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-10600 | Sine in terms of Secant | {{begin-eqn}}
{{eqn | l = \sin x
| r = + \frac {\sqrt{\sec ^2 x - 1} } {\sec x}
| c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
}}
{{eqn | l = \sin x
| r = - \frac {\sqrt{\sec ^2 x - 1} } {\sec x}
| c = if there exists an integer $n$ such that $\paren {n ... | For the first part, if there exists integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$:
{{begin-eqn}}
{{eqn | l = \tan x
| r = +\sqrt {\sec^2 x - 1}
| c = Tangent in terms of Secant
}}
{{eqn | ll= \leadsto
| l = \frac {\sin x} {\cos x}
| r = +\sqrt {\sec^2 x - 1}
| c = Tangent... | {{begin-eqn}}
{{eqn | l = \sin x
| r = + \frac {\sqrt{\sec ^2 x - 1} } {\sec x}
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
}}
{{eqn | l = \sin x
| r = - \frac {\sqrt{\sec ^2 x - 1} } {\sec x}
| c = if there exists an [[Definiti... | For the first part, if there exists [[Definition:Integer|integer]] $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$:
{{begin-eqn}}
{{eqn | l = \tan x
| r = +\sqrt {\sec^2 x - 1}
| c = [[Tangent in terms of Secant]]
}}
{{eqn | ll= \leadsto
| l = \frac {\sin x} {\cos x}
| r = +\sqrt {\sec^... | Sine in terms of Secant | https://proofwiki.org/wiki/Sine_in_terms_of_Secant | https://proofwiki.org/wiki/Sine_in_terms_of_Secant | [
"Sine Function",
"Secant Function"
] | [
"Definition:Integer",
"Definition:Integer"
] | [
"Definition:Integer",
"Tangent in terms of Secant",
"Tangent is Sine divided by Cosine",
"Secant is Reciprocal of Cosine",
"Definition:Integer",
"Tangent in terms of Secant",
"Tangent is Sine divided by Cosine",
"Secant is Reciprocal of Cosine"
] |
proofwiki-10601 | Sign of Sine | Let $x$ be a real number.
{{begin-eqn}}
{{eqn | l = \sin x
| o = >
| r = 0
| c = if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$
}}
{{eqn | l = \sin x
| o = <
| r = 0
| c = if there exists an integer $n$ such that $\paren {2 n + 1} \pi < x < \paren {2 n ... | First the case where $n \ge 0$ is addressed.
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\forall x \in \R:$
::$2 n \pi < x < \paren {2 n + 1} \pi \implies \sin x > 0$
::$\paren {2 n + 1} \pi < x < \paren {2 n + 2} \pi \implies \sin x < 0$ | Let $x$ be a [[Definition:Real Number|real number]].
{{begin-eqn}}
{{eqn | l = \sin x
| o = >
| r = 0
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$
}}
{{eqn | l = \sin x
| o = <
| r = 0
| c = if there exists an [[Definitio... | First the case where $n \ge 0$ is addressed.
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\forall x \in \R:$
::$2 n \pi < x < \paren {2 n + 1} \pi \implies \sin x > 0$
::$\paren {2 n + 1} \pi < x... | Sign of Sine | https://proofwiki.org/wiki/Sign_of_Sine | https://proofwiki.org/wiki/Sign_of_Sine | [
"Sine Function"
] | [
"Definition:Real Number",
"Definition:Integer",
"Definition:Integer",
"Definition:Sine/Real Function"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-10602 | Second Principle of Mathematical Induction | Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is true
:$(2): \quad \forall k \in \Z: k \ge n_0: \map P {n_0} \land \map P {n_0 + 1} \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is tru... | For each $n \ge n_0$, let $\map {P'} n$ be defined as:
:$\map {P'} n := \map P {n_0} \land \dots \land \map P n$
It suffices to show that $\map {P'} n$ is true for all $n \ge n_0$.
It is immediate from the assumption $\map P {n_0}$ that $\map {P'} {n_0}$ is true.
Now suppose that $\map {P'} n$ holds.
By $(2)$, this imp... | Let $\map P n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is [[Definition:True|true]]
:$(2): \quad \forall k \in \Z: k \ge n_0: \map P {n_0} \land \map P {n_0 + 1} \land \ldots \land \map P {k - 1} \... | For each $n \ge n_0$, let $\map {P'} n$ be defined as:
:$\map {P'} n := \map P {n_0} \land \dots \land \map P n$
It suffices to show that $\map {P'} n$ is true for all $n \ge n_0$.
It is immediate from the assumption $\map P {n_0}$ that $\map {P'} {n_0}$ is [[Definition:True|true]].
Now suppose that $\map {P'} n$ ... | Second Principle of Mathematical Induction | https://proofwiki.org/wiki/Second_Principle_of_Mathematical_Induction | https://proofwiki.org/wiki/Second_Principle_of_Mathematical_Induction | [
"Second Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Mathematical Induction",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:True",
"Definition:True"
] | [
"Definition:True",
"Principle of Mathematical Induction"
] |
proofwiki-10603 | Second Principle of Mathematical Induction | Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is true
:$(2): \quad \forall k \in \Z: k \ge n_0: \map P {n_0} \land \map P {n_0 + 1} \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is tru... | For each $n \in \N_{>0}$, let $\map {P'} n$ be defined as:
:$\map {P'} n := \map P 1 \land \dots \land \map P n$
It suffices to show that $\map {P'} n$ is true for all $n \in \N_{>0}$.
It is immediate from the assumption $\map P 1$ that $\map {P'} 1$ is true.
Now suppose that $\map {P'} n$ holds.
By $(2)$, this implies... | Let $\map P n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is [[Definition:True|true]]
:$(2): \quad \forall k \in \Z: k \ge n_0: \map P {n_0} \land \map P {n_0 + 1} \land \ldots \land \map P {k - 1} \... | For each $n \in \N_{>0}$, let $\map {P'} n$ be defined as:
:$\map {P'} n := \map P 1 \land \dots \land \map P n$
It suffices to show that $\map {P'} n$ is true for all $n \in \N_{>0}$.
It is immediate from the assumption $\map P 1$ that $\map {P'} 1$ is [[Definition:True|true]].
Now suppose that $\map {P'} n$ hold... | Second Principle of Mathematical Induction/One-Based/Proof 1 | https://proofwiki.org/wiki/Second_Principle_of_Mathematical_Induction | https://proofwiki.org/wiki/Second_Principle_of_Mathematical_Induction/One-Based/Proof_1 | [
"Second Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Mathematical Induction",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:True",
"Definition:True"
] | [
"Definition:True",
"Principle of Mathematical Induction"
] |
proofwiki-10604 | Second Principle of Mathematical Induction | Let $\map P n$ be a propositional function depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is true
:$(2): \quad \forall k \in \Z: k \ge n_0: \map P {n_0} \land \map P {n_0 + 1} \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$
Then:
:$\map P n$ is tru... | Let $S \subseteq \N_{>0}$ containing those $n \in \N_{>0}$ for which $\map P n$ does not hold.
{{AimForCont}} $S \ne \O$.
Then by the Well-Ordering Principle $S$ contains a minimal element $s$.
We have that $s \ne 1$ because $\map P 1$ is true from $(1)$.
Thus there must exist some $k \in \N_{>0}$ such that $s = k + 1$... | Let $\map P n$ be a [[Definition:Propositional Function|propositional function]] depending on $n \in \Z$.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad \map P {n_0}$ is [[Definition:True|true]]
:$(2): \quad \forall k \in \Z: k \ge n_0: \map P {n_0} \land \map P {n_0 + 1} \land \ldots \land \map P {k - 1} \... | Let $S \subseteq \N_{>0}$ containing those $n \in \N_{>0}$ for which $\map P n$ does not hold.
{{AimForCont}} $S \ne \O$.
Then by the [[Well-Ordering Principle]] $S$ contains a [[Definition:Minimal Element|minimal element]] $s$.
We have that $s \ne 1$ because $\map P 1$ is true from $(1)$.
Thus there must exist som... | Second Principle of Mathematical Induction/One-Based/Proof 2 | https://proofwiki.org/wiki/Second_Principle_of_Mathematical_Induction | https://proofwiki.org/wiki/Second_Principle_of_Mathematical_Induction/One-Based/Proof_2 | [
"Second Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Mathematical Induction",
"Proof Techniques"
] | [
"Definition:Propositional Function",
"Definition:True",
"Definition:True"
] | [
"Well-Ordering Principle",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Contradiction",
"Proof by Contradiction"
] |
proofwiki-10605 | Second Principle of Finite Induction | Let $S \subseteq \Z$ be a subset of the integers.
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: \paren {\forall k: n_0 \le k \le n \implies k \in S} \implies n + 1 \in S$
Then:
:$\forall n \ge n_0: n \in S$ | Define $T$ as:
:$T = \set {n \in \Z : \forall k: n_0 \le k \le n: k \in S}$
Since $n \le n$, it follows that $T \subseteq S$.
Therefore, it will suffice to show that:
:$\forall n \ge n_0: n \in T$
Firstly, we have that $n_0 \in T$ {{iff}} the following condition holds:
:$\forall k: n_0 \le k \le n_0 \implies k \in S$
S... | Let $S \subseteq \Z$ be a [[Definition:Subset|subset]] of the [[Definition:Integer|integers]].
Let $n_0 \in \Z$ be given.
Suppose that:
:$(1): \quad n_0 \in S$
:$(2): \quad \forall n \ge n_0: \paren {\forall k: n_0 \le k \le n \implies k \in S} \implies n + 1 \in S$
Then:
:$\forall n \ge n_0: n \in S$ | Define $T$ as:
:$T = \set {n \in \Z : \forall k: n_0 \le k \le n: k \in S}$
Since $n \le n$, it follows that $T \subseteq S$.
Therefore, it will suffice to show that:
:$\forall n \ge n_0: n \in T$
Firstly, we have that $n_0 \in T$ {{iff}} the following condition holds:
:$\forall k: n_0 \le k \le n_0 \implies k \... | Second Principle of Finite Induction | https://proofwiki.org/wiki/Second_Principle_of_Finite_Induction | https://proofwiki.org/wiki/Second_Principle_of_Finite_Induction | [
"Second Principle of Finite Induction",
"Principle of Finite Induction",
"Mathematical Induction",
"Proof Techniques",
"Named Theorems"
] | [
"Definition:Subset",
"Definition:Integer"
] | [
"Closed Interval of Naturally Ordered Semigroup with Successor equals Union with Successor",
"Principle of Finite Induction"
] |
proofwiki-10606 | Sign of Cosine | Let $x$ be a real number.
Then:
{{begin-eqn}}
{{eqn | l = \cos x
| o = >
| r = 0
| c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \cos x
| o = <
| r = 0
| c = if there exists an integer $n$ such that $\p... | Proof by induction: | Let $x$ be a [[Definition:Real Number|real number]].
Then:
{{begin-eqn}}
{{eqn | l = \cos x
| o = >
| r = 0
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \cos x
| o = <
| r = 0
... | Proof by [[Principle of Mathematical Induction|induction]]: | Sign of Cosine | https://proofwiki.org/wiki/Sign_of_Cosine | https://proofwiki.org/wiki/Sign_of_Cosine | [
"Cosine Function"
] | [
"Definition:Real Number",
"Definition:Integer",
"Definition:Integer",
"Definition:Cosine/Real Function"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-10607 | Sign of Tangent | Let $x$ be a real number.
Then:
{{begin-eqn}}
{{eqn | l = \tan x
| o = >
| r = 0
| c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
}}
{{eqn | l = \tan x
| o = <
| r = 0
| c = if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi ... | From Tangent is Sine divided by Cosine:
:$\tan x = \dfrac {\sin x} {\cos x}$
Since $n$ is an integer, $n$ is either odd or even. | Let $x$ be a [[Definition:Real Number|real number]].
Then:
{{begin-eqn}}
{{eqn | l = \tan x
| o = >
| r = 0
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
}}
{{eqn | l = \tan x
| o = <
| r = 0
| c = if there exists an... | From [[Tangent is Sine divided by Cosine]]:
:$\tan x = \dfrac {\sin x} {\cos x}$
Since $n$ is an integer, $n$ is either odd or even. | Sign of Tangent | https://proofwiki.org/wiki/Sign_of_Tangent | https://proofwiki.org/wiki/Sign_of_Tangent | [
"Tangent Function"
] | [
"Definition:Real Number",
"Definition:Integer",
"Definition:Integer",
"Definition:Tangent Function"
] | [
"Tangent is Sine divided by Cosine"
] |
proofwiki-10608 | Reciprocal of Strictly Negative Real Number is Strictly Negative | :$\forall x \in \R: x < 0 \implies \dfrac 1 x < 0$ | Let $x < 0$.
{{AimForCont}} $\dfrac 1 x > 0$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = <
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = x \times \dfrac 1 x
| o = <
| r = 0 \times 0
| c = Real Number Ordering is Compatible with Multiplication: Negative Factor
}}
{{eqn | ll= \leadst... | :$\forall x \in \R: x < 0 \implies \dfrac 1 x < 0$ | Let $x < 0$.
{{AimForCont}} $\dfrac 1 x > 0$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = <
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = x \times \dfrac 1 x
| o = <
| r = 0 \times 0
| c = [[Real Number Ordering is Compatible with Multiplication/Negative Factor|Real Number Orderi... | Reciprocal of Strictly Negative Real Number is Strictly Negative | https://proofwiki.org/wiki/Reciprocal_of_Strictly_Negative_Real_Number_is_Strictly_Negative | https://proofwiki.org/wiki/Reciprocal_of_Strictly_Negative_Real_Number_is_Strictly_Negative | [
"Real Numbers",
"Reciprocals"
] | [] | [
"Real Number Ordering is Compatible with Multiplication/Negative Factor",
"Real Zero is Less than Real One",
"Proof by Contradiction"
] |
proofwiki-10609 | Sign of Cosecant | Let $x$ be a real number.
Then:
{{begin-eqn}}
{{eqn | l = \csc x
| o = >
| r = 0
| c = if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$
}}
{{eqn | l = \csc x
| o = <
| r = 0
| c = if there exists an integer $n$ such that $\paren {2 n + 1} \pi < x < \paren... | For the first part:
{{begin-eqn}}
{{eqn | l = \sin x
| o = >
| r = 0
| c = if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$
| cc= Sign of Sine
}}
{{eqn | ll= \leadsto
| l = \frac 1 {\sin x}
| o = >
| r = 0
| c = if there exists an integer $n$ ... | Let $x$ be a [[Definition:Real Number|real number]].
Then:
{{begin-eqn}}
{{eqn | l = \csc x
| o = >
| r = 0
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$
}}
{{eqn | l = \csc x
| o = <
| r = 0
| c = if there exists an [[De... | For the first part:
{{begin-eqn}}
{{eqn | l = \sin x
| o = >
| r = 0
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$
| cc= [[Sign of Sine]]
}}
{{eqn | ll= \leadsto
| l = \frac 1 {\sin x}
| o = >
| r = 0
| c = if ... | Sign of Cosecant | https://proofwiki.org/wiki/Sign_of_Cosecant | https://proofwiki.org/wiki/Sign_of_Cosecant | [
"Cosecant Function"
] | [
"Definition:Real Number",
"Definition:Integer",
"Definition:Integer",
"Definition:Cosecant/Real Function"
] | [
"Definition:Integer",
"Sign of Sine",
"Definition:Integer",
"Reciprocal of Strictly Positive Real Number is Strictly Positive",
"Definition:Integer",
"Cosecant is Reciprocal of Sine",
"Definition:Integer",
"Sign of Sine",
"Definition:Integer",
"Reciprocal of Strictly Negative Real Number is Strict... |
proofwiki-10610 | Sign of Secant | Let $x$ be a real number.
{{begin-eqn}}
{{eqn | l = \sec x
| o = >
| r = 0
| c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \sec x
| o = <
| r = 0
| c = if there exists an integer $n$ such that $\paren {... | For the first part:
{{begin-eqn}}
{{eqn | l = \cos x
| o = >
| r = 0
| c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
| cc= Sign of Cosine
}}
{{eqn | ll= \leadsto
| l = \frac 1 {\cos x}
| o = >
| r = 0
| c ... | Let $x$ be a [[Definition:Real Number|real number]].
{{begin-eqn}}
{{eqn | l = \sec x
| o = >
| r = 0
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \sec x
| o = <
| r = 0
| c =... | For the first part:
{{begin-eqn}}
{{eqn | l = \cos x
| o = >
| r = 0
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
| cc= [[Sign of Cosine]]
}}
{{eqn | ll= \leadsto
| l = \frac 1 {\cos x}
| o ... | Sign of Secant | https://proofwiki.org/wiki/Sign_of_Secant | https://proofwiki.org/wiki/Sign_of_Secant | [
"Secant Function"
] | [
"Definition:Real Number",
"Definition:Integer",
"Definition:Integer",
"Definition:Secant Function/Real"
] | [
"Definition:Integer",
"Sign of Cosine",
"Definition:Integer",
"Reciprocal of Strictly Positive Real Number is Strictly Positive",
"Definition:Integer",
"Secant is Reciprocal of Cosine",
"Definition:Integer",
"Sign of Cosine",
"Definition:Integer",
"Reciprocal of Strictly Negative Real Number is St... |
proofwiki-10611 | Sign of Cotangent | Let $x$ be a real number.
Then:
{{begin-eqn}}
{{eqn | l = \cot x
| o = >
| r = 0
| c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
}}
{{eqn | l = \cot x
| o = <
| r = 0
| c = if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi ... | For the first part:
{{begin-eqn}}
{{eqn | l = \tan x
| o = >
| r = 0
| c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
| cc= Sign of Tangent
}}
{{eqn | ll= \leadsto
| l = \frac 1 \tan x
| o = >
| r = 0
| c = if there exists an intege... | Let $x$ be a [[Definition:Real Number|real number]].
Then:
{{begin-eqn}}
{{eqn | l = \cot x
| o = >
| r = 0
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
}}
{{eqn | l = \cot x
| o = <
| r = 0
| c = if there exists an... | For the first part:
{{begin-eqn}}
{{eqn | l = \tan x
| o = >
| r = 0
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
| cc= [[Sign of Tangent]]
}}
{{eqn | ll= \leadsto
| l = \frac 1 \tan x
| o = >
| r = 0
| c... | Sign of Cotangent | https://proofwiki.org/wiki/Sign_of_Cotangent | https://proofwiki.org/wiki/Sign_of_Cotangent | [
"Cotangent Function"
] | [
"Definition:Real Number",
"Definition:Integer",
"Definition:Integer",
"Definition:Cotangent/Real Function"
] | [
"Definition:Integer",
"Sign of Tangent",
"Definition:Integer",
"Reciprocal of Strictly Positive Real Number is Strictly Positive",
"Definition:Integer",
"Cotangent is Reciprocal of Tangent",
"Definition:Integer",
"Sign of Tangent",
"Definition:Integer",
"Reciprocal of Strictly Negative Real Number... |
proofwiki-10612 | Cosine in terms of Tangent | {{begin-eqn}}
{{eqn | l = \cos x
| r = +\frac 1 {\sqrt {1 + \tan^2 x} }
| c = if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \cos x
| r = -\frac 1 {\sqrt {1 + \tan^2 x} }
| c = if there exists an integer $n$ such that $... | {{begin-eqn}}
{{eqn | l = \sec^2 x - \tan^2 x
| r = 1
| c = Difference of Squares of Secant and Tangent
}}
{{eqn | ll= \leadsto
| l = \sec^2 x
| r = 1 + \tan ^2 x
}}
{{eqn | ll= \leadsto
| l = \frac 1 {\cos^2 x}
| r = 1 + \tan^2 x
| c = Secant is Reciprocal of Cosine
}}
{{eqn |... | {{begin-eqn}}
{{eqn | l = \cos x
| r = +\frac 1 {\sqrt {1 + \tan^2 x} }
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$
}}
{{eqn | l = \cos x
| r = -\frac 1 {\sqrt {1 + \tan^2 x} }
| c = if there exists an ... | {{begin-eqn}}
{{eqn | l = \sec^2 x - \tan^2 x
| r = 1
| c = [[Difference of Squares of Secant and Tangent]]
}}
{{eqn | ll= \leadsto
| l = \sec^2 x
| r = 1 + \tan ^2 x
}}
{{eqn | ll= \leadsto
| l = \frac 1 {\cos^2 x}
| r = 1 + \tan^2 x
| c = [[Secant is Reciprocal of Cosine]]
}}... | Cosine in terms of Tangent | https://proofwiki.org/wiki/Cosine_in_terms_of_Tangent | https://proofwiki.org/wiki/Cosine_in_terms_of_Tangent | [
"Cosine Function",
"Tangent Function"
] | [
"Definition:Integer",
"Definition:Integer"
] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Secant is Reciprocal of Cosine",
"Sign of Cosine",
"Definition:Integer",
"Definition:Integer"
] |
proofwiki-10613 | Tangent in terms of Secant | {{begin-eqn}}
{{eqn | l = \tan x
| r = +\sqrt {\sec^2 x - 1}
| c = if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
}}
{{eqn | l = \tan x
| r = -\sqrt {\sec^2 x - 1}
| c = if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1}... | {{begin-eqn}}
{{eqn | l = \sec^2 x - \tan^2 x
| r = 1
| c = Difference of Squares of Secant and Tangent
}}
{{eqn | ll= \leadsto
| l = \tan^2 x
| r = \sec^2 x - 1
}}
{{eqn | ll= \leadsto
| l = \tan x
| r = \pm \sqrt {\sec^2 x - 1}
}}
{{end-eqn}}
Also, from Sign of Tangent:
:If there e... | {{begin-eqn}}
{{eqn | l = \tan x
| r = +\sqrt {\sec^2 x - 1}
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$
}}
{{eqn | l = \tan x
| r = -\sqrt {\sec^2 x - 1}
| c = if there exists an [[Definition:Integer|integer]] $n$ such that $\p... | {{begin-eqn}}
{{eqn | l = \sec^2 x - \tan^2 x
| r = 1
| c = [[Difference of Squares of Secant and Tangent]]
}}
{{eqn | ll= \leadsto
| l = \tan^2 x
| r = \sec^2 x - 1
}}
{{eqn | ll= \leadsto
| l = \tan x
| r = \pm \sqrt {\sec^2 x - 1}
}}
{{end-eqn}}
Also, from [[Sign of Tangent]]:
:... | Tangent in terms of Secant | https://proofwiki.org/wiki/Tangent_in_terms_of_Secant | https://proofwiki.org/wiki/Tangent_in_terms_of_Secant | [
"Tangent Function",
"Secant Function"
] | [
"Definition:Integer",
"Definition:Integer"
] | [
"Sum of Squares of Sine and Cosine/Corollary 1",
"Sign of Tangent",
"Definition:Integer",
"Definition:Integer"
] |
proofwiki-10614 | Union of Left-Total Relations is Left-Total | Let $S_1, S_2, T_1, T_2$ be sets or classes.
Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be left-total relations.
Then $\RR_1 \cup \RR_2$ is left-total. | Let both $\RR_1$ and $\RR_2$ be left-total.
Let $\RR = \RR_1 \cup \RR_2$.
Let $s \in S_1 \cup S_2$.
By the definition of union:
:$s \in S_1 \lor s \in S_2$
Thus $s \in S_i$ for $i \in \set {1, 2}$.
By definition of left-total relation, there is a $t \in T_i$ such that $\tuple {s, t} \in \RR_i$.
We have that $\RR$ is a ... | Let $S_1, S_2, T_1, T_2$ be [[Definition:Set|sets]] or [[Definition:Class (Class Theory)|classes]].
Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be [[Definition:Left-Total Relation|left-total relations]].
Then $\RR_1 \cup \RR_2$ is [[Definition:Left-Total Relation|left-total]]. | Let both $\RR_1$ and $\RR_2$ be [[Definition:Left-Total Relation|left-total]].
Let $\RR = \RR_1 \cup \RR_2$.
Let $s \in S_1 \cup S_2$.
By the definition of [[Definition:Set Union|union]]:
:$s \in S_1 \lor s \in S_2$
Thus $s \in S_i$ for $i \in \set {1, 2}$.
By definition of [[Definition:Left-Total Relation|left-to... | Union of Left-Total Relations is Left-Total | https://proofwiki.org/wiki/Union_of_Left-Total_Relations_is_Left-Total | https://proofwiki.org/wiki/Union_of_Left-Total_Relations_is_Left-Total | [
"Relation Theory"
] | [
"Definition:Set",
"Definition:Class (Class Theory)",
"Definition:Left-Total Relation",
"Definition:Left-Total Relation"
] | [
"Definition:Left-Total Relation",
"Definition:Set Union",
"Definition:Left-Total Relation",
"Definition:Subset/Superset",
"Union is Smallest Superset"
] |
proofwiki-10615 | Union of Inverse of Relations is Inverse of their Union | For $i \in \set {1, 2}$, let $\RR_i \subseteq S_i \times T_i$ be relations on $S_i \times T_i$.
Let ${\RR_i}^{-1} \subseteq T_i \times S_i$ be the inverse of $\RR_i$.
Then:
:${\RR_1}^{-1} \cup {\RR_2}^{-1} = \paren {\RR_1 \cup \RR_2}^{-1}$ | Let $\tuple {t, s} \in {\RR_1}^{-1} \cup {\RR_2}^{-1}$.
By definition of union:
:$\tuple {t, s} \in {\RR_1}^{-1} \lor \tuple {t, s} \in {\RR_2}^{-1}$.
For $i \in \set {1, 2}$, let $\tuple {t, s} \in {\RR_i}^{-1}$.
By definition of inverse:
:$\tuple {s, t} \in \RR_i$
That is:
:$\tuple {s, t} \in \RR_1 \lor \tuple {s, t}... | For $i \in \set {1, 2}$, let $\RR_i \subseteq S_i \times T_i$ be [[Definition:Relation|relations]] on $S_i \times T_i$.
Let ${\RR_i}^{-1} \subseteq T_i \times S_i$ be the [[Definition:Inverse Relation|inverse]] of $\RR_i$.
Then:
:${\RR_1}^{-1} \cup {\RR_2}^{-1} = \paren {\RR_1 \cup \RR_2}^{-1}$ | Let $\tuple {t, s} \in {\RR_1}^{-1} \cup {\RR_2}^{-1}$.
By definition of [[Definition:Set Union|union]]:
:$\tuple {t, s} \in {\RR_1}^{-1} \lor \tuple {t, s} \in {\RR_2}^{-1}$.
For $i \in \set {1, 2}$, let $\tuple {t, s} \in {\RR_i}^{-1}$.
By definition of [[Definition:Inverse_Relation|inverse]]:
:$\tuple {s, t} \... | Union of Inverse of Relations is Inverse of their Union | https://proofwiki.org/wiki/Union_of_Inverse_of_Relations_is_Inverse_of_their_Union | https://proofwiki.org/wiki/Union_of_Inverse_of_Relations_is_Inverse_of_their_Union | [
"Inverse Relations",
"Set Union"
] | [
"Definition:Relation",
"Definition:Inverse Relation"
] | [
"Definition:Set Union",
"Definition:Inverse_Relation",
"Definition:Set Union",
"Definition:Inverse_Relation",
"Category:Inverse Relations",
"Category:Set Union"
] |
proofwiki-10616 | Condition for Darboux Integrability | Let $\closedint a b$ be a closed real interval.
Let $f$ be a bounded real function defined on $\closedint a b$.
Then $f$ is Darboux integrable {{iff}}:
:for every $\epsilon \in \R_{>0}$, there exists a finite subdivision $S$ of $\closedint a b$ such that $\map U S - \map L S < \epsilon$
where
:$\map U S$ is the upper D... | === Necessary Condition ===
Let $f$ be Darboux integrable.
Let $\epsilon \in \R_{>0}$ be given.
It is to be proved that a finite subdivision $S$ of $\closedint a b$ exists such that:
:$\map U S - \map L S < \epsilon$
As $f$ is Darboux integrable:
:$\ds \int_a^b \map f x \rd x$ exists.
By the definition of the Darboux i... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $f$ be a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Real Function|real function]] defined on $\closedint a b$.
Then $f$ is [[Definition:Darboux Integrable Function|Darboux integrable]] {{iff}}:
:for every $\... | === Necessary Condition ===
Let $f$ be [[Definition:Darboux Integrable Function|Darboux integrable]].
Let $\epsilon \in \R_{>0}$ be given.
It is to be proved that a [[Definition:Finite Subdivision|finite subdivision]] $S$ of $\closedint a b$ exists such that:
:$\map U S - \map L S < \epsilon$
As $f$ is [[Definitio... | Condition for Darboux Integrability | https://proofwiki.org/wiki/Condition_for_Darboux_Integrability | https://proofwiki.org/wiki/Condition_for_Darboux_Integrability | [
"Integral Calculus",
"Darboux Integrable Functions"
] | [
"Definition:Real Interval/Closed",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Definition:Darboux Integrable Function",
"Definition:Subdivision of Interval/Finite",
"Definition:Upper Darboux Sum",
"Definition:Lower Darboux Sum"
] | [
"Definition:Darboux Integrable Function",
"Definition:Subdivision of Interval/Finite",
"Definition:Darboux Integrable Function",
"Definition:Definite Integral/Darboux",
"Definition:Lower Darboux Integral",
"Definition:Lower Darboux Integral",
"Definition:Lower Darboux Sum",
"Definition:Subdivision of ... |
proofwiki-10617 | Open Set minus Closed Set is Open | Let $T = \struct {S, \tau}$ be a topological space.
For $A \subseteq S$ denote by $\relcomp S A$ the relative complement of $A$ in $S$.
Let $U \in \tau$ and $\relcomp S V \in \tau$.
Then:
:$U \setminus V \in \tau$
and:
:$\relcomp S {V \setminus U} \in \tau$ | From Set Difference as Intersection with Relative Complement:
:$U \setminus V = U \cap \relcomp S V$
Since $\tau$ is a topology:
:$U, \relcomp S V \in \tau \implies U \cap \relcomp S V \in \tau \implies U \setminus V \in \tau$
The other statement follows {{mutatis}}.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
For $A \subseteq S$ denote by $\relcomp S A$ the [[Definition:Relative Complement|relative complement]] of $A$ in $S$.
Let $U \in \tau$ and $\relcomp S V \in \tau$.
Then:
:$U \setminus V \in \tau$
and:
:$\relcomp S {V \setminus U} ... | From [[Set Difference as Intersection with Relative Complement]]:
:$U \setminus V = U \cap \relcomp S V$
Since $\tau$ is a [[Definition:Topology|topology]]:
:$U, \relcomp S V \in \tau \implies U \cap \relcomp S V \in \tau \implies U \setminus V \in \tau$
The other statement follows {{mutatis}}.
{{qed}} | Open Set minus Closed Set is Open | https://proofwiki.org/wiki/Open_Set_minus_Closed_Set_is_Open | https://proofwiki.org/wiki/Open_Set_minus_Closed_Set_is_Open | [
"Open Sets",
"Closed Sets"
] | [
"Definition:Topological Space",
"Definition:Relative Complement"
] | [
"Set Difference as Intersection with Relative Complement",
"Definition:Topology"
] |
proofwiki-10618 | Union of Right-Total Relations is Right-Total | Let $S_1, S_2, T_1, T_2$ be sets or classes.
Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be right-total relations.
Then $\RR_1 \cup \RR_2$ is right-total. | Define the predicates $L$ and $R$ by:
:$\map L X \iff \text {$X$ is left-total}$
:$\map R X \iff \text {$X$ is right-total}$
{{begin-eqn}}
{{eqn | l = \map R {\RR_1} \land \map R {\RR_2}
| o = \leadsto
| r = \map L {\RR_1^{-1} } \land \map L {\RR_2^{-1} }
| c = Inverse of Right-Total Relation is Left-... | Let $S_1, S_2, T_1, T_2$ be [[Definition:Set|sets]] or [[Definition:Class (Class Theory)|classes]].
Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be [[Definition:Right-Total Relation|right-total]] [[Definition:Relation|relations]].
Then $\RR_1 \cup \RR_2$ is [[Definition:Right-Total Relat... | Define the [[Definition:Predicate|predicates]] $L$ and $R$ by:
:$\map L X \iff \text {$X$ is left-total}$
:$\map R X \iff \text {$X$ is right-total}$
{{begin-eqn}}
{{eqn | l = \map R {\RR_1} \land \map R {\RR_2}
| o = \leadsto
| r = \map L {\RR_1^{-1} } \land \map L {\RR_2^{-1} }
| c = [[Inverse of ... | Union of Right-Total Relations is Right-Total | https://proofwiki.org/wiki/Union_of_Right-Total_Relations_is_Right-Total | https://proofwiki.org/wiki/Union_of_Right-Total_Relations_is_Right-Total | [
"Relation Theory"
] | [
"Definition:Set",
"Definition:Class (Class Theory)",
"Definition:Right-Total Relation",
"Definition:Relation",
"Definition:Right-Total Relation"
] | [
"Definition:Predicate",
"Inverse of Right-Total Relation is Left-Total",
"Union of Left-Total Relations is Left-Total",
"Union of Inverse of Relations is Inverse of their Union",
"Inverse of Right-Total Relation is Left-Total"
] |
proofwiki-10619 | Primitive of x over a x + b squared by p x + q/Corollary | :$\ds \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } + \frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} }
| r = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } - \frac b {a \paren {a x + b} } } + C
| c = Primitive of $\dfrac x {\paren {a x + b}^2 \paren {p x + q} }$
}}
{{eqn | r = \frac 1... | :$\ds \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } + \frac x {a x + b} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} }
| r = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } - \frac b {a \paren {a x + b} } } + C
| c = [[Primitive of x over a x + b squared by p x + q|Primitive of $\dfrac x {\paren {a x ... | Primitive of x over a x + b squared by p x + q/Corollary | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared_by_p_x_+_q/Corollary | https://proofwiki.org/wiki/Primitive_of_x_over_a_x_+_b_squared_by_p_x_+_q/Corollary | [
"Primitive of x over a x + b squared by p x + q",
"Primitives involving a x + b and p x + q"
] | [] | [
"Primitive of x over a x + b squared by p x + q",
"Category:Primitive of x over a x + b squared by p x + q",
"Category:Primitives involving a x + b and p x + q"
] |
proofwiki-10620 | Arctangent of Imaginary Number | Let $x$ belong to the open real interval $\openint {-1} 1$.
Then:
:$\map {\tan^{-1} } {i x} = \dfrac i 2 \map \ln {\dfrac {1 + x} {1 - x} }$
where $\tan$ is the complex tangent function, $\ln$ is the real natural logarithm, and $i$ is the imaginary unit. | Let $y = \map {\tan^{-1} } {i x}$.
Let $x = \tanh \theta$, then $\theta = \tanh^{-1} x$.
{{begin-eqn}}
{{eqn | l = \tan y
| r = i x
| c =
}}
{{eqn | l = \tan y
| r = i \tanh \theta
| c =
}}
{{eqn | l = \tan y
| r = \map \tan {i \theta}
| c = Hyperbolic Tangent in terms of Tangent
}... | Let $x$ belong to the [[Definition:Open Real Interval|open real interval]] $\openint {-1} 1$.
Then:
:$\map {\tan^{-1} } {i x} = \dfrac i 2 \map \ln {\dfrac {1 + x} {1 - x} }$
where $\tan$ is the [[Definition:Complex Tangent Function|complex tangent function]], $\ln$ is the [[Definition:Real Natural Logarithm|real natu... | Let $y = \map {\tan^{-1} } {i x}$.
Let $x = \tanh \theta$, then $\theta = \tanh^{-1} x$.
{{begin-eqn}}
{{eqn | l = \tan y
| r = i x
| c =
}}
{{eqn | l = \tan y
| r = i \tanh \theta
| c =
}}
{{eqn | l = \tan y
| r = \map \tan {i \theta}
| c = [[Hyperbolic Tangent in terms of Tange... | Arctangent of Imaginary Number | https://proofwiki.org/wiki/Arctangent_of_Imaginary_Number | https://proofwiki.org/wiki/Arctangent_of_Imaginary_Number | [
"Complex Numbers",
"Tangent Function"
] | [
"Definition:Real Interval/Open",
"Definition:Tangent Function/Complex",
"Definition:Natural Logarithm/Positive Real",
"Definition:Complex Number/Imaginary Unit"
] | [
"Hyperbolic Tangent in terms of Tangent",
"Category:Complex Numbers",
"Category:Tangent Function"
] |
proofwiki-10621 | Equivalence of Definitions of Real Area Hyperbolic Tangent | Let $S$ denote the open real interval:
:$S := \openint {-1} 1$
{{TFAE|def = Real Area Hyperbolic Tangent}} | === Definition 1 implies Definition 2 ===
Let $x = \tanh y$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {e^{2 y} - 1} {e^{2 y} + 1}
| c = {{Defof|Hyperbolic Tangent|index = 3}}
}}
{{eqn | ll= \leadsto
| l = x e^{2 y} + x
| r = e^{2 y} - 1
| c =
}}
{{eqn | ll= \leadsto
| l = e^{2 ... | Let $S$ denote the [[Definition:Open Real Interval|open real interval]]:
:$S := \openint {-1} 1$
{{TFAE|def = Real Area Hyperbolic Tangent}} | === Definition 1 implies Definition 2 ===
Let $x = \tanh y$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \frac {e^{2 y} - 1} {e^{2 y} + 1}
| c = {{Defof|Hyperbolic Tangent|index = 3}}
}}
{{eqn | ll= \leadsto
| l = x e^{2 y} + x
| r = e^{2 y} - 1
| c =
}}
{{eqn | ll= \leadsto
| l = e^{... | Equivalence of Definitions of Real Area Hyperbolic Tangent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Area_Hyperbolic_Tangent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Area_Hyperbolic_Tangent | [
"Inverse Hyperbolic Tangent"
] | [
"Definition:Real Interval/Open"
] | [] |
proofwiki-10622 | Square Root is Strictly Increasing | The positive square root function is strictly increasing, that is:
:$\forall x, y \in \R_{>0}: x < y \implies \sqrt x < \sqrt y$ | Let $x$ and $y$ be positive real numbers such that $x < y$.
{{AimForCont}} $\sqrt x \ge \sqrt y$.
{{begin-eqn}}
{{eqn | n = 1
| l = \sqrt x
| o = \ge
| r = \sqrt y
| c =
}}
{{eqn | n = 2
| l = \sqrt x
| o = \ge
| r = \sqrt y
| c =
}}
{{eqn | l = x
| o = \ge
... | The [[Definition:Positive Square Root|positive square root function]] is [[Definition:Strictly Increasing Real Function|strictly increasing]], that is:
:$\forall x, y \in \R_{>0}: x < y \implies \sqrt x < \sqrt y$ | Let $x$ and $y$ be [[Definition:Positive/Real Number|positive real numbers]] such that $x < y$.
{{AimForCont}} $\sqrt x \ge \sqrt y$.
{{begin-eqn}}
{{eqn | n = 1
| l = \sqrt x
| o = \ge
| r = \sqrt y
| c =
}}
{{eqn | n = 2
| l = \sqrt x
| o = \ge
| r = \sqrt y
| c =
}... | Square Root is Strictly Increasing | https://proofwiki.org/wiki/Square_Root_is_Strictly_Increasing | https://proofwiki.org/wiki/Square_Root_is_Strictly_Increasing | [
"Real Numbers",
"Square Roots",
"Examples of Strictly Increasing Real Functions"
] | [
"Definition:Square Root/Positive",
"Definition:Strictly Increasing/Real Function"
] | [
"Definition:Positive/Real Number",
"Proof by Contradiction",
"Category:Real Numbers",
"Category:Square Roots",
"Category:Examples of Strictly Increasing Real Functions"
] |
proofwiki-10623 | Minimum of Real Hyperbolic Cosine Function | Let $x$ be a real number.
Then:
:$\cosh x \ge 1$
where $\cosh$ denotes the hyperbolic cosine function. | {{begin-eqn}}
{{eqn | l = \cosh^2 x - \sinh^2 x
| r = 1
| c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | ll= \leadsto
| l = \cosh^2 x
| r = 1 + \sinh^2 x
| c =
}}
{{eqn | o = \ge
| r = 1
| c = Square of Real Number is Non-Negative
}}
{{end-eqn}}
Furthermore... | Let $x$ be a [[Definition:Real Number|real number]].
Then:
:$\cosh x \ge 1$
where $\cosh$ denotes the [[Definition:Hyperbolic Cosine|hyperbolic cosine function]]. | {{begin-eqn}}
{{eqn | l = \cosh^2 x - \sinh^2 x
| r = 1
| c = [[Difference of Squares of Hyperbolic Cosine and Sine]]
}}
{{eqn | ll= \leadsto
| l = \cosh^2 x
| r = 1 + \sinh^2 x
| c =
}}
{{eqn | o = \ge
| r = 1
| c = [[Square of Real Number is Non-Negative]]
}}
{{end-eqn}}
Fu... | Minimum of Real Hyperbolic Cosine Function | https://proofwiki.org/wiki/Minimum_of_Real_Hyperbolic_Cosine_Function | https://proofwiki.org/wiki/Minimum_of_Real_Hyperbolic_Cosine_Function | [
"Hyperbolic Cosine Function"
] | [
"Definition:Real Number",
"Definition:Hyperbolic Cosine"
] | [
"Difference of Squares of Hyperbolic Cosine and Sine",
"Square of Real Number is Non-Negative",
"Category:Hyperbolic Cosine Function"
] |
proofwiki-10624 | Exponential of Real Number is Strictly Positive | Let $x$ be a real number.
Let $\exp$ denote the (real) exponential function.
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the series definition of $\exp$.
That is, let:
:$\ds \exp x = \sum_{n \mathop = 0}^\infty \dfrac {x^n} {n!}$
First, suppose $0 < x$.
Then:
{{begin-eqn}}
{{eqn | l = 0
| o = <
| r = x^n
| c = Power Function is Strictly Increasing over Positive Reals: Natural Exponent
}}
{{eqn | ll= \... | Let $x$ be a [[Definition:Real Number|real number]].
Let $\exp$ denote the [[Definition:Real Exponential Function|(real) exponential function]].
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the [[Definition:Exponential Function/Real/Power Series Expansion|series definition of $\exp$]].
That is, let:
:$\ds \exp x = \sum_{n \mathop = 0}^\infty \dfrac {x^n} {n!}$
First, suppose $0 < x$.
Then:
{{begin-eqn}}
{{eqn | l = 0
| o = <
| r = x^n
| c = [[Power Function is Str... | Exponential of Real Number is Strictly Positive/Proof 1 | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive/Proof_1 | [
"Exponential Function",
"Exponential of Real Number is Strictly Positive"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Power Series Expansion",
"Power Function is Strictly Increasing over Positive Reals/Natural Exponent",
"Real Number Ordering is Compatible with Multiplication",
"Ordering of Series of Ordered Sequences",
"Definition:Strictly Positive Real Function",
"Exponential of Ze... |
proofwiki-10625 | Exponential of Real Number is Strictly Positive | Let $x$ be a real number.
Let $\exp$ denote the (real) exponential function.
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the limit definition of $\exp$.
That is, let:
:$\ds \exp x = \lim_{n \mathop \to \infty} \map {f_n} x$
where $\map {f_n} x = \paren {1 + \dfrac x n}^n$
First, fix $x \in \R$.
Let $N = \ceiling {\size x}$, where $\ceiling {\, \cdot \,}$ denotes the ceiling function.
Then:
{{begin-eqn}}
{{eqn | l = \ex... | Let $x$ be a [[Definition:Real Number|real number]].
Let $\exp$ denote the [[Definition:Real Exponential Function|(real) exponential function]].
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the [[Definition:Exponential Function/Real/Limit of Sequence|limit definition of $\exp$]].
That is, let:
:$\ds \exp x = \lim_{n \mathop \to \infty} \map {f_n} x$
where $\map {f_n} x = \paren {1 + \dfrac x n}^n$
First, fix $x \in \R$.
Let $N = \ceiling {\size x}$, where $\ceiling {\, \cdot \,}$ de... | Exponential of Real Number is Strictly Positive/Proof 2 | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive/Proof_2 | [
"Exponential Function",
"Exponential of Real Number is Strictly Positive"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Limit of Sequence",
"Definition:Ceiling Function",
"Tail of Convergent Sequence",
"Exponential Sequence is Eventually Increasing",
"Limit of Bounded Convergent Sequence is Bounded",
"Exponential Sequence is Eventually Increasing",
"Exponential Sequence is Eventually... |
proofwiki-10626 | Exponential of Real Number is Strictly Positive | Let $x$ be a real number.
Let $\exp$ denote the (real) exponential function.
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the definition of $\exp x$ as the unique continuous extension of $e^x$.
Since $e > 0$, the result follows immediately from Power of Positive Real Number is Positive over Rationals.
{{qed}} | Let $x$ be a [[Definition:Real Number|real number]].
Let $\exp$ denote the [[Definition:Real Exponential Function|(real) exponential function]].
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the [[Definition:Exponential Function/Real/Extension of Rational Exponential|definition of $\exp x$ as the unique continuous extension of $e^x$]].
Since $e > 0$, the result follows immediately from [[Power of Positive Real Number is Positive/Rational Number|Power of Positive Real Number is Positive... | Exponential of Real Number is Strictly Positive/Proof 3 | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive/Proof_3 | [
"Exponential Function",
"Exponential of Real Number is Strictly Positive"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Extension of Rational Exponential",
"Power of Positive Real Number is Positive/Rational Number"
] |
proofwiki-10627 | Exponential of Real Number is Strictly Positive | Let $x$ be a real number.
Let $\exp$ denote the (real) exponential function.
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the definition of $\exp$ as the inverse mapping of extension of $\ln$, where $\ln$ denotes the natural logarithm.
Recall that the domain of $\ln$ is $\R_{>0}$.
From the definition of inverse mapping, the image of $\exp$ is the domain of $\ln$.
That is, the image of $\exp$ is $\R_{>0}$.
Hence the resu... | Let $x$ be a [[Definition:Real Number|real number]].
Let $\exp$ denote the [[Definition:Real Exponential Function|(real) exponential function]].
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the [[Definition:Exponential Function/Real/Inverse of Natural Logarithm|definition of $\exp$ as the inverse mapping of extension of $\ln$]], where $\ln$ denotes the [[Definition:Natural Logarithm|natural logarithm]].
Recall that the [[Definition:Domain of Mapping|domain]] of $\ln$ is $\R_{>0}$.
Fr... | Exponential of Real Number is Strictly Positive/Proof 4 | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive/Proof_4 | [
"Exponential Function",
"Exponential of Real Number is Strictly Positive"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Inverse of Natural Logarithm",
"Definition:Natural Logarithm",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Inverse of Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Image (Set Theory)/Mappin... |
proofwiki-10628 | Exponential of Real Number is Strictly Positive | Let $x$ be a real number.
Let $\exp$ denote the (real) exponential function.
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the definition of $\exp$ as the solution to an initial value problem.
That is, suppose $\exp$ satisfies:
:$ (1): \quad D_x \exp x = \exp x$
:$ (2): \quad \map \exp 0 = 1$
on $\R$.
=== Lemma ===
{{:Exponential of Real Number is Strictly Positive/Proof 5/Lemma}}{{qed|lemma}}
{{AimForCont}} that $\exist... | Let $x$ be a [[Definition:Real Number|real number]].
Let $\exp$ denote the [[Definition:Real Exponential Function|(real) exponential function]].
Then:
:$\forall x \in \R : \exp x > 0$ | This proof assumes the [[Definition:Exponential Function/Real/Differential Equation|definition of $\exp$ as the solution to an initial value problem]].
That is, suppose $\exp$ satisfies:
:$ (1): \quad D_x \exp x = \exp x$
:$ (2): \quad \map \exp 0 = 1$
on $\R$.
=== [[Exponential of Real Number is Strictly Positive/P... | Exponential of Real Number is Strictly Positive/Proof 5 | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive | https://proofwiki.org/wiki/Exponential_of_Real_Number_is_Strictly_Positive/Proof_5 | [
"Exponential Function",
"Exponential of Real Number is Strictly Positive"
] | [
"Definition:Real Number",
"Definition:Exponential Function/Real"
] | [
"Definition:Exponential Function/Real/Differential Equation",
"Exponential of Real Number is Strictly Positive/Proof 5/Lemma",
"Intermediate Value Theorem",
"Definition:Contradiction",
"Exponential of Real Number is Strictly Positive/Proof 5/Lemma"
] |
proofwiki-10629 | Real Power of Strictly Positive Real Number is Strictly Positive | Let $x$ be a strictly positive real number.
Let $y$ be a real number.
Then:
:$x^y > 0$
where $x^y$ denotes $x$ raised to the $y$th power. | From the definition of power:
:$x^y = \exp \left({y \ln x}\right)$
From Exponential of Real Number is Strictly Positive:
:$x^y = \exp \left({y \ln x}\right) > 0$
{{qed}}
Category:Real Analysis
oa8bjlxwrg5kas1mcyixm085z7x58vm | Let $x$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $y$ be a [[Definition:Real Number|real number]].
Then:
:$x^y > 0$
where $x^y$ denotes $x$ [[Definition:Power to Real Number|raised to the $y$th power]]. | From the definition of [[Definition:Power to Real Number|power]]:
:$x^y = \exp \left({y \ln x}\right)$
From [[Exponential of Real Number is Strictly Positive]]:
:$x^y = \exp \left({y \ln x}\right) > 0$
{{qed}}
[[Category:Real Analysis]]
oa8bjlxwrg5kas1mcyixm085z7x58vm | Real Power of Strictly Positive Real Number is Strictly Positive | https://proofwiki.org/wiki/Real_Power_of_Strictly_Positive_Real_Number_is_Strictly_Positive | https://proofwiki.org/wiki/Real_Power_of_Strictly_Positive_Real_Number_is_Strictly_Positive | [
"Real Analysis"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real Number",
"Definition:Power (Algebra)/Real Number"
] | [
"Definition:Power (Algebra)/Real Number",
"Exponential of Real Number is Strictly Positive",
"Category:Real Analysis"
] |
proofwiki-10630 | Derivative of Power of Function | Let $\map u x$ be a differentiable real function of $x$.
Let $n$ be a real number such that $n \ne -1$.
Then:
:$\map {\dfrac \d {\d x} } {\map u x^n} = n \map u x^{n - 1} \map {\dfrac \d {\d x} } {\map u x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\map u x^n}
| r = \map {\frac \d {\d u} } {\map u x^n} \map {\frac \d {\d x} } {\map u x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = n \map u x^{n - 1} \map {\frac {\d u} {\d x} } {\map u x}
| c = Derivative of Hyperbolic Sine
}}
{{end-eqn}}
{{qe... | Let $\map u x$ be a [[Definition:Differentiable Real Function|differentiable real function]] of $x$.
Let $n$ be a [[Definition:Real Number|real number]] such that $n \ne -1$.
Then:
:$\map {\dfrac \d {\d x} } {\map u x^n} = n \map u x^{n - 1} \map {\dfrac \d {\d x} } {\map u x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\map u x^n}
| r = \map {\frac \d {\d u} } {\map u x^n} \map {\frac \d {\d x} } {\map u x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = n \map u x^{n - 1} \map {\frac {\d u} {\d x} } {\map u x}
| c = [[Derivative of Hyperbolic Sine]]
}}
{{end-eq... | Derivative of Power of Function/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Power_of_Function | https://proofwiki.org/wiki/Derivative_of_Power_of_Function/Proof_1 | [
"Derivative of Power of Function",
"Differential Calculus"
] | [
"Definition:Differentiable Mapping/Real Function",
"Definition:Real Number"
] | [
"Derivative of Composite Function",
"Derivative of Hyperbolic Sine"
] |
proofwiki-10631 | Derivative of Power of Function | Let $\map u x$ be a differentiable real function of $x$.
Let $n$ be a real number such that $n \ne -1$.
Then:
:$\map {\dfrac \d {\d x} } {\map u x^n} = n \map u x^{n - 1} \map {\dfrac \d {\d x} } {\map u x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\map u x^n}
| r = \lim_{h \mathop \to 0} \frac {\paren {\map u {x + h} }^n - \paren {\map u x}^n} h
| c =
}}
{{eqn | r = \paren {\map u x}^n \lim_{h \mathop \to 0} \frac {\paren {\frac {\map u {x + h} } {\map u x} }^n - 1} h
| c = Power of Product
}... | Let $\map u x$ be a [[Definition:Differentiable Real Function|differentiable real function]] of $x$.
Let $n$ be a [[Definition:Real Number|real number]] such that $n \ne -1$.
Then:
:$\map {\dfrac \d {\d x} } {\map u x^n} = n \map u x^{n - 1} \map {\dfrac \d {\d x} } {\map u x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\map u x^n}
| r = \lim_{h \mathop \to 0} \frac {\paren {\map u {x + h} }^n - \paren {\map u x}^n} h
| c =
}}
{{eqn | r = \paren {\map u x}^n \lim_{h \mathop \to 0} \frac {\paren {\frac {\map u {x + h} } {\map u x} }^n - 1} h
| c = [[Power of Product... | Derivative of Power of Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Power_of_Function | https://proofwiki.org/wiki/Derivative_of_Power_of_Function/Proof_2 | [
"Derivative of Power of Function",
"Differential Calculus"
] | [
"Definition:Differentiable Mapping/Real Function",
"Definition:Real Number"
] | [
"Exponent Combination Laws/Power of Product",
"Derivative of Exponential at Zero",
"Derivative of Logarithm at One",
"Exponent Combination Laws/Product of Powers"
] |
proofwiki-10632 | Natural Logarithm of e is 1 | :$\ln e = 1$ | The definition of the Euler's number as the Base of Logarithm will be used.
Then the result follows directly.
{{qed}} | :$\ln e = 1$ | The [[Definition:Euler's Number/Base of Logarithm|definition of the Euler's number as the Base of Logarithm]] will be used.
Then the result follows directly.
{{qed}} | Natural Logarithm of e is 1 | https://proofwiki.org/wiki/Natural_Logarithm_of_e_is_1 | https://proofwiki.org/wiki/Natural_Logarithm_of_e_is_1 | [
"Examples of Natural Logarithms"
] | [] | [
"Definition:Euler's Number/Base of Logarithm"
] |
proofwiki-10633 | Real Area Hyperbolic Cosine is Strictly Increasing | The real area hyperbolic cosine function is strictly increasing, that is:
:$\forall x, y \ge 1 : x < y \implies \arcosh x < \arcosh y$ | {{begin-eqn}}
{{eqn | n = 1
| l = x
| o = <
| r = y
| c = Assumption
}}
{{eqn | ll= \leadsto
| l = x^2
| o = <
| r = y^2
| c = {{Real-number-axiom|O2}}
}}
{{eqn | ll= \leadsto
| l = x^2 - 1
| o = <
| r = y^2 - 1
| c =
}}
{{eqn | n = 2
| ll= ... | The [[Definition:Real Area Hyperbolic Cosine|real area hyperbolic cosine]] function is [[Definition:Strictly Increasing Real Function|strictly increasing]], that is:
:$\forall x, y \ge 1 : x < y \implies \arcosh x < \arcosh y$ | {{begin-eqn}}
{{eqn | n = 1
| l = x
| o = <
| r = y
| c = Assumption
}}
{{eqn | ll= \leadsto
| l = x^2
| o = <
| r = y^2
| c = {{Real-number-axiom|O2}}
}}
{{eqn | ll= \leadsto
| l = x^2 - 1
| o = <
| r = y^2 - 1
| c =
}}
{{eqn | n = 2
| ll= ... | Real Area Hyperbolic Cosine is Strictly Increasing | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cosine_is_Strictly_Increasing | https://proofwiki.org/wiki/Real_Area_Hyperbolic_Cosine_is_Strictly_Increasing | [
"Inverse Hyperbolic Cosine",
"Examples of Strictly Increasing Real Functions"
] | [
"Definition:Inverse Hyperbolic Cosine/Real/Principal Branch",
"Definition:Strictly Increasing/Real Function"
] | [
"Square Root is Strictly Increasing",
"Category:Inverse Hyperbolic Cosine",
"Category:Examples of Strictly Increasing Real Functions"
] |
proofwiki-10634 | Laplace Transform of Function of Constant Multiple | Let $a \in \C$ or $\R$ be constant.
Then:
:$a \laptrans {\map f {a t} } = \map F {\dfrac s a}$ | {{begin-eqn}}
{{eqn | l = a \laptrans {\map f {a t} }
| r = a \int_0^{\to + \infty} e^{-s t} \map f {a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = a \paren {\frac 1 a} \int_0^{\to + \infty} e^{-s t} \map f {a t} \rd \paren {a t}
| c = Primitive of Function of Constant Multiple
}}
{{eqn |... | Let $a \in \C$ or $\R$ be [[Definition:Constant|constant]].
Then:
:$a \laptrans {\map f {a t} } = \map F {\dfrac s a}$ | {{begin-eqn}}
{{eqn | l = a \laptrans {\map f {a t} }
| r = a \int_0^{\to + \infty} e^{-s t} \map f {a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = a \paren {\frac 1 a} \int_0^{\to + \infty} e^{-s t} \map f {a t} \rd \paren {a t}
| c = [[Primitive of Function of Constant Multiple]]
}}
{{e... | Laplace Transform of Function of Constant Multiple | https://proofwiki.org/wiki/Laplace_Transform_of_Function_of_Constant_Multiple | https://proofwiki.org/wiki/Laplace_Transform_of_Function_of_Constant_Multiple | [
"Laplace Transform of Function of Constant Multiple",
"Properties of Laplace Transforms"
] | [
"Definition:Constant"
] | [
"Primitive of Function of Constant Multiple"
] |
proofwiki-10635 | Second Translation Property of Laplace Transforms | Let $g$ be the function defined as:
:$\map g t = \begin {cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end {cases}$
Then:
:$\laptrans {\map g t} = e^{-a s} \map F s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \exp \dfrac {-2 \pi s} 3 \laptrans {\cos t}
| c = Second Translation Property of Laplace Transforms
}}
{{eqn | r = \exp \dfrac {-2 \pi s} 3 \dfrac s {s^2 + 1}
| c = Laplace Transform of Cosine
}}
{{end-eqn}}
and the result follows.
{{qed}} | Let $g$ be the [[Definition:Function|function]] defined as:
:$\map g t = \begin {cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end {cases}$
Then:
:$\laptrans {\map g t} = e^{-a s} \map F s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \exp \dfrac {-2 \pi s} 3 \laptrans {\cos t}
| c = [[Second Translation Property of Laplace Transforms]]
}}
{{eqn | r = \exp \dfrac {-2 \pi s} 3 \dfrac s {s^2 + 1}
| c = [[Laplace Transform of Cosine]]
}}
{{end-eqn}}
and the result follows.
{{qed}} | Second Translation Property of Laplace Transforms/Examples/Example 2/Proof 1 | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms/Examples/Example_2/Proof_1 | [
"Second Translation Property of Laplace Transforms",
"Properties of Laplace Transforms",
"Laplace Transforms",
"Exponential Function"
] | [
"Definition:Function"
] | [
"Second Translation Property of Laplace Transforms",
"Laplace Transform of Cosine"
] |
proofwiki-10636 | Second Translation Property of Laplace Transforms | Let $g$ be the function defined as:
:$\map g t = \begin {cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end {cases}$
Then:
:$\laptrans {\map g t} = e^{-a s} \map F s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \int_0^\infty e^{-s t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\frac {-2 \pi s} 3} e^{-s t} \map f t \rd t + \int_{\frac {-2 \pi s} 3}^\infty e^{-s t} \map f t \rd t
| c =
}}
{{eqn | r = \int_0^{\frac {-2 \pi s} 3} e... | Let $g$ be the [[Definition:Function|function]] defined as:
:$\map g t = \begin {cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end {cases}$
Then:
:$\laptrans {\map g t} = e^{-a s} \map F s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \int_0^\infty e^{-s t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\frac {-2 \pi s} 3} e^{-s t} \map f t \rd t + \int_{\frac {-2 \pi s} 3}^\infty e^{-s t} \map f t \rd t
| c =
}}
{{eqn | r = \int_0^{\frac {-2 \pi s} 3} e... | Second Translation Property of Laplace Transforms/Examples/Example 2/Proof 2 | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms/Examples/Example_2/Proof_2 | [
"Second Translation Property of Laplace Transforms",
"Properties of Laplace Transforms",
"Laplace Transforms",
"Exponential Function"
] | [
"Definition:Function"
] | [
"Integration by Substitution",
"Laplace Transform of Cosine"
] |
proofwiki-10637 | Second Translation Property of Laplace Transforms | Let $g$ be the function defined as:
:$\map g t = \begin {cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end {cases}$
Then:
:$\laptrans {\map g t} = e^{-a s} \map F s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map f {t - a} }
| r = \int_0^{\to + \infty} e^{-s t} \map f {t - a} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to + \infty} e^{-s \paren {t - a} } e^{-a s} \map f {t - a} \rd \paren {t - a}
| c =
}}
{{eqn | r = e^{-a s} \int_0^{\to + \infty} ... | Let $g$ be the [[Definition:Function|function]] defined as:
:$\map g t = \begin {cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end {cases}$
Then:
:$\laptrans {\map g t} = e^{-a s} \map F s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map f {t - a} }
| r = \int_0^{\to + \infty} e^{-s t} \map f {t - a} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to + \infty} e^{-s \paren {t - a} } e^{-a s} \map f {t - a} \rd \paren {t - a}
| c =
}}
{{eqn | r = e^{-a s} \int_0^{\to + \infty} ... | Second Translation Property of Laplace Transforms/Proof 1 | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms/Proof_1 | [
"Second Translation Property of Laplace Transforms",
"Properties of Laplace Transforms",
"Laplace Transforms",
"Exponential Function"
] | [
"Definition:Function"
] | [] |
proofwiki-10638 | Second Translation Property of Laplace Transforms | Let $g$ be the function defined as:
:$\map g t = \begin {cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end {cases}$
Then:
:$\laptrans {\map g t} = e^{-a s} \map F s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map g t}
| r = \int_0^\infty e^{-s t} \map g t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^a e^{-s t} \map g t \rd t + \int_a^\infty e^{-s t} \map g t \rd t
| c =
}}
{{eqn | r = \int_0^a 0 \times e^{-s t} \rd t + \int_a^\infty e^{-s t} \map f {t... | Let $g$ be the [[Definition:Function|function]] defined as:
:$\map g t = \begin {cases} \map f {t - a} & : t > a \\ 0 & : t \le a \end {cases}$
Then:
:$\laptrans {\map g t} = e^{-a s} \map F s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map g t}
| r = \int_0^\infty e^{-s t} \map g t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^a e^{-s t} \map g t \rd t + \int_a^\infty e^{-s t} \map g t \rd t
| c =
}}
{{eqn | r = \int_0^a 0 \times e^{-s t} \rd t + \int_a^\infty e^{-s t} \map f {t... | Second Translation Property of Laplace Transforms/Proof 2 | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms/Proof_2 | [
"Second Translation Property of Laplace Transforms",
"Properties of Laplace Transforms",
"Laplace Transforms",
"Exponential Function"
] | [
"Definition:Function"
] | [
"Integration by Substitution"
] |
proofwiki-10639 | Seifert-van Kampen Theorem | The functor $\pi_1 : \mathbf{Top_\bullet} \to \mathbf{Grp}$ preserves pushouts of inclusions. | Let $\struct {X, \tau}$ be a topological space.
Let $U_1, U_2 \in \tau$ such that:
: $U_1 \cup U_2 = X$
: $U_1 \cap U_2 \ne \O$ is connected
Let $\ast \in U_1 \cap U_2$.
Let:
: $i_k : U_1 \cap U_2 \hookrightarrow U_k$
: $j_k : U_k \hookrightarrow U_1 \cup U_2$
be inclusions.
For simplicity, let:
:$\map {\pi_1} X = \map... | The functor $\pi_1 : \mathbf{Top_\bullet} \to \mathbf{Grp}$ preserves pushouts of inclusions. | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U_1, U_2 \in \tau$ such that:
: $U_1 \cup U_2 = X$
: $U_1 \cap U_2 \ne \O$ is connected
Let $\ast \in U_1 \cap U_2$.
Let:
: $i_k : U_1 \cap U_2 \hookrightarrow U_k$
: $j_k : U_k \hookrightarrow U_1 \cup U_2$
be inclusions.
For sim... | Seifert-van Kampen Theorem | https://proofwiki.org/wiki/Seifert-van_Kampen_Theorem | https://proofwiki.org/wiki/Seifert-van_Kampen_Theorem | [
"Category Theory"
] | [] | [
"Definition:Topological Space",
"Definition:Amalgamated Free Product",
"Category:Category Theory"
] |
proofwiki-10640 | Functions of Independent Random Variables are Independent | Let $X$ and $Y$ be independent random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $g$ and $h$ be real-valued functions defined on the codomains of $X$ and $Y$ respectively.
Then $\map g X$ and $\map h Y$ are independent random variables. | Let $A$ and $B$ be subsets of the real numbers $\R$.
Let $g^{-1} \sqbrk A$ and $h^{-1} \sqbrk B$ denote the preimages of $A$ and $B$ under $g$ and $h$ respectively.
Applying the definition of independent random variables:
{{begin-eqn}}
{{eqn | l = \map \Pr {\map g X \in A, \map h Y \in B}
| r = \map \Pr {X \in g... | Let $X$ and $Y$ be [[Definition:Independent Random Variables|independent random variables]] on a [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$.
Let $g$ and $h$ be [[Definition:Real-Valued Function|real-valued functions]] defined on the [[Definition:Codomain of Mapping|codomains]] o... | Let $A$ and $B$ be [[Definition:Subset|subsets]] of the [[Definition:Real Number|real numbers]] $\R$.
Let $g^{-1} \sqbrk A$ and $h^{-1} \sqbrk B$ denote the [[Definition:Preimage of Subset under Mapping|preimages]] of $A$ and $B$ under $g$ and $h$ respectively.
Applying the definition of [[Definition:Independent Ran... | Functions of Independent Random Variables are Independent | https://proofwiki.org/wiki/Functions_of_Independent_Random_Variables_are_Independent | https://proofwiki.org/wiki/Functions_of_Independent_Random_Variables_are_Independent | [
"Independent Random Variables"
] | [
"Definition:Independent Random Variables",
"Definition:Probability Space",
"Definition:Real-Valued Function",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Independent Random Variables"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Preimage/Mapping/Subset",
"Definition:Independent Random Variables",
"Definition:Independent Random Variables"
] |
proofwiki-10641 | Multiplication Property of Characteristic Functions | Let $X$ and $Y$ be independent random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $\phi_X$ and $\phi_Y$ denote the characteristic functions of $X$ and $Y$ respectively.
Then:
:$\phi_{X + Y} = \phi_X \phi_Y$ | Let $i = \sqrt{-1}$.
Let $\expect X$ denote the expectation of $X$.
{{begin-eqn}}
{{eqn | l = \map {\phi_{X + Y} } t
| r = \expect {e^{i t \paren {X + Y} } }
| c = {{Defof|Characteristic Function of Random Variable}}
}}
{{eqn | r = \expect {e^{i t X} e^{i t Y} }
| c =
}}
{{eqn | r = \expect {e^{i t X... | Let $X$ and $Y$ be [[Definition:Independent Random Variables|independent random variables]] on a [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$.
Let $\phi_X$ and $\phi_Y$ denote the [[Definition:Characteristic Function of Random Variable|characteristic functions]] of $X$ and $Y$ res... | Let $i = \sqrt{-1}$.
Let $\expect X$ denote the [[Definition:Expectation|expectation]] of $X$.
{{begin-eqn}}
{{eqn | l = \map {\phi_{X + Y} } t
| r = \expect {e^{i t \paren {X + Y} } }
| c = {{Defof|Characteristic Function of Random Variable}}
}}
{{eqn | r = \expect {e^{i t X} e^{i t Y} }
| c =
}}
... | Multiplication Property of Characteristic Functions | https://proofwiki.org/wiki/Multiplication_Property_of_Characteristic_Functions | https://proofwiki.org/wiki/Multiplication_Property_of_Characteristic_Functions | [
"Probability Theory"
] | [
"Definition:Independent Random Variables",
"Definition:Probability Space",
"Definition:Characteristic Function of Random Variable"
] | [
"Definition:Expectation",
"Functions of Independent Random Variables are Independent",
"Expected Value of Product is Product of Expected Value"
] |
proofwiki-10642 | Relationship between Limit Inferior and Lower Limit | Let $\struct {S, \tau}$ be a topological space.
Let $f: S \to \R \cup \set {-\infty, \infty}$ be an extended real-valued function.
Let $\sequence {s_n}_{n \mathop \in \N}$ be a convergent sequence in $S$ such that $s_n \to \bar s$.
Then the lower limit of $f$ at $\bar s$ is bounded above by the limit inferior of $\sequ... | Let $\NN_{\bar s}$ denote the neighborhood filter of $\bar s$.
By definition of the lower limit, there exists a sequence of open neighborhoods $\sequence {V_k}_{k \mathop \in \N} \in \NN_{\bar s}$ such that:
:$\ds \lim_{k \mathop \to \infty} \set {\inf_{s \mathop \in V_k} \map f s} = \liminf_{s \mathop \to \bar s} \map... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f: S \to \R \cup \set {-\infty, \infty}$ be an [[Definition:Extended Real-Valued Function|extended real-valued function]].
Let $\sequence {s_n}_{n \mathop \in \N}$ be a [[Definition:Convergent Sequence|convergent sequence]] in $S$ s... | Let $\NN_{\bar s}$ denote the [[Definition:Neighborhood Filter|neighborhood filter]] of $\bar s$.
By definition of the [[Definition:Lower Limit (Topological Space)|lower limit]], there exists a [[Definition:Sequence|sequence]] of [[Definition:Neighborhood (Topology)|open neighborhoods]] $\sequence {V_k}_{k \mathop \in... | Relationship between Limit Inferior and Lower Limit | https://proofwiki.org/wiki/Relationship_between_Limit_Inferior_and_Lower_Limit | https://proofwiki.org/wiki/Relationship_between_Limit_Inferior_and_Lower_Limit | [
"Topology"
] | [
"Definition:Topological Space",
"Definition:Extended Real-Valued Function",
"Definition:Convergent Sequence",
"Definition:Lower Limit (Topological Space)",
"Definition:Bounded Above Mapping/Real-Valued",
"Definition:Limit Inferior"
] | [
"Definition:Neighborhood Filter",
"Definition:Lower Limit (Topological Space)",
"Definition:Sequence",
"Definition:Neighborhood (Topology)",
"Category:Topology"
] |
proofwiki-10643 | Group is Abelian iff Opposite Group is Itself | Let $\struct {G, \circ}$ be a group.
Let $\struct {G, *}$ be the opposite group to $\struct {G, \circ}$.
$\struct {G, \circ}$ is an Abelian group {{iff}}:
:$\struct {G, \circ} = \struct {G, *}$ | By definition of opposite group:
:$(1): \quad \forall a, b \in G : a \circ b = b * a$ | Let $\struct {G, \circ}$ be a [[Definition:Group| group]].
Let $\struct {G, *}$ be the [[Definition:Opposite Group|opposite group]] to $\struct {G, \circ}$.
$\struct {G, \circ}$ is an [[Definition: Abelian Group|Abelian group]] {{iff}}:
:$\struct {G, \circ} = \struct {G, *}$ | By definition of [[Definition:Opposite Group|opposite group]]:
:$(1): \quad \forall a, b \in G : a \circ b = b * a$ | Group is Abelian iff Opposite Group is Itself | https://proofwiki.org/wiki/Group_is_Abelian_iff_Opposite_Group_is_Itself | https://proofwiki.org/wiki/Group_is_Abelian_iff_Opposite_Group_is_Itself | [
"Abelian Groups",
"Opposite Groups"
] | [
"Definition:Group",
"Definition:Opposite Group",
"Definition: Abelian Group"
] | [
"Definition:Opposite Group"
] |
proofwiki-10644 | Sequence on Product Space Converges to Point iff Projections Converge to Projections of Point | Let $I$ be an arbitrary index set.
For all $i \in I$, let $T_i = \struct {X_i, \tau_i}$ be topological spaces.
Let $\ds X = \prod_{i \mathop \in I} X_i$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$.
Let $\tau$ be the product topology on $X$.
Denote by $\pr_i : X \to X_i$ the projection from $X$ onto $X... | === Necessary Condition ===
Let $x_n \to x$.
Let $i \in I$.
From Projection from Product Topology is Continuous it follows that $\pr_i$ is continuous.
By Continuous Mapping is Sequentially Continuous, $\pr_i$ is also sequentially continuous.
Hence $\map {\pr_i} {x_n} \to \map {\pr_i} x$.
{{qed|lemma}} | Let $I$ be an arbitrary [[Definition:Indexing Set|index set]].
For all $i \in I$, let $T_i = \struct {X_i, \tau_i}$ be [[Definition:Topological Space|topological spaces]].
Let $\ds X = \prod_{i \mathop \in I} X_i$ be the [[Definition:Cartesian Product|cartesian product]] of $\family {X_i}_{i \mathop \in I}$.
Let $\t... | === Necessary Condition ===
Let $x_n \to x$.
Let $i \in I$.
From [[Projection from Product Topology is Continuous]] it follows that $\pr_i$ is continuous.
By [[Continuous Mapping is Sequentially Continuous]], $\pr_i$ is also [[Definition:Sequential Continuity|sequentially continuous]].
Hence $\map {\pr_i} {x_n} \t... | Sequence on Product Space Converges to Point iff Projections Converge to Projections of Point | https://proofwiki.org/wiki/Sequence_on_Product_Space_Converges_to_Point_iff_Projections_Converge_to_Projections_of_Point | https://proofwiki.org/wiki/Sequence_on_Product_Space_Converges_to_Point_iff_Projections_Converge_to_Projections_of_Point | [
"Topology",
"Convergence",
"Sequences",
"Projections"
] | [
"Definition:Indexing Set",
"Definition:Topological Space",
"Definition:Cartesian Product",
"Definition:Product Topology",
"Definition:Projection (Mapping Theory)",
"Definition:Sequence",
"Definition:Convergent Sequence/Topology",
"Definition:Sequence",
"Definition:Convergent Sequence/Topology"
] | [
"Projection from Product Topology is Continuous",
"Continuous Mapping is Sequentially Continuous",
"Definition:Sequential Continuity"
] |
proofwiki-10645 | Cauchy's Convergence Criterion/Complex Numbers | Let $\sequence {z_n}$ be a complex sequence.
Then $\sequence {z_n}$ is a Cauchy sequence {{iff}} it is convergent. | === Lemma ===
{{:Cauchy's Convergence Criterion/Complex Numbers/Lemma 1}}{{qed|lemma}}
Let $\sequence {x_n}$ be a real sequence where:
:$x_n = \map \Re {z_n}$ for every $n$
:$\map \Re {z_n}$ is the real part of $z_n$
Let $\sequence {y_n}$ be a real sequence where:
:$y_n = \map \Im {z_n}$ for every $n$
:$\map \Im {z_n}$... | Let $\sequence {z_n}$ be a [[Definition:Complex Sequence|complex sequence]].
Then $\sequence {z_n}$ is a [[Definition:Cauchy Sequence|Cauchy sequence]] {{iff}} it is [[Definition:Convergent Complex Sequence|convergent]]. | === [[Cauchy's Convergence Criterion/Complex Numbers/Lemma 1|Lemma]] ===
{{:Cauchy's Convergence Criterion/Complex Numbers/Lemma 1}}{{qed|lemma}}
Let $\sequence {x_n}$ be a [[Definition:Real Sequence|real sequence]] where:
:$x_n = \map \Re {z_n}$ for every $n$
:$\map \Re {z_n}$ is the [[Definition:Real Part|real part... | Cauchy's Convergence Criterion/Complex Numbers/Proof 1 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Complex_Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Complex_Numbers/Proof_1 | [
"Convergent Complex Sequences",
"Cauchy Sequences",
"Cauchy's Convergence Criterion"
] | [
"Definition:Complex Sequence",
"Definition:Cauchy Sequence",
"Definition:Convergent Sequence/Complex Numbers"
] | [
"Cauchy's Convergence Criterion/Complex Numbers/Lemma 1",
"Definition:Real Sequence",
"Definition:Complex Number/Real Part",
"Definition:Real Sequence",
"Definition:Complex Number/Imaginary Part",
"Definition:Cauchy Sequence",
"Definition:Convergent Sequence/Complex Numbers",
"Definition:Cauchy Sequen... |
proofwiki-10646 | Cauchy's Convergence Criterion/Complex Numbers | Let $\sequence {z_n}$ be a complex sequence.
Then $\sequence {z_n}$ is a Cauchy sequence {{iff}} it is convergent. | === Lemma ===
{{:Cauchy's Convergence Criterion/Complex Numbers/Lemma 1}}{{qed|lemma}}
Let $\sequence {x_n}$ be a real sequence where:
:$x_n = \map \Re {z_n}$ for every $n$
:$\map \Re {z_n}$ is the real part of $z_n$
Let $\sequence {y_n}$ be a real sequence where
:$y_n = \map \Im {z_n}$ for every $n$
:$\map \Im {z_n}$ ... | Let $\sequence {z_n}$ be a [[Definition:Complex Sequence|complex sequence]].
Then $\sequence {z_n}$ is a [[Definition:Cauchy Sequence|Cauchy sequence]] {{iff}} it is [[Definition:Convergent Complex Sequence|convergent]]. | === [[Cauchy's Convergence Criterion/Complex Numbers/Lemma 1|Lemma]] ===
{{:Cauchy's Convergence Criterion/Complex Numbers/Lemma 1}}{{qed|lemma}}
Let $\sequence {x_n}$ be a [[Definition:Real Sequence|real sequence]] where:
:$x_n = \map \Re {z_n}$ for every $n$
:$\map \Re {z_n}$ is the [[Definition:Real Part|real part... | Cauchy's Convergence Criterion/Complex Numbers/Proof 2 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Complex_Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Complex_Numbers/Proof_2 | [
"Convergent Complex Sequences",
"Cauchy Sequences",
"Cauchy's Convergence Criterion"
] | [
"Definition:Complex Sequence",
"Definition:Cauchy Sequence",
"Definition:Convergent Sequence/Complex Numbers"
] | [
"Cauchy's Convergence Criterion/Complex Numbers/Lemma 1",
"Definition:Real Sequence",
"Definition:Complex Number/Real Part",
"Definition:Real Sequence",
"Definition:Complex Number/Imaginary Part",
"Definition:Cauchy Sequence/Complex Numbers",
"Definition:Cauchy Sequence/Real Numbers",
"Cauchy's Conver... |
proofwiki-10647 | Sum of Arctangents | Let $\arctan a + \arctan b \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$
Then:
:$\arctan a + \arctan b = \map \arctan {\dfrac {a + b} {1 - a b} }$
where $\arctan$ denotes the arctangent. | Let $x = \arctan a$ and $y = \arctan b$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = \tan x
| r = a
| c =
}}
{{eqn | n = 2
| l = \tan y
| r = b
| c =
}}
{{eqn | l = \map \tan {\arctan a + \arctan b}
| r = \map \tan {x + y}
| c =
}}
{{eqn | r = \frac {\tan x + \tan y} {1 - \... | Let $\arctan a + \arctan b \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$
Then:
:$\arctan a + \arctan b = \map \arctan {\dfrac {a + b} {1 - a b} }$
where $\arctan$ denotes the [[Definition:Real Arctangent|arctangent]]. | Let $x = \arctan a$ and $y = \arctan b$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = \tan x
| r = a
| c =
}}
{{eqn | n = 2
| l = \tan y
| r = b
| c =
}}
{{eqn | l = \map \tan {\arctan a + \arctan b}
| r = \map \tan {x + y}
| c =
}}
{{eqn | r = \frac {\tan x + \tan y} {1 - ... | Sum of Arctangents/Proof | https://proofwiki.org/wiki/Sum_of_Arctangents | https://proofwiki.org/wiki/Sum_of_Arctangents/Proof | [
"Sum of Arctangents",
"Arctangent Function"
] | [
"Definition:Inverse Tangent/Real/Arctangent"
] | [
"Tangent of Sum"
] |
proofwiki-10648 | Sum of Arccotangents | :$\arccot a + \arccot b = \arccot \dfrac {a b - 1} {a + b}$
where $\arccot$ denotes the arccotangent. | Let $x = \arccot a$ and $y = \arccot b$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = \cot x
| r = a
| c =
}}
{{eqn | n = 2
| l = \cot y
| r = b
| c =
}}
{{eqn | l = \map \cot {\arccot a + \arccot b}
| r = \map \cot {x + y}
| c =
}}
{{eqn | r = \frac {\cot x \cot y - 1} {\co... | :$\arccot a + \arccot b = \arccot \dfrac {a b - 1} {a + b}$
where $\arccot$ denotes the [[Definition:Arccotangent|arccotangent]]. | Let $x = \arccot a$ and $y = \arccot b$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = \cot x
| r = a
| c =
}}
{{eqn | n = 2
| l = \cot y
| r = b
| c =
}}
{{eqn | l = \map \cot {\arccot a + \arccot b}
| r = \map \cot {x + y}
| c =
}}
{{eqn | r = \frac {\cot x \cot y - 1} {\c... | Sum of Arccotangents | https://proofwiki.org/wiki/Sum_of_Arccotangents | https://proofwiki.org/wiki/Sum_of_Arccotangents | [
"Arccotangent Function"
] | [
"Definition:Inverse Cotangent/Real/Arccotangent"
] | [
"Cotangent of Sum"
] |
proofwiki-10649 | Difference of Arccotangents | :$\arccot a - \arccot b = \arccot \dfrac {a b + 1} {a - b}$
where $\arccot$ denotes the arccotangent. | Let $x = \arccot a$ and $y = \arccot b$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = \cot x
| r = a
| c =
}}
{{eqn | n = 2
| l = \cot y
| r = b
| c =
}}
{{eqn | l = \map \cot {\arccot a - \arccot b}
| r = \map \cot {x - y}
| c =
}}
{{eqn | r = \frac {\cot x \cot y + 1} {\co... | :$\arccot a - \arccot b = \arccot \dfrac {a b + 1} {a - b}$
where $\arccot$ denotes the [[Definition:Arccotangent|arccotangent]]. | Let $x = \arccot a$ and $y = \arccot b$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = \cot x
| r = a
| c =
}}
{{eqn | n = 2
| l = \cot y
| r = b
| c =
}}
{{eqn | l = \map \cot {\arccot a - \arccot b}
| r = \map \cot {x - y}
| c =
}}
{{eqn | r = \frac {\cot x \cot y + 1} {\c... | Difference of Arccotangents | https://proofwiki.org/wiki/Difference_of_Arccotangents | https://proofwiki.org/wiki/Difference_of_Arccotangents | [
"Arccotangent Function"
] | [
"Definition:Inverse Cotangent/Real/Arccotangent"
] | [
"Cotangent of Difference"
] |
proofwiki-10650 | Multiple Angle Formula for Tangent | :$\ds \map \tan {n \theta} = \frac {\ds \sum_{i \mathop = 0}^{\floor{\frac {n - 1} 2} } \paren {-1}^i \binom n {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\floor {\frac n 2} } \paren {-1}^i \binom n {2 i} \tan^{2 i}\theta}$ | Proof by induction:
For all $n \in \N_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \map \tan {n \theta} = \frac {\ds \sum_{i \mathop = 0}^{\floor{\frac {n - 1} 2} } \paren {-1}^i \binom n {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\floor {\frac n 2} } \paren {-1}^i \binom n {2 i} \tan^{2 i}\theta}... | :$\ds \map \tan {n \theta} = \frac {\ds \sum_{i \mathop = 0}^{\floor{\frac {n - 1} 2} } \paren {-1}^i \binom n {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\floor {\frac n 2} } \paren {-1}^i \binom n {2 i} \tan^{2 i}\theta}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \map \tan {n \theta} = \frac {\ds \sum_{i \mathop = 0}^{\floor{\frac {n - 1} 2} } \paren {-1}^i \binom n {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^... | Multiple Angle Formula for Tangent | https://proofwiki.org/wiki/Multiple_Angle_Formula_for_Tangent | https://proofwiki.org/wiki/Multiple_Angle_Formula_for_Tangent | [
"Tangent Function"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-10651 | Laplace Transform of Dirac Delta Function by Function | Let $\map f t: \R \to \R$ or $\R \to \C$ be a function.
Let $\map \delta t$ denote the Dirac delta function.
Let $c$ be a positive constant real number.
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.
Then:
:$\laptrans {\map \delta {t - c} \map f t} = e^{- s c} \map f c$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map \delta {t - c} \map f t}
| r = \int^{\to+\infty}_0 e^{-s t} \map \delta {t - c} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int^{c^+}_{c^-} e^{-s t} \map \delta {t - c} \map f t \rd t
| c = Integrand elsewhere zero by {{Defof|Dirac Delta F... | Let $\map f t: \R \to \R$ or $\R \to \C$ be a [[Definition:Function|function]].
Let $\map \delta t$ denote the [[Definition:Dirac Delta Function|Dirac delta function]].
Let $c$ be a [[Definition:Positive Real Number|positive]] [[Definition:Constant|constant]] [[Definition:Real Number|real number]].
Let $\laptrans {\... | {{begin-eqn}}
{{eqn | l = \laptrans {\map \delta {t - c} \map f t}
| r = \int^{\to+\infty}_0 e^{-s t} \map \delta {t - c} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int^{c^+}_{c^-} e^{-s t} \map \delta {t - c} \map f t \rd t
| c = Integrand elsewhere zero by {{Defof|Dirac Delta F... | Laplace Transform of Dirac Delta Function by Function | https://proofwiki.org/wiki/Laplace_Transform_of_Dirac_Delta_Function_by_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Dirac_Delta_Function_by_Function | [
"Dirac Delta Function",
"Examples of Laplace Transforms"
] | [
"Definition:Function",
"Definition:Dirac Delta Function",
"Definition:Positive/Real Number",
"Definition:Constant",
"Definition:Real Number",
"Definition:Laplace Transform"
] | [
"Definition:Real Interval/Closed",
"Primitive of Constant Multiple of Function",
"Integration by Substitution",
"Category:Dirac Delta Function",
"Category:Examples of Laplace Transforms"
] |
proofwiki-10652 | Modulus of Complex Number equals its Distance from Origin | The modulus of a complex number equals its distance from the origin on the complex plane. | Let $z = x + y i$ be a complex number and $O = 0 + 0 i$ be the origin on the complex plane.
We have its modulus:
{{begin-eqn}}
{{eqn | l = \cmod z
| r = \cmod {x + y i}
| c =
}}
{{eqn | r = \sqrt {x^2 + y^2}
| c = {{Defof|Complex Modulus}}
}}
{{end-eqn}}
and its distance from the origin on the complex... | The [[Definition:Complex Modulus|modulus]] of a [[Definition:Complex Number|complex number]] equals its [[Definition:Distance|distance]] from the [[Definition:Origin|origin]] on the [[Definition:Complex Plane|complex plane]]. | Let $z = x + y i$ be a [[Definition:Complex Number|complex number]] and $O = 0 + 0 i$ be the [[Definition:Origin|origin]] on the [[Definition:Complex Plane|complex plane]].
We have its [[Definition:Complex Modulus|modulus]]:
{{begin-eqn}}
{{eqn | l = \cmod z
| r = \cmod {x + y i}
| c =
}}
{{eqn | r = \sq... | Modulus of Complex Number equals its Distance from Origin | https://proofwiki.org/wiki/Modulus_of_Complex_Number_equals_its_Distance_from_Origin | https://proofwiki.org/wiki/Modulus_of_Complex_Number_equals_its_Distance_from_Origin | [
"Complex Analysis"
] | [
"Definition:Complex Modulus",
"Definition:Complex Number",
"Definition:Distance",
"Definition:Coordinate System/Origin",
"Definition:Complex Number/Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Coordinate System/Origin",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Modulus",
"Definition:Distance",
"Definition:Coordinate System/Origin",
"Definition:Complex Number/Complex Plane",
"Distance Formula",
"Category:Complex Analysis"
] |
proofwiki-10653 | Empty Set is Countable | The empty set $\O$ is countable. | By Peano's Axioms, $\N_0 \sim \O$, where $\N_n$ denotes the initial segment of natural number $n$.
By definition, $\O$ is finite.
By definition, $\O$ is a countable set.
{{qed}}
Category:Set Theory
Category:Empty Set
Category:Countable Sets
ly0tg8512aadn5cgyxnqkjdd9niomfe | The [[Definition:Empty Set|empty set]] $\O$ is [[Definition:Countable Set|countable]]. | By [[Axiom:Peano's Axioms|Peano's Axioms]], $\N_0 \sim \O$, where $\N_n$ denotes the [[Definition:Initial Segment of Natural Numbers|initial segment of natural number $n$]].
By definition, $\O$ is [[Definition:Finite Set|finite]].
By definition, $\O$ is a [[Definition:Countable Set/Definition 2|countable set]].
{{qed... | Empty Set is Countable | https://proofwiki.org/wiki/Empty_Set_is_Countable | https://proofwiki.org/wiki/Empty_Set_is_Countable | [
"Set Theory",
"Empty Set",
"Countable Sets"
] | [
"Definition:Empty Set",
"Definition:Countable Set"
] | [
"Axiom:Peano's Axioms",
"Definition:Initial Segment of Natural Numbers",
"Definition:Finite Set",
"Definition:Countable Set/Definition 2",
"Category:Set Theory",
"Category:Empty Set",
"Category:Countable Sets"
] |
proofwiki-10654 | Lindelöf's Lemma | Let $C$ be a set of open real sets.
Let $S \subseteq \R$ be a subset of the real numbers that is covered by $C$.
Then there exists a countable subset of $C$ that covers $S$. | === Lemma $1$ ===
{{:Lindelöf's Lemma/Lemma 1}}{{qed|lemma}}
We have that $S$ is covered by $C$.
This means that $S$ is a subset of $\ds \bigcup_{O \mathop \in C} O$.
From {{Lemma|Lindelöf's Lemma|1}}:
:$\ds \bigcup_{O \mathop \in D} O = \bigcup_{O \mathop \in C} O$
where $D$ is a countable subset of $C$.
Hence $S$ is ... | Let $C$ be a [[Definition:Set|set]] of [[Definition:Open Set of Real Numbers|open real sets]].
Let $S \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] that is [[Definition:Cover of Set|covered]] by $C$.
Then there exists a [[Definition:Countable Set|countable]] [[Definit... | === [[Lindelöf's Lemma/Lemma 1|Lemma $1$]] ===
{{:Lindelöf's Lemma/Lemma 1}}{{qed|lemma}}
We have that $S$ is [[Definition:Cover of Set|covered]] by $C$.
This means that $S$ is a [[Definition:Subset|subset]] of $\ds \bigcup_{O \mathop \in C} O$.
From {{Lemma|Lindelöf's Lemma|1}}:
:$\ds \bigcup_{O \mathop \in D} O =... | Lindelöf's Lemma | https://proofwiki.org/wiki/Lindelöf's_Lemma | https://proofwiki.org/wiki/Lindelöf's_Lemma | [
"Lindelöf's Lemma",
"Real Analysis"
] | [
"Definition:Set",
"Definition:Open Set/Real Analysis/Real Numbers",
"Definition:Subset",
"Definition:Real Number",
"Definition:Cover of Set",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Cover of Set"
] | [
"Lindelöf's Lemma/Lemma 1",
"Definition:Cover of Set",
"Definition:Subset",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Subset",
"Definition:Cover of Set",
"Definition:Cover of Set",
"Definition:Countable Set",
"Definition:Subset",
"Category:Lindelöf's Lemma",
"Category:Real A... |
proofwiki-10655 | Area of Isosceles Triangle in terms of Sides | Let $\triangle ABC$ be an isosceles triangle whose apex is $A$.
Let $r$ be the length of a leg of $\triangle ABC$.
Let $b$ be the length of the base of $\triangle ABC$.
Then the area $\AA$ of $\triangle ABC$ is given by:
:$\AA = \dfrac b 4 \sqrt {4 r^2 - b^2}$ | :300px
Let $h$ be the height of $\triangle ABC$.
{{begin-eqn}}
{{eqn | l = \AA
| r = \frac 1 2 b h
| c = Area of Triangle in Terms of Side and Altitude
}}
{{eqn | r = \frac b 2 \sqrt {r^2 - \paren {\frac b 2}^2}
| c = Pythagoras's Theorem
}}
{{eqn | r = \frac b 2 \sqrt {\frac {4 r^2 - b^2} 4}
| ... | Let $\triangle ABC$ be an [[Definition:Isosceles Triangle|isosceles triangle]] whose [[Definition:Apex of Isosceles Triangle|apex]] is $A$.
Let $r$ be the [[Definition:Length (Linear Measure)|length]] of a [[Definition:Legs of Isosceles Triangle|leg]] of $\triangle ABC$.
Let $b$ be the [[Definition:Length (Linear Mea... | :[[File:IsoscelesTriangleArea.png|300px]]
Let $h$ be the [[Definition:Height of Triangle|height]] of $\triangle ABC$.
{{begin-eqn}}
{{eqn | l = \AA
| r = \frac 1 2 b h
| c = [[Area of Triangle in Terms of Side and Altitude]]
}}
{{eqn | r = \frac b 2 \sqrt {r^2 - \paren {\frac b 2}^2}
| c = [[Pythag... | Area of Isosceles Triangle in terms of Sides | https://proofwiki.org/wiki/Area_of_Isosceles_Triangle_in_terms_of_Sides | https://proofwiki.org/wiki/Area_of_Isosceles_Triangle_in_terms_of_Sides | [
"Areas of Triangles",
"Isosceles Triangles"
] | [
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles/Apex",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Isosceles/Legs",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Isosceles/Base",
"Definition:Area"
] | [
"File:IsoscelesTriangleArea.png",
"Definition:Triangle (Geometry)/Height",
"Area of Triangle in Terms of Side and Altitude",
"Pythagoras's Theorem",
"Category:Areas of Triangles",
"Category:Isosceles Triangles"
] |
proofwiki-10656 | Even Function Times Even Function is Even | Let $X \subset \R$ be a symmetric set of real numbers:
:$\forall x \in X: -x \in X$
Let $f, g: X \to \R$ be two even functions.
Let $f \cdot g$ denote the pointwise product of $f$ and $g$.
Then $\paren {f \cdot g}: X \to \R$ is also an even function. | {{begin-eqn}}
{{eqn | l = \map {\paren {f \cdot g} } {-x}
| r = \map f {-x} \cdot \map g {-x}
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = \map f x \cdot \map g x
| c = {{Defof|Even Function}}
}}
{{eqn | r = \map {\paren {f \cdot g} } x
| c = {{Defof|Pointwise... | Let $X \subset \R$ be a [[Definition:Symmetric Set of Real Numbers|symmetric set of real numbers]]:
:$\forall x \in X: -x \in X$
Let $f, g: X \to \R$ be two [[Definition:Even Function|even functions]].
Let $f \cdot g$ denote the [[Definition:Pointwise Multiplication of Real-Valued Functions|pointwise product]] of $f$... | {{begin-eqn}}
{{eqn | l = \map {\paren {f \cdot g} } {-x}
| r = \map f {-x} \cdot \map g {-x}
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = \map f x \cdot \map g x
| c = {{Defof|Even Function}}
}}
{{eqn | r = \map {\paren {f \cdot g} } x
| c = {{Defof|Pointwise... | Even Function Times Even Function is Even | https://proofwiki.org/wiki/Even_Function_Times_Even_Function_is_Even | https://proofwiki.org/wiki/Even_Function_Times_Even_Function_is_Even | [
"Even Functions"
] | [
"Definition:Symmetric Set/Real Numbers",
"Definition:Even Function",
"Definition:Pointwise Multiplication of Real-Valued Functions",
"Definition:Even Function"
] | [
"Definition:Even Function"
] |
proofwiki-10657 | Odd Function Times Even Function is Odd | Let $X \subset \R$ be a symmetric set of real numbers:
:$\forall x \in X: -x \in X$
Let $f: X \to \R$ be an odd function.
Let $g: X \to \R$ be an even function.
Let $f \cdot g$ denote the pointwise product of $f$ and $g$.
Then $\paren {f \cdot g}: X \to \R$ is an odd function. | {{begin-eqn}}
{{eqn | l = \map {\paren {f \cdot g} } {-x}
| r = \map f {-x} \cdot \map g {-x}
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = \paren {-\map f x} \cdot \map g x
| c = {{Defof|Odd Function}} and {{Defof|Even Function}}
}}
{{eqn | r = -\map f x \cdot \map ... | Let $X \subset \R$ be a [[Definition:Symmetric Set of Real Numbers|symmetric set of real numbers]]:
:$\forall x \in X: -x \in X$
Let $f: X \to \R$ be an [[Definition:Odd Function|odd function]].
Let $g: X \to \R$ be an [[Definition:Even Function|even function]].
Let $f \cdot g$ denote the [[Definition:Pointwise Mult... | {{begin-eqn}}
{{eqn | l = \map {\paren {f \cdot g} } {-x}
| r = \map f {-x} \cdot \map g {-x}
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = \paren {-\map f x} \cdot \map g x
| c = {{Defof|Odd Function}} and {{Defof|Even Function}}
}}
{{eqn | r = -\map f x \cdot \map ... | Odd Function Times Even Function is Odd | https://proofwiki.org/wiki/Odd_Function_Times_Even_Function_is_Odd | https://proofwiki.org/wiki/Odd_Function_Times_Even_Function_is_Odd | [
"Even Functions",
"Odd Functions"
] | [
"Definition:Symmetric Set/Real Numbers",
"Definition:Odd Function",
"Definition:Even Function",
"Definition:Pointwise Multiplication of Real-Valued Functions",
"Definition:Odd Function"
] | [
"Definition:Odd Function"
] |
proofwiki-10658 | Odd Function Times Odd Function is Even | Let $S \subset \R$ be a symmetric set of real numbers:
:$\forall x \in S: -x \in X$
Let $f, g: X \to \R$ be two odd functions.
Let $f \cdot g$ denote the pointwise product of $f$ and $g$.
Then $\paren {f \cdot g}: S \to \R$ is an even function. | {{begin-eqn}}
{{eqn | l = \map {\paren {f \cdot g} } {-x}
| r = \map f {-x} \cdot \map g {-x}
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = \paren {-\map f x} \cdot \paren {-\map g x}
| c = {{Defof|Odd Function}}
}}
{{eqn | r = \map f x \cdot \map g x
| c =
}}... | Let $S \subset \R$ be a [[Definition:Symmetric Set of Real Numbers|symmetric set of real numbers]]:
:$\forall x \in S: -x \in X$
Let $f, g: X \to \R$ be two [[Definition:Odd Function|odd functions]].
Let $f \cdot g$ denote the [[Definition:Pointwise Multiplication of Real-Valued Functions|pointwise product]] of $f$ a... | {{begin-eqn}}
{{eqn | l = \map {\paren {f \cdot g} } {-x}
| r = \map f {-x} \cdot \map g {-x}
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = \paren {-\map f x} \cdot \paren {-\map g x}
| c = {{Defof|Odd Function}}
}}
{{eqn | r = \map f x \cdot \map g x
| c =
}}... | Odd Function Times Odd Function is Even | https://proofwiki.org/wiki/Odd_Function_Times_Odd_Function_is_Even | https://proofwiki.org/wiki/Odd_Function_Times_Odd_Function_is_Even | [
"Even Functions",
"Odd Functions"
] | [
"Definition:Symmetric Set/Real Numbers",
"Definition:Odd Function",
"Definition:Pointwise Multiplication of Real-Valued Functions",
"Definition:Even Function"
] | [
"Definition:Even Function"
] |
proofwiki-10659 | Mellin Transform of Exponential | Let $a$ be a complex constant and $e^t$ be the complex exponential.
Let $\MM$ be the Mellin transform.
Then:
:$\map {\MM \set {e^{-a t} } } s = a^{-s} \, \map \Gamma s$
where $\map \Re a, \map \Re s > 0$ | {{begin-eqn}}
{{eqn | l = \map {\MM \set {e^{-a t} } } s
| r = \int_0^{\to +\infty} t^{s - 1} e^{-a t} \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} \paren {\dfrac t a}^{s - 1} e^{-a \paren {\frac t a} } \frac {\d t} a
| c = Integration by Substitution, $t \mapsto \dfrac t... | Let $a$ be a [[Definition:Complex Number|complex]] [[Definition:Constant|constant]] and $e^t$ be the [[Definition:Complex Exponential Function|complex exponential]].
Let $\MM$ be the [[Definition:Mellin Transform|Mellin transform]].
Then:
:$\map {\MM \set {e^{-a t} } } s = a^{-s} \, \map \Gamma s$
where $\map \Re a... | {{begin-eqn}}
{{eqn | l = \map {\MM \set {e^{-a t} } } s
| r = \int_0^{\to +\infty} t^{s - 1} e^{-a t} \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} \paren {\dfrac t a}^{s - 1} e^{-a \paren {\frac t a} } \frac {\d t} a
| c = [[Integration by Substitution]], $t \mapsto \dfr... | Mellin Transform of Exponential | https://proofwiki.org/wiki/Mellin_Transform_of_Exponential | https://proofwiki.org/wiki/Mellin_Transform_of_Exponential | [
"Mellin Transforms"
] | [
"Definition:Complex Number",
"Definition:Constant",
"Definition:Exponential Function/Complex",
"Definition:Mellin Transform"
] | [
"Integration by Substitution",
"Primitive of Constant Multiple of Function"
] |
proofwiki-10660 | Mellin Transform of Dirac Delta Function | Let $c \in \R_{>0}$ be a (strictly) positive real number.
Let $\map {\delta_c} t$ be the Dirac delta function.
Let $\MM$ be the Mellin transform.
Then:
:$\map {\MM \set {\map {\delta_c} t} } s = c^{s - 1}$ | {{begin-eqn}}
{{eqn | l = \map {\MM \set {\map {\delta_c} t} } s
| r = \int_0^{\to +\infty} t^{s - 1} \map {\delta_c} t \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_{c^-}^{c^+} t^{s - 1} \map {\delta_c} t \rd t
| c = {{Defof|Dirac Delta Function}}: integrand is elsewhere zero
}}
{{eqn | ... | Let $c \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $\map {\delta_c} t$ be the [[Definition:Dirac Delta Function|Dirac delta function]].
Let $\MM$ be the [[Definition:Mellin Transform|Mellin transform]].
Then:
:$\map {\MM \set {\map {\delta_c} t} } s = c^{s - 1... | {{begin-eqn}}
{{eqn | l = \map {\MM \set {\map {\delta_c} t} } s
| r = \int_0^{\to +\infty} t^{s - 1} \map {\delta_c} t \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_{c^-}^{c^+} t^{s - 1} \map {\delta_c} t \rd t
| c = {{Defof|Dirac Delta Function}}: [[Definition:Integrand|integrand]] is e... | Mellin Transform of Dirac Delta Function | https://proofwiki.org/wiki/Mellin_Transform_of_Dirac_Delta_Function | https://proofwiki.org/wiki/Mellin_Transform_of_Dirac_Delta_Function | [
"Mellin Transforms",
"Dirac Delta Function"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Dirac Delta Function",
"Definition:Mellin Transform"
] | [
"Definition:Integration/Integrand",
"Definition:Constant",
"Definition:Interval/Ordered Set/Closed",
"Primitive of Constant Multiple of Function"
] |
proofwiki-10661 | Mellin Transform of Heaviside Step Function | Let $c$ be a constant real number.
Let $\map {u_c} t$ be the Heaviside step function.
Let $\MM$ be the Mellin transform.
Then:
:$\map {\MM \set {\map {u_c} t} } s = -\dfrac {c^s} s$
for $c > 0, \map \Re s < 0$. | === Lemma ===
{{:Mellin Transform of Heaviside Step Function/Lemma}}
{{begin-eqn}}
{{eqn | l = \map {\MM \set {\map {u_c} t} } s
| r = \int_0^{\to +\infty} t^{s - 1} \map {u_c} t \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_c^{\to +\infty} t^{s - 1} \rd t
| c = {{Defof|Heaviside Step Fun... | Let $c$ be a [[Definition:Constant|constant]] [[Definition:Real Number|real number]].
Let $\map {u_c} t$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $\MM$ be the [[Definition:Mellin Transform|Mellin transform]].
Then:
:$\map {\MM \set {\map {u_c} t} } s = -\dfrac {c^s} s$
for $c > 0, ... | === [[Mellin Transform of Heaviside Step Function/Lemma|Lemma]] ===
{{:Mellin Transform of Heaviside Step Function/Lemma}}
{{begin-eqn}}
{{eqn | l = \map {\MM \set {\map {u_c} t} } s
| r = \int_0^{\to +\infty} t^{s - 1} \map {u_c} t \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_c^{\to +\infty}... | Mellin Transform of Heaviside Step Function | https://proofwiki.org/wiki/Mellin_Transform_of_Heaviside_Step_Function | https://proofwiki.org/wiki/Mellin_Transform_of_Heaviside_Step_Function | [
"Mellin Transforms",
"Heaviside Step Function"
] | [
"Definition:Constant",
"Definition:Real Number",
"Definition:Heaviside Step Function",
"Definition:Mellin Transform"
] | [
"Mellin Transform of Heaviside Step Function/Lemma",
"Definition:Integration/Integrand",
"Primitive of Power",
"Category:Mellin Transforms",
"Category:Heaviside Step Function"
] |
proofwiki-10662 | Mellin Transform of Heaviside Step Function/Corollary | :$\map {\MM \set {\map u {c - t} } } s = \dfrac {c^s} s$
for $c > 0, \map \Re s > 0$ | {{begin-eqn}}
{{eqn | l = \map {\MM \set {\map u {c - t} } } s
| r = \int_0^{\to +\infty} t^{s - 1} \map u {c - t} \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_0^c t^{s - 1} \rd t
| c = {{Defof|Heaviside Step Function}}: integrand is elsewhere zero
}}
{{eqn | r = \bigintlimits {\dfrac {... | :$\map {\MM \set {\map u {c - t} } } s = \dfrac {c^s} s$
for $c > 0, \map \Re s > 0$ | {{begin-eqn}}
{{eqn | l = \map {\MM \set {\map u {c - t} } } s
| r = \int_0^{\to +\infty} t^{s - 1} \map u {c - t} \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_0^c t^{s - 1} \rd t
| c = {{Defof|Heaviside Step Function}}: [[Definition:Integrand|integrand]] is elsewhere zero
}}
{{eqn | r ... | Mellin Transform of Heaviside Step Function/Corollary | https://proofwiki.org/wiki/Mellin_Transform_of_Heaviside_Step_Function/Corollary | https://proofwiki.org/wiki/Mellin_Transform_of_Heaviside_Step_Function/Corollary | [
"Mellin Transforms",
"Heaviside Step Function"
] | [] | [
"Definition:Integration/Integrand",
"Primitive of Power",
"Category:Mellin Transforms",
"Category:Heaviside Step Function"
] |
proofwiki-10663 | Mellin Transform of Power Times Function | Let $t^n: \R \to \R$ be $t$ to the $n$th power for some $n \in \N_{\ge 0}$.
Let $\MM$ be the Mellin transform.
Then:
:$\map {\MM \set {t^n \map f t} } s = \map {\MM \set {\map f t} } {s + n}$
given that both transforms exist. | {{begin-eqn}}
{{eqn | l = \map {\MM \set {t^n \map f t} } s
| r = \int_0^{\to +\infty} t^{s - 1} t^n \map f t \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} t^{\paren {s + n} - 1} \map f t \rd t
| c = Exponent Combination Laws
}}
{{eqn | r = \map {\MM \set {\map f t} } {s +... | Let $t^n: \R \to \R$ be [[Definition:Integer Power|$t$ to the $n$th power]] for some $n \in \N_{\ge 0}$.
Let $\MM$ be the [[Definition:Mellin Transform|Mellin transform]].
Then:
:$\map {\MM \set {t^n \map f t} } s = \map {\MM \set {\map f t} } {s + n}$
given that both transforms exist. | {{begin-eqn}}
{{eqn | l = \map {\MM \set {t^n \map f t} } s
| r = \int_0^{\to +\infty} t^{s - 1} t^n \map f t \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} t^{\paren {s + n} - 1} \map f t \rd t
| c = [[Exponent Combination Laws]]
}}
{{eqn | r = \map {\MM \set {\map f t} } ... | Mellin Transform of Power Times Function | https://proofwiki.org/wiki/Mellin_Transform_of_Power_Times_Function | https://proofwiki.org/wiki/Mellin_Transform_of_Power_Times_Function | [
"Mellin Transforms"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Mellin Transform"
] | [
"Exponent Combination Laws",
"Category:Mellin Transforms"
] |
proofwiki-10664 | Mellin Transform of Dirac Delta Function by Function | Let $f: \R \to \R$ be a function.
Let $c \in \R_{>0}$ be a positive constant real number.
Let $\map {\delta_c} t$ be the Dirac delta function.
Let $\MM$ be the Mellin transform.
Then:
:$\map {\MM \set {\map {\delta_c} t \map f t} } s = c^{s - 1} \map f c$ | {{begin-eqn}}
{{eqn | l = \map {\MM \set {\map {\delta_c} t \map f t} } s
| r = \int_0^{\to +\infty} t^{s - 1} \map {\delta_c} t \map f t \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_{c^-}^{c^+} t^{s - 1} \map {\delta_c} t \map f t \rd t
| c = {{Defof|Dirac Delta Function}}: integrand is... | Let $f: \R \to \R$ be a [[Definition:Function|function]].
Let $c \in \R_{>0}$ be a [[Definition:Positive Real Number|positive]] [[Definition:Constant|constant]] [[Definition:Real Number|real number]].
Let $\map {\delta_c} t$ be the [[Definition:Dirac Delta Function|Dirac delta function]].
Let $\MM$ be the [[Definiti... | {{begin-eqn}}
{{eqn | l = \map {\MM \set {\map {\delta_c} t \map f t} } s
| r = \int_0^{\to +\infty} t^{s - 1} \map {\delta_c} t \map f t \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_{c^-}^{c^+} t^{s - 1} \map {\delta_c} t \map f t \rd t
| c = {{Defof|Dirac Delta Function}}: [[Definition... | Mellin Transform of Dirac Delta Function by Function | https://proofwiki.org/wiki/Mellin_Transform_of_Dirac_Delta_Function_by_Function | https://proofwiki.org/wiki/Mellin_Transform_of_Dirac_Delta_Function_by_Function | [
"Mellin Transforms",
"Dirac Delta Function"
] | [
"Definition:Function",
"Definition:Positive/Real Number",
"Definition:Constant",
"Definition:Real Number",
"Definition:Dirac Delta Function",
"Definition:Mellin Transform"
] | [
"Definition:Integration/Integrand",
"Definition:Constant",
"Definition:Interval/Ordered Set/Closed",
"Primitive of Constant Multiple of Function",
"Category:Mellin Transforms",
"Category:Dirac Delta Function"
] |
proofwiki-10665 | Unity of Ring is Idempotent | Let $\left({R, +, \circ}\right)$ be a ring with unity whose unity is $1_R$.
Then $1_R$ is an idempotent element of $R$ under the ring product $\circ$:
:$1_R \circ 1_R = 1_R$ | By definition of ring with unity, $\left({R, \circ}\right)$ is a monoid whose identity element is $1_R$.
From Identity Element is Idempotent (applied to $1_R$):
:$1_R \circ 1_R = 1_R$
which was to be proven.
{{qed}}
Category:Ring Theory
sykxawsy95ekov7hb6d6eq70yxiv8oq | Let $\left({R, +, \circ}\right)$ be a [[Definition:Ring with Unity|ring with unity]] whose [[Definition:Unity of Ring|unity]] is $1_R$.
Then $1_R$ is an [[Definition:Idempotent Element|idempotent element]] of $R$ under the [[Definition:Ring Product|ring product]] $\circ$:
:$1_R \circ 1_R = 1_R$ | By definition of [[Definition:Ring with Unity|ring with unity]], $\left({R, \circ}\right)$ is a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity element]] is $1_R$.
From [[Identity Element is Idempotent]] (applied to $1_R$):
:$1_R \circ 1_R = 1_R$
which was to be proven.
{{qed}}
[[Category:R... | Unity of Ring is Idempotent | https://proofwiki.org/wiki/Unity_of_Ring_is_Idempotent | https://proofwiki.org/wiki/Unity_of_Ring_is_Idempotent | [
"Ring Theory"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Idempotence/Element",
"Definition:Ring (Abstract Algebra)/Product"
] | [
"Definition:Ring with Unity",
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Identity Element is Idempotent",
"Category:Ring Theory"
] |
proofwiki-10666 | Mellin Transform of Higher Order Exponential | Let $a$ be a complex constant.
Let $n$ be a natural number.
Let $e^t$ be the complex exponential of $t$.
Let $\MM$ be the Mellin transform.
Then:
:$\map {\MM \set {e^{-a t^n} } } s = \dfrac {a^{-s/n} } n \map \Gamma {\dfrac s n}$
where $\map \Gamma z$ is the Gamma function and $\map \Re a$, $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \map {\MM \set {e^{-a t^n} } } s
| r = \int_0^{\to +\infty} t^{s-1} e^{-a t^n} \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} \paren {\dfrac t {\sqrt[n] a} }^{s - 1} e^{-a \paren {\dfrac t {\sqrt[n] {a} } }^n} \dfrac {\d t} {\sqrt[n] a}
| c = Integ... | Let $a$ be a [[Definition:Complex Number|complex]] [[Definition:Constant|constant]].
Let $n$ be a [[Definition:Natural Number|natural number]].
Let $e^t$ be the [[Definition:Complex Exponential Function|complex exponential of $t$]].
Let $\MM$ be the [[Definition:Mellin Transform|Mellin transform]].
Then:
:$\map {\... | {{begin-eqn}}
{{eqn | l = \map {\MM \set {e^{-a t^n} } } s
| r = \int_0^{\to +\infty} t^{s-1} e^{-a t^n} \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} \paren {\dfrac t {\sqrt[n] a} }^{s - 1} e^{-a \paren {\dfrac t {\sqrt[n] {a} } }^n} \dfrac {\d t} {\sqrt[n] a}
| c = [[Int... | Mellin Transform of Higher Order Exponential | https://proofwiki.org/wiki/Mellin_Transform_of_Higher_Order_Exponential | https://proofwiki.org/wiki/Mellin_Transform_of_Higher_Order_Exponential | [
"Mellin Transforms"
] | [
"Definition:Complex Number",
"Definition:Constant",
"Definition:Natural Numbers",
"Definition:Exponential Function/Complex",
"Definition:Mellin Transform",
"Definition:Gamma Function"
] | [
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Exponent Combination Laws",
"Category:Mellin Transforms"
] |
proofwiki-10667 | Linear Combination of Mellin Transforms | Let $\MM$ be the Mellin transform.
Let $\map f t$, $g \left({t}\right)$ be functions such that $\MM \left\{ {\map f t}\right\} \left({s}\right)$ and $\MM \left\{ {\map f t}\right\} \left({s}\right)$ exist.
Let $\lambda \in \C$ be a constant.
Then:
:$\map {\MM \set {\lambda \map f t + \map g t} } s = \lambda \map {\MM \... | {{begin-eqn}}
{{eqn | l = \map {\MM \set {\lambda \map f t + \map g t} } s
| r = \int_0^{\to +\infty} t^{s - 1} \paren {\lambda \map f t + \map g t} \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \lambda \int_0^{\to +\infty} t^{s - 1} \map f t \rd t + \int_0^{\to +\infty} t^{s - 1} \map g t \rd t
... | Let $\MM$ be the [[Definition:Mellin Transform|Mellin transform]].
Let $\map f t$, $g \left({t}\right)$ be [[Definition:Function|functions]] such that $\MM \left\{ {\map f t}\right\} \left({s}\right)$ and $\MM \left\{ {\map f t}\right\} \left({s}\right)$ exist.
Let $\lambda \in \C$ be a [[Definition:Constant|constant... | {{begin-eqn}}
{{eqn | l = \map {\MM \set {\lambda \map f t + \map g t} } s
| r = \int_0^{\to +\infty} t^{s - 1} \paren {\lambda \map f t + \map g t} \rd t
| c = {{Defof|Mellin Transform}}
}}
{{eqn | r = \lambda \int_0^{\to +\infty} t^{s - 1} \map f t \rd t + \int_0^{\to +\infty} t^{s - 1} \map g t \rd t
... | Linear Combination of Mellin Transforms | https://proofwiki.org/wiki/Linear_Combination_of_Mellin_Transforms | https://proofwiki.org/wiki/Linear_Combination_of_Mellin_Transforms | [
"Mellin Transforms"
] | [
"Definition:Mellin Transform",
"Definition:Function",
"Definition:Constant"
] | [
"Linear Combination of Complex Integrals",
"Category:Mellin Transforms"
] |
proofwiki-10668 | Complex Numbers as External Direct Product | Let $\struct {\C_{\ne 0}, \times}$ be the group of non-zero complex numbers under multiplication.
Let $\struct {\R_{> 0}, \times}$ be the group of positive real numbers under multiplication.
Let $\struct {K, \times}$ be the circle group.
Then:
:$\struct {\C_{\ne 0}, \times} \cong \struct {\R_{> 0}, \times} \times \stru... | Let $\phi: \C_{\ne 0} \to \R_{> 0} \times K$ be the mapping:
:$\map \phi {r e^{i \theta} } = \paren {r, e^{i \theta} }$
$\forall \tuple {a, b} \in \R_{> 0} \times K:\exists z = a \times b \in \C$ such that:
:$\map \phi z = \tuple {a, b}$
by Complex Multiplication is Closed and $\R \subset \C$.
So $\phi$ is surjective.
... | Let $\struct {\C_{\ne 0}, \times}$ be the [[Definition:Group|group]] of non-[[Definition:Complex Zero|zero]] [[Definition:Complex Number|complex numbers]] under [[Definition:Complex Multiplication|multiplication]].
Let $\struct {\R_{> 0}, \times}$ be the [[Definition:Group|group]] of [[Definition:Positive Real Number|... | Let $\phi: \C_{\ne 0} \to \R_{> 0} \times K$ be the [[Definition:Mapping|mapping]]:
:$\map \phi {r e^{i \theta} } = \paren {r, e^{i \theta} }$
$\forall \tuple {a, b} \in \R_{> 0} \times K:\exists z = a \times b \in \C$ such that:
:$\map \phi z = \tuple {a, b}$
by [[Complex Multiplication is Closed]] and $\R \subset \... | Complex Numbers as External Direct Product | https://proofwiki.org/wiki/Complex_Numbers_as_External_Direct_Product | https://proofwiki.org/wiki/Complex_Numbers_as_External_Direct_Product | [
"Complex Numbers",
"External Direct Products"
] | [
"Definition:Group",
"Definition:Zero (Number)/Complex",
"Definition:Complex Number",
"Definition:Multiplication/Complex Numbers",
"Definition:Group",
"Definition:Positive/Real Number",
"Definition:Multiplication/Real Numbers",
"Definition:Circle Group"
] | [
"Definition:Mapping",
"Complex Multiplication is Closed",
"Definition:Surjection",
"Definition:Injective",
"Definition:Injective",
"Definition:Bijection",
"Product of Complex Numbers in Exponential Form",
"Exponential of Sum",
"Definition:Group Homomorphism",
"Definition:Bijection",
"Definition:... |
proofwiki-10669 | Complex Numbers as Quotient Ring of Real Polynomial | Let $\C$ be the set of complex numbers.
Let $P \sqbrk x$ be the set of polynomials over real numbers, where the coefficients of the polynomials are real.
Let $\ideal {x^2 + 1} = \set {\map Q x \paren {x^2 + 1}: \map Q x \in P \sqbrk x}$ be the ideal generated by $x^2 + 1$ in $P \sqbrk x$.
Let $D = P \sqbrk x / \ideal {... | By Division Algorithm of Polynomial, any set in $D$ has an element in the form $a + b x$.
Define $\phi: D \to \C$ as a mapping:
:$\map \phi {\eqclass {a + b x} {x^2 + 1} } = a + b i$
We have that:
:$\forall z = a + b i \in \C : \exists \eqclass {a + b x} {x^2 + 1} \in D$
such that:
:$\map \phi {\eqclass {a + b x} {x^2 ... | Let $\C$ be the [[Definition:Complex Number|set of complex numbers]].
Let $P \sqbrk x$ be the set of [[Definition:Polynomial over Real Numbers|polynomials over real numbers]], where the [[Definition:Polynomial Coefficient|coefficients]] of the [[Definition:Polynomial over Real Numbers|polynomials]] are [[Definition:Re... | By [[Division Algorithm of Polynomial]], any set in $D$ has an [[Definition:Element|element]] in the form $a + b x$.
Define $\phi: D \to \C$ as a [[Definition:Mapping|mapping]]:
:$\map \phi {\eqclass {a + b x} {x^2 + 1} } = a + b i$
We have that:
:$\forall z = a + b i \in \C : \exists \eqclass {a + b x} {x^2 + 1} \in... | Complex Numbers as Quotient Ring of Real Polynomial | https://proofwiki.org/wiki/Complex_Numbers_as_Quotient_Ring_of_Real_Polynomial | https://proofwiki.org/wiki/Complex_Numbers_as_Quotient_Ring_of_Real_Polynomial | [
"Complex Numbers",
"Quotient Rings"
] | [
"Definition:Complex Number",
"Definition:Polynomial/Real Numbers",
"Definition:Coefficient of Polynomial",
"Definition:Polynomial/Real Numbers",
"Definition:Real Number",
"Definition:Ideal of Ring",
"Definition:Generator of Ideal of Ring",
"Definition:Quotient Ring"
] | [
"Division Algorithm of Polynomial",
"Definition:Element",
"Definition:Mapping",
"Definition:Surjection",
"Definition:injection",
"Equality of Complex Numbers",
"Definition:Injection",
"Definition:Bijection",
"Definition:Homomorphism",
"Definition:Quotient Ring",
"Definition:Bijection",
"Defini... |
proofwiki-10670 | Quaternion Modulus in Terms of Conjugate | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
Let $\size {\mathbf x}$ be the modulus of $\mathbf x$.
Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.
Then:
:$\size {\mathbf x}^2 \mathbf 1 = \mathbf x \overline {\mathbf x}$ | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf x \overline {\mathbf x}
| r = \paren {a^2 + b^2 + c^2 + d^2} \mathbf 1
| c = Product of Quaternion with Conjugate
}}
{{eqn | r = \size {\mathbf x}^2 \mathbf 1
| c = {{Defof|Quaternion Modul... | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a [[Definition:Quaternion|quaternion]].
Let $\size {\mathbf x}$ be the [[Definition:Quaternion Modulus|modulus]] of $\mathbf x$.
Let $\overline {\mathbf x}$ be the [[Definition:Conjugate Quaternion|conjugate]] of $\mathbf x$.
Then:
:$\size {... | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf x \overline {\mathbf x}
| r = \paren {a^2 + b^2 + c^2 + d^2} \mathbf 1
| c = [[Product of Quaternion with Conjugate]]
}}
{{eqn | r = \size {\mathbf x}^2 \mathbf 1
| c = {{Defof|Quaternion ... | Quaternion Modulus in Terms of Conjugate | https://proofwiki.org/wiki/Quaternion_Modulus_in_Terms_of_Conjugate | https://proofwiki.org/wiki/Quaternion_Modulus_in_Terms_of_Conjugate | [
"Quaternion Modulus"
] | [
"Definition:Quaternion",
"Definition:Quaternion Modulus",
"Definition:Conjugate Quaternion"
] | [
"Product of Quaternion with Conjugate",
"Category:Quaternion Modulus"
] |
proofwiki-10671 | Sufficient Condition for Quaternion Multiplication to Commute | In general, quaternion multiplication does not commute.
But, for $\mathbf x,\mathbf y \in \H$, $\mathbf x \times \mathbf y = \mathbf y \times \mathbf x$ if any one of the following conditions hold:
{{begin-eqn}}
{{eqn | n = 1a
| l = \mathbf x, \mathbf y
| o = \in
| r = \set {a \mathbf 1 + b \mathbf i ... | === Proof of $\paren 1$ ===
It follows directly from Complex Numbers form Subfield of Quaternions and Complex Multiplication is Commutative.
{{qed|lemma}} | In general, [[Definition:Quaternion/Multiplication|quaternion multiplication]] does not [[Definition:Commutative Operation|commute]].
But, for $\mathbf x,\mathbf y \in \H$, $\mathbf x \times \mathbf y = \mathbf y \times \mathbf x$ if any one of the following conditions hold:
{{begin-eqn}}
{{eqn | n = 1a
| l = \... | === Proof of $\paren 1$ ===
It follows directly from [[Complex Numbers form Subfield of Quaternions]] and [[Complex Multiplication is Commutative]].
{{qed|lemma}} | Sufficient Condition for Quaternion Multiplication to Commute | https://proofwiki.org/wiki/Sufficient_Condition_for_Quaternion_Multiplication_to_Commute | https://proofwiki.org/wiki/Sufficient_Condition_for_Quaternion_Multiplication_to_Commute | [
"Quaternions"
] | [
"Definition:Quaternion/Multiplication",
"Definition:Commutative/Operation"
] | [
"Complex Numbers form Subfield of Quaternions",
"Complex Multiplication is Commutative"
] |
proofwiki-10672 | Complex Conjugation is Involution | Let $z = x + i y$ be a complex number.
Let $\overline z$ denote the complex conjugate of $z$.
Then the operation of complex conjugation is an involution:
:$\overline {\paren {\overline z} } = z$ | {{begin-eqn}}
{{eqn | l = \overline {\paren {\overline z} }
| r = \overline {\paren {\overline {x + i y} } }
| c = Definition of $z$
}}
{{eqn | r = \overline {x - i y}
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = x + i y
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = z
| c = Definiti... | Let $z = x + i y$ be a [[Definition:Complex Number|complex number]].
Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
Then the [[Definition:Unary Operation|operation]] of [[Definition:Complex Conjugation|complex conjugation]] is an [[Definition:Involution (Mapping)|involution]]... | {{begin-eqn}}
{{eqn | l = \overline {\paren {\overline z} }
| r = \overline {\paren {\overline {x + i y} } }
| c = Definition of $z$
}}
{{eqn | r = \overline {x - i y}
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = x + i y
| c = {{Defof|Complex Conjugate}}
}}
{{eqn | r = z
| c = Definiti... | Complex Conjugation is Involution | https://proofwiki.org/wiki/Complex_Conjugation_is_Involution | https://proofwiki.org/wiki/Complex_Conjugation_is_Involution | [
"Complex Conjugates",
"Involutions"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Operation/Unary Operation",
"Definition:Complex Conjugate/Complex Conjugation",
"Definition:Involution (Mapping)"
] | [] |
proofwiki-10673 | Quaternion Modulus of Conjugate | Let $z = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
Let $\overline z$ be the conjugate of $z$.
Let $\cmod z$ be the quaternion modulus of $z$.
Then:
:$\cmod {\overline z} = \cmod z$ | {{begin-eqn}}
{{eqn | l = \cmod z
| r = a^2 + b^2 + c^2 + d^2
| c = {{Defof|Quaternion Modulus}}
}}
{{eqn | l = \cmod {\overline z}
| r = \cmod {a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k}
| c = {{Defof|Quaternion Conjugate}}
}}
{{eqn | r = a^2 + \paren {-b}^2 + \paren {-c}^2 + \paren... | Let $z = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a [[Definition:Quaternion|quaternion]].
Let $\overline z$ be the [[Definition:Quaternion Conjugate|conjugate]] of $z$.
Let $\cmod z$ be the [[Definition:Quaternion Modulus|quaternion modulus]] of $z$.
Then:
:$\cmod {\overline z} = \cmod z$ | {{begin-eqn}}
{{eqn | l = \cmod z
| r = a^2 + b^2 + c^2 + d^2
| c = {{Defof|Quaternion Modulus}}
}}
{{eqn | l = \cmod {\overline z}
| r = \cmod {a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k}
| c = {{Defof|Quaternion Conjugate}}
}}
{{eqn | r = a^2 + \paren {-b}^2 + \paren {-c}^2 + \paren... | Quaternion Modulus of Conjugate | https://proofwiki.org/wiki/Quaternion_Modulus_of_Conjugate | https://proofwiki.org/wiki/Quaternion_Modulus_of_Conjugate | [
"Quaternion Modulus"
] | [
"Definition:Quaternion",
"Definition:Conjugate Quaternion",
"Definition:Quaternion Modulus"
] | [
"Category:Quaternion Modulus"
] |
proofwiki-10674 | Quaternion Conjugation is Involution | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion.
Let $\overline {\mathbf x}$ denote the quaternion conjugate of $\mathbf x$.
Then the operation of quaternion conjugation is an involution:
:$\overline {\paren {\overline {\mathbf x} } } = \mathbf x$ | {{begin-eqn}}
{{eqn | l = \overline {\paren {\overline {\mathbf x} } }
| r = \overline {\paren {\overline {a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k} } }
| c = Definition of $\mathbf x$
}}
{{eqn | r = \overline {a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k}
| c = {{Defof|Quaternio... | Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a [[Definition:Quaternion|quaternion]].
Let $\overline {\mathbf x}$ denote the [[Definition:Quaternion Conjugate|quaternion conjugate]] of $\mathbf x$.
Then the [[Definition:Unary Operation|operation]] of [[Definition:Quaternion Conjugate/Qua... | {{begin-eqn}}
{{eqn | l = \overline {\paren {\overline {\mathbf x} } }
| r = \overline {\paren {\overline {a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k} } }
| c = Definition of $\mathbf x$
}}
{{eqn | r = \overline {a \mathbf 1 - b \mathbf i - c \mathbf j - d \mathbf k}
| c = {{Defof|Quaternio... | Quaternion Conjugation is Involution | https://proofwiki.org/wiki/Quaternion_Conjugation_is_Involution | https://proofwiki.org/wiki/Quaternion_Conjugation_is_Involution | [
"Complex Conjugates",
"Involutions"
] | [
"Definition:Quaternion",
"Definition:Conjugate Quaternion",
"Definition:Operation/Unary Operation",
"Definition:Quaternion Conjugate/Quaternion Conjugation",
"Definition:Involution (Mapping)"
] | [
"Category:Complex Conjugates",
"Category:Involutions"
] |
proofwiki-10675 | Sum of Quaternion Conjugates | Let $\mathbf x, \mathbf y \in \mathbb H$ be quaternions.
Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.
Then:
:$\overline {\mathbf x + \mathbf y} = \overline {\mathbf x} + \overline {\mathbf y}$ | Let:
:$\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$
:$\mathbf y = e \mathbf 1 + f \mathbf i + g \mathbf j + h \mathbf k$
Then:
{{begin-eqn}}
{{eqn | l = \overline {\mathbf x + \mathbf y}
| r = \overline {\paren {a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k} + \paren {e \mathbf 1 + f... | Let $\mathbf x, \mathbf y \in \mathbb H$ be [[Definition:Quaternion|quaternions]].
Let $\overline {\mathbf x}$ be the [[Definition:Quaternion Conjugate|conjugate]] of $\mathbf x$.
Then:
:$\overline {\mathbf x + \mathbf y} = \overline {\mathbf x} + \overline {\mathbf y}$ | Let:
:$\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$
:$\mathbf y = e \mathbf 1 + f \mathbf i + g \mathbf j + h \mathbf k$
Then:
{{begin-eqn}}
{{eqn | l = \overline {\mathbf x + \mathbf y}
| r = \overline {\paren {a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k} + \paren {e \mathbf 1 + ... | Sum of Quaternion Conjugates | https://proofwiki.org/wiki/Sum_of_Quaternion_Conjugates | https://proofwiki.org/wiki/Sum_of_Quaternion_Conjugates | [
"Quaternions"
] | [
"Definition:Quaternion",
"Definition:Conjugate Quaternion"
] | [
"Category:Quaternions"
] |
proofwiki-10676 | Principle of Recursive Definition for Peano Structure | Let $\struct {P, 0, s}$ be a Peano structure.
Let $T$ be a set.
Let $a \in T$.
Let $g: T \to T$ be a mapping.
Then there exists exactly one mapping $f: P \to T$ such that:
:<nowiki>$\forall x \in P: \map f x = \begin{cases}
a & : x = 0 \\
\map g {\map f n} & : x = \map s n
\end{cases}$</nowiki> | For each $n \in P$, define $\map A n$ as:
:$\map A n = \set {h: P \to T \mid \map h 0 = a \land \forall m < n: \map h {\map s n} = \map g {\map h m} }$
{{MissingLinks|Ordering $<$ on Peano Structure}}
First, we prove for all $n \in P$ that $\map A n$ is not empty.
More formally, we prove that $A = \set {n \in P: \map A... | Let $\struct {P, 0, s}$ be a [[Definition:Peano Structure|Peano structure]].
Let $T$ be a [[Definition:Set|set]].
Let $a \in T$.
Let $g: T \to T$ be a [[Definition:Mapping|mapping]].
Then there exists exactly one [[Definition:Mapping|mapping]] $f: P \to T$ such that:
:<nowiki>$\forall x \in P: \map f x = \begin{c... | For each $n \in P$, define $\map A n$ as:
:$\map A n = \set {h: P \to T \mid \map h 0 = a \land \forall m < n: \map h {\map s n} = \map g {\map h m} }$
{{MissingLinks|Ordering $<$ on Peano Structure}}
First, we prove for all $n \in P$ that $\map A n$ is not [[Definition:Empty Set|empty]].
More formally, we prove tha... | Principle of Recursive Definition for Peano Structure | https://proofwiki.org/wiki/Principle_of_Recursive_Definition_for_Peano_Structure | https://proofwiki.org/wiki/Principle_of_Recursive_Definition_for_Peano_Structure | [
"Principle of Recursive Definition",
"Mapping Theory",
"Peano's Axioms"
] | [
"Definition:Peano Structure",
"Definition:Set",
"Definition:Mapping",
"Definition:Mapping"
] | [
"Definition:Empty Set",
"Definition:Mapping",
"Definition:Element",
"Definition:Constant Mapping",
"Axiom:Peano's Axioms",
"Axiom:Peano's Axioms",
"Category:Principle of Recursive Definition",
"Category:Mapping Theory",
"Category:Peano's Axioms"
] |
proofwiki-10677 | Principle of Recursive Definition for Minimally Inductive Set | Let $\omega$ be the minimally inductive set.
Let $T$ be a set.
Let $a \in T$.
Let $g: T \to T$ be a mapping.
Then there exists exactly one mapping $f: \omega \to T$ such that:
:$\forall x \in \omega: \map f x = \begin {cases}
a & : x = \O \\
\map g {\map f n} & : x = n^+
\end {cases}$
where $n^+$ is the successor set o... | {{questionable|The second principle of transfinite recursion that is linked to is not easily applicable to this situation}}
Take the function $F$ generated in Second Principle of Transfinite Recursion.
Set $f = F {\restriction_\omega}$.
{{begin-eqn}}
{{eqn | l = \map f \O
| r = \map F \O
| c = $\O \in \omeg... | Let $\omega$ be the [[Definition:Minimally Inductive Set|minimally inductive set]].
Let $T$ be a [[Definition:Set|set]].
Let $a \in T$.
Let $g: T \to T$ be a [[Definition:Mapping|mapping]].
Then there exists exactly one [[Definition:Mapping|mapping]] $f: \omega \to T$ such that:
:$\forall x \in \omega: \map f x =... | {{questionable|The second principle of transfinite recursion that is linked to is not easily applicable to this situation}}
Take the function $F$ generated in [[Second Principle of Transfinite Recursion]].
Set $f = F {\restriction_\omega}$.
{{begin-eqn}}
{{eqn | l = \map f \O
| r = \map F \O
| c = $\O \in... | Principle of Recursive Definition for Minimally Inductive Set | https://proofwiki.org/wiki/Principle_of_Recursive_Definition_for_Minimally_Inductive_Set | https://proofwiki.org/wiki/Principle_of_Recursive_Definition_for_Minimally_Inductive_Set | [
"Mapping Theory",
"Minimally Inductive Set",
"Principle of Recursive Definition"
] | [
"Definition:Minimally Inductive Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Successor Mapping/Successor Set"
] | [
"Transfinite Recursion Theorem/Theorem 2",
"Transfinite Recursion Theorem/Theorem 2"
] |
proofwiki-10678 | Ordering on 1-Based Natural Numbers is Trichotomy | Let $\N_{> 0}$ be the $1$-based natural numbers.
Let $<$ be the strict ordering on $\N_{>0}$.
Then exactly one of the following is true:
:$(1): \quad a = b$
:$(2): \quad a > b$
:$(3): \quad a < b$
That is, $<$ is a trichotomy on $\N_{> 0}$. | Using the following axioms:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
Axiom $E$ states:
:$\forall a, b \in \N_{>0}$, either:
::$a = b$, in which case $(1)$ holds
::$\exists x \in \N_{> 0}: a = b + x$, in which case, by definition of the ordering defined, $a > b$, in which case $(2)$ holds
::$\exists x \in \N... | Let $\N_{> 0}$ be the [[Definition:1-Based Natural Numbers|$1$-based natural numbers]].
Let $<$ be the [[Definition:Ordering on 1-Based Natural Numbers|strict ordering on $\N_{>0}$]].
Then exactly one of the following is true:
:$(1): \quad a = b$
:$(2): \quad a > b$
:$(3): \quad a < b$
That is, $<$ is a [[Definitio... | Using the [[Axiom:Axiomatization of 1-Based Natural Numbers|following axioms]]:
{{:Axiom:Axiomatization of 1-Based Natural Numbers}}
[[Axiom:Axiomatization of 1-Based Natural Numbers|Axiom $E$]] states:
:$\forall a, b \in \N_{>0}$, either:
::$a = b$, in which case $(1)$ holds
::$\exists x \in \N_{> 0}: a = b + x$, in... | Ordering on 1-Based Natural Numbers is Trichotomy | https://proofwiki.org/wiki/Ordering_on_1-Based_Natural_Numbers_is_Trichotomy | https://proofwiki.org/wiki/Ordering_on_1-Based_Natural_Numbers_is_Trichotomy | [
"Natural Numbers/1-Based"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Ordering on Natural Numbers/1-Based",
"Definition:Trichotomy"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Ordering on Natural Numbers",
"Definition:Ordering on Natural Numbers"
] |
proofwiki-10679 | Product of Quaternion Conjugates | Let $\mathbf x, \mathbf y \in \mathbb H$ be quaternions.
Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.
Then:
:$\overline {\mathbf x \times \mathbf y} = \overline {\mathbf y} \times \overline {\mathbf x}$
but in general:
:$\overline {\mathbf x \times \mathbf y} \ne \overline {\mathbf x} \times \overline {... | Consider the matrix form of $\mathbf x$ and $\mathbf y$:
{{begin-eqn}}
{{eqn | l = \mathbf x
| r = \begin {bmatrix} a & b \\ -\overline b & \overline a \end {bmatrix}
}}
{{eqn | l = \mathbf y
| r = \begin {bmatrix} c & d \\ -\overline d & \overline c \end {bmatrix}
}}
{{end-eqn}}
where $a, b, c, d \in \C$.
... | Let $\mathbf x, \mathbf y \in \mathbb H$ be [[Definition:Quaternion|quaternions]].
Let $\overline {\mathbf x}$ be the [[Definition:Quaternion Conjugate|conjugate]] of $\mathbf x$.
Then:
:$\overline {\mathbf x \times \mathbf y} = \overline {\mathbf y} \times \overline {\mathbf x}$
but in general:
:$\overline {\math... | Consider the [[Matrix Form of Quaternion|matrix form]] of $\mathbf x$ and $\mathbf y$:
{{begin-eqn}}
{{eqn | l = \mathbf x
| r = \begin {bmatrix} a & b \\ -\overline b & \overline a \end {bmatrix}
}}
{{eqn | l = \mathbf y
| r = \begin {bmatrix} c & d \\ -\overline d & \overline c \end {bmatrix}
}}
{{end-eq... | Product of Quaternion Conjugates | https://proofwiki.org/wiki/Product_of_Quaternion_Conjugates | https://proofwiki.org/wiki/Product_of_Quaternion_Conjugates | [
"Quaternions"
] | [
"Definition:Quaternion",
"Definition:Conjugate Quaternion"
] | [
"Matrix Form of Quaternion",
"Complex Conjugation is Field Automorphism of Complex Numbers",
"Complex Conjugation is Involution"
] |
proofwiki-10680 | Quaternion Modulus of Product of Quaternions | Let $\mathbf x, \mathbf y$ be quaternions.
Let $\size {\mathbf x}$ be the modulus of $\mathbf x$.
Then:
:$\size {\mathbf {x y} } = \size {\mathbf x} \size {\mathbf y}$ | Let $\mathbf x, \mathbf y$ be in their matrix form.
Then:
{{begin-eqn}}
{{eqn | l = \size {\mathbf {x y} }
| r = \sqrt {\map \det {\mathbf {x y} } }
| c = {{Defof|Quaternion Modulus}}
}}
{{eqn | r = \sqrt {\map \det {\mathbf x} \map \det {\mathbf y} }
| c = Determinant of Matrix Product
}}
{{eqn | r =... | Let $\mathbf x, \mathbf y$ be [[Definition:Quaternion|quaternions]].
Let $\size {\mathbf x}$ be the [[Definition:Quaternion Modulus|modulus]] of $\mathbf x$.
Then:
:$\size {\mathbf {x y} } = \size {\mathbf x} \size {\mathbf y}$ | Let $\mathbf x, \mathbf y$ be in their [[Matrix Form of Quaternion|matrix form]].
Then:
{{begin-eqn}}
{{eqn | l = \size {\mathbf {x y} }
| r = \sqrt {\map \det {\mathbf {x y} } }
| c = {{Defof|Quaternion Modulus}}
}}
{{eqn | r = \sqrt {\map \det {\mathbf x} \map \det {\mathbf y} }
| c = [[Determinant... | Quaternion Modulus of Product of Quaternions | https://proofwiki.org/wiki/Quaternion_Modulus_of_Product_of_Quaternions | https://proofwiki.org/wiki/Quaternion_Modulus_of_Product_of_Quaternions | [
"Quaternion Modulus"
] | [
"Definition:Quaternion",
"Definition:Quaternion Modulus"
] | [
"Matrix Form of Quaternion",
"Determinant of Matrix Product",
"Exponent Combination Laws",
"Category:Quaternion Modulus"
] |
proofwiki-10681 | Octonion Conjugation is Involution | Let $x = \tuple {a, b}: a, b \in \mathbb H$ be a octonion.
Let $\overline x$ be the conjugate of $x$.
Then:
:$\overline \cdot: x \mapsto \overline x$
is an involution.
That is:
:$\overline {\paren {\overline x} } = x$ | {{begin-eqn}}
{{eqn | l = \overline {\paren {\overline x} }
| r = \overline {\paren {\overline {\tuple {a, b} } } }
| c = {{Defof|Octonion}}
}}
{{eqn | r = \overline {\tuple {\overline a, -b} }
| c = {{Defof|Conjugate of Octonion}}
}}
{{eqn | r = \tuple {\overline {\paren {\overline a} }, -\paren {-b}... | Let $x = \tuple {a, b}: a, b \in \mathbb H$ be a [[Definition:Octonion|octonion]].
Let $\overline x$ be the [[Definition:Conjugate of Octonion|conjugate]] of $x$.
Then:
:$\overline \cdot: x \mapsto \overline x$
is an [[Definition:Involution (Mapping)|involution]].
That is:
:$\overline {\paren {\overline x} } = x$ | {{begin-eqn}}
{{eqn | l = \overline {\paren {\overline x} }
| r = \overline {\paren {\overline {\tuple {a, b} } } }
| c = {{Defof|Octonion}}
}}
{{eqn | r = \overline {\tuple {\overline a, -b} }
| c = {{Defof|Conjugate of Octonion}}
}}
{{eqn | r = \tuple {\overline {\paren {\overline a} }, -\paren {-b}... | Octonion Conjugation is Involution | https://proofwiki.org/wiki/Octonion_Conjugation_is_Involution | https://proofwiki.org/wiki/Octonion_Conjugation_is_Involution | [
"Octonions",
"Involutions"
] | [
"Definition:Octonion",
"Definition:Conjugate of Octonion",
"Definition:Involution (Mapping)"
] | [
"Quaternion Conjugation is Involution",
"Category:Octonions",
"Category:Involutions"
] |
proofwiki-10682 | Ordering on 1-Based Natural Numbers is Compatible with Addition | Let $\N_{> 0}$ be the $1$-based natural numbers.
Let $+$ denote addition on $\N_{>0}$.
Let $<$ be the strict ordering on $\N_{>0}$.
Then:
:$\forall a, b, n \in \N_{>0}: a < b \implies a + n < b + n$
That is, $>$ is compatible with $+$ on $\N_{>0}$. | {{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
| q = \exists c \in \N_{>0}
| l = a
| r = b + c
| c = {{Defof|Ordering on 1-Based Natural Numbers|Ordering on $1$-Based Natural Numbers}}
}}
{{eqn | ll= \leadsto
| l = a + n
| r = \paren {b +... | Let $\N_{> 0}$ be the [[Definition:1-Based Natural Numbers|$1$-based natural numbers]].
Let $+$ denote [[Definition:Addition on 1-Based Natural Numbers|addition]] on $\N_{>0}$.
Let $<$ be the [[Definition:Ordering on 1-Based Natural Numbers|strict ordering on $\N_{>0}$]].
Then:
:$\forall a, b, n \in \N_{>0}: a < b ... | {{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
| q = \exists c \in \N_{>0}
| l = a
| r = b + c
| c = {{Defof|Ordering on 1-Based Natural Numbers|Ordering on $1$-Based Natural Numbers}}
}}
{{eqn | ll= \leadsto
| l = a + n
| r = \paren {b +... | Ordering on 1-Based Natural Numbers is Compatible with Addition | https://proofwiki.org/wiki/Ordering_on_1-Based_Natural_Numbers_is_Compatible_with_Addition | https://proofwiki.org/wiki/Ordering_on_1-Based_Natural_Numbers_is_Compatible_with_Addition | [
"Natural Numbers/1-Based"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Addition on 1-Based Natural Numbers",
"Definition:Ordering on Natural Numbers/1-Based",
"Definition:Relation Compatible with Operation"
] | [
"Natural Number Addition is Associative",
"Natural Number Addition is Commutative",
"Natural Number Addition is Associative"
] |
proofwiki-10683 | Ordering on 1-Based Natural Numbers is Compatible with Multiplication | Let $\N_{> 0}$ be the $1$-based natural numbers.
Let $\times$ denote multiplication on $\N_{>0}$.
Let $<$ be the strict ordering on $\N_{>0}$.
Then:
:$\forall a, b, n \in \N_{>0}: a < b \implies a \times n < b \times n$
That is, $<$ is compatible with $\times$ on $\N_{>0}$. | {{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
| q = \exists c \in \N_{>0}
| l = a
| r = b + c
| c = {{Defof|Ordering on 1-Based Natural Numbers|$<$ on $\N_{>0}$}}
}}
{{eqn | ll= \leadsto
| l = a \times n
| r = \paren {b + c} \times n
... | Let $\N_{> 0}$ be the [[Definition:1-Based Natural Numbers|$1$-based natural numbers]].
Let $\times$ denote [[Definition:Multiplication on 1-Based Natural Numbers|multiplication]] on $\N_{>0}$.
Let $<$ be the [[Definition:Ordering on 1-Based Natural Numbers|strict ordering on $\N_{>0}$]].
Then:
:$\forall a, b, n \i... | {{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
| q = \exists c \in \N_{>0}
| l = a
| r = b + c
| c = {{Defof|Ordering on 1-Based Natural Numbers|$<$ on $\N_{>0}$}}
}}
{{eqn | ll= \leadsto
| l = a \times n
| r = \paren {b + c} \times n
... | Ordering on 1-Based Natural Numbers is Compatible with Multiplication | https://proofwiki.org/wiki/Ordering_on_1-Based_Natural_Numbers_is_Compatible_with_Multiplication | https://proofwiki.org/wiki/Ordering_on_1-Based_Natural_Numbers_is_Compatible_with_Multiplication | [
"Natural Numbers/1-Based"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Natural Number Multiplication/1-Based",
"Definition:Ordering on Natural Numbers/1-Based",
"Definition:Relation Compatible with Operation"
] | [
"Natural Number Multiplication Distributes over Addition"
] |
proofwiki-10684 | Integral between Limits is Independent of Direction | Let $f$ be a real function which is integrable on the interval $\openint a b$.
Then:
:$\ds \int_a^b \map f x \rd x = \int_a^b \map f {a + b - x} \rd x$ | Let $z = a + b - x$.
Then:
:$\dfrac {\d z} {\d x} = -1$
and:
:$x = a \implies z = a + b - a = b$
:$x = b \implies z = a + b - b = a$
So:
{{begin-eqn}}
{{eqn | l = \int_a^b \map f {a + b - x} \rd x
| r = \int_b^a \map f z \paren {-1} \rd z
| c = Integration by Substitution
}}
{{eqn | r = \int_a^b \map f z \r... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]] on the [[Definition:Open Real Interval|interval]] $\openint a b$.
Then:
:$\ds \int_a^b \map f x \rd x = \int_a^b \map f {a + b - x} \rd x$ | Let $z = a + b - x$.
Then:
:$\dfrac {\d z} {\d x} = -1$
and:
:$x = a \implies z = a + b - a = b$
:$x = b \implies z = a + b - b = a$
So:
{{begin-eqn}}
{{eqn | l = \int_a^b \map f {a + b - x} \rd x
| r = \int_b^a \map f z \paren {-1} \rd z
| c = [[Integration by Substitution]]
}}
{{eqn | r = \int_a^b \m... | Integral between Limits is Independent of Direction | https://proofwiki.org/wiki/Integral_between_Limits_is_Independent_of_Direction | https://proofwiki.org/wiki/Integral_between_Limits_is_Independent_of_Direction | [
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Open"
] | [
"Integration by Substitution",
"Reversal of Limits of Definite Integral",
"Category:Definite Integrals"
] |
proofwiki-10685 | Countable Set equals Range of Sequence | Let $S$ be a set.
Then $S$ is countable {{iff}} there exists a sequence $\sequence {s_i}_{i \mathop \in N}$ where $N$ is a subset of $\N$ such that $S$ equals the range of $\sequence {s_i}_{i \mathop \in N}$. | === Necessary Condition ===
Assume that $S$ is countable.
We need to prove that there exists a sequence $\sequence {s_i}_{i \mathop \in N}$, $N \subseteq \N$, such that $S$ equals the range of $\sequence {s_i}_{i \mathop \in N}$.
The range of $\sequence {s_i}_{i \mathop \in N}$ is defined as $\set {s_i: i \in N}$.
;Cas... | Let $S$ be a [[Definition:Set|set]].
Then $S$ is [[Definition:Countable Set|countable]] {{iff}} there exists a [[Definition:Sequence|sequence]] $\sequence {s_i}_{i \mathop \in N}$ where $N$ is a [[Definition:Subset|subset]] of $\N$ such that $S$ equals the [[Definition:Range of Sequence|range]] of $\sequence {s_i}_{i... | === Necessary Condition ===
Assume that $S$ is [[Definition:Countable Set/Definition 2|countable]].
We need to prove that there exists a [[Definition:Sequence|sequence]] $\sequence {s_i}_{i \mathop \in N}$, $N \subseteq \N$, such that $S$ equals the [[Definition:Range of Sequence|range]] of $\sequence {s_i}_{i \matho... | Countable Set equals Range of Sequence | https://proofwiki.org/wiki/Countable_Set_equals_Range_of_Sequence | https://proofwiki.org/wiki/Countable_Set_equals_Range_of_Sequence | [
"Countable Sets"
] | [
"Definition:Set",
"Definition:Countable Set",
"Definition:Sequence",
"Definition:Subset",
"Definition:Range of Sequence"
] | [
"Definition:Countable Set/Definition 2",
"Definition:Sequence",
"Definition:Range of Sequence",
"Definition:Range of Sequence",
"Definition:Empty Set",
"Empty Set is Countable",
"Definition:Countable Set/Definition 2",
"Definition:Sequence/Empty Sequence",
"Definition:Range of Sequence",
"Definiti... |
proofwiki-10686 | Equivalence of Definitions of Connected Set | {{TFAE|def = Connected Set (Topology)|view = Connected Set|context = Topology (Mathematical Branch)|contextview = Topology}}
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a non-empty subset of $S$. | In order to prove equivalence, it is to be shown that if a subset of a topological space is not connected by one of the given definitions above, then it will likewise not be connected by one of the other definitions.
Equivalence of definitions for connected set will then follow by the Rule of Transposition. | {{TFAE|def = Connected Set (Topology)|view = Connected Set|context = Topology (Mathematical Branch)|contextview = Topology}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$. | In order to prove equivalence, it is to be shown that if a [[Definition:Subset|subset]] of a [[Definition:Topological Space|topological space]] is not [[Definition:Connected Set (Topology)|connected]] by one of the given definitions above, then it will likewise not be [[Definition:Connected Set (Topology)|connected]] b... | Equivalence of Definitions of Connected Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Set | [
"Connected Sets (Topology)"
] | [
"Definition:Topological Space",
"Definition:Non-Empty Set",
"Definition:Subset"
] | [
"Definition:Subset",
"Definition:Topological Space",
"Definition:Connected Set (Topology)",
"Definition:Connected Set (Topology)",
"Definition:Connected Set (Topology)",
"Rule of Transposition",
"Definition:Connected Set (Topology)"
] |
proofwiki-10687 | Initial Segment of Natural Numbers determined by Zero is Empty | Let $\N_k$ denote the initial segment of the natural numbers determined by $k$:
:$\N_k = \set {0, 1, 2, 3, \ldots, k - 1}$
Then $\N_0 = \O$. | From the definition of $\N_0$:
:$\N_0 = \set {n \in \N: n < 0}$
From the definition of zero, $0$ is the minimal element of $\N$.
So there is no element $n$ of $\N$ such that $n < 0$.
Thus $\N_0 = \O$.
{{qed}}
Category:Natural Numbers
qbigqp7k4plig58dwsn1gh24fsrjyqb | Let $\N_k$ denote the [[Definition:Initial Segment of Natural Numbers|initial segment of the natural numbers]] determined by $k$:
:$\N_k = \set {0, 1, 2, 3, \ldots, k - 1}$
Then $\N_0 = \O$. | From the [[Definition:Initial Segment of Natural Numbers|definition of $\N_0$]]:
:$\N_0 = \set {n \in \N: n < 0}$
From the definition of [[Definition:Zero (Number)|zero]], $0$ is the [[Definition:Minimal Element|minimal element of $\N$]].
So there is no [[Definition:Element|element]] $n$ of $\N$ such that $n < 0$.
... | Initial Segment of Natural Numbers determined by Zero is Empty | https://proofwiki.org/wiki/Initial_Segment_of_Natural_Numbers_determined_by_Zero_is_Empty | https://proofwiki.org/wiki/Initial_Segment_of_Natural_Numbers_determined_by_Zero_is_Empty | [
"Natural Numbers"
] | [
"Definition:Initial Segment of Natural Numbers"
] | [
"Definition:Initial Segment of Natural Numbers",
"Definition:Zero (Number)",
"Definition:Minimal/Element",
"Definition:Element",
"Category:Natural Numbers"
] |
proofwiki-10688 | Initial Segment of One-Based Natural Numbers determined by Zero is Empty | Let $\N^*_k$ denote the initial segment of the one-based natural numbers determined by $k$:
:$\N^*_k = \set {1, 2, 3, \ldots, k - 1, k}$
Then $\N^*_0 = \O$. | From the definition of $\N^*_0$:
:$\N^*_0 = \set {n \in \N_{>0}: n \le 0}$
From the definition of one, the minimal element of $\N_{>0}$ is $1$.
From Zero Strictly Precedes One we know that $0 < 1$.
So there is no element $n$ of $\N_{>0}$ such that $n \le 0$.
Thus $\N^*_0 = \O$.
{{qed}} | Let $\N^*_k$ denote the [[Definition:Initial Segment of One-Based Natural Numbers|initial segment of the one-based natural numbers]] determined by $k$:
:$\N^*_k = \set {1, 2, 3, \ldots, k - 1, k}$
Then $\N^*_0 = \O$. | From the [[Definition:Initial Segment of One-Based Natural Numbers|definition of $\N^*_0$]]:
:$\N^*_0 = \set {n \in \N_{>0}: n \le 0}$
From the definition of [[Definition:One|one]], the [[Definition:Minimal Element|minimal element]] of $\N_{>0}$ is $1$.
From [[Zero Strictly Precedes One]] we know that $0 < 1$.
So ... | Initial Segment of One-Based Natural Numbers determined by Zero is Empty | https://proofwiki.org/wiki/Initial_Segment_of_One-Based_Natural_Numbers_determined_by_Zero_is_Empty | https://proofwiki.org/wiki/Initial_Segment_of_One-Based_Natural_Numbers_determined_by_Zero_is_Empty | [
"Natural Numbers"
] | [
"Definition:Initial Segment of Natural Numbers/One-Based"
] | [
"Definition:Initial Segment of Natural Numbers/One-Based",
"Definition:One",
"Definition:Minimal/Element",
"Zero Strictly Precedes One",
"Definition:Element"
] |
proofwiki-10689 | Heine-Borel Theorem/Real Line/Closed and Bounded Interval | Let $\closedint a b$, $a < b$, be a closed and bounded real interval.
Let $S$ be a set of open real sets.
Let $S$ be a cover of $\closedint a b$.
Then there is a finite subset of $S$ that covers $\closedint a b$. | Consider the set $T = \set {x \in \closedint a b: \closedint a x \text { is covered by a finite subset of } S}$. | Let $\closedint a b$, $a < b$, be a [[Definition:Closed Real Interval|closed and bounded real interval]].
Let $S$ be a [[Definition:Set|set]] of [[Definition:Open Set (Real Analysis)|open real sets]].
Let $S$ be a [[Definition:Cover of Set|cover]] of $\closedint a b$.
Then there is a [[Definition:Finite Set|finite]... | Consider the [[Definition:Set|set]] $T = \set {x \in \closedint a b: \closedint a x \text { is covered by a finite subset of } S}$. | Heine-Borel Theorem/Real Line/Closed and Bounded Interval | https://proofwiki.org/wiki/Heine-Borel_Theorem/Real_Line/Closed_and_Bounded_Interval | https://proofwiki.org/wiki/Heine-Borel_Theorem/Real_Line/Closed_and_Bounded_Interval | [
"Real Analysis",
"Direct Proofs"
] | [
"Definition:Real Interval/Closed",
"Definition:Set",
"Definition:Open Set/Real Analysis",
"Definition:Cover of Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Cover of Set"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Set",
"Definition:Set"
] |
proofwiki-10690 | Addition on 1-Based Natural Numbers is Cancellable | Let $\N_{> 0}$ be the $1$-based natural numbers.
Let $+$ be addition on $\N_{>0}$.
Then:
:$\forall a, b, c \in \N_{>0}: a + c = b + c \implies a = b$
:$\forall a, b, c \in \N_{>0}: a + b = a + c \implies b = c$
That is, $+$ is cancellable on $\N_{>0}$. | By Ordering on $1$-Based Natural Numbers is Trichotomy, one and only one of the following holds:
:$a = b$
:$a < b$
:$b < a$
Suppose $a < b$.
Then by Ordering on $1$-Based Natural Numbers is Compatible with Addition:
:$a + c < b + c$
By Ordering on $1$-Based Natural Numbers is Trichotomy, this contradicts the fact that ... | Let $\N_{> 0}$ be the [[Axiom:Axiomatization of 1-Based Natural Numbers|$1$-based natural numbers]].
Let $+$ be [[Definition:Addition on 1-Based Natural Numbers|addition]] on $\N_{>0}$.
Then:
:$\forall a, b, c \in \N_{>0}: a + c = b + c \implies a = b$
:$\forall a, b, c \in \N_{>0}: a + b = a + c \implies b = c$
T... | By [[Ordering on 1-Based Natural Numbers is Trichotomy|Ordering on $1$-Based Natural Numbers is Trichotomy]], one and only one of the following holds:
:$a = b$
:$a < b$
:$b < a$
Suppose $a < b$.
Then by [[Ordering on 1-Based Natural Numbers is Compatible with Addition|Ordering on $1$-Based Natural Numbers is Compati... | Addition on 1-Based Natural Numbers is Cancellable | https://proofwiki.org/wiki/Addition_on_1-Based_Natural_Numbers_is_Cancellable | https://proofwiki.org/wiki/Addition_on_1-Based_Natural_Numbers_is_Cancellable | [
"Natural Numbers/1-Based"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Addition on 1-Based Natural Numbers",
"Definition:Cancellable Operation"
] | [
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Ordering on 1-Based Natural Numbers is Compatible with Addition",
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Ordering on 1-Based Natural Numbers is Compatible with Addition",
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Definition:R... |
proofwiki-10691 | Addition on 1-Based Natural Numbers is Cancellable for Ordering | Let $\N_{> 0}$ be the $1$-based natural numbers.
Let $<$ be the strict ordering on $\N_{>0}$.
Let $+$ be addition on $\N_{>0}$.
Then:
:$\forall a, b, c \in \N_{>0}: a + c < b + c \implies a < b$
:$\forall a, b, c \in \N_{>0}: a + b < a + c \implies b < c$
That is, $+$ is cancellable on $\N_{>0}$ for $<$. | By Ordering on $1$-Based Natural Numbers is Trichotomy, one and only one of the following holds:
:$a = b$
:$a < b$
:$b < a$
Let $a + c < b + c$.
Suppose $a = b$.
Then by Ordering on $1$-Based Natural Numbers is Compatible with Addition:
:$a + c = b + c$
By Ordering on $1$-Based Natural Numbers is Trichotomy, this contr... | Let $\N_{> 0}$ be the [[Definition:1-Based Natural Numbers|$1$-based natural numbers]].
Let $<$ be the [[Definition:Ordering on 1-Based Natural Numbers|strict ordering on $\N_{>0}$]].
Let $+$ be [[Definition:Addition on 1-Based Natural Numbers|addition]] on $\N_{>0}$.
Then:
:$\forall a, b, c \in \N_{>0}: a + c < b ... | By [[Ordering on 1-Based Natural Numbers is Trichotomy|Ordering on $1$-Based Natural Numbers is Trichotomy]], one and only one of the following holds:
:$a = b$
:$a < b$
:$b < a$
Let $a + c < b + c$.
Suppose $a = b$.
Then by [[Ordering on 1-Based Natural Numbers is Compatible with Addition|Ordering on $1$-Based Natu... | Addition on 1-Based Natural Numbers is Cancellable for Ordering | https://proofwiki.org/wiki/Addition_on_1-Based_Natural_Numbers_is_Cancellable_for_Ordering | https://proofwiki.org/wiki/Addition_on_1-Based_Natural_Numbers_is_Cancellable_for_Ordering | [
"Natural Numbers/1-Based"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Ordering on Natural Numbers/1-Based",
"Definition:Addition on 1-Based Natural Numbers",
"Definition:Cancellable Operation"
] | [
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Ordering on 1-Based Natural Numbers is Compatible with Addition",
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Ordering on 1-Based Natural Numbers is Compatible with Addition",
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Definition:R... |
proofwiki-10692 | Multiplication on 1-Based Natural Numbers is Cancellable | Let $\N_{> 0}$ be the $1$-based natural numbers.
Let $\times$ be multiplication on $\N_{>0}$.
Then:
:$\forall a, b, c \in \N_{>0}: a \times c = b \times c \implies a = b$
:$\forall a, b, c \in \N_{>0}: a \times b = a \times c \implies b = c$
That is, $\times$ is cancellable on $\N_{>0}$. | By Ordering on $1$-Based Natural Numbers is Trichotomy, one and only one of the following holds:
:$a = b$
:$a < b$
:$b < a$
Suppose $a < b$.
Then by Ordering on $1$-Based Natural Numbers is Compatible with Multiplication:
:$a \times c < b \times c$
By Ordering on $1$-Based Natural Numbers is Trichotomy, this contradict... | Let $\N_{> 0}$ be the [[Definition:1-Based Natural Numbers|$1$-based natural numbers]].
Let $\times$ be [[Definition:Multiplication on 1-Based Natural Numbers|multiplication]] on $\N_{>0}$.
Then:
:$\forall a, b, c \in \N_{>0}: a \times c = b \times c \implies a = b$
:$\forall a, b, c \in \N_{>0}: a \times b = a \tim... | By [[Ordering on 1-Based Natural Numbers is Trichotomy|Ordering on $1$-Based Natural Numbers is Trichotomy]], one and only one of the following holds:
:$a = b$
:$a < b$
:$b < a$
Suppose $a < b$.
Then by [[Ordering on 1-Based Natural Numbers is Compatible with Multiplication|Ordering on $1$-Based Natural Numbers is C... | Multiplication on 1-Based Natural Numbers is Cancellable | https://proofwiki.org/wiki/Multiplication_on_1-Based_Natural_Numbers_is_Cancellable | https://proofwiki.org/wiki/Multiplication_on_1-Based_Natural_Numbers_is_Cancellable | [
"Natural Numbers/1-Based",
"Natural Number Multiplication"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Natural Number Multiplication/1-Based",
"Definition:Cancellable Operation"
] | [
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Ordering on 1-Based Natural Numbers is Compatible with Multiplication",
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Ordering on 1-Based Natural Numbers is Compatible with Multiplication",
"Ordering on 1-Based Natural Numbers is Trichotomy",
"... |
proofwiki-10693 | Multiplication on 1-Based Natural Numbers is Cancellable for Ordering | Let $\N_{> 0}$ be the $1$-based natural numbers.
Let $\times$ be multiplication on $\N_{>0}$.
Let $<$ be the strict ordering on $\N_{>0}$.
Then:
:$\forall a, b, c \in \N_{>0}: a \times c < b \times c \implies a < b$
:$\forall a, b, c \in \N_{>0}: a \times b < a \times c \implies b < c$
That is, $\times$ is cancellable ... | By Ordering on $1$-Based Natural Numbers is Trichotomy, one and only one of the following holds:
:$a = b$
:$a < b$
:$b < a$
Let $a \times c < b \times c$.
Suppose $a = b$.
Then by Ordering on $1$-Based Natural Numbers is Compatible with Multiplication:
:$a \times c = b \times c$
By Ordering on $1$-Based Natural Numbers... | Let $\N_{> 0}$ be the [[Definition:1-Based Natural Numbers|$1$-based natural numbers]].
Let $\times$ be [[Definition:Multiplication on 1-Based Natural Numbers|multiplication]] on $\N_{>0}$.
Let $<$ be the [[Definition:Ordering on 1-Based Natural Numbers|strict ordering on $\N_{>0}$]].
Then:
:$\forall a, b, c \in \N... | By [[Ordering on 1-Based Natural Numbers is Trichotomy|Ordering on $1$-Based Natural Numbers is Trichotomy]], one and only one of the following holds:
:$a = b$
:$a < b$
:$b < a$
Let $a \times c < b \times c$.
Suppose $a = b$.
Then by [[Ordering on 1-Based Natural Numbers is Compatible with Multiplication|Ordering o... | Multiplication on 1-Based Natural Numbers is Cancellable for Ordering | https://proofwiki.org/wiki/Multiplication_on_1-Based_Natural_Numbers_is_Cancellable_for_Ordering | https://proofwiki.org/wiki/Multiplication_on_1-Based_Natural_Numbers_is_Cancellable_for_Ordering | [
"Natural Numbers/1-Based",
"Natural Number Multiplication"
] | [
"Axiom:Axiomatization of 1-Based Natural Numbers",
"Definition:Natural Number Multiplication/1-Based",
"Definition:Ordering on Natural Numbers/1-Based",
"Definition:Cancellable Operation"
] | [
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Ordering on 1-Based Natural Numbers is Compatible with Multiplication",
"Ordering on 1-Based Natural Numbers is Trichotomy",
"Ordering on 1-Based Natural Numbers is Compatible with Multiplication",
"Ordering on 1-Based Natural Numbers is Trichotomy",
"... |
proofwiki-10694 | Index of Trivial Subgroup is Cardinality of Group | Let $G$ be a group whose identity element is $e$.
Let $\set e$ be the trivial subgroup of $G$.
Then:
:$\index G {\set e} = \order G$
where:
:$\index G {\set e}$ denotes the index of $\set e$ in $G$
:$\order G$ denotes the cardinality of $G$. | By definition of cardinality and the trivial subgroup:
:$\order {\set e} = 1$
From Lagrange's Theorem:
:$\index G {\set e} = \dfrac {\order G} {\order {\set e} } = \dfrac {\order G} 1 = \order G$
{{qed}} | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity element]] is $e$.
Let $\set e$ be the [[Definition:Trivial Subgroup|trivial subgroup]] of $G$.
Then:
:$\index G {\set e} = \order G$
where:
:$\index G {\set e}$ denotes the [[Definition:Index of Subgroup|index]] of $\set e$ in $G$
:... | By definition of [[Definition:Cardinality|cardinality]] and the [[Definition:Trivial Subgroup|trivial subgroup]]:
:$\order {\set e} = 1$
From [[Lagrange's Theorem (Group Theory)|Lagrange's Theorem]]:
:$\index G {\set e} = \dfrac {\order G} {\order {\set e} } = \dfrac {\order G} 1 = \order G$
{{qed}} | Index of Trivial Subgroup is Cardinality of Group | https://proofwiki.org/wiki/Index_of_Trivial_Subgroup_is_Cardinality_of_Group | https://proofwiki.org/wiki/Index_of_Trivial_Subgroup_is_Cardinality_of_Group | [
"Order of Groups",
"Index of Subgroups"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Trivial Subgroup",
"Definition:Index of Subgroup",
"Definition:Cardinality"
] | [
"Definition:Cardinality",
"Definition:Trivial Subgroup",
"Lagrange's Theorem (Group Theory)"
] |
proofwiki-10695 | Index is One iff Subgroup equals Group | Let $G$ be a group whose identity element is $e$.
Let $H$ be a subgroup of $G$.
Then:
:$\index G H = 1 \iff G = H$
where $\index G H$ denotes the index of $H$ in $G$. | For finite groups, we can apply Lagrange's Theorem:
:$\index G H = \dfrac {\order G} {\order H}$
But then:
:$\dfrac {\order G} {\order H} = 1 \iff \order G = \order H$
Hence the result.
For the general case (including infinite groups) we need to consider the (left) coset space $G / H$.
Note that we must have $e H = H \... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity element]] is $e$.
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Then:
:$\index G H = 1 \iff G = H$
where $\index G H$ denotes the [[Definition:Index of Subgroup|index]] of $H$ in $G$. | For [[Definition:Finite Group|finite groups]], we can apply [[Lagrange's Theorem (Group Theory)|Lagrange's Theorem]]:
:$\index G H = \dfrac {\order G} {\order H}$
But then:
:$\dfrac {\order G} {\order H} = 1 \iff \order G = \order H$
Hence the result.
For the general case (including [[Definition:Infinite Group|infi... | Index is One iff Subgroup equals Group | https://proofwiki.org/wiki/Index_is_One_iff_Subgroup_equals_Group | https://proofwiki.org/wiki/Index_is_One_iff_Subgroup_equals_Group | [
"Order of Groups",
"Index of Subgroups"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Index of Subgroup"
] | [
"Definition:Finite Group",
"Lagrange's Theorem (Group Theory)",
"Definition:Infinite Group",
"Definition:Coset Space/Left Coset Space"
] |
proofwiki-10696 | Set Equality is Equivalence Relation | Let $S$ be a set.
Set equality is an equivalence relation on the power set $\powerset S$ of $S$. | Checking in turn each of the criteria for equivalence: | Let $S$ be a [[Definition:Set|set]].
[[Definition:Set Equality|Set equality]] is an [[Definition:Equivalence Relation|equivalence relation]] on the [[Definition:Power Set|power set]] $\powerset S$ of $S$. | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Set Equality is Equivalence Relation | https://proofwiki.org/wiki/Set_Equality_is_Equivalence_Relation | https://proofwiki.org/wiki/Set_Equality_is_Equivalence_Relation | [
"Set Theory",
"Examples of Equivalence Relations"
] | [
"Definition:Set",
"Definition:Set Equality",
"Definition:Equivalence Relation",
"Definition:Power Set"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-10697 | Subset Relation is Antisymmetric | The subset relation is '''antisymmetric''':
:$\paren {x \subseteq y} \land \paren {y \subseteq x} \iff x = y$
where $x$ and $y$ are sets. | This is a direct statement of the definition of set equality:
:$x = y := \paren {x \subseteq y} \land \paren {y \subseteq x}$
{{qed}} | The [[Definition:Subset Relation|subset relation]] is '''[[Definition:Antisymmetric Relation|antisymmetric]]''':
:$\paren {x \subseteq y} \land \paren {y \subseteq x} \iff x = y$
where $x$ and $y$ are [[Definition:Set|sets]]. | This is a direct statement of the definition of [[Definition:Set Equality/Definition 2|set equality]]:
:$x = y := \paren {x \subseteq y} \land \paren {y \subseteq x}$
{{qed}} | Subset Relation is Antisymmetric | https://proofwiki.org/wiki/Subset_Relation_is_Antisymmetric | https://proofwiki.org/wiki/Subset_Relation_is_Antisymmetric | [
"Subset Relation",
"Examples of Antisymmetric Relations"
] | [
"Definition:Subset Relation",
"Definition:Antisymmetric Relation",
"Definition:Set"
] | [
"Definition:Set Equality/Definition 2"
] |
proofwiki-10698 | There Exists No Universal Set | There exists no set which is an absolutely universal set.
That is:
:$\map \neg {\exists \, \UU: \forall T: T \in \UU}$
where $T$ is any arbitrary object at all.
That is, a set that contains ''everything'' cannot exist. | {{AimForCont}} such a $\UU$ exists.
Using the Axiom of Specification, we can create the set:
:$R = \set {x \in \UU: x \notin x}$
But from Russell's Paradox, this set cannot exist.
Thus:
:$R \notin \UU$
and so $\UU$ cannot contain everything.
{{qed}} | There exists no [[Definition:Set|set]] which is an absolutely [[Definition:Universal Set|universal set]].
That is:
:$\map \neg {\exists \, \UU: \forall T: T \in \UU}$
where $T$ is any arbitrary [[Definition:Object|object]] at all.
That is, a [[Definition:Set|set]] that contains ''everything'' cannot exist. | {{AimForCont}} such a $\UU$ exists.
Using the [[Axiom:Axiom of Specification (Sets)|Axiom of Specification]], we can create the [[Definition:Set|set]]:
:$R = \set {x \in \UU: x \notin x}$
But from [[Russell's Paradox]], this [[Definition:Set|set]] cannot exist.
Thus:
:$R \notin \UU$
and so $\UU$ cannot contain ever... | There Exists No Universal Set/Proof 1 | https://proofwiki.org/wiki/There_Exists_No_Universal_Set | https://proofwiki.org/wiki/There_Exists_No_Universal_Set/Proof_1 | [
"There Exists No Universal Set",
"Universal Set",
"Naive Set Theory"
] | [
"Definition:Set",
"Definition:Universal Set",
"Definition:Object",
"Definition:Set"
] | [
"Axiom:Axiom of Specification/Set Theory",
"Definition:Set",
"Russell's Paradox",
"Definition:Set"
] |
proofwiki-10699 | There Exists No Universal Set | There exists no set which is an absolutely universal set.
That is:
:$\map \neg {\exists \, \UU: \forall T: T \in \UU}$
where $T$ is any arbitrary object at all.
That is, a set that contains ''everything'' cannot exist. | Let $\SS$ be the set of all sets.
Then $\SS$ must be an element of itself:
:$\SS \owns \SS$
Thus we have an infinite descending sequence of membership:
:$\SS \owns \SS \owns \SS \owns \cdots$
But by No Infinitely Descending Membership Chains, no such sequence exists, a contradiction.
{{qed}} | There exists no [[Definition:Set|set]] which is an absolutely [[Definition:Universal Set|universal set]].
That is:
:$\map \neg {\exists \, \UU: \forall T: T \in \UU}$
where $T$ is any arbitrary [[Definition:Object|object]] at all.
That is, a [[Definition:Set|set]] that contains ''everything'' cannot exist. | Let $\SS$ be the [[Definition:Set|set]] of all [[Definition:Set|sets]].
Then $\SS$ must be an [[Definition:Element|element]] of itself:
:$\SS \owns \SS$
Thus we have an infinite descending sequence of membership:
:$\SS \owns \SS \owns \SS \owns \cdots$
But by [[No Infinitely Descending Membership Chains]], no such ... | There Exists No Universal Set/Proof 3 | https://proofwiki.org/wiki/There_Exists_No_Universal_Set | https://proofwiki.org/wiki/There_Exists_No_Universal_Set/Proof_3 | [
"There Exists No Universal Set",
"Universal Set",
"Naive Set Theory"
] | [
"Definition:Set",
"Definition:Universal Set",
"Definition:Object",
"Definition:Set"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Element",
"No Infinitely Descending Membership Chains",
"Definition:Contradiction"
] |
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