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-4500 | Tangent Function is Odd | :$\map \tan {-x} = -\tan x$
That is, the tangent function is odd. | {{begin-eqn}}
{{eqn | l = \map \tan {-x}
| r = \frac {\map \sin {-x} } {\map \cos {-x} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {-\sin x} {\cos x}
| c = Sine Function is Odd; Cosine Function is Even
}}
{{eqn | r = -\tan x
| c = Tangent is Sine divided by Cosine
}}
{{end-eqn... | :$\map \tan {-x} = -\tan x$
That is, the [[Definition:Real Tangent Function|tangent function]] is [[Definition:Odd Function|odd]]. | {{begin-eqn}}
{{eqn | l = \map \tan {-x}
| r = \frac {\map \sin {-x} } {\map \cos {-x} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {-\sin x} {\cos x}
| c = [[Sine Function is Odd]]; [[Cosine Function is Even]]
}}
{{eqn | r = -\tan x
| c = [[Tangent is Sine divided by Cosin... | Tangent Function is Odd | https://proofwiki.org/wiki/Tangent_Function_is_Odd | https://proofwiki.org/wiki/Tangent_Function_is_Odd | [
"Tangent Function",
"Examples of Odd Functions"
] | [
"Definition:Tangent Function/Real",
"Definition:Odd Function"
] | [
"Tangent is Sine divided by Cosine",
"Sine Function is Odd",
"Cosine Function is Even",
"Tangent is Sine divided by Cosine"
] |
proofwiki-4501 | Tangent of Sum | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | First we note:
{{begin-eqn}}
{{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j
| r = \prod_j \paren {\cos \theta_j + i \sin \theta_j}
| c = Product of Complex Numbers in Polar Form
}}
{{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j}
| c =
}}
{{eqn | r = \prod_j \cos \the... | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | First we note:
{{begin-eqn}}
{{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j
| r = \prod_j \paren {\cos \theta_j + i \sin \theta_j}
| c = [[Product of Complex Numbers in Polar Form]]
}}
{{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j}
| c =
}}
{{eqn | r = \prod_j \cos ... | Tangent of Sum of Series of Angles/Proof 1 | https://proofwiki.org/wiki/Tangent_of_Sum | https://proofwiki.org/wiki/Tangent_of_Sum_of_Series_of_Angles/Proof_1 | [
"Tangent of Sum",
"Tangent Function",
"Trigonometric Addition Formulas"
] | [] | [
"Product of Complex Numbers in Polar Form",
"Product of Complex Numbers in Polar Form",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-4502 | Tangent of Sum | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | First we note:
{{begin-eqn}}
{{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j
| r = \prod_j \paren {\cos \theta_j + i \sin \theta_j}
| c = Product of Complex Numbers in Polar Form
}}
{{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j}
| c =
}}
{{eqn | r = \prod_j \cos \the... | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | First we note:
{{begin-eqn}}
{{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j
| r = \prod_j \paren {\cos \theta_j + i \sin \theta_j}
| c = [[Product of Complex Numbers in Polar Form]]
}}
{{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j}
| c =
}}
{{eqn | r = \prod_j \cos ... | Tangent of Sum of Series of Angles/Proof 2 | https://proofwiki.org/wiki/Tangent_of_Sum | https://proofwiki.org/wiki/Tangent_of_Sum_of_Series_of_Angles/Proof_2 | [
"Tangent of Sum",
"Tangent Function",
"Trigonometric Addition Formulas"
] | [] | [
"Product of Complex Numbers in Polar Form"
] |
proofwiki-4503 | Tangent of Sum | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | {{begin-eqn}}
{{eqn | l = \map \tan {A + B + C}
| r = \dfrac {\tan A + \map \tan {B + C} } {1 - \tan A \tan {B + C} }
| c = Tangent of Sum
}}
{{eqn | r = \dfrac {\tan A + \frac {\tan B + \tan C} {1 - \tan B \tan C} } {1 - \tan A \frac {\tan B + \tan C} {1 - \tan B \tan C} }
| c = Tangent of Sum
}}
{{e... | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | {{begin-eqn}}
{{eqn | l = \map \tan {A + B + C}
| r = \dfrac {\tan A + \map \tan {B + C} } {1 - \tan A \tan {B + C} }
| c = [[Tangent of Sum]]
}}
{{eqn | r = \dfrac {\tan A + \frac {\tan B + \tan C} {1 - \tan B \tan C} } {1 - \tan A \frac {\tan B + \tan C} {1 - \tan B \tan C} }
| c = [[Tangent of Sum]... | Tangent of Sum of Three Angles/Proof 2 | https://proofwiki.org/wiki/Tangent_of_Sum | https://proofwiki.org/wiki/Tangent_of_Sum_of_Three_Angles/Proof_2 | [
"Tangent of Sum",
"Tangent Function",
"Trigonometric Addition Formulas"
] | [] | [
"Tangent of Sum",
"Tangent of Sum",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-4504 | Tangent of Sum | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | This is a special case of Tangent of Sum of Series of Angles, for $n = 3$.
{{qed}} | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | This is a special case of [[Tangent of Sum of Series of Angles]], for $n = 3$.
{{qed}} | Tangent of Sum of Three Angles/Proof 3 | https://proofwiki.org/wiki/Tangent_of_Sum | https://proofwiki.org/wiki/Tangent_of_Sum_of_Three_Angles/Proof_3 | [
"Tangent of Sum",
"Tangent Function",
"Trigonometric Addition Formulas"
] | [] | [
"Tangent of Sum of Series of Angles"
] |
proofwiki-4505 | Tangent of Sum | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | {{begin-eqn}}
{{eqn | l = \map \tan {a + b}
| r = \frac {\map \sin {a + b} } {\map \cos {a + b} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\sin a \cos b + \cos a \sin b} {\cos a \cos b - \sin a \sin b}
| c = Sine of Sum and Cosine of Sum
}}
{{eqn | r = \frac {\frac {\sin a} {\cos ... | :$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$ | {{begin-eqn}}
{{eqn | l = \map \tan {a + b}
| r = \frac {\map \sin {a + b} } {\map \cos {a + b} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\sin a \cos b + \cos a \sin b} {\cos a \cos b - \sin a \sin b}
| c = [[Sine of Sum]] and [[Cosine of Sum]]
}}
{{eqn | r = \frac {\frac {\s... | Tangent of Sum/Proof | https://proofwiki.org/wiki/Tangent_of_Sum | https://proofwiki.org/wiki/Tangent_of_Sum/Proof | [
"Tangent of Sum",
"Tangent Function",
"Trigonometric Addition Formulas"
] | [] | [
"Tangent is Sine divided by Cosine",
"Sine of Sum",
"Cosine of Sum",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Tangent is Sine divided by Cosine"
] |
proofwiki-4506 | Natural Logarithm Function is Differentiable | The (real) natural logarithm function is differentiable. | This proof assumes the definition of $\ln$ as:
:$\ln x = \ds \int_1^x \frac 1 t \rd t$
As $\ln$ is defined as a definite integral, the result follows from the {{Corollary|Fundamental Theorem of Calculus/First Part|disp = Fundamental Theorem of Calculus}}.
{{qed}} | The [[Definition:Real Natural Logarithm|(real) natural logarithm function]] is [[Definition:Differentiable Real Function|differentiable]]. | This proof assumes the [[Definition:Natural Logarithm/Positive Real/Definition 1|definition of $\ln$]] as:
:$\ln x = \ds \int_1^x \frac 1 t \rd t$
As $\ln$ is defined as a [[Definition:Definite Integral|definite integral]], the result follows from the {{Corollary|Fundamental Theorem of Calculus/First Part|disp = Fund... | Natural Logarithm Function is Differentiable/Proof 1 | https://proofwiki.org/wiki/Natural_Logarithm_Function_is_Differentiable | https://proofwiki.org/wiki/Natural_Logarithm_Function_is_Differentiable/Proof_1 | [
"Differential Calculus",
"Natural Logarithms",
"Natural Logarithm Function is Differentiable"
] | [
"Definition:Natural Logarithm/Positive Real",
"Definition:Differentiable Mapping/Real Function"
] | [
"Definition:Natural Logarithm/Positive Real/Definition 1",
"Definition:Definite Integral"
] |
proofwiki-4507 | Natural Logarithm Function is Differentiable | The (real) natural logarithm function is differentiable. | This proof assumes the definition of $\ln$ as the inverse of the exponential function.
As the Exponential Function is Differentiable, the result follows from the differentiability of inverse functions.
{{qed}} | The [[Definition:Real Natural Logarithm|(real) natural logarithm function]] is [[Definition:Differentiable Real Function|differentiable]]. | This proof assumes the definition of $\ln$ as the [[Definition:Inverse Mapping|inverse]] of the [[Definition:Real Exponential Function|exponential function]].
As the [[Exponential Function is Differentiable]], the result follows from the [[Derivative of Inverse Function|differentiability of inverse functions]].
{{qed}... | Natural Logarithm Function is Differentiable/Proof 2 | https://proofwiki.org/wiki/Natural_Logarithm_Function_is_Differentiable | https://proofwiki.org/wiki/Natural_Logarithm_Function_is_Differentiable/Proof_2 | [
"Differential Calculus",
"Natural Logarithms",
"Natural Logarithm Function is Differentiable"
] | [
"Definition:Natural Logarithm/Positive Real",
"Definition:Differentiable Mapping/Real Function"
] | [
"Definition:Inverse Mapping",
"Definition:Exponential Function/Real",
"Derivative of Exponential Function",
"Derivative of Inverse Function"
] |
proofwiki-4508 | Countable Basis of Real Number Line | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\BB$ be the set of subsets of $\R$ defined as:
:$\BB = \set {\openint a b: a, b \in \Q,\ a < b}$
That is, $\BB$ is the set of open intervals of $\R$ whose endpoints are rational numbers.
Then $\BB$ forms a countable basis of $\... | Let $U \in \tau_d$.
Let $x \in U$.
Let:
:$\BB' = \set {\openint c d: c, d \in \R,\ c < d}$
By Basis for Euclidean Topology on Real Number Line, $\BB'$ is an analytic basis for the Euclidean topology on $\R$.
Then let $\openint c d \in \BB'$ such that $x \in \openint c d \subseteq U$.
That is, $c < x < d$.
By Between tw... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $\BB$ be the [[Definition:Set of Sets|set of subsets]] of $\R$ defined as:
:$\BB = \set {\openint a b: a, b \in \Q,\ a < b}$
That is, $\BB$ is the [[Definition:Set|set]] ... | Let $U \in \tau_d$.
Let $x \in U$.
Let:
:$\BB' = \set {\openint c d: c, d \in \R,\ c < d}$
By [[Basis for Euclidean Topology on Real Number Line]], $\BB'$ is an [[Definition:Analytic Basis|analytic basis]] for the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] on $\R$.
Then let $\openint c... | Countable Basis of Real Number Line | https://proofwiki.org/wiki/Countable_Basis_of_Real_Number_Line | https://proofwiki.org/wiki/Countable_Basis_of_Real_Number_Line | [
"Real Number Line with Euclidean Topology",
"Examples of Countable Bases",
"Examples of Topological Bases"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Set of Sets",
"Definition:Set",
"Definition:Real Interval/Open",
"Definition:Real Interval/Endpoints",
"Definition:Rational Number",
"Definition:Countable Basis"
] | [
"Basis for Euclidean Topology on Real Number Line",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Between two Real Numbers exists Rational Number",
"Between two Real Numbers exists Rational Number",
"Definition:Basis (Topology)",
"Defini... |
proofwiki-4509 | Second-Countable Space is Separable | Let $T = \struct {S, \tau}$ be a second-countable topological space.
Then $T$ is also a separable space. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
Let $T = \struct {S, \tau}$ be a second-countable space.
{{Recall|Second-Countable Space|second-countable space}}
{{:Definition:Second-Countable Space}}
By definition, there exists a countable basis $\BB$ for $\tau$.
Using the {{Axiom-link|Count... | Let $T = \struct {S, \tau}$ be a [[Definition:Second-Countable Space|second-countable]] [[Definition:Topological Space|topological space]].
Then $T$ is also a [[Definition:Separable Space|separable space]]. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Second-Countable Space|second-countable space]].
{{Recall|Second-Countable Space|second-countable space}}
{{:Definition:Second-Countable Space}}
By definition, there exists a [[Definition:Countabl... | Second-Countable Space is Separable | https://proofwiki.org/wiki/Second-Countable_Space_is_Separable | https://proofwiki.org/wiki/Second-Countable_Space_is_Separable | [
"Second-Countable Spaces",
"Separable Spaces"
] | [
"Definition:Second-Countable Space",
"Definition:Topological Space",
"Definition:Separable Space"
] | [
"Definition:Second-Countable Space",
"Definition:Countable Basis",
"Definition:Choice Function",
"Definition:Element",
"Definition:Element",
"Image of Countable Set under Mapping is Countable",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Equivalence of Definitions of Analytic Basis",... |
proofwiki-4510 | Real Number Line is Second-Countable | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is a second-countable space. | From Countable Basis of Real Number Line we have that $\struct {\R, \tau_d}$ has a countable basis.
The result follows directly from the definition of a second-countable space.
{{qed}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is a [[Definition:Second-Countable Space|second-countable space]]. | From [[Countable Basis of Real Number Line]] we have that $\struct {\R, \tau_d}$ has a [[Definition:Countable Basis|countable basis]].
The result follows directly from the definition of a [[Definition:Second-Countable Space|second-countable space]].
{{qed}} | Real Number Line is Second-Countable | https://proofwiki.org/wiki/Real_Number_Line_is_Second-Countable | https://proofwiki.org/wiki/Real_Number_Line_is_Second-Countable | [
"Real Number Line with Euclidean Topology",
"Examples of Second-Countable Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Second-Countable Space"
] | [
"Countable Basis of Real Number Line",
"Definition:Countable Basis",
"Definition:Second-Countable Space"
] |
proofwiki-4511 | Real Number Line is not Countably Compact | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is not a countably compact space. | Let $\CC$ be the set of subsets of $\R$ defined as:
:$\CC = \set {\openint n {n + 2}: n \in \Z}$
Then $\CC$ is an open cover of $\R$ which is countable.
However, there is no finite subcover for $\R$ of $\CC$.
Hence the result, by definition of countably compact.
{{qed}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is not a [[Definition:Countably Compact Space|countably compact space]]. | Let $\CC$ be the [[Definition:Set of Sets|set of subsets]] of $\R$ defined as:
:$\CC = \set {\openint n {n + 2}: n \in \Z}$
Then $\CC$ is an [[Definition:Open Cover|open cover]] of $\R$ which is [[Definition:Countable Set|countable]].
However, there is no [[Definition:Finite Subcover|finite subcover]] for $\R$ of $\C... | Real Number Line is not Countably Compact | https://proofwiki.org/wiki/Real_Number_Line_is_not_Countably_Compact | https://proofwiki.org/wiki/Real_Number_Line_is_not_Countably_Compact | [
"Real Number Line with Euclidean Topology",
"Examples of Countably Compact Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Countably Compact Space"
] | [
"Definition:Set of Sets",
"Definition:Open Cover",
"Definition:Countable Set",
"Definition:Subcover/Finite",
"Definition:Countably Compact Space"
] |
proofwiki-4512 | Real Number Line is Locally Compact Hausdorff Space | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is a locally compact Hausdorff space. | We have that a Real Number Line satisfies all Separation Axioms.
Specifically, $\struct {\R, \tau_d}$ is a Hausdorff space.
Consider $\CC$ the set of subsets of $\R$ defined as:
:$\CC = \set {\closedint n {n + 1}: n \in \Z}$
where $\closedint n {n + 1}$ is the closed real interval between successive integers.
By the He... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is a [[Definition:Locally Compact Hausdorff Space|locally compact Hausdorff space]]. | We have that a [[Real Number Line satisfies all Separation Axioms]].
Specifically, $\struct {\R, \tau_d}$ is a [[Definition:Hausdorff Space|Hausdorff space]].
Consider $\CC$ the [[Definition:Set of Sets|set of subsets]] of $\R$ defined as:
:$\CC = \set {\closedint n {n + 1}: n \in \Z}$
where $\closedint n {n + 1}$ i... | Real Number Line is Locally Compact Hausdorff Space | https://proofwiki.org/wiki/Real_Number_Line_is_Locally_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Real_Number_Line_is_Locally_Compact_Hausdorff_Space | [
"Real Number Line with Euclidean Topology",
"Examples of Locally Compact Hausdorff Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Locally Compact Hausdorff Space"
] | [
"Real Number Line satisfies all Separation Axioms",
"Definition:T2 Space",
"Definition:Set of Sets",
"Definition:Real Interval/Closed",
"Definition:Integer",
"Heine-Borel Theorem",
"Definition:Compact Topological Space",
"Definition:Locally Compact Hausdorff Space"
] |
proofwiki-4513 | Real Number Line is Sigma-Compact | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is a $\sigma$-compact space. | We have that a Real Number Line satisfies all Separation Axioms.
Specifically, $\struct {\R, \tau_d}$ is a Hausdorff space.
Consider $\CC$ the set of subsets of $\R$ defined as:
:$\CC = \set {\closedint n {n + 1}: n \in \Z}$
where $\closedint n {n + 1}$ is the closed real interval between successive integers.
By the He... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is a [[Definition:Sigma-Compact Space|$\sigma$-compact space]]. | We have that a [[Real Number Line satisfies all Separation Axioms]].
Specifically, $\struct {\R, \tau_d}$ is a [[Definition:Hausdorff Space|Hausdorff space]].
Consider $\CC$ the [[Definition:Set of Sets|set of subsets]] of $\R$ defined as:
:$\CC = \set {\closedint n {n + 1}: n \in \Z}$
where $\closedint n {n + 1}$ i... | Real Number Line is Sigma-Compact | https://proofwiki.org/wiki/Real_Number_Line_is_Sigma-Compact | https://proofwiki.org/wiki/Real_Number_Line_is_Sigma-Compact | [
"Real Number Line with Euclidean Topology",
"Examples of Sigma-Compact Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Sigma-Compact Space"
] | [
"Real Number Line satisfies all Separation Axioms",
"Definition:T2 Space",
"Definition:Set of Sets",
"Definition:Real Interval/Closed",
"Definition:Integer",
"Heine-Borel Theorem",
"Definition:Element",
"Definition:Compact Topological Space/Subspace",
"Definition:Countable Set",
"Definition:Biject... |
proofwiki-4514 | Closed Subset of Real Number Line is G-Delta | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $H \subseteq \R$ be a closed subset of $\R$.
Then $H$ is a $G_\delta$ set. | We have:
:Real Number Line is Metric Space
:Closed Set in Metric Space is $G_\delta$
Hence the result.
{{qed}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Let $H \subseteq \R$ be a [[Definition:Closed Set (Topology)|closed subset]] of $\R$.
Then $H$ is a [[Definition:G-Delta Set|$G_\delta$ set]]. | We have:
:[[Real Number Line is Metric Space]]
:[[Closed Set in Metric Space is G-Delta|Closed Set in Metric Space is $G_\delta$]]
Hence the result.
{{qed}} | Closed Subset of Real Number Line is G-Delta | https://proofwiki.org/wiki/Closed_Subset_of_Real_Number_Line_is_G-Delta | https://proofwiki.org/wiki/Closed_Subset_of_Real_Number_Line_is_G-Delta | [
"Real Number Line with Euclidean Topology",
"Examples of G-Delta Sets"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Closed Set/Topology",
"Definition:G-Delta Set"
] | [
"Real Number Line is Metric Space",
"Closed Set in Metric Space is G-Delta"
] |
proofwiki-4515 | Real Number Line is Paracompact | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is paracompact. | Let $\CC$ be an open cover for $\R$.
Then $\CC$ covers each of the closed real intervals $\closedint n {n + 1}$ for all $n \in \Z$.
By the Heine-Borel Theorem, each of $\closedint n {n + 1}$ is compact.
So, for each of these intervals $\closedint n {n + 1}$, it follows that $\CC$ can be reduced to a sequence $\sequence... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is [[Definition:Paracompact Space|paracompact]]. | Let $\CC$ be an [[Definition:Open Cover|open cover]] for $\R$.
Then $\CC$ covers each of the [[Definition:Closed Real Interval|closed real intervals]] $\closedint n {n + 1}$ for all $n \in \Z$.
By the [[Heine-Borel Theorem]], each of $\closedint n {n + 1}$ is [[Definition:Compact Topological Space|compact]].
So, for... | Real Number Line is Paracompact | https://proofwiki.org/wiki/Real_Number_Line_is_Paracompact | https://proofwiki.org/wiki/Real_Number_Line_is_Paracompact | [
"Real Number Line with Euclidean Topology",
"Examples of Paracompact Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Paracompact Space"
] | [
"Definition:Open Cover",
"Definition:Real Interval/Closed",
"Heine-Borel Theorem",
"Definition:Compact Topological Space",
"Definition:Sequence",
"Definition:Subcover/Finite",
"Definition:Refinement of Cover",
"Definition:Locally Finite Cover",
"Definition:Paracompact Space"
] |
proofwiki-4516 | Real Number Line with Off-Center Distance Function is Quasimetric Space | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\tau_d$ can be given by a quasimetric defined as:
:<nowiki>$\map d {x, y} = \begin {cases}
y - x & : y \ge x \\
2 \paren {x - y} & : y < x \end {cases}$</nowiki>
Thus $\struct {\R, \tau_d}$ is a quasimetric space. | To show that $\map d {x, y}$ is a quasimetric, we need to show that $d: \R \times \R \to \R$ satisfies the following conditions for all $x, y, z \in \R$:
{{begin-itemize}}
{{item|(\text M 1):|$\map d {x, x} {{=}} 0$}}
{{item|(\text M 2):|$\map d {x, y} + \map d {y, z} \ge \map d {x, z}$}}
{{item|(\text M 4):|$x \ne y \... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\tau_d$ can be given by a [[Definition:Quasimetric|quasimetric]] defined as:
:<nowiki>$\map d {x, y} = \begin {cases}
y - x & : y \ge x \\
2 \paren {x - y} & : y < x \e... | To show that $\map d {x, y}$ is a [[Definition:Quasimetric|quasimetric]], we need to show that $d: \R \times \R \to \R$ satisfies the following conditions for all $x, y, z \in \R$:
{{begin-itemize}}
{{item|(\text M 1):|$\map d {x, x} {{=}} 0$}}
{{item|(\text M 2):|$\map d {x, y} + \map d {y, z} \ge \map d {x, z}$}}
{{... | Real Number Line with Off-Center Distance Function is Quasimetric Space | https://proofwiki.org/wiki/Real_Number_Line_with_Off-Center_Distance_Function_is_Quasimetric_Space | https://proofwiki.org/wiki/Real_Number_Line_with_Off-Center_Distance_Function_is_Quasimetric_Space | [
"Real Number Line with Euclidean Topology",
"Examples of Quasimetric Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Quasimetric",
"Definition:Quasimetric/Quasimetric Space"
] | [
"Definition:Quasimetric"
] |
proofwiki-4517 | NAND with Equal Arguments | :$p \uparrow p \dashv \vdash \neg p$
That is, the NAND of a proposition with itself corresponds to the negation operation. | {{BeginTableau|p \uparrow p \vdash \neg p}}
{{Premise|1|p \uparrow p}}
{{SequentIntro|2|1|\neg \paren {p \land p}|1|Definition of Logical NAND}}
{{Idempotence|3|1|\neg p|2|Conjunction}}
{{EndTableau|lemma}}
{{BeginTableau|\neg p \vdash p \uparrow p}}
{{Premise|1|\neg p}}
{{Idempotence|2|1|\neg \paren {p \land p}|1|Conj... | :$p \uparrow p \dashv \vdash \neg p$
That is, the [[Definition:Logical NAND|NAND]] of a [[Definition:Proposition|proposition]] with itself corresponds to the [[Definition:Logical Not|negation]] operation. | {{BeginTableau|p \uparrow p \vdash \neg p}}
{{Premise|1|p \uparrow p}}
{{SequentIntro|2|1|\neg \paren {p \land p}|1|Definition of [[Definition:Logical NAND|Logical NAND]]}}
{{Idempotence|3|1|\neg p|2|Conjunction}}
{{EndTableau|lemma}}
{{BeginTableau|\neg p \vdash p \uparrow p}}
{{Premise|1|\neg p}}
{{Idempotence|2|1|... | NAND with Equal Arguments/Proof 1 | https://proofwiki.org/wiki/NAND_with_Equal_Arguments | https://proofwiki.org/wiki/NAND_with_Equal_Arguments/Proof_1 | [
"Logical NAND",
"NAND with Equal Arguments"
] | [
"Definition:Logical NAND",
"Definition:Proposition",
"Definition:Logical Not"
] | [
"Definition:Logical NAND",
"Definition:Logical NAND"
] |
proofwiki-4518 | NAND with Equal Arguments | :$p \uparrow p \dashv \vdash \neg p$
That is, the NAND of a proposition with itself corresponds to the negation operation. | We apply the Method of Truth Tables:
:$\begin{array}{|ccc||cc|} \hline
p & \uparrow & p & \neg & p \\
\hline
\F & \T & \F & \T & \F \\
\T & \F & \T & \F & \T \\
\hline
\end{array}$
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
{{qed}} | :$p \uparrow p \dashv \vdash \neg p$
That is, the [[Definition:Logical NAND|NAND]] of a [[Definition:Proposition|proposition]] with itself corresponds to the [[Definition:Logical Not|negation]] operation. | We apply the [[Method of Truth Tables]]:
:$\begin{array}{|ccc||cc|} \hline
p & \uparrow & p & \neg & p \\
\hline
\F & \T & \F & \T & \F \\
\T & \F & \T & \F & \T \\
\hline
\end{array}$
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logi... | NAND with Equal Arguments/Proof by Truth Table | https://proofwiki.org/wiki/NAND_with_Equal_Arguments | https://proofwiki.org/wiki/NAND_with_Equal_Arguments/Proof_by_Truth_Table | [
"Logical NAND",
"NAND with Equal Arguments"
] | [
"Definition:Logical NAND",
"Definition:Proposition",
"Definition:Logical Not"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-4519 | NAND is Commutative | :$p \uparrow q \dashv \vdash q \uparrow p$ | {{BeginTableau|p \uparrow q \vdash q \uparrow p}}
{{Premise|1|p \uparrow q}}
{{SequentIntro|2|1|\neg \paren {p \land q}|1|{{Defof|Logical NAND}} }}
{{Commutation|3|1|\neg \paren {q \land p}|2|Conjunction}}
{{SequentIntro|4|1|q \uparrow p|3|{{Defof|Logical NAND}} }}
{{EndTableau|lemma}}
{{BeginTableau|q \uparrow p \vdas... | :$p \uparrow q \dashv \vdash q \uparrow p$ | {{BeginTableau|p \uparrow q \vdash q \uparrow p}}
{{Premise|1|p \uparrow q}}
{{SequentIntro|2|1|\neg \paren {p \land q}|1|{{Defof|Logical NAND}} }}
{{Commutation|3|1|\neg \paren {q \land p}|2|Conjunction}}
{{SequentIntro|4|1|q \uparrow p|3|{{Defof|Logical NAND}} }}
{{EndTableau|lemma}}
{{BeginTableau|q \uparrow p \vd... | NAND is Commutative/Proof 1 | https://proofwiki.org/wiki/NAND_is_Commutative | https://proofwiki.org/wiki/NAND_is_Commutative/Proof_1 | [
"Logical NAND",
"NAND is Commutative",
"Examples of Commutative Operations"
] | [] | [] |
proofwiki-4520 | NAND is Commutative | :$p \uparrow q \dashv \vdash q \uparrow p$ | We apply the Method of Truth Tables:
:$\begin{array}{|ccc||ccc|} \hline
p & \uparrow & q & q & \uparrow & p \\
\hline
\F & \T & \F & \F & \T & \F \\
\F & \T & \T & \T & \T & \F \\
\T & \T & \F & \F & \T & \T \\
\T & \F & \T & \T & \F & \T \\
\hline
\end{array}$
As can be seen by inspection, the truth values under the m... | :$p \uparrow q \dashv \vdash q \uparrow p$ | We apply the [[Method of Truth Tables]]:
:$\begin{array}{|ccc||ccc|} \hline
p & \uparrow & q & q & \uparrow & p \\
\hline
\F & \T & \F & \F & \T & \F \\
\F & \T & \T & \T & \T & \F \\
\T & \T & \F & \F & \T & \T \\
\T & \F & \T & \T & \F & \T \\
\hline
\end{array}$
As can be seen by inspection, the [[Definition:Trut... | NAND is Commutative/Proof by Truth Table | https://proofwiki.org/wiki/NAND_is_Commutative | https://proofwiki.org/wiki/NAND_is_Commutative/Proof_by_Truth_Table | [
"Logical NAND",
"NAND is Commutative",
"Examples of Commutative Operations"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-4521 | NAND is not Associative | :$p \uparrow \paren {q \uparrow r} \not \vdash \paren {p \uparrow q} \uparrow r$ | {{BeginTableau|\neg \paren {\paren {p \uparrow q} \uparrow r \implies p \uparrow \paren {q \uparrow r} } }}
{{Assumption|1|\paren {p \uparrow q} \uparrow r \implies p \uparrow \paren {q \uparrow r} }}
{{Assumption|2|p \land \neg r}}
{{Simplification|3|2|p|2|1}}
{{Simplification|4|2|\neg r|2|2}}
{{Addition|5|2|\paren {\... | :$p \uparrow \paren {q \uparrow r} \not \vdash \paren {p \uparrow q} \uparrow r$ | {{BeginTableau|\neg \paren {\paren {p \uparrow q} \uparrow r \implies p \uparrow \paren {q \uparrow r} } }}
{{Assumption|1|\paren {p \uparrow q} \uparrow r \implies p \uparrow \paren {q \uparrow r} }}
{{Assumption|2|p \land \neg r}}
{{Simplification|3|2|p|2|1}}
{{Simplification|4|2|\neg r|2|2}}
{{Addition|5|2|\paren {\... | NAND is not Associative/Proof 1 | https://proofwiki.org/wiki/NAND_is_not_Associative | https://proofwiki.org/wiki/NAND_is_not_Associative/Proof_1 | [
"Logical NAND",
"NAND is not Associative"
] | [] | [
"Disjunction and Conditional"
] |
proofwiki-4522 | NAND is not Associative | :$p \uparrow \paren {q \uparrow r} \not \vdash \paren {p \uparrow q} \uparrow r$ | We apply the Method of Truth Tables:
:$\begin{array}{|ccccc||ccccc|} \hline
p & \uparrow & (q & \uparrow & r) & (p & \uparrow & q) & \uparrow & r \\
\hline
\F & \T & \F & \T & \F & \F & \T & \F & \T & \F \\
\F & \T & \F & \T & \T & \F & \T & \F & \F & \T \\
\F & \T & \T & \T & \F & \F & \T & \T & \T & \F \\
\F & \T & \... | :$p \uparrow \paren {q \uparrow r} \not \vdash \paren {p \uparrow q} \uparrow r$ | We apply the [[Method of Truth Tables]]:
:$\begin{array}{|ccccc||ccccc|} \hline
p & \uparrow & (q & \uparrow & r) & (p & \uparrow & q) & \uparrow & r \\
\hline
\F & \T & \F & \T & \F & \F & \T & \F & \T & \F \\
\F & \T & \F & \T & \T & \F & \T & \F & \F & \T \\
\F & \T & \T & \T & \F & \F & \T & \T & \T & \F \\
\F & \... | NAND is not Associative/Proof by Truth Table | https://proofwiki.org/wiki/NAND_is_not_Associative | https://proofwiki.org/wiki/NAND_is_not_Associative/Proof_by_Truth_Table | [
"Logical NAND",
"NAND is not Associative"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-4523 | NOR with Equal Arguments | :$p \downarrow p \dashv \vdash \neg p$
That is, the NOR of a proposition with itself corresponds to the negation operator. | {{BeginTableau|p \downarrow p \vdash \neg p}}
{{Premise|1|p \downarrow p}}
{{SequentIntro|2|1|\neg \paren {p \lor p}|1|Definition of Logical NOR}}
{{Idempotence|3|1|\neg p|2|Disjunction}}
{{EndTableau|lemma}}
{{BeginTableau|\neg p \vdash p \downarrow p}}
{{Premise|1|\neg p}}
{{Idempotence|2|1|\neg \paren {p \lor p}|1|D... | :$p \downarrow p \dashv \vdash \neg p$
That is, the [[Definition:Logical NOR|NOR]] of a proposition with itself corresponds to the [[Definition:Logical Not|negation]] operator. | {{BeginTableau|p \downarrow p \vdash \neg p}}
{{Premise|1|p \downarrow p}}
{{SequentIntro|2|1|\neg \paren {p \lor p}|1|Definition of [[Definition:Logical NOR|Logical NOR]]}}
{{Idempotence|3|1|\neg p|2|Disjunction}}
{{EndTableau|lemma}}
{{BeginTableau|\neg p \vdash p \downarrow p}}
{{Premise|1|\neg p}}
{{Idempotence|2... | NOR with Equal Arguments/Proof 1 | https://proofwiki.org/wiki/NOR_with_Equal_Arguments | https://proofwiki.org/wiki/NOR_with_Equal_Arguments/Proof_1 | [
"Logical NOR",
"NOR with Equal Arguments"
] | [
"Definition:Logical NOR",
"Definition:Logical Not"
] | [
"Definition:Logical NOR",
"Definition:Logical NOR"
] |
proofwiki-4524 | NOR with Equal Arguments | :$p \downarrow p \dashv \vdash \neg p$
That is, the NOR of a proposition with itself corresponds to the negation operator. | Apply the Method of Truth Tables:
:$\begin {array} {|ccc||cc|} \hline
p & \downarrow & p & \neg & p \\
\hline
\F & \T & \F & \T & \F \\
\T & \F & \T & \F & \T \\
\hline
\end{array}$
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
{{qed}} | :$p \downarrow p \dashv \vdash \neg p$
That is, the [[Definition:Logical NOR|NOR]] of a proposition with itself corresponds to the [[Definition:Logical Not|negation]] operator. | Apply the [[Method of Truth Tables]]:
:$\begin {array} {|ccc||cc|} \hline
p & \downarrow & p & \neg & p \\
\hline
\F & \T & \F & \T & \F \\
\T & \F & \T & \F & \T \\
\hline
\end{array}$
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Log... | NOR with Equal Arguments/Proof by Truth Table | https://proofwiki.org/wiki/NOR_with_Equal_Arguments | https://proofwiki.org/wiki/NOR_with_Equal_Arguments/Proof_by_Truth_Table | [
"Logical NOR",
"NOR with Equal Arguments"
] | [
"Definition:Logical NOR",
"Definition:Logical Not"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-4525 | NOR is Commutative | :$p \downarrow q \dashv \vdash q \downarrow p$ | {{BeginTableau|p \downarrow q \vdash q \downarrow p}}
{{Premise|1|p \downarrow q}}
{{SequentIntro|2|1|\neg \paren {p \lor q}|1|{{Defof|Logical NOR}} }}
{{Commutation|3|1|\neg \paren {q \lor p}|2|Disjunction}}
{{SequentIntro|4|1|q \uparrow p|3|{{Defof|Logical NOR}} }}
{{EndTableau|lemma}}
{{BeginTableau|q \downarrow p \... | :$p \downarrow q \dashv \vdash q \downarrow p$ | {{BeginTableau|p \downarrow q \vdash q \downarrow p}}
{{Premise|1|p \downarrow q}}
{{SequentIntro|2|1|\neg \paren {p \lor q}|1|{{Defof|Logical NOR}} }}
{{Commutation|3|1|\neg \paren {q \lor p}|2|Disjunction}}
{{SequentIntro|4|1|q \uparrow p|3|{{Defof|Logical NOR}} }}
{{EndTableau|lemma}}
{{BeginTableau|q \downarrow p... | NOR is Commutative/Proof 1 | https://proofwiki.org/wiki/NOR_is_Commutative | https://proofwiki.org/wiki/NOR_is_Commutative/Proof_1 | [
"Logical NOR",
"NOR is Commutative",
"Examples of Commutative Operations"
] | [] | [] |
proofwiki-4526 | NOR is Commutative | :$p \downarrow q \dashv \vdash q \downarrow p$ | Apply the Method of Truth Tables:
:$\begin{array}{|ccc||ccc|} \hline
p & \downarrow & q & q & \downarrow & p \\
\hline
\F & \T & \F & \F & \T & \F \\
\F & \F & \T & \T & \F & \F \\
\T & \F & \F & \F & \F & \T \\
\T & \F & \T & \T & \F & \T \\
\hline
\end{array}$
As can be seen by inspection, the truth values under the ... | :$p \downarrow q \dashv \vdash q \downarrow p$ | Apply the [[Method of Truth Tables]]:
:$\begin{array}{|ccc||ccc|} \hline
p & \downarrow & q & q & \downarrow & p \\
\hline
\F & \T & \F & \F & \T & \F \\
\F & \F & \T & \T & \F & \F \\
\T & \F & \F & \F & \F & \T \\
\T & \F & \T & \T & \F & \T \\
\hline
\end{array}$
As can be seen by inspection, the [[Definition:Tru... | NOR is Commutative/Proof by Truth Table | https://proofwiki.org/wiki/NOR_is_Commutative | https://proofwiki.org/wiki/NOR_is_Commutative/Proof_by_Truth_Table | [
"Logical NOR",
"NOR is Commutative",
"Examples of Commutative Operations"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-4527 | NOR is not Associative | :$p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$ | {{BeginTableau|\neg \paren {p \downarrow \paren {q \downarrow r} \implies \paren {p \downarrow q} \downarrow r} }}
{{Assumption|1|\neg p \land r}}
{{Simplification|2|1|\neg p|1|1}}
{{Simplification|3|1|r|1|2}}
{{Addition|4|1|q \lor r|3|2}}
{{DoubleNegIntro|5|1|\neg \neg \paren {q \lor r}|4}}
{{SequentIntro|6|1|\neg \pa... | :$p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$ | {{BeginTableau|\neg \paren {p \downarrow \paren {q \downarrow r} \implies \paren {p \downarrow q} \downarrow r} }}
{{Assumption|1|\neg p \land r}}
{{Simplification|2|1|\neg p|1|1}}
{{Simplification|3|1|r|1|2}}
{{Addition|4|1|q \lor r|3|2}}
{{DoubleNegIntro|5|1|\neg \neg \paren {q \lor r}|4}}
{{SequentIntro|6|1|\neg \pa... | NOR is not Associative/Proof 1 | https://proofwiki.org/wiki/NOR_is_not_Associative | https://proofwiki.org/wiki/NOR_is_not_Associative/Proof_1 | [
"Logical NOR",
"NOR is not Associative"
] | [] | [
"Definition:Logical NOR",
"Definition:Logical NOR",
"Definition:Logical NOR",
"Rule of Material Implication"
] |
proofwiki-4528 | NOR is not Associative | :$p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$ | Apply the Method of Truth Tables:
:$\begin{array}{|ccccc||ccccc|} \hline
p & \downarrow & (q & \downarrow & r) & (p & \downarrow & q) & \downarrow & r \\
\hline
\F & \F & \F & \T & \F & \F & \T & \F & \F & \F \\
\F & \T & \F & \F & \T & \F & \T & \F & \F & \T \\
\F & \T & \T & \F & \F & \F & \F & \T & \T & \F \\
\F & \... | :$p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$ | Apply the [[Method of Truth Tables]]:
:$\begin{array}{|ccccc||ccccc|} \hline
p & \downarrow & (q & \downarrow & r) & (p & \downarrow & q) & \downarrow & r \\
\hline
\F & \F & \F & \T & \F & \F & \T & \F & \F & \F \\
\F & \T & \F & \F & \T & \F & \T & \F & \F & \T \\
\F & \T & \T & \F & \F & \F & \F & \T & \T & \F \\
\... | NOR is not Associative/Proof by Truth Table | https://proofwiki.org/wiki/NOR_is_not_Associative | https://proofwiki.org/wiki/NOR_is_not_Associative/Proof_by_Truth_Table | [
"Logical NOR",
"NOR is not Associative"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-4529 | Representation of Ternary Expansions | Let $x \in \R$ be a real number.
Let $x$ be represented in base $3$ notation.
While it may be possible for $x$ to have two different such representations, for example:
:$\dfrac 1 3 = 0.100000 \ldots_3 = 0.022222 \ldots_3$
it is not possible for $x$ be written in more than one way without using the digit $1$. | It is sufficient to show that two distinct representations represents two distinct numbers.
Let $a$ and $b$ two real numbers representable as the form above.
Their signs are easy to distinguish, so we consider $\size a$ and $\size b$.
There is some $n$ such that:
:$\size a, \size b < 3^n$
In that case, $\dfrac {\size a... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $x$ be represented in [[Definition:Ternary Notation|base $3$ notation]].
While it may be possible for $x$ to have two different such representations, for example:
:$\dfrac 1 3 = 0.100000 \ldots_3 = 0.022222 \ldots_3$
it is not possible for $x$ be written... | It is sufficient to show that two distinct representations represents two distinct numbers.
Let $a$ and $b$ two [[Definition:Real Number|real numbers]] representable as the form above.
Their signs are easy to distinguish, so we consider $\size a$ and $\size b$.
There is some $n$ such that:
:$\size a, \size b < 3^n$
... | Representation of Ternary Expansions | https://proofwiki.org/wiki/Representation_of_Ternary_Expansions | https://proofwiki.org/wiki/Representation_of_Ternary_Expansions | [
"Ternary Notation"
] | [
"Definition:Real Number",
"Definition:Ternary Notation"
] | [
"Definition:Real Number",
"Definition:Integer",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-4530 | Equivalence of Definitions of Cantor Set | {{TFAE|def = Cantor Set|view = the Cantor Set $\CC$}}: | Let $\CC_n$ be defined as in $(1)$.
Let $x \in \closedint 0 1$.
We need to show that:
:$x$ can be written in base $3$ without using the digit $1$ {{iff}}:
::$\forall n \in \Z, n \ge 1: x \in C_n$
First we note that from Sum of Infinite Geometric Sequence:
:$\ds 1 = \sum_{n \mathop = 0}^\infty \frac 2 3 \paren {\frac 1 ... | {{TFAE|def = Cantor Set|view = the Cantor Set $\CC$}}: | Let $\CC_n$ be defined as in $(1)$.
Let $x \in \closedint 0 1$.
We need to show that:
:$x$ can be written in [[Definition:Ternary Notation|base $3$]] without using the digit $1$ {{iff}}:
::$\forall n \in \Z, n \ge 1: x \in C_n$
First we note that from [[Sum of Infinite Geometric Sequence]]:
:$\ds 1 = \sum_{n \matho... | Equivalence of Definitions of Cantor Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cantor_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Cantor_Set | [
"Cantor Set"
] | [] | [
"Definition:Ternary Notation",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-4531 | Vertices in Locally Finite Graph | Let $G$ be a locally finite graph.
Then if $G$ is infinite, it contains an infinite number of vertices. | Suppose $G = \struct {V, E}$ has a finite number of vertices.
Let $V = \set {v_1, v_2, \ldots, v_n}$ where $n = \card V$ is the cardinality of $V$.
As $G$ is locally finite, each element of $V$ has a finite number of incident edges.
For each $v_k \in V$, let $r_k$ be the degree of $v_k$.
Then:
:$\ds \card E \le \sum_{i... | Let $G$ be a [[Definition:Locally Finite Graph|locally finite graph]].
Then if $G$ is [[Definition:Infinite Graph|infinite]], it contains an [[Definition:Infinite|infinite]] number of [[Definition:Vertex of Graph|vertices]]. | Suppose $G = \struct {V, E}$ has a [[Definition:Finite Set|finite]] number of [[Definition:Vertex of Graph|vertices]].
Let $V = \set {v_1, v_2, \ldots, v_n}$ where $n = \card V$ is the [[Definition:Cardinality|cardinality]] of $V$.
As $G$ is [[Definition:Locally Finite Graph|locally finite]], each element of $V$ has ... | Vertices in Locally Finite Graph | https://proofwiki.org/wiki/Vertices_in_Locally_Finite_Graph | https://proofwiki.org/wiki/Vertices_in_Locally_Finite_Graph | [
"Graph Theory"
] | [
"Definition:Locally Finite Graph",
"Definition:Infinite Graph",
"Definition:Infinite",
"Definition:Graph (Graph Theory)/Vertex"
] | [
"Definition:Finite Set",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Cardinality",
"Definition:Locally Finite Graph",
"Definition:Finite Set",
"Definition:Incident (Graph Theory)",
"Definition:Degree of Vertex",
"Definition:Cardinality",
"Definition:Graph (Graph Theory)/Edge",
"Definitio... |
proofwiki-4532 | Cantor Set is Closed in Real Number Space | Let $\CC$ be the Cantor set.
Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$.
Then $\CC$ is a closed subset of $\struct {\R, \tau_d}$. | By definition, the Cantor set is the complement of a union of open sets relative to the closed interval $\closedint 0 1$.
By the definition of a topology, that union is itself open in $\R$.
The closed interval $\closedint 0 1$ is itself the complement of a union of open sets $\openint \gets 0 \cup \openint 1 \to$.
Henc... | Let $\CC$ be the [[Definition:Cantor Set|Cantor set]].
Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number space]] $\R$ under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\CC$ is a [[Definition:Closed Set (Topology)|closed subset]] of $\struct {... | By [[Definition:Cantor Set/Limit of Intersections|definition]], the [[Definition:Cantor Set|Cantor set]] is the [[Definition:Relative Complement|complement]] of a [[Definition:Set Union|union]] of [[Definition:Open Set (Topology)|open sets]] relative to the [[Definition:Closed Real Interval|closed interval]] $\closedin... | Cantor Set is Closed in Real Number Space | https://proofwiki.org/wiki/Cantor_Set_is_Closed_in_Real_Number_Space | https://proofwiki.org/wiki/Cantor_Set_is_Closed_in_Real_Number_Space | [
"Cantor Set",
"Real Number Line with Euclidean Topology",
"Examples of Closed Sets"
] | [
"Definition:Cantor Set",
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Closed Set/Topology"
] | [
"Definition:Cantor Set/Limit of Intersections",
"Definition:Cantor Set",
"Definition:Relative Complement",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Real Interval/Closed",
"Definition:Topology",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:Real Inte... |
proofwiki-4533 | Word Metric is Metric | Let $\struct {G, \circ}$ be a group.
Let $S$ be a generating set for $G$ which is closed under inverses (that is, $x^{-1} \in S \iff x \in S$).
Let $d_S$ be the associated word metric.
Then $d_S$ is a metric on $G$. | Let $g, h \in G$.
It is given that $S$ is a generating set for $G$.
It follows that there exist $s_1, \ldots, s_n \in S$ such that $g^{-1} \circ h = s_1 \circ \cdots \circ s_n$.
Therefore $\map {d_S} {g, h} \le n$, establishing that $\R$ is a valid codomain for the mapping $d_S$ with domain $G \times G$.
This is the fo... | Let $\struct {G, \circ}$ be a [[Definition:Group|group]].
Let $S$ be a [[Definition:Generator of Group|generating set]] for $G$ which is closed under inverses (that is, $x^{-1} \in S \iff x \in S$).
Let $d_S$ be the associated [[Definition:Word Metric|word metric]].
Then $d_S$ is a [[Definition:Metric|metric]] on $... | Let $g, h \in G$.
It is given that $S$ is a [[Definition:Generator of Group|generating set]] for $G$.
It follows that there exist $s_1, \ldots, s_n \in S$ such that $g^{-1} \circ h = s_1 \circ \cdots \circ s_n$.
Therefore $\map {d_S} {g, h} \le n$, establishing that $\R$ is a valid codomain for the [[Definition:Mapp... | Word Metric is Metric | https://proofwiki.org/wiki/Word_Metric_is_Metric | https://proofwiki.org/wiki/Word_Metric_is_Metric | [
"Group Theory",
"Word Metric"
] | [
"Definition:Group",
"Definition:Generator of Group",
"Definition:Word Metric",
"Definition:Metric Space/Metric"
] | [
"Definition:Generator of Group",
"Definition:Mapping",
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric"
] |
proofwiki-4534 | Cantor Space is Compact | Let $\CC$ be the Cantor set.
Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$.
Then $\CC$ is a compact subset of $\struct {\R, \tau_d}$. | We have Cantor Set is Closed in Real Number Space.
Taking, for example, $0 \in \CC$ and $1 \in \R$ it is clear that:
:$\forall x \in \CC: \map d {0, x} \le 1$
and so $\CC$ is bounded.
Hence the result from the Heine-Borel Theorem.
{{qed}} | Let $\CC$ be the [[Definition:Cantor Set|Cantor set]].
Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line|real number space]] $\R$ under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\CC$ is a [[Definition:Compact Topological Subspace|compact subset]] of $\... | We have [[Cantor Set is Closed in Real Number Space]].
Taking, for example, $0 \in \CC$ and $1 \in \R$ it is clear that:
:$\forall x \in \CC: \map d {0, x} \le 1$
and so $\CC$ is [[Definition:Bounded Metric Space|bounded]].
Hence the result from the [[Heine-Borel Theorem]].
{{qed}} | Cantor Space is Compact | https://proofwiki.org/wiki/Cantor_Space_is_Compact | https://proofwiki.org/wiki/Cantor_Space_is_Compact | [
"Cantor Space",
"Examples of Compact Topological Spaces"
] | [
"Definition:Cantor Set",
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Compact Topological Space/Subspace"
] | [
"Cantor Set is Closed in Real Number Space",
"Definition:Bounded Metric Space",
"Heine-Borel Theorem"
] |
proofwiki-4535 | Cantor Space is Complete Metric Space | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is a complete metric space. | We have that the Cantor space is a metric subspace of the real number space $\R$, and hence a metric space.
We also have Cantor Space is Compact.
The result follows from Compact Metric Space is Complete.
{{qed}} | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is a [[Definition:Complete Metric Space|complete metric space]]. | We have that the [[Definition:Cantor Space|Cantor space]] is a [[Definition:Metric Subspace|metric subspace]] of the [[Definition:Real Number Line|real number space]] $\R$, and hence a [[Definition:Metric Space|metric space]].
We also have [[Cantor Space is Compact]].
The result follows from [[Compact Metric Space is... | Cantor Space is Complete Metric Space | https://proofwiki.org/wiki/Cantor_Space_is_Complete_Metric_Space | https://proofwiki.org/wiki/Cantor_Space_is_Complete_Metric_Space | [
"Cantor Space",
"Examples of Complete Metric Spaces"
] | [
"Definition:Cantor Space",
"Definition:Complete Metric Space"
] | [
"Definition:Cantor Space",
"Definition:Metric Subspace",
"Definition:Real Number/Real Number Line",
"Definition:Metric Space",
"Cantor Space is Compact",
"Compact Metric Space is Complete"
] |
proofwiki-4536 | Cantor Space satisfies all Separation Axioms | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ satisfies all the separation axioms. | We have that the Cantor space is a metric subspace of the real number space $\R$, and hence a metric space.
The result follows from Metric Space fulfils all Separation Axioms.
{{qed}} | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ satisfies all the [[Definition:Separation Axioms|separation axioms]]. | We have that the [[Definition:Cantor Space|Cantor space]] is a [[Definition:Metric Subspace|metric subspace]] of the [[Definition:Real Number Line|real number space]] $\R$, and hence a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space fulfils all Separation Axioms]].
{{qed}} | Cantor Space satisfies all Separation Axioms | https://proofwiki.org/wiki/Cantor_Space_satisfies_all_Separation_Axioms | https://proofwiki.org/wiki/Cantor_Space_satisfies_all_Separation_Axioms | [
"Cantor Space",
"Examples of Separation Axioms"
] | [
"Definition:Cantor Space",
"Definition:Tychonoff Separation Axioms"
] | [
"Definition:Cantor Space",
"Definition:Metric Subspace",
"Definition:Real Number/Real Number Line",
"Definition:Metric Space",
"Metric Space fulfils all Separation Axioms"
] |
proofwiki-4537 | Cantor Space is Second-Countable | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is a second-countable space. | We have that the Cantor space is a topological subspace of the real number space with the usual (Euclidean) topology $\struct {\R, \tau_d}$.
We also have that the Real Number Line is Second-Countable.
The result follows from Second-Countability is Hereditary.
{{qed}} | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is a [[Definition:Second-Countable Space|second-countable space]]. | We have that the [[Definition:Cantor Space|Cantor space]] is a [[Definition:Topological Subspace|topological subspace]] of the [[Definition:Real Number Line with Euclidean Topology|real number space with the usual (Euclidean) topology]] $\struct {\R, \tau_d}$.
We also have that the [[Real Number Line is Second-Countab... | Cantor Space is Second-Countable | https://proofwiki.org/wiki/Cantor_Space_is_Second-Countable | https://proofwiki.org/wiki/Cantor_Space_is_Second-Countable | [
"Cantor Space",
"Examples of Second-Countable Spaces"
] | [
"Definition:Cantor Space",
"Definition:Second-Countable Space"
] | [
"Definition:Cantor Space",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Real Number Line is Second-Countable",
"Second-Countability is Hereditary"
] |
proofwiki-4538 | Cantor Space is Dense-in-itself | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is dense-in-itself. | {{Recall|Dense-in-itself|dense-in-itself}}
{{:Definition:Dense-in-itself}}
Let $U \in \tau_d$ be open in $T$.
Let $p \in U$.
Then:
:$\exists x \in U: \exists \epsilon \in \R: \map d {x, p} < \epsilon$
Thus $p$ is not an isolated point of $\T$.
Hence the result by definition of dense-in-itself.
{{qed}} | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is [[Definition:Dense-in-itself|dense-in-itself]]. | {{Recall|Dense-in-itself|dense-in-itself}}
{{:Definition:Dense-in-itself}}
Let $U \in \tau_d$ be [[Definition:Open Set (Topology)|open]] in $T$.
Let $p \in U$.
Then:
:$\exists x \in U: \exists \epsilon \in \R: \map d {x, p} < \epsilon$
Thus $p$ is not an [[Definition:Isolated Point of Subset|isolated point]] of $\T... | Cantor Space is Dense-in-itself | https://proofwiki.org/wiki/Cantor_Space_is_Dense-in-itself | https://proofwiki.org/wiki/Cantor_Space_is_Dense-in-itself | [
"Cantor Space",
"Examples of Dense-in-itself"
] | [
"Definition:Cantor Space",
"Definition:Dense-in-itself"
] | [
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Dense-in-itself"
] |
proofwiki-4539 | Cantor Space is not Scattered | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is not scattered. | {{Recall|Scattered Space|scattered space}}
{{:Definition:Scattered Space/Definition 1}}
We have that Cantor Space is Dense-in-itself.
Hence the result by definition of a scattered space.
{{qed}} | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is not [[Definition:Scattered Space|scattered]]. | {{Recall|Scattered Space|scattered space}}
{{:Definition:Scattered Space/Definition 1}}
We have that [[Cantor Space is Dense-in-itself]].
Hence the result by definition of a [[Definition:Scattered Space|scattered space]].
{{qed}} | Cantor Space is not Scattered | https://proofwiki.org/wiki/Cantor_Space_is_not_Scattered | https://proofwiki.org/wiki/Cantor_Space_is_not_Scattered | [
"Cantor Space",
"Examples of Scattered Spaces"
] | [
"Definition:Cantor Space",
"Definition:Scattered Space"
] | [
"Cantor Space is Dense-in-itself",
"Definition:Scattered Space"
] |
proofwiki-4540 | Cantor Space is Perfect | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $\CC$ is a perfect set of the real number space $\R$ under the usual (Euclidean) topology $\tau_d$. | {{Recall|Perfect Set|perfect set}}
{{:Definition:Perfect Set/Definition 2}}
From Cantor Space is Dense-in-itself, $\CC$ contains no isolated points.
We also have that the Cantor Set is Closed in Real Number Space.
The result follows from the definition of perfect set.
{{qed}} | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $\CC$ is a [[Definition:Perfect Set|perfect set]] of the [[Definition:Real Number Line|real number space]] $\R$ under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$. | {{Recall|Perfect Set|perfect set}}
{{:Definition:Perfect Set/Definition 2}}
From [[Cantor Space is Dense-in-itself]], $\CC$ contains no [[Definition:Isolated Point of Topological Space|isolated points]].
We also have that the [[Cantor Set is Closed in Real Number Space]].
The result follows from the definition of [[... | Cantor Space is Perfect | https://proofwiki.org/wiki/Cantor_Space_is_Perfect | https://proofwiki.org/wiki/Cantor_Space_is_Perfect | [
"Cantor Space",
"Examples of Perfect Sets"
] | [
"Definition:Cantor Space",
"Definition:Perfect Set",
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line"
] | [
"Cantor Space is Dense-in-itself",
"Definition:Isolated Point (Topology)/Space",
"Cantor Set is Closed in Real Number Space",
"Definition:Perfect Set"
] |
proofwiki-4541 | Cantor Space is Nowhere Dense | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is nowhere dense in $\closedint 0 1$. | From Cantor Set is Closed in Real Number Space, $\CC$ is closed.
So from Closed Set equals its Closure:
:$\CC^- = \CC$
where $\CC^-$ denotes the closure of $\CC$.
Let $0 \le a < b \le 1$.
Then $I = \openint a b$ is an open interval of $\closedint 0 1$.
Let $\epsilon = b - a$.
Clearly $\epsilon > 0$.
Let $n \in \N$ such... | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is [[Definition:Nowhere Dense|nowhere dense]] in $\closedint 0 1$. | From [[Cantor Set is Closed in Real Number Space]], $\CC$ is [[Definition:Closed Set (Topology)|closed]].
So from [[Closed Set equals its Closure]]:
:$\CC^- = \CC$
where $\CC^-$ denotes the [[Definition:Closure (Topology)|closure]] of $\CC$.
Let $0 \le a < b \le 1$.
Then $I = \openint a b$ is an [[Definition:Open R... | Cantor Space is Nowhere Dense/Proof 1 | https://proofwiki.org/wiki/Cantor_Space_is_Nowhere_Dense | https://proofwiki.org/wiki/Cantor_Space_is_Nowhere_Dense/Proof_1 | [
"Cantor Space is Nowhere Dense",
"Cantor Space",
"Examples of Nowhere Dense"
] | [
"Definition:Cantor Space",
"Definition:Nowhere Dense"
] | [
"Cantor Set is Closed in Real Number Space",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Definition:Closure (Topology)",
"Definition:Real Interval/Open",
"Definition:Real Interval/Open",
"Definition:Real Interval/Open",
"Definition:Disjoint Sets",
"Definition:R... |
proofwiki-4542 | Cantor Space is Nowhere Dense | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is nowhere dense in $\closedint 0 1$. | Let $S_n$ and $C_n$ be as in the definition of the Cantor set as a limit of a decreasing sequence.
Then the length of every interval in $S_n$ is seen to be $\dfrac 1 {3^n} = 3^{-n}$.
Let $0 \le a < b \le 1$.
Then $\openint a b \subseteq \closedint 0 1$ is an open interval.
Let $n \in \N$ such that $3^{-n} < b - a$, so ... | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is [[Definition:Nowhere Dense|nowhere dense]] in $\closedint 0 1$. | Let $S_n$ and $C_n$ be as in the definition of the [[Definition:Cantor Set/Limit of Decreasing Sequence|Cantor set as a limit of a decreasing sequence]].
Then the [[Definition:Length of Real Interval|length]] of every [[Definition:Real Interval|interval]] in $S_n$ is seen to be $\dfrac 1 {3^n} = 3^{-n}$.
Let $0 \le ... | Cantor Space is Nowhere Dense/Proof 2 | https://proofwiki.org/wiki/Cantor_Space_is_Nowhere_Dense | https://proofwiki.org/wiki/Cantor_Space_is_Nowhere_Dense/Proof_2 | [
"Cantor Space is Nowhere Dense",
"Cantor Space",
"Examples of Nowhere Dense"
] | [
"Definition:Cantor Space",
"Definition:Nowhere Dense"
] | [
"Definition:Cantor Set/Limit of Decreasing Sequence",
"Definition:Real Interval/Length",
"Definition:Real Interval",
"Definition:Real Interval/Open",
"Definition:Real Interval/Length",
"Definition:Real Interval",
"Definition:Real Interval",
"Definition:Real Interval",
"Definition:Real Interval/Lengt... |
proofwiki-4543 | Cantor Space is Meager in Closed Unit Interval | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is meager in $\closedint 0 1$. | From Cantor Space is Nowhere Dense, $T$ is nowhere dense in $\closedint 0 1$.
So, trivially, $\CC$ is the union of nowhere dense subsets of $\closedint 0 1$.
Hence the result from definition of meager.
{{qed}} | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is [[Definition:Meager Space|meager]] in $\closedint 0 1$. | From [[Cantor Space is Nowhere Dense]], $T$ is [[Definition:Nowhere Dense|nowhere dense]] in $\closedint 0 1$.
So, trivially, $\CC$ is the [[Definition:Set Union|union]] of [[Definition:Nowhere Dense|nowhere dense]] subsets of $\closedint 0 1$.
Hence the result from definition of [[Definition:Meager Space|meager]].
{... | Cantor Space is Meager in Closed Unit Interval | https://proofwiki.org/wiki/Cantor_Space_is_Meager_in_Closed_Unit_Interval | https://proofwiki.org/wiki/Cantor_Space_is_Meager_in_Closed_Unit_Interval | [
"Cantor Space",
"Examples of Meager Spaces"
] | [
"Definition:Cantor Space",
"Definition:Meager Space"
] | [
"Cantor Space is Nowhere Dense",
"Definition:Nowhere Dense",
"Definition:Set Union",
"Definition:Nowhere Dense",
"Definition:Meager Space"
] |
proofwiki-4544 | Cantor Space is Non-Meager in Itself | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is non-meager in itself. | We have that the Cantor Space is Complete Metric Space.
By Baire Category Theorem, a complete metric space is also a Baire space.
The result then follows by Baire Space is Non-Meager.
{{qed}} | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]] in itself. | We have that the [[Cantor Space is Complete Metric Space]].
By [[Baire Category Theorem]], a [[Definition:Complete Metric Space|complete metric space]] is also a [[Definition:Baire Space|Baire space]].
The result then follows by [[Baire Space is Non-Meager]].
{{qed}} | Cantor Space is Non-Meager in Itself | https://proofwiki.org/wiki/Cantor_Space_is_Non-Meager_in_Itself | https://proofwiki.org/wiki/Cantor_Space_is_Non-Meager_in_Itself | [
"Cantor Space",
"Examples of Non-Meager Spaces"
] | [
"Definition:Cantor Space",
"Definition:Meager Space/Non-Meager"
] | [
"Cantor Space is Complete Metric Space",
"Baire Category Theorem",
"Definition:Complete Metric Space",
"Definition:Baire Space",
"Baire Space is Non-Meager"
] |
proofwiki-4545 | Cantor Set is Uncountable | The Cantor set $\CC$ is uncountable. | From the definition as a ternary representation, $\CC$ consists of all the elements of $\closedint 0 1$ which can be written without using a $1$.
So let $x \in \CC$. Then in base $3$ notation, we have (as $0 \le x \le 1$):
:$\ds x = \sum_{i \mathop = 1}^\infty r_j 3^{-j}$
From the definition of the Cantor set, we have ... | The [[Definition:Cantor Set|Cantor set]] $\CC$ is [[Definition:Uncountable Set|uncountable]]. | From the definition as a [[Definition:Cantor Set/Ternary Representation|ternary representation]], $\CC$ consists of all the elements of $\closedint 0 1$ which can be written without using a $1$.
So let $x \in \CC$. Then in [[Definition:Ternary Notation|base $3$ notation]], we have (as $0 \le x \le 1$):
:$\ds x = \sum... | Cantor Set is Uncountable/Proof 1 | https://proofwiki.org/wiki/Cantor_Set_is_Uncountable | https://proofwiki.org/wiki/Cantor_Set_is_Uncountable/Proof_1 | [
"Cantor Set is Uncountable",
"Cantor Set",
"Examples of Uncountable Sets"
] | [
"Definition:Cantor Set",
"Definition:Uncountable/Set"
] | [
"Definition:Cantor Set/Ternary Representation",
"Definition:Ternary Notation",
"Definition:Cantor Set/Ternary Representation",
"Representation of Ternary Expansions",
"Definition:Element",
"Definition:Binary Notation",
"Existence of Base-N Representation",
"Definition:Surjection",
"Closed Interval i... |
proofwiki-4546 | Cantor Set is Uncountable | The Cantor set $\CC$ is uncountable. | It follows from Representation of Ternary Expansions that every string of the form $0.nnnnn \ldots$ where $n \in \set {0, 2}$ is an element of $\CC$.
We also have that every string of the form $0.nnnnn \ldots$ where $n \in \set {0, 1}$ is an element of $\closedint 0 1 \subset \R$ expressed in binary notation.
Let $f: \... | The [[Definition:Cantor Set|Cantor set]] $\CC$ is [[Definition:Uncountable Set|uncountable]]. | It follows from [[Representation of Ternary Expansions]] that every string of the form $0.nnnnn \ldots$ where $n \in \set {0, 2}$ is an element of $\CC$.
We also have that every string of the form $0.nnnnn \ldots$ where $n \in \set {0, 1}$ is an element of $\closedint 0 1 \subset \R$ expressed in [[Definition:Binary N... | Cantor Set is Uncountable/Proof 2 | https://proofwiki.org/wiki/Cantor_Set_is_Uncountable | https://proofwiki.org/wiki/Cantor_Set_is_Uncountable/Proof_2 | [
"Cantor Set is Uncountable",
"Cantor Set",
"Examples of Uncountable Sets"
] | [
"Definition:Cantor Set",
"Definition:Uncountable/Set"
] | [
"Representation of Ternary Expansions",
"Definition:Binary Notation",
"Definition:Real Function",
"Definition:Ternary Notation",
"Definition:Surjection"
] |
proofwiki-4547 | Equality is Symmetric | :$\forall a, b: a = b \implies b = a$ | {{begin-eqn}}
{{eqn | l = a
| r = b
| c =
}}
{{eqn | ll= \vdash
| l = \map P a
| o = \iff
| r = \map P b
| c = Leibniz's Law
}}
{{eqn | ll= \vdash
| l = \map P b
| o = \iff
| r = \map P a
| c = Biconditional is Commutative
}}
{{eqn | ll= \vdash
| l = b
... | :$\forall a, b: a = b \implies b = a$ | {{begin-eqn}}
{{eqn | l = a
| r = b
| c =
}}
{{eqn | ll= \vdash
| l = \map P a
| o = \iff
| r = \map P b
| c = [[Axiom:Leibniz's Law|Leibniz's Law]]
}}
{{eqn | ll= \vdash
| l = \map P b
| o = \iff
| r = \map P a
| c = [[Biconditional is Commutative]]
}}
{{eqn... | Equality is Symmetric | https://proofwiki.org/wiki/Equality_is_Symmetric | https://proofwiki.org/wiki/Equality_is_Symmetric | [
"Logic",
"Equality"
] | [] | [
"Axiom:Leibniz's Law",
"Biconditional is Commutative",
"Axiom:Leibniz's Law"
] |
proofwiki-4548 | Derivative of Hyperbolic Sine Function | :$\map {\dfrac \d {\d x} } {\sinh u} = \cosh u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sinh u}
| r = \map {\frac \d {\d u} } {\sinh u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \cosh u \frac {\d u} {\d x}
| c = Derivative of Hyperbolic Sine
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\sinh u} = \cosh u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sinh u}
| r = \map {\frac \d {\d u} } {\sinh u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \cosh u \frac {\d u} {\d x}
| c = [[Derivative of Hyperbolic Sine]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Sine Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine_Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Sine_Function | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Sine Function",
"Derivative of Hyperbolic Sine Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Hyperbolic Sine"
] |
proofwiki-4549 | Derivative of Hyperbolic Cosine Function | :$\map {\dfrac \d {\d x} } {\cosh u} = \sinh u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cosh u}
| r = \map {\frac \d {\d u} } {\cosh u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \sinh u \frac {\d u} {\d x}
| c = Derivative of Hyperbolic Cosine
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\cosh u} = \sinh u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\cosh u}
| r = \map {\frac \d {\d u} } {\cosh u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \sinh u \frac {\d u} {\d x}
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Cosine Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosine_Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosine_Function | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Cosine Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Hyperbolic Cosine"
] |
proofwiki-4550 | Cantor Space is Totally Separated | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is totally separated. | Let $a, b \in \CC$ such that $a < b$.
Then $b - a = \epsilon$.
Consider $n \in \N$ such that $3^{-n} < \epsilon$.
{{wtd|To be proved that between $a + 3^{-\paren {n + 1} }$ and $a + 2 \times 3^{-\paren {n + 1} }$ there exists an interval $I$ which is excluded from $\CC$. Or something like that.<br/>Then we pick some $r... | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is [[Definition:Totally Separated Space|totally separated]]. | Let $a, b \in \CC$ such that $a < b$.
Then $b - a = \epsilon$.
Consider $n \in \N$ such that $3^{-n} < \epsilon$.
{{wtd|To be proved that between $a + 3^{-\paren {n + 1} }$ and $a + 2 \times 3^{-\paren {n + 1} }$ there exists an interval $I$ which is excluded from $\CC$. Or something like that.<br/>Then we pick some... | Cantor Space is Totally Separated | https://proofwiki.org/wiki/Cantor_Space_is_Totally_Separated | https://proofwiki.org/wiki/Cantor_Space_is_Totally_Separated | [
"Cantor Space",
"Examples of Totally Separated Spaces"
] | [
"Definition:Cantor Space",
"Definition:Totally Separated Space"
] | [
"Definition:Separation (Topology)",
"Definition:Separation (Topology)",
"Definition:Totally Separated Space"
] |
proofwiki-4551 | Cantor Space is not Extremally Disconnected | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is not extremally disconnected. | {{Recall|Extremally Disconnected Space|extremally disconnected space}}
{{:Definition:Extremally Disconnected Space/Definition 3}}
Consider the real number $\dfrac 1 4 = 0.020202 \ldots_3$.
We have that:
:$C_1 := \CC \cap \hointr 0 {\dfrac 1 4}$
:$C_2 := \CC \cap \hointl {\dfrac 1 4} 1$
are disjoint sets both of which a... | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is not [[Definition:Extremally Disconnected Space|extremally disconnected]]. | {{Recall|Extremally Disconnected Space|extremally disconnected space}}
{{:Definition:Extremally Disconnected Space/Definition 3}}
Consider the [[Definition:Real Number|real number]] $\dfrac 1 4 = 0.020202 \ldots_3$.
We have that:
:$C_1 := \CC \cap \hointr 0 {\dfrac 1 4}$
:$C_2 := \CC \cap \hointl {\dfrac 1 4} 1$
are ... | Cantor Space is not Extremally Disconnected | https://proofwiki.org/wiki/Cantor_Space_is_not_Extremally_Disconnected | https://proofwiki.org/wiki/Cantor_Space_is_not_Extremally_Disconnected | [
"Cantor Space",
"Examples of Extremally Disconnected Spaces"
] | [
"Definition:Cantor Space",
"Definition:Extremally Disconnected Space"
] | [
"Definition:Real Number",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Closure (Topology)"
] |
proofwiki-4552 | Cantor Space as Countably Infinite Product | Let $A_n = \struct {\set {0, 2}, \tau_n}$ be the discrete space of the two points $0$ and $2$.
Let $\ds A = \prod_{n \mathop = 1}^\infty A_n$.
Let $\struct {A, \tau}$ be the product space where $\tau$ is the product topology on $A$.
Then $A$ is homeomorphic to the Cantor space. | In $\CC$, basis elements are sets of the form $\set {y: \size {x - y} < \epsilon}$ for $x \in \CC$ and some $\epsilon \in \R_{>0}$.
In $\ds \prod_{n \mathop = 1}^\infty A_n$, sets of the form $\set {\sequence {a_i} \in \prod A_n: a_i \text { is fixed for } 1 \le i \le n}$ forms a basis for the product topology.
Conside... | Let $A_n = \struct {\set {0, 2}, \tau_n}$ be the [[Definition:Discrete Space|discrete space]] of the two points $0$ and $2$.
Let $\ds A = \prod_{n \mathop = 1}^\infty A_n$.
Let $\struct {A, \tau}$ be the [[Definition:Product Space|product space]] where $\tau$ is the [[Definition:Product Topology|product topology]] on... | In $\CC$, [[Definition:Basis (Topology)|basis]] elements are sets of the form $\set {y: \size {x - y} < \epsilon}$ for $x \in \CC$ and some $\epsilon \in \R_{>0}$.
In $\ds \prod_{n \mathop = 1}^\infty A_n$, sets of the form $\set {\sequence {a_i} \in \prod A_n: a_i \text { is fixed for } 1 \le i \le n}$ forms a [[Defi... | Cantor Space as Countably Infinite Product/Proof 1 | https://proofwiki.org/wiki/Cantor_Space_as_Countably_Infinite_Product | https://proofwiki.org/wiki/Cantor_Space_as_Countably_Infinite_Product/Proof_1 | [
"Cantor Space as Countably Infinite Product",
"Cantor Space",
"Examples of Product Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Product Space",
"Definition:Product Topology",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Cantor Space"
] | [
"Definition:Basis (Topology)",
"Definition:Basis (Topology)",
"Definition:Product Topology",
"Definition:Real Function",
"Definition:Basis (Topology)",
"Definition:Basis (Topology)",
"Definition:Continuous Mapping (Topology)/Everywhere"
] |
proofwiki-4553 | Cantor Space as Countably Infinite Product | Let $A_n = \struct {\set {0, 2}, \tau_n}$ be the discrete space of the two points $0$ and $2$.
Let $\ds A = \prod_{n \mathop = 1}^\infty A_n$.
Let $\struct {A, \tau}$ be the product space where $\tau$ is the product topology on $A$.
Then $A$ is homeomorphic to the Cantor space. | Since $\CC$ is a metric space, $\CC$ is Hausdorff. By Tychonoff's Theorem, $\ds A = \prod_{n \mathop = 1}^\infty A_n$ is compact.
Consider the function $f$ taking each point from $\tuple {a_1, a_2, \ldots, a_n, \ldots}$ in $\prod A_n$ to the point $0 \cdotp a_1 a_2 \ldots a_n \ldots_3$ in $\CC$.
$f$ is a bijection.
In ... | Let $A_n = \struct {\set {0, 2}, \tau_n}$ be the [[Definition:Discrete Space|discrete space]] of the two points $0$ and $2$.
Let $\ds A = \prod_{n \mathop = 1}^\infty A_n$.
Let $\struct {A, \tau}$ be the [[Definition:Product Space|product space]] where $\tau$ is the [[Definition:Product Topology|product topology]] on... | Since $\CC$ is a metric space, $\CC$ is Hausdorff. By [[Tychonoff's Theorem]], $\ds A = \prod_{n \mathop = 1}^\infty A_n$ is compact.
Consider the [[Definition:Real Function|function]] $f$ taking each point from $\tuple {a_1, a_2, \ldots, a_n, \ldots}$ in $\prod A_n$ to the point $0 \cdotp a_1 a_2 \ldots a_n \ldots_3$... | Cantor Space as Countably Infinite Product/Proof 2 | https://proofwiki.org/wiki/Cantor_Space_as_Countably_Infinite_Product | https://proofwiki.org/wiki/Cantor_Space_as_Countably_Infinite_Product/Proof_2 | [
"Cantor Space as Countably Infinite Product",
"Cantor Space",
"Examples of Product Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Product Space",
"Definition:Product Topology",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Cantor Space"
] | [
"Tychonoff's Theorem",
"Definition:Real Function",
"Definition:Basis (Topology)",
"Product Space is T2 iff Factor Spaces are T2",
"Continuous Bijection from Compact to Hausdorff is Homeomorphism"
] |
proofwiki-4554 | Cantor Space is not Locally Connected | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Then $T$ is not locally connected. | Let $\BB$ be a basis of $T$.
Let $A \in \BB$.
By definition of $\BB$, $A$ is an open set of $T$.
But the Cantor Space is Totally Separated.
Therefore $A$ is not a connected set.
Hence the result from definition of a locally connected space.
{{qed}} | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Then $T$ is not [[Definition:Locally Connected Space|locally connected]]. | Let $\BB$ be a [[Definition:Basis (Topology)|basis]] of $T$.
Let $A \in \BB$.
By definition of $\BB$, $A$ is an [[Definition:Open Set (Topology)|open set]] of $T$.
But the [[Cantor Space is Totally Separated]].
Therefore $A$ is not a [[Definition:Connected Set (Topology)|connected set]].
Hence the result from defi... | Cantor Space is not Locally Connected | https://proofwiki.org/wiki/Cantor_Space_is_not_Locally_Connected | https://proofwiki.org/wiki/Cantor_Space_is_not_Locally_Connected | [
"Cantor Space",
"Examples of Locally Connected Spaces"
] | [
"Definition:Cantor Space",
"Definition:Locally Connected Space"
] | [
"Definition:Basis (Topology)",
"Definition:Open Set/Topology",
"Cantor Space is Totally Separated",
"Definition:Connected Set (Topology)",
"Definition:Locally Connected Space"
] |
proofwiki-4555 | Local Connectedness is not Preserved under Infinite Product | The property of local connectedness is not preserved under the operation of forming an infinite product space. | Let $T = \struct {\CC, \tau_d}$ be the Cantor space.
Let $A_n = \struct {\set {0, 2}, \tau_n}$ be the discrete space of the two points $0$ and $2$.
Let $\ds A = \prod_{n \mathop = 1}^\infty A_n$.
Let $T' = \struct {A, \tau}$ be the product space where $\tau$ is the product topology on $A$.
From Cantor Space as Countabl... | The property of [[Definition:Locally Connected Space|local connectedness]] is not preserved under the operation of forming an [[Definition:Infinite Set|infinite]] [[Definition:Product Space|product space]]. | Let $T = \struct {\CC, \tau_d}$ be the [[Definition:Cantor Space|Cantor space]].
Let $A_n = \struct {\set {0, 2}, \tau_n}$ be the [[Definition:Discrete Space|discrete space]] of the two points $0$ and $2$.
Let $\ds A = \prod_{n \mathop = 1}^\infty A_n$.
Let $T' = \struct {A, \tau}$ be the [[Definition:Product Space|... | Local Connectedness is not Preserved under Infinite Product | https://proofwiki.org/wiki/Local_Connectedness_is_not_Preserved_under_Infinite_Product | https://proofwiki.org/wiki/Local_Connectedness_is_not_Preserved_under_Infinite_Product | [
"Locally Connected Spaces",
"Product Spaces"
] | [
"Definition:Locally Connected Space",
"Definition:Infinite Set",
"Definition:Product Space"
] | [
"Definition:Cantor Space",
"Definition:Discrete Topology",
"Definition:Product Space",
"Definition:Product Topology",
"Cantor Space as Countably Infinite Product",
"Definition:Homeomorphism/Topological Spaces",
"Totally Disconnected and Locally Connected Space is Discrete",
"Definition:Locally Connect... |
proofwiki-4556 | Derivative of Hyperbolic Tangent Function | :$\map {\dfrac \d {\d x} } {\tanh u} = \sech^2 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\tanh u}
| r = \map {\frac \d {\d u} } {\tanh u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \sech^2 u \frac {\d u} {\d x}
| c = Derivative of Hyperbolic Tangent
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\tanh u} = \sech^2 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\tanh u}
| r = \map {\frac \d {\d u} } {\tanh u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \sech^2 u \frac {\d u} {\d x}
| c = [[Derivative of Hyperbolic Tangent]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Tangent Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Tangent_Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Tangent_Function | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Tangent Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Hyperbolic Tangent"
] |
proofwiki-4557 | Difference of Squares of Hyperbolic Cosine and Sine | :$\cosh^2 x - \sinh^2 x = 1$
where $\cosh$ and $\sinh$ are hyperbolic cosine and hyperbolic sine. | {{begin-eqn}}
{{eqn | l = \cosh^2 x - \sinh^2 x
| r = \paren {\frac {e^x + e^{-x} } 2}^2 - \paren {\frac {e^x - e^{-x} } 2}^2
| c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \paren {\frac {\paren {e^x}^2 + 2 \paren {e^x} \paren {e^{-x} } + \paren {e^{-x} }^2} 4} - \paren {\f... | :$\cosh^2 x - \sinh^2 x = 1$
where $\cosh$ and $\sinh$ are [[Definition:Hyperbolic Cosine|hyperbolic cosine]] and [[Definition:Hyperbolic Sine|hyperbolic sine]]. | {{begin-eqn}}
{{eqn | l = \cosh^2 x - \sinh^2 x
| r = \paren {\frac {e^x + e^{-x} } 2}^2 - \paren {\frac {e^x - e^{-x} } 2}^2
| c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \paren {\frac {\paren {e^x}^2 + 2 \paren {e^x} \paren {e^{-x} } + \paren {e^{-x} }^2} 4} - \paren {\f... | Difference of Squares of Hyperbolic Cosine and Sine | https://proofwiki.org/wiki/Difference_of_Squares_of_Hyperbolic_Cosine_and_Sine | https://proofwiki.org/wiki/Difference_of_Squares_of_Hyperbolic_Cosine_and_Sine | [
"Difference of Squares of Hyperbolic Cosine and Sine",
"Hyperbolic Cosine Function",
"Hyperbolic Sine Function"
] | [
"Definition:Hyperbolic Cosine",
"Definition:Hyperbolic Sine"
] | [
"Square of Sum",
"Exponential of Sum"
] |
proofwiki-4558 | Derivative of Arccosecant Function | :<nowiki>$\dfrac {\map \d {\arccsc x} } {\d x} = \dfrac {-1} {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {-1} {x \sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\
\dfrac {+1} {x \sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc x < 0 \ (\text {that is: $x < -1$}) \\
\end{cases}$</no... | {{:Graph of Arccosecant Function}}
Let $y = \arccsc x$ where $x < -1$ or $x > 1$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = \arccsc x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \csc y
| c = where $y \in \closedint 0 \pi \land y \ne \dfrac pi 2$
}}
{{eqn | ll= \leadsto
| l = \frac {\... | :<nowiki>$\dfrac {\map \d {\arccsc x} } {\d x} = \dfrac {-1} {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {-1} {x \sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\
\dfrac {+1} {x \sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc x < 0 \ (\text {that is: $x < -1$}) \\
\end{cases}$</no... | {{:Graph of Arccosecant Function}}
Let $y = \arccsc x$ where $x < -1$ or $x > 1$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = \arccsc x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \csc y
| c = where $y \in \closedint 0 \pi \land y \ne \dfrac pi 2$
}}
{{eqn | ll= \leadsto
| l = \frac ... | Derivative of Arccosecant Function | https://proofwiki.org/wiki/Derivative_of_Arccosecant_Function | https://proofwiki.org/wiki/Derivative_of_Arccosecant_Function | [
"Derivatives of Inverse Trigonometric Functions",
"Arccosecant Function"
] | [] | [
"Derivative of Cosecant Function",
"Derivative of Inverse Function",
"Sum of Squares of Sine and Cosine/Corollary 2",
"Sine and Cosine are Periodic on Reals",
"Definition:Inverse Cosecant/Real/Arccosecant"
] |
proofwiki-4559 | Derivative of Hyperbolic Secant Function | :$\map {\dfrac \d {\d x} } {\sech u} = -\sech u \tanh u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech u}
| r = \map {\frac \d {\d u} } {\sech u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = -\sech u \tanh u \frac {\d u} {\d x}
| c = Derivative of Hyperbolic Secant
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\sech u} = -\sech u \tanh u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\sech u}
| r = \map {\frac \d {\d u} } {\sech u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\sech u \tanh u \frac {\d u} {\d x}
| c = [[Derivative of Hyperbolic Secant]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Secant Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Secant_Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Secant_Function | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Secant Function",
"Derivative of Hyperbolic Secant Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Hyperbolic Secant"
] |
proofwiki-4560 | Power of Sum Modulo Prime | Let $p$ be a prime number.
Then:
:$\paren {a + b}^p \equiv a^p + b^p \pmod p$ | From the Binomial Theorem:
:$\ds \paren {a + b}^p = \sum_{k \mathop = 0}^p \binom p k a^k b^{p - k}$
Also note that:
:$\ds \sum_{k \mathop = 0}^p \binom p k a^k b^{p-k} = a^p + \sum_{k \mathop = 1}^{p - 1} \binom p k a^k b^{p - k} + b^p$
So:
{{begin-eqn}}
{{eqn | q = \forall k: 0 < k < p
| l = \binom p k
| ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Then:
:$\paren {a + b}^p \equiv a^p + b^p \pmod p$ | From the [[Binomial Theorem]]:
:$\ds \paren {a + b}^p = \sum_{k \mathop = 0}^p \binom p k a^k b^{p - k}$
Also note that:
:$\ds \sum_{k \mathop = 0}^p \binom p k a^k b^{p-k} = a^p + \sum_{k \mathop = 1}^{p - 1} \binom p k a^k b^{p - k} + b^p$
So:
{{begin-eqn}}
{{eqn | q = \forall k: 0 < k < p
| l = \binom p k
... | Power of Sum Modulo Prime | https://proofwiki.org/wiki/Power_of_Sum_Modulo_Prime | https://proofwiki.org/wiki/Power_of_Sum_Modulo_Prime | [
"Number Theory",
"Combinatorics",
"Prime Numbers"
] | [
"Definition:Prime Number"
] | [
"Binomial Theorem",
"Binomial Coefficient of Prime"
] |
proofwiki-4561 | Volume of Right Circular Cone | The volume $V$ of a right circular cone is given by:
:$V = \dfrac 1 3 \pi r^2 h$
where:
:$r$ is the radius of the base
:$h$ is the height of the cone, that is, the distance between the apex and the center of the base. | This proof utilizes the Method of Disks and thus is dependent on Volume of Right Circular Cylinder.
From the Method of Disks, the volume of the cone can be found by the definite integral:
:$\ds (1): \quad V = \pi \int_0^{AC} \paren {\map R x}^2 \rd x$
where $\map R x$ is the function describing the line which is to be ... | The [[Definition:Volume|volume]] $V$ of a [[Definition:Right Circular Cone|right circular cone]] is given by:
:$V = \dfrac 1 3 \pi r^2 h$
where:
:$r$ is the [[Definition:Radius of Circle|radius]] of the [[Definition:Base of Cone|base]]
:$h$ is the [[Definition:Height of Cone|height]] of the cone, that is, the [[Defini... | This proof utilizes the [[Method of Disks]] and thus is dependent on [[Volume of Right Circular Cylinder]].
From the [[Method of Disks]], the volume of the cone can be found by the [[Definition:Definite Integral|definite integral]]:
:$\ds (1): \quad V = \pi \int_0^{AC} \paren {\map R x}^2 \rd x$
where $\map R x$ is ... | Volume of Right Circular Cone | https://proofwiki.org/wiki/Volume_of_Right_Circular_Cone | https://proofwiki.org/wiki/Volume_of_Right_Circular_Cone | [
"Right Circular Cones",
"Volume Formulas",
"Analytic Geometry",
"Integral Calculus"
] | [
"Definition:Volume",
"Definition:Right Circular Cone",
"Definition:Circle/Radius",
"Definition:Cone (Geometry)/Base",
"Definition:Cone (Geometry)/Height",
"Definition:Linear Measure",
"Definition:Cone (Geometry)/Apex",
"Definition:Circle/Center",
"Definition:Cone (Geometry)/Base"
] | [
"Method of Disks",
"Volume of Right Circular Cylinder",
"Method of Disks",
"Definition:Definite Integral",
"Definition:Real Function",
"Definition:Solid of Revolution",
"Definition:Line/Segment",
"Definition:Right Circular Cone/Axis",
"Definition:Linear Measure",
"Definition:Coordinate System/Orig... |
proofwiki-4562 | Intersection of Topologies is Topology | Let $\family {\tau_i}_{i \mathop \in I}$ be an arbitrary indexed family of topologies on a set $S$.
Then $\tau := \ds \bigcap_{i \mathop \in I} \tau_i$ is also a topology on $S$. | Each of the open set axioms are examined in turn: | Let $\family {\tau_i}_{i \mathop \in I}$ be an arbitrary [[Definition:Indexed Family|indexed family]] of [[Definition:Topology|topologies]] on a [[Definition:Set|set]] $S$.
Then $\tau := \ds \bigcap_{i \mathop \in I} \tau_i$ is also a [[Definition:Topology|topology]] on $S$. | Each of the [[Axiom:Open Set Axioms|open set axioms]] are examined in turn: | Intersection of Topologies is Topology | https://proofwiki.org/wiki/Intersection_of_Topologies_is_Topology | https://proofwiki.org/wiki/Intersection_of_Topologies_is_Topology | [
"Topology"
] | [
"Definition:Indexing Set/Family",
"Definition:Topology",
"Definition:Set",
"Definition:Topology"
] | [
"Axiom:Open Set Axioms"
] |
proofwiki-4563 | Power Set is Closed under Set Difference | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then:
:$\forall A, B \in \powerset S: A \setminus B \in \powerset S$
where $A \setminus B$ denotes the set difference of $A$ and $B$. | Let $A, B \in \powerset S$.
Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.
We also have $A \setminus B \subseteq A$ from Set Difference is Subset.
Thus by Subset Relation is Transitive, $A \setminus B \subseteq S$.
Thus $A \setminus B \in \powerset S$, and closure is proved.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Then:
:$\forall A, B \in \powerset S: A \setminus B \in \powerset S$
where $A \setminus B$ denotes the [[Definition:Set Difference|set difference]] of $A$ and $B$. | Let $A, B \in \powerset S$.
Then by the definition of [[Definition:Power Set|power set]], $A \subseteq S$ and $B \subseteq S$.
We also have $A \setminus B \subseteq A$ from [[Set Difference is Subset]].
Thus by [[Subset Relation is Transitive]], $A \setminus B \subseteq S$.
Thus $A \setminus B \in \powerset S$, and... | Power Set is Closed under Set Difference | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Set_Difference | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Set_Difference | [
"Power Set",
"Set Difference"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Set Difference"
] | [
"Definition:Power Set",
"Set Difference is Subset",
"Subset Relation is Transitive",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-4564 | Powers of Semigroup Element Commute | Let $\struct {S, \odot}$ be a semigroup.
Let $a \in S$.
Let $m, n \in \Z_{>0}$.
Then:
:$\forall m, n \in \Z_{>0}: a^n \odot a^m = a^m \odot a^n$ | {{begin-eqn}}
{{eqn | l = a^n \odot a^m
| r = a^{n + m}
| c = Index Laws for Semigroup: Sum of Indices
}}
{{eqn | r = a^{m + n}
| c = Integer Addition is Commutative
}}
{{eqn | r = a^m \odot a^n
| c = Index Laws for Semigroup: Sum of Indices
}}
{{end-eqn}}
{{Qed}} | Let $\struct {S, \odot}$ be a [[Definition:Semigroup|semigroup]].
Let $a \in S$.
Let $m, n \in \Z_{>0}$.
Then:
:$\forall m, n \in \Z_{>0}: a^n \odot a^m = a^m \odot a^n$ | {{begin-eqn}}
{{eqn | l = a^n \odot a^m
| r = a^{n + m}
| c = [[Index Laws for Semigroup/Sum of Indices|Index Laws for Semigroup: Sum of Indices]]
}}
{{eqn | r = a^{m + n}
| c = [[Integer Addition is Commutative]]
}}
{{eqn | r = a^m \odot a^n
| c = [[Index Laws for Semigroup/Sum of Indices|Index... | Powers of Semigroup Element Commute | https://proofwiki.org/wiki/Powers_of_Semigroup_Element_Commute | https://proofwiki.org/wiki/Powers_of_Semigroup_Element_Commute | [
"Semigroups",
"Powers (Abstract Algebra)",
"Commutativity"
] | [
"Definition:Semigroup"
] | [
"Index Laws/Sum of Indices/Semigroup",
"Integer Addition is Commutative",
"Index Laws/Sum of Indices/Semigroup"
] |
proofwiki-4565 | Equivalence of Definitions of Derivative | {{TFAE|def = Derivative of Real Function at Point}}
Let $I$ be an open real interval.
Let $f: I \to \R$ be a real function defined on $I$.
Let $\xi \in I$ be a point in $I$. | {{begin-eqn}}
{{eqn | l = f' \left({\xi}\right)
| r = \lim_{h \mathop \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h
| c =
}}
{{eqn | r = \lim_{x - \xi \mathop \to 0} \frac {f \left({x}\right) - f \left({\xi}\right)} {\xi + h - \xi}
| c = substituting $x = \xi + h$
}}
{{eqn | r = \l... | {{TFAE|def = Derivative of Real Function at Point}}
Let $I$ be an [[Definition:Open Real Interval|open real interval]].
Let $f: I \to \R$ be a [[Definition:Real Function|real function]] defined on $I$.
Let $\xi \in I$ be a point in $I$. | {{begin-eqn}}
{{eqn | l = f' \left({\xi}\right)
| r = \lim_{h \mathop \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h
| c =
}}
{{eqn | r = \lim_{x - \xi \mathop \to 0} \frac {f \left({x}\right) - f \left({\xi}\right)} {\xi + h - \xi}
| c = substituting $x = \xi + h$
}}
{{eqn | r = \l... | Equivalence of Definitions of Derivative | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Derivative | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Derivative | [
"Differential Calculus",
"Analysis"
] | [
"Definition:Real Interval/Open",
"Definition:Real Function"
] | [] |
proofwiki-4566 | Derivative of x to the x | :$\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$ | Note that the Power Rule cannot be used because the index is not a constant.
Let $y := x^x$.
As $x$ was stipulated to be positive, we can take the natural logarithm of both sides:
{{begin-eqn}}
{{eqn | l = \ln y
| r = \ln x^x
}}
{{eqn | r = x \ln x
| c = Laws of Logarithms
}}
{{eqn | l = \map {\frac \d {\d ... | :$\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$ | Note that the [[Power Rule for Derivatives|Power Rule]] cannot be used because the index is not a constant.
Let $y := x^x$.
As $x$ was stipulated to be positive, we can take the [[Definition:Natural Logarithm|natural logarithm]] of both sides:
{{begin-eqn}}
{{eqn | l = \ln y
| r = \ln x^x
}}
{{eqn | r = x \ln ... | Derivative of x to the x/Proof 1 | https://proofwiki.org/wiki/Derivative_of_x_to_the_x | https://proofwiki.org/wiki/Derivative_of_x_to_the_x/Proof_1 | [
"Derivative of x to the x",
"Derivatives involving Exponential Function"
] | [] | [
"Power Rule for Derivatives",
"Definition:Natural Logarithm",
"Laws of Logarithms",
"Derivative of Composite Function",
"Derivative of Natural Logarithm Function",
"Product Rule for Derivatives",
"Derivative of Identity Function",
"Derivative of Natural Logarithm Function"
] |
proofwiki-4567 | Derivative of x to the x | :$\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$ | Note that the Power Rule cannot be used because the index is not a constant.
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} x^x
| r = \frac \d {\d x} \map \exp {x \ln x}
| c = {{Defof|Power (Algebra)|subdef = Real Number|index = 1}}
}}
{{eqn | r = \paren {\frac \d {\map \d {x \ln x} } \map \exp {x \ln x} } \pare... | :$\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$ | Note that the [[Power Rule for Derivatives|Power Rule]] cannot be used because the index is not a constant.
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} x^x
| r = \frac \d {\d x} \map \exp {x \ln x}
| c = {{Defof|Power (Algebra)|subdef = Real Number|index = 1}}
}}
{{eqn | r = \paren {\frac \d {\map \d {x \ln ... | Derivative of x to the x/Proof 2 | https://proofwiki.org/wiki/Derivative_of_x_to_the_x | https://proofwiki.org/wiki/Derivative_of_x_to_the_x/Proof_2 | [
"Derivative of x to the x",
"Derivatives involving Exponential Function"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Derivative of Exponential Function",
"Product Rule for Derivatives",
"Derivative of Identity Function",
"Derivative of Natural Logarithm Function",
"Real Multiplication Identity is One",
"Inverse for Real Multiplication"
] |
proofwiki-4568 | Derivative of x to the x | :$\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$ | From Derivative of $x^{a x}$ we have:
:$\dfrac \d {\d x} x^{a x} = a x^{a x} \paren {\ln x + 1}$
The result follows on setting $a = 1$.
{{qed}} | :$\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$ | From [[Derivative of x to the a x|Derivative of $x^{a x}$]] we have:
:$\dfrac \d {\d x} x^{a x} = a x^{a x} \paren {\ln x + 1}$
The result follows on setting $a = 1$.
{{qed}} | Derivative of x to the x/Proof 3 | https://proofwiki.org/wiki/Derivative_of_x_to_the_x | https://proofwiki.org/wiki/Derivative_of_x_to_the_x/Proof_3 | [
"Derivative of x to the x",
"Derivatives involving Exponential Function"
] | [] | [
"Derivative of x to the a x"
] |
proofwiki-4569 | Commutativity of Powers in Semigroup | :$\forall m, n \in \N_{>0}: a \circ b = b \circ a \implies a^m \circ b^n = b^n \circ a^m$
but it is not necessarily the case that:
:$\forall m, n \in \N_{>0}: a^m \circ b^n = b^n \circ a^m \implies a \circ b = b \circ a$ | Let $a, b \in S: a \circ b = b \circ a$.
Then from Powers of Commuting Elements of Semigroup Commute:
:$\forall m, n \in \N_{>0}: a^m \circ b^n = b^n \circ a^m$
{{qed|lemma}}
However, consider the dihedral group $D_3$ $= \gen {a, b: a^3 = b^2 = e, b a b = a^{−1} }$.
A group is a semigroup.
Moreover, the Cancellation La... | :$\forall m, n \in \N_{>0}: a \circ b = b \circ a \implies a^m \circ b^n = b^n \circ a^m$
but it is not necessarily the case that:
:$\forall m, n \in \N_{>0}: a^m \circ b^n = b^n \circ a^m \implies a \circ b = b \circ a$ | Let $a, b \in S: a \circ b = b \circ a$.
Then from [[Powers of Commuting Elements of Semigroup Commute]]:
:$\forall m, n \in \N_{>0}: a^m \circ b^n = b^n \circ a^m$
{{qed|lemma}}
However, consider [[Definition:Dihedral Group D3|the dihedral group $D_3$]] $= \gen {a, b: a^3 = b^2 = e, b a b = a^{−1} }$.
A [[Definiti... | Commutativity of Powers in Semigroup | https://proofwiki.org/wiki/Commutativity_of_Powers_in_Semigroup | https://proofwiki.org/wiki/Commutativity_of_Powers_in_Semigroup | [
"Semigroups",
"Commutativity"
] | [] | [
"Powers of Commuting Elements of Semigroup Commute",
"Definition:Dihedral Group D3",
"Definition:Group",
"Definition:Semigroup",
"Cancellation Laws",
"Definition:Group",
"Definition:Cancellable Element",
"Proof by Counterexample",
"Category:Semigroups",
"Category:Commutativity"
] |
proofwiki-4570 | Power of Product of Commutative Elements in Semigroup | :$\forall n \in \N_{>1}: \paren {x \circ y}^n = x^n \circ y^n \iff x \circ y = y \circ x$ | === Necessary Condition ===
Let $x \circ y = y \circ x$.
Then by Power of Product of Commuting Elements in Semigroup equals Product of Powers:
:$\forall n \in \N_{>1}: \paren {x \circ y}^n = x^n \circ y^n$
{{qed|lemma}} | :$\forall n \in \N_{>1}: \paren {x \circ y}^n = x^n \circ y^n \iff x \circ y = y \circ x$ | === Necessary Condition ===
Let $x \circ y = y \circ x$.
Then by [[Power of Product of Commuting Elements in Semigroup equals Product of Powers]]:
:$\forall n \in \N_{>1}: \paren {x \circ y}^n = x^n \circ y^n$
{{qed|lemma}} | Power of Product of Commutative Elements in Semigroup | https://proofwiki.org/wiki/Power_of_Product_of_Commutative_Elements_in_Semigroup | https://proofwiki.org/wiki/Power_of_Product_of_Commutative_Elements_in_Semigroup | [
"Semigroups",
"Commutativity",
"Power of Product of Commutative Elements in Semigroup"
] | [] | [
"Power of Product of Commuting Elements in Semigroup equals Product of Powers"
] |
proofwiki-4571 | Sum of Arcsecant and Arccosecant | Let $x \in \R$ be a real number such that $\size x \ge 1$.
Then:
: $\arcsec x + \arccsc x = \dfrac \pi 2$
where $\arcsec$ and $\arccsc$ denote arcsecant and arccosecant respectively. | Let $y \in \R$ such that:
:$\exists x \in \R: \size x \ge 1$ and $x = \map \csc {y + \dfrac \pi 2}$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \map \sec {y + \frac \pi 2}
| c =
}}
{{eqn | r = -\csc y
| c = Secant of Angle plus Right Angle
}}
{{eqn | r = \map \csc {-y}
| c = Cosecant Function is Od... | Let $x \in \R$ be a [[Definition:Real Number|real number]] such that $\size x \ge 1$.
Then:
: $\arcsec x + \arccsc x = \dfrac \pi 2$
where $\arcsec$ and $\arccsc$ denote [[Definition:Real Arcsecant|arcsecant]] and [[Definition:Real Arccosecant|arccosecant]] respectively. | Let $y \in \R$ such that:
:$\exists x \in \R: \size x \ge 1$ and $x = \map \csc {y + \dfrac \pi 2}$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \map \sec {y + \frac \pi 2}
| c =
}}
{{eqn | r = -\csc y
| c = [[Secant of Angle plus Right Angle]]
}}
{{eqn | r = \map \csc {-y}
| c = [[Cosecant Functio... | Sum of Arcsecant and Arccosecant | https://proofwiki.org/wiki/Sum_of_Arcsecant_and_Arccosecant | https://proofwiki.org/wiki/Sum_of_Arcsecant_and_Arccosecant | [
"Arcsecant Function",
"Arccosecant Function"
] | [
"Definition:Real Number",
"Definition:Inverse Secant/Real/Arcsecant",
"Definition:Inverse Cosecant/Real/Arccosecant"
] | [
"Secant of Angle plus Right Angle",
"Cosecant Function is Odd"
] |
proofwiki-4572 | Equation of Straight Line Tangent to Circle | Let $\tuple {a, b}$ be the center of a circle $\CC$ whose radius is $r$.
Let $P_n = \tuple {x_n, y_n}$ be any point on $\CC$.
The equation of a non-vertical tangent line $\TT$ to $\CC$ is given by:
:$y - y_n = \dfrac {a - x_n} {y_n - b} \paren {x - x_n}$
The equations of the vertical tangent lines to $\CC$ are:
:$x = r... | === Non-Vertical Tangent Lines ===
From Equation of Circle, $\CC$ can be described on the $x y$-plane in the form:
:$\paren {x - a}^2 + \paren {y - b}^2 = r^2$
where $P = \tuple {a, b}$ is the center of the circle and $r$ is the radius.
We use the definition of the derivative as the gradient of the tangent line $\TT$.... | Let $\tuple {a, b}$ be the [[Definition:Center of Circle|center]] of a [[Definition:Circle|circle]] $\CC$ whose [[Definition:Radius of Circle|radius]] is $r$.
Let $P_n = \tuple {x_n, y_n}$ be any [[Definition:Point|point]] on $\CC$.
The equation of a non-vertical [[Definition:Tangent to Circle|tangent line]] $\TT$ t... | === Non-Vertical Tangent Lines ===
From [[Equation of Circle]], $\CC$ can be described on the [[Definition:Cartesian Plane|$x y$-plane]] in the form:
:$\paren {x - a}^2 + \paren {y - b}^2 = r^2$
where $P = \tuple {a, b}$ is the [[Definition:Center of Circle|center of the circle]] and $r$ is the [[Definition:Radius o... | Equation of Straight Line Tangent to Circle | https://proofwiki.org/wiki/Equation_of_Straight_Line_Tangent_to_Circle | https://proofwiki.org/wiki/Equation_of_Straight_Line_Tangent_to_Circle | [
"Analytic Geometry",
"Equations of Straight Lines in Plane",
"Tangents to Circles",
"Circles"
] | [
"Definition:Circle/Center",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Point",
"Definition:Tangent Line/Circle",
"Definition:Vertical Tangent Line"
] | [
"Equation of Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Derivative",
"Definition:Gradient",
"Definition:Tangent Line/Circle",
"Definition:Derivative",
"Derivative of Constant",
"Derivative of Composite Function",
"Power Rule for Der... |
proofwiki-4573 | Index Laws for Monoids/Negative Index | :$\forall n \in \Z: \paren {a^n}^{-1} = a^{-n} = \paren {a^{-1} }^n$ | We have $a^0 = e$ so it follows trivially that $a^{-0} = \paren {a^{-1} }^0$.
From the general inverse of product, we have:
:$\paren {a_1 \circ a_2 \circ \cdots \circ a_n}^{-1} = a_n^{-1} \circ \cdots \circ a_2^{-1} \circ a_1^{-1}$
where $a_1, a_2, \ldots, a_n \in S$ are all invertible for $\circ$.
Hence we have:
:$a_1... | :$\forall n \in \Z: \paren {a^n}^{-1} = a^{-n} = \paren {a^{-1} }^n$ | We have $a^0 = e$ so it follows trivially that $a^{-0} = \paren {a^{-1} }^0$.
From the [[Inverse of Product/Monoid/General Result|general inverse of product]], we have:
:$\paren {a_1 \circ a_2 \circ \cdots \circ a_n}^{-1} = a_n^{-1} \circ \cdots \circ a_2^{-1} \circ a_1^{-1}$
where $a_1, a_2, \ldots, a_n \in S$ are ... | Index Laws for Monoids/Negative Index | https://proofwiki.org/wiki/Index_Laws_for_Monoids/Negative_Index | https://proofwiki.org/wiki/Index_Laws_for_Monoids/Negative_Index | [
"Monoids",
"Index Laws"
] | [] | [
"Inverse of Product/Monoid/General Result",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element",
"Definition:Invertible Element"
] |
proofwiki-4574 | Infinite Limit Theorem | Let $f$ be a real function of $x$ of the form
:$\map f x = \dfrac {\map g x} {\map h x}$
Further, let $g$ and $h$ be continuous on some open interval $\mathbb I$, where $c$ is a constant in $\mathbb I$.
If:
:$(1): \quad \map g c \ne 0$
:$(2): \quad \map h c = 0$
:$(3): \quad \forall x \in \mathbb I: x \ne c \implies \... | To prove the claim, it will suffice to show that for each $N \in \R_{>0}$, one can find an $\epsilon > 0$ such that:
:$\size {x - c} < \epsilon \implies \size {\map f x} \ge N$
So fix $N \in \R_{>0}$.
First, by continuity of $g$, find an $\epsilon_1 > 0$ such that:
:$\size {x - c} < \epsilon_1 \implies \size {\map g x ... | Let $f$ be a [[Definition:Real Function|real function]] of $x$ of the form
:$\map f x = \dfrac {\map g x} {\map h x}$
Further, let $g$ and $h$ be [[Definition:Continuous on Interval|continuous]] on some [[Definition:Open Real Interval|open interval]] $\mathbb I$, where $c$ is a constant in $\mathbb I$.
If:
:$(1): ... | To prove the claim, it will suffice to show that for each $N \in \R_{>0}$, one can find an $\epsilon > 0$ such that:
:$\size {x - c} < \epsilon \implies \size {\map f x} \ge N$
So fix $N \in \R_{>0}$.
First, by [[Definition:Continuous on Interval|continuity]] of $g$, find an $\epsilon_1 > 0$ such that:
:$\size {x ... | Infinite Limit Theorem | https://proofwiki.org/wiki/Infinite_Limit_Theorem | https://proofwiki.org/wiki/Infinite_Limit_Theorem | [
"Real Analysis"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Limit of Real Function"
] | [
"Definition:Continuous Real Function/Interval",
"Definition:Continuous Real Function/Interval",
"Ordering of Reciprocals"
] |
proofwiki-4575 | Index Laws for Monoids/Product of Indices | :$\forall m, n \in \Z: a^{n m} = \paren {a^m}^n = \paren {a^n}^m$ | Let $m \in \N, c = a^m, d = \paren {a^{-1}}^m$.
We define the mapping $g_c: \Z \to S$ as:
:$\forall n \in \Z: \map {g_c} n = \map {\circ^n} c$
as defined in the proof of the Index Law for Sum of Indices.
Let $h: \Z \to \Z$ be the mapping defined as:
:$\forall z \in \Z: \map h z = z m$
Then:
{{begin-eqn}}
{{eqn | l = a^... | :$\forall m, n \in \Z: a^{n m} = \paren {a^m}^n = \paren {a^n}^m$ | Let $m \in \N, c = a^m, d = \paren {a^{-1}}^m$.
We define the [[Definition:Mapping|mapping]] $g_c: \Z \to S$ as:
:$\forall n \in \Z: \map {g_c} n = \map {\circ^n} c$
as defined in the proof of the [[Index Laws for Monoids/Sum of Indices|Index Law for Sum of Indices]].
Let $h: \Z \to \Z$ be the [[Definition:Mapping|m... | Index Laws for Monoids/Product of Indices | https://proofwiki.org/wiki/Index_Laws_for_Monoids/Product_of_Indices | https://proofwiki.org/wiki/Index_Laws_for_Monoids/Product_of_Indices | [
"Monoids",
"Index Laws"
] | [] | [
"Definition:Mapping",
"Index Laws for Monoids/Sum of Indices",
"Definition:Mapping",
"Index Laws for Monoids/Sum of Indices",
"Index Laws/Product of Indices/Semigroup",
"Definition:Homomorphism (Abstract Algebra)",
"Extension Theorem for Homomorphisms"
] |
proofwiki-4576 | Powers of Group Elements/Negative Index | :$\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$ | All elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Sum of Indices:
:$\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$
{{qed}} | :$\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$ | All elements of a [[Definition:Group|group]] are [[Definition:Invertible Element|invertible]], so we can directly use the result from [[Index Laws for Monoids/Sum of Indices|Index Laws for Monoids: Sum of Indices]]:
:$\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$
{{qed}} | Powers of Group Elements/Negative Index | https://proofwiki.org/wiki/Powers_of_Group_Elements/Negative_Index | https://proofwiki.org/wiki/Powers_of_Group_Elements/Negative_Index | [
"Group Theory",
"Index Laws"
] | [] | [
"Definition:Group",
"Definition:Invertible Element",
"Index Laws for Monoids/Sum of Indices"
] |
proofwiki-4577 | Powers of Group Elements/Sum of Indices | :$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$ | All elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Sum of Indices:
:$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$
{{qed}} | :$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$ | All elements of a [[Definition:Group|group]] are [[Definition:Invertible Element|invertible]], so we can directly use the result from [[Index Laws for Monoids/Sum of Indices|Index Laws for Monoids: Sum of Indices]]:
:$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$
{{qed}} | Powers of Group Elements/Sum of Indices | https://proofwiki.org/wiki/Powers_of_Group_Elements/Sum_of_Indices | https://proofwiki.org/wiki/Powers_of_Group_Elements/Sum_of_Indices | [
"Group Theory",
"Powers (Abstract Algebra)",
"Index Laws"
] | [] | [
"Definition:Group",
"Definition:Invertible Element",
"Index Laws for Monoids/Sum of Indices"
] |
proofwiki-4578 | Powers of Group Elements/Product of Indices | :$\forall m, n \in \Z: \paren {g^m}^n = g^{m n} = \paren {g^n}^m$ | All elements of a group are invertible, so we can directly use the result from Index Laws for Monoids: Product of Indices:
:$\forall m, n \in \Z: g^{m n} = \paren {g^m}^n = \paren {g^n}^m$
{{qed}} | :$\forall m, n \in \Z: \paren {g^m}^n = g^{m n} = \paren {g^n}^m$ | All elements of a [[Definition:Group|group]] are [[Definition:Invertible Element|invertible]], so we can directly use the result from [[Index Laws for Monoids/Product of Indices|Index Laws for Monoids: Product of Indices]]:
:$\forall m, n \in \Z: g^{m n} = \paren {g^m}^n = \paren {g^n}^m$
{{qed}} | Powers of Group Elements/Product of Indices | https://proofwiki.org/wiki/Powers_of_Group_Elements/Product_of_Indices | https://proofwiki.org/wiki/Powers_of_Group_Elements/Product_of_Indices | [
"Group Theory",
"Powers (Abstract Algebra)",
"Index Laws"
] | [] | [
"Definition:Group",
"Definition:Invertible Element",
"Index Laws for Monoids/Product of Indices"
] |
proofwiki-4579 | Group Element Commutes with Inverse | Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $x \in G$.
Then:
:$x \circ x^{-1} = x^{-1} \circ x$
That is, $x$ commutes with its inverse $x^{-1}$. | By definition of inverse element:
:$x \circ x^{-1} = e = x^{-1} \circ x$
Hence the result by definition. | Let $\struct {G, \circ}$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $x \in G$.
Then:
:$x \circ x^{-1} = x^{-1} \circ x$
That is, $x$ [[Definition:Commute|commutes]] with its [[Definition:Inverse Element|inverse]] $x^{-1}$. | By definition of [[Definition:Inverse Element|inverse element]]:
:$x \circ x^{-1} = e = x^{-1} \circ x$
Hence the result by definition. | Group Element Commutes with Inverse | https://proofwiki.org/wiki/Group_Element_Commutes_with_Inverse | https://proofwiki.org/wiki/Group_Element_Commutes_with_Inverse | [
"Group Theory",
"Commutativity"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutative/Elements",
"Definition:Inverse (Abstract Algebra)/Inverse"
] | [
"Definition:Inverse (Abstract Algebra)/Inverse"
] |
proofwiki-4580 | Commutativity of Powers in Monoid | :$\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$ | By Powers of Commuting Elements of Semigroup Commute, if $m > 0$ and $n > 0$ then $a^m$ commutes with $b^n$.
By Commutation with Inverse in Monoid, again if $m > 0$ and $n > 0$ then $a^m$ commutes with $\paren {b^n}^{-1} = b^{-n}$.
Similarly $b^n$ commutes with $a^{-m}$.
But as $a^{-m}$ commutes with $b^n$, it also com... | :$\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$ | By [[Powers of Commuting Elements of Semigroup Commute]], if $m > 0$ and $n > 0$ then $a^m$ [[Definition:Commute|commutes]] with $b^n$.
By [[Commutation with Inverse in Monoid]], again if $m > 0$ and $n > 0$ then $a^m$ [[Definition:Commute|commutes]] with $\paren {b^n}^{-1} = b^{-n}$.
Similarly $b^n$ [[Definition:Com... | Commutativity of Powers in Monoid | https://proofwiki.org/wiki/Commutativity_of_Powers_in_Monoid | https://proofwiki.org/wiki/Commutativity_of_Powers_in_Monoid | [
"Monoids",
"Commutativity"
] | [] | [
"Powers of Commuting Elements of Semigroup Commute",
"Definition:Commutative/Elements",
"Commutation with Inverse in Monoid",
"Definition:Commutative/Elements",
"Definition:Commutative/Elements",
"Definition:Commutative/Elements",
"Definition:Commutative/Elements",
"Commutation with Inverse in Monoid"... |
proofwiki-4581 | Power of Product of Commutative Elements in Monoid | :$\forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$ | From Power of Product of Commutative Elements in Semigroup, this result holds if $n \ge 0$.
Since $a$ and $b$ commute, then so do $a^{-1}$ and $b^{-1}$ by Commutation of Inverses in Monoid.
Hence, if $n > 0$:
{{begin-eqn}}
{{eqn | l = \paren {a \circ b}^{-n}
| r = \paren {\paren {a \circ b}^{-1} }^n
}}
{{eqn | r ... | :$\forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$ | From [[Power of Product of Commutative Elements in Semigroup]], this result holds if $n \ge 0$.
Since $a$ and $b$ [[Definition:Commute|commute]], then so do $a^{-1}$ and $b^{-1}$ by [[Commutation of Inverses in Monoid]].
Hence, if $n > 0$:
{{begin-eqn}}
{{eqn | l = \paren {a \circ b}^{-n}
| r = \paren {\paren ... | Power of Product of Commutative Elements in Monoid | https://proofwiki.org/wiki/Power_of_Product_of_Commutative_Elements_in_Monoid | https://proofwiki.org/wiki/Power_of_Product_of_Commutative_Elements_in_Monoid | [
"Monoids",
"Commutativity"
] | [] | [
"Power of Product of Commutative Elements in Semigroup",
"Definition:Commutative/Elements",
"Commutation of Inverses in Monoid",
"Inverse of Product",
"Commutation of Inverses in Monoid"
] |
proofwiki-4582 | Commutativity of Powers in Group | : $\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$ | By definition, all elements of a group are invertible.
Therefore Commutativity of Powers in Monoid can be applied directly.
{{Qed}}
Category:Group Theory
Category:Commutativity
s8ogf503onvplgoz7ir9lu0b4wljf2f | : $\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$ | By definition, all [[Definition:Element|elements]] of a [[Definition:Group|group]] are [[Definition:Invertible Element|invertible]].
Therefore [[Commutativity of Powers in Monoid]] can be applied directly.
{{Qed}}
[[Category:Group Theory]]
[[Category:Commutativity]]
s8ogf503onvplgoz7ir9lu0b4wljf2f | Commutativity of Powers in Group | https://proofwiki.org/wiki/Commutativity_of_Powers_in_Group | https://proofwiki.org/wiki/Commutativity_of_Powers_in_Group | [
"Group Theory",
"Commutativity"
] | [] | [
"Definition:Element",
"Definition:Group",
"Definition:Invertible Element",
"Commutativity of Powers in Monoid",
"Category:Group Theory",
"Category:Commutativity"
] |
proofwiki-4583 | Power of Product of Commutative Elements in Group | :$a \circ b = b \circ a \iff \forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$ | === Necessary Condition ===
Let $a \circ b = b \circ a$.
By definition, all elements of a group are invertible.
Therefore the results in Power of Product of Commutative Elements in Monoid can be applied directly.
{{qed|lemma}} | :$a \circ b = b \circ a \iff \forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$ | === Necessary Condition ===
Let $a \circ b = b \circ a$.
By definition, all [[Definition:Element|elements]] of a [[Definition:Group|group]] are [[Definition:Invertible Element|invertible]].
Therefore the results in [[Power of Product of Commutative Elements in Monoid]] can be applied directly.
{{qed|lemma}} | Power of Product of Commutative Elements in Group | https://proofwiki.org/wiki/Power_of_Product_of_Commutative_Elements_in_Group | https://proofwiki.org/wiki/Power_of_Product_of_Commutative_Elements_in_Group | [
"Group Theory",
"Commutativity"
] | [] | [
"Definition:Element",
"Definition:Group",
"Definition:Invertible Element",
"Power of Product of Commutative Elements in Monoid"
] |
proofwiki-4584 | Primitive of Tangent Function/Cosine Form | :$\ds \int \tan x \rd x = -\ln \size {\cos x} + C$
where $\cos x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \tan x \rd x
| r = \int \frac {\sin x} {\cos x} \rd x
| c = {{Defof|Real Tangent Function}}
}}
{{eqn | r = -\int \frac {-\sin x} {\cos x} \rd x
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = -\int \frac {\paren {\cos x}'} {\cos x} \rd x
| c = Deriv... | :$\ds \int \tan x \rd x = -\ln \size {\cos x} + C$
where $\cos x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \tan x \rd x
| r = \int \frac {\sin x} {\cos x} \rd x
| c = {{Defof|Real Tangent Function}}
}}
{{eqn | r = -\int \frac {-\sin x} {\cos x} \rd x
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = -\int \frac {\paren {\cos x}'} {\cos x} \rd x
| c = [... | Primitive of Tangent Function/Cosine Form/Proof | https://proofwiki.org/wiki/Primitive_of_Tangent_Function/Cosine_Form | https://proofwiki.org/wiki/Primitive_of_Tangent_Function/Cosine_Form/Proof | [
"Primitive of Tangent Function"
] | [] | [
"Primitive of Constant Multiple of Function",
"Derivative of Cosine Function",
"Primitive of Function under its Derivative"
] |
proofwiki-4585 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\arctan x}
| r = \dfrac 1 {1 + x^2}
| c = Derivative of Arctangent Function
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d x} {1 + x^2}
| r = \arctan x + C
| c = {{Defof|Primitive (Calculus)}}
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\arctan x}
| r = \dfrac 1 {1 + x^2}
| c = [[Derivative of Arctangent Function]]
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d x} {1 + x^2}
| r = \arctan x + C
| c = {{Defof|Primitive (Calculus)}}
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of 1 plus x squared/Arctangent Form/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_plus_x_squared/Arctangent_Form/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Arctangent Function"
] |
proofwiki-4586 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From Primitive of $\dfrac {x^m} {\ln x}$:
:$\ds \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$
The result follows by setting $m = 0$.
{{qed}} | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From [[Primitive of Power of x over Logarithm of x|Primitive of $\dfrac {x^m} {\ln x}$]]:
:$\ds \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$
The result follows by setting $m = 0$.
{{qed}} | Primitive of Reciprocal of Logarithm of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Logarithm_of_x/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Power of x over Logarithm of x"
] |
proofwiki-4587 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From Primitive of $\dfrac {1 + x^2} {1 + x^4}$, we have:
:$\ds \int \frac {x^2 + 1} {x^4 + 1} \rd x = \frac 1 {\sqrt 2} \map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + C$
From Primitive of $\dfrac {-1 + x^2} {1 + x^4}$, we have:
:$\ds \int \frac {x^2 - 1} {x^4 + 1} \rd x = \frac 1 {2 \sqrt 2} \ln \size {\fra... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From [[Primitive of One plus x Squared over One plus Fourth Power of x|Primitive of $\dfrac {1 + x^2} {1 + x^4}$]], we have:
:$\ds \int \frac {x^2 + 1} {x^4 + 1} \rd x = \frac 1 {\sqrt 2} \map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + C$
From [[Primitive of Minus One plus x Squared over One plus Fourth Po... | Primitive of Reciprocal of One plus Fourth Power of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_One_plus_Fourth_Power_of_x/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of One plus x Squared over One plus Fourth Power of x",
"Primitive of Minus One plus x Squared over One plus Fourth Power of x",
"Logarithm of Reciprocal"
] |
proofwiki-4588 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | A special case of Primitive of $\dfrac 1 {x^4 + a^4}$, setting $a = 1$.
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^4 + a^4}
| r = \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arcta... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | A special case of [[Primitive of Reciprocal of x fourth plus a fourth|Primitive of $\dfrac 1 {x^4 + a^4}$]], setting $a = 1$.
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^4 + a^4}
| r = \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \pare... | Primitive of Reciprocal of One plus Fourth Power of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_One_plus_Fourth_Power_of_x/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x fourth plus a fourth",
"Difference of Arctangents",
"Difference of Two Squares",
"Inverse Tangent is Odd Function",
"Sum of Arctangent and Arccotangent",
"Inverse Tangent is Odd Function"
] |
proofwiki-4589 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$: Arcsine Form:
:$\ds \int \frac {\d x} {\sqrt {a^2 - x^2} } = \arcsin \frac x a + C$
The result follows by setting $a = 1$.
{{Qed}} | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From [[Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form|Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$: Arcsine Form]]:
:$\ds \int \frac {\d x} {\sqrt {a^2 - x^2} } = \arcsin \frac x a + C$
The result follows by setting $a = 1$.
{{Qed}} | Primitive of Reciprocal of Root of 1 minus x squared/Arcsine Form/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_1_minus_x_squared/Arcsine_Form/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-4590 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\arcsin x}
| r = \dfrac 1 {\sqrt {1 - x^2} }
| c = Derivative of Arcsine Function
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d x} {\sqrt {1 - x^2} }
| r = \arcsin x + C
| c = {{Defof|Primitive (Calculus)}}
}}
{{end-eqn}}
{{qed}} | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\arcsin x}
| r = \dfrac 1 {\sqrt {1 - x^2} }
| c = [[Derivative of Arcsine Function]]
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d x} {\sqrt {1 - x^2} }
| r = \arcsin x + C
| c = {{Defof|Primitive (Calculus)}}
}}
{{end-eqn}}
{{qed}} | Primitive of Reciprocal of Root of 1 minus x squared/Arcsine Form/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_1_minus_x_squared/Arcsine_Form/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Arcsine Function"
] |
proofwiki-4591 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cos^2 \theta + b \sin^2 \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d \theta}
| r = 2 a \cos \theta \paren {-\sin \theta} + 2 b \sin \theta \cos \theta
| c = Chain Rule for Derivatives, Derivative of Cosine Function, Derivative of S... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cos^2 \theta + b \sin^2 \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d x} {\d \theta}
| r = 2 a \cos \theta \paren {-\sin \theta} + 2 b \sin \theta \cos \theta
| c = [[Chain Rule for Derivatives]], [[Derivative of Cosine Function]], [[Deri... | Primitive of Reciprocal of Root of a minus x by Cube of Root of x minus b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_minus_x_by_Cube_of_Root_of_x_minus_b/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Composite Function",
"Derivative of Cosine Function",
"Derivative of Sine Function",
"Sum of Squares of Sine and Cosine",
"Sum of Squares of Sine and Cosine",
"Primitive of Square of Secant Function"
] |
proofwiki-4592 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First let us express the integrand in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{end-eqn}}
Recall the defin... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First let us express the [[Definition:Integrand|integrand]] in the following form:
{{begin-eqn}}
{{eqn | n = 1
| l = \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }
| r = \int \frac {\d x} {\sqrt {a p \paren {x - \paren {-\frac b a} } \paren {x - \paren {-\frac q p} } } }
| c =
}}
{{... | Primitive of Reciprocal of Root of a x + b by Root of p x + q/a p less than 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_+_b_by_Root_of_p_x_+_q/a_p_less_than_0/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Definition:Integration/Integrand",
"Definition:Euler Substitution/Third",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Arccotangent of Reciprocal equals Arctangent",
"Sum of A... |
proofwiki-4593 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} }
| r = \dfrac 1 {\sqrt 3} \int \dfrac {\d x} {\sqrt {\frac 2 3 + \frac 4 3 x - x^2} }
| c =
}}
{{eqn | r = \dfrac 1 {\sqrt 3} \int \dfrac {\d x} {\sqrt {\frac {10} 9 - \paren {x - \frac 2 3}^2} }
| c = Completing the Square
}}
{{e... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} }
| r = \dfrac 1 {\sqrt 3} \int \dfrac {\d x} {\sqrt {\frac 2 3 + \frac 4 3 x - x^2} }
| c =
}}
{{eqn | r = \dfrac 1 {\sqrt 3} \int \dfrac {\d x} {\sqrt {\frac {10} 9 - \paren {x - \frac 2 3}^2} }
| c = [[Completing the Square]]
}}... | Primitive of Reciprocal of Root of a x squared plus b x plus c/Examples/2 + 4 x - 3 x^2/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_a_x_squared_plus_b_x_plus_c/Examples/2_+_4_x_-_3_x^2/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Completing the Square",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-4594 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 - a^2} }
| r = \cosh^{-1} {\frac x a} + C'
| c = Primitive of Reciprocal of $\sqrt {x^2 - a^2}$: $\cosh^{-1}$ form
}}
{{eqn | r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} } + C'
| c = {{Defof|Real Inverse Hyperbolic Cosine}}
}}
{{eq... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 - a^2} }
| r = \cosh^{-1} {\frac x a} + C'
| c = [[Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form|Primitive of Reciprocal of $\sqrt {x^2 - a^2}$: $\cosh^{-1}$ form]]
}}
{{eqn | r = \map \ln {\frac x a + \sqr... | Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared/Logarithm_Form/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form",
"Difference of Logarithms"
] |
proofwiki-4595 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sec^2 \theta
| c = Derivative of Tangent Function
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {a \sec^2 \theta \rd \the... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sec^2 \theta
| c = [[Derivative of Tangent Function]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {\paren {\sqrt {x^2 + a^2} }^3}
| r = \int \frac {a \sec^2 \theta \rd ... | Primitive of Reciprocal of Root of x squared plus a squared cubed/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared_cubed/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Tangent Function",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Primitive of Cosine Function",
"Tangent is Sine divided by Cosine",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-4596 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 + a^2} }
| r = \arsinh {\frac x a} + C
| c = Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ in $\arsinh$ form
}}
{{eqn | r = \map \ln {x + \sqrt {x^2 + a^2} } - \ln a + C
| c = $\arsinh \dfrac x a$ in Logarithm Form
}}
{{eqn | r = \map \ln {x + \sq... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 + a^2} }
| r = \arsinh {\frac x a} + C
| c = [[Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form|Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ in $\arsinh$ form]]
}}
{{eqn | r = \map \ln {x + \sqrt {x^2 + a^2} } - ... | Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Logarithm_Form/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of Root of x squared plus a squared/Inverse Hyperbolic Sine Form",
"Real Area Hyperbolic Sine of x over a in Logarithm Form",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-4597 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $y^2 = a^2 + x^2$.
Then:
{{begin-eqn}}
{{eqn | l = 2 y \frac {\d y} {\d x}
| r = 2 x
| c = Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = y \frac {\d y} {\d x}
| r = x
| c = simplification
}}
{{eqn | ll= \leadsto
| l = \frac {\d y} x
| ... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $y^2 = a^2 + x^2$.
Then:
{{begin-eqn}}
{{eqn | l = 2 y \frac {\d y} {\d x}
| r = 2 x
| c = [[Power Rule for Derivatives]], [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = y \frac {\d y} {\d x}
| r = x
| c = simplification
}}
{{eqn | ll= \leadsto
| l = \frac {\d y} ... | Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Logarithm_Form/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Primitive of Function under its Derivative"
] |
proofwiki-4598 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x + \cos a x}
| r = \int \frac {\d x} {\sqrt 2 \map \cos {a x - \dfrac \pi 4} }
| c = Sine of x plus Cosine of x: Cosine Form
}}
{{eqn | r = \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac \pi 4} \rd x
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x + \cos a x}
| r = \int \frac {\d x} {\sqrt 2 \map \cos {a x - \dfrac \pi 4} }
| c = [[Sine of x plus Cosine of x/Cosine Form|Sine of x plus Cosine of x: Cosine Form]]
}}
{{eqn | r = \frac 1 {\sqrt 2} \int \map \sec {a x - \dfrac \pi 4} \rd x
| c = ... | Primitive of Reciprocal of Sine of a x plus Cosine of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_plus_Cosine_of_a_x/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Sine of x plus Cosine of x/Cosine Form",
"Secant is Reciprocal of Cosine",
"Derivative of Identity Function",
"Derivatives of Function of a x + b",
"Integration by Substitution",
"Primitive of Secant of a x/Tangent Form"
] |
proofwiki-4599 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x + \cos a x}
| r = \frac 1 a \int \frac {\dfrac {2 \rd u} {1 + u^2} } {\dfrac {2 u} {1 + u^2} + \dfrac {1 - u^2} {1 + u^2} }
| c = Weierstrass Substitution: $u = \tan \dfrac {a x} 2$
}}
{{eqn | r = \frac 2 a \int \frac {\d u} {- u^2 + 2 u + 1}
| c =... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin a x + \cos a x}
| r = \frac 1 a \int \frac {\dfrac {2 \rd u} {1 + u^2} } {\dfrac {2 u} {1 + u^2} + \dfrac {1 - u^2} {1 + u^2} }
| c = [[Weierstrass Substitution]]: $u = \tan \dfrac {a x} 2$
}}
{{eqn | r = \frac 2 a \int \frac {\d u} {- u^2 + 2 u + 1}
|... | Primitive of Reciprocal of Sine of a x plus Cosine of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Sine_of_a_x_plus_Cosine_of_a_x/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Weierstrass Substitution",
"Primitive of Reciprocal of a x squared plus b x plus c",
"Tangent of 22.5 Degrees",
"Tangent of 67.5 Degrees"
] |
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