiki.org/wiki/Expectation_of_Exponential_Distribution | https://proofwiki.org/wiki/Expectation_of_Exponential_Distribution/Proof_2 | [
"Expectation of Exponential Distribution",
"Expectation",
"Exponential Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Exponential Distribution",
"Definition:Expectation/Continuous"
] | [
"Moment Generating Function of Exponential Distribution",
"Definition:Moment Generating Function",
"Moment in terms of Moment Generating Function",
"Derivative of Composite Function",
"Power Rule for Derivatives"
] |
proofwiki-8549 | Finite Group is p-Group iff Order is Power of p | Let $p$ be a prime number.
Let $G$ be a finite group.
Then $G$ is a $p$-group {{iff}} the order of $G$ is a power of $p$. | === Necessary Condition ===
Let $G$ be a finite group whose order is $p^n$ for some $n \in \Z_{>0}$.
Let $g \in G$.
From Order of Element Divides Order of Finite Group, the order of $g$ is a divisor of $p^n$.
That is, $x$ is a $p$-element by definition.
As $x$ is arbitrary, it follows that all elements of $G$ are $p$-e... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $G$ be a [[Definition:Finite Group|finite group]].
Then $G$ is a [[Definition:P-Group|$p$-group]] {{iff}} the [[Definition:Order of Structure|order]] of $G$ is a [[Definition:Prime Power|power of $p$.]] | === Necessary Condition ===
Let $G$ be a [[Definition:Finite Group|finite group]] whose [[Definition:Order of Structure|order]] is $p^n$ for some $n \in \Z_{>0}$.
Let $g \in G$.
From [[Order of Element Divides Order of Finite Group]], the [[Definition:Order of Group Element|order]] of $g$ is a [[Definition:Divisor o... | Finite Group is p-Group iff Order is Power of p | https://proofwiki.org/wiki/Finite_Group_is_p-Group_iff_Order_is_Power_of_p | https://proofwiki.org/wiki/Finite_Group_is_p-Group_iff_Order_is_Power_of_p | [
"P-Groups",
"Finite Groups"
] | [
"Definition:Prime Number",
"Definition:Finite Group",
"Definition:P-Group",
"Definition:Order of Structure",
"Definition:Prime Power"
] | [
"Definition:Finite Group",
"Definition:Order of Structure",
"Order of Element Divides Order of Finite Group",
"Definition:Order of Group Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:P-Element",
"Definition:Element",
"Definition:P-Element",
"Definition:P-Group",
"Definition:Finite ... |
proofwiki-8550 | Equivalence of Definitions of Semantic Equivalence for Boolean Interpretations | {{TFAE|def = Semantic Equivalence for Boolean Interpretations}}
Let $\mathbf A, \mathbf B$ be WFFs of propositional logic. | === Definition 1 implies Definition 2 ===
Let $\mathbf A, \mathbf B$ be equivalent according to definition 1.
Let $v$ be a boolean interpretation.
By definition. either $\map v {\mathbf A} = \T$ or $\map v {\mathbf A} = \F$.
In the first case, it follows {{hypothesis}} that $\map v {\mathbf B} = \T$.
In particular, the... | {{TFAE|def = Semantic Equivalence for Boolean Interpretations}}
Let $\mathbf A, \mathbf B$ be [[Definition:WFF of Propositional Logic|WFFs of propositional logic]]. | === Definition 1 implies Definition 2 ===
Let $\mathbf A, \mathbf B$ be [[Definition:Semantic Equivalence/Boolean Interpretations/Definition 1|equivalent]] according to definition 1.
Let $v$ be a [[Definition:Boolean Interpretation|boolean interpretation]].
By definition. either $\map v {\mathbf A} = \T$ or $\map v... | Equivalence of Definitions of Semantic Equivalence for Boolean Interpretations | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Semantic_Equivalence_for_Boolean_Interpretations | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Semantic_Equivalence_for_Boolean_Interpretations | [
"Semantic Equivalence for Boolean Interpretations"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF"
] | [
"Definition:Semantic Equivalence/Boolean Interpretations/Definition 1",
"Definition:Boolean Interpretation",
"Definition:Semantic Equivalence/Boolean Interpretations/Definition 2",
"Definition:Semantic Equivalence/Boolean Interpretations/Definition 2",
"Definition:Boolean Interpretation",
"Definition:Sema... |
proofwiki-8551 | Von Mangoldt Equivalence | For $n \in \N_{>0}$, let $\map \Lambda n$ be the von Mangoldt function.
Then:
:$\ds \lim_{N \mathop \to \infty} \frac 1 N \sum_{n \mathop = 1}^N \map \Lambda n = 1$
is logically equivalent to the Prime Number Theorem. | Observe:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^N \map \Lambda n
| r = \map \Lambda 1 + \map \Lambda 2 + \cdots + \map \Lambda n
| c=
}}
{{eqn | r = 0 + \map \ln 2 + \map \ln 3 + \map \ln 2 + \map \ln 5 + 0 + \map \ln 7 + \map \ln 2 + \map \ln 3 + 0 + \cdots
| c =
}}
{{end-eqn}}
Notice this su... | For $n \in \N_{>0}$, let $\map \Lambda n$ be the [[Definition:Von Mangoldt Function|von Mangoldt function]].
Then:
:$\ds \lim_{N \mathop \to \infty} \frac 1 N \sum_{n \mathop = 1}^N \map \Lambda n = 1$
is [[Definition:Logical Equivalence|logically equivalent]] to the [[Prime Number Theorem]]. | Observe:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^N \map \Lambda n
| r = \map \Lambda 1 + \map \Lambda 2 + \cdots + \map \Lambda n
| c=
}}
{{eqn | r = 0 + \map \ln 2 + \map \ln 3 + \map \ln 2 + \map \ln 5 + 0 + \map \ln 7 + \map \ln 2 + \map \ln 3 + 0 + \cdots
| c =
}}
{{end-eqn}}
Notice this ... | Von Mangoldt Equivalence | https://proofwiki.org/wiki/Von_Mangoldt_Equivalence | https://proofwiki.org/wiki/Von_Mangoldt_Equivalence | [
"Prime Numbers"
] | [
"Definition:Von Mangoldt Function",
"Definition:Logical Equivalence",
"Prime Number Theorem"
] | [
"Definition:Term of Sequence",
"Definition:Power (Algebra)/Integer",
"Definition:Term of Sequence",
"Definition:Power (Algebra)/Integer",
"Prime Number Theorem",
"Definition:Von Mangoldt Function",
"Definition:Logical Equivalence",
"Prime Number Theorem",
"Category:Prime Numbers"
] |
proofwiki-8552 | Zeta Equivalence to Prime Number Theorem | Let $\map \zeta z$ be the Riemann $\zeta$ function.
The Prime Number Theorem is logically equivalent to the statement that the average of the first $N$ coefficients of $\dfrac {\zeta'} {\zeta}$ tend to $-1$ as $N$ goes to infinity.
{{explain|What does $z$ range over, and what does it mean by "first $N$ coefficients" of... | The Von Mangoldt Equivalence is equivalent (clearly) to the statement that the average of the coefficients of the function of $z$ defined as:
:$(1): \quad \ds \sum_{n \mathop = 1}^\infty \frac {\map \Lambda n} {n^z}$
tend to $1$.
{{handwaving|Needs to be explained in more detail.}}
Let $ \set {p_1, p_2, p_3, \dots}$ be... | Let $\map \zeta z$ be the [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]].
The [[Prime Number Theorem]] is [[Definition:Logical Equivalence|logically equivalent]] to the statement that the average of the first $N$ coefficients of $\dfrac {\zeta'} {\zeta}$ tend to $-1$ as $N$ goes to infinity.
{{explain|... | The [[Von Mangoldt Equivalence]] is equivalent (clearly) to the statement that the average of the coefficients of the [[Definition:Function|function]] of $z$ defined as:
:$(1): \quad \ds \sum_{n \mathop = 1}^\infty \frac {\map \Lambda n} {n^z}$
tend to $1$.
{{handwaving|Needs to be explained in more detail.}}
Let $ \... | Zeta Equivalence to Prime Number Theorem | https://proofwiki.org/wiki/Zeta_Equivalence_to_Prime_Number_Theorem | https://proofwiki.org/wiki/Zeta_Equivalence_to_Prime_Number_Theorem | [
"Prime Numbers"
] | [
"Definition:Riemann Zeta Function",
"Prime Number Theorem",
"Definition:Logical Equivalence"
] | [
"Von Mangoldt Equivalence",
"Definition:Function",
"Definition:Enumeration",
"Definition:Prime Number",
"Von Mangoldt Equivalence",
"Definition:Von Mangoldt Function",
"Sum of Infinite Geometric Sequence",
"Definition:Riemann Zeta Function",
"Category:Prime Numbers"
] |
proofwiki-8553 | Divisor of Product may not be Divisor of Factors | Let $a, b, c \in \Z_{>0}$ be (strictly) positive integers.
Let:
:$c \divides a b$
where $\divides$ expresses the relation of divisibility.
Then it is not necessarily the case that either $c \divides a$ or $c \divides b$. | Proof by Counterexample:
Let $c = 6, a = 3, b = 4$.
Then $6 \times 2 = 12$ so $c \divides a b$.
But neither $6 \divides 4$ nor $6 \divides 3$.
{{qed}} | Let $a, b, c \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let:
:$c \divides a b$
where $\divides$ expresses the [[Definition:Relation|relation]] of [[Definition:Divisor of Integer|divisibility]].
Then it is not necessarily the case that either $c \divides a$ or $c \divides b... | [[Proof by Counterexample]]:
Let $c = 6, a = 3, b = 4$.
Then $6 \times 2 = 12$ so $c \divides a b$.
But neither $6 \divides 4$ nor $6 \divides 3$.
{{qed}} | Divisor of Product may not be Divisor of Factors | https://proofwiki.org/wiki/Divisor_of_Product_may_not_be_Divisor_of_Factors | https://proofwiki.org/wiki/Divisor_of_Product_may_not_be_Divisor_of_Factors | [
"Number Theory"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Relation",
"Definition:Divisor (Algebra)/Integer"
] | [
"Proof by Counterexample"
] |
proofwiki-8554 | Existence of Prime between Prime and Factorial | Let $p$ be a prime number.
Then there exists a prime number $q$ such that:
:$p < q \le p! + 1$
where $p!$ denotes the factorial of $p$. | Let $N = p! + 1$.
If $N$ is prime, then the proof is complete.
Otherwise, from Positive Integer Greater than 1 has Prime Divisor, $N$ has a prime divisor $q$.
From Absolute Value of Integer is not less than Divisors:
:$q < N$
{{AimForCont}} $q \le p$.
Then by the definition of factorial:
:$q \divides p!$
But $q$ was ch... | Let $p$ be a [[Definition:Prime Number|prime number]].
Then there exists a [[Definition:Prime Number|prime number]] $q$ such that:
:$p < q \le p! + 1$
where $p!$ denotes the [[Definition:Factorial|factorial]] of $p$. | Let $N = p! + 1$.
If $N$ is [[Definition:Prime Number|prime]], then the proof is complete.
Otherwise, from [[Positive Integer Greater than 1 has Prime Divisor]], $N$ has a [[Definition:Prime Factor|prime divisor]] $q$.
From [[Absolute Value of Integer is not less than Divisors]]:
:$q < N$
{{AimForCont}} $q \le p$.
... | Existence of Prime between Prime and Factorial | https://proofwiki.org/wiki/Existence_of_Prime_between_Prime_and_Factorial | https://proofwiki.org/wiki/Existence_of_Prime_between_Prime_and_Factorial | [
"Prime Numbers"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Factorial"
] | [
"Definition:Prime Number",
"Positive Integer Greater than 1 has Prime Divisor",
"Definition:Prime Factor",
"Absolute Value of Integer is not less than Divisors",
"Definition:Factorial",
"Absolute Value of Integer is not less than Divisors",
"Definition:Prime Number",
"Proof by Contradiction"
] |
proofwiki-8555 | Substitution Rule for Matrices | Let $\mathbf A$ be a square matrix of order $n$.
Then:
:$(1): \quad \ds \sum_{j \mathop = 1}^n \delta_{i j} a_{j k} = a_{i k}$
:$(2): \quad \ds \sum_{j \mathop = 1}^n \delta_{i j} a_{k j} = a_{k i}$
where:
:$\delta_{i j}$ is the Kronecker delta
:$a_{j k}$ is element $\tuple {j, k}$ of $\mathbf A$. | By definition of Kronecker delta:
:$\delta_{i j} = \begin {cases} 1 & : i = j \\ 0 & : i \ne j \end {cases}$
Thus:
:$\delta_{i j} a_{j k} = \begin {cases} a_{i k} & : i = j \\ 0 & : i \ne j \end {cases}$
and:
:$\delta_{i j} a_{k j} = \begin {cases} a_{k i} & : i = j \\ 0 & : i \ne j \end {cases}$
from which the result ... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Then:
:$(1): \quad \ds \sum_{j \mathop = 1}^n \delta_{i j} a_{j k} = a_{i k}$
:$(2): \quad \ds \sum_{j \mathop = 1}^n \delta_{i j} a_{k j} = a_{k i}$
where:
:$\delta_{i j}$ is the [[Definition:Kronecker ... | By definition of [[Definition:Kronecker Delta|Kronecker delta]]:
:$\delta_{i j} = \begin {cases} 1 & : i = j \\ 0 & : i \ne j \end {cases}$
Thus:
:$\delta_{i j} a_{j k} = \begin {cases} a_{i k} & : i = j \\ 0 & : i \ne j \end {cases}$
and:
:$\delta_{i j} a_{k j} = \begin {cases} a_{k i} & : i = j \\ 0 & : i \ne j \end... | Substitution Rule for Matrices | https://proofwiki.org/wiki/Substitution_Rule_for_Matrices | https://proofwiki.org/wiki/Substitution_Rule_for_Matrices | [
"Matrix Algebra"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Kronecker Delta",
"Definition:Matrix/Element"
] | [
"Definition:Kronecker Delta"
] |
proofwiki-8556 | Trace of Unit Matrix | Let $\mathbf I_n$ be the unit matrix of order $n$.
Then:
:$\map \tr {\mathbf I_n} = n$
where $\map \tr {\mathbf I_n}$ denotes the trace of $\mathbf I_n$. | By definition:
:$\mathbf I_n := \sqbrk a_n: a_{i j} = \delta_{i j}$
That is: each of the elements on the main diagonal is equal to $1$.
There are $n$ such elements.
Hence the result.
{{qed}} | Let $\mathbf I_n$ be the [[Definition:Unit Matrix|unit matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Then:
:$\map \tr {\mathbf I_n} = n$
where $\map \tr {\mathbf I_n}$ denotes the [[Definition:Trace of Matrix|trace]] of $\mathbf I_n$. | By definition:
:$\mathbf I_n := \sqbrk a_n: a_{i j} = \delta_{i j}$
That is: each of the elements on the [[Definition:Main Diagonal|main diagonal]] is equal to $1$.
There are $n$ such elements.
Hence the result.
{{qed}} | Trace of Unit Matrix | https://proofwiki.org/wiki/Trace_of_Unit_Matrix | https://proofwiki.org/wiki/Trace_of_Unit_Matrix | [
"Matrix Algebra"
] | [
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Trace (Linear Algebra)/Matrix"
] | [
"Definition:Matrix/Diagonal/Main"
] |
proofwiki-8557 | Absorption Laws (Set Theory)/Corollary | :$S \cup \paren {S \cap T} = S \cap \paren {S \cup T}$ | {{begin-eqn}}
{{eqn | o =
| r = S \cup \paren {S \cap T}
}}
{{eqn | o = \leadstoandfrom
| r = \paren {S \cup S} \cap \paren {S \cup T}
| c = Union Distributes over Intersection
}}
{{eqn | o = \leadstoandfrom
| r = S \cap \paren {S \cup T}
| c = Set Union is Idempotent
}}
{{end-eqn}}
{{qed... | :$S \cup \paren {S \cap T} = S \cap \paren {S \cup T}$ | {{begin-eqn}}
{{eqn | o =
| r = S \cup \paren {S \cap T}
}}
{{eqn | o = \leadstoandfrom
| r = \paren {S \cup S} \cap \paren {S \cup T}
| c = [[Union Distributes over Intersection]]
}}
{{eqn | o = \leadstoandfrom
| r = S \cap \paren {S \cup T}
| c = [[Set Union is Idempotent]]
}}
{{end-eqn... | Absorption Laws (Set Theory)/Corollary | https://proofwiki.org/wiki/Absorption_Laws_(Set_Theory)/Corollary | https://proofwiki.org/wiki/Absorption_Laws_(Set_Theory)/Corollary | [
"Set Intersection",
"Set Union",
"Absorption Laws"
] | [] | [
"Union Distributes over Intersection",
"Set Union is Idempotent",
"Category:Set Intersection",
"Category:Set Union",
"Category:Absorption Laws"
] |
proofwiki-8558 | Matrix is Nonsingular iff Determinant has Multiplicative Inverse/Necessary Condition | Let $\struct {R, +, \circ}$ be a commutative ring with unity.
Let $\mathbf A \in R^{n \times n}$ be an nonsingular square matrix of order $n$.
Let $\mathbf B = \mathbf A^{-1}$ be the inverse of $\mathbf A$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Then:
:$\map \det {\mathbf B} = \dfrac 1 {\map \de... | Let $\mathbf A$ be nonsingular with $\mathbf B = \mathbf A^{-1}$.
Let $1_R$ denote the unity of $R$.
Let $\mathbf I_n$ denote the unit matrix of order $n$.
Then:
{{begin-eqn}}
{{eqn | l = 1_R
| r = \map \det {\mathbf I_n}
| c = Determinant of Unit Matrix
}}
{{eqn | r = \map \det {\mathbf A \mathbf B}
... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\mathbf A \in R^{n \times n}$ be an [[Definition:Nonsingular Matrix|nonsingular]] [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$.
Let $\mathbf B = \mathbf A^{-1... | Let $\mathbf A$ be [[Definition:Nonsingular Matrix|nonsingular]] with $\mathbf B = \mathbf A^{-1}$.
Let $1_R$ denote the [[Definition:Unity of Ring|unity]] of $R$.
Let $\mathbf I_n$ denote the [[Definition:Unit Matrix|unit matrix of order $n$]].
Then:
{{begin-eqn}}
{{eqn | l = 1_R
| r = \map \det {\mathbf I_... | Matrix is Nonsingular iff Determinant has Multiplicative Inverse/Necessary Condition | https://proofwiki.org/wiki/Matrix_is_Nonsingular_iff_Determinant_has_Multiplicative_Inverse/Necessary_Condition | https://proofwiki.org/wiki/Matrix_is_Nonsingular_iff_Determinant_has_Multiplicative_Inverse/Necessary_Condition | [
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Nonsingular Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Inverse Matrix",
"Definition:Determinant/Matrix"
] | [
"Definition:Nonsingular Matrix",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Unit Matrix",
"Determinant of Unit Matrix",
"Determinant of Matrix Product"
] |
proofwiki-8559 | Product of Orthogonal Matrix with Transpose is Identity | Let $\mathbf Q$ be an orthogonal matrix.
Then:
:$\mathbf Q \mathbf Q^\intercal = \mathbf I = \mathbf Q^\intercal \mathbf Q$
where:
:$\mathbf Q^\intercal$ is the transpose of $\mathbf Q$
:$\mathbf I$ is a unit (identity) matrix | By definition, an orthogonal matrix is one such that:
:$\mathbf Q^\intercal = \mathbf Q^{-1}$
and so the result follows by definition of inverse.
{{qed}} | Let $\mathbf Q$ be an [[Definition:Orthogonal Matrix|orthogonal matrix]].
Then:
:$\mathbf Q \mathbf Q^\intercal = \mathbf I = \mathbf Q^\intercal \mathbf Q$
where:
:$\mathbf Q^\intercal$ is the [[Definition:Transpose of Matrix|transpose]] of $\mathbf Q$
:$\mathbf I$ is a [[Definition:Unit Matrix|unit (identity) matrix... | By definition, an [[Definition:Orthogonal Matrix|orthogonal matrix]] is one such that:
:$\mathbf Q^\intercal = \mathbf Q^{-1}$
and so the result follows by definition of [[Definition:Inverse Matrix|inverse]].
{{qed}} | Product of Orthogonal Matrix with Transpose is Identity | https://proofwiki.org/wiki/Product_of_Orthogonal_Matrix_with_Transpose_is_Identity | https://proofwiki.org/wiki/Product_of_Orthogonal_Matrix_with_Transpose_is_Identity | [
"Orthogonal Matrices",
"Transposes of Matrices",
"Unit Matrices"
] | [
"Definition:Orthogonal Matrix",
"Definition:Transpose of Matrix",
"Definition:Unit Matrix"
] | [
"Definition:Orthogonal Matrix",
"Definition:Inverse Matrix"
] |
proofwiki-8560 | Determinant of Orthogonal Matrix is Plus or Minus One | Let $\mathbf Q$ be an orthogonal matrix.
Then:
:$\det \mathbf Q = \pm 1$
where $\det \mathbf Q$ is the determinant of $\mathbf Q$. | By Determinant of Transpose:
:$\det \mathbf Q^\intercal = \det \mathbf Q$
Then:
{{begin-eqn}}
{{eqn | l = \mathbf Q \mathbf Q^\intercal
| r = \mathbf I
| c = Product of Orthogonal Matrix with Transpose is Identity
}}
{{eqn | ll= \leadsto
| l = \map \det {\mathbf Q \mathbf Q^\intercal}
| r = \det... | Let $\mathbf Q$ be an [[Definition:Orthogonal Matrix|orthogonal matrix]].
Then:
:$\det \mathbf Q = \pm 1$
where $\det \mathbf Q$ is the [[Definition:Determinant of Matrix|determinant]] of $\mathbf Q$. | By [[Determinant of Transpose]]:
:$\det \mathbf Q^\intercal = \det \mathbf Q$
Then:
{{begin-eqn}}
{{eqn | l = \mathbf Q \mathbf Q^\intercal
| r = \mathbf I
| c = [[Product of Orthogonal Matrix with Transpose is Identity]]
}}
{{eqn | ll= \leadsto
| l = \map \det {\mathbf Q \mathbf Q^\intercal}
... | Determinant of Orthogonal Matrix is Plus or Minus One | https://proofwiki.org/wiki/Determinant_of_Orthogonal_Matrix_is_Plus_or_Minus_One | https://proofwiki.org/wiki/Determinant_of_Orthogonal_Matrix_is_Plus_or_Minus_One | [
"Orthogonal Matrices",
"Determinants"
] | [
"Definition:Orthogonal Matrix",
"Definition:Determinant/Matrix"
] | [
"Determinant of Transpose",
"Product of Orthogonal Matrix with Transpose is Identity",
"Determinant of Unit Matrix",
"Determinant of Matrix Product",
"Determinant of Transpose"
] |
proofwiki-8561 | Product of Proper Orthogonal Matrices is Proper Orthogonal Matrix | Let $\mathbf P$ and $\mathbf Q$ be proper orthogonal matrices.
Let $\mathbf {P Q}$ be the (conventional) matrix product of $\mathbf P$ and $\mathbf Q$.
Then $\mathbf {P Q}$ is a proper orthogonal matrix. | By definition, $\mathbf {P Q}$ is a proper orthogonal matrix {{iff}} it is an orthogonal matrix with a determinant of $1$.
From Product of Orthogonal Matrices is Orthogonal Matrix, $\mathbf {P Q}$ is an orthogonal matrix.
By definition, $\mathbf P$ and $\mathbf Q$ both have a determinant of $1$.
From Determinant of Mat... | Let $\mathbf P$ and $\mathbf Q$ be [[Definition:Proper Orthogonal Matrix|proper orthogonal matrices]].
Let $\mathbf {P Q}$ be the [[Definition:Matrix Product (Conventional)|(conventional) matrix product]] of $\mathbf P$ and $\mathbf Q$.
Then $\mathbf {P Q}$ is a [[Definition:Proper Orthogonal Matrix|proper orthogona... | By definition, $\mathbf {P Q}$ is a [[Definition:Proper Orthogonal Matrix|proper orthogonal matrix]] {{iff}} it is an [[Definition:Orthogonal Matrix|orthogonal matrix]] with a [[Definition:Determinant of Matrix|determinant]] of $1$.
From [[Product of Orthogonal Matrices is Orthogonal Matrix]], $\mathbf {P Q}$ is an [... | Product of Proper Orthogonal Matrices is Proper Orthogonal Matrix | https://proofwiki.org/wiki/Product_of_Proper_Orthogonal_Matrices_is_Proper_Orthogonal_Matrix | https://proofwiki.org/wiki/Product_of_Proper_Orthogonal_Matrices_is_Proper_Orthogonal_Matrix | [
"Proper Orthogonal Matrices"
] | [
"Definition:Proper Orthogonal Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Proper Orthogonal Matrix"
] | [
"Definition:Proper Orthogonal Matrix",
"Definition:Orthogonal Matrix",
"Definition:Determinant/Matrix",
"Product of Orthogonal Matrices is Orthogonal Matrix",
"Definition:Orthogonal Matrix",
"Determinant of Matrix Product",
"Category:Proper Orthogonal Matrices"
] |
proofwiki-8562 | Product of Orthogonal Matrices is Orthogonal Matrix | Let $\mathbf P$ and $\mathbf Q$ be orthogonal matrices.
Let $\mathbf P \mathbf Q$ be the (conventional) matrix product of $\mathbf P$ and $\mathbf Q$.
Then $\mathbf P \mathbf Q$ is an orthogonal matrix. | From Determinant of Orthogonal Matrix is Plus or Minus One and Matrix is Nonsingular iff Determinant has Multiplicative Inverse it follows that both $\mathbf P$ and $\mathbf Q$ are nonsingular.
Thus:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf P \mathbf Q}^{-1}
| r = \mathbf Q^{-1} \mathbf P^{-1}
| c = Invers... | Let $\mathbf P$ and $\mathbf Q$ be [[Definition:Orthogonal Matrix|orthogonal matrices]].
Let $\mathbf P \mathbf Q$ be the [[Definition:Matrix Product (Conventional)|(conventional) matrix product]] of $\mathbf P$ and $\mathbf Q$.
Then $\mathbf P \mathbf Q$ is an [[Definition:Orthogonal Matrix|orthogonal matrix]]. | From [[Determinant of Orthogonal Matrix is Plus or Minus One]] and [[Matrix is Nonsingular iff Determinant has Multiplicative Inverse]] it follows that both $\mathbf P$ and $\mathbf Q$ are [[Definition:Nonsingular Matrix|nonsingular]].
Thus:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf P \mathbf Q}^{-1}
| r = \ma... | Product of Orthogonal Matrices is Orthogonal Matrix | https://proofwiki.org/wiki/Product_of_Orthogonal_Matrices_is_Orthogonal_Matrix | https://proofwiki.org/wiki/Product_of_Orthogonal_Matrices_is_Orthogonal_Matrix | [
"Orthogonal Matrices"
] | [
"Definition:Orthogonal Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Orthogonal Matrix"
] | [
"Determinant of Orthogonal Matrix is Plus or Minus One",
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse",
"Definition:Nonsingular Matrix",
"Inverse of Matrix Product",
"Transpose of Matrix Product",
"Definition:Orthogonal Matrix"
] |
proofwiki-8563 | Internal Group Direct Product is Injective/General Result | Let $G$ be a group whose identity is $e$.
Let $\sequence {H_n}$ be a sequence of subgroups of $G$.
Let $\ds \phi_n: \prod_{j \mathop = 1}^n H_j \to G$ be a mapping defined by:
:$\ds \map {\phi_n} {h_1, h_2, \ldots, h_n} = \prod_{j \mathop = 1}^n h_j$
Then $\phi_n$ is injective {{iff}}:
:$\forall i, j \in \set {1, 2, \l... | === Necessary Condition ===
Let $\ds \phi_n: \prod_{j \mathop = 1}^n H_j \to G$ be a mapping defined by:
:$\ds \map {\phi_n} {h_1, h_2, \ldots, h_n} = \prod_{j \mathop = 1}^n h_j$
Let $\phi_n$ be an injection.
Let $\map {\phi_n} {h_1, h_2, \ldots, h_n} = \map {\phi_n} {g_1, g_2, \ldots, g_n}$.
As $\phi_n$ is injective:... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $\sequence {H_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Subgroup|subgroups]] of $G$.
Let $\ds \phi_n: \prod_{j \mathop = 1}^n H_j \to G$ be a [[Definition:Mapping|mapping]] defined by:
:$\ds \map {\phi_n... | === Necessary Condition ===
Let $\ds \phi_n: \prod_{j \mathop = 1}^n H_j \to G$ be a [[Definition:Mapping|mapping]] defined by:
:$\ds \map {\phi_n} {h_1, h_2, \ldots, h_n} = \prod_{j \mathop = 1}^n h_j$
Let $\phi_n$ be an [[Definition:Injection|injection]].
Let $\map {\phi_n} {h_1, h_2, \ldots, h_n} = \map {\phi_n}... | Internal Group Direct Product is Injective/General Result | https://proofwiki.org/wiki/Internal_Group_Direct_Product_is_Injective/General_Result | https://proofwiki.org/wiki/Internal_Group_Direct_Product_is_Injective/General_Result | [
"Internal Group Direct Products"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Sequence",
"Definition:Subgroup",
"Definition:Mapping",
"Definition:Injection",
"Definition:Sequence",
"Definition:Independent Subgroups"
] | [
"Definition:Mapping",
"Definition:Injection",
"Definition:Injection",
"Definition:Integer",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection"
] |
proofwiki-8564 | Trace of Matrix Product | Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$.
Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$.
Then:
:$\ds \map \tr {\mathbf A \mathbf B} = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n a_{i j} b_{j i}$
where $\map \tr {\mathbf A \mathbf B}$ denotes the t... | Let $\mathbf C := \mathbf A \mathbf B$.
By definition of matrix product:
:$\ds c_{i k} = \sum_{j \mathop = 1}^n a_{i j} b_{j k}$
Thus for the diagonal elements:
:$\ds c_{i i} = \sum_{j \mathop = 1}^n a_{i j} b_{j i}$
By definition of trace:
:$\ds \map \tr {\mathbf C} = \sum_{i \mathop = 1}^n c_{i i}$
Hence the result.
... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\mathbf A \mathbf B$ be the [[Definition:Matrix Product (Conventional)|(conventional) matrix product]] of $\mathbf A$ and $\mathbf B$.
Then:
:$\ds \map \tr {\mathbf A \mathbf B} = ... | Let $\mathbf C := \mathbf A \mathbf B$.
By definition of [[Definition:Matrix Product (Conventional)|matrix product]]:
:$\ds c_{i k} = \sum_{j \mathop = 1}^n a_{i j} b_{j k}$
Thus for the [[Definition:Diagonal Element|diagonal elements]]:
:$\ds c_{i i} = \sum_{j \mathop = 1}^n a_{i j} b_{j i}$
By definition of [[De... | Trace of Matrix Product | https://proofwiki.org/wiki/Trace_of_Matrix_Product | https://proofwiki.org/wiki/Trace_of_Matrix_Product | [
"Matrix Algebra"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix Product (Conventional)",
"Definition:Trace (Linear Algebra)/Matrix",
"Definition:Einstein Summation Convention"
] | [
"Definition:Matrix Product (Conventional)",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Trace (Linear Algebra)/Matrix"
] |
proofwiki-8565 | Trace of Matrix Product/General Result | Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_m$ be square matrices of order $n$.
Let $\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m$ be the (conventional) matrix product of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_m$.
Then:
:$(1): \quad \ds \map \tr {\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m} = \map {a_1} {i_1, i... | Let $\mathbf C = \mathbf A_1 \mathbf A_2 \cdots \mathbf A_m$
From Product of Finite Sequence of Matrices, the general element of $\mathbf C$ is given in the Einstein summation convention by:
:$\map c {i_1, j} = \map {a_1} {i_1, i_2} \map {a_2} {i_2, i_3} \cdots \map {a_{m - 1} } {i_{m - 1}, i_m} \map {a_m} {i_m, j}$
T... | Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_m$ be [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m$ be the [[Definition:Matrix Product (Conventional)|(conventional) matrix product]] of $\mathbf A_1, \mathbf A_2, \ldots... | Let $\mathbf C = \mathbf A_1 \mathbf A_2 \cdots \mathbf A_m$
From [[Product of Finite Sequence of Matrices]], the general element of $\mathbf C$ is given in the [[Definition:Einstein Summation Convention|Einstein summation convention]] by:
:$\map c {i_1, j} = \map {a_1} {i_1, i_2} \map {a_2} {i_2, i_3} \cdots \map {a... | Trace of Matrix Product/General Result | https://proofwiki.org/wiki/Trace_of_Matrix_Product/General_Result | https://proofwiki.org/wiki/Trace_of_Matrix_Product/General_Result | [
"Matrix Algebra"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix Product (Conventional)",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Trace (Linear Algebra)/Matrix",
"Definition:Einstein Summation Convention",
"Definition:Summation",
"Definitio... | [
"Product of Finite Sequence of Matrices",
"Definition:Einstein Summation Convention",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Einstein Summation Convention",
"Definition:Trace (Linear Algebra)/Matrix"
] |
proofwiki-8566 | Product of Finite Sequence of Matrices | Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_n$ be matrices.
Let the order of $\mathbf A_j$ be $d_j \times d_{j + 1}$.
Let $\ds \mathbf C := \prod_{j \mathop = 1}^n \mathbf A_j = \mathbf A_1 \mathbf A_2 \cdots \mathbf A_n$ be the (conventional) matrix product of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_n$.
Then... | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \map c {i_1, i_{n + 1} } = \sum_{i_n \mathop = 1}^{d_n} \dotsm \sum_{i_3 \mathop = 1}^{d_3} \sum_{i_2 \mathop = 1}^{d_2} \map {a_1} {i_1, i_2} \map {a_2} {i_2, i_3} \dotsm \map {a_{n - 1} } {i_{n - 1}, i_n} \map {a_n} {i_n, i_{n + 1}... | Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_n$ be [[Definition:Matrix|matrices]].
Let the [[Definition:Order of Matrix|order]] of $\mathbf A_j$ be $d_j \times d_{j + 1}$.
Let $\ds \mathbf C := \prod_{j \mathop = 1}^n \mathbf A_j = \mathbf A_1 \mathbf A_2 \cdots \mathbf A_n$ be the [[Definition:Matrix Product (Co... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \map c {i_1, i_{n + 1} } = \sum_{i_n \mathop = 1}^{d_n} \dotsm \sum_{i_3 \mathop = 1}^{d_3} \sum_{i_2 \mathop = 1}^{d_2} \map {a_1} {i_1, i_2} \map {a_2} {i_2, i_3} ... | Product of Finite Sequence of Matrices | https://proofwiki.org/wiki/Product_of_Finite_Sequence_of_Matrices | https://proofwiki.org/wiki/Product_of_Finite_Sequence_of_Matrices | [
"Conventional Matrix Multiplication"
] | [
"Definition:Matrix",
"Definition:Matrix/Order",
"Definition:Matrix Product (Conventional)",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Matrix/Order"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-8567 | Characteristic of Galois Field is Prime | Let $\GF$ be a Galois field.
Then the characteristic of $\GF$ is a prime number. | By Characteristic of Field is Zero or Prime, it follows that $\Char \GF$ is $0$ or a prime number.
By Finite Field has Non-Zero Characteristic:
:$\Char \GF \ne 0$
Thus $\Char \GF$ is a prime number.
{{qed}}
Category:Galois Fields
j1df1dpfgmn7ywm6w4um79xpevopykt | Let $\GF$ be a [[Definition:Galois Field|Galois field]].
Then the [[Definition:Characteristic of Ring|characteristic]] of $\GF$ is a [[Definition:Prime Number|prime number]]. | By [[Characteristic of Field is Zero or Prime]], it follows that $\Char \GF$ is $0$ or a [[Definition:Prime Number|prime number]].
By [[Finite Field has Non-Zero Characteristic]]:
:$\Char \GF \ne 0$
Thus $\Char \GF$ is a [[Definition:Prime Number|prime number]].
{{qed}}
[[Category:Galois Fields]]
j1df1dpfgmn7ywm6w... | Characteristic of Galois Field is Prime | https://proofwiki.org/wiki/Characteristic_of_Galois_Field_is_Prime | https://proofwiki.org/wiki/Characteristic_of_Galois_Field_is_Prime | [
"Galois Fields"
] | [
"Definition:Galois Field",
"Definition:Characteristic of Ring",
"Definition:Prime Number"
] | [
"Characteristic of Field is Zero or Prime",
"Definition:Prime Number",
"Galois Field has Non-Zero Characteristic",
"Definition:Prime Number",
"Category:Galois Fields"
] |
proofwiki-8568 | Finite Ring with No Proper Zero Divisors is Field | Let $\struct {R, +, \circ}$ be a finite non-null ring with no proper zero divisors.
Then $R$ is a field. | As $R$ is non-null, there is at least one nonzero element in $R$.
Consider the two maps from $R$ to itself, for each nonzero $a \in R$:
:$\varphi_R: x \mapsto a \circ x$
:$\varphi_L: x \mapsto x \circ a$
By Ring Element is Zero Divisor iff not Cancellable, all nonzero elements in $R$ are cancellable. Thus:
:$a \circ x... | Let $\struct {R, +, \circ}$ be a [[Definition:Finite Set|finite]] [[Definition:Non-Null Ring|non-null ring]] with no [[Definition:Proper Zero Divisor|proper zero divisors]].
Then $R$ is a [[Definition:Field (Abstract Algebra)|field]]. | As $R$ is [[Definition:Non-Null Ring|non-null]], there is at least one [[Definition:Ring Zero|nonzero]] element in $R$.
Consider the two [[Definition:Mapping|maps]] from $R$ to itself, for each [[Definition:Ring Zero|nonzero]] $a \in R$:
:$\varphi_R: x \mapsto a \circ x$
:$\varphi_L: x \mapsto x \circ a$
By [[Ring ... | Finite Ring with No Proper Zero Divisors is Field | https://proofwiki.org/wiki/Finite_Ring_with_No_Proper_Zero_Divisors_is_Field | https://proofwiki.org/wiki/Finite_Ring_with_No_Proper_Zero_Divisors_is_Field | [
"Ring Theory",
"Field Theory",
"Galois Fields"
] | [
"Definition:Finite Set",
"Definition:Non-Null Ring",
"Definition:Proper Zero Divisor",
"Definition:Field (Abstract Algebra)"
] | [
"Definition:Non-Null Ring",
"Definition:Ring Zero",
"Definition:Mapping",
"Definition:Ring Zero",
"Ring Element is Zero Divisor iff not Cancellable",
"Definition:Cancellable Element",
"Definition:Injection",
"Equivalence of Mappings between Finite Sets of Same Cardinality",
"Definition:Surjection",
... |
proofwiki-8569 | Equivalence of Definitions of Dot Product | {{TFAE|def = Dot Product|context = Real Euclidean Space}}
Let $\mathbf a$ and $\mathbf b$ be vectors in the real Euclidean space $\R^n$. | === General Context implies Definition by Cosine ===
Let $\mathbf a \cdot \mathbf b$ be a dot product in its general context.
From Cosine Formula for Dot Product:
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$
where $\theta$ is the angle between $\mathbf v$ and $\mathbf w$.
Thus $\cdot$ ... | {{TFAE|def = Dot Product|context = Real Euclidean Space}}
Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector (Real Euclidean Space)|vectors]] in the [[Definition:Real Euclidean Space|real Euclidean space]] $\R^n$. | === General Context implies Definition by Cosine ===
Let $\mathbf a \cdot \mathbf b$ be a [[Definition:Dot Product/General Context|dot product in its general context]].
From [[Cosine Formula for Dot Product]]:
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$
where $\theta$ is the [[Defin... | Equivalence of Definitions of Dot Product | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Dot_Product | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Dot_Product | [
"Dot Product"
] | [
"Definition:Vector/Real Euclidean Space",
"Definition:Euclidean Space/Real"
] | [
"Definition:Dot Product/General Context",
"Cosine Formula for Dot Product",
"Definition:Angle between Vectors",
"Definition:Dot Product/Real Euclidean Space",
"Definition:Dot Product/Real Euclidean Space",
"Definition:Dot Product/General Context"
] |
proofwiki-8570 | Row Equivalent Matrix for Homogeneous System has same Solutions/Corollary | :$\set {\mathbf x: \mathbf A \mathbf x = \mathbf 0} = \set {\mathbf x: \map {\mathrm {ref} } {\mathbf A} \mathbf x = \mathbf 0}$
where $\map {\mathrm {ref} } {\mathbf A}$ is the reduced echelon form of $\mathbf A$. | Follows from Row Equivalent Matrix for Homogeneous System has same Solutions and from Matrix is Row Equivalent to Reduced Echelon Matrix.
{{qed}}
Category:Matrix Theory
Category:Linear Algebra
dla471omeqku92w6a8j8ujatufa5qe7 | :$\set {\mathbf x: \mathbf A \mathbf x = \mathbf 0} = \set {\mathbf x: \map {\mathrm {ref} } {\mathbf A} \mathbf x = \mathbf 0}$
where $\map {\mathrm {ref} } {\mathbf A}$ is the [[Definition:Reduced Echelon Form|reduced echelon form]] of $\mathbf A$. | Follows from [[Row Equivalent Matrix for Homogeneous System has same Solutions]] and from [[Matrix is Row Equivalent to Reduced Echelon Matrix]].
{{qed}}
[[Category:Matrix Theory]]
[[Category:Linear Algebra]]
dla471omeqku92w6a8j8ujatufa5qe7 | Row Equivalent Matrix for Homogeneous System has same Solutions/Corollary | https://proofwiki.org/wiki/Row_Equivalent_Matrix_for_Homogeneous_System_has_same_Solutions/Corollary | https://proofwiki.org/wiki/Row_Equivalent_Matrix_for_Homogeneous_System_has_same_Solutions/Corollary | [
"Matrix Theory",
"Linear Algebra"
] | [
"Definition:Echelon Matrix/Reduced Echelon Form"
] | [
"Row Equivalent Matrix for Homogeneous System has same Solutions",
"Matrix is Row Equivalent to Reduced Echelon Matrix",
"Category:Matrix Theory",
"Category:Linear Algebra"
] |
proofwiki-8571 | Continuous Mapping is Sequentially Continuous/Corollary | Let $f$ be continuous (everywhere) on $X$.
Then $f$ is sequentially continuous on $X$. | This follows immediately from the definitions:
:$(1): \quad$ A mapping is sequentially continuous everywhere in $X$ {{iff}} if it is sequentially continuous at each point
:$(2): \quad$ A mapping is continuous everywhere on $X$ {{iff}} it is continuous at each point.
{{Qed}}
Category:Continuous Mappings
Category:Sequent... | Let $f$ be [[Definition:Everywhere Continuous Mapping (Topology)|continuous (everywhere)]] on $X$.
Then $f$ is [[Definition:Sequential Continuity on Domain|sequentially continuous on $X$]]. | This follows immediately from the definitions:
:$(1): \quad$ A [[Definition:Mapping|mapping]] is [[Definition:Sequential Continuity on Domain|sequentially continuous]] everywhere in $X$ {{iff}} if it is [[Definition:Sequential Continuity at Point|sequentially continuous at each point]]
:$(2): \quad$ A [[Definition:Mapp... | Continuous Mapping is Sequentially Continuous/Corollary | https://proofwiki.org/wiki/Continuous_Mapping_is_Sequentially_Continuous/Corollary | https://proofwiki.org/wiki/Continuous_Mapping_is_Sequentially_Continuous/Corollary | [
"Continuous Mappings",
"Sequential Continuity"
] | [
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Sequential Continuity/Domain"
] | [
"Definition:Mapping",
"Definition:Sequential Continuity/Domain",
"Definition:Sequential Continuity/Point",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Continuous Mapping (Topology)/Point",
"Category:Continuous Mappings",
"Category:Sequential Continuity"
] |
proofwiki-8572 | Overflow Theorem/Corollary | The class of finite models is not $\Delta$-elementary.
That is:
:there is no set of formulas $F$ such that $F$ is satisfied by a model $\MM$
{{iff}}:
:$\MM$ is finite. | Follows directly from the Overflow Theorem.
{{qed}}
Category:Mathematical Logic
Category:Named Theorems
m2k1rzobtxxsdh4s6etfbvlsnaq57s4 | The [[Definition:Class (Class Theory)|class]] of [[Definition:Finite Model|finite]] [[Definition:Model (Predicate Logic)|models]] is not $\Delta$-elementary.
That is:
:there is no [[Definition:Set|set]] of formulas $F$ such that $F$ is [[Definition:Satisfiable Set of Formulas|satisfied]] by a [[Definition:Model (Predi... | Follows directly from the [[Overflow Theorem]].
{{qed}}
[[Category:Mathematical Logic]]
[[Category:Named Theorems]]
m2k1rzobtxxsdh4s6etfbvlsnaq57s4 | Overflow Theorem/Corollary | https://proofwiki.org/wiki/Overflow_Theorem/Corollary | https://proofwiki.org/wiki/Overflow_Theorem/Corollary | [
"Mathematical Logic",
"Named Theorems"
] | [
"Definition:Class (Class Theory)",
"Definition:Finite Model",
"Definition:Model (Predicate Logic)",
"Definition:Set",
"Definition:Satisfiable/Set of Formulas",
"Definition:Model (Predicate Logic)",
"Definition:Finite Model"
] | [
"Overflow Theorem",
"Category:Mathematical Logic",
"Category:Named Theorems"
] |
proofwiki-8573 | Product of Complex Numbers in Polar Form | Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be complex numbers expressed in polar form.
Then:
:$z_1 z_2 = r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} }$ | {{begin-eqn}}
{{eqn | l = z_1 z_2
| r = r_1 \paren {\cos \theta_1 + i \sin \theta_1} r_2 \paren {\cos \theta_2 + i \sin \theta_2}
| c = {{Defof|Polar Form of Complex Number}}
}}
{{eqn | r = r_1 r_2 \paren {\paren {\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2} + i \paren {\cos \theta_1 \sin \the... | Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be [[Definition:Polar Form of Complex Number|complex numbers expressed in polar form]].
Then:
:$z_1 z_2 = r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} }$ | {{begin-eqn}}
{{eqn | l = z_1 z_2
| r = r_1 \paren {\cos \theta_1 + i \sin \theta_1} r_2 \paren {\cos \theta_2 + i \sin \theta_2}
| c = {{Defof|Polar Form of Complex Number}}
}}
{{eqn | r = r_1 r_2 \paren {\paren {\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2} + i \paren {\cos \theta_1 \sin \the... | Product of Complex Numbers in Polar Form | https://proofwiki.org/wiki/Product_of_Complex_Numbers_in_Polar_Form | https://proofwiki.org/wiki/Product_of_Complex_Numbers_in_Polar_Form | [
"Product of Complex Numbers in Polar Form",
"Complex Multiplication",
"Polar Form of Complex Number"
] | [
"Definition:Complex Number/Polar Form"
] | [
"Cosine of Sum",
"Sine of Sum"
] |
proofwiki-8574 | Equality of Complex Numbers | Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers.
Then $z_1 = z_2$ {{iff}} $a_1 = a_2$ and $b_1 = b_2$. | By definition of a complex number, $z_1$ and $z_2$ can be expressed in the form:
:$z_1 = \tuple {a_1, b_1}$
:$z_2 = \tuple {a_2, b_2}$
where $\tuple {a, b}$ denotes an ordered pair.
The result follows from Equality of Ordered Pairs. | Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be [[Definition:Complex Number|complex numbers]].
Then $z_1 = z_2$ {{iff}} $a_1 = a_2$ and $b_1 = b_2$. | By definition of a [[Definition:Complex Number/Definition 2|complex number]], $z_1$ and $z_2$ can be expressed in the form:
:$z_1 = \tuple {a_1, b_1}$
:$z_2 = \tuple {a_2, b_2}$
where $\tuple {a, b}$ denotes an [[Definition:Ordered Pair|ordered pair]].
The result follows from [[Equality of Ordered Pairs]]. | Equality of Complex Numbers | https://proofwiki.org/wiki/Equality_of_Complex_Numbers | https://proofwiki.org/wiki/Equality_of_Complex_Numbers | [
"Equality of Complex Numbers",
"Complex Numbers",
"Equality"
] | [
"Definition:Complex Number"
] | [
"Definition:Complex Number/Definition 2",
"Definition:Ordered Pair",
"Equality of Ordered Pairs"
] |
proofwiki-8575 | Subtraction of Complex Numbers | Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers.
The subtraction operation on $z_1$ and $z_2$ is:
:$z_1 - z_2 = \paren {a_1 - a_2} + i \paren {b_1 - b_2}$ | {{begin-eqn}}
{{eqn | l = z_1 - z_2
| r = z_1 + \paren {- z_2}
| c = {{Defof|Complex Subtraction}}
}}
{{eqn | r = a_1 + \paren {-a_2} + i \paren {b_1 + \paren {-b_2} }
| c = Inverse for Complex Addition
}}
{{eqn | r = \paren {a_1 - a_2} + i \paren {b_1 - b_2}
| c = {{Defof|Complex Subtraction}}
... | Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be [[Definition:Complex Number|complex numbers]].
The [[Definition:Complex Subtraction|subtraction operation]] on $z_1$ and $z_2$ is:
:$z_1 - z_2 = \paren {a_1 - a_2} + i \paren {b_1 - b_2}$ | {{begin-eqn}}
{{eqn | l = z_1 - z_2
| r = z_1 + \paren {- z_2}
| c = {{Defof|Complex Subtraction}}
}}
{{eqn | r = a_1 + \paren {-a_2} + i \paren {b_1 + \paren {-b_2} }
| c = [[Inverse for Complex Addition]]
}}
{{eqn | r = \paren {a_1 - a_2} + i \paren {b_1 - b_2}
| c = {{Defof|Complex Subtractio... | Subtraction of Complex Numbers | https://proofwiki.org/wiki/Subtraction_of_Complex_Numbers | https://proofwiki.org/wiki/Subtraction_of_Complex_Numbers | [
"Complex Subtraction"
] | [
"Definition:Complex Number",
"Definition:Subtraction/Complex Numbers"
] | [
"Inverse for Complex Addition"
] |
proofwiki-8576 | Division of Complex Numbers | Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers such that $z_2 \ne 0$.
The operation of division is performed on $z_1$ by $z_2$ as follows:
:$\dfrac {z_1} {z_2} = \dfrac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2} + i \dfrac {a_2 b_1 - a_1 b_2} {a_2^2 + b_2^2}$ | {{begin-eqn}}
{{eqn | l = \frac {z_1} {z_2}
| r = z_1 \paren {z_2}^{-1}
| c = {{Defof|Complex Division}}
}}
{{eqn | r = \paren {a_1 + i b_1} \dfrac {a_2 - i b_2} {a_2^2 + b_2^2}
| c = Inverse for Complex Multiplication
}}
{{eqn | r = \frac {\paren {a_1 a_2 + b_1 b_2} + i \paren {a_2 b_1 - a_1 b_2} } {... | Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be [[Definition:Complex Number|complex numbers]] such that $z_2 \ne 0$.
The [[Definition:Binary Operation|operation]] of [[Definition:Complex Division|division]] is performed on $z_1$ by $z_2$ as follows:
:$\dfrac {z_1} {z_2} = \dfrac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^... | {{begin-eqn}}
{{eqn | l = \frac {z_1} {z_2}
| r = z_1 \paren {z_2}^{-1}
| c = {{Defof|Complex Division}}
}}
{{eqn | r = \paren {a_1 + i b_1} \dfrac {a_2 - i b_2} {a_2^2 + b_2^2}
| c = [[Inverse for Complex Multiplication]]
}}
{{eqn | r = \frac {\paren {a_1 a_2 + b_1 b_2} + i \paren {a_2 b_1 - a_1 b_2}... | Division of Complex Numbers/Proof 1 | https://proofwiki.org/wiki/Division_of_Complex_Numbers | https://proofwiki.org/wiki/Division_of_Complex_Numbers/Proof_1 | [
"Division of Complex Numbers",
"Complex Division"
] | [
"Definition:Complex Number",
"Definition:Operation/Binary Operation",
"Definition:Division/Field/Complex Numbers"
] | [
"Inverse for Complex Multiplication"
] |
proofwiki-8577 | De Moivre's Formula/Proof 1 | Let $z \in \C$ be a complex number expressed in complex form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \map \cos {\omega x} + i \map \sin {\omega x}$ | {{begin-eqn}}
{{eqn | l = \paren {r \paren {\cos x + i \sin x} }^\omega
| r = \paren {r e^{i x} }^\omega
| c = Euler's Formula
}}
{{eqn | r = r^\omega e^{i \omega x}
| c = Power of Power
}}
{{eqn | r = r^\omega \paren {\map \cos {\omega x} + i \map \sin {\omega x} }
| c = Euler's Formula
}}
{{en... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|complex form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \map \cos {\omega x} + i \map \sin {\omega x}$ | {{begin-eqn}}
{{eqn | l = \paren {r \paren {\cos x + i \sin x} }^\omega
| r = \paren {r e^{i x} }^\omega
| c = [[Euler's Formula]]
}}
{{eqn | r = r^\omega e^{i \omega x}
| c = [[Power of Power]]
}}
{{eqn | r = r^\omega \paren {\map \cos {\omega x} + i \map \sin {\omega x} }
| c = [[Euler's Formu... | De Moivre's Formula/Proof 1 | https://proofwiki.org/wiki/De_Moivre's_Formula/Proof_1 | https://proofwiki.org/wiki/De_Moivre's_Formula/Proof_1 | [
"De Moivre's Formula"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Euler's Formula",
"Exponent Combination Laws/Power of Power",
"Euler's Formula"
] |
proofwiki-8578 | Division of Complex Numbers in Polar Form | Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be complex numbers expressed in polar form, such that $z_2 \ne 0$.
Then:
:$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \paren {\map \cos {\theta_1 - \theta_2} + i \map \sin {\theta_1 - \theta_2} }$
or:
:$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \map \ci... | {{begin-eqn}}
{{eqn | l = \frac {z_1} {z_2}
| r = \frac {r_1 \paren {\cos \theta_1 + i \sin \theta_1} } {r_2 \paren {\cos \theta_2 + i \sin \theta_2} }
| c = {{Defof|Polar Form of Complex Number}}
}}
{{eqn | r = \frac {\paren {r_1 \paren {\cos \theta_1 + i \sin \theta_1} } \paren {r_2 \paren {\cos \theta_2 ... | Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be [[Definition:Polar Form of Complex Number|complex numbers expressed in polar form]], such that $z_2 \ne 0$.
Then:
:$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \paren {\map \cos {\theta_1 - \theta_2} + i \map \sin {\theta_1 - \theta_2} }$
or:
:$\... | {{begin-eqn}}
{{eqn | l = \frac {z_1} {z_2}
| r = \frac {r_1 \paren {\cos \theta_1 + i \sin \theta_1} } {r_2 \paren {\cos \theta_2 + i \sin \theta_2} }
| c = {{Defof|Polar Form of Complex Number}}
}}
{{eqn | r = \frac {\paren {r_1 \paren {\cos \theta_1 + i \sin \theta_1} } \paren {r_2 \paren {\cos \theta_2 ... | Division of Complex Numbers in Polar Form | https://proofwiki.org/wiki/Division_of_Complex_Numbers_in_Polar_Form | https://proofwiki.org/wiki/Division_of_Complex_Numbers_in_Polar_Form | [
"Complex Division",
"Polar Form of Complex Number",
"Division of Complex Numbers in Polar Form"
] | [
"Definition:Complex Number/Polar Form"
] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Product of Complex Numbers in Polar Form"
] |
proofwiki-8579 | Roots of Complex Number | Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then the $n$th roots of $z$ are given by:
:$z^{1 / n} = \set {r^{1 / n} \paren {\map \cos {\dfrac {\theta + 2 \pi k} n} + i \, \map \sin {\dfrac {\theta + 2 \pi k} n} }: ... | Let:
: $w := r^{1 / n} \paren {\map \cos {\dfrac {\theta + 2 \pi k} n} + i \, \map \sin {\dfrac {\theta + 2 \pi k} n} }$
for $k \in \Z_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = w^n
| r = \paren {r^{1 / n} \paren {\map \cos {\dfrac {\theta + 2 \pi k} n} + i \, \sin {\dfrac {\theta + 2 \pi k} n} } }^n
| c =
}}
... | Let $z := \polar {r, \theta}$ be a [[Definition:Polar Form of Complex Number|complex number expressed in polar form]], such that $z \ne 0$.
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then the [[Definition:Complex Root|$n$th roots]] of $z$ are given by:
:$z^{1 / n} ... | Let:
: $w := r^{1 / n} \paren {\map \cos {\dfrac {\theta + 2 \pi k} n} + i \, \map \sin {\dfrac {\theta + 2 \pi k} n} }$
for $k \in \Z_{>0}$.
Then:
{{begin-eqn}}
{{eqn | l = w^n
| r = \paren {r^{1 / n} \paren {\map \cos {\dfrac {\theta + 2 \pi k} n} + i \, \sin {\dfrac {\theta + 2 \pi k} n} } }^n
| c =
}... | Roots of Complex Number | https://proofwiki.org/wiki/Roots_of_Complex_Number | https://proofwiki.org/wiki/Roots_of_Complex_Number | [
"Complex Analysis",
"Complex Roots"
] | [
"Definition:Complex Number/Polar Form",
"Definition:Strictly Positive/Integer",
"Definition:Complex Root",
"Definition:Distinct/Plural",
"Definition:Complex Root"
] | [
"De Moivre's Formula",
"Cosine of Angle plus Multiple of Full Angle",
"Sine of Angle plus Multiple of Full Angle",
"Sine and Cosine are Periodic on Reals"
] |
proofwiki-8580 | Zeroth Power of Real Number equals One | Let $a \in \R_{>0}$ be a (strictly) positive real number.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
:$a^0 = 1$ | {{begin-eqn}}
{{eqn | l = a^0
| r = \map \exp {0 \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp 0
| c =
}}
{{eqn | r = 1
| c = Exponential of Zero
}}
{{end-eqn}}
{{qed}} | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $a^x$ be defined as [[Definition:Power to Real Number|$a$ to the power of $x$]].
Then:
:$a^0 = 1$ | {{begin-eqn}}
{{eqn | l = a^0
| r = \map \exp {0 \ln a}
| c = {{Defof|Power to Real Number}}
}}
{{eqn | r = \map \exp 0
| c =
}}
{{eqn | r = 1
| c = [[Exponential of Zero]]
}}
{{end-eqn}}
{{qed}} | Zeroth Power of Real Number equals One | https://proofwiki.org/wiki/Zeroth_Power_of_Real_Number_equals_One | https://proofwiki.org/wiki/Zeroth_Power_of_Real_Number_equals_One | [
"Powers"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Power (Algebra)/Real Number"
] | [
"Exponential of Zero"
] |
proofwiki-8581 | Exponential of One | :$\exp 1 = e$
where $e$ is Euler's number: $e = 2.718281828\ldots$ | We have that the exponential function is the inverse of the natural logarithm function:
:$\ln e = 1$
Hence the result.
{{qed}} | :$\exp 1 = e$
where $e$ is [[Definition:Euler's Number|Euler's number]]: $e = 2.718281828\ldots$ | We have that the [[Definition:Real Exponential Function|exponential function]] is the [[Definition:Inverse Mapping|inverse]] of the [[Definition:Natural Logarithm|natural logarithm function]]:
:$\ln e = 1$
Hence the result.
{{qed}} | Exponential of One | https://proofwiki.org/wiki/Exponential_of_One | https://proofwiki.org/wiki/Exponential_of_One | [
"Exponential Function"
] | [
"Definition:Euler's Number"
] | [
"Definition:Exponential Function/Real",
"Definition:Inverse Mapping",
"Definition:Natural Logarithm"
] |
proofwiki-8582 | Exponential of Zero | : $\exp 0 = 1$ | We have that the exponential function is the inverse of the natural logarithm function:
: $\ln 1 = 0$
Hence the result.
{{qed}} | : $\exp 0 = 1$ | We have that the [[Definition:Real Exponential Function|exponential function]] is the [[Definition:Inverse Mapping|inverse]] of the [[Definition:Natural Logarithm|natural logarithm function]]:
: $\ln 1 = 0$
Hence the result.
{{qed}} | Exponential of Zero/Proof 1 | https://proofwiki.org/wiki/Exponential_of_Zero | https://proofwiki.org/wiki/Exponential_of_Zero/Proof_1 | [
"Exponential Function",
"Exponential of Zero"
] | [] | [
"Definition:Exponential Function/Real",
"Definition:Inverse Mapping",
"Definition:Natural Logarithm"
] |
proofwiki-8583 | Exponential of Zero | : $\exp 0 = 1$ | Using the definition of the exponential as a limit of a sequence:
{{begin-eqn}}
{{eqn | l = \exp 0
| r = \lim_{n \mathop \to \infty} \left({1 + \frac 0 n}\right)^n
}}
{{eqn | r = 1
}}
{{end-eqn}}
{{qed}} | : $\exp 0 = 1$ | Using the definition of the exponential [[Definition:Exponential Function/Real/Limit of Sequence|as a limit of a sequence]]:
{{begin-eqn}}
{{eqn | l = \exp 0
| r = \lim_{n \mathop \to \infty} \left({1 + \frac 0 n}\right)^n
}}
{{eqn | r = 1
}}
{{end-eqn}}
{{qed}} | Exponential of Zero/Proof 2 | https://proofwiki.org/wiki/Exponential_of_Zero | https://proofwiki.org/wiki/Exponential_of_Zero/Proof_2 | [
"Exponential Function",
"Exponential of Zero"
] | [] | [
"Definition:Exponential Function/Real/Limit of Sequence"
] |
proofwiki-8584 | Exponential of Zero | : $\exp 0 = 1$ | This proof assumes the power series definition of $\exp$.
That is, let:
:$\ds \exp x = \sum_{k \mathop = 0}^\infty \frac {x^k} {k!}$
Then:
{{begin-eqn}}
{{eqn | l = \exp 0
| r = \sum_{k \mathop = 0}^\infty \frac {0^k} {k!}
}}
{{eqn | r = 1
| c = {{Defof|Power of Zero}}
}}
{{end-eqn}}
{{qed}} | : $\exp 0 = 1$ | This proof assumes the [[Definition:Exponential Function/Real/Power Series Expansion|power series definition of $\exp$]].
That is, let:
:$\ds \exp x = \sum_{k \mathop = 0}^\infty \frac {x^k} {k!}$
Then:
{{begin-eqn}}
{{eqn | l = \exp 0
| r = \sum_{k \mathop = 0}^\infty \frac {0^k} {k!}
}}
{{eqn | r = 1
|... | Exponential of Zero/Proof 3 | https://proofwiki.org/wiki/Exponential_of_Zero | https://proofwiki.org/wiki/Exponential_of_Zero/Proof_3 | [
"Exponential Function",
"Exponential of Zero"
] | [] | [
"Definition:Exponential Function/Real/Power Series Expansion"
] |
proofwiki-8585 | Exponential of Zero | : $\exp 0 = 1$ | This proof assumes the Definition of $\exp x$ as the unique continuous extension of $e^x$.
{{begin-eqn}}
{{eqn | l = \exp 0
| r = e^0
}}
{{eqn | r = 1
| c = {{Defof|Power (Algebra)|$x^0$}}, where $x \ne 0$
}}
{{end-eqn}}
{{qed}} | : $\exp 0 = 1$ | This proof assumes the [[Definition:Exponential Function/Real/Extension of Rational Exponential|Definition of $\exp x$ as the unique continuous extension of $e^x$]].
{{begin-eqn}}
{{eqn | l = \exp 0
| r = e^0
}}
{{eqn | r = 1
| c = {{Defof|Power (Algebra)|$x^0$}}, where $x \ne 0$
}}
{{end-eqn}}
{{qed}} | Exponential of Zero/Proof 4 | https://proofwiki.org/wiki/Exponential_of_Zero | https://proofwiki.org/wiki/Exponential_of_Zero/Proof_4 | [
"Exponential Function",
"Exponential of Zero"
] | [] | [
"Definition:Exponential Function/Real/Extension of Rational Exponential"
] |
proofwiki-8586 | Exponential of Zero | : $\exp 0 = 1$ | {{begin-eqn}}
{{eqn | l = \map \exp {z + \paren {-z} }
| r = \exp z \, \map \exp {-z}
| c = Exponential of Sum
}}
{{eqn | ll= \leadsto
| l = \map \exp {z - z}
| r = \dfrac {\exp z} {\exp z}
| c =
}}
{{eqn | ll= \leadsto
| l = \exp 0
| r = 1
| c =
}}
{{end-eqn}}
{{qed}} | : $\exp 0 = 1$ | {{begin-eqn}}
{{eqn | l = \map \exp {z + \paren {-z} }
| r = \exp z \, \map \exp {-z}
| c = [[Exponential of Sum]]
}}
{{eqn | ll= \leadsto
| l = \map \exp {z - z}
| r = \dfrac {\exp z} {\exp z}
| c =
}}
{{eqn | ll= \leadsto
| l = \exp 0
| r = 1
| c =
}}
{{end-eqn}}
{{qe... | Exponential of Zero/Proof 5 | https://proofwiki.org/wiki/Exponential_of_Zero | https://proofwiki.org/wiki/Exponential_of_Zero/Proof_5 | [
"Exponential Function",
"Exponential of Zero"
] | [] | [
"Exponential of Sum"
] |
proofwiki-8587 | Difference of Logarithms | :$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$ | {{begin-eqn}}
{{eqn | l = \log_b x - \log_b y
| r = \map {\log_b} {b^{\log_b x - \log_b y} }
| c = {{Defof|General Logarithm}}
}}
{{eqn | r = \map {\log_b} {\frac {\paren {b^{\log_b x} } } {\paren {b^{\log_b y} } } }
| c = Quotient of Powers
}}
{{eqn | r = \map {\log_b} {\frac x y}
| c = {{Defof... | :$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$ | {{begin-eqn}}
{{eqn | l = \log_b x - \log_b y
| r = \map {\log_b} {b^{\log_b x - \log_b y} }
| c = {{Defof|General Logarithm}}
}}
{{eqn | r = \map {\log_b} {\frac {\paren {b^{\log_b x} } } {\paren {b^{\log_b y} } } }
| c = [[Quotient of Powers]]
}}
{{eqn | r = \map {\log_b} {\frac x y}
| c = {{D... | Difference of Logarithms/Proof 1 | https://proofwiki.org/wiki/Difference_of_Logarithms | https://proofwiki.org/wiki/Difference_of_Logarithms/Proof_1 | [
"Difference of Logarithms",
"Logarithms"
] | [] | [
"Exponent Combination Laws/Quotient of Powers"
] |
proofwiki-8588 | Difference of Logarithms | :$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$ | {{begin-eqn}}
{{eqn | l = \log_b x - \log_b y
| r = \frac {\log_e x} {\log_e b} - \frac {\log_e y} {\log_e b}
| c = Change of Base of Logarithm
}}
{{eqn | r = \frac {\log_e x - \log_e y} {\log_e b}
| c =
}}
{{eqn | r = \frac {\log_e \left({\frac x y}\right)} {\log_e b}
| c = Difference of Logar... | :$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$ | {{begin-eqn}}
{{eqn | l = \log_b x - \log_b y
| r = \frac {\log_e x} {\log_e b} - \frac {\log_e y} {\log_e b}
| c = [[Change of Base of Logarithm]]
}}
{{eqn | r = \frac {\log_e x - \log_e y} {\log_e b}
| c =
}}
{{eqn | r = \frac {\log_e \left({\frac x y}\right)} {\log_e b}
| c = [[Difference of... | Difference of Logarithms/Proof 2 | https://proofwiki.org/wiki/Difference_of_Logarithms | https://proofwiki.org/wiki/Difference_of_Logarithms/Proof_2 | [
"Difference of Logarithms",
"Logarithms"
] | [] | [
"Change of Base of Logarithm",
"Difference of Logarithms/Natural Logarithm",
"Change of Base of Logarithm"
] |
proofwiki-8589 | Difference of Logarithms | :$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$ | {{begin-eqn}}
{{eqn | l = \map {\log_b} {\frac x y} + \log_b y
| r = \map {\log_b} {\frac x y \times y}
| c = Sum of Logarithms
}}
{{eqn | r = \log_b x
| c =
}}
{{eqn | ll= \leadsto
| l = \map {\log_b} {\frac x y}
| r = \log_b x - \log_b y
| c = subtracting $\log_b y$ from both side... | :$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$ | {{begin-eqn}}
{{eqn | l = \map {\log_b} {\frac x y} + \log_b y
| r = \map {\log_b} {\frac x y \times y}
| c = [[Sum of Logarithms]]
}}
{{eqn | r = \log_b x
| c =
}}
{{eqn | ll= \leadsto
| l = \map {\log_b} {\frac x y}
| r = \log_b x - \log_b y
| c = subtracting $\log_b y$ from both ... | Difference of Logarithms/Proof 3 | https://proofwiki.org/wiki/Difference_of_Logarithms | https://proofwiki.org/wiki/Difference_of_Logarithms/Proof_3 | [
"Difference of Logarithms",
"Logarithms"
] | [] | [
"Sum of Logarithms"
] |
proofwiki-8590 | Power of Identity is Identity | Let $\struct {M, \circ}$ be a monoid whose identity element is $e$.
Then:
:$\forall n \in \Z: e^n = e$ | Since $e$ is invertible, the power of $e$ is defined for all $n \in \Z$.
We prove the case $n \ge 0$ by induction. | Let $\struct {M, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity element]] is $e$.
Then:
:$\forall n \in \Z: e^n = e$ | Since [[Inverse of Identity Element is Itself|$e$ is invertible]], the [[Definition:Power of Element of Monoid|power of $e$]] is defined for all $n \in \Z$.
We prove the case $n \ge 0$ by [[Principle of Mathematical Induction|induction]]. | Power of Identity is Identity | https://proofwiki.org/wiki/Power_of_Identity_is_Identity | https://proofwiki.org/wiki/Power_of_Identity_is_Identity | [
"Monoids",
"Powers (Abstract Algebra)",
"Proofs by Induction"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Inverse of Identity Element is Itself",
"Definition:Power of Element/Monoid",
"Principle of Mathematical Induction",
"Definition:Power of Element/Monoid",
"Principle of Mathematical Induction",
"Inverse of Identity Element is Itself"
] |
proofwiki-8591 | Euler's Tangent Identity/Formulation 1 | :$\tan z = i \dfrac {1 - e^{2 i z} } {1 + e^{2 i z} }$ | {{begin-eqn}}
{{eqn | l = \tan z
| r = \frac {\sin z} {\cos z}
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {\frac 1 2 i \paren {e^{-i z} - e^{i z} } } {\frac 1 2 \paren {e^{-i z} + e^{i z} } }
| c = Euler's Sine Identity and Euler's Cosine Identity
}}
{{eqn | r = i \frac {e^{-i z} - ... | :$\tan z = i \dfrac {1 - e^{2 i z} } {1 + e^{2 i z} }$ | {{begin-eqn}}
{{eqn | l = \tan z
| r = \frac {\sin z} {\cos z}
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {\frac 1 2 i \paren {e^{-i z} - e^{i z} } } {\frac 1 2 \paren {e^{-i z} + e^{i z} } }
| c = [[Euler's Sine Identity]] and [[Euler's Cosine Identity]]
}}
{{eqn | r = i \frac {e^{... | Euler's Tangent Identity/Formulation 1 | https://proofwiki.org/wiki/Euler's_Tangent_Identity/Formulation_1 | https://proofwiki.org/wiki/Euler's_Tangent_Identity/Formulation_1 | [
"Euler's Tangent Identity"
] | [] | [
"Euler's Sine Identity",
"Euler's Cosine Identity",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Category:Euler's Tangent Identity"
] |
proofwiki-8592 | Euler's Tangent Identity/Formulation 2 | :$\tan z = \dfrac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }$ | {{begin-eqn}}
{{eqn | l = \tan z
| r = \frac {\sin z} {\cos z}
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {e^{i z} - e^{-i z} } {2 i} / \frac {e^{i z} + e^{-i z} } 2
| c = Euler's Sine Identity and Euler's Cosine Identity
}}
{{eqn | r = \frac {e^{i z} - e^{-i z} } {i \paren {e^{i z}... | :$\tan z = \dfrac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }$ | {{begin-eqn}}
{{eqn | l = \tan z
| r = \frac {\sin z} {\cos z}
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {e^{i z} - e^{-i z} } {2 i} / \frac {e^{i z} + e^{-i z} } 2
| c = [[Euler's Sine Identity]] and [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac {e^{i z} - e^{-i z} } {i \paren ... | Euler's Tangent Identity/Formulation 2 | https://proofwiki.org/wiki/Euler's_Tangent_Identity/Formulation_2 | https://proofwiki.org/wiki/Euler's_Tangent_Identity/Formulation_2 | [
"Euler's Tangent Identity"
] | [] | [
"Euler's Sine Identity",
"Euler's Cosine Identity",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8593 | Euler's Tangent Identity/Formulation 3 | :$\tan z = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$ | {{begin-eqn}}
{{eqn | l = \tan z
| r = \frac {\sin z} {\cos z}
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }
| c = Euler's Tangent Identity: Formulation 2
}}
{{eqn | r = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} ... | :$\tan z = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$ | {{begin-eqn}}
{{eqn | l = \tan z
| r = \frac {\sin z} {\cos z}
| c = {{Defof|Complex Tangent Function}}
}}
{{eqn | r = \frac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }
| c = [[Euler's Tangent Identity/Formulation 2|Euler's Tangent Identity: Formulation 2]]
}}
{{eqn | r = -i \paren {\dfrac... | Euler's Tangent Identity/Formulation 3 | https://proofwiki.org/wiki/Euler's_Tangent_Identity/Formulation_3 | https://proofwiki.org/wiki/Euler's_Tangent_Identity/Formulation_3 | [
"Euler's Tangent Identity"
] | [] | [
"Euler's Tangent Identity/Formulation 2",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8594 | Euler's Cotangent Identity | :$\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$ | We have, {{hypothesis}}, that $z$ is a complex number such that:
:$\forall k \in \Z: z \ne k \pi$
Therefore:
:$\sin z \ne 0$
It follows from the definition of the complex cotangent function that:
:$\cot z$
is well-defined.
Hence:
{{begin-eqn}}
{{eqn | l = \cot z
| r = \frac {\cos z} {\sin z}
| c = {{Defof|C... | :$\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$ | We have, {{hypothesis}}, that $z$ is a [[Definition:Complex Number|complex number]] such that:
:$\forall k \in \Z: z \ne k \pi$
Therefore:
:$\sin z \ne 0$
It follows from the definition of the [[Definition:Complex Cotangent Function|complex cotangent function]] that:
:$\cot z$
is [[Definition:Well-Defined|well-define... | Euler's Cotangent Identity/Proof 1 | https://proofwiki.org/wiki/Euler's_Cotangent_Identity | https://proofwiki.org/wiki/Euler's_Cotangent_Identity/Proof_1 | [
"Euler's Cotangent Identity",
"Euler's Identities",
"Cotangent Function"
] | [] | [
"Definition:Complex Number",
"Definition:Cotangent/Complex Function",
"Definition:Well-Defined",
"Euler's Sine Identity",
"Euler's Cosine Identity",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8595 | Euler's Cotangent Identity | :$\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$ | We have, {{hypothesis}}, that $z$ is a complex number such that:
:$\forall k \in \Z: z \ne k \pi$
Therefore:
:$\sin z \ne 0$
It follows from the definition of the complex cotangent function that:
:$\cot z$
is well-defined.
Hence:
{{begin-eqn}}
{{eqn | l = \cot z
| r = \frac 1 {\tan z}
| c = {{Defof|Complex ... | :$\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$ | We have, {{hypothesis}}, that $z$ is a [[Definition:Complex Number|complex number]] such that:
:$\forall k \in \Z: z \ne k \pi$
Therefore:
:$\sin z \ne 0$
It follows from the definition of the [[Definition:Complex Cotangent Function|complex cotangent function]] that:
:$\cot z$
is [[Definition:Well-Defined|well-define... | Euler's Cotangent Identity/Proof 2 | https://proofwiki.org/wiki/Euler's_Cotangent_Identity | https://proofwiki.org/wiki/Euler's_Cotangent_Identity/Proof_2 | [
"Euler's Cotangent Identity",
"Euler's Identities",
"Cotangent Function"
] | [] | [
"Definition:Complex Number",
"Definition:Cotangent/Complex Function",
"Definition:Well-Defined",
"Euler's Tangent Identity"
] |
proofwiki-8596 | Euler's Cosecant Identity | :$\csc z = \dfrac {2 i} {e^{i z} - e^{-i z} }$ | {{begin-eqn}}
{{eqn | l = \csc z
| r = \frac 1 {\sin z}
| c = {{Defof|Complex Cosecant Function}}
}}
{{eqn | r = 1 / \frac {e^{i z} - e^{-i z} } {2 i}
| c = Euler's Sine Identity
}}
{{eqn | r = \frac {2 i} {e^{i z} - e^{-i z} }
| c = multiplying top and bottom by $2 i$
}}
{{end-eqn}}
{{qed}} | :$\csc z = \dfrac {2 i} {e^{i z} - e^{-i z} }$ | {{begin-eqn}}
{{eqn | l = \csc z
| r = \frac 1 {\sin z}
| c = {{Defof|Complex Cosecant Function}}
}}
{{eqn | r = 1 / \frac {e^{i z} - e^{-i z} } {2 i}
| c = [[Euler's Sine Identity]]
}}
{{eqn | r = \frac {2 i} {e^{i z} - e^{-i z} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:... | Euler's Cosecant Identity | https://proofwiki.org/wiki/Euler's_Cosecant_Identity | https://proofwiki.org/wiki/Euler's_Cosecant_Identity | [
"Euler's Identities",
"Cosecant Function"
] | [] | [
"Euler's Sine Identity",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8597 | Euler's Secant Identity | :$\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$ | {{begin-eqn}}
{{eqn | l = \sec z
| r = \frac 1 {\cos z}
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = 1 / \frac {e^{i z} + e^{-i z} } 2
| c = Euler's Sine Identity and Euler's Cosine Identity
}}
{{eqn | r = \frac 2 {e^{i z} + e^{-i z} }
| c = multiplying top and bottom by $2$
}}
{{end-e... | :$\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$ | {{begin-eqn}}
{{eqn | l = \sec z
| r = \frac 1 {\cos z}
| c = {{Defof|Complex Secant Function}}
}}
{{eqn | r = 1 / \frac {e^{i z} + e^{-i z} } 2
| c = [[Euler's Sine Identity]] and [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac 2 {e^{i z} + e^{-i z} }
| c = multiplying [[Definition:Numerator|t... | Euler's Secant Identity | https://proofwiki.org/wiki/Euler's_Secant_Identity | https://proofwiki.org/wiki/Euler's_Secant_Identity | [
"Euler's Identities",
"Secant Function"
] | [] | [
"Euler's Sine Identity",
"Euler's Cosine Identity",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-8598 | Product of Complex Numbers in Exponential Form | Let $z_1 := r_1 e^{i \theta_1}$ and $z_2 := r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form.
Then:
:$z_1 z_2 = r_1 r_2 e^{i \paren {\theta_1 + \theta_2} }$ | {{begin-eqn}}
{{eqn | l = z_1 z_2
| r = r_1 e^{i \theta_1} r_2 e^{i \theta_2}
| c = {{Defof|Exponential Form of Complex Number}}
}}
{{eqn | r = r_1 \paren {\cos \theta_1 + i \sin \theta_1 } r_2 \paren {\cos \theta_2 + i \sin \theta_2}
| c = Euler's Formula
}}
{{eqn | r = r_1 r_2 \paren {\map \cos {\th... | Let $z_1 := r_1 e^{i \theta_1}$ and $z_2 := r_2 e^{i \theta_2}$ be [[Definition:Exponential Form of Complex Number|complex numbers expressed in exponential form]].
Then:
:$z_1 z_2 = r_1 r_2 e^{i \paren {\theta_1 + \theta_2} }$ | {{begin-eqn}}
{{eqn | l = z_1 z_2
| r = r_1 e^{i \theta_1} r_2 e^{i \theta_2}
| c = {{Defof|Exponential Form of Complex Number}}
}}
{{eqn | r = r_1 \paren {\cos \theta_1 + i \sin \theta_1 } r_2 \paren {\cos \theta_2 + i \sin \theta_2}
| c = [[Euler's Formula]]
}}
{{eqn | r = r_1 r_2 \paren {\map \cos ... | Product of Complex Numbers in Exponential Form | https://proofwiki.org/wiki/Product_of_Complex_Numbers_in_Exponential_Form | https://proofwiki.org/wiki/Product_of_Complex_Numbers_in_Exponential_Form | [
"Complex Multiplication",
"Exponential Form of Complex Number"
] | [
"Definition:Complex Number/Polar Form/Exponential Form"
] | [
"Euler's Formula",
"Product of Complex Numbers in Polar Form",
"Euler's Formula"
] |
proofwiki-8599 | Division of Complex Numbers in Exponential Form | Let $z_1 := r_1 e^{i \theta_1}$ and $z_2 := r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form.
Then:
:$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} e^{i \paren {\theta_1 - \theta_2} }$ | {{begin-eqn}}
{{eqn | l = \frac {z_1} {z_2}
| r = \frac {r_1 e^{i \theta_1} } {r_2 e^{i \theta_2} }
| c = {{Defof|Exponential Form of Complex Number}}
}}
{{eqn | r = \frac {r_1 \paren {\cos \theta_1 + i \sin \theta_1 } } {r_2 \paren {\cos \theta_2 + i \sin \theta_2} }
| c = Euler's Formula
}}
{{eqn | ... | Let $z_1 := r_1 e^{i \theta_1}$ and $z_2 := r_2 e^{i \theta_2}$ be [[Definition:Exponential Form of Complex Number|complex numbers expressed in exponential form]].
Then:
:$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} e^{i \paren {\theta_1 - \theta_2} }$ | {{begin-eqn}}
{{eqn | l = \frac {z_1} {z_2}
| r = \frac {r_1 e^{i \theta_1} } {r_2 e^{i \theta_2} }
| c = {{Defof|Exponential Form of Complex Number}}
}}
{{eqn | r = \frac {r_1 \paren {\cos \theta_1 + i \sin \theta_1 } } {r_2 \paren {\cos \theta_2 + i \sin \theta_2} }
| c = [[Euler's Formula]]
}}
{{eq... | Division of Complex Numbers in Exponential Form | https://proofwiki.org/wiki/Division_of_Complex_Numbers_in_Exponential_Form | https://proofwiki.org/wiki/Division_of_Complex_Numbers_in_Exponential_Form | [
"Complex Division"
] | [
"Definition:Complex Number/Polar Form/Exponential Form"
] | [
"Euler's Formula",
"Division of Complex Numbers in Polar Form",
"Euler's Formula"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.