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-16300 | Set Consisting of Empty Set is not Empty | Let $S$ be the set defined as:
:$S = \set \O$
Then $S$ is not the empty set.
That is:
:$\O \ne \set \O$ | We have:
:$\O \in \set \O$
and so:
:$\neg \paren {\forall x: x \notin \O}$
The result follows by definition of the empty set.
{{qed}} | Let $S$ be the [[Definition:Set|set]] defined as:
:$S = \set \O$
Then $S$ is not the [[Definition:Empty Set|empty set]].
That is:
:$\O \ne \set \O$ | We have:
:$\O \in \set \O$
and so:
:$\neg \paren {\forall x: x \notin \O}$
The result follows by definition of the [[Definition:Empty Set|empty set]].
{{qed}} | Set Consisting of Empty Set is not Empty | https://proofwiki.org/wiki/Set_Consisting_of_Empty_Set_is_not_Empty | https://proofwiki.org/wiki/Set_Consisting_of_Empty_Set_is_not_Empty | [
"Empty Set"
] | [
"Definition:Set",
"Definition:Empty Set"
] | [
"Definition:Empty Set"
] |
proofwiki-16301 | Elements of Ordered Pair do not Commute | Let $\set {a, b}$ be a doubleton, so that $a$ and $b$ are distinct objects.
Let $\tuple {a, b}$ denote the ordered pair such that the first coordinate is $a$ and the second coordinate is $b$.
Then:
:$\tuple {a, b} \ne \tuple {b, a}$ | By the Kuratowski formalization of $\tuple {a, b}$:
:$\tuple {a, b} = \set {\set a, \set {a, b} }$
and by Equality of Ordered Pairs:
:$\tuple {a, b} = \tuple {b, a} \iff a = b$
But $a \ne b$ and so:
:$\tuple {a, b} \ne \tuple {b, a}$
{{qed}} | Let $\set {a, b}$ be a [[Definition:Doubleton|doubleton]], so that $a$ and $b$ are [[Definition:Distinct Objects|distinct objects]].
Let $\tuple {a, b}$ denote the [[Definition:Ordered Pair|ordered pair]] such that the [[Definition:Coordinate of Ordered Pair|first coordinate]] is $a$ and the [[Definition:Coordinate of... | By the [[Definition:Kuratowski Formalization of Ordered Pair|Kuratowski formalization of $\tuple {a, b}$]]:
:$\tuple {a, b} = \set {\set a, \set {a, b} }$
and by [[Equality of Ordered Pairs]]:
:$\tuple {a, b} = \tuple {b, a} \iff a = b$
But $a \ne b$ and so:
:$\tuple {a, b} \ne \tuple {b, a}$
{{qed}} | Elements of Ordered Pair do not Commute | https://proofwiki.org/wiki/Elements_of_Ordered_Pair_do_not_Commute | https://proofwiki.org/wiki/Elements_of_Ordered_Pair_do_not_Commute | [
"Ordered Pairs"
] | [
"Definition:Doubleton",
"Definition:Distinct/Plural",
"Definition:Ordered Pair",
"Definition:Coordinate System/Coordinate/Element of Ordered Pair",
"Definition:Coordinate System/Coordinate/Element of Ordered Pair"
] | [
"Definition:Ordered Pair/Kuratowski Formalization",
"Equality of Ordered Pairs"
] |
proofwiki-16302 | Definite Integral to Infinity of Cosine m x over x Squared plus a Squared | :$\ds \int_0^\infty \frac {\cos m x} {x^2 + a^2} \rd x = \frac \pi {2 a} e^{-m a}$ | From Definite Integral of Even Function:
:$\ds \frac 1 2 \int_{-\infty}^\infty \frac {\cos m x} {x^2 + a^2} \rd x = \int_0^\infty \frac {\cos m x} {x^2 + a^2} \rd x$
Let $R$ be a positive real number with $R > a$.
Let $C_1$ be the straight line segment from $-R$ to $R$.
Let $C_2$ be the arc of the circle of radius $R$ ... | :$\ds \int_0^\infty \frac {\cos m x} {x^2 + a^2} \rd x = \frac \pi {2 a} e^{-m a}$ | From [[Definite Integral of Even Function]]:
:$\ds \frac 1 2 \int_{-\infty}^\infty \frac {\cos m x} {x^2 + a^2} \rd x = \int_0^\infty \frac {\cos m x} {x^2 + a^2} \rd x$
Let $R$ be a [[Definition:Positive Real Number|positive real number]] with $R > a$.
Let $C_1$ be the [[Definition:Straight Line Segment|straight li... | Definite Integral to Infinity of Cosine m x over x Squared plus a Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_m_x_over_x_Squared_plus_a_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_m_x_over_x_Squared_plus_a_Squared | [
"Definite Integrals involving Cosine Function"
] | [] | [
"Definite Integral of Even Function",
"Definition:Positive/Real Number",
"Definition:Line/Straight Line Segment",
"Definition:Circle/Arc",
"Definition:Anticlockwise",
"Euler's Formula",
"Linear Combination of Integrals/Definite",
"Definite Integral of Odd Function",
"Definition:Integration/Integrand... |
proofwiki-16303 | Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule | :$f + g: \struct {S, \tau_{_S} } \to \struct{R, \tau_{_R} }$ is continuous. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct {R, +, *, \tau_{_R} }$ is a topological division ring.
From Sum Rule for Continuous Mappings to Topological Division Ring:
:$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$f + g: \struct {S, \tau_{_S} } \to \struct{R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct {R, +, *, \tau_{_R} }$ is a [[Definition:Topological Division Ring|topological division ring]].
From [[Sum Rule for Continuous Mappings to Topological Division Ring]]:
:$f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[Defin... | Combination Theorem for Continuous Mappings/Normed Division Ring/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Sum_Rule | [
"Combination Theorem for Continuous Mappings to Normed Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Combination Theorem for Continuous Mappings/Topological Division Ring/Sum Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16304 | Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple Rule | :$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous
:$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct {R, +, *, \tau_{_R} }$ is a topological division ring.
From Multiple Rule for Continuous Mappings to Topological Division Ring:
:$\lambda * f, f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ are continuous mappings.
{{qed}} | :$\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]
:$f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct {R, +, *, \tau_{_R} }$ is a [[Definition:Topological Division Ring|topological division ring]].
From [[Multiple Rule for Continuous Mappings to Topological Division Ring]]:
:$\lambda * f, f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \... | Combination Theorem for Continuous Mappings/Normed Division Ring/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Multiple_Rule | [
"Combination Theorem for Continuous Mappings to Normed Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set",
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Combination Theorem for Continuous Mappings/Topological Division Ring/Multiple Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16305 | Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule | :$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct {R, +, *, \tau_{_R} }$ is a topological division ring.
From Product Rule for Continuous Mappings to Topological Division Ring:
:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct {R, +, *, \tau_{_R} }$ is a [[Definition:Topological Division Ring|topological division ring]].
From [[Product Rule for Continuous Mappings to Topological Division Ring]]:
:$f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a [[D... | Combination Theorem for Continuous Mappings/Normed Division Ring/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Product_Rule | [
"Combination Theorem for Continuous Mappings to Normed Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Combination Theorem for Continuous Mappings/Topological Division Ring/Product Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16306 | Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule | :$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is continuous. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct {R, +, *, \tau_{_R} }$ is a topological division ring.
From Inverse Rule for Continuous Mappings to Topological Division Ring:
:$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
{{qed}} | :$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is [[Definition:Continuous Mapping on Set|continuous]]. | From {{Corollary|Normed Division Ring Operations are Continuous}}:
:$\struct {R, +, *, \tau_{_R} }$ is a [[Definition:Topological Division Ring|topological division ring]].
From [[Inverse Rule for Continuous Mappings to Topological Division Ring]]:
:$g^{-1}: \struct {U, \tau_{_U} } \to \struct {R, \tau_{_R} }$ is a [[... | Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Inverse_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Normed_Division_Ring/Inverse_Rule | [
"Combination Theorem for Continuous Mappings to Normed Division Ring"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Division Ring",
"Combination Theorem for Continuous Mappings/Topological Division Ring/Inverse Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16307 | Definite Integral to Infinity of x by Sine m x over x Squared plus a Squared | :$\ds \int_0^\infty \frac {x \sin m x} {x^2 + a^2} \rd x = \frac \pi 2 e^{-m a}$ | From Definite Integral of Even Function:
:$\ds \frac 1 2 \int_{-\infty}^\infty \frac {x \sin m x} {x^2 + a^2} \rd x = \int_0^\infty \frac {x \sin m x} {x^2 + a^2} \rd x$
With the aim of integrating over the domain, we split the domain up into $2$ components as follows:
Let $R$ be a positive real number with $R > a$.
Le... | :$\ds \int_0^\infty \frac {x \sin m x} {x^2 + a^2} \rd x = \frac \pi 2 e^{-m a}$ | From [[Definite Integral of Even Function]]:
:$\ds \frac 1 2 \int_{-\infty}^\infty \frac {x \sin m x} {x^2 + a^2} \rd x = \int_0^\infty \frac {x \sin m x} {x^2 + a^2} \rd x$
With the aim of integrating over the [[Definition:Domain of Mapping|domain]], we split the [[Definition:Domain of Mapping|domain]] up into $2$ ... | Definite Integral to Infinity of x by Sine m x over x Squared plus a Squared/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_by_Sine_m_x_over_x_Squared_plus_a_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_by_Sine_m_x_over_x_Squared_plus_a_Squared/Proof_1 | [
"Definite Integrals involving Sine Function",
"Definite Integral to Infinity of x by Sine m x over x Squared plus a Squared"
] | [] | [
"Definite Integral of Even Function",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Component (Topology)",
"Definition:Positive/Real Number",
"Definition:Line/Straight Line",
"Definition:Circle/Arc",
"Definition:Circle/Radius",
"Definition:Coordinate... |
proofwiki-16308 | Definite Integral to Infinity of x by Sine m x over x Squared plus a Squared | :$\ds \int_0^\infty \frac {x \sin m x} {x^2 + a^2} \rd x = \frac \pi 2 e^{-m a}$ | From Definite Integral to Infinity of $\dfrac {\cos m x} {x^2 + a^2}$:
:$\ds \int_0^\infty \frac {\cos m x} {x^2 + a^2} \rd x = \frac \pi {2 a} e^{-m a}$
We have:
{{begin-eqn}}
{{eqn | l = \frac \d {\d m} \int_0^\infty \frac {\cos m x} {x^2 + a^2} \rd x
| r = \int_0^\infty \frac \partial {\partial m} \paren {\frac {\... | :$\ds \int_0^\infty \frac {x \sin m x} {x^2 + a^2} \rd x = \frac \pi 2 e^{-m a}$ | From [[Definite Integral to Infinity of Cosine m x over x Squared plus a Squared|Definite Integral to Infinity of $\dfrac {\cos m x} {x^2 + a^2}$]]:
:$\ds \int_0^\infty \frac {\cos m x} {x^2 + a^2} \rd x = \frac \pi {2 a} e^{-m a}$
We have:
{{begin-eqn}}
{{eqn | l = \frac \d {\d m} \int_0^\infty \frac {\cos m x} {x... | Definite Integral to Infinity of x by Sine m x over x Squared plus a Squared/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_by_Sine_m_x_over_x_Squared_plus_a_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_by_Sine_m_x_over_x_Squared_plus_a_Squared/Proof_2 | [
"Definite Integrals involving Sine Function",
"Definite Integral to Infinity of x by Sine m x over x Squared plus a Squared"
] | [] | [
"Definite Integral to Infinity of Cosine m x over x Squared plus a Squared",
"Definite Integral of Partial Derivative",
"Derivative of Cosine Function/Corollary",
"Derivative of Exponential Function/Corollary 1"
] |
proofwiki-16309 | Power Series Expansion for Cosine Integral Function | {{begin-eqn}}
{{eqn | l = \map \Ci x
| r = -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {x^{2 n} } {\paren {2 n} \times \paren {2 n}!}
| c =
}}
{{eqn | r = -\gamma - \ln x + \dfrac {x^2} {2 \cdot 2!} - \dfrac {x^4} {4 \cdot 4!} + \dfrac {x^6} {6 \cdot 6!} - \dfrac {x^8} {8 \cdot ... | {{begin-eqn}}
{{eqn | l = \map \Ci x
| r = -\gamma - \ln x + \int_0^x \frac {1 - \cos u} u \rd u
| c = Characterization of Cosine Integral Function
}}
{{eqn | r = -\gamma - \ln x + \int_0^x \frac 1 u \paren {1 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {\paren {2 n}!} } \rd u
| c = P... | {{begin-eqn}}
{{eqn | l = \map \Ci x
| r = -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {x^{2 n} } {\paren {2 n} \times \paren {2 n}!}
| c =
}}
{{eqn | r = -\gamma - \ln x + \dfrac {x^2} {2 \cdot 2!} - \dfrac {x^4} {4 \cdot 4!} + \dfrac {x^6} {6 \cdot 6!} - \dfrac {x^8} {8 \cdot ... | {{begin-eqn}}
{{eqn | l = \map \Ci x
| r = -\gamma - \ln x + \int_0^x \frac {1 - \cos u} u \rd u
| c = [[Characterization of Cosine Integral Function]]
}}
{{eqn | r = -\gamma - \ln x + \int_0^x \frac 1 u \paren {1 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {\paren {2 n}!} } \rd u
| c... | Power Series Expansion for Cosine Integral Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cosine_Integral_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cosine_Integral_Function | [
"Cosine Integral Function",
"Examples of Power Series"
] | [] | [
"Characterization of Cosine Integral Function",
"Power Series Expansion for Cosine Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Primitive of Power"
] |
proofwiki-16310 | Derivative of Cosine Integral Function | :$\dfrac \d {\d x} \paren {\map \Ci x} = -\dfrac {\cos x} x$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} \paren {\map \Ci x}
| r = \frac \d {\d x} \paren {-\gamma - \ln x + \int_0^x \frac {1 - \cos t} t \rd t}
| c = Characterization of Cosine Integral Function
}}
{{eqn | r = -\frac 1 x + \frac 1 x - \frac {\cos x} x
| c = Derivative of Constant, Derivative of Natural Logarithm,... | :$\dfrac \d {\d x} \paren {\map \Ci x} = -\dfrac {\cos x} x$ | {{begin-eqn}}
{{eqn | l = \frac \d {\d x} \paren {\map \Ci x}
| r = \frac \d {\d x} \paren {-\gamma - \ln x + \int_0^x \frac {1 - \cos t} t \rd t}
| c = [[Characterization of Cosine Integral Function]]
}}
{{eqn | r = -\frac 1 x + \frac 1 x - \frac {\cos x} x
| c = [[Derivative of Constant]], [[Derivative of Natural ... | Derivative of Cosine Integral Function | https://proofwiki.org/wiki/Derivative_of_Cosine_Integral_Function | https://proofwiki.org/wiki/Derivative_of_Cosine_Integral_Function | [
"Cosine Integral Function",
"Derivatives"
] | [] | [
"Characterization of Cosine Integral Function",
"Derivative of Constant",
"Derivative of Natural Logarithm Function",
"Fundamental Theorem of Calculus/First Part/Corollary"
] |
proofwiki-16311 | Derivative of Exponential Integral Function | Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\dfrac \d {\d x} \paren {\map \Ei x} = -\dfrac {e^{-x} } x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\map \Ei x}
| r = \map {\frac \d {\d x} } {-\gamma - \ln x + \int_0^x \frac {1 - e^{-t} } t \rd t}
| c = Characterization of Exponential Integral Function
}}
{{eqn | r = -\frac 1 x + \frac 1 x - \frac {e^{-x} } x
| c = Derivative of Constant, Derivati... | Let $\Ei: \R_{>0} \to \R$ denote the [[Definition:Exponential Integral Function/Formulation 1|exponential integral function]]:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\dfrac \d {\d x} \paren {\map \Ei x} = -\dfrac {e^{-x} } x$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\map \Ei x}
| r = \map {\frac \d {\d x} } {-\gamma - \ln x + \int_0^x \frac {1 - e^{-t} } t \rd t}
| c = [[Characterization of Exponential Integral Function/Formulation 1|Characterization of Exponential Integral Function]]
}}
{{eqn | r = -\frac 1 x + \frac ... | Derivative of Exponential Integral Function | https://proofwiki.org/wiki/Derivative_of_Exponential_Integral_Function | https://proofwiki.org/wiki/Derivative_of_Exponential_Integral_Function | [
"Derivatives",
"Exponential Integral Function"
] | [
"Definition:Exponential Integral Function/Formulation 1"
] | [
"Characterization of Exponential Integral Function/Formulation 1",
"Derivative of Constant",
"Derivative of Natural Logarithm Function",
"Fundamental Theorem of Calculus/First Part/Corollary",
"Category:Derivatives",
"Category:Exponential Integral Function"
] |
proofwiki-16312 | Derivative of Sine Integral Function | :$\dfrac \d {\d x} \paren {\map \Si x} = \dfrac {\sin x} x$ | We have, by the definition of the sine integral function:
:$\ds \map \Si x = \int_0^x \frac {\sin t} t \rd t$
By {{Corollary|Fundamental Theorem of Calculus/First Part|disp = Fundamental Theorem of Calculus (First Part)}}, we have:
:$\ds \frac \d {\d x} \paren {\map \Si x} = \frac {\sin x} x$
{{qed}} | :$\dfrac \d {\d x} \paren {\map \Si x} = \dfrac {\sin x} x$ | We have, by the definition of the [[Definition:Sine Integral Function|sine integral function]]:
:$\ds \map \Si x = \int_0^x \frac {\sin t} t \rd t$
By {{Corollary|Fundamental Theorem of Calculus/First Part|disp = Fundamental Theorem of Calculus (First Part)}}, we have:
:$\ds \frac \d {\d x} \paren {\map \Si x} = ... | Derivative of Sine Integral Function | https://proofwiki.org/wiki/Derivative_of_Sine_Integral_Function | https://proofwiki.org/wiki/Derivative_of_Sine_Integral_Function | [
"Sine Integral Function",
"Derivatives"
] | [] | [
"Definition:Sine Integral Function"
] |
proofwiki-16313 | Primitive of Cosine Integral Function | :$\ds \int \map \Ci x \rd x = x \map \Ci x + \sin x + C$ | By Derivative of Cosine Integral Function, we have:
:$\ds \frac \d {\d x} \paren {\map \Ci x} = -\frac {\cos x} x$
So:
{{begin-eqn}}
{{eqn | l = \int \map \Ci x \rd x
| r = \int 1 \times \map \Ci \rd x
}}
{{eqn | r = x \map \Ci x - \int \paren {-x \frac {\cos x} x} \rd x
| c = Integration by Parts
}}
{{eqn | r = x \... | :$\ds \int \map \Ci x \rd x = x \map \Ci x + \sin x + C$ | By [[Derivative of Cosine Integral Function]], we have:
:$\ds \frac \d {\d x} \paren {\map \Ci x} = -\frac {\cos x} x$
So:
{{begin-eqn}}
{{eqn | l = \int \map \Ci x \rd x
| r = \int 1 \times \map \Ci \rd x
}}
{{eqn | r = x \map \Ci x - \int \paren {-x \frac {\cos x} x} \rd x
| c = [[Integration by Parts]]
}}
{{eq... | Primitive of Cosine Integral Function | https://proofwiki.org/wiki/Primitive_of_Cosine_Integral_Function | https://proofwiki.org/wiki/Primitive_of_Cosine_Integral_Function | [
"Primitives",
"Cosine Integral Function"
] | [] | [
"Derivative of Cosine Integral Function",
"Integration by Parts",
"Primitive of Cosine Function",
"Category:Primitives",
"Category:Cosine Integral Function"
] |
proofwiki-16314 | Primitive of Exponential Integral Function | Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\ds \int \map \Ei x \rd x = x \map \Ei x - e^{-x} + C$ | By Derivative of Exponential Integral Function, we have:
:$\ds \frac \d {\d x} \paren {\map \Ei x} = -\frac {e^{-x} } x$
So:
{{begin-eqn}}
{{eqn | l = \int \map \Ei x \rd x
| r = \int 1 \times \map \Ei x \rd x
}}
{{eqn | r = x \map \Ei x - \int \paren {-x \frac {e^{-x} } x} \rd x
| c = Integration by Part... | Let $\Ei: \R_{>0} \to \R$ denote the [[Definition:Exponential Integral Function/Formulation 1|exponential integral function]]:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\ds \int \map \Ei x \rd x = x \map \Ei x - e^{-x} + C$ | By [[Derivative of Exponential Integral Function]], we have:
:$\ds \frac \d {\d x} \paren {\map \Ei x} = -\frac {e^{-x} } x$
So:
{{begin-eqn}}
{{eqn | l = \int \map \Ei x \rd x
| r = \int 1 \times \map \Ei x \rd x
}}
{{eqn | r = x \map \Ei x - \int \paren {-x \frac {e^{-x} } x} \rd x
| c = [[Integratio... | Primitive of Exponential Integral Function | https://proofwiki.org/wiki/Primitive_of_Exponential_Integral_Function | https://proofwiki.org/wiki/Primitive_of_Exponential_Integral_Function | [
"Primitives",
"Exponential Integral Function"
] | [
"Definition:Exponential Integral Function/Formulation 1"
] | [
"Derivative of Exponential Integral Function",
"Integration by Parts",
"Primitive of Exponential of a x",
"Category:Primitives",
"Category:Exponential Integral Function"
] |
proofwiki-16315 | Direct Image Mapping of Domain is Image Set of Mapping | Let $S$ and $T$ be sets.
Let $\powerset S$ and $\powerset T$ be their power sets.
Let $f \subseteq S \times T$ be a mapping from $S$ to $T$.
Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$:
:$\forall X \in \powerset S: \map {f^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \map f s ... | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \map {f^\to} S
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists x \in S
| l = \map f x
| r = y
| c = {{Defof|Direct Image Mapping of Mapping}}
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \in
| r = \Img f
| ... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\powerset S$ and $\powerset T$ be their [[Definition:Power Set|power sets]].
Let $f \subseteq S \times T$ be a [[Definition:Mapping|mapping]] from $S$ to $T$.
Let $f^\to: \powerset S \to \powerset T$ be the [[Definition:Direct Image Mapping of Mapping|direct image map... | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \map {f^\to} S
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists x \in S
| l = \map f x
| r = y
| c = {{Defof|Direct Image Mapping of Mapping}}
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o = \in
| r = \Img f
| ... | Direct Image Mapping of Domain is Image Set of Mapping | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Domain_is_Image_Set_of_Mapping | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Domain_is_Image_Set_of_Mapping | [
"Direct Image Mappings"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Category:Direct Image Mappings"
] |
proofwiki-16316 | Direct Image Mapping of Domain is Image Set of Relation | Let $S$ and $T$ be sets.
Let $\powerset S$ and $\powerset T$ be their power sets.
Let $\RR \subseteq S \times T$ be a relation on $S \times T$.
Let $\RR^\to: \powerset S \to \powerset T$ be the direct image mapping of $\RR$:
:$\forall X \in \powerset S: \map {\RR^\to} X = \begin {cases} \set {t \in T: \exists s \in X: ... | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \map {\RR^\to} {\Dom \RR}
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists x \in S
| l = \tuple {x, y}
| o = \in
| r = \RR
| c = {{Defof|Direct Image Mapping of Mapping}}
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\powerset S$ and $\powerset T$ be their [[Definition:Power Set|power sets]].
Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]] on $S \times T$.
Let $\RR^\to: \powerset S \to \powerset T$ be the [[Definition:Direct Image Mapping of Relation|direct im... | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \map {\RR^\to} {\Dom \RR}
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists x \in S
| l = \tuple {x, y}
| o = \in
| r = \RR
| c = {{Defof|Direct Image Mapping of Mapping}}
}}
{{eqn | ll= \leadstoandfrom
| l = y
| o... | Direct Image Mapping of Domain is Image Set of Relation | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Domain_is_Image_Set_of_Relation | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Domain_is_Image_Set_of_Relation | [
"Direct Image Mappings"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Relation",
"Definition:Direct Image Mapping/Relation",
"Definition:Domain (Set Theory)/Relation",
"Definition:Image (Set Theory)/Relation/Relation"
] | [
"Category:Direct Image Mappings"
] |
proofwiki-16317 | Inverse Image Mapping of Codomain is Preimage Set of Mapping | Let $S$ and $T$ be sets.
Let $\powerset S$ and $\powerset T$ be their power sets.
Let $f \subseteq S \times T$ be a mapping from $S$ to $T$.
Let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$:
:$\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {f^\gets} T
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists y \in T
| l = \map f x
| r = y
| c = {{Defof|Inverse Image Mapping of Mapping}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \in
| r = \Preimg f
... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\powerset S$ and $\powerset T$ be their [[Definition:Power Set|power sets]].
Let $f \subseteq S \times T$ be a [[Definition:Mapping|mapping]] from $S$ to $T$.
Let $f^\gets: \powerset T \to \powerset S$ be the [[Definition:Inverse Image Mapping of Mapping|inverse image... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {f^\gets} T
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists y \in T
| l = \map f x
| r = y
| c = {{Defof|Inverse Image Mapping of Mapping}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \in
| r = \Preimg f
... | Inverse Image Mapping of Codomain is Preimage Set of Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Codomain_is_Preimage_Set_of_Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Codomain_is_Preimage_Set_of_Mapping | [
"Inverse Image Mappings"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Preimage/Mapping/Mapping"
] | [
"Category:Inverse Image Mappings"
] |
proofwiki-16318 | Inverse Image Mapping of Codomain is Preimage Set of Relation | Let $S$ and $T$ be sets.
Let $\powerset S$ and $\powerset T$ be their power sets.
Let $\RR \subseteq S \times T$ be a relation on $S \times T$.
Let $\RR^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $\RR$:
:$\forall X \in \powerset S: \map {\RR^\to} X = \begin {cases} \set {t \in T: \exists s \in ... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {\RR^\gets} T
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists x \in S
| l = \tuple {x, y}
| o = \in
| r = \RR
| c = {{Defof|Inverse Image Mapping of Mapping}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \in... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\powerset S$ and $\powerset T$ be their [[Definition:Power Set|power sets]].
Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]] on $S \times T$.
Let $\RR^\gets: \powerset T \to \powerset S$ be the [[Definition:Inverse Image Mapping of Relation|invers... | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \map {\RR^\gets} T
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \exists x \in S
| l = \tuple {x, y}
| o = \in
| r = \RR
| c = {{Defof|Inverse Image Mapping of Mapping}}
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \in... | Inverse Image Mapping of Codomain is Preimage Set of Relation | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Codomain_is_Preimage_Set_of_Relation | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Codomain_is_Preimage_Set_of_Relation | [
"Inverse Image Mappings"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Relation",
"Definition:Inverse Image Mapping/Relation",
"Definition:Preimage/Relation/Relation"
] | [
"Category:Inverse Image Mappings"
] |
proofwiki-16319 | Direct Image Mapping of Mapping is Mapping | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping on $S \times T$.
Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$:
:$\forall X \in \powerset S: \map {f^\to} X = \set {t \in T: \exists s \in X: \map f s = t}$
Then $f^\to$ is indeed a mapping. | $f$, being a mapping, is also a relation.
Hence Direct Image Mapping of Relation is Mapping can be applied directly.
{{qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] on $S \times T$.
Let $f^\to: \powerset S \to \powerset T$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$:
:$\forall X \in \powerset S: \map {f^\to} X = \set {t \in T: \exists s \in X: \... | $f$, being a [[Definition:Mapping|mapping]], is also a [[Definition:Relation|relation]].
Hence [[Direct Image Mapping of Relation is Mapping]] can be applied directly.
{{qed}} | Direct Image Mapping of Mapping is Mapping | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Mapping_is_Mapping | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Mapping_is_Mapping | [
"Direct Image Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Mapping"
] | [
"Definition:Mapping",
"Definition:Relation",
"Direct Image Mapping of Relation is Mapping"
] |
proofwiki-16320 | Inverse Image Mapping of Mapping is Mapping | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping from $S$ to $T$.
Let $f^\gets$ be the inverse image mapping of $f$:
:$f^\gets: \powerset T \to \powerset S: \map {f^\gets} Y = f^{-1} \sqbrk Y$
Then $f^\gets$ is indeed a mapping. | $f^{-1}$ is a relation.
So Inverse Image Mapping of Relation is Mapping applies directly.
{{qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] from $S$ to $T$.
Let $f^\gets$ be the [[Definition:Inverse Image Mapping of Mapping|inverse image mapping]] of $f$:
:$f^\gets: \powerset T \to \powerset S: \map {f^\gets} Y = f^{-1} \sqbrk Y$
Then $f^\gets$ is indeed ... | $f^{-1}$ is a [[Definition:Relation|relation]].
So [[Inverse Image Mapping of Relation is Mapping]] applies directly.
{{qed}} | Inverse Image Mapping of Mapping is Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Mapping_is_Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Mapping_is_Mapping | [
"Inverse Image Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Mapping"
] | [
"Definition:Relation",
"Inverse Image Mapping of Relation is Mapping"
] |
proofwiki-16321 | Direct Image Mapping of Mapping is Empty iff Argument is Empty | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping from $S$ to $T$.
Let $f^\to$ be the direct image mapping of $f$:
:$f^\to: \powerset S \to \powerset T: \map {f^\to} X = \set {t \in T: \exists s \in X: \map f s = t}$
Then:
:$\map {f^\to} X = \O \iff X = \O$ | By definition, a mapping is a left-total relation.
The result then follows from Direct Image Mapping of Left-Total Relation is Empty iff Argument is Empty.
{{qed}} | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] from $S$ to $T$.
Let $f^\to$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$:
:$f^\to: \powerset S \to \powerset T: \map {f^\to} X = \set {t \in T: \exists s \in X: \map f s = t}$
The... | By definition, a [[Definition:Mapping|mapping]] is a [[Definition:Left-Total Relation|left-total relation]].
The result then follows from [[Direct Image Mapping of Left-Total Relation is Empty iff Argument is Empty]].
{{qed}} | Direct Image Mapping of Mapping is Empty iff Argument is Empty | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Mapping_is_Empty_iff_Argument_is_Empty | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Mapping_is_Empty_iff_Argument_is_Empty | [
"Direct Image Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping"
] | [
"Definition:Mapping",
"Definition:Left-Total Relation",
"Direct Image Mapping of Left-Total Relation is Empty iff Argument is Empty"
] |
proofwiki-16322 | Derivative of Fresnel Sine Integral Function | :$\dfrac {\d \mathrm S} {\d x} = \sqrt {\dfrac 2 \pi} \sin x^2$ | We have, by the definition of the Fresnel sine integral function:
:$\ds \map {\mathrm S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$
By Fundamental Theorem of Calculus (First Part): Corollary, we therefore have:
:$\dfrac {\d \mathrm S} {\d x} = \sqrt {\dfrac 2 \pi} \sin x^2$
{{qed}}
Category:Fresnel Sine Integral ... | :$\dfrac {\d \mathrm S} {\d x} = \sqrt {\dfrac 2 \pi} \sin x^2$ | We have, by the definition of the [[Definition:Fresnel Sine Integral Function|Fresnel sine integral function]]:
:$\ds \map {\mathrm S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$
By [[Fundamental Theorem of Calculus/First Part/Corollary|Fundamental Theorem of Calculus (First Part): Corollary]], we therefore have... | Derivative of Fresnel Sine Integral Function | https://proofwiki.org/wiki/Derivative_of_Fresnel_Sine_Integral_Function | https://proofwiki.org/wiki/Derivative_of_Fresnel_Sine_Integral_Function | [
"Fresnel Sine Integral Function",
"Derivatives"
] | [] | [
"Definition:Fresnel Integral/Sine",
"Fundamental Theorem of Calculus/First Part/Corollary",
"Category:Fresnel Sine Integral Function",
"Category:Derivatives"
] |
proofwiki-16323 | Derivative of Fresnel Cosine Integral Function | :$\dfrac {\d \mathrm C} {\d x} = \sqrt {\dfrac 2 \pi} \cos x^2$ | We have, by the definition of the Fresnel cosine integral function:
:$\ds \map {\mathrm C} x = \sqrt {\dfrac 2 \pi} \int_0^x \cos u^2 \rd u$
By Fundamental Theorem of Calculus (First Part): Corollary, we therefore have:
:$\dfrac {\d \mathrm C} {\d x} = \sqrt {\dfrac 2 \pi} \cos x^2$
{{qed}}
Category:Fresnel Cosine Inte... | :$\dfrac {\d \mathrm C} {\d x} = \sqrt {\dfrac 2 \pi} \cos x^2$ | We have, by the definition of the [[Definition:Fresnel Cosine Integral Function|Fresnel cosine integral function]]:
:$\ds \map {\mathrm C} x = \sqrt {\dfrac 2 \pi} \int_0^x \cos u^2 \rd u$
By [[Fundamental Theorem of Calculus/First Part/Corollary|Fundamental Theorem of Calculus (First Part): Corollary]], we therefore... | Derivative of Fresnel Cosine Integral Function | https://proofwiki.org/wiki/Derivative_of_Fresnel_Cosine_Integral_Function | https://proofwiki.org/wiki/Derivative_of_Fresnel_Cosine_Integral_Function | [
"Fresnel Cosine Integral Function",
"Derivatives"
] | [] | [
"Definition:Fresnel Integral/Cosine",
"Fundamental Theorem of Calculus/First Part/Corollary",
"Category:Fresnel Cosine Integral Function",
"Category:Derivatives"
] |
proofwiki-16324 | Power Series Expansion for Fresnel Sine Integral Function | {{begin-eqn}}
{{eqn | l = \map {\operatorname S} x
| r = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 3} } {\paren {4 n + 3} \paren {2 n + 1}!}
| c =
}}
{{eqn | r = \sqrt {\frac 2 \pi} \paren {\dfrac {x^3} {3 \cdot 1!} - \dfrac {x^7} {7 \cdot 3!} + \dfrac {x^{11} } {11 \cdo... | {{begin-eqn}}
{{eqn | l = \map {\operatorname S} x
| r = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u
| c = {{Defof|Fresnel Sine Integral Function}}
}}
{{eqn | r = \sqrt {\frac 2 \pi} \int_0^x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {u^2}^{2 n + 1} } {\paren {2 n + 1}!} } \rd u
... | {{begin-eqn}}
{{eqn | l = \map {\operatorname S} x
| r = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 3} } {\paren {4 n + 3} \paren {2 n + 1}!}
| c =
}}
{{eqn | r = \sqrt {\frac 2 \pi} \paren {\dfrac {x^3} {3 \cdot 1!} - \dfrac {x^7} {7 \cdot 3!} + \dfrac {x^{11} } {11 \cdo... | {{begin-eqn}}
{{eqn | l = \map {\operatorname S} x
| r = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u
| c = {{Defof|Fresnel Sine Integral Function}}
}}
{{eqn | r = \sqrt {\frac 2 \pi} \int_0^x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {u^2}^{2 n + 1} } {\paren {2 n + 1}!} } \rd u
... | Power Series Expansion for Fresnel Sine Integral Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Fresnel_Sine_Integral_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Fresnel_Sine_Integral_Function | [
"Fresnel Sine Integral Function",
"Examples of Power Series"
] | [] | [
"Power Series Expansion for Sine Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Primitive of Power"
] |
proofwiki-16325 | Power Series Expansion for Fresnel Cosine Integral Function | {{begin-eqn}}
{{eqn | l = \map {\operatorname C} x
| r = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 1} } {\paren {4 n + 1} \paren {2 n}!}
| c =
}}
{{eqn | r = \sqrt {\frac 2 \pi} \paren {\dfrac x {1!} - \dfrac {x^5} {5 \cdot 2!} + \dfrac {x^9} {9 \cdot 4!} - \dfrac {x^{13... | {{begin-eqn}}
{{eqn | l = \map {\operatorname C} x
| r = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u
| c = {{Defof|Fresnel Cosine Integral Function}}
}}
{{eqn | r = \sqrt {\frac 2 \pi} \int_0^x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {u^2}^{2 n} } {\paren {2 n}!} } \rd u
| c =... | {{begin-eqn}}
{{eqn | l = \map {\operatorname C} x
| r = \sqrt {\frac 2 \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{4 n + 1} } {\paren {4 n + 1} \paren {2 n}!}
| c =
}}
{{eqn | r = \sqrt {\frac 2 \pi} \paren {\dfrac x {1!} - \dfrac {x^5} {5 \cdot 2!} + \dfrac {x^9} {9 \cdot 4!} - \dfrac {x^{13... | {{begin-eqn}}
{{eqn | l = \map {\operatorname C} x
| r = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u
| c = {{Defof|Fresnel Cosine Integral Function}}
}}
{{eqn | r = \sqrt {\frac 2 \pi} \int_0^x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {u^2}^{2 n} } {\paren {2 n}!} } \rd u
| c =... | Power Series Expansion for Fresnel Cosine Integral Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Fresnel_Cosine_Integral_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Fresnel_Cosine_Integral_Function | [
"Fresnel Cosine Integral Function",
"Examples of Power Series"
] | [] | [
"Power Series Expansion for Cosine Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Primitive of Power"
] |
proofwiki-16326 | Fresnel Sine Integral Function is Odd | :$\map {\operatorname S} {-x} = -\map {\operatorname S} x$ | {{begin-eqn}}
{{eqn | l = \map {\operatorname S} {-x}
| r = \sqrt {\frac 2 \pi} \int_0^{-x} \sin u^2 \rd u
| c = {{Defof|Fresnel Sine Integral Function}}
}}
{{eqn | r = -\sqrt {\frac 2 \pi} \int_0^{-\paren {-x} } \map \sin {\paren {-u}^2} \rd u
| c = substituting $u \mapsto -u$
}}
{{eqn | r = -\sqrt {... | :$\map {\operatorname S} {-x} = -\map {\operatorname S} x$ | {{begin-eqn}}
{{eqn | l = \map {\operatorname S} {-x}
| r = \sqrt {\frac 2 \pi} \int_0^{-x} \sin u^2 \rd u
| c = {{Defof|Fresnel Sine Integral Function}}
}}
{{eqn | r = -\sqrt {\frac 2 \pi} \int_0^{-\paren {-x} } \map \sin {\paren {-u}^2} \rd u
| c = [[Integration by Substitution|substituting]] $u \ma... | Fresnel Sine Integral Function is Odd | https://proofwiki.org/wiki/Fresnel_Sine_Integral_Function_is_Odd | https://proofwiki.org/wiki/Fresnel_Sine_Integral_Function_is_Odd | [
"Fresnel Sine Integral Function",
"Examples of Odd Functions"
] | [] | [
"Integration by Substitution"
] |
proofwiki-16327 | Fresnel Sine Integral Function of Zero | :$\map {\operatorname S} 0 = 0$ | By Fresnel Sine Integral Function is Odd, $\operatorname S$ is an odd function.
Therefore, by Odd Function of Zero is Zero:
:$\map {\operatorname S} 0 = 0$
{{qed}} | :$\map {\operatorname S} 0 = 0$ | By [[Fresnel Sine Integral Function is Odd]], $\operatorname S$ is an [[Definition:Odd Function|odd function]].
Therefore, by [[Odd Function of Zero is Zero]]:
:$\map {\operatorname S} 0 = 0$
{{qed}} | Fresnel Sine Integral Function of Zero | https://proofwiki.org/wiki/Fresnel_Sine_Integral_Function_of_Zero | https://proofwiki.org/wiki/Fresnel_Sine_Integral_Function_of_Zero | [
"Fresnel Sine Integral Function"
] | [] | [
"Fresnel Sine Integral Function is Odd",
"Definition:Odd Function",
"Odd Function of Zero is Zero"
] |
proofwiki-16328 | Limit to Infinity of Fresnel Sine Integral Function | :$\ds \lim_{x \mathop \to \infty} \map {\mathrm S} x = \frac 1 2$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map {\mathrm S} x
| r = \sqrt {\frac 2 \pi} \lim_{x \mathop \to \infty} \int_0^x \sin u^2 \rd u
| c = Multiple Rule for Limits of Real Functions, {{Defof|Fresnel Sine Integral Function}}
}}
{{eqn | r = \sqrt {\frac 2 \pi} \int_0^\infty \sin u^2 \rd u
}}... | :$\ds \lim_{x \mathop \to \infty} \map {\mathrm S} x = \frac 1 2$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map {\mathrm S} x
| r = \sqrt {\frac 2 \pi} \lim_{x \mathop \to \infty} \int_0^x \sin u^2 \rd u
| c = [[Multiple Rule for Limits of Real Functions]], {{Defof|Fresnel Sine Integral Function}}
}}
{{eqn | r = \sqrt {\frac 2 \pi} \int_0^\infty \sin u^2 \rd ... | Limit to Infinity of Fresnel Sine Integral Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Fresnel_Sine_Integral_Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Fresnel_Sine_Integral_Function | [
"Fresnel Sine Integral Function"
] | [] | [
"Combination Theorem for Limits of Functions/Real/Multiple Rule",
"Definite Integral to Infinity of Sine of a x^2"
] |
proofwiki-16329 | Asymptotic Expansion for Fresnel Sine Integral Function | :$\map {\operatorname S} x \sim \dfrac 1 2 - \dfrac 1 {\sqrt {2 \pi} } \paren {\map \cos {x^2} \paren {\dfrac 1 x - \dfrac {1 \times 3} {2^2 x^5} + \dfrac {1 \times 3 \times 5 \times 7} {2^4 x^9} - \ldots} + \map \sin {x^2} \paren {\dfrac 1 {2 x^3} - \dfrac {1 \times 3 \times 5} {2^3 x^7} + \ldots} }$ | {{ProofWanted|also establish general term and write sums with $\sum$}} | :$\map {\operatorname S} x \sim \dfrac 1 2 - \dfrac 1 {\sqrt {2 \pi} } \paren {\map \cos {x^2} \paren {\dfrac 1 x - \dfrac {1 \times 3} {2^2 x^5} + \dfrac {1 \times 3 \times 5 \times 7} {2^4 x^9} - \ldots} + \map \sin {x^2} \paren {\dfrac 1 {2 x^3} - \dfrac {1 \times 3 \times 5} {2^3 x^7} + \ldots} }$ | {{ProofWanted|also establish general term and write sums with $\sum$}} | Asymptotic Expansion for Fresnel Sine Integral Function | https://proofwiki.org/wiki/Asymptotic_Expansion_for_Fresnel_Sine_Integral_Function | https://proofwiki.org/wiki/Asymptotic_Expansion_for_Fresnel_Sine_Integral_Function | [
"Fresnel Sine Integral Function",
"Asymptotic Expansions"
] | [] | [] |
proofwiki-16330 | Fresnel Cosine Integral Function is Odd | :$\map {\operatorname C} {-x} = -\map {\operatorname C} x$ | {{begin-eqn}}
{{eqn | l = \map {\operatorname C} {-x}
| r = \sqrt {\frac 2 \pi} \int_0^{-x} \cos u^2 \rd u
| c = {{Defof|Fresnel Cosine Integral Function}}
}}
{{eqn | r = -\sqrt {\frac 2 \pi} \int_0^{-\paren {-x} } \map \cos {\paren {-u}^2} \rd u
| c = substituting $u \mapsto -u$
}}
{{eqn | r = -\sqrt... | :$\map {\operatorname C} {-x} = -\map {\operatorname C} x$ | {{begin-eqn}}
{{eqn | l = \map {\operatorname C} {-x}
| r = \sqrt {\frac 2 \pi} \int_0^{-x} \cos u^2 \rd u
| c = {{Defof|Fresnel Cosine Integral Function}}
}}
{{eqn | r = -\sqrt {\frac 2 \pi} \int_0^{-\paren {-x} } \map \cos {\paren {-u}^2} \rd u
| c = [[Integration by Substitution|substituting]] $u \... | Fresnel Cosine Integral Function is Odd | https://proofwiki.org/wiki/Fresnel_Cosine_Integral_Function_is_Odd | https://proofwiki.org/wiki/Fresnel_Cosine_Integral_Function_is_Odd | [
"Fresnel Cosine Integral Function",
"Examples of Odd Functions"
] | [] | [
"Integration by Substitution"
] |
proofwiki-16331 | Fresnel Cosine Integral Function of Zero | :$\map {\operatorname C} 0 = 0$ | By Fresnel Cosine Integral Function is Odd, $\operatorname C$ is an odd function.
Therefore, by Odd Function of Zero is Zero:
:$\map {\operatorname C} 0 = 0$
{{qed}} | :$\map {\operatorname C} 0 = 0$ | By [[Fresnel Cosine Integral Function is Odd]], $\operatorname C$ is an [[Definition:Odd Function|odd function]].
Therefore, by [[Odd Function of Zero is Zero]]:
:$\map {\operatorname C} 0 = 0$
{{qed}} | Fresnel Cosine Integral Function of Zero | https://proofwiki.org/wiki/Fresnel_Cosine_Integral_Function_of_Zero | https://proofwiki.org/wiki/Fresnel_Cosine_Integral_Function_of_Zero | [
"Fresnel Cosine Integral Function"
] | [] | [
"Fresnel Cosine Integral Function is Odd",
"Definition:Odd Function",
"Odd Function of Zero is Zero"
] |
proofwiki-16332 | Direct Image Mapping of Left-Total Relation is Empty iff Argument is Empty | Let $S$ and $T$ be sets.
Let $\RR: S \to T$ be a left-total relation on $S \times T$.
Let $\RR^\to$ be the direct image mapping of $\RR$:
:$\RR^\to: \powerset S \to \powerset T: \map {\RR^\to} X = \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR}$
Then:
:$\map {\RR^\to} X = \O \iff X = \O$ | === Sufficient Condition ===
Let $\map {\RR^\to} X = \O$.
By definition of direct image mapping:
:$\set {t \in T: \exists s \in X: \tuple {s, t} \in \RR} = \O$
That is:
:$\neg \exists s \in X: \tuple {s, t} \in \RR$
But as $\RR$ is a left-total relation:
:$\forall s \in X: \exists t \in T: \tuple {s, t} \in \RR$
Thus:
... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\RR: S \to T$ be a [[Definition:Left-Total Relation|left-total relation]] on $S \times T$.
Let $\RR^\to$ be the [[Definition:Direct Image Mapping of Relation|direct image mapping]] of $\RR$:
:$\RR^\to: \powerset S \to \powerset T: \map {\RR^\to} X = \set {t \in T: \e... | === Sufficient Condition ===
Let $\map {\RR^\to} X = \O$.
By definition of [[Definition:Direct Image Mapping of Relation|direct image mapping]]:
:$\set {t \in T: \exists s \in X: \tuple {s, t} \in \RR} = \O$
That is:
:$\neg \exists s \in X: \tuple {s, t} \in \RR$
But as $\RR$ is a [[Definition:Left-Total Relation|l... | Direct Image Mapping of Left-Total Relation is Empty iff Argument is Empty | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Left-Total_Relation_is_Empty_iff_Argument_is_Empty | https://proofwiki.org/wiki/Direct_Image_Mapping_of_Left-Total_Relation_is_Empty_iff_Argument_is_Empty | [
"Direct Image Mappings"
] | [
"Definition:Set",
"Definition:Left-Total Relation",
"Definition:Direct Image Mapping/Relation"
] | [
"Definition:Direct Image Mapping/Relation",
"Definition:Left-Total Relation",
"Definition:Direct Image Mapping/Relation"
] |
proofwiki-16333 | Limit to Infinity of Fresnel Cosine Integral Function | :$\ds \lim_{x \mathop \to \infty} \map {\mathrm C} x = \frac 1 2$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map {\mathrm C} x
| r = \sqrt {\frac 2 \pi} \lim_{x \mathop \to \infty} \int_0^x \cos u^2 \rd u
| c = Multiple Rule for Limits of Real Functions, {{Defof|Fresnel Cosine Integral Function}}
}}
{{eqn | r = \sqrt {\frac 2 \pi} \int_0^\infty \cos u^2 \rd u
... | :$\ds \lim_{x \mathop \to \infty} \map {\mathrm C} x = \frac 1 2$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map {\mathrm C} x
| r = \sqrt {\frac 2 \pi} \lim_{x \mathop \to \infty} \int_0^x \cos u^2 \rd u
| c = [[Multiple Rule for Limits of Real Functions]], {{Defof|Fresnel Cosine Integral Function}}
}}
{{eqn | r = \sqrt {\frac 2 \pi} \int_0^\infty \cos u^2 \r... | Limit to Infinity of Fresnel Cosine Integral Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Fresnel_Cosine_Integral_Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Fresnel_Cosine_Integral_Function | [
"Fresnel Cosine Integral Function"
] | [] | [
"Combination Theorem for Limits of Functions/Real/Multiple Rule",
"Definite Integral to Infinity of Cosine of a x^2"
] |
proofwiki-16334 | Successor Mapping on Natural Numbers is not Surjection | Let $f: \N \to \N$ be the successor mapping on the natural numbers $\N$:
:$\forall n \in \N: \map f n = n + 1$
Then $f$ is not a surjection. | There exists no $n \in \N$ such that $n + 1 = 0$.
Thus $\map f 0$ has no preimage.
The result follows by definition of surjection.
{{qed}} | Let $f: \N \to \N$ be the [[Definition:Successor Mapping on Natural Numbers|successor mapping]] on the [[Definition:Natural Numbers|natural numbers]] $\N$:
:$\forall n \in \N: \map f n = n + 1$
Then $f$ is not a [[Definition:Surjection|surjection]]. | There exists no $n \in \N$ such that $n + 1 = 0$.
Thus $\map f 0$ has no [[Definition:Preimage of Element under Mapping|preimage]].
The result follows by definition of [[Definition:Surjection|surjection]].
{{qed}} | Successor Mapping on Natural Numbers is not Surjection | https://proofwiki.org/wiki/Successor_Mapping_on_Natural_Numbers_is_not_Surjection | https://proofwiki.org/wiki/Successor_Mapping_on_Natural_Numbers_is_not_Surjection | [
"Surjections"
] | [
"Definition:Successor Mapping on Natural Numbers",
"Definition:Natural Numbers",
"Definition:Surjection"
] | [
"Definition:Preimage/Mapping/Element",
"Definition:Surjection"
] |
proofwiki-16335 | Composition of Direct Image Mappings of Mappings | Let $A, B, C$ be non-empty sets.
Let $f: A \to B$ and $g: B \to C$ be mappings.
Let:
:$f^\to: \powerset A \to \powerset B$
and
:$g^\to: \powerset B \to \powerset C$
be the direct image mappings of $f$ and $g$.
Then:
:$\paren {g \circ f}^\to = g^\to \circ f^\to$ | Let $S \subseteq A$ such that $S \ne \O$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {g^\to \circ f^\to} } S
| r = \map {g^\to} {\map {f^\to} S}
| c =
}}
{{eqn | r = \set {\map g x: x \in \map {f^\to} S}
| c =
}}
{{eqn | r = \set {\map g x: \exists y \in S: x = \map f y}
| c =
}}
{{eqn | r ... | Let $A, B, C$ be [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|sets]].
Let $f: A \to B$ and $g: B \to C$ be [[Definition:Mapping|mappings]].
Let:
:$f^\to: \powerset A \to \powerset B$
and
:$g^\to: \powerset B \to \powerset C$
be the [[Definition:Direct Image Mapping of Mapping|direct image mappings]] of $f... | Let $S \subseteq A$ such that $S \ne \O$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {g^\to \circ f^\to} } S
| r = \map {g^\to} {\map {f^\to} S}
| c =
}}
{{eqn | r = \set {\map g x: x \in \map {f^\to} S}
| c =
}}
{{eqn | r = \set {\map g x: \exists y \in S: x = \map f y}
| c =
}}
{{eqn |... | Composition of Direct Image Mappings of Mappings/Proof 1 | https://proofwiki.org/wiki/Composition_of_Direct_Image_Mappings_of_Mappings | https://proofwiki.org/wiki/Composition_of_Direct_Image_Mappings_of_Mappings/Proof_1 | [
"Composite Mappings",
"Direct Image Mappings",
"Composition of Direct Image Mappings of Mappings"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping"
] | [] |
proofwiki-16336 | Composition of Inverse Image Mappings of Mappings | Let $A, B, C$ be non-empty sets.
Let $f: A \to B, g: B \to C$ be mappings.
Let:
:$f^\gets: \powerset B \to \powerset A$
and
:$g^\gets: \powerset C \to \powerset B$
be the inverse image mappings of $f$ and $g$.
Then:
:$\paren {g \circ f}^\gets = f^\gets \circ g^\gets$ | Let $T \subseteq C$.
We have:
{{begin-eqn}}
{{eqn | l = \map {\paren {f \circ g}^\gets} T
| r = \begin {cases} \set {x \in A: \map g {\map f x} \in T} & : \Img {g \circ f} \cap T \ne \O \\ \O & : \Img {g \circ f} \cap T = \O \end {cases}
| c =
}}
{{eqn-intertext|and}}
{{eqn | l = \map {f^\gets \circ g^\get... | Let $A, B, C$ be [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|sets]].
Let $f: A \to B, g: B \to C$ be [[Definition:Mapping|mappings]].
Let:
:$f^\gets: \powerset B \to \powerset A$
and
:$g^\gets: \powerset C \to \powerset B$
be the [[Definition:Inverse Image Mapping of Mapping|inverse image mappings]] of $... | Let $T \subseteq C$.
We have:
{{begin-eqn}}
{{eqn | l = \map {\paren {f \circ g}^\gets} T
| r = \begin {cases} \set {x \in A: \map g {\map f x} \in T} & : \Img {g \circ f} \cap T \ne \O \\ \O & : \Img {g \circ f} \cap T = \O \end {cases}
| c =
}}
{{eqn-intertext|and}}
{{eqn | l = \map {f^\gets \circ g^\g... | Composition of Inverse Image Mappings of Mappings | https://proofwiki.org/wiki/Composition_of_Inverse_Image_Mappings_of_Mappings | https://proofwiki.org/wiki/Composition_of_Inverse_Image_Mappings_of_Mappings | [
"Composite Relations",
"Composite Mappings",
"Inverse Image Mappings"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Inverse Image Mapping/Mapping"
] | [
"Empty Intersection iff Subset of Complement",
"Complement of Preimage equals Preimage of Complement",
"Intersection with Complement is Empty iff Subset",
"Subset of Preimage under Relation is Preimage of Subset/Corollary",
"Empty Intersection iff Subset of Complement"
] |
proofwiki-16337 | Unit of Ring of Mappings iff Image is Subset of Ring Units | Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.
Let $U_R$ be the set of units in $R$.
Let $S$ be a set.
Let $\struct {R^S, +', \circ'}$ be the ring of mappings on the set of mappings $R^S$.
Then:
:$f \in R^S$ is a unit of $R^S$ {{iff}} $\Img f \subseteq U_R$
where $\Img f$ is the image of $f$.
In this case, the... | From Structure Induced by Ring with Unity Operations is Ring with Unity, $\struct {R^S, +', \circ'}$ has a unity $f_{1_R}$ defined by:
:$\forall x \in S: \map {f_{1_R}} x = 1_R$ | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
Let $U_R$ be the [[Definition:Set|set]] of [[Definition:Unit of Ring|units]] in $R$.
Let $S$ be a [[Definition:Set|set]].
Let $\struct {R^S, +', \circ'}$ be the [[Definition:Ring of Mappings|ring of ma... | From [[Structure Induced by Ring with Unity Operations is Ring with Unity]], $\struct {R^S, +', \circ'}$ has a [[Definition:Unity of Ring|unity]] $f_{1_R}$ defined by:
:$\forall x \in S: \map {f_{1_R}} x = 1_R$ | Unit of Ring of Mappings iff Image is Subset of Ring Units | https://proofwiki.org/wiki/Unit_of_Ring_of_Mappings_iff_Image_is_Subset_of_Ring_Units | https://proofwiki.org/wiki/Unit_of_Ring_of_Mappings_iff_Image_is_Subset_of_Ring_Units | [
"Unit of Ring of Mappings iff Image is Subset of Ring Units",
"Rings of Mappings",
"Rings with Unity"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Set",
"Definition:Unit of Ring",
"Definition:Set",
"Definition:Ring of Mappings",
"Definition:Set",
"Definition:Mapping",
"Definition:Unit of Ring",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:... | [
"Structure Induced by Ring with Unity Operations is Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring"
] |
proofwiki-16338 | Mapping is Bijection iff Composite with Direct Image Mapping with Complementation Commutes | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Then:
:$f$ is a bijection
{{iff}}:
:$f^\to \circ \complement_S = \complement_T \circ f^\to$
where:
:$f^\to: \powerset S \to \powerset T$ denotes the direct image mapping of $f$
:$\complement_S: \powerset S \to \powerset S$ denotes the complement relative to $S$
:$... | === Sufficient Condition ===
Let $f$ be a bijection.
Thus a fortiori $f$ is:
:a surjection
:a one-to-many relation
From One-to-Many Image of Set Difference: Corollary 2 we have:
:$\forall X \in \powerset S: \map {\paren {f^\to \circ \complement_S} } X = \map {\paren {\complement_{\Img f} \circ f^\to} } X$
By definition... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
:$f$ is a [[Definition:Bijection|bijection]]
{{iff}}:
:$f^\to \circ \complement_S = \complement_T \circ f^\to$
where:
:$f^\to: \powerset S \to \powerset T$ denotes the [[Definition:Direct Image Mapping of Mapping|... | === Sufficient Condition ===
Let $f$ be a [[Definition:Bijection|bijection]].
Thus [[Definition:A Fortiori|a fortiori]] $f$ is:
:a [[Definition:Surjection|surjection]]
:a [[Definition:One-to-Many Relation|one-to-many relation]]
From [[One-to-Many Image of Set Difference/Corollary 2|One-to-Many Image of Set Differen... | Mapping is Bijection iff Composite with Direct Image Mapping with Complementation Commutes | https://proofwiki.org/wiki/Mapping_is_Bijection_iff_Composite_with_Direct_Image_Mapping_with_Complementation_Commutes | https://proofwiki.org/wiki/Mapping_is_Bijection_iff_Composite_with_Direct_Image_Mapping_with_Complementation_Commutes | [
"Set Complement",
"Bijections",
"Direct Image Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Direct Image Mapping/Mapping",
"Definition:Relative Complement",
"Definition:Relative Complement",
"Definition:Power Set"
] | [
"Definition:Bijection",
"Definition:A Fortiori",
"Definition:Surjection",
"Definition:One-to-Many Relation",
"One-to-Many Image of Set Difference/Corollary 2",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Bijection"
] |
proofwiki-16339 | Relative Complement Mapping on Powerset is Bijection | Let $S$ be a set.
Let $\complement_S: \powerset S \to \powerset S$ denote the relative complement mapping on the power set of $S$.
Then $\complement_S$ is a bijection.
Thus each $T \subseteq S$ is in one-to-one correspondence with its relative complement. | Let $f: \powerset S \to \powerset S$ be a mapping defined as:
:$\forall T \in \powerset S: \map f T = \relcomp S T$
It is to be demonstrated that $f$ is a bijection.
By definition of relative complement:
:$\relcomp S T = S \setminus T = \set {x \in S: x \notin T}$
and so it can be seen that $f$ is well-defined.
Let $T_... | Let $S$ be a [[Definition:Set|set]].
Let $\complement_S: \powerset S \to \powerset S$ denote the [[Definition:Relative Complement|relative complement mapping]] on the [[Definition:Power Set|power set]] of $S$.
Then $\complement_S$ is a [[Definition:Bijection|bijection]].
Thus each $T \subseteq S$ is in [[Definition... | Let $f: \powerset S \to \powerset S$ be a [[Definition:Mapping|mapping]] defined as:
:$\forall T \in \powerset S: \map f T = \relcomp S T$
It is to be demonstrated that $f$ is a [[Definition:Bijection|bijection]].
By definition of [[Definition:Relative Complement|relative complement]]:
:$\relcomp S T = S \setminus T... | Relative Complement Mapping on Powerset is Bijection/Proof 2 | https://proofwiki.org/wiki/Relative_Complement_Mapping_on_Powerset_is_Bijection | https://proofwiki.org/wiki/Relative_Complement_Mapping_on_Powerset_is_Bijection/Proof_2 | [
"Relative Complement",
"Examples of Bijections",
"Relative Complement Mapping on Powerset is Bijection"
] | [
"Definition:Set",
"Definition:Relative Complement",
"Definition:Power Set",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Relative Complement"
] | [
"Definition:Mapping",
"Definition:Bijection",
"Definition:Relative Complement",
"Definition:Well-Defined/Mapping",
"Relative Complement of Relative Complement",
"Definition:Injection",
"Relative Complement of Relative Complement",
"Definition:Surjection",
"Definition:Bijection"
] |
proofwiki-16340 | Power of One plus x in terms of Gaussian Hypergeometric Function | :$\map F {-p, 1; 1; -x} = \paren {1 + x}^p$ | {{begin-eqn}}
{{eqn | l = \map F {-p, 1; 1; -x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-p}^{\overline n} 1^{\overline n} } {1^{\overline n} } \frac {\paren {-x}^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {\prod_{j \matho... | :$\map F {-p, 1; 1; -x} = \paren {1 + x}^p$ | {{begin-eqn}}
{{eqn | l = \map F {-p, 1; 1; -x}
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-p}^{\overline n} 1^{\overline n} } {1^{\overline n} } \frac {\paren {-x}^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {\prod_{j \matho... | Power of One plus x in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Power_of_One_plus_x_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Power_of_One_plus_x_in_terms_of_Gaussian_Hypergeometric_Function | [
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"Product of Products",
"Binomial Theorem/General Binomial Theorem"
] |
proofwiki-16341 | Logarithm of One plus x in terms of Gaussian Hypergeometric Function | :$\map \ln {1 + x} = x \map F {1, 1; 2; -x}$ | {{begin-eqn}}
{{eqn | l = x \map F {1, 1; 2; -x}
| r = x \sum_{n \mathop = 0}^\infty \frac {\paren {1^{\overline n} }^2} {2^{\overline n} } \frac {\paren {-x}^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {n!}^2 \paren {\frac {\paren... | :$\map \ln {1 + x} = x \map F {1, 1; 2; -x}$ | {{begin-eqn}}
{{eqn | l = x \map F {1, 1; 2; -x}
| r = x \sum_{n \mathop = 0}^\infty \frac {\paren {1^{\overline n} }^2} {2^{\overline n} } \frac {\paren {-x}^n} {n!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {n!}^2 \paren {\frac {\paren... | Logarithm of One plus x in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Logarithm_of_One_plus_x_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Logarithm_of_One_plus_x_in_terms_of_Gaussian_Hypergeometric_Function | [
"Gaussian Hypergeometric Function",
"Hypergeometric Functions",
"Logarithms"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"One to Integer Rising is Integer Factorial",
"Power Series Expansion for Logarithm of 1 + x"
] |
proofwiki-16342 | Laplace Transform of Exponential times Sine | :$\map {\laptrans {e^{b t} \sin a t} } s = \dfrac a {\paren {s - b}^2 + a^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{b t} \sin a t} } s
| r = \map {\laptrans {\sin a t} } {s - b}
| c = First Translation Property of Laplace Transforms
}}
{{eqn | r = \frac a {\paren {s - b}^2 + a^2}
| c = Laplace Transform of Sine
}}
{{end-eqn}}
{{qed}} | :$\map {\laptrans {e^{b t} \sin a t} } s = \dfrac a {\paren {s - b}^2 + a^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{b t} \sin a t} } s
| r = \map {\laptrans {\sin a t} } {s - b}
| c = [[First Translation Property of Laplace Transforms]]
}}
{{eqn | r = \frac a {\paren {s - b}^2 + a^2}
| c = [[Laplace Transform of Sine]]
}}
{{end-eqn}}
{{qed}} | Laplace Transform of Exponential times Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_times_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_times_Sine | [
"Laplace Transform of Exponential times Sine",
"Laplace Transforms involving Exponential Function",
"Laplace Transforms involving Sine Function",
"Examples of Laplace Transforms"
] | [] | [
"First Translation Property of Laplace Transforms",
"Laplace Transform of Sine"
] |
proofwiki-16343 | Laplace Transform of Exponential times Cosine | :$\map {\laptrans {e^{b t} \cos a t} } s = \dfrac {s - b} {\paren {s - b}^2 + a^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{b t} \cos a t} } s
| r = \map {\laptrans {\cos a t} } {s - b}
| c = First Translation Property of Laplace Transforms
}}
{{eqn | r = \frac {s - b} {\paren {s - b}^2 + a^2}
| c = Laplace Transform of Cosine
}}
{{end-eqn}}
{{qed}} | :$\map {\laptrans {e^{b t} \cos a t} } s = \dfrac {s - b} {\paren {s - b}^2 + a^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{b t} \cos a t} } s
| r = \map {\laptrans {\cos a t} } {s - b}
| c = [[First Translation Property of Laplace Transforms]]
}}
{{eqn | r = \frac {s - b} {\paren {s - b}^2 + a^2}
| c = [[Laplace Transform of Cosine]]
}}
{{end-eqn}}
{{qed}} | Laplace Transform of Exponential times Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_times_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_times_Cosine | [
"Laplace Transforms involving Exponential Function",
"Laplace Transforms involving Cosine Function",
"Examples of Laplace Transforms"
] | [] | [
"First Translation Property of Laplace Transforms",
"Laplace Transform of Cosine"
] |
proofwiki-16344 | Complement of Direct Image Mapping of Injection equals Direct Image of Complement | Let $f: S \to T$ be an injection.
Let $f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$.
Then:
:$\forall A \in \powerset S: \map {\paren {\complement_{\Img f} \circ f^\to} } A = \map {\paren {f^\to \circ \complement_S} } A$
where $\circ$ denotes composition of mappings. | As $f$ is an injection, it is a fortiori a one-to-many relation.
From Image of Set Difference under Relation: Corollary 2:
:$\forall A \in \powerset S: \map {\paren {\complement_{\Img \RR} \circ \RR^\to} } A = \map {\paren {\RR^\to \circ \complement_S} } A$
where $\RR \subseteq S \times T$ is a one-to-many relation on ... | Let $f: S \to T$ be an [[Definition:Injection|injection]].
Let $f^\to: \powerset S \to \powerset T$ denote the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$.
Then:
:$\forall A \in \powerset S: \map {\paren {\complement_{\Img f} \circ f^\to} } A = \map {\paren {f^\to \circ \complement_S} ... | As $f$ is an [[Definition:Injection|injection]], it is [[Definition:A Fortiori|a fortiori]] a [[Definition:One-to-Many Relation|one-to-many relation]].
From [[Image of Set Difference under Relation/Corollary 2|Image of Set Difference under Relation: Corollary 2]]:
:$\forall A \in \powerset S: \map {\paren {\complemen... | Complement of Direct Image Mapping of Injection equals Direct Image of Complement | https://proofwiki.org/wiki/Complement_of_Direct_Image_Mapping_of_Injection_equals_Direct_Image_of_Complement | https://proofwiki.org/wiki/Complement_of_Direct_Image_Mapping_of_Injection_equals_Direct_Image_of_Complement | [
"Direct Image Mappings",
"Injections",
"Relative Complement"
] | [
"Definition:Injection",
"Definition:Direct Image Mapping/Mapping",
"Definition:Composition of Mappings"
] | [
"Definition:Injection",
"Definition:A Fortiori",
"Definition:One-to-Many Relation",
"Image of Set Difference under Relation/Corollary 2",
"Definition:One-to-Many Relation"
] |
proofwiki-16345 | Direct Image of Inverse Image of Direct Image equals Direct Image Mapping | Let $f: S \to T$ be a mapping.
Let:
:$f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$
:$f^\gets: \powerset T \to \powerset S$ denote the inverse image mapping of $f$
where $\powerset S$ denotes the power set of $S$.
Then:
:$f^\to \circ f^\gets \circ f^\to = f^\to$
where $\circ$ denotes compos... | {{begin-eqn}}
{{eqn | q = \forall A \in \powerset S
| l = A
| o = \subseteq
| r = \map {\paren {f^\gets \circ f^\to} } A
| c = Subset of Domain is Subset of Preimage of Image
}}
{{eqn | ll= \leadsto
| q = \forall A \in \powerset S
| l = \map {f^\to} A
| o = \subseteq
| r ... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let:
:$f^\to: \powerset S \to \powerset T$ denote the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$
:$f^\gets: \powerset T \to \powerset S$ denote the [[Definition:Inverse Image Mapping of Mapping|inverse image mapping]] of $f$
where $\... | {{begin-eqn}}
{{eqn | q = \forall A \in \powerset S
| l = A
| o = \subseteq
| r = \map {\paren {f^\gets \circ f^\to} } A
| c = [[Subset of Domain is Subset of Preimage of Image]]
}}
{{eqn | ll= \leadsto
| q = \forall A \in \powerset S
| l = \map {f^\to} A
| o = \subseteq
... | Direct Image of Inverse Image of Direct Image equals Direct Image Mapping | https://proofwiki.org/wiki/Direct_Image_of_Inverse_Image_of_Direct_Image_equals_Direct_Image_Mapping | https://proofwiki.org/wiki/Direct_Image_of_Inverse_Image_of_Direct_Image_equals_Direct_Image_Mapping | [
"Direct Image Mappings",
"Inverse Image Mappings"
] | [
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Power Set",
"Definition:Composition of Mappings"
] | [
"Subset of Domain is Subset of Preimage of Image",
"Image of Subset under Mapping is Subset of Image",
"Subset of Codomain is Superset of Image of Preimage",
"Image of Subset under Mapping is Subset of Image"
] |
proofwiki-16346 | Inverse Image of Direct Image of Inverse Image equals Inverse Image Mapping | Let $f: S \to T$ be a mapping.
Let:
:$f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$
:$f^\gets: \powerset T \to \powerset S$ denote the inverse image mapping of $f$
where $\powerset S$ denotes the power set of $S$.
Then:
:$f^\gets \circ f^\to \circ f^\gets = f^\gets$
where $\circ$ denotes co... | {{begin-eqn}}
{{eqn | q = \forall A \in \powerset S
| l = A
| o = \subseteq
| r = \map {\paren {f^\gets \circ f^\to} } A
| c = Subset of Domain is Subset of Preimage of Image
}}
{{eqn | ll= \leadsto
| q = \forall B \in \powerset T
| l = \map {f^\gets} B
| o = \subseteq
| ... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let:
:$f^\to: \powerset S \to \powerset T$ denote the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$
:$f^\gets: \powerset T \to \powerset S$ denote the [[Definition:Inverse Image Mapping of Mapping|inverse image mapping]] of $f$
where $\... | {{begin-eqn}}
{{eqn | q = \forall A \in \powerset S
| l = A
| o = \subseteq
| r = \map {\paren {f^\gets \circ f^\to} } A
| c = [[Subset of Domain is Subset of Preimage of Image]]
}}
{{eqn | ll= \leadsto
| q = \forall B \in \powerset T
| l = \map {f^\gets} B
| o = \subseteq
... | Inverse Image of Direct Image of Inverse Image equals Inverse Image Mapping | https://proofwiki.org/wiki/Inverse_Image_of_Direct_Image_of_Inverse_Image_equals_Inverse_Image_Mapping | https://proofwiki.org/wiki/Inverse_Image_of_Direct_Image_of_Inverse_Image_equals_Inverse_Image_Mapping | [
"Direct Image Mappings",
"Inverse Image Mappings"
] | [
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Power Set",
"Definition:Composition of Mappings"
] | [
"Subset of Domain is Subset of Preimage of Image",
"Image of Subset under Mapping is Subset of Image",
"Subset of Codomain is Superset of Image of Preimage",
"Image of Subset under Mapping is Subset of Image"
] |
proofwiki-16347 | Direct Image of Intersection with Inverse Image | Then:
:$\forall A \in \powerset S, B \in \powerset T: \map {f^\to} {A \cap \map {f^\gets} B} = \map {f^\to} A \cap B$ | Let $A \in \powerset S, B \in \powerset T$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = \map {f^\to} {A \cap \map {f^\gets} B}
| o = \subseteq
| r = \map {f^\to} A \cap \map {f^\to} {\map {f^\gets} B}
| c = Image of Intersection under Mapping
}}
{{eqn | o = \subseteq
| r = \map {f^\to} A \cap... | Then:
:$\forall A \in \powerset S, B \in \powerset T: \map {f^\to} {A \cap \map {f^\gets} B} = \map {f^\to} A \cap B$ | Let $A \in \powerset S, B \in \powerset T$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = \map {f^\to} {A \cap \map {f^\gets} B}
| o = \subseteq
| r = \map {f^\to} A \cap \map {f^\to} {\map {f^\gets} B}
| c = [[Image of Intersection under Mapping]]
}}
{{eqn | o = \subseteq
| r = \map {f^\to} ... | Direct Image of Intersection with Inverse Image | https://proofwiki.org/wiki/Direct_Image_of_Intersection_with_Inverse_Image | https://proofwiki.org/wiki/Direct_Image_of_Intersection_with_Inverse_Image | [
"Direct Image Mappings",
"Inverse Image Mappings",
"Set Intersection"
] | [] | [
"Image of Intersection under Mapping",
"Subset of Codomain is Superset of Image of Preimage",
"Definition:Set Equality"
] |
proofwiki-16348 | Graph of Real Function in Cartesian Plane intersects Vertical at One Point | Let $f: \R \to \R$ be a real function.
Let its graph be embedded in the Cartesian plane $\CC$:
:520px
Every vertical line through a point $a$ in the domain of $f$ intersects the graph of $f$ at exactly one point $P = \tuple {a, \map f a}$. | From Equation of Vertical Line, a vertical line in $\CC$ through the point $\tuple {a, 0}$ on the $x$-axis has an equation $x = a$.
A real function is by definition a mapping.
Hence:
:$\forall a_1, a_2 \in \Dom f: a_1 = a_2 \implies \map f {a_1} = \map f {a_2}$
where $\Dom f$ denotes the domain of $f$.
Thus for each $a... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let its [[Definition:Graph of Mapping|graph]] be embedded in the [[Definition:Cartesian Plane|Cartesian plane]] $\CC$:
:[[File:Graph-of-function-intersect-vertical.png|520px]]
Every [[Definition:Vertical Line|vertical line]] through a [[Definition... | From [[Equation of Vertical Line]], a [[Definition:Vertical Line|vertical line]] in $\CC$ through the [[Definition:Point|point]] $\tuple {a, 0}$ on the [[Definition:X-Axis|$x$-axis]] has an [[Definition:Equation of Geometric Figure|equation]] $x = a$.
A [[Definition:Real Function|real function]] is by definition a [[... | Graph of Real Function in Cartesian Plane intersects Vertical at One Point | https://proofwiki.org/wiki/Graph_of_Real_Function_in_Cartesian_Plane_intersects_Vertical_at_One_Point | https://proofwiki.org/wiki/Graph_of_Real_Function_in_Cartesian_Plane_intersects_Vertical_at_One_Point | [
"Real Functions",
"Graphs of Mappings"
] | [
"Definition:Real Function",
"Definition:Graph of Mapping",
"Definition:Cartesian Plane",
"File:Graph-of-function-intersect-vertical.png",
"Definition:Vertical Line",
"Definition:Point",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Intersection (Geometry)",
"Definition:Graph of Mapping",
"... | [
"Equation of Vertical Line",
"Definition:Vertical Line",
"Definition:Point",
"Definition:Axis/X-Axis",
"Definition:Equation of Geometric Figure",
"Definition:Real Function",
"Definition:Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Unique",
"Definition:Ordered Pair",
"Definitio... |
proofwiki-16349 | Equation of Vertical Line | Let $\LL$ be a vertical line embedded in the Cartesian plane $\CC$.
Then the equation of $\LL$ can be given by:
:$x = a$
where $\tuple {a, 0}$ is the point at which $\LL$ intersects the $x$-axis.
:520px | From the Normal Form of Equation of Straight Line in Plane, a general straight line can be expressed in the form:
:$x \cos \alpha + y \sin \alpha = p$
where:
:$p$ is the length of a perpendicular $\PP$ from $\LL$ to the origin.
:$\alpha$ is the angle made between $\PP$ and the $x$-axis.
As $\LL$ is vertical, then by de... | Let $\LL$ be a [[Definition:Vertical Line|vertical line]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]] $\CC$.
Then the [[Definition:Equation of Geometric Figure|equation]] of $\LL$ can be given by:
:$x = a$
where $\tuple {a, 0}$ is the [[Definition:Point|point]] at which $\LL$ [[Definition:Intersec... | From the [[Equation of Straight Line in Plane/Normal Form|Normal Form of Equation of Straight Line in Plane]], a general [[Definition:Straight Line|straight line]] can be expressed in the form:
:$x \cos \alpha + y \sin \alpha = p$
where:
:$p$ is the [[Definition:Length of Line|length]] of a [[Definition:Perpendicular... | Equation of Vertical Line | https://proofwiki.org/wiki/Equation_of_Vertical_Line | https://proofwiki.org/wiki/Equation_of_Vertical_Line | [
"Equations of Straight Lines in Plane"
] | [
"Definition:Vertical Line",
"Definition:Cartesian Plane",
"Definition:Equation of Geometric Figure",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Axis/X-Axis",
"File:Graph-of-vertical-line.png"
] | [
"Equation of Straight Line in Plane/Normal Form",
"Definition:Line/Straight Line",
"Definition:Linear Measure/Length",
"Definition:Right Angle/Perpendicular",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Vertical Line",
"Definition:Horizontal Line"... |
proofwiki-16350 | Equation of Horizontal Line | Let $\LL$ be a horizontal line embedded in the Cartesian plane $\CC$.
Then the equation of $\LL$ can be given by:
:$y = b$
where $\tuple {0, b}$ is the point at which $\LL$ intersects the $y$-axis.
:520px | From the Normal Form of Equation of Straight Line in Plane, a general straight line can be expressed in the form:
:$x \cos \alpha + y \sin \alpha = p$
where:
:$p$ is the length of a perpendicular $\PP$ from $\LL$ to the origin
:$\alpha$ is the angle made between $\PP$ and the $x$-axis.
As $\LL$ is horizontal, then by d... | Let $\LL$ be a [[Definition:Horizontal Line|horizontal line]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]] $\CC$.
Then the [[Definition:Equation of Geometric Figure|equation]] of $\LL$ can be given by:
:$y = b$
where $\tuple {0, b}$ is the [[Definition:Point|point]] at which $\LL$ [[Definition:Inte... | From the [[Equation of Straight Line in Plane/Normal Form|Normal Form of Equation of Straight Line in Plane]], a general [[Definition:Straight Line|straight line]] can be expressed in the form:
:$x \cos \alpha + y \sin \alpha = p$
where:
:$p$ is the [[Definition:Length of Line|length]] of a [[Definition:Perpendicular... | Equation of Horizontal Line | https://proofwiki.org/wiki/Equation_of_Horizontal_Line | https://proofwiki.org/wiki/Equation_of_Horizontal_Line | [
"Equations of Straight Lines in Plane"
] | [
"Definition:Horizontal Line",
"Definition:Cartesian Plane",
"Definition:Equation of Geometric Figure",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Axis/Y-Axis",
"File:Graph-of-horizontal-line.png"
] | [
"Equation of Straight Line in Plane/Normal Form",
"Definition:Line/Straight Line",
"Definition:Linear Measure/Length",
"Definition:Right Angle/Perpendicular",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Horizontal Line",
"Definition:Vertical Line"... |
proofwiki-16351 | Unit of Ring of Mappings iff Image is Subset of Ring Units/Image is Subset of Ring Units implies Unit of Ring of Mappings | Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.
Let $U_R$ be the set of units in $R$.
Let $S$ be a set.
Let $\struct {R^S, +', \circ'}$ be the ring of mappings on the set of mappings $R^S$.
Let $\Img f \subseteq U_R$ where $\Img f$ is the image of $f$.
Then:
:$f \in R^S$ is a unit of $R^S$
and the inverse of $f... | By assumption:
:$\forall x \in S: \exists \map f x^{-1} : \map f x \circ \map f x^{-1} = \map f x^{-1} \circ \map f x = 1_R$
Let $f^{-1} : S \to U_R$ be defined by:
:$\forall x \in S : \map {f^{-1}} {x} = \map f x^{-1}$
Consider the mapping $f \circ’ f^{-1}$.
For all $x \in S$:
{{begin-eqn}}
{{eqn | l = \map {\paren... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
Let $U_R$ be the [[Definition:Set|set]] of [[Definition:Unit of Ring|units]] in $R$.
Let $S$ be a [[Definition:Set|set]].
Let $\struct {R^S, +', \circ'}$ be the [[Definition:Ring of Mappings|ring of ma... | By assumption:
:$\forall x \in S: \exists \map f x^{-1} : \map f x \circ \map f x^{-1} = \map f x^{-1} \circ \map f x = 1_R$
Let $f^{-1} : S \to U_R$ be defined by:
:$\forall x \in S : \map {f^{-1}} {x} = \map f x^{-1}$
Consider the [[Definition:Mapping|mapping]] $f \circ’ f^{-1}$.
For all $x \in S$:
{{begin-eqn... | Unit of Ring of Mappings iff Image is Subset of Ring Units/Image is Subset of Ring Units implies Unit of Ring of Mappings | https://proofwiki.org/wiki/Unit_of_Ring_of_Mappings_iff_Image_is_Subset_of_Ring_Units/Image_is_Subset_of_Ring_Units_implies_Unit_of_Ring_of_Mappings | https://proofwiki.org/wiki/Unit_of_Ring_of_Mappings_iff_Image_is_Subset_of_Ring_Units/Image_is_Subset_of_Ring_Units_implies_Unit_of_Ring_of_Mappings | [
"Unit of Ring of Mappings iff Image is Subset of Ring Units"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Set",
"Definition:Unit of Ring",
"Definition:Set",
"Definition:Ring of Mappings",
"Definition:Set",
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Unit of Ring",
"Definition:... | [
"Definition:Mapping",
"Definition:Unit of Ring",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Mapping"
] |
proofwiki-16352 | Unit of Ring of Mappings iff Image is Subset of Ring Units/Unit of Ring of Mappings implies Image is Subset of Ring Units | Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.
Let $U_R$ be the set of units in $R$.
Let $S$ be a set.
Let $\struct {R^S, +', \circ'}$ be the ring of mappings on the set of mappings $R^S$.
Let $f \in R^S$ be a unit of $R^S$.
Then:
:$\Img f \subseteq U_R$
where $\Img f$ is the image of $f$.
In which case, the i... | Let $f^{-1}$ be the product inverse of $f$.
Let $x \in R$.
Then:
{{begin-eqn}}
{{eqn | l = 1_R
| r = \map {f_{1_R} } x
| c = Structure Induced by Ring with Unity Operations is Ring with Unity
}}
{{eqn | r = \map {\paren {f \circ' f^{-1} } } x
| c = {{Defof|Product Inverse|product inverse}}
}}
{{eqn |... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
Let $U_R$ be the [[Definition:Set|set]] of [[Definition:Unit of Ring|units]] in $R$.
Let $S$ be a [[Definition:Set|set]].
Let $\struct {R^S, +', \circ'}$ be the [[Definition:Ring of Mappings|ring of ma... | Let $f^{-1}$ be the [[Definition:Product Inverse|product inverse]] of $f$.
Let $x \in R$.
Then:
{{begin-eqn}}
{{eqn | l = 1_R
| r = \map {f_{1_R} } x
| c = [[Structure Induced by Ring with Unity Operations is Ring with Unity]]
}}
{{eqn | r = \map {\paren {f \circ' f^{-1} } } x
| c = {{Defof|Produc... | Unit of Ring of Mappings iff Image is Subset of Ring Units/Unit of Ring of Mappings implies Image is Subset of Ring Units | https://proofwiki.org/wiki/Unit_of_Ring_of_Mappings_iff_Image_is_Subset_of_Ring_Units/Unit_of_Ring_of_Mappings_implies_Image_is_Subset_of_Ring_Units | https://proofwiki.org/wiki/Unit_of_Ring_of_Mappings_iff_Image_is_Subset_of_Ring_Units/Unit_of_Ring_of_Mappings_implies_Image_is_Subset_of_Ring_Units | [
"Unit of Ring of Mappings iff Image is Subset of Ring Units"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Set",
"Definition:Unit of Ring",
"Definition:Set",
"Definition:Ring of Mappings",
"Definition:Set",
"Definition:Mapping",
"Definition:Unit of Ring",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:... | [
"Definition:Product Inverse",
"Structure Induced by Ring with Unity Operations is Ring with Unity",
"Definition:Product Inverse",
"Definition:Unit of Ring"
] |
proofwiki-16353 | Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line | Let $f: \R \to \R$ be a real function which is surjective.
Let its graph be embedded in the Cartesian plane $\CC$:
:520px
Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on at least one point $P = \tuple {a, b}$ where $b = \map f a$. | From Equation of Horizontal Line, a horizontal line in $\CC$ through the point $\tuple {0, b}$ on the $y$-axis has an equation $y = b$.
By hypothesis, $f$ is a surjection.
Hence:
:$\forall b \in \R: \exists a \in \R: b = \map f a$
Thus for each $b \in \R$ there exists at least one ordered pair $\tuple {a, b}$ such that... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] which is [[Definition:Surjection|surjective]].
Let its [[Definition:Graph of Mapping|graph]] be embedded in the [[Definition:Cartesian Plane|Cartesian plane]] $\CC$:
:[[File:Graph-of-surjection-intersect-horizontal.png|520px]]
Every [[Definition:Ho... | From [[Equation of Horizontal Line]], a [[Definition:Horizontal Line|horizontal line]] in $\CC$ through the [[Definition:Point|point]] $\tuple {0, b}$ on the [[Definition:Y-Axis|$y$-axis]] has an [[Definition:Equation of Geometric Figure|equation]] $y = b$.
[[Definition:By Hypothesis|By hypothesis]], $f$ is a [[Defin... | Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line | https://proofwiki.org/wiki/Graph_of_Real_Surjection_in_Coordinate_Plane_intersects_Every_Horizontal_Line | https://proofwiki.org/wiki/Graph_of_Real_Surjection_in_Coordinate_Plane_intersects_Every_Horizontal_Line | [
"Real Functions",
"Surjections",
"Graphs of Mappings"
] | [
"Definition:Real Function",
"Definition:Surjection",
"Definition:Graph of Mapping",
"Definition:Cartesian Plane",
"File:Graph-of-surjection-intersect-horizontal.png",
"Definition:Horizontal Line",
"Definition:Point",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Intersection (Geometry)",
... | [
"Equation of Horizontal Line",
"Definition:Horizontal Line",
"Definition:Point",
"Definition:Axis/Y-Axis",
"Definition:Equation of Geometric Figure",
"Definition:By Hypothesis",
"Definition:Surjection",
"Definition:Ordered Pair",
"Definition:Point",
"Definition:Graph of Mapping"
] |
proofwiki-16354 | Graph of Real Injection in Coordinate Plane intersects Horizontal Line at most Once | Let $f: \R \to \R$ be a real function which is injective.
Let its graph be embedded in the Cartesian plane $\CC$:
:520px
Let $\LL$ be a horizontal line through a point $b$ in the codomain of $f$.
Then $\LL$ intersects the graph of $f$ on at most one point $P = \tuple {a, b}$ where $b = \map f a$. | From Equation of Horizontal Line, a horizontal line in $\CC$ through the point $\tuple {0, b}$ on the $y$-axis has an equation $y = b$.
By hypothesis, $f$ is a injection.
Hence:
:$\forall a_1, a_2 \in \Dom f: \map f {a_1} = \map f {a_2} \implies a_1 = a_2$
where $\Dom f$ denotes the domain of $f$.
Thus for each $b \in ... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] which is [[Definition:Injection|injective]].
Let its [[Definition:Graph of Mapping|graph]] be embedded in the [[Definition:Cartesian Plane|Cartesian plane]] $\CC$:
:[[File:Graph-of-injection-intersect-horizontal.png|520px]]
Let $\LL$ be a [[Definit... | From [[Equation of Horizontal Line]], a [[Definition:Horizontal Line|horizontal line]] in $\CC$ through the [[Definition:Point|point]] $\tuple {0, b}$ on the [[Definition:Y-Axis|$y$-axis]] has an [[Definition:Equation of Geometric Figure|equation]] $y = b$.
[[Definition:By Hypothesis|By hypothesis]], $f$ is a [[Defin... | Graph of Real Injection in Coordinate Plane intersects Horizontal Line at most Once | https://proofwiki.org/wiki/Graph_of_Real_Injection_in_Coordinate_Plane_intersects_Horizontal_Line_at_most_Once | https://proofwiki.org/wiki/Graph_of_Real_Injection_in_Coordinate_Plane_intersects_Horizontal_Line_at_most_Once | [
"Real Functions",
"Injections",
"Graphs of Mappings"
] | [
"Definition:Real Function",
"Definition:Injection",
"Definition:Graph of Mapping",
"Definition:Cartesian Plane",
"File:Graph-of-injection-intersect-horizontal.png",
"Definition:Horizontal Line",
"Definition:Point",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Intersection (Geometry)",
"... | [
"Equation of Horizontal Line",
"Definition:Horizontal Line",
"Definition:Point",
"Definition:Axis/Y-Axis",
"Definition:Equation of Geometric Figure",
"Definition:By Hypothesis",
"Definition:Injection",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Unique",
"Definition:Ordered Pair",
"Def... |
proofwiki-16355 | Graph of Real Bijection in Coordinate Plane intersects Horizontal Line at One Point | Let $f: \R \to \R$ be a real function which is bijective.
Let its graph be embedded in the Cartesian plane $\CC$:
:520px
Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on exactly one point $P = \tuple {a, b}$ where $b = \map f a$. | By definition, a bijection is a mapping which is both an injection and a surjection.
Let $\LL$ be a horizontal line through a point $b$ in the codomain of $f$.
From Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line:
:$\LL$ intersects the graph of $f$ on at least one point $P = \tuple {a, b}$... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] which is [[Definition:Bijection|bijective]].
Let its [[Definition:Graph of Mapping|graph]] be embedded in the [[Definition:Cartesian Plane|Cartesian plane]] $\CC$:
:[[File:Graph-of-bijection-intersect-horizontal.png|520px]]
Every [[Definition:Horiz... | By definition, a [[Definition:Bijection|bijection]] is a [[Definition:Mapping|mapping]] which is both an [[Definition:Injection|injection]] and a [[Definition:Surjection|surjection]].
Let $\LL$ be a [[Definition:Horizontal Line|horizontal line]] through a [[Definition:Point|point]] $b$ in the [[Definition:Codomain of... | Graph of Real Bijection in Coordinate Plane intersects Horizontal Line at One Point | https://proofwiki.org/wiki/Graph_of_Real_Bijection_in_Coordinate_Plane_intersects_Horizontal_Line_at_One_Point | https://proofwiki.org/wiki/Graph_of_Real_Bijection_in_Coordinate_Plane_intersects_Horizontal_Line_at_One_Point | [
"Real Functions",
"Bijections",
"Graphs of Mappings"
] | [
"Definition:Real Function",
"Definition:Bijection",
"Definition:Graph of Mapping",
"Definition:Cartesian Plane",
"File:Graph-of-bijection-intersect-horizontal.png",
"Definition:Horizontal Line",
"Definition:Point",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Intersection (Geometry)",
"... | [
"Definition:Bijection",
"Definition:Mapping",
"Definition:Injection",
"Definition:Surjection",
"Definition:Horizontal Line",
"Definition:Point",
"Definition:Codomain (Set Theory)/Mapping",
"Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line",
"Definition:Intersection (Geom... |
proofwiki-16356 | Composite of Surjection on Injection is not necessarily Either | Let $f$ be an injection.
Let $g$ be a surjection.
Let $g \circ f$ denote the composition of $g$ with $f$.
Then it is not necessarily the case that $g \circ f$ is either a surjection or an injection. | Let $X, Y, Z$ be sets defined as:
{{begin-eqn}}
{{eqn | l = X
| r = \set {a, b, c}
| c =
}}
{{eqn | l = Y
| r = \set {1, 2, 3, 4}
| c =
}}
{{eqn | l = Z
| r = \set {z, y, z}
| c =
}}
{{end-eqn}}
Let $f: X \to Y$ be defined in two-row notation as:
:$\dbinom {a \ b \ c } {1 \ 2 \ 3}... | Let $f$ be an [[Definition:Injection|injection]].
Let $g$ be a [[Definition:Surjection|surjection]].
Let $g \circ f$ denote the [[Definition:Composition of Mappings|composition]] of $g$ with $f$.
Then it is not necessarily the case that $g \circ f$ is either a [[Definition:Surjection|surjection]] or an [[Definition... | Let $X, Y, Z$ be [[Definition:Set|sets]] defined as:
{{begin-eqn}}
{{eqn | l = X
| r = \set {a, b, c}
| c =
}}
{{eqn | l = Y
| r = \set {1, 2, 3, 4}
| c =
}}
{{eqn | l = Z
| r = \set {z, y, z}
| c =
}}
{{end-eqn}}
Let $f: X \to Y$ be defined in [[Definition:Two-Row Notation|tw... | Composite of Surjection on Injection is not necessarily Either | https://proofwiki.org/wiki/Composite_of_Surjection_on_Injection_is_not_necessarily_Either | https://proofwiki.org/wiki/Composite_of_Surjection_on_Injection_is_not_necessarily_Either | [
"Injections",
"Surjections",
"Composite Mappings"
] | [
"Definition:Injection",
"Definition:Surjection",
"Definition:Composition of Mappings",
"Definition:Surjection",
"Definition:Injection"
] | [
"Definition:Set",
"Definition:Permutation on n Letters/Two-Row Notation",
"Definition:Injection",
"Definition:Permutation on n Letters/Two-Row Notation",
"Definition:Surjection",
"Definition:Composition of Mappings",
"Definition:Injection",
"Definition:Surjection",
"Category:Injections",
"Category... |
proofwiki-16357 | Composition of 3 Mappings where Pairs of Mappings are Bijections | Let $A$, $B$, $C$ and $D$ be sets.
Let:
:$f: A \to B$
:$g: B \to C$
:$h: C \to D$
be mappings.
Let $g \circ f$ and $h \circ g$ be bijections.
Then $f$, $g$ and $h$ are all bijections. | We note that both $g \circ f$ and $h \circ g$ are both injections and surjections by definition of bijection.
First it is shown that $g$ is a bijection.
We are given that:
:$g \circ f$ is a bijection.
From Injection if Composite is Injection it follows that $f$ is an injection.
From Surjection if Composite is Surjectio... | Let $A$, $B$, $C$ and $D$ be [[Definition:Set|sets]].
Let:
:$f: A \to B$
:$g: B \to C$
:$h: C \to D$
be [[Definition:Mapping|mappings]].
Let $g \circ f$ and $h \circ g$ be [[Definition:Bijection|bijections]].
Then $f$, $g$ and $h$ are all [[Definition:Bijection|bijections]]. | We note that both $g \circ f$ and $h \circ g$ are both [[Definition:Injection|injections]] and [[Definition:Surjection|surjections]] by definition of [[Definition:Bijection|bijection]].
First it is shown that $g$ is a [[Definition:Bijection|bijection]].
We are given that:
:$g \circ f$ is a [[Definition:Bijection|bij... | Composition of 3 Mappings where Pairs of Mappings are Bijections | https://proofwiki.org/wiki/Composition_of_3_Mappings_where_Pairs_of_Mappings_are_Bijections | https://proofwiki.org/wiki/Composition_of_3_Mappings_where_Pairs_of_Mappings_are_Bijections | [
"Bijections",
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Bijection"
] | [
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Bijection",
"Injection if Composite is Injection",
"Definition:Injection",
"Surjection if Composite is Surjection",
"Definition:Surjection",
"Definition:Bijection",
"Injection if Composit... |
proofwiki-16358 | Mapping Composed with Bijection forming Bijection is Bijection | Let $A$, $B$ and $C$ be sets.
Let $f: A \to B$ and $g: B \to C$ be mappings.
Let the composite mapping $g \circ f$ be a bijection.
Let either $f$ or $g$ be a bijection.
Then both $f$ and $g$ are bijections. | We note that $g \circ f$ is both an injection and a surjection by definition of bijection.
We are given that:
:$g \circ f$ is a bijection.
From Injection if Composite is Injection it follows that $f$ is an injection.
From Surjection if Composite is Surjection it follows that $g$ is a surjection.
{{qed|lemma}}
First sup... | Let $A$, $B$ and $C$ be [[Definition:Set|sets]].
Let $f: A \to B$ and $g: B \to C$ be [[Definition:Mapping|mappings]].
Let the [[Definition:Composition of Mappings|composite mapping]] $g \circ f$ be a [[Definition:Bijection|bijection]].
Let either $f$ or $g$ be a [[Definition:Bijection|bijection]].
Then both $f$ a... | We note that $g \circ f$ is both an [[Definition:Injection|injection]] and a [[Definition:Surjection|surjection]] by definition of [[Definition:Bijection|bijection]].
We are given that:
:$g \circ f$ is a [[Definition:Bijection|bijection]].
From [[Injection if Composite is Injection]] it follows that $f$ is an [[Defi... | Mapping Composed with Bijection forming Bijection is Bijection | https://proofwiki.org/wiki/Mapping_Composed_with_Bijection_forming_Bijection_is_Bijection | https://proofwiki.org/wiki/Mapping_Composed_with_Bijection_forming_Bijection_is_Bijection | [
"Bijections",
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Bijection"
] | [
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Bijection",
"Injection if Composite is Injection",
"Definition:Injection",
"Surjection if Composite is Surjection",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Injection",
"Definition:Surjection... |
proofwiki-16359 | Composite of Three Mappings in Cycle forming Injections and Surjection | Let $A$, $B$ and $C$ be non-empty sets.
Let $f: A \to B$, $g: B \to C$ and $h: C \to A$ be mappings.
Let the following hold:
:$h \circ g \circ f$ is an injection
:$f \circ h \circ g$ is an injection
:$g \circ f \circ h$ is a surjection.
where:
:$g \circ f$ (and so on) denote composition of mappings.
Then each of $f$, $... | First note that from Composition of Mappings is Associative:
:$\paren {h \circ g} \circ f = h \circ \paren {g \circ f}$
and so on.
However, while there is no need to use parenthesis to establish the order of composition of mappings, in the following the technique will be used in order to clarify what is being done.
We ... | Let $A$, $B$ and $C$ be [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|sets]].
Let $f: A \to B$, $g: B \to C$ and $h: C \to A$ be [[Definition:Mapping|mappings]].
Let the following hold:
:$h \circ g \circ f$ is an [[Definition:Injection|injection]]
:$f \circ h \circ g$ is an [[Definition:Injection|injectio... | First note that from [[Composition of Mappings is Associative]]:
:$\paren {h \circ g} \circ f = h \circ \paren {g \circ f}$
and so on.
However, while there is no need to use [[Definition:Parenthesis|parenthesis]] to establish the order of [[Definition:Composition of Mappings|composition of mappings]], in the followin... | Composite of Three Mappings in Cycle forming Injections and Surjection | https://proofwiki.org/wiki/Composite_of_Three_Mappings_in_Cycle_forming_Injections_and_Surjection | https://proofwiki.org/wiki/Composite_of_Three_Mappings_in_Cycle_forming_Injections_and_Surjection | [
"Injections",
"Surjections",
"Bijections",
"Composite Mappings"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Injection",
"Definition:Injection",
"Definition:Surjection",
"Definition:Composition of Mappings",
"Definition:Bijection"
] | [
"Composition of Mappings is Associative",
"Definition:Parenthesis",
"Definition:Composition of Mappings",
"Definition:Injection",
"Injection if Composite is Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Defini... |
proofwiki-16360 | Component Mappings of Set Coproduct are Injective | Let $S_1$ and $S_2$ be sets.
Let $\struct {C, i_1, i_2}$ be a coproduct of $S_1$ and $S_2$.
Then $i_1$ and $i_2$ are injections. | By definition of coproduct:
:for all sets $X$ and mappings $f_1: S_1 \to X$ and $f_1: S_1 \to X$
::there exists a unique mapping $h: C \to X$ such that:
:::$h \circ i_1 = f_1$
:::$h \circ i_2 = f_2$
Let $X := S_1$ and $f_1: S_1 \to X = I_{S_1}$ where $I_{S_1}$ denotes the identity mapping.
We have:
:$h \circ i_1 = I_{S... | Let $S_1$ and $S_2$ be [[Definition:Set|sets]].
Let $\struct {C, i_1, i_2}$ be a [[Definition:Set Coproduct|coproduct]] of $S_1$ and $S_2$.
Then $i_1$ and $i_2$ are [[Definition:Injection|injections]]. | By definition of [[Definition:Set Coproduct|coproduct]]:
:for all [[Definition:Set|sets]] $X$ and [[Definition:Mapping|mappings]] $f_1: S_1 \to X$ and $f_1: S_1 \to X$
::there exists a unique [[Definition:Mapping|mapping]] $h: C \to X$ such that:
:::$h \circ i_1 = f_1$
:::$h \circ i_2 = f_2$
Let $X := S_1$ and $f_1:... | Component Mappings of Set Coproduct are Injective | https://proofwiki.org/wiki/Component_Mappings_of_Set_Coproduct_are_Injective | https://proofwiki.org/wiki/Component_Mappings_of_Set_Coproduct_are_Injective | [
"Set Coproducts"
] | [
"Definition:Set",
"Definition:Coproduct/Sets",
"Definition:Injection"
] | [
"Definition:Coproduct/Sets",
"Definition:Set",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Identity Mapping",
"Identity Mapping is Injection",
"Definition:Injection",
"Injection if Composite is Injection",
"Definition:Injection",
"Definition:Injection"
] |
proofwiki-16361 | Coproduct on Disjoint Union | Let $S_1$ and $S_2$ be sets.
Let $S_1 \sqcup S_2 := \paren {S_1 \times \set 1} \cup \paren {S_2 \times \set 2}$ be the disjoint union of $S_1$ and $S_2$.
Let $i_1: S_1 \to S_1 \sqcup S_2$ and $i_2: S_2 \to S_1 \sqcup S_2$ be the mappings defined as:
:$\forall s_1 \in S_1: \map {i_1} {s_1} = \tuple {s_1, 1}$
:$\forall s... | For $\struct {S_1 \sqcup S_2, i_1, i_2}$ to be a coproduct, it is necessary that:
:for all sets $X$ and all mappings $f_1: S_1 \to X$ and $f_2: S_2 \to X$
::there exists a unique mapping $h: S_1 \sqcup S_2 \to X$ such that:
:::$h \circ i_1 = f_1$
:::$h \circ i_2 = f_2$
Let $h$ be the mapping defined as:
:$\forall \tupl... | Let $S_1$ and $S_2$ be [[Definition:Set|sets]].
Let $S_1 \sqcup S_2 := \paren {S_1 \times \set 1} \cup \paren {S_2 \times \set 2}$ be the [[Definition:Disjoint Union (Set Theory)|disjoint union]] of $S_1$ and $S_2$.
Let $i_1: S_1 \to S_1 \sqcup S_2$ and $i_2: S_2 \to S_1 \sqcup S_2$ be the [[Definition:Mapping|mappi... | For $\struct {S_1 \sqcup S_2, i_1, i_2}$ to be a [[Definition:Set Coproduct|coproduct]], it is necessary that:
:for all [[Definition:Set|sets]] $X$ and all [[Definition:Mapping|mappings]] $f_1: S_1 \to X$ and $f_2: S_2 \to X$
::there exists a [[Definition:Unique|unique]] [[Definition:Mapping|mapping]] $h: S_1 \sqcup S... | Coproduct on Disjoint Union | https://proofwiki.org/wiki/Coproduct_on_Disjoint_Union | https://proofwiki.org/wiki/Coproduct_on_Disjoint_Union | [
"Set Coproducts",
"Disjoint Unions"
] | [
"Definition:Set",
"Definition:Disjoint Union (Set Theory)",
"Definition:Mapping",
"Definition:Coproduct/Sets"
] | [
"Definition:Coproduct/Sets",
"Definition:Set",
"Definition:Mapping",
"Definition:Unique",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Unique",
"Definition:Mapping",
"Definition:Unique",
"Definition:Coproduct/Sets"
] |
proofwiki-16362 | Existence of Bijection between Coproducts of two Sets | Let $S_1$ and $S_2$ be sets.
Let $\struct {C, i_1, i_2}$ and $\struct {D, j_1, j_2}$ be two coproducts on $S_1$ and $S_2$.
Then there exists a unique bijection $\theta: D \to C$ such that:
:$\theta \circ j_i = i_1$
:$\theta \circ j_2 = i_2$ | Let $X$ be an arbitrary set.
Let $f_1: S_1 \to X$ and $f_2: S_2 \to X$ be arbitrary mappings.
Let $h_C: C \to X$ be the unique mapping such that:
:$h_C \circ i_1 = f_1$
:$h_C \circ i_2 = f_2$
Let $h_D: D \to X$ be the unique mapping such that:
:$h_D \circ j_1 = f_1$
:$h_D \circ j_2 = f_2$
The existence and uniqueness o... | Let $S_1$ and $S_2$ be [[Definition:Set|sets]].
Let $\struct {C, i_1, i_2}$ and $\struct {D, j_1, j_2}$ be two [[Definition:Set Coproduct|coproducts]] on $S_1$ and $S_2$.
Then there exists a [[Definition:Unique|unique]] [[Definition:Bijection|bijection]] $\theta: D \to C$ such that:
:$\theta \circ j_i = i_1$
:$\theta... | Let $X$ be an arbitrary [[Definition:Set|set]].
Let $f_1: S_1 \to X$ and $f_2: S_2 \to X$ be arbitrary [[Definition:Mapping|mappings]].
Let $h_C: C \to X$ be the [[Definition:Unique|unique]] [[Definition:Mapping|mapping]] such that:
:$h_C \circ i_1 = f_1$
:$h_C \circ i_2 = f_2$
Let $h_D: D \to X$ be the [[Definiti... | Existence of Bijection between Coproducts of two Sets | https://proofwiki.org/wiki/Existence_of_Bijection_between_Coproducts_of_two_Sets | https://proofwiki.org/wiki/Existence_of_Bijection_between_Coproducts_of_two_Sets | [
"Set Coproducts"
] | [
"Definition:Set",
"Definition:Coproduct/Sets",
"Definition:Unique",
"Definition:Bijection"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Unique",
"Definition:Mapping",
"Definition:Unique",
"Definition:Mapping",
"Definition:Unique",
"Definition:Coproduct/Sets"
] |
proofwiki-16363 | Number of Friday 13ths in a Year | In any given year, there are between $1$ and $3$ (inclusive) months in which the $13$th falls on a Friday. | The day of the week on which the $13$th falls is directly dependent upon the day of the week that the $1$st of the month falls.
From Months that Start on the Same Day of the Week, the months can be grouped into equivalence classes according to which day of the week the month starts:
For a non-leap year, the set of equi... | In any given year, there are between $1$ and $3$ (inclusive) [[Definition:Month|months]] in which the $13$th falls on a Friday. | The [[Definition:Day|day]] of the [[Definition:Week|week]] on which the $13$th falls is directly dependent upon the [[Definition:Day|day]] of the [[Definition:Week|week]] that the $1$st of the [[Definition:Month|month]] falls.
From [[Equivalence Class/Examples/Months that Start on the Same Day of the Week|Months that ... | Number of Friday 13ths in a Year | https://proofwiki.org/wiki/Number_of_Friday_13ths_in_a_Year | https://proofwiki.org/wiki/Number_of_Friday_13ths_in_a_Year | [
"Calendars"
] | [
"Definition:Time/Unit/Month"
] | [
"Definition:Time/Unit/Day",
"Definition:Time/Unit/Week",
"Definition:Time/Unit/Day",
"Definition:Time/Unit/Week",
"Definition:Time/Unit/Month",
"Equivalence Class/Examples/Months that Start on the Same Day of the Week",
"Definition:Time/Unit/Month",
"Definition:Equivalence Class",
"Definition:Time/U... |
proofwiki-16364 | Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class Contains 1 Odd Number | Let $\eqclass n \alpha$ be the $\alpha$-equivalence class of a natural number $n$.
Then $\eqclass n \alpha$ contains exactly $1$ odd number. | That $\alpha$ is an equivalence relation is proved in Equivalence Relation on Natural Numbers such that Quotient is Power of Two.
Let $x$ be a natural number whose $\alpha$-equivalence class is $\eqclass x \alpha$.
If $x$ is odd, then $\eqclass x \alpha$ contains that odd number $x$
We have that $x$ is of the form $x =... | Let $\eqclass n \alpha$ be the [[Definition:Equivalence Class|$\alpha$-equivalence class]] of a [[Definition:Natural Number|natural number]] $n$.
Then $\eqclass n \alpha$ contains [[Definition:Unique|exactly $1$]] [[Definition:Odd Number|odd number]]. | That $\alpha$ is an [[Definition:Equivalence Relation|equivalence relation]] is proved in [[Equivalence Relation on Natural Numbers such that Quotient is Power of Two]].
Let $x$ be a [[Definition:Natural Number|natural number]] whose [[Definition:Equivalence Class|$\alpha$-equivalence class]] is $\eqclass x \alpha$.
... | Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class Contains 1 Odd Number | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two/Equivalence_Class_Contains_1_Odd_Number | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two/Equivalence_Class_Contains_1_Odd_Number | [
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two",
"Examples of Equivalence Relations"
] | [
"Definition:Equivalence Class",
"Definition:Natural Numbers",
"Definition:Unique",
"Definition:Odd Integer"
] | [
"Definition:Equivalence Relation",
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two",
"Definition:Natural Numbers",
"Definition:Equivalence Class",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Odd Integ... |
proofwiki-16365 | Equivalence Relation on Natural Numbers such that Quotient is Power of Two/One of Pair of Equivalent Elements is Divisor of the Other | Let $c, d \in \N$ such that $c \mathrel \alpha d$.
Then either:
:$c \divides d$
or:
:$d \divides c$
where $\divides$ denotes divisibility. | That $\alpha$ is an equivalence relation is proved in Equivalence Relation on Natural Numbers such that Quotient is Power of Two.
We are given that $c \mathrel \alpha d$.
{{WLOG}}, suppose $c < d$.
If $d < c$ then the same argument holds, {{mutatis}}.
By definition of $\alpha$, we have that:
:$c = 2^n d$
for some $n \i... | Let $c, d \in \N$ such that $c \mathrel \alpha d$.
Then either:
:$c \divides d$
or:
:$d \divides c$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | That $\alpha$ is an [[Definition:Equivalence Relation|equivalence relation]] is proved in [[Equivalence Relation on Natural Numbers such that Quotient is Power of Two]].
We are given that $c \mathrel \alpha d$.
{{WLOG}}, suppose $c < d$.
If $d < c$ then the same argument holds, {{mutatis}}.
By definition of $\alp... | Equivalence Relation on Natural Numbers such that Quotient is Power of Two/One of Pair of Equivalent Elements is Divisor of the Other | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two/One_of_Pair_of_Equivalent_Elements_is_Divisor_of_the_Other | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two/One_of_Pair_of_Equivalent_Elements_is_Divisor_of_the_Other | [
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two"
] | [
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Equivalence Relation",
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-16366 | Exists Divisor in Set of n+1 Natural Numbers no greater than 2n | Let $S$ be a set of $n + 1$ non-non-zero natural numbers all less than or equal to $2 n$.
Then there exists $a, b \in S$ such that
:$a \divides b$
where $\divides$ denotes divisibility. | Let $\alpha$ denote the relation defined on the natural numbers $\N$ by:
:$\forall x, y \in \N: x \mathrel \alpha y \iff \exists n \in \Z: x = 2^n y$
From Equivalence Relation on Natural Numbers such that Quotient is Power of Two, $\alpha$ is an equivalence relation.
From Equivalence Class under $\alpha$ Contains $1$ O... | Let $S$ be a [[Definition:Set|set]] of $n + 1$ non-[[Definition:Zero (Number)|non-zero]] [[Definition:Natural Number|natural numbers]] all less than or equal to $2 n$.
Then there exists $a, b \in S$ such that
:$a \divides b$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]]. | Let $\alpha$ denote the [[Definition:Relation|relation]] defined on the [[Definition:Natural Numbers|natural numbers]] $\N$ by:
:$\forall x, y \in \N: x \mathrel \alpha y \iff \exists n \in \Z: x = 2^n y$
From [[Equivalence Relation on Natural Numbers such that Quotient is Power of Two]], $\alpha$ is an [[Definition:E... | Exists Divisor in Set of n+1 Natural Numbers no greater than 2n | https://proofwiki.org/wiki/Exists_Divisor_in_Set_of_n+1_Natural_Numbers_no_greater_than_2n | https://proofwiki.org/wiki/Exists_Divisor_in_Set_of_n+1_Natural_Numbers_no_greater_than_2n | [
"Divisibility"
] | [
"Definition:Set",
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Relation",
"Definition:Natural Numbers",
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two",
"Definition:Equivalence Relation",
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class Contains 1 Odd Number",
"Definition:Odd Intege... |
proofwiki-16367 | Equality of Squares Modulo Integer is Equivalence Relation | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\RR_n$ be the relation on the set of integers $\Z$ defined as:
:$\forall x, y \in \Z: x \mathrel {\RR_n} y \iff x^2 \equiv y^2 \pmod n$
Then $\RR_n$ is an equivalence relation. | Checking in turn each of the criteria for equivalence: | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\RR_n$ be the [[Definition:Endorelation|relation]] on the [[Definition:Integer|set of integers]] $\Z$ defined as:
:$\forall x, y \in \Z: x \mathrel {\RR_n} y \iff x^2 \equiv y^2 \pmod n$
Then $\RR_n$ is an [[Definition... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Equality of Squares Modulo Integer is Equivalence Relation | https://proofwiki.org/wiki/Equality_of_Squares_Modulo_Integer_is_Equivalence_Relation | https://proofwiki.org/wiki/Equality_of_Squares_Modulo_Integer_is_Equivalence_Relation | [
"Examples of Equivalence Relations",
"Examples of Equivalence Relations"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Endorelation",
"Definition:Integer",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-16368 | Left Ideal is Left Module over Ring | Let $\struct {R, +, \times}$ be a ring.
Let $J \subseteq R$ be a left ideal of $R$.
Let $\circ : R \times J \to J$ be the restriction of $\times$ to $R \times J$.
Then $\struct {J, +, \circ}$ is a left module over $\struct {R, +, \times}$. | By definition of a left ideal then $\circ$ is well-defined. | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $J \subseteq R$ be a [[Definition:Left Ideal of Ring|left ideal]] of $R$.
Let $\circ : R \times J \to J$ be the [[Definition:Restriction of Mapping|restriction]] of $\times$ to $R \times J$.
Then $\struct {J, +, \circ}$ is a [[Defini... | By definition of a [[Definition:Left Ideal of Ring|left ideal]] then $\circ$ is [[Definition:Well-Defined|well-defined]]. | Left Ideal is Left Module over Ring | https://proofwiki.org/wiki/Left_Ideal_is_Left_Module_over_Ring | https://proofwiki.org/wiki/Left_Ideal_is_Left_Module_over_Ring | [
"Left Ideal is Left Module over Ring",
"Left Modules over Rings",
"Ideal Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring/Left Ideal",
"Definition:Restriction/Mapping",
"Definition:Left Module over Ring"
] | [
"Definition:Ideal of Ring/Left Ideal",
"Definition:Well-Defined"
] |
proofwiki-16369 | Right Ideal is Right Module over Ring | Let $\struct {R, +, \times}$ be a ring.
Let $J \subseteq R$ be a right ideal of $R$.
Let $\circ : J \times R \to J$ be the restriction of $\times$ to $J \times R$.
Then $\struct {J, +, \circ}$ is a right module over $\struct {R, +, \times}$. | By definition of a right ideal then $\circ$ is well-defined. | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $J \subseteq R$ be a [[Definition:Right Ideal of Ring|right ideal]] of $R$.
Let $\circ : J \times R \to J$ be the [[Definition:Restriction of Mapping|restriction]] of $\times$ to $J \times R$.
Then $\struct {J, +, \circ}$ is a [[Defi... | By definition of a [[Definition:Right Ideal of Ring|right ideal]] then $\circ$ is [[Definition:Well-Defined|well-defined]]. | Right Ideal is Right Module over Ring | https://proofwiki.org/wiki/Right_Ideal_is_Right_Module_over_Ring | https://proofwiki.org/wiki/Right_Ideal_is_Right_Module_over_Ring | [
"Right Ideal is Right Module over Ring",
"Module Theory",
"Ideal Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring/Right Ideal",
"Definition:Restriction/Mapping",
"Definition:Right Module over Ring"
] | [
"Definition:Ideal of Ring/Right Ideal",
"Definition:Well-Defined"
] |
proofwiki-16370 | Opposite Ring is Ring | Let $\struct {R, +, \times}$ be a ring.
Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$.
Then $\struct {R, +, *}$ is a ring. | By definition of the opposite ring:
:$\forall x, y \in R: x * y = y \times x$.
By definition of the ring $R$, $\struct {R, +}$ is an abelian group.
To complete the proof, it remains to be shown that $\struct {R, *}$ is a semigroup.
That is, it remains to show that $\struct{R, *}$ is associative.
Let $a, b, c \in R$.
{{... | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {R, +, *}$ be the [[Definition:Opposite Ring|opposite ring]] of $\struct {R, +, \times}$.
Then $\struct {R, +, *}$ is a [[Definition:Ring (Abstract Algebra)|ring]]. | By definition of the [[Definition:Opposite Ring|opposite ring]]:
:$\forall x, y \in R: x * y = y \times x$.
By definition of the [[Definition:Ring (Abstract Algebra)|ring]] $R$, $\struct {R, +}$ is an [[Definition:Abelian Group|abelian group]].
To complete the proof, it remains to be shown that $\struct {R, *}$ is a... | Opposite Ring is Ring | https://proofwiki.org/wiki/Opposite_Ring_is_Ring | https://proofwiki.org/wiki/Opposite_Ring_is_Ring | [
"Examples of Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Opposite Ring",
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Opposite Ring",
"Definition:Ring (Abstract Algebra)",
"Definition:Abelian Group",
"Definition:Semigroup",
"Definition:Associative Operation"
] |
proofwiki-16371 | Opposite Ring of Opposite Ring | Let $\struct {R, +, \times}$ be a ring.
Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$.
Let $\struct {R, +, \circ}$ be the opposite ring of $\struct {R, +, *}$.
Then $\struct {R, +, \circ} = \struct {R, +, \times}$. | By definition of the opposite ring:
:$\forall x, y \in S: x * y = y \times x$
:$\forall x, y \in S: x \circ y = y * x$
Hence for all $x,y \in S$:
:$x \circ y = y * x = x \times y$
The result follows.
{{qed}}
Category:Ring Theory
k14myncy6bkp8xbic8ndvx15pkuigex | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {R, +, *}$ be the [[Definition:Opposite Ring|opposite ring]] of $\struct {R, +, \times}$.
Let $\struct {R, +, \circ}$ be the [[Definition:Opposite Ring|opposite ring]] of $\struct {R, +, *}$.
Then $\struct {R, +, \circ} = \s... | By definition of the [[Definition:Opposite Ring|opposite ring]]:
:$\forall x, y \in S: x * y = y \times x$
:$\forall x, y \in S: x \circ y = y * x$
Hence for all $x,y \in S$:
:$x \circ y = y * x = x \times y$
The result follows.
{{qed}}
[[Category:Ring Theory]]
k14myncy6bkp8xbic8ndvx15pkuigex | Opposite Ring of Opposite Ring | https://proofwiki.org/wiki/Opposite_Ring_of_Opposite_Ring | https://proofwiki.org/wiki/Opposite_Ring_of_Opposite_Ring | [
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Opposite Ring",
"Definition:Opposite Ring"
] | [
"Definition:Opposite Ring",
"Category:Ring Theory"
] |
proofwiki-16372 | Left Module over Ring Induces Right Module over Opposite Ring | Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {R, +_R, *_R}$ be the opposite ring of $\struct {R, +_R, \times_R}$.
Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$.
Let $\circ' : G \times R \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: x \... | It is shown that $\struct{G, +_G, \circ'}$ satisfies the right module axioms.
By definition of the opposite ring:
:$\forall x, y \in R: x *_R y = y \times_R x$ | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {R, +_R, *_R}$ be the [[Definition:Opposite Ring|opposite ring]] of $\struct {R, +_R, \times_R}$.
Let $\struct{G, +_G, \circ}$ be a [[Definition:Left Module over Ring|left module]] over $\struct {R, +_R, \times_R}$.
Let $... | It is shown that $\struct{G, +_G, \circ'}$ satisfies the [[Axiom:Right Module Axioms|right module axioms]].
By definition of the [[Definition:Opposite Ring|opposite ring]]:
:$\forall x, y \in R: x *_R y = y \times_R x$ | Left Module over Ring Induces Right Module over Opposite Ring | https://proofwiki.org/wiki/Left_Module_over_Ring_Induces_Right_Module_over_Opposite_Ring | https://proofwiki.org/wiki/Left_Module_over_Ring_Induces_Right_Module_over_Opposite_Ring | [
"Right Modules over Rings",
"Left Modules over Rings",
"Modules over Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Opposite Ring",
"Definition:Left Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Right Module over Ring"
] | [
"Axiom:Right Module Axioms",
"Definition:Opposite Ring"
] |
proofwiki-16373 | Right Module over Ring Induces Left Module over Opposite Ring | Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {R, +_R, *_R}$ be the opposite ring of $\struct {R, +_R, \times_R}$.
Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: \l... | It is shown that $\struct {G, +_G, \circ'}$ satisfies the left module axioms.
By definition of the opposite ring:
:$\forall x, y \in S: x *_R y = y \times_R x$. | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {R, +_R, *_R}$ be the [[Definition:Opposite Ring|opposite ring]] of $\struct {R, +_R, \times_R}$.
Let $\struct{G, +_G, \circ}$ be a [[Definition:Right Module over Ring|right module]] over $\struct {R, +_R, \times_R}$.
Let... | It is shown that $\struct {G, +_G, \circ'}$ satisfies the [[Axiom:Left Module Axioms|left module axioms]].
By definition of the [[Definition:Opposite Ring|opposite ring]]:
:$\forall x, y \in S: x *_R y = y \times_R x$. | Right Module over Ring Induces Left Module over Opposite Ring | https://proofwiki.org/wiki/Right_Module_over_Ring_Induces_Left_Module_over_Opposite_Ring | https://proofwiki.org/wiki/Right_Module_over_Ring_Induces_Left_Module_over_Opposite_Ring | [
"Right Modules over Rings",
"Left Modules over Rings",
"Modules over Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Opposite Ring",
"Definition:Right Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Left Module over Ring"
] | [
"Axiom:Left Module Axioms",
"Definition:Opposite Ring"
] |
proofwiki-16374 | Ring is Commutative iff Opposite Ring is Itself | Let $\struct {R, +, \times}$ be a ring.
Let $\struct {R, +, *}$ be the opposite ring of $\struct {R, +, \times}$.
Then $\struct {R, +, \times}$ is a commutative ring {{iff}}:
:$\struct {R, +, \times} = \struct {R, +, *}$ | By definition of the opposite ring:
:$\forall x, y \in R: x * y = y \times x$ | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {R, +, *}$ be the [[Definition:Opposite Ring|opposite ring]] of $\struct {R, +, \times}$.
Then $\struct {R, +, \times}$ is a [[Definition:Commutative Ring|commutative ring]] {{iff}}:
:$\struct {R, +, \times} = \struct {R, +, ... | By definition of the [[Definition:Opposite Ring|opposite ring]]:
:$\forall x, y \in R: x * y = y \times x$ | Ring is Commutative iff Opposite Ring is Itself | https://proofwiki.org/wiki/Ring_is_Commutative_iff_Opposite_Ring_is_Itself | https://proofwiki.org/wiki/Ring_is_Commutative_iff_Opposite_Ring_is_Itself | [
"Examples of Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Opposite Ring",
"Definition:Commutative Ring"
] | [
"Definition:Opposite Ring"
] |
proofwiki-16375 | Ceva's Theorem | Let $\triangle ABC$ be a triangle.
Let $L$, $M$ and $N$ be points on the sides $BC$, $AC$ and $AB$ respectively.
Then the lines $AL$, $BM$ and $CN$ are concurrent {{iff}}:
:$\dfrac {BL} {LC} \times \dfrac {CM} {MA} \times \dfrac {AN} {NB} = 1$ | :400px
=== Necessary Condition ===
Let $AL$, $BM$ and $CN$ be concurrent.
Let the point of concurrency be $P$.
Consider the triangles $\triangle ALB$ and $\triangle ALC$.
They have the same altitude from the common base $BC$.
Hence:
:$\dfrac {\map \Area {ALB} } {\map \Area {ALC} } = \dfrac {BL} {LC}$
Similarly, conside... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $L$, $M$ and $N$ be [[Definition:Point|points]] on the [[Definition:Side of Polygon|sides]] $BC$, $AC$ and $AB$ respectively.
Then the [[Definition:Line Segment|lines]] $AL$, $BM$ and $CN$ are [[Definition:Concurrent Lines|concurrent]] {{iff}}... | :[[File:Cevas-Theorem.png|400px]]
=== Necessary Condition ===
Let $AL$, $BM$ and $CN$ be [[Definition:Concurrent Lines|concurrent]].
Let the point of [[Definition:Concurrent Lines|concurrency]] be $P$.
Consider the [[Definition:Triangle (Geometry)|triangles]] $\triangle ALB$ and $\triangle ALC$.
They have the sam... | Ceva's Theorem/Proof 1 | https://proofwiki.org/wiki/Ceva's_Theorem | https://proofwiki.org/wiki/Ceva's_Theorem/Proof_1 | [
"Ceva's Theorem",
"Triangles",
"Concurrency"
] | [
"Definition:Triangle (Geometry)",
"Definition:Point",
"Definition:Polygon/Side",
"Definition:Line/Segment",
"Definition:Concurrent Lines"
] | [
"File:Cevas-Theorem.png",
"Definition:Concurrent Lines",
"Definition:Concurrent Lines",
"Definition:Triangle (Geometry)",
"Definition:Altitude of Triangle",
"Definition:Triangle (Geometry)/Base",
"Definition:Triangle (Geometry)",
"Definition:Altitude of Triangle",
"Definition:Triangle (Geometry)/Bas... |
proofwiki-16376 | Ceva's Theorem | Let $\triangle ABC$ be a triangle.
Let $L$, $M$ and $N$ be points on the sides $BC$, $AC$ and $AB$ respectively.
Then the lines $AL$, $BM$ and $CN$ are concurrent {{iff}}:
:$\dfrac {BL} {LC} \times \dfrac {CM} {MA} \times \dfrac {AN} {NB} = 1$ | === Necessary Condition ===
We have {{hypothesis}}:
:$AL$, $BM$ and $CN$ are concurrent in $\triangle ABC$ at point $P$.
:400px
Following the sides anticlockwise in $\triangle LAB$:
{{begin-eqn}}
{{eqn | n = 1
| l = \dfrac {LP} {PA} \cdot \dfrac {AN} {NB} \cdot \dfrac {BC} {CL}
| r = -1
| c = Menelaus... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $L$, $M$ and $N$ be [[Definition:Point|points]] on the [[Definition:Side of Polygon|sides]] $BC$, $AC$ and $AB$ respectively.
Then the [[Definition:Line Segment|lines]] $AL$, $BM$ and $CN$ are [[Definition:Concurrent Lines|concurrent]] {{iff}}... | === Necessary Condition ===
We have {{hypothesis}}:
:$AL$, $BM$ and $CN$ are [[Definition:Concurrent|concurrent]] in $\triangle ABC$ at [[Definition:Point|point]] $P$.
:[[File:Cevas-Theorem.png|400px]]
Following the [[Definition:Side|sides]] [[Definition:Anticlockwise|anticlockwise]] in $\triangle LAB$:
{{begin-eqn... | Ceva's Theorem/Proof 2 | https://proofwiki.org/wiki/Ceva's_Theorem | https://proofwiki.org/wiki/Ceva's_Theorem/Proof_2 | [
"Ceva's Theorem",
"Triangles",
"Concurrency"
] | [
"Definition:Triangle (Geometry)",
"Definition:Point",
"Definition:Polygon/Side",
"Definition:Line/Segment",
"Definition:Concurrent Lines"
] | [
"Definition:Concurrent",
"Definition:Point",
"File:Cevas-Theorem.png",
"Definition:Side",
"Definition:Anticlockwise",
"Menelaus's Theorem",
"Definition:Side",
"Definition:Clockwise",
"Menelaus's Theorem",
"Definition:Directed Line Segment",
"Definition:Concurrent",
"Definition:Point",
"Defin... |
proofwiki-16377 | Left Module induces Right Module over same Ring iff Actions are Commutative | Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$.
Let $\circ': G \times R \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: x \circ' \lambda = \lambda \circ x$
Then $\struct {G, +_G, \circ'}$ is a right module... | === Necessary Condition ===
Let $\struct {G, +_G, \circ'}$ be a right module over $\struct {R, +_R, \times_R}$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {\lambda \times_R \mu} \circ x
| r = x \circ' \paren {\lambda \times_R \mu}
| c = Definition of $\circ'$
}}
{{eqn | r = \paren {x \circ' \lambda} \circ' \mu
... | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {G, +_G, \circ}$ be a [[Definition:Left Module over Ring|left module]] over $\struct {R, +_R, \times_R}$.
Let $\circ': G \times R \to G$ be the [[Definition:Binary Operation|binary operation]] defined by:
:$\forall \lambda... | === Necessary Condition ===
Let $\struct {G, +_G, \circ'}$ be a [[Definition:Right Module over Ring|right module]] over $\struct {R, +_R, \times_R}$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {\lambda \times_R \mu} \circ x
| r = x \circ' \paren {\lambda \times_R \mu}
| c = Definition of $\circ'$
}}
{{eqn | r... | Left Module induces Right Module over same Ring iff Actions are Commutative | https://proofwiki.org/wiki/Left_Module_induces_Right_Module_over_same_Ring_iff_Actions_are_Commutative | https://proofwiki.org/wiki/Left_Module_induces_Right_Module_over_same_Ring_iff_Actions_are_Commutative | [
"Module Theory",
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Left Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Right Module over Ring"
] | [
"Definition:Right Module over Ring"
] |
proofwiki-16378 | Right Module induces Left Module over same Ring iff Actions are Commutative | Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct {G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: \lambda \circ' x = x \circ \lambda $
Then $\struct {G, +_G, \circ'}$ is a left modu... | === Necessary Condition ===
Let $\struct {G, +_G, \circ'}$ be a left module over $\struct {R, +_R, \times_R}$.
Then:
{{begin-eqn}}
{{eqn | l = x \circ \paren {\lambda \times_R \mu}
| r = \paren {\lambda \times_R \mu} \circ' x
| c = Definition of $\circ'$
}}
{{eqn | r = \lambda \circ' \paren {\mu \circ' x}
... | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {G, +_G, \circ}$ be a [[Definition:Right Module over Ring|right module]] over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the [[Definition:Binary Operation|binary operation]] defined by:
:$\forall \lam... | === Necessary Condition ===
Let $\struct {G, +_G, \circ'}$ be a [[Definition:Left Module over Ring|left module]] over $\struct {R, +_R, \times_R}$.
Then:
{{begin-eqn}}
{{eqn | l = x \circ \paren {\lambda \times_R \mu}
| r = \paren {\lambda \times_R \mu} \circ' x
| c = Definition of $\circ'$
}}
{{eqn | r =... | Right Module induces Left Module over same Ring iff Actions are Commutative | https://proofwiki.org/wiki/Right_Module_induces_Left_Module_over_same_Ring_iff_Actions_are_Commutative | https://proofwiki.org/wiki/Right_Module_induces_Left_Module_over_same_Ring_iff_Actions_are_Commutative | [
"Module Theory",
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Right Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Left Module over Ring"
] | [
"Definition:Left Module over Ring"
] |
proofwiki-16379 | Fermat Problem | Let $\triangle ABC$ be a triangle
Let the vertices of $\triangle ABC$ all have angles less than $120 \degrees$.
Let $\triangle ABG$, $\triangle BCE$ and $\triangle ACF$ be equilateral triangles constructed on the sides of $ABC$.
Let $AE$, $BF$ and $CG$ be constructed.
Let $P$ be the point at which $AE$, $BF$ and $CG$ m... | The sum of the distances will be a minimum when the lines $PA$, $PB$ and $PC$ all meet at an angle of $120 \degrees$.
This is a consequence of the '''Fermat problem''' being a special case of the Steiner tree problem.
Consider the circles which circumscribe the $3$ equilateral triangles $\triangle ABG$, $\triangle BCE$... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]]
Let the [[Definition:Vertex of Polygon|vertices]] of $\triangle ABC$ all have [[Definition:Plane Angle|angles]] less than $120 \degrees$.
Let $\triangle ABG$, $\triangle BCE$ and $\triangle ACF$ be [[Definition:Equilateral Triangle|equilateral tria... | The sum of the [[Definition:Distance between Points|distances]] will be a minimum when the [[Definition:Line Segment|lines]] $PA$, $PB$ and $PC$ all meet at an [[Definition:Plane Angle|angle]] of $120 \degrees$.
This is a consequence of the '''[[Fermat Problem|Fermat problem]]''' being a special case of the [[Steiner ... | Fermat Problem | https://proofwiki.org/wiki/Fermat_Problem | https://proofwiki.org/wiki/Fermat_Problem | [
"Fermat Problem",
"Fermat-Torricelli Point",
"Triangles",
"Extremum Problems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Vertex",
"Definition:Angle",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Polygon/Side",
"Definition:Point",
"File:FermatPointConstruction.png",
"Definition:Fermat-Torricelli Point",
"Definition:Polygon/Vertex",
"Definition:Polygon... | [
"Definition:Distance between Points",
"Definition:Line/Segment",
"Definition:Angle",
"Fermat Problem",
"Steiner Tree Problem",
"Definition:Circle",
"Definition:Circumscribe/Circle around Polygon",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Quadrilateral",
"Definition:Point",
"Defi... |
proofwiki-16380 | Left Module over Commutative Ring induces Right Module | Let $\struct {R, +_R, \times_R}$ be a commutative ring.
Let $\struct {G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$.
Let $\circ' : G \times R \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: x \circ' \lambda = \lambda \circ x$
Then $\struct {G, +_G, \circ'}$ is a... | From Ring is Commutative iff Opposite Ring is Itself, $\struct {R, +_R, \times_R}$ is its own opposite ring.
From Left Module over Ring Induces Right Module over Opposite Ring, $\struct {G, +_G, \circ'}$ is a right module over $\struct {R, +_R, \times_R}$.
{{qed}} | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {G, +_G, \circ}$ be a [[Definition:Left Module over Ring|left module]] over $\struct {R, +_R, \times_R}$.
Let $\circ' : G \times R \to G$ be the [[Definition:Binary Operation|binary operation]] defined by:
:$\forall \... | From [[Ring is Commutative iff Opposite Ring is Itself]], $\struct {R, +_R, \times_R}$ is its own [[Definition:Opposite Ring|opposite ring]].
From [[Left Module over Ring Induces Right Module over Opposite Ring]], $\struct {G, +_G, \circ'}$ is a [[Definition:Right Module over Ring|right module]] over $\struct {R, +_R,... | Left Module over Commutative Ring induces Right Module | https://proofwiki.org/wiki/Left_Module_over_Commutative_Ring_induces_Right_Module | https://proofwiki.org/wiki/Left_Module_over_Commutative_Ring_induces_Right_Module | [
"Right Modules over Rings",
"Left Modules over Rings",
"Modules over Rings"
] | [
"Definition:Commutative Ring",
"Definition:Left Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Right Module over Ring"
] | [
"Ring is Commutative iff Opposite Ring is Itself",
"Definition:Opposite Ring",
"Left Module over Ring Induces Right Module over Opposite Ring",
"Definition:Right Module over Ring"
] |
proofwiki-16381 | Right Module over Commutative Ring induces Left Module | Let $\struct {R, +_R, \times_R}$ be a commutative ring.
Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: \lambda \circ’ x = x \circ \lambda$
Then $\struct{G, +_G, \circ'}$ is a ... | From Ring is Commutative iff Opposite Ring is Itself, $\struct {R, +_R, \times_R}$ is its own opposite ring.
From Right Module over Ring Induces Left Module over Opposite Ring, $\struct{G, +_G, \circ'}$ is a left module over $\struct {R, +_R, \times_R}$.
{{qed}} | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct{G, +_G, \circ}$ be a [[Definition:Right Module over Ring|right module]] over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the [[Definition:Binary Operation|binary operation]] defined by:
:$\forall ... | From [[Ring is Commutative iff Opposite Ring is Itself]], $\struct {R, +_R, \times_R}$ is its own [[Definition:Opposite Ring|opposite ring]].
From [[Right Module over Ring Induces Left Module over Opposite Ring]], $\struct{G, +_G, \circ'}$ is a [[Definition:Left Module over Ring|left module]] over $\struct {R, +_R, \t... | Right Module over Commutative Ring induces Left Module | https://proofwiki.org/wiki/Right_Module_over_Commutative_Ring_induces_Left_Module | https://proofwiki.org/wiki/Right_Module_over_Commutative_Ring_induces_Left_Module | [
"Right Modules over Rings",
"Left Modules over Rings",
"Modules over Rings"
] | [
"Definition:Commutative Ring",
"Definition:Right Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Left Module over Ring"
] | [
"Ring is Commutative iff Opposite Ring is Itself",
"Definition:Opposite Ring",
"Right Module over Ring Induces Left Module over Opposite Ring",
"Definition:Left Module over Ring"
] |
proofwiki-16382 | Right Ideal is Right Module over Ring/Ring is Right Module over Ring | Let $\struct {R, +, \times}$ be a ring.
Then $\struct {R, +, \times}$ is a right module over $\struct {R, +, \times}$. | From Ring is Ideal of Itself, $R$ is a right ideal.
From Right Ideal is Right Module over Ring, $\struct {R, +, \times}$ is a right module over $\struct {R, +, \times}$.
{{qed}} | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Then $\struct {R, +, \times}$ is a [[Definition:Right Module over Ring|right module]] over $\struct {R, +, \times}$. | From [[Ring is Ideal of Itself]], $R$ is a [[Definition:Right Ideal of Ring|right ideal]].
From [[Right Ideal is Right Module over Ring]], $\struct {R, +, \times}$ is a [[Definition:Right Module over Ring|right module]] over $\struct {R, +, \times}$.
{{qed}} | Right Ideal is Right Module over Ring/Ring is Right Module over Ring | https://proofwiki.org/wiki/Right_Ideal_is_Right_Module_over_Ring/Ring_is_Right_Module_over_Ring | https://proofwiki.org/wiki/Right_Ideal_is_Right_Module_over_Ring/Ring_is_Right_Module_over_Ring | [
"Right Ideal is Right Module over Ring"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Right Module over Ring"
] | [
"Ring is Ideal of Itself",
"Definition:Ideal of Ring/Right Ideal",
"Right Ideal is Right Module over Ring",
"Definition:Right Module over Ring"
] |
proofwiki-16383 | Left Ideal is Left Module over Ring/Ring is Left Module over Ring | Let $\struct {R, +, \times}$ be a ring.
Then $\struct {R, +, \times}$ is a left module over $\struct {R, +, \times}$. | From Ring is Ideal of Itself, $R$ is a left ideal.
From Left Ideal is Left Module over Ring, $\struct {R, +, \times}$ is a left module over $\struct {R, +, \times}$.
{{qed}} | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Then $\struct {R, +, \times}$ is a [[Definition:Left Module over Ring|left module]] over $\struct {R, +, \times}$. | From [[Ring is Ideal of Itself]], $R$ is a [[Definition:Left Ideal of Ring|left ideal]].
From [[Left Ideal is Left Module over Ring]], $\struct {R, +, \times}$ is a [[Definition:Left Module over Ring|left module]] over $\struct {R, +, \times}$.
{{qed}} | Left Ideal is Left Module over Ring/Ring is Left Module over Ring | https://proofwiki.org/wiki/Left_Ideal_is_Left_Module_over_Ring/Ring_is_Left_Module_over_Ring | https://proofwiki.org/wiki/Left_Ideal_is_Left_Module_over_Ring/Ring_is_Left_Module_over_Ring | [
"Left Ideal is Left Module over Ring"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Left Module over Ring"
] | [
"Ring is Ideal of Itself",
"Definition:Ideal of Ring/Left Ideal",
"Left Ideal is Left Module over Ring",
"Definition:Left Module over Ring"
] |
proofwiki-16384 | Integration by Inversion | :$\ds \int_0^{+\infty} \map f x \rd x = \int_0^{+\infty} \dfrac {\map f {\frac 1 x} } {x^2} \rd x$ | {{begin-eqn}}
{{eqn | l = \int_0^{+\infty} \map f x \rd x
| r = \int_{x \mathop \to 0}^{x \mathop \to +\infty} \map f x \rd x
| c = {{Defof|Improper Integral}}
}}
{{eqn | r = \int_{\frac 1 x \mathop \to 0}^{\frac 1 x \mathop \to +\infty} \map f {\frac 1 x} \map \rd {\frac 1 x}
| c = Integration by Sub... | :$\ds \int_0^{+\infty} \map f x \rd x = \int_0^{+\infty} \dfrac {\map f {\frac 1 x} } {x^2} \rd x$ | {{begin-eqn}}
{{eqn | l = \int_0^{+\infty} \map f x \rd x
| r = \int_{x \mathop \to 0}^{x \mathop \to +\infty} \map f x \rd x
| c = {{Defof|Improper Integral}}
}}
{{eqn | r = \int_{\frac 1 x \mathop \to 0}^{\frac 1 x \mathop \to +\infty} \map f {\frac 1 x} \map \rd {\frac 1 x}
| c = [[Integration by S... | Integration by Inversion | https://proofwiki.org/wiki/Integration_by_Inversion | https://proofwiki.org/wiki/Integration_by_Inversion | [
"Definite Integrals"
] | [] | [
"Integration by Substitution",
"Power Rule for Derivatives",
"Reversal of Limits of Definite Integral",
"Category:Definite Integrals"
] |
proofwiki-16385 | Left Module Does Not Necessarily Induce Right Module over Ring | Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$.
Let $\circ' : G \times R \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: x \circ' \lambda = \lambda \circ x$
Then $\struct{G, +_G, \circ'}$ is not necessarily... | Proof by Counterexample:
Let $\struct {S, +, \times}$ be a ring with unity
Let $\struct {\map {\MM_S} 2, +, \times}$ denote the ring of square matrices of order $2$ over $S$.
From Ring of Square Matrices over Ring is Ring, $\struct {\map {\MM_S} 2, +, \times}$ is a ring.
Let:
:$G := \set {\begin {bmatrix} x & 0 \\ y & ... | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct{G, +_G, \circ}$ be a [[Definition:Left Module over Ring|left module]] over $\struct {R, +_R, \times_R}$.
Let $\circ' : G \times R \to G$ be the [[Definition:Binary Operation|binary operation]] defined by:
:$\forall \lambda... | [[Proof by Counterexample]]:
Let $\struct {S, +, \times}$ be a [[Definition:Ring with Unity|ring with unity]]
Let $\struct {\map {\MM_S} 2, +, \times}$ denote the [[Definition:Ring of Square Matrices|ring of square matrices of order $2$ over $S$]].
From [[Ring of Square Matrices over Ring is Ring]], $\struct {\map {... | Left Module Does Not Necessarily Induce Right Module over Ring | https://proofwiki.org/wiki/Left_Module_Does_Not_Necessarily_Induce_Right_Module_over_Ring | https://proofwiki.org/wiki/Left_Module_Does_Not_Necessarily_Induce_Right_Module_over_Ring | [
"Left Module Does Not Necessarily Induce Right Module over Ring",
"Right Modules over Rings",
"Left Modules over Rings",
"Modules over Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Left Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Right Module over Ring"
] | [
"Proof by Counterexample",
"Definition:Ring with Unity",
"Definition:Ring of Square Matrices",
"Ring of Square Matrices over Ring is Ring",
"Definition:Ring (Abstract Algebra)"
] |
proofwiki-16386 | Right Module Does Not Necessarily Induce Left Module over Ring | Let $\struct {R, +_R, \times_R}$ be a ring.
Let $\struct{G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: \lambda \circ' x = x \circ \lambda$
Then $\struct{G, +_G, \circ'}$ is not necessaril... | Proof by Counterexample:
Let $\struct {S, +, \times}$ be a ring with unity
Let $\struct {\map {\MM_S} 2, +, \times}$ denote the ring of square matrices of order $2$ over $S$.
From Ring of Square Matrices over Ring is Ring, $\struct {\map {\MM_S} 2, +, \times}$ is a ring.
Let:
:$G := \set {\begin{bmatrix} x & y \\ 0 & 0... | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct{G, +_G, \circ}$ be a [[Definition:Right Module over Ring|right module]] over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the [[Definition:Binary Operation|binary operation]] defined by:
:$\forall \lamb... | [[Proof by Counterexample]]:
Let $\struct {S, +, \times}$ be a [[Definition:Ring with Unity|ring with unity]]
Let $\struct {\map {\MM_S} 2, +, \times}$ denote the [[Definition:Ring of Square Matrices|ring of square matrices of order $2$ over $S$]].
From [[Ring of Square Matrices over Ring is Ring]], $\struct {\map {... | Right Module Does Not Necessarily Induce Left Module over Ring | https://proofwiki.org/wiki/Right_Module_Does_Not_Necessarily_Induce_Left_Module_over_Ring | https://proofwiki.org/wiki/Right_Module_Does_Not_Necessarily_Induce_Left_Module_over_Ring | [
"Right Module Does Not Necessarily Induce Left Module over Ring",
"Right Modules over Rings",
"Left Modules over Rings",
"Modules over Rings"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Right Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Left Module over Ring"
] | [
"Proof by Counterexample",
"Definition:Ring with Unity",
"Definition:Ring of Square Matrices",
"Ring of Square Matrices over Ring is Ring",
"Definition:Ring (Abstract Algebra)"
] |
proofwiki-16387 | Left Module Does Not Necessarily Induce Right Module over Ring/Lemma | :$G$ is a left ideal of $\struct {\map {\MM_S} 2, +, \times}$. | From Test for Left Ideal, the following need to be proved:
:$(1): \quad G \ne \O$
:$(2): \quad \forall \mathop {\mathbf X}, \mathop {\mathbf Y} \in G: \mathbf X + \paren {-\mathbf Y} \in G$
:$(3): \quad \forall \mathop{\mathbf J} \in G, \mathop {\mathbf R} \in \map {\MM_S} 2: \mathbf R \times \mathbf J \in G$ | :$G$ is a [[Definition:Left Ideal of Ring|left ideal]] of $\struct {\map {\MM_S} 2, +, \times}$. | From [[Test for Left Ideal]], the following need to be proved:
:$(1): \quad G \ne \O$
:$(2): \quad \forall \mathop {\mathbf X}, \mathop {\mathbf Y} \in G: \mathbf X + \paren {-\mathbf Y} \in G$
:$(3): \quad \forall \mathop{\mathbf J} \in G, \mathop {\mathbf R} \in \map {\MM_S} 2: \mathbf R \times \mathbf J \in G$ | Left Module Does Not Necessarily Induce Right Module over Ring/Lemma | https://proofwiki.org/wiki/Left_Module_Does_Not_Necessarily_Induce_Right_Module_over_Ring/Lemma | https://proofwiki.org/wiki/Left_Module_Does_Not_Necessarily_Induce_Right_Module_over_Ring/Lemma | [
"Left Module Does Not Necessarily Induce Right Module over Ring"
] | [
"Definition:Ideal of Ring/Left Ideal"
] | [
"Test for Left Ideal"
] |
proofwiki-16388 | Right Module Does Not Necessarily Induce Left Module over Ring/Lemma | :$G$ is a right ideal of $\struct {\map {\MM_S} 2, +, \times}$. | From Test for Right Ideal, the following need to be proved:
:$(1): \quad G \ne \O$
:$(2): \quad \forall \mathop {\mathbf X}, \mathop{\mathbf Y} \in G: \mathbf X + \paren {-\mathbf Y} \in G$
:$(3): \quad \forall \mathop{\mathbf J} \in G, \mathop{\mathbf R} \in \map {\MM_S} 2: \mathbf J \times \mathbf R \in G$ | :$G$ is a [[Definition:Right Ideal of Ring|right ideal]] of $\struct {\map {\MM_S} 2, +, \times}$. | From [[Test for Right Ideal]], the following need to be proved:
:$(1): \quad G \ne \O$
:$(2): \quad \forall \mathop {\mathbf X}, \mathop{\mathbf Y} \in G: \mathbf X + \paren {-\mathbf Y} \in G$
:$(3): \quad \forall \mathop{\mathbf J} \in G, \mathop{\mathbf R} \in \map {\MM_S} 2: \mathbf J \times \mathbf R \in G$ | Right Module Does Not Necessarily Induce Left Module over Ring/Lemma | https://proofwiki.org/wiki/Right_Module_Does_Not_Necessarily_Induce_Left_Module_over_Ring/Lemma | https://proofwiki.org/wiki/Right_Module_Does_Not_Necessarily_Induce_Left_Module_over_Ring/Lemma | [
"Right Module Does Not Necessarily Induce Left Module over Ring"
] | [
"Definition:Ideal of Ring/Right Ideal"
] | [
"Test for Right Ideal"
] |
proofwiki-16389 | Ideal is Bimodule over Ring | Let $\struct {R, +, \times}$ be a ring.
Let $J \subseteq R$ be an ideal of $R$.
Let $\circ_l : R \times J \to J$ be the restriction of $\times$ to $R \times J$.
Let $\circ_r : J \times R \to J$ be the restriction of $\times$ to $J \times R$.
Then $\struct {J, +, \circ_l, \circ_r}$ is a bimodule over $\struct {R, +, \ti... | By definition of an ideal, $J$ is both a left ideal and a right ideal.
From Left Ideal is Left Module over Ring then $\struct {J, +, \circ_l}$ is a left module.
From Right Ideal is Right Module over Ring then $\struct {J, +, \circ_r}$ is a right module.
Then:
{{begin-eqn}}
{{eqn | q = \forall x, y \in R: \forall j \in ... | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $J \subseteq R$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
Let $\circ_l : R \times J \to J$ be the [[Definition:Restriction of Mapping|restriction]] of $\times$ to $R \times J$.
Let $\circ_r : J \times R \to J$ be the [[Definiti... | By definition of an [[Definition:Ideal of Ring|ideal]], $J$ is both a [[Definition:Left Ideal of Ring|left ideal]] and a [[Definition:Right Ideal of Ring|right ideal]].
From [[Left Ideal is Left Module over Ring]] then $\struct {J, +, \circ_l}$ is a [[Definition:Left Module over Ring|left module]].
From [[Right Ideal... | Ideal is Bimodule over Ring | https://proofwiki.org/wiki/Ideal_is_Bimodule_over_Ring | https://proofwiki.org/wiki/Ideal_is_Bimodule_over_Ring | [
"Bimodules",
"Ideal Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring",
"Definition:Restriction/Mapping",
"Definition:Restriction/Mapping",
"Definition:Bimodule"
] | [
"Definition:Ideal of Ring",
"Definition:Ideal of Ring/Left Ideal",
"Definition:Ideal of Ring/Right Ideal",
"Left Ideal is Left Module over Ring",
"Definition:Left Module over Ring",
"Right Ideal is Right Module over Ring",
"Definition:Right Module over Ring",
"Definition:Bimodule"
] |
proofwiki-16390 | Ideal is Bimodule over Ring/Ring is Bimodule over Ring | Let $\struct {R, +, \times}$ be a ring.
Then $\struct {R, +, \times, \times}$ is a bimodule over $\struct {R, +, \times}$. | From Ring is Ideal of Itself and Ideal is Bimodule over Ring, $\struct {R, +, \times, \times}$ is a bimodule over $\struct {R, +, \times}$.
{{qed}} | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Then $\struct {R, +, \times, \times}$ is a [[Definition:Bimodule|bimodule]] over $\struct {R, +, \times}$. | From [[Ring is Ideal of Itself]] and [[Ideal is Bimodule over Ring]], $\struct {R, +, \times, \times}$ is a [[Definition:Bimodule|bimodule]] over $\struct {R, +, \times}$.
{{qed}} | Ideal is Bimodule over Ring/Ring is Bimodule over Ring | https://proofwiki.org/wiki/Ideal_is_Bimodule_over_Ring/Ring_is_Bimodule_over_Ring | https://proofwiki.org/wiki/Ideal_is_Bimodule_over_Ring/Ring_is_Bimodule_over_Ring | [
"Bimodules",
"Module Theory",
"Ring Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Bimodule"
] | [
"Ring is Ideal of Itself",
"Ideal is Bimodule over Ring",
"Definition:Bimodule"
] |
proofwiki-16391 | Left Module over Commutative Ring induces Bimodule | Let $\struct {R, +_R, \times_R}$ be a commutative ring.
Let $\struct{G, +_G, \circ}$ be a left module over $\struct {R, +_R, \times_R}$.
Let $\circ' : G \times R \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: x \circ' \lambda = \lambda \circ x$
Then $\struct{G, +_G, \circ, \circ'}$... | From Left Module over Commutative Ring induces Right Module, $\struct{G, +_G, \circ'}$ is a right module.
Let $\lambda, \mu \in R$ and $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \lambda \circ \paren{x \circ' \mu}
| r = \lambda \circ \paren{\mu \circ x}
| c = Definition of $\circ’$
}}
{{eqn | r = \paren {\la... | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct{G, +_G, \circ}$ be a [[Definition:Left Module over Ring|left module]] over $\struct {R, +_R, \times_R}$.
Let $\circ' : G \times R \to G$ be the [[Definition:Binary Operation|binary operation]] defined by:
:$\forall \l... | From [[Left Module over Commutative Ring induces Right Module]], $\struct{G, +_G, \circ'}$ is a [[Definition:Right Module over Ring|right module]].
Let $\lambda, \mu \in R$ and $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \lambda \circ \paren{x \circ' \mu}
| r = \lambda \circ \paren{\mu \circ x}
| c = Defin... | Left Module over Commutative Ring induces Bimodule | https://proofwiki.org/wiki/Left_Module_over_Commutative_Ring_induces_Bimodule | https://proofwiki.org/wiki/Left_Module_over_Commutative_Ring_induces_Bimodule | [
"Module Theory",
"Ring Theory"
] | [
"Definition:Commutative Ring",
"Definition:Left Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Bimodule"
] | [
"Left Module over Commutative Ring induces Right Module",
"Definition:Right Module over Ring",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Commutative/Operation",
"Definition:Bimodule"
] |
proofwiki-16392 | Right Module over Commutative Ring induces Bimodule | Let $\struct {R, +_R, \times_R}$ be a commutative ring.
Let $\struct {G, +_G, \circ}$ be a right module over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the binary operation defined by:
:$\forall \lambda \in R: \forall x \in G: \lambda \circ' x = x \circ \lambda $
Then $\struct {G, +_G, \circ', \ci... | From Right Module over Commutative Ring induces Left Module, $\struct {G, +_G, \circ'}$ is a left module.
Let $\lambda, \mu \in R$ and $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \lambda \circ' \paren {x \circ \mu}
| r = \paren {x \circ \mu} \circ \lambda
| c = Definition of $\circ’$
}}
{{eqn | r = x \circ \... | Let $\struct {R, +_R, \times_R}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\struct {G, +_G, \circ}$ be a [[Definition:Right Module over Ring|right module]] over $\struct {R, +_R, \times_R}$.
Let $\circ' : R \times G \to G$ be the [[Definition:Binary Operation|binary operation]] defined by:
:$\forall... | From [[Right Module over Commutative Ring induces Left Module]], $\struct {G, +_G, \circ'}$ is a [[Definition:Left Module over Ring|left module]].
Let $\lambda, \mu \in R$ and $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \lambda \circ' \paren {x \circ \mu}
| r = \paren {x \circ \mu} \circ \lambda
| c = Defi... | Right Module over Commutative Ring induces Bimodule | https://proofwiki.org/wiki/Right_Module_over_Commutative_Ring_induces_Bimodule | https://proofwiki.org/wiki/Right_Module_over_Commutative_Ring_induces_Bimodule | [
"Module Theory",
"Ring Theory"
] | [
"Definition:Commutative Ring",
"Definition:Right Module over Ring",
"Definition:Operation/Binary Operation",
"Definition:Bimodule"
] | [
"Right Module over Commutative Ring induces Left Module",
"Definition:Left Module over Ring",
"Axiom:Right Module Axioms",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Commutative/Operation",
"Axiom:Right Module Axioms",
"Definition:Bimodule"
] |
proofwiki-16393 | Diagonals of Kite are Perpendicular | Let $ABCD$ be a kite such that $AC$ and $BD$ are its diagonals.
Then $AC$ and $BD$ are perpendicular. | :450px
Let $AC$ and $BD$ meet at $E$.
Consider the triangles $\triangle ABD$ and $\triangle CBD$.
We have that:
:$AB = CB$
:$AD = CD$
:$BD$ is common.
Hence by Triangle Side-Side-Side Congruence, $\triangle ABD$ and $\triangle CBD$ are congruent.
Consider the triangles $\triangle ABE$ and $\triangle CBE$.
We have from ... | Let $ABCD$ be a [[Definition:Kite|kite]] such that $AC$ and $BD$ are its [[Definition:Diagonal of Quadrilateral|diagonals]].
Then $AC$ and $BD$ are [[Definition:Perpendicular|perpendicular]]. | :[[File:Diagonals-of-Kite.png|450px]]
Let $AC$ and $BD$ [[Definition:Intersection (Geometry)|meet]] at $E$.
Consider the [[Definition:Triangle (Geometry)|triangles]] $\triangle ABD$ and $\triangle CBD$.
We have that:
:$AB = CB$
:$AD = CD$
:$BD$ is common.
Hence by [[Triangle Side-Side-Side Congruence]], $\triangl... | Diagonals of Kite are Perpendicular | https://proofwiki.org/wiki/Diagonals_of_Kite_are_Perpendicular | https://proofwiki.org/wiki/Diagonals_of_Kite_are_Perpendicular | [
"Kites"
] | [
"Definition:Quadrilateral/Kite",
"Definition:Diameter of Quadrilateral",
"Definition:Right Angle/Perpendicular"
] | [
"File:Diagonals-of-Kite.png",
"Definition:Intersection (Geometry)",
"Definition:Triangle (Geometry)",
"Triangle Side-Side-Side Congruence",
"Definition:Congruence (Geometry)",
"Definition:Triangle (Geometry)",
"Definition:Congruence (Geometry)",
"Triangle Side-Angle-Side Congruence",
"Definition:Con... |
proofwiki-16394 | One Diagonal of Kite Bisects the Other | Let $ABCD$ be a kite such that:
:$AC$ and $BD$ are its diagonals
:$AB = BC$
:$AD = DC$
Then $BD$ is the perpendicular bisector of $AC$. | :450px
Let $AC$ and $BD$ meet at $E$.
From Diagonals of Kite are Perpendicular, $AC$ and $BD$ are perpendicular.
That is:
:$\angle AEB = \angle CEB$
both being right angles.
Consider the triangles $\triangle ABE$ and $\triangle CBE$.
We have that:
:$\angle AEB = \angle CEB$ are both right angles.
:$AB = BC$
:$BE$ is co... | Let $ABCD$ be a [[Definition:Kite|kite]] such that:
:$AC$ and $BD$ are its [[Definition:Diagonal of Quadrilateral|diagonals]]
:$AB = BC$
:$AD = DC$
Then $BD$ is the [[Definition:Perpendicular Bisector|perpendicular bisector]] of $AC$. | :[[File:Diagonals-of-Kite.png|450px]]
Let $AC$ and $BD$ [[Definition:Intersection (Geometry)|meet]] at $E$.
From [[Diagonals of Kite are Perpendicular]], $AC$ and $BD$ are [[Definition:Perpendicular|perpendicular]].
That is:
:$\angle AEB = \angle CEB$
both being [[Definition:Right Angle|right angles]].
Consider t... | One Diagonal of Kite Bisects the Other | https://proofwiki.org/wiki/One_Diagonal_of_Kite_Bisects_the_Other | https://proofwiki.org/wiki/One_Diagonal_of_Kite_Bisects_the_Other | [
"Kites"
] | [
"Definition:Quadrilateral/Kite",
"Definition:Diameter of Quadrilateral",
"Definition:Perpendicular Bisector"
] | [
"File:Diagonals-of-Kite.png",
"Definition:Intersection (Geometry)",
"Diagonals of Kite are Perpendicular",
"Definition:Right Angle/Perpendicular",
"Definition:Right Angle",
"Definition:Triangle (Geometry)",
"Definition:Right Angle",
"Triangle Right-Angle-Hypotenuse-Side Congruence",
"Definition:Cong... |
proofwiki-16395 | Combination Theorem for Continuous Mappings/Topological Group/Product Rule | :$f * g : \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping. | By definition, a topological group is a topological semigroup.
Hence $\struct {G, *, \tau_{_G} }$ is a topological semigroup.
From Product Rule for Continuous Mappings to Topological Semigroup:
:$f * g: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping.
{{qed}} | :$f * g : \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a [[Definition:Continuous Mapping on Set|continuous mapping]]. | By definition, a [[Definition:Topological Group|topological group]] is a [[Definition:Topological Semigroup|topological semigroup]].
Hence $\struct {G, *, \tau_{_G} }$ is a [[Definition:Topological Semigroup|topological semigroup]].
From [[Product Rule for Continuous Mappings to Topological Semigroup]]:
:$f * g: \str... | Combination Theorem for Continuous Mappings/Topological Group/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Group/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Group/Product_Rule | [
"Combination Theorem for Continuous Mappings to Topological Group"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Group",
"Definition:Topological Semigroup",
"Definition:Topological Semigroup",
"Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16396 | Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule | :$\lambda * f: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping
:$f * \lambda: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping. | By definition, a topological group is a topological semigroup.
Hence $\struct {G, *, \tau_{_G}}$ is a topological semigroup.
From Multiple Rule for Continuous Mappings to Topological Semigroup:
:$\lambda * f, f * \lambda: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ are continuous mappings.
{{qed}} | :$\lambda * f: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a [[Definition:Continuous Mapping on Set|continuous mapping]]
:$f * \lambda: \struct {S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a [[Definition:Continuous Mapping on Set|continuous mapping]]. | By definition, a [[Definition:Topological Group|topological group]] is a [[Definition:Topological Semigroup|topological semigroup]].
Hence $\struct {G, *, \tau_{_G}}$ is a [[Definition:Topological Semigroup|topological semigroup]].
From [[Multiple Rule for Continuous Mappings to Topological Semigroup]]:
:$\lambda * f... | Combination Theorem for Continuous Mappings/Topological Group/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Group/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Group/Multiple_Rule | [
"Combination Theorem for Continuous Mappings to Topological Group"
] | [
"Definition:Continuous Mapping (Topology)/Set",
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Group",
"Definition:Topological Semigroup",
"Definition:Topological Semigroup",
"Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule",
"Definition:Continuous Mapping (Topology)/Set"
] |
proofwiki-16397 | Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule | :$g^{-1}: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping. | By definition of a topological group:
:$\phi: \struct {G, \tau_G} \to \struct {G, \tau_G}$ such that $\forall x \in G: \map \phi x = x^{-1}$ is a continuous mapping
From Composite of Continuous Mappings is Continuous:
:the composition $\phi \circ g: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is continuous.
Now:
{{beg... | :$g^{-1}: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a [[Definition:Continuous Mapping on Set|continuous mapping]]. | By definition of a [[Definition:Topological Group|topological group]]:
:$\phi: \struct {G, \tau_G} \to \struct {G, \tau_G}$ such that $\forall x \in G: \map \phi x = x^{-1}$ is a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]]
From [[Composite of Continuous Mappings is Continuous]]:
:the [[D... | Combination Theorem for Continuous Mappings/Topological Group/Inverse Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Group/Inverse_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Group/Inverse_Rule | [
"Combination Theorem for Continuous Mappings to Topological Group"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Topological Group",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Composite of Continuous Mappings is Continuous",
"Definition:Composition of Mappings",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Equality of Mappings",
"Category:Combination Theorem for Continuous Mappi... |
proofwiki-16398 | Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule | :$f * g: \struct{S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a continuous mapping. | Let $f \times g: S \to G \times G$ be the mapping defined by:
:$\forall x \in S : \map {\paren {f \times g} } x = \tuple {\map f x, \map g x}$
From Pointwise Operation is Composite of Operation with Mapping to Cartesian Product:
:$f * g = * \circ \paren {f \times g}$
Let $\tau$ be the product topology on $G \times G$.
... | :$f * g: \struct{S, \tau_{_S} } \to \struct {G, \tau_{_G} }$ is a [[Definition:Continuous Mapping on Set|continuous mapping]]. | Let $f \times g: S \to G \times G$ be the [[Definition:Mapping|mapping]] defined by:
:$\forall x \in S : \map {\paren {f \times g} } x = \tuple {\map f x, \map g x}$
From [[Pointwise Operation is Composite of Operation with Mapping to Cartesian Product]]:
:$f * g = * \circ \paren {f \times g}$
Let $\tau$ be the [[De... | Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Semigroup/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Semigroup/Product_Rule | [
"Combination Theorem for Continuous Mappings to Topological Semigroup"
] | [
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Mapping",
"Pointwise Operation is Composite of Operation with Mapping to Cartesian Product",
"Definition:Product Topology",
"Definition:Continuous Mapping (Topology)/Set",
"Definition:Continuous Mapping (Topology)/Set",
"Definition:Topological Semigroup",
"Composite of Continuous Mappings is... |
proofwiki-16399 | Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule | :$\lambda * f: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping
:$f * \lambda: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a continuous mapping. | Let $c_\lambda : S \to G$ be the constant mapping defined by:
:$\forall x \in S: \map {c_\lambda} x = \lambda$
From Constant Mapping is Continuous, $c_\lambda$ is continuous.
From Product Rule for Continuous Mappings to Topological Semigroup:
:$c_\lambda * f$ and $f * c_\lambda$ are continuous.
Now:
{{begin-eqn}}
{{eqn... | :$\lambda * f: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a [[Definition:Continuous Mapping on Set|continuous mapping]]
:$f * \lambda: \struct {S, \tau_S} \to \struct {G, \tau_G}$ is a [[Definition:Continuous Mapping on Set|continuous mapping]]. | Let $c_\lambda : S \to G$ be the [[Definition:Constant Mapping|constant mapping]] defined by:
:$\forall x \in S: \map {c_\lambda} x = \lambda$
From [[Constant Mapping is Continuous]], $c_\lambda$ is [[Definition:Continuous Mapping on Set|continuous]].
From [[Product Rule for Continuous Mappings to Topological Semigro... | Combination Theorem for Continuous Mappings/Topological Semigroup/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Semigroup/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Mappings/Topological_Semigroup/Multiple_Rule | [
"Combination Theorem for Continuous Mappings to Topological Semigroup"
] | [
"Definition:Continuous Mapping (Topology)/Set",
"Definition:Continuous Mapping (Topology)/Set"
] | [
"Definition:Constant Mapping",
"Constant Mapping is Continuous",
"Definition:Continuous Mapping (Topology)/Set",
"Combination Theorem for Continuous Mappings/Topological Semigroup/Product Rule",
"Definition:Continuous Mapping (Topology)/Set",
"Equality of Mappings",
"Category:Combination Theorem for Con... |
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