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-8700 | Rational Division is Closed | The set of rational numbers less zero is closed under division:
:$\forall a, b \in \Q_{\ne 0}: a / b \in \Q_{\ne 0}$ | From the definition of division:
:$a / b := a \times \paren {b^{-1} }$
where $b^{-1}$ is the inverse for rational multiplication.
From Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group, the algebraic structure $\struct {\Q_{\ne 0}, \times}$ is a group.
From {{Group-axiom|3}} it follows that eve... | The [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] less [[Definition:Zero (Number)|zero]] is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Rational Division|division]]:
:$\forall a, b \in \Q_{\ne 0}: a / b \in \Q_{\ne 0}$ | From the definition of [[Definition:Rational Division|division]]:
:$a / b := a \times \paren {b^{-1} }$
where $b^{-1}$ is the [[Definition:Inverse Element|inverse]] for [[Definition:Rational Multiplication|rational multiplication]].
From [[Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group]], t... | Rational Division is Closed | https://proofwiki.org/wiki/Rational_Division_is_Closed | https://proofwiki.org/wiki/Rational_Division_is_Closed | [
"Rational Division",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Zero (Number)",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Division/Field/Rational Numbers"
] | [
"Definition:Division/Field/Rational Numbers",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Multiplication/Rational Numbers",
"Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group",
"Definition:Algebraic Structure",
"Definition:Group",
"Definition:Inverse (Abstract A... |
proofwiki-8701 | Complex Modulus of Product of Complex Numbers/General Result | Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers.
Let $\cmod z$ be the modulus of $z$.
Then:
: $\cmod {z_1 z_2 \cdots z_n} = \cmod {z_1} \cdot \cmod {z_2} \cdots \cmod {z_n}$ | Proof by induction:
For all $n \in \N_{> 0}$, let $P \left({n}\right)$ be the proposition:
: $\cmod {z_1 z_2 \cdots z_n} = \cmod {z_1} \cdot \cmod {z_2} \cdots \cmod {z_n}$
$P \left({1}\right)$ is trivially true:
:$\cmod {z_1} = \cmod {z_1}$ | Let $z_1, z_2, \ldots, z_n \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\cmod z$ be the [[Definition:Complex Modulus|modulus]] of $z$.
Then:
: $\cmod {z_1 z_2 \cdots z_n} = \cmod {z_1} \cdot \cmod {z_2} \cdots \cmod {z_n}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $P \left({n}\right)$ be the [[Definition:Proposition|proposition]]:
: $\cmod {z_1 z_2 \cdots z_n} = \cmod {z_1} \cdot \cmod {z_2} \cdots \cmod {z_n}$
$P \left({1}\right)$ is trivially true:
:$\cmod {z_1} = \cmod {z_1}$ | Complex Modulus of Product of Complex Numbers/General Result | https://proofwiki.org/wiki/Complex_Modulus_of_Product_of_Complex_Numbers/General_Result | https://proofwiki.org/wiki/Complex_Modulus_of_Product_of_Complex_Numbers/General_Result | [
"Complex Modulus of Product of Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-8702 | Complex Modulus of Quotient of Complex Numbers | Let $z_1, z_2 \in \C$ be complex numbers such that $z_2 \ne 0$.
Let $\cmod z$ denote the modulus of $z$.
Then:
:$\cmod {\dfrac {z_1} {z_2} } = \dfrac {\cmod {z_1} } {\cmod {z_2} }$ | {{begin-eqn}}
{{eqn | l = \cmod {\dfrac {z_1} {z_2} }
| r = \cmod {z_1 \times z_2^{-1} }
| c = {{Defof|Complex Division}}
}}
{{eqn | r = \cmod {z_1} \times \cmod {z_2^{-1} }
| c = Complex Modulus of Product of Complex Numbers
}}
{{eqn | r = \cmod {z_1} \times \cmod {z_2}^{-1}
| c = Complex Modul... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]] such that $z_2 \ne 0$.
Let $\cmod z$ denote the [[Definition:Complex Modulus|modulus]] of $z$.
Then:
:$\cmod {\dfrac {z_1} {z_2} } = \dfrac {\cmod {z_1} } {\cmod {z_2} }$ | {{begin-eqn}}
{{eqn | l = \cmod {\dfrac {z_1} {z_2} }
| r = \cmod {z_1 \times z_2^{-1} }
| c = {{Defof|Complex Division}}
}}
{{eqn | r = \cmod {z_1} \times \cmod {z_2^{-1} }
| c = [[Complex Modulus of Product of Complex Numbers]]
}}
{{eqn | r = \cmod {z_1} \times \cmod {z_2}^{-1}
| c = [[Complex... | Complex Modulus of Quotient of Complex Numbers | https://proofwiki.org/wiki/Complex_Modulus_of_Quotient_of_Complex_Numbers | https://proofwiki.org/wiki/Complex_Modulus_of_Quotient_of_Complex_Numbers | [
"Complex Modulus",
"Complex Division"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [
"Complex Modulus of Product of Complex Numbers",
"Complex Modulus of Reciprocal of Complex Number"
] |
proofwiki-8703 | Complex Modulus of Reciprocal of Complex Number | Let $z \in \C$ be a complex number such that $z \ne 0$.
Let $\cmod z$ denote the complex modulus of $z$.
Then:
:$\cmod {\dfrac 1 z} = \dfrac 1 {\cmod z}$ | Let $z = a + i b$.
{{begin-eqn}}
{{eqn | l = \cmod {\frac 1 z}
| r = \cmod {\frac {a - i b} {a^2 + b^2} }
| c = Reciprocal of Complex Number
}}
{{eqn | r = \cmod {\frac a {a^2 + b^2} + i \frac {-b} {a^2 + b^2} }
| c =
}}
{{eqn | r = \sqrt {\paren {\frac a {a^2 + b^2} }^2 + \paren {\frac {-b} {a^2 + b... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] such that $z \ne 0$.
Let $\cmod z$ denote the [[Definition:Complex Modulus|complex modulus]] of $z$.
Then:
:$\cmod {\dfrac 1 z} = \dfrac 1 {\cmod z}$ | Let $z = a + i b$.
{{begin-eqn}}
{{eqn | l = \cmod {\frac 1 z}
| r = \cmod {\frac {a - i b} {a^2 + b^2} }
| c = [[Reciprocal of Complex Number]]
}}
{{eqn | r = \cmod {\frac a {a^2 + b^2} + i \frac {-b} {a^2 + b^2} }
| c =
}}
{{eqn | r = \sqrt {\paren {\frac a {a^2 + b^2} }^2 + \paren {\frac {-b} {a^... | Complex Modulus of Reciprocal of Complex Number | https://proofwiki.org/wiki/Complex_Modulus_of_Reciprocal_of_Complex_Number | https://proofwiki.org/wiki/Complex_Modulus_of_Reciprocal_of_Complex_Number | [
"Complex Modulus",
"Reciprocals"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [
"Reciprocal of Complex Number"
] |
proofwiki-8704 | Triangle Inequality/Complex Numbers/General Result | Let $z_1, z_2, \dotsc, z_n \in \C$ be complex numbers.
Let $\cmod z$ be the modulus of $z$.
Then:
:$\cmod {z_1 + z_2 + \dotsb + z_n} \le \cmod {z_1} + \cmod {z_2} + \dotsb + \cmod {z_n}$ | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\cmod {z_1 + z_2 + \dotsb + z_n} \le \cmod {z_1} + \cmod {z_2} + \dotsb + \cmod {z_n}$
$\map P 1$ is true by definition of the usual ordering on real numbers:
:$\cmod {z_1} \le \cmod {z_1}$ | Let $z_1, z_2, \dotsc, z_n \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\cmod z$ be the [[Definition:Modulus of Complex Number|modulus]] of $z$.
Then:
:$\cmod {z_1 + z_2 + \dotsb + z_n} \le \cmod {z_1} + \cmod {z_2} + \dotsb + \cmod {z_n}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\cmod {z_1 + z_2 + \dotsb + z_n} \le \cmod {z_1} + \cmod {z_2} + \dotsb + \cmod {z_n}$
$\map P 1$ is true by definition of the [[Definition:Usual Ordering|usual orderi... | Triangle Inequality/Complex Numbers/General Result | https://proofwiki.org/wiki/Triangle_Inequality/Complex_Numbers/General_Result | https://proofwiki.org/wiki/Triangle_Inequality/Complex_Numbers/General_Result | [
"Triangle Inequality"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Usual Ordering",
"Principle of Mathematical Induction"
] |
proofwiki-8705 | Triangle Inequality for Complex Numbers/Corollary 1 | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\cmod z$ be the modulus of $z$.
Then:
: $\cmod {z_1 + z_2} \ge \cmod {z_1} - \cmod {z_2}$ | Let $z_3 := z_1 + z_2$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_3} + \cmod {\paren {-z_2} }
| o = \ge
| r = \cmod {z_3 + \paren {-z_2} }
| c = Triangle Inequality for Complex Numbers
}}
{{eqn | ll= \leadsto
| l = \cmod {z_3} + \cmod {z_2}
| o = \ge
| r = \cmod {z_3 - z_2}
| c ... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\cmod z$ be the [[Definition:Modulus of Complex Number|modulus]] of $z$.
Then:
: $\cmod {z_1 + z_2} \ge \cmod {z_1} - \cmod {z_2}$ | Let $z_3 := z_1 + z_2$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_3} + \cmod {\paren {-z_2} }
| o = \ge
| r = \cmod {z_3 + \paren {-z_2} }
| c = [[Triangle Inequality for Complex Numbers]]
}}
{{eqn | ll= \leadsto
| l = \cmod {z_3} + \cmod {z_2}
| o = \ge
| r = \cmod {z_3 - z_2}
... | Triangle Inequality for Complex Numbers/Corollary 1 | https://proofwiki.org/wiki/Triangle_Inequality_for_Complex_Numbers/Corollary_1 | https://proofwiki.org/wiki/Triangle_Inequality_for_Complex_Numbers/Corollary_1 | [
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [
"Triangle Inequality/Complex Numbers",
"Complex Modulus of Additive Inverse"
] |
proofwiki-8706 | Complex Modulus of Additive Inverse | Let $z \in \C$ be a complex number.
Let $-z$ be the negative of $z$:
:$z + \paren {-z} = 0$
Then:
:$\cmod z = \cmod {\paren {-z} }$
where $\cmod z$ denotes the modulus of $z$. | Let $z = a + i b$.
{{begin-eqn}}
{{eqn | l = \cmod {\paren {-z} }
| r = \cmod {\paren {-a - i b} }
| c = {{Defof|Negative of Complex Number}}
}}
{{eqn | r = \sqrt {\paren {-a}^2 + \paren {-b}^2}
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = \sqrt {a^2 + b^2}
| c = Even Power of Negative Real Nu... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $-z$ be the [[Definition:Negative of Complex Number|negative]] of $z$:
:$z + \paren {-z} = 0$
Then:
:$\cmod z = \cmod {\paren {-z} }$
where $\cmod z$ denotes the [[Definition:Complex Modulus|modulus]] of $z$. | Let $z = a + i b$.
{{begin-eqn}}
{{eqn | l = \cmod {\paren {-z} }
| r = \cmod {\paren {-a - i b} }
| c = {{Defof|Negative of Complex Number}}
}}
{{eqn | r = \sqrt {\paren {-a}^2 + \paren {-b}^2}
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = \sqrt {a^2 + b^2}
| c = [[Even Power of Negative Real... | Complex Modulus of Additive Inverse | https://proofwiki.org/wiki/Complex_Modulus_of_Additive_Inverse | https://proofwiki.org/wiki/Complex_Modulus_of_Additive_Inverse | [
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Negative/Complex Number",
"Definition:Complex Modulus"
] | [
"Even Power of Negative Real Number",
"Category:Complex Modulus"
] |
proofwiki-8707 | Even Power of Negative Real Number | Let $x \in \R$ be a real number.
Let $n \in \Z$ be an even integer.
Then:
:$\paren {-x}^n = x^n$ | From Real Numbers form Totally Ordered Field, $\R$ is a field {{afortiori}}.
By definition, $\R$ is therefore a ring.
The result follows from Power of Ring Negative.
{{qed}}
Category:Real Numbers
fypi6553dimnhzhw1mbscas78azp2zy | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $n \in \Z$ be an [[Definition:Even Integer|even integer]].
Then:
:$\paren {-x}^n = x^n$ | From [[Real Numbers form Totally Ordered Field]], $\R$ is a [[Definition:Field (Abstract Algebra)|field]] {{afortiori}}.
By definition, $\R$ is therefore a [[Definition:Ring (Abstract Algebra)|ring]].
The result follows from [[Power of Ring Negative]].
{{qed}}
[[Category:Real Numbers]]
fypi6553dimnhzhw1mbscas78azp2z... | Even Power of Negative Real Number | https://proofwiki.org/wiki/Even_Power_of_Negative_Real_Number | https://proofwiki.org/wiki/Even_Power_of_Negative_Real_Number | [
"Real Numbers"
] | [
"Definition:Real Number",
"Definition:Even Integer"
] | [
"Real Numbers form Totally Ordered Field",
"Definition:Field (Abstract Algebra)",
"Definition:Ring (Abstract Algebra)",
"Power of Ring Negative",
"Category:Real Numbers"
] |
proofwiki-8708 | Complex Addition is Closed/Proof 1 | The set of complex numbers $\C$ is closed under addition:
:$\forall z, w \in \C: z + w \in \C$ | From the informal definition of complex numbers, we define the following:
:$z = x_1 + i y_1$
:$w = x_2 + i y_2$
where $i = \sqrt {-1}$ and $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of complex addition:
:$z + w = \paren {x_1 + x_2} + i \paren {y_1 + y_2}$
From Real Numbers under Addition form Group, real add... | The [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Addition|addition]]:
:$\forall z, w \in \C: z + w \in \C$ | From the informal definition of [[Definition:Complex Number/Definition 1|complex numbers]], we define the following:
:$z = x_1 + i y_1$
:$w = x_2 + i y_2$
where $i = \sqrt {-1}$ and $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of [[Definition:Complex Addition|complex addition]]:
:$z + w = \paren {x_1 + x_2}... | Complex Addition is Closed/Proof 1 | https://proofwiki.org/wiki/Complex_Addition_is_Closed/Proof_1 | https://proofwiki.org/wiki/Complex_Addition_is_Closed/Proof_1 | [
"Complex Addition is Closed"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Addition/Complex Numbers"
] | [
"Definition:Complex Number/Definition 1",
"Definition:Addition/Complex Numbers",
"Real Numbers under Addition form Group",
"Definition:Addition/Real Numbers",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-8709 | Complex Addition is Closed/Proof 2 | The set of complex numbers $\C$ is closed under addition:
:$\forall z, w \in \C: z + w \in \C$ | From the formal definition of complex numbers, we have:
:$z = \tuple {x_1, y_1}$
:$w = \tuple {x_2, y_2}$
where $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of complex addition:
:$z + w = \tuple {x_1 + x_2, y_1 + y_2}$
From Real Numbers under Addition form Group, real addition is closed.
So:
:$\paren {x_1 + x_... | The [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$ is [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Complex Addition|addition]]:
:$\forall z, w \in \C: z + w \in \C$ | From the formal definition of [[Definition:Complex Number/Definition 2|complex numbers]], we have:
:$z = \tuple {x_1, y_1}$
:$w = \tuple {x_2, y_2}$
where $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of [[Definition:Complex Number/Definition 2/Addition|complex addition]]:
:$z + w = \tuple {x_1 + x_2, y_1 + ... | Complex Addition is Closed/Proof 2 | https://proofwiki.org/wiki/Complex_Addition_is_Closed/Proof_2 | https://proofwiki.org/wiki/Complex_Addition_is_Closed/Proof_2 | [
"Complex Addition is Closed"
] | [
"Definition:Set",
"Definition:Complex Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Addition/Complex Numbers"
] | [
"Definition:Complex Number/Definition 2",
"Definition:Complex Number/Definition 2/Addition",
"Real Numbers under Addition form Group",
"Definition:Addition/Real Numbers",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-8710 | Cardinality of Cartesian Product of Finite Sets/General Result | Let $\ds \prod_{k \mathop = 1}^n S_k$ be the cartesian product of a (finite) sequence of sets $\sequence {S_n}$.
Then:
:$\ds \card {\prod_{k \mathop = 1}^n S_k} = \prod_{k \mathop = 1}^n \card {S_k}$ | {{tidy|I'll get to it}}
Proof by mathematical induction.
Let:
:$\ds \map P n = \paren {\card {\prod_{k \mathop = 1}^n S_k} = \prod_{k \mathop = 1}^n \card {S_k} }$
For $n = 1$:
:$\ds \card {\prod_{k \mathop = 1}^1 S_k} = \card{S_1} = \prod_{k \mathop = 1}^1 \card {S_k}$
These equalities follow directly from the definit... | Let $\ds \prod_{k \mathop = 1}^n S_k$ be the [[Definition:Finite Cartesian Product|cartesian product]] of a [[Definition:Finite Sequence|(finite) sequence]] of [[Definition:Set|sets]] $\sequence {S_n}$.
Then:
:$\ds \card {\prod_{k \mathop = 1}^n S_k} = \prod_{k \mathop = 1}^n \card {S_k}$ | {{tidy|I'll get to it}}
Proof by [[Principle of Mathematical Induction|mathematical induction]].
Let:
:$\ds \map P n = \paren {\card {\prod_{k \mathop = 1}^n S_k} = \prod_{k \mathop = 1}^n \card {S_k} }$
For $n = 1$:
:$\ds \card {\prod_{k \mathop = 1}^1 S_k} = \card{S_1} = \prod_{k \mathop = 1}^1 \card {S_k}$
The... | Cardinality of Cartesian Product of Finite Sets/General Result | https://proofwiki.org/wiki/Cardinality_of_Cartesian_Product_of_Finite_Sets/General_Result | https://proofwiki.org/wiki/Cardinality_of_Cartesian_Product_of_Finite_Sets/General_Result | [
"Cardinality of Cartesian Product"
] | [
"Definition:Cartesian Product/Finite",
"Definition:Finite Sequence",
"Definition:Set"
] | [
"Principle of Mathematical Induction",
"Definition:Cartesian Product/Finite",
"Cardinality of Cartesian Product of Finite Sets",
"Category:Cardinality of Cartesian Product"
] |
proofwiki-8711 | Cardinality of Cartesian Product of Finite Sets/General Result/Corollary | Let $S$ be a finite set.
Let $S^n$ be a cartesian space on $S$.
Then:
:$\card {S^n} = \card S^n$ | This is an instance of Cardinality of Cartesian Product of Finite Sets: General Result, where each set is equal.
{{qed}}
Category:Cardinality of Cartesian Product
tb04pnvlwn6zextob4zydc2tkh7xy6k | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $S^n$ be a [[Definition:Cartesian Space|cartesian space]] on $S$.
Then:
:$\card {S^n} = \card S^n$ | This is an instance of [[Cardinality of Cartesian Product of Finite Sets/General Result|Cardinality of Cartesian Product of Finite Sets: General Result]], where each [[Definition:Set|set]] is equal.
{{qed}}
[[Category:Cardinality of Cartesian Product]]
tb04pnvlwn6zextob4zydc2tkh7xy6k | Cardinality of Cartesian Product of Finite Sets/General Result/Corollary | https://proofwiki.org/wiki/Cardinality_of_Cartesian_Product_of_Finite_Sets/General_Result/Corollary | https://proofwiki.org/wiki/Cardinality_of_Cartesian_Product_of_Finite_Sets/General_Result/Corollary | [
"Cardinality of Cartesian Product"
] | [
"Definition:Finite Set",
"Definition:Cartesian Product/Cartesian Space"
] | [
"Cardinality of Cartesian Product of Finite Sets/General Result",
"Definition:Set",
"Category:Cardinality of Cartesian Product"
] |
proofwiki-8712 | Product of Complex Numbers in Polar Form/General Result | Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers.
Let $z_j = \polar {r_j, \theta_j}$ be $z_j$ expressed in polar form for each $j \in \set {1, 2, \ldots, n}$.
Then:
:$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \map \sin {\theta_1 + \theta_2 + \cdots + \t... | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$
Let this be expressed as:
:$\ds \prod_{j \mathop = 1}^n z_j = \prod_{j \mathop ... | Let $z_1, z_2, \ldots, z_n \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $z_j = \polar {r_j, \theta_j}$ be [[Definition:Polar Form of Complex Number|$z_j$ expressed in polar form]] for each $j \in \set {1, 2, \ldots, n}$.
Then:
:$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$
Let this b... | Product of Complex Numbers in Polar Form/General Result | https://proofwiki.org/wiki/Product_of_Complex_Numbers_in_Polar_Form/General_Result | https://proofwiki.org/wiki/Product_of_Complex_Numbers_in_Polar_Form/General_Result | [
"Complex Multiplication"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-8713 | Substitution for Equivalent Subformula is Equivalent | Let $\mathbf B$ a WFF of propositional logic.
Let $\mathbf A, \mathbf A'$ be equivalent WFFs.
Let $\mathbf A$ be a subformula of $\mathbf B$.
Let $\mathbf B' = \map {\mathbf B} {\mathbf A \,//\, \mathbf A'}$ be the substitution of $\mathbf A'$ for $\mathbf A$ in $\mathbf B$.
Then $\mathbf B$ and $\mathbf B'$ are equiva... | Let $v$ be an arbitrary boolean interpretation.
Then $\map v {\mathbf A} = \map v {\mathbf A'}$.
It is to be shown that $\map v {\mathbf B} = \map v {\mathbf B'}$.
We proceed by induction.
Let $\map n {\mathbf B}$ be the number of WFFs $\mathbf C$ such that:
:$\mathbf A$ is a subformula of $\mathbf C$, and $\mathbf C$ ... | Let $\mathbf B$ a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Let $\mathbf A, \mathbf A'$ be [[Definition:Semantic Equivalence for Boolean Interpretations|equivalent]] [[Definition:WFF of Propositional Logic|WFFs]].
Let $\mathbf A$ be a [[Definition:Subformula|subformula]] of $\mathbf B$.
L... | Let $v$ be an arbitrary [[Definition:Boolean Interpretation|boolean interpretation]].
Then $\map v {\mathbf A} = \map v {\mathbf A'}$.
It is to be shown that $\map v {\mathbf B} = \map v {\mathbf B'}$.
We proceed by [[Second Principle of Mathematical Induction|induction]].
Let $\map n {\mathbf B}$ be the number of... | Substitution for Equivalent Subformula is Equivalent | https://proofwiki.org/wiki/Substitution_for_Equivalent_Subformula_is_Equivalent | https://proofwiki.org/wiki/Substitution_for_Equivalent_Subformula_is_Equivalent | [
"Boolean Interpretations",
"Propositional Logic"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Semantic Equivalence/Boolean Interpretations",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Well-Formed Part",
"Definition:Substitution (Formal Systems)/Well-Formed Part",
"Definition:Semantic Equi... | [
"Definition:Boolean Interpretation",
"Second Principle of Mathematical Induction",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Well-Formed Part",
"Definition:Well-Formed Part",
"Definition:Substitution (Formal Systems)/Well-Formed Part",
"Definition:Boolean Interpretatio... |
proofwiki-8714 | Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle | Let $n \in \Z$ be an integer such that $n \ge 3$.
Let $z \in \C$ be a complex number such that $z^n = 1$.
Let $U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$ be the set of $n$th roots of unity.
Let $U_n$ be plotted on the complex plane.
Then the elements of $U_n$ are located at the vertices of a regular $n$-sided polygon ... | {{Proofread}}
600px
{{tidy}}
{{MissingLinks}}
The above diagram illustrates the $7$th roots of unity.
Let $O$ be the origin.
Consider $A_0, \cdots, A_{n - 1}$ the points $e^{0 \times 2i\pi / n}, \cdots, e^{(n - 1) \times 2 i\pi / n}$ in the complex plane.
Note $A_n = A_0$ and $A_{n+1} = A_1$.
Consider $P$ the polygon w... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ge 3$.
Let $z \in \C$ be a [[Definition:Complex Number|complex number]] such that $z^n = 1$.
Let $U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$ be the [[Definition:Complex Roots of Unity|set of $n$th roots of unity]].
Let $U_n$ be plotted on the [[Defin... | {{Proofread}}
[[File:Roots-of-Unity-as-Polygon.png|600px]]
{{tidy}}
{{MissingLinks}}
The above diagram illustrates the [[Definition:Complex Roots of Unity|$7$th roots of unity]].
Let $O$ be the origin.
Consider $A_0, \cdots, A_{n - 1}$ the points $e^{0 \times 2i\pi / n}, \cdots, e^{(n - 1) \times 2 i\pi / n}$ in the... | Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle | https://proofwiki.org/wiki/Complex_Roots_of_Unity_are_Vertices_of_Regular_Polygon_Inscribed_in_Circle | https://proofwiki.org/wiki/Complex_Roots_of_Unity_are_Vertices_of_Regular_Polygon_Inscribed_in_Circle | [
"Geometry of Complex Plane",
"Complex Roots of Unity"
] | [
"Definition:Integer",
"Definition:Complex Number",
"Definition:Root of Unity/Complex",
"Definition:Complex Number/Complex Plane",
"Definition:Polygon/Vertex",
"Definition:Polygon/Regular",
"Definition:Circumscribe/Circle around Polygon",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Defin... | [
"File:Roots-of-Unity-as-Polygon.png",
"Definition:Root of Unity/Complex"
] |
proofwiki-8715 | Equation of Unit Circle in Complex Plane | Consider the unit circle $C$ whose center is at $\tuple {0, 0}$ on the complex plane.
Its equation is given by:
:$\cmod z = 1$
where $\cmod z$ denotes the complex modulus of $z$. | From Equation of Unit Circle, the unit circle whose center is at the origin of the Cartesian $xy$ plane has the equation:
:$x^2 + y^2 = 1$
Identifying the Cartesian $xy$ plane with the complex plane:
{{ProofWanted|Can't think of the precise words I need for this at the moment}} | Consider the [[Definition:Unit Circle|unit circle]] $C$ whose [[Definition:Center of Circle|center]] is at $\tuple {0, 0}$ on the [[Definition:Complex Plane|complex plane]].
Its [[Definition:Equation of Geometric Figure|equation]] is given by:
:$\cmod z = 1$
where $\cmod z$ denotes the [[Definition:Complex Modulus|com... | From [[Equation of Unit Circle]], the [[Definition:Unit Circle|unit circle]] whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian $xy$ plane]] has the [[Definition:Equation of Geometric Figure|equation]]:
:$x^2 + y^2 = 1$
Identifying the [[De... | Equation of Unit Circle in Complex Plane/Proof 1 | https://proofwiki.org/wiki/Equation_of_Unit_Circle_in_Complex_Plane | https://proofwiki.org/wiki/Equation_of_Unit_Circle_in_Complex_Plane/Proof_1 | [
"Geometry of Complex Plane",
"Equation of Unit Circle in Complex Plane"
] | [
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Complex Number/Complex Plane",
"Definition:Equation of Geometric Figure",
"Definition:Complex Modulus"
] | [
"Equation of Unit Circle",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Equation of Geometric Figure",
"Definition:Cartesian Plane",
"Definition:Complex Number/Complex Plane"
] |
proofwiki-8716 | Equation of Unit Circle in Complex Plane | Consider the unit circle $C$ whose center is at $\tuple {0, 0}$ on the complex plane.
Its equation is given by:
:$\cmod z = 1$
where $\cmod z$ denotes the complex modulus of $z$. | Let $C$ be the set of points on the unit circle defined above.
Let $P$ be the set:
:$\set {z \in \C: \cmod z = 1}$
Let $z = x + i y \in C$.
$z$ can be expressed in polar form as:
:$z = \polar {r, \theta}$
where:
:$x = r \cos \theta$
:$y = r \sin \theta$
By definition of unit circle, $z$ is $1$ unit away from $\tuple {0... | Consider the [[Definition:Unit Circle|unit circle]] $C$ whose [[Definition:Center of Circle|center]] is at $\tuple {0, 0}$ on the [[Definition:Complex Plane|complex plane]].
Its [[Definition:Equation of Geometric Figure|equation]] is given by:
:$\cmod z = 1$
where $\cmod z$ denotes the [[Definition:Complex Modulus|com... | Let $C$ be the [[Definition:Set|set]] of [[Definition:Point|points]] on the [[Definition:Unit Circle|unit circle]] defined above.
Let $P$ be the set:
:$\set {z \in \C: \cmod z = 1}$
Let $z = x + i y \in C$.
$z$ can be expressed in [[Definition:Polar Form of Complex Number|polar form]] as:
:$z = \polar {r, \theta}$
w... | Equation of Unit Circle in Complex Plane/Proof 2 | https://proofwiki.org/wiki/Equation_of_Unit_Circle_in_Complex_Plane | https://proofwiki.org/wiki/Equation_of_Unit_Circle_in_Complex_Plane/Proof_2 | [
"Geometry of Complex Plane",
"Equation of Unit Circle in Complex Plane"
] | [
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Complex Number/Complex Plane",
"Definition:Equation of Geometric Figure",
"Definition:Complex Modulus"
] | [
"Definition:Set",
"Definition:Point",
"Definition:Unit Circle",
"Definition:Complex Number/Polar Form",
"Definition:Unit Circle",
"Pythagoras's Theorem",
"Definition:Complex Modulus",
"Modulus of Complex Number equals its Distance from Origin",
"Definition:Set Equality"
] |
proofwiki-8717 | Equation of Unit Circle | Let the unit circle have its center at the origin of the Cartesian plane.
Its equation is given by:
:$x^2 + y^2 = 1$
{{expand|Present it in polar coordinates as well}} | From Equation of Circle, the equation of a circle with radius $R$ and center $\tuple {a, b}$ is:
:$\paren {x - a}^2 + \paren {y - b}^2 = R^2$
Substituting $\tuple {0, 0}$ for $\tuple {a, b}$ and $1$ for $R$ gives the result.
{{qed}} | Let the [[Definition:Unit Circle|unit circle]] have its [[Definition:Center of Circle|center]] at the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian plane]].
Its [[Definition:Equation of Geometric Figure|equation]] is given by:
:$x^2 + y^2 = 1$
{{expand|Present it in polar coordinates as ... | From [[Equation of Circle]], the [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Circle|circle]] with [[Definition:Radius of Circle|radius]] $R$ and [[Definition:Center of Circle|center]] $\tuple {a, b}$ is:
:$\paren {x - a}^2 + \paren {y - b}^2 = R^2$
Substituting $\tuple {0, 0}$ for $\tuple {... | Equation of Unit Circle | https://proofwiki.org/wiki/Equation_of_Unit_Circle | https://proofwiki.org/wiki/Equation_of_Unit_Circle | [
"Unit Circles",
"Equation of Circle"
] | [
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Equation of Geometric Figure"
] | [
"Equation of Circle",
"Definition:Equation of Geometric Figure",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center"
] |
proofwiki-8718 | Equation of Unit Circle | Let the unit circle have its center at the origin of the Cartesian plane.
Its equation is given by:
:$x^2 + y^2 = 1$
{{expand|Present it in polar coordinates as well}} | From Equation of Unit Circle, the unit circle whose center is at the origin of the Cartesian $xy$ plane has the equation:
:$x^2 + y^2 = 1$
Identifying the Cartesian $xy$ plane with the complex plane:
{{ProofWanted|Can't think of the precise words I need for this at the moment}} | Let the [[Definition:Unit Circle|unit circle]] have its [[Definition:Center of Circle|center]] at the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian plane]].
Its [[Definition:Equation of Geometric Figure|equation]] is given by:
:$x^2 + y^2 = 1$
{{expand|Present it in polar coordinates as ... | From [[Equation of Unit Circle]], the [[Definition:Unit Circle|unit circle]] whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian $xy$ plane]] has the [[Definition:Equation of Geometric Figure|equation]]:
:$x^2 + y^2 = 1$
Identifying the [[De... | Equation of Unit Circle in Complex Plane/Proof 1 | https://proofwiki.org/wiki/Equation_of_Unit_Circle | https://proofwiki.org/wiki/Equation_of_Unit_Circle_in_Complex_Plane/Proof_1 | [
"Unit Circles",
"Equation of Circle"
] | [
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Equation of Geometric Figure"
] | [
"Equation of Unit Circle",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Equation of Geometric Figure",
"Definition:Cartesian Plane",
"Definition:Complex Number/Complex Plane"
] |
proofwiki-8719 | Equation of Unit Circle | Let the unit circle have its center at the origin of the Cartesian plane.
Its equation is given by:
:$x^2 + y^2 = 1$
{{expand|Present it in polar coordinates as well}} | Let $C$ be the set of points on the unit circle defined above.
Let $P$ be the set:
:$\set {z \in \C: \cmod z = 1}$
Let $z = x + i y \in C$.
$z$ can be expressed in polar form as:
:$z = \polar {r, \theta}$
where:
:$x = r \cos \theta$
:$y = r \sin \theta$
By definition of unit circle, $z$ is $1$ unit away from $\tuple {0... | Let the [[Definition:Unit Circle|unit circle]] have its [[Definition:Center of Circle|center]] at the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian plane]].
Its [[Definition:Equation of Geometric Figure|equation]] is given by:
:$x^2 + y^2 = 1$
{{expand|Present it in polar coordinates as ... | Let $C$ be the [[Definition:Set|set]] of [[Definition:Point|points]] on the [[Definition:Unit Circle|unit circle]] defined above.
Let $P$ be the set:
:$\set {z \in \C: \cmod z = 1}$
Let $z = x + i y \in C$.
$z$ can be expressed in [[Definition:Polar Form of Complex Number|polar form]] as:
:$z = \polar {r, \theta}$
w... | Equation of Unit Circle in Complex Plane/Proof 2 | https://proofwiki.org/wiki/Equation_of_Unit_Circle | https://proofwiki.org/wiki/Equation_of_Unit_Circle_in_Complex_Plane/Proof_2 | [
"Unit Circles",
"Equation of Circle"
] | [
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Equation of Geometric Figure"
] | [
"Definition:Set",
"Definition:Point",
"Definition:Unit Circle",
"Definition:Complex Number/Polar Form",
"Definition:Unit Circle",
"Pythagoras's Theorem",
"Definition:Complex Modulus",
"Modulus of Complex Number equals its Distance from Origin",
"Definition:Set Equality"
] |
proofwiki-8720 | Tautological Consequent | :$p \implies \top \dashv \vdash \top$ | {{BeginTableau|p \implies \top \vdash \top}}
{{Premise|1|p \implies \top}}
{{TopIntro|2}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|\top \vdash p \implies \top}}
{{Assumption|1|p}}
{{Premise|2|\top}}
{{Implication|3|1|p \implies \top|1|2}}
{{EndTableau}}
{{qed}} | :$p \implies \top \dashv \vdash \top$ | {{BeginTableau|p \implies \top \vdash \top}}
{{Premise|1|p \implies \top}}
{{TopIntro|2}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|\top \vdash p \implies \top}}
{{Assumption|1|p}}
{{Premise|2|\top}}
{{Implication|3|1|p \implies \top|1|2}}
{{EndTableau}}
{{qed}} | Tautological Consequent/Proof 1 | https://proofwiki.org/wiki/Tautological_Consequent | https://proofwiki.org/wiki/Tautological_Consequent/Proof_1 | [
"Tautological Consequent",
"Tautology",
"Conditional"
] | [] | [] |
proofwiki-8721 | Tautological Consequent | :$p \implies \top \dashv \vdash \top$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
$\begin{array}{|c|ccc||c|ccc|} \hline
p & p & \implies & \top & \top \\
\hline
F & F & T & T & T \\
T & T & T & T & T \\
\hline
\end{arra... | :$p \implies \top \dashv \vdash \top$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] in the appropriate columns match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|c|ccc||c|ccc|} \hline
p & p & \implies & \top & \t... | Tautological Consequent/Proof 2 | https://proofwiki.org/wiki/Tautological_Consequent | https://proofwiki.org/wiki/Tautological_Consequent/Proof_2 | [
"Tautological Consequent",
"Tautology",
"Conditional"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8722 | Tautological Antecedent | :$\top \implies p \dashv \vdash p$ | {{BeginTableau|\top \implies p \vdash p}}
{{Premise|1|\top \implies p}}
{{TopIntro|2}}
{{ModusPonens|3|1|p|1|2}}
{{EndTableau|lemma}}
{{BeginTableau|p \vdash \top \implies p}}
{{Premise|1|p}}
{{Assumption|2|\top}}
{{Implication|3|1|\top \implies p|2|1}}
{{EndTableau|qed}} | :$\top \implies p \dashv \vdash p$ | {{BeginTableau|\top \implies p \vdash p}}
{{Premise|1|\top \implies p}}
{{TopIntro|2}}
{{ModusPonens|3|1|p|1|2}}
{{EndTableau|lemma}}
{{BeginTableau|p \vdash \top \implies p}}
{{Premise|1|p}}
{{Assumption|2|\top}}
{{Implication|3|1|\top \implies p|2|1}}
{{EndTableau|qed}} | Tautological Antecedent/Proof 1 | https://proofwiki.org/wiki/Tautological_Antecedent | https://proofwiki.org/wiki/Tautological_Antecedent/Proof_1 | [
"Tautological Antecedent",
"Tautology",
"Conditional"
] | [] | [] |
proofwiki-8723 | Tautological Antecedent | :$\top \implies p \dashv \vdash p$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
$\begin{array}{|c|ccc||c|ccc|} \hline
p & \top & \implies & p \\
\hline
F & T & F & F \\
T & T & T & T \\
\hline
\end{array}$
{{qed}} | :$\top \implies p \dashv \vdash p$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] in the appropriate columns match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|c|ccc||c|ccc|} \hline
p & \top & \implies & p \\
\... | Tautological Antecedent/Proof 2 | https://proofwiki.org/wiki/Tautological_Antecedent | https://proofwiki.org/wiki/Tautological_Antecedent/Proof_2 | [
"Tautological Antecedent",
"Tautology",
"Conditional"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8724 | Contradictory Consequent | :$p \implies \bot \dashv \vdash \neg p$ | {{BeginTableau|p \implies \bot \vdash \neg p}}
{{Premise|1|p \implies \bot}}
{{Assumption|2|p}}
{{ModusPonens|3|1, 2|\bot|1|2}}
{{Contradiction|4|1|\neg p|2|3}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|\neg p \vdash p \implies \bot}}
{{Premise|1|\neg p}}
{{Assumption|2|p}}
{{NonContradiction|3|1,2|1|2}}
{{Implicatio... | :$p \implies \bot \dashv \vdash \neg p$ | {{BeginTableau|p \implies \bot \vdash \neg p}}
{{Premise|1|p \implies \bot}}
{{Assumption|2|p}}
{{ModusPonens|3|1, 2|\bot|1|2}}
{{Contradiction|4|1|\neg p|2|3}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|\neg p \vdash p \implies \bot}}
{{Premise|1|\neg p}}
{{Assumption|2|p}}
{{NonContradiction|3|1,2|1|2}}
{{Implicat... | Contradictory Consequent/Proof 1 | https://proofwiki.org/wiki/Contradictory_Consequent | https://proofwiki.org/wiki/Contradictory_Consequent/Proof_1 | [
"Contradictory Consequent",
"Contradiction",
"Conditional"
] | [] | [] |
proofwiki-8725 | Contradictory Consequent | :$p \implies \bot \dashv \vdash \neg p$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
$\begin{array}{|c|ccc||cc|} \hline
p & p & \implies & \bot & \neg & p \\
\hline
\F & \F & \T & \F & \T & \F \\
\T & \T & \F & \F & \F & \... | :$p \implies \bot \dashv \vdash \neg p$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] in the appropriate columns match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|c|ccc||cc|} \hline
p & p & \implies & \bot & \neg ... | Contradictory Consequent/Proof by Truth Table | https://proofwiki.org/wiki/Contradictory_Consequent | https://proofwiki.org/wiki/Contradictory_Consequent/Proof_by_Truth_Table | [
"Contradictory Consequent",
"Contradiction",
"Conditional"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8726 | Contradictory Antecedent | :$\bot \implies p \dashv \vdash \top$ | {{BeginTableau|\bot \implies p \vdash \top}}
{{Premise|1|\bot \implies p}}
{{TopIntro|2}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|\top \vdash \bot \implies p}}
{{Assumption|1|\bot}}
{{Premise|2|\top}}
{{Explosion|3|1|p|1}}
{{Implication|4||\bot \implies p|1|3}}
{{EndTableau}}
{{qed}} | :$\bot \implies p \dashv \vdash \top$ | {{BeginTableau|\bot \implies p \vdash \top}}
{{Premise|1|\bot \implies p}}
{{TopIntro|2}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|\top \vdash \bot \implies p}}
{{Assumption|1|\bot}}
{{Premise|2|\top}}
{{Explosion|3|1|p|1}}
{{Implication|4||\bot \implies p|1|3}}
{{EndTableau}}
{{qed}} | Contradictory Antecedent/Proof 1 | https://proofwiki.org/wiki/Contradictory_Antecedent | https://proofwiki.org/wiki/Contradictory_Antecedent/Proof_1 | [
"Contradictory Antecedent",
"Contradiction",
"Conditional"
] | [] | [] |
proofwiki-8727 | Contradictory Antecedent | :$\bot \implies p \dashv \vdash \top$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
:<nowiki>$\begin{array}{|c|ccc||c|} \hline
p & \bot & \implies & p & \top \\
\hline
\F & \F & \T & \F & \T \\
\T & \F & \T & \T & \T \\
\... | :$\bot \implies p \dashv \vdash \top$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] in the appropriate columns match for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|c|ccc||c|} \hline
p & \bot & \implies & p... | Contradictory Antecedent/Proof by Truth Table | https://proofwiki.org/wiki/Contradictory_Antecedent | https://proofwiki.org/wiki/Contradictory_Antecedent/Proof_by_Truth_Table | [
"Contradictory Antecedent",
"Contradiction",
"Conditional"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8728 | Biconditional with Tautology | :$p \iff \top \dashv \vdash p$ | {{BeginTableau|p \iff \top \vdash p}}
{{Premise|1|p \iff \top}}
{{TopIntro|2}}
{{BiconditionalElimination|3|1|\top \implies p|1|2}}
{{ModusPonens|4|1|p|2|3}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|p \vdash p \iff \top}}
{{Premise|1|\top}}
{{Assumption|2|p}}
{{TopIntro|3}}
{{Implication|4||p \implies \top|2|3}}
{{... | :$p \iff \top \dashv \vdash p$ | {{BeginTableau|p \iff \top \vdash p}}
{{Premise|1|p \iff \top}}
{{TopIntro|2}}
{{BiconditionalElimination|3|1|\top \implies p|1|2}}
{{ModusPonens|4|1|p|2|3}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|p \vdash p \iff \top}}
{{Premise|1|\top}}
{{Assumption|2|p}}
{{TopIntro|3}}
{{Implication|4||p \implies \top|2|3}}
... | Biconditional with Tautology/Proof 1 | https://proofwiki.org/wiki/Biconditional_with_Tautology | https://proofwiki.org/wiki/Biconditional_with_Tautology/Proof_1 | [
"Tautology",
"Biconditional",
"Biconditional with Tautology"
] | [] | [] |
proofwiki-8729 | Biconditional with Tautology | :$p \iff \top \dashv \vdash p$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
$\begin{array}{|c|ccc||c|ccc|} \hline
p & p & \iff & \top & p \\
\hline
F & F & F & T & F \\
T & T & T & T & T \\
\hline
\end{array}$
{{q... | :$p \iff \top \dashv \vdash p$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] in the appropriate columns match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|c|ccc||c|ccc|} \hline
p & p & \iff & \top & p \\
\... | Biconditional with Tautology/Proof 2 | https://proofwiki.org/wiki/Biconditional_with_Tautology | https://proofwiki.org/wiki/Biconditional_with_Tautology/Proof_2 | [
"Tautology",
"Biconditional",
"Biconditional with Tautology"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8730 | Exclusive Or with Tautology | :$p \oplus \top \dashv \vdash \neg p$ | {{BeginTableau|p \oplus \top \vdash \neg p}}
{{Premise|1|p \oplus \top}}
{{SequentIntro|2|1|\left({p \lor \top} \right) \land \neg \left({p \land \top}\right)|1|Definition of Exclusive Or}}
{{SequentIntro|3|1|\top \land \neg \left({p \land \top}\right)|1|Disjunction with Tautology}}
{{SequentIntro|4|1|\neg \left({p \la... | :$p \oplus \top \dashv \vdash \neg p$ | {{BeginTableau|p \oplus \top \vdash \neg p}}
{{Premise|1|p \oplus \top}}
{{SequentIntro|2|1|\left({p \lor \top} \right) \land \neg \left({p \land \top}\right)|1|Definition of [[Definition:Exclusive Or|Exclusive Or]]}}
{{SequentIntro|3|1|\top \land \neg \left({p \land \top}\right)|1|[[Disjunction with Tautology]]}}
{{Se... | Exclusive Or with Tautology/Proof 1 | https://proofwiki.org/wiki/Exclusive_Or_with_Tautology | https://proofwiki.org/wiki/Exclusive_Or_with_Tautology/Proof_1 | [
"Tautology",
"Exclusive Or",
"Exclusive Or with Tautology"
] | [] | [
"Definition:Exclusive Or",
"Disjunction with Tautology",
"Conjunction with Tautology",
"Conjunction with Tautology",
"Disjunction with Tautology",
"Conjunction with Tautology",
"Non-Equivalence"
] |
proofwiki-8731 | Exclusive Or with Tautology | :$p \oplus \top \dashv \vdash \neg p$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
$\begin{array}{|c|ccc||cc|} \hline
p & p & \oplus & \top & \neg & p \\
\hline
F & F & T & T & T & F \\
T & T & F & T & F & T \\
\hline
\e... | :$p \oplus \top \dashv \vdash \neg p$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] in the appropriate columns match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|c|ccc||cc|} \hline
p & p & \oplus & \top & \neg & ... | Exclusive Or with Tautology/Proof 2 | https://proofwiki.org/wiki/Exclusive_Or_with_Tautology | https://proofwiki.org/wiki/Exclusive_Or_with_Tautology/Proof_2 | [
"Tautology",
"Exclusive Or",
"Exclusive Or with Tautology"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8732 | Biconditional with Contradiction | :$p \iff \bot \dashv \vdash \neg p$ | {{BeginTableau|p \iff \bot \vdash \neg p}}
{{Premise|1|p \iff \bot}}
{{BiconditionalElimination|2|1|p \implies \bot|1|1}}
{{SequentIntro|3|1|\neg p|2|Contradictory Consequent}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|\neg p \vdash p \iff \bot}}
{{Assumption|1|\neg p}}
{{SequentIntro|2|1|p \implies \bot|1|Contradict... | :$p \iff \bot \dashv \vdash \neg p$ | {{BeginTableau|p \iff \bot \vdash \neg p}}
{{Premise|1|p \iff \bot}}
{{BiconditionalElimination|2|1|p \implies \bot|1|1}}
{{SequentIntro|3|1|\neg p|2|[[Contradictory Consequent]]}}
{{EndTableau}}
{{qed|lemma}}
{{BeginTableau|\neg p \vdash p \iff \bot}}
{{Assumption|1|\neg p}}
{{SequentIntro|2|1|p \implies \bot|1|[[Co... | Biconditional with Contradiction/Proof 1 | https://proofwiki.org/wiki/Biconditional_with_Contradiction | https://proofwiki.org/wiki/Biconditional_with_Contradiction/Proof_1 | [
"Contradiction",
"Biconditional",
"Biconditional with Contradiction"
] | [] | [
"Contradictory Consequent",
"Contradictory Consequent",
"Contradictory Antecedent"
] |
proofwiki-8733 | Biconditional with Contradiction | :$p \iff \bot \dashv \vdash \neg p$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
$\begin{array}{|c|ccc||cc|} \hline
p & p & \iff & \bot & \neg & p \\
\hline
F & F & T & F & T & F \\
T & T & F & F & F & T \\
\hline
\end... | :$p \iff \bot \dashv \vdash \neg p$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] in the appropriate columns match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|c|ccc||cc|} \hline
p & p & \iff & \bot & \neg & p ... | Biconditional with Contradiction/Proof 2 | https://proofwiki.org/wiki/Biconditional_with_Contradiction | https://proofwiki.org/wiki/Biconditional_with_Contradiction/Proof_2 | [
"Contradiction",
"Biconditional",
"Biconditional with Contradiction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8734 | Unique Representation of Complex Number in Spherical Form | Let $\PP$ be the complex plane.
Let $\mathbb S$ be the unit sphere which is tangent to $\PP$ at $\tuple {0, 0}$ (that is, where $z = 0$).
Let the diameter of $\mathbb S$ perpendicular to $\PP$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$.
Let the point $N$ be referred to as the '''north pole''... | {{ProofWanted|Should be standard Euclidean 3-d stuff but I haven't got that far in Euclid yet.}} | Let $\PP$ be the [[Definition:Complex Plane|complex plane]].
Let $\mathbb S$ be the [[Definition:Unit Sphere|unit sphere]] which is [[Definition:Tangent Plane|tangent]] to $\PP$ at $\tuple {0, 0}$ (that is, where $z = 0$).
Let the [[Definition:Diameter of Sphere|diameter]] of $\mathbb S$ [[Definition:Perpendicular|pe... | {{ProofWanted|Should be standard Euclidean 3-d stuff but I haven't got that far in Euclid yet.}} | Unique Representation of Complex Number in Spherical Form | https://proofwiki.org/wiki/Unique_Representation_of_Complex_Number_in_Spherical_Form | https://proofwiki.org/wiki/Unique_Representation_of_Complex_Number_in_Spherical_Form | [
"Complex Analysis"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Unit Sphere",
"Definition:Tangent Plane",
"Definition:Sphere/Geometry/Diameter",
"Definition:Right Angle/Perpendicular",
"File:Spherical-Representation-of-Complex-Number.png",
"Definition:Point",
"Definition:Point",
"Definition:Point",
"Defini... | [] |
proofwiki-8735 | Exclusive Or with Contradiction | :$p \oplus \bot \dashv \vdash p$ | {{BeginTableau|p \oplus \bot \vdash p}}
{{Premise|1|p \oplus \bot}}
{{SequentIntro|2|1|\left({p \lor \bot} \right) \land \neg \left({p \land \bot}\right)|1| {{Defof|Exclusive Or}} }}
{{SequentIntro|3|1|p \land \neg \left({p \land \bot}\right)|2|Disjunction with Contradiction}}
{{SequentIntro|4|1|p \land \neg \bot|3|Con... | :$p \oplus \bot \dashv \vdash p$ | {{BeginTableau|p \oplus \bot \vdash p}}
{{Premise|1|p \oplus \bot}}
{{SequentIntro|2|1|\left({p \lor \bot} \right) \land \neg \left({p \land \bot}\right)|1| {{Defof|Exclusive Or}} }}
{{SequentIntro|3|1|p \land \neg \left({p \land \bot}\right)|2|[[Disjunction with Contradiction]]}}
{{SequentIntro|4|1|p \land \neg \bot|3... | Exclusive Or with Contradiction/Proof 1 | https://proofwiki.org/wiki/Exclusive_Or_with_Contradiction | https://proofwiki.org/wiki/Exclusive_Or_with_Contradiction/Proof_1 | [
"Contradiction",
"Exclusive Or",
"Exclusive Or with Contradiction"
] | [] | [
"Disjunction with Contradiction",
"Conjunction with Contradiction",
"Tautology is Negation of Contradiction",
"Conjunction with Tautology",
"Conjunction with Tautology",
"Disjunction with Contradiction",
"Tautology is Negation of Contradiction",
"Conjunction with Contradiction"
] |
proofwiki-8736 | Exclusive Or with Contradiction | :$p \oplus \bot \dashv \vdash p$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations.
:<nowiki>$\begin{array}{|c|ccc||c|ccc|} \hline
p & p & \oplus & \bot & p \\
\hline
\F & \F & \F & \F & \F \\
\T & \T & \T & \F & \T \\
\h... | :$p \oplus \bot \dashv \vdash p$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] in the appropriate columns match for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|c|ccc||c|ccc|} \hline
p & p & \oplus & \b... | Exclusive Or with Contradiction/Proof 2 | https://proofwiki.org/wiki/Exclusive_Or_with_Contradiction | https://proofwiki.org/wiki/Exclusive_Or_with_Contradiction/Proof_2 | [
"Contradiction",
"Exclusive Or",
"Exclusive Or with Contradiction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8737 | Product of Complex Number with Conjugate by Dot and Cross Product | Let $z_1$ and $z_2$ be complex numbers.
Then:
:$\overline {z_1} z_2 = \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}$
where:
:$\overline {z_1}$ denotes the complex conjugate of $z_1$
:$z_1 \circ z_2$ denotes the complex dot product of $z_1$ with $z_2$
:$z_1 \times z_2$ denotes the complex cross product of $z_1$ wit... | {{begin-eqn}}
{{eqn | o =
| r = \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}
| c =
}}
{{eqn | r = \paren {z_1 \circ z_2} + i \frac {\overline {z_1} z_2 - z_1 \overline {z_2} } {2 i}
| c = {{Defof|Vector Cross Product|subdef = Complex|index = 4}}
}}
{{eqn | r = \frac {\overline {z_1} z_2 + z_1 \... | Let $z_1$ and $z_2$ be [[Definition:Complex Number|complex numbers]].
Then:
:$\overline {z_1} z_2 = \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}$
where:
:$\overline {z_1}$ denotes the [[Definition:Complex Conjugate|complex conjugate]] of $z_1$
:$z_1 \circ z_2$ denotes the [[Definition:Complex Dot Product|comple... | {{begin-eqn}}
{{eqn | o =
| r = \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}
| c =
}}
{{eqn | r = \paren {z_1 \circ z_2} + i \frac {\overline {z_1} z_2 - z_1 \overline {z_2} } {2 i}
| c = {{Defof|Vector Cross Product|subdef = Complex|index = 4}}
}}
{{eqn | r = \frac {\overline {z_1} z_2 + z_1 \... | Product of Complex Number with Conjugate by Dot and Cross Product | https://proofwiki.org/wiki/Product_of_Complex_Number_with_Conjugate_by_Dot_and_Cross_Product | https://proofwiki.org/wiki/Product_of_Complex_Number_with_Conjugate_by_Dot_and_Cross_Product | [
"Complex Conjugates",
"Complex Dot Product",
"Complex Cross Product"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Dot Product/Complex",
"Definition:Vector Cross Product/Complex"
] | [] |
proofwiki-8738 | Product of Complex Number with Conjugate in Exponential Form | Let $z_1$ and $z_2$ be complex numbers.
Then:
:$\overline {z_1} z_2 = \cmod {z_1} \, \cmod {z_2} e^{i \theta}$
where:
:$\overline {z_1}$ denotes the complex conjugate of $z_1$
:$\cmod {z_1}$ denotes the complex modulus of $z_1$
:$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction. | {{begin-eqn}}
{{eqn | l =\overline {z_1} z_2
| r = \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}
| c = Product of Complex Number with Conjugate by Dot and Cross Product
}}
{{eqn | r = \cmod {z_1} \, \cmod {z_2} \cos \theta + i \paren {z_1 \times z_2}
| c = {{Defof|Dot Product|subdef = Complex|in... | Let $z_1$ and $z_2$ be [[Definition:Complex Number|complex numbers]].
Then:
:$\overline {z_1} z_2 = \cmod {z_1} \, \cmod {z_2} e^{i \theta}$
where:
:$\overline {z_1}$ denotes the [[Definition:Complex Conjugate|complex conjugate]] of $z_1$
:$\cmod {z_1}$ denotes the [[Definition:Complex Modulus|complex modulus]] of $z... | {{begin-eqn}}
{{eqn | l =\overline {z_1} z_2
| r = \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}
| c = [[Product of Complex Number with Conjugate by Dot and Cross Product]]
}}
{{eqn | r = \cmod {z_1} \, \cmod {z_2} \cos \theta + i \paren {z_1 \times z_2}
| c = {{Defof|Dot Product|subdef = Comple... | Product of Complex Number with Conjugate in Exponential Form | https://proofwiki.org/wiki/Product_of_Complex_Number_with_Conjugate_in_Exponential_Form | https://proofwiki.org/wiki/Product_of_Complex_Number_with_Conjugate_in_Exponential_Form | [
"Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Modulus",
"Definition:Angle",
"Definition:Axis/Positive Direction"
] | [
"Product of Complex Number with Conjugate by Dot and Cross Product",
"Euler's Formula"
] |
proofwiki-8739 | Complex Numbers are Perpendicular iff Dot Product is Zero | Let $z_1$ and $z_2$ be complex numbers in vector form such that $z_1 \ne 0$ and $z_2 \ne 0$.
Then $z_1$ and $z_2$ are perpendicular {{iff}}:
:$z_1 \circ z_2 = 0$
where $z_1 \circ z_2$ denotes the complex dot product of $z_1$ with $z_2$. | By definition of complex dot product:
:$z_1 \circ z_2 = \cmod {z_1} \, \cmod {z_2} \cos \theta$
:$\cmod {z_1}$ denotes the complex modulus of $z_1$
:$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction. | Let $z_1$ and $z_2$ be [[Definition:Complex Number as Vector|complex numbers in vector form]] such that $z_1 \ne 0$ and $z_2 \ne 0$.
Then $z_1$ and $z_2$ are [[Definition:Perpendicular|perpendicular]] {{iff}}:
:$z_1 \circ z_2 = 0$
where $z_1 \circ z_2$ denotes the [[Definition:Complex Dot Product|complex dot product]]... | By definition of [[Definition:Dot Product/Complex/Definition 2|complex dot product]]:
:$z_1 \circ z_2 = \cmod {z_1} \, \cmod {z_2} \cos \theta$
:$\cmod {z_1}$ denotes the [[Definition:Complex Modulus|complex modulus]] of $z_1$
:$\theta$ denotes the [[Definition:Angle|angle]] from $z_1$ to $z_2$, measured in the [[Defin... | Complex Numbers are Perpendicular iff Dot Product is Zero | https://proofwiki.org/wiki/Complex_Numbers_are_Perpendicular_iff_Dot_Product_is_Zero | https://proofwiki.org/wiki/Complex_Numbers_are_Perpendicular_iff_Dot_Product_is_Zero | [
"Geometry of Complex Plane",
"Complex Dot Product"
] | [
"Definition:Complex Number as Vector",
"Definition:Right Angle/Perpendicular",
"Definition:Dot Product/Complex"
] | [
"Definition:Dot Product/Complex/Definition 2",
"Definition:Complex Modulus",
"Definition:Angle",
"Definition:Axis/Positive Direction"
] |
proofwiki-8740 | Complex Numbers are Parallel iff Cross Product is Zero | Let $z_1$ and $z_2$ be complex numbers in vector form such that $z_1 \ne 0$ and $z_2 \ne 0$.
Then $z_1$ and $z_2$ are parallel {{iff}}:
:$z_1 \times z_2 = 0$
where $z_1 \times z_2$ denotes the complex cross product of $z_1$ with $z_2$. | By definition of complex cross product:
:$z_1 \times z_2 = \cmod {z_1} \, \cmod {z_2} \sin \theta$
:$\cmod {z_1}$ denotes the complex modulus of $z_1$
:$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction. | Let $z_1$ and $z_2$ be [[Definition:Complex Number as Vector|complex numbers in vector form]] such that $z_1 \ne 0$ and $z_2 \ne 0$.
Then $z_1$ and $z_2$ are [[Definition:Parallel Vectors|parallel]] {{iff}}:
:$z_1 \times z_2 = 0$
where $z_1 \times z_2$ denotes the [[Definition:Complex Cross Product|complex cross produ... | By definition of [[Definition:Vector Cross Product/Complex/Definition 2|complex cross product]]:
:$z_1 \times z_2 = \cmod {z_1} \, \cmod {z_2} \sin \theta$
:$\cmod {z_1}$ denotes the [[Definition:Complex Modulus|complex modulus]] of $z_1$
:$\theta$ denotes the [[Definition:Angle|angle]] from $z_1$ to $z_2$, measured in... | Complex Numbers are Parallel iff Cross Product is Zero | https://proofwiki.org/wiki/Complex_Numbers_are_Parallel_iff_Cross_Product_is_Zero | https://proofwiki.org/wiki/Complex_Numbers_are_Parallel_iff_Cross_Product_is_Zero | [
"Geometry of Complex Plane",
"Complex Cross Product"
] | [
"Definition:Complex Number as Vector",
"Definition:Parallel Vectors",
"Definition:Vector Cross Product/Complex"
] | [
"Definition:Vector Cross Product/Complex/Definition 2",
"Definition:Complex Modulus",
"Definition:Angle",
"Definition:Axis/Positive Direction"
] |
proofwiki-8741 | Tautology iff Negation is Unsatisfiable | Let $\mathbf A$ be a WFF of propositional logic.
Then $\mathbf A$ is a tautology {{iff}} its negation $\neg \mathbf A$ is unsatisfiable. | === Necessary Condition ===
Let $\mathbf A$ be a tautology.
Let $v$ be a boolean interpretation of $\mathbf A$.
Then $\map v {\mathbf A} = \T$.
Hence, by definition of the boolean interpretation of negation:
:$\map v {\neg \mathbf A} = \F$
Since $v$ was arbitrary, it follows that $\neg \mathbf A$ is unsatisfiable.
{{qe... | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Then $\mathbf A$ is a [[Definition:Tautology (Boolean Interpretations)|tautology]] {{iff}} its [[Definition:Logical Not|negation]] $\neg \mathbf A$ is [[Definition:Unsatisfiable (Boolean Interpretations)|unsatisfiable]]. | === Necessary Condition ===
Let $\mathbf A$ be a [[Definition:Tautology (Boolean Interpretations)|tautology]].
Let $v$ be a [[Definition:Boolean Interpretation|boolean interpretation]] of $\mathbf A$.
Then $\map v {\mathbf A} = \T$.
Hence, by definition of the [[Definition:Logical Not/Boolean Interpretation|boolea... | Tautology iff Negation is Unsatisfiable | https://proofwiki.org/wiki/Tautology_iff_Negation_is_Unsatisfiable | https://proofwiki.org/wiki/Tautology_iff_Negation_is_Unsatisfiable | [
"Boolean Interpretations",
"Formal Semantics"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Tautology/Formal Semantics/Boolean Interpretations",
"Definition:Logical Not",
"Definition:Unsatisfiable/Boolean Interpretations"
] | [
"Definition:Tautology/Formal Semantics/Boolean Interpretations",
"Definition:Boolean Interpretation",
"Definition:Logical Not/Boolean Interpretation",
"Definition:Unsatisfiable/Boolean Interpretations",
"Definition:Unsatisfiable/Boolean Interpretations",
"Definition:Boolean Interpretation",
"Definition:... |
proofwiki-8742 | Satisfiable iff Negation is Falsifiable | Let $\mathbf A$ be a WFF of propositional logic.
Then $\mathbf A$ is satisfiable {{iff}} its negation $\neg \mathbf A$ is falsifiable. | === Necessary Condition ===
Let $\mathbf A$ be satisfiable.
Then there exists a boolean interpretation $v$ of $\mathbf A$ such that:
:$\map v {\mathbf A} = \T$
Hence, by definition of the boolean interpretation of negation:
:$\map v {\neg \mathbf A} = \F$
It follows that $\neg \mathbf A$ is falsifiable.
{{qed|lemma}} | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Then $\mathbf A$ is [[Definition:Satisfiable (Boolean Interpretations)|satisfiable]] {{iff}} its [[Definition:Logical Not|negation]] $\neg \mathbf A$ is [[Definition:Falsifiable (Boolean Interpretations)|falsifiable]]. | === Necessary Condition ===
Let $\mathbf A$ be [[Definition:Satisfiable (Boolean Interpretations)|satisfiable]].
Then there exists a [[Definition:Boolean Interpretation|boolean interpretation]] $v$ of $\mathbf A$ such that:
:$\map v {\mathbf A} = \T$
Hence, by definition of the [[Definition:Logical Not/Boolean Inte... | Satisfiable iff Negation is Falsifiable | https://proofwiki.org/wiki/Satisfiable_iff_Negation_is_Falsifiable | https://proofwiki.org/wiki/Satisfiable_iff_Negation_is_Falsifiable | [
"Boolean Interpretations",
"Formal Semantics"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Satisfiable/Boolean Interpretations",
"Definition:Logical Not",
"Definition:Falsifiable/Boolean Interpretations"
] | [
"Definition:Satisfiable/Boolean Interpretations",
"Definition:Boolean Interpretation",
"Definition:Logical Not/Boolean Interpretation",
"Definition:Falsifiable/Boolean Interpretations",
"Definition:Falsifiable/Boolean Interpretations",
"Definition:Boolean Interpretation",
"Definition:Logical Not/Boolean... |
proofwiki-8743 | Satisfiable Set minus Formula is Satisfiable | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be an $\mathscr M$-satisfiable set of formulas from $\LL$.
Let $\phi \in \FF$.
Then $\FF \setminus \set \phi$ is also $\mathscr M$-satisfiable. | This is an immediate consequence of Subset of Satisfiable Set is Satisfiable.
{{qed}} | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF$ be an [[Definition:Satisfiable Set of Formulas|$\mathscr M$-satisfiable set of formulas]] from $\LL$.
Let $\phi \in \FF$.
Then $\FF \setminus \set \phi$ is al... | This is an immediate consequence of [[Subset of Satisfiable Set is Satisfiable]].
{{qed}} | Satisfiable Set minus Formula is Satisfiable | https://proofwiki.org/wiki/Satisfiable_Set_minus_Formula_is_Satisfiable | https://proofwiki.org/wiki/Satisfiable_Set_minus_Formula_is_Satisfiable | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Satisfiable/Set of Formulas",
"Definition:Satisfiable/Set of Formulas"
] | [
"Subset of Satisfiable Set is Satisfiable"
] |
proofwiki-8744 | Subset of Satisfiable Set is Satisfiable | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be an $\mathscr M$-satisfiable set of formulas from $\LL$.
Let $\FF'$ be a subset of $\FF$.
Then $\FF'$ is also $\mathscr M$-satisfiable. | Since $\FF$ is $\mathscr M$-satisfiable, there exists some model $\MM$ of $\FF$:
:$\MM \models_{\mathscr M} \FF$
Thus for every $\psi \in \FF$:
:$\MM \models_{\mathscr M} \psi$
Now, for every $\psi$ in $\FF'$:
:$\psi \in \FF$
by definition of subset.
Hence:
:$\forall \psi \in \FF': \MM \models_{\mathscr M} \psi$
that i... | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF$ be an [[Definition:Satisfiable Set of Formulas|$\mathscr M$-satisfiable set of formulas]] from $\LL$.
Let $\FF'$ be a [[Definition:Subset|subset]] of $\FF$.
T... | Since $\FF$ is [[Definition:Satisfiable Set of Formulas|$\mathscr M$-satisfiable]], there exists some [[Definition:Model of Set of Formulas|model]] $\MM$ of $\FF$:
:$\MM \models_{\mathscr M} \FF$
Thus for every $\psi \in \FF$:
:$\MM \models_{\mathscr M} \psi$
Now, for every $\psi$ in $\FF'$:
:$\psi \in \FF$
by d... | Subset of Satisfiable Set is Satisfiable | https://proofwiki.org/wiki/Subset_of_Satisfiable_Set_is_Satisfiable | https://proofwiki.org/wiki/Subset_of_Satisfiable_Set_is_Satisfiable | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Satisfiable/Set of Formulas",
"Definition:Subset",
"Definition:Satisfiable/Set of Formulas"
] | [
"Definition:Satisfiable/Set of Formulas",
"Definition:Model (Logic)/Set of Logical Formulas",
"Definition:Subset",
"Definition:Model (Logic)/Set of Logical Formulas",
"Definition:Satisfiable/Set of Formulas",
"Category:Formal Semantics"
] |
proofwiki-8745 | Satisfiable Set Union Tautology is Satisfiable | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be an $\mathscr M$-satisfiable set of formulas from $\LL$.
Let $\phi$ be a tautology for $\mathscr M$.
Then $\FF \cup \set \phi$ is also $\mathscr M$-satisfiable. | Since $\FF$ is $\mathscr M$-satisfiable, there exists some model $\MM$ of $\FF$:
:$\MM \models_{\mathscr M} \FF$
Since $\psi$ is a tautology, also:
:$\MM \models_{\mathscr M} \psi$
Therefore, we conclude that:
:$\MM \models_{\mathscr M} \FF \cup \set \phi$
that is, $\FF \cup \set \phi$ is satisfiable. | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF$ be an [[Definition:Satisfiable Set of Formulas|$\mathscr M$-satisfiable set of formulas]] from $\LL$.
Let $\phi$ be a [[Definition:Tautology (Formal Semantics)|... | Since $\FF$ is [[Definition:Satisfiable Set of Formulas|$\mathscr M$-satisfiable]], there exists some [[Definition:Model of Set of Formulas|model]] $\MM$ of $\FF$:
:$\MM \models_{\mathscr M} \FF$
Since $\psi$ is a [[Definition:Tautology (Formal Semantics)|tautology]], also:
:$\MM \models_{\mathscr M} \psi$
Theref... | Satisfiable Set Union Tautology is Satisfiable | https://proofwiki.org/wiki/Satisfiable_Set_Union_Tautology_is_Satisfiable | https://proofwiki.org/wiki/Satisfiable_Set_Union_Tautology_is_Satisfiable | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Satisfiable/Set of Formulas",
"Definition:Tautology/Formal Semantics",
"Definition:Satisfiable/Set of Formulas"
] | [
"Definition:Satisfiable/Set of Formulas",
"Definition:Model (Logic)/Set of Logical Formulas",
"Definition:Tautology/Formal Semantics",
"Definition:Satisfiable/Set of Formulas"
] |
proofwiki-8746 | Unsatisfiable Set Union Formula is Unsatisfiable | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be an $\mathscr M$-unsatisfiable set of formulas from $\LL$.
Let $\phi$ be a logical formula.
Then $\FF \cup \set \phi$ is also $\mathscr M$-unsatisfiable. | This is an immediate consequence of Superset of Unsatisfiable Set is Unsatisfiable.
{{qed}} | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF$ be an [[Definition:Unsatisfiable Set of Formulas|$\mathscr M$-unsatisfiable set of formulas]] from $\LL$.
Let $\phi$ be a [[Definition:Logical Formula|logical f... | This is an immediate consequence of [[Superset of Unsatisfiable Set is Unsatisfiable]].
{{qed}} | Unsatisfiable Set Union Formula is Unsatisfiable | https://proofwiki.org/wiki/Unsatisfiable_Set_Union_Formula_is_Unsatisfiable | https://proofwiki.org/wiki/Unsatisfiable_Set_Union_Formula_is_Unsatisfiable | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Unsatisfiable/Set of Formulas",
"Definition:Logical Formula",
"Definition:Unsatisfiable/Set of Formulas"
] | [
"Superset of Unsatisfiable Set is Unsatisfiable"
] |
proofwiki-8747 | Superset of Unsatisfiable Set is Unsatisfiable | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be an $\mathscr M$-unsatisfiable set of formulas from $\LL$.
Let $\FF'$ be a superset of $\FF$.
Then $\FF'$ is also $\mathscr M$-unsatisfiable. | By assumption, $\FF$ is unsatisfiable.
Suppose now $\FF'$ were satisfiable.
Then it would follow from Subset of Satisfiable Set is Satisfiable that $\FF$ were also satisfiable.
We conclude that $\FF'$ must be unsatisfiable.
{{qed}}
Category:Formal Semantics
fca2oi6sx6lh7lv4yg1ci7rqrwdwlqq | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF$ be an [[Definition:Unsatisfiable Set of Formulas|$\mathscr M$-unsatisfiable set of formulas]] from $\LL$.
Let $\FF'$ be a [[Definition:Superset|superset]] of $\... | By assumption, $\FF$ is [[Definition:Unsatisfiable Set of Formulas|unsatisfiable]].
Suppose now $\FF'$ were [[Definition:Satisfiable Set of Formulas|satisfiable]].
Then it would follow from [[Subset of Satisfiable Set is Satisfiable]] that $\FF$ were also [[Definition:Satisfiable Set of Formulas|satisfiable]].
We ... | Superset of Unsatisfiable Set is Unsatisfiable | https://proofwiki.org/wiki/Superset_of_Unsatisfiable_Set_is_Unsatisfiable | https://proofwiki.org/wiki/Superset_of_Unsatisfiable_Set_is_Unsatisfiable | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Unsatisfiable/Set of Formulas",
"Definition:Subset/Superset",
"Definition:Unsatisfiable/Set of Formulas"
] | [
"Definition:Unsatisfiable/Set of Formulas",
"Definition:Satisfiable/Set of Formulas",
"Subset of Satisfiable Set is Satisfiable",
"Definition:Satisfiable/Set of Formulas",
"Definition:Unsatisfiable/Set of Formulas",
"Category:Formal Semantics"
] |
proofwiki-8748 | Unsatisfiable Set minus Tautology is Unsatisfiable | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be an $\mathscr M$-unsatisfiable set of formulas from $\LL$.
Let $\phi \in \FF$ be a tautology.
Then $\FF \setminus \set {\phi}$ is also $\mathscr M$-unsatisfiable. | Suppose $\FF \setminus \set {\phi}$ were satisfiable.
Then by Satisfiable Set Union Tautology is Satisfiable, so would $\FF$ be, because:
:$\FF = \paren {\FF \setminus \set {\phi} } \cup \set {\phi}$
by Set Difference Union Intersection and Intersection with Subset is Subset.
Therefore, $\FF \setminus \set {\phi}$ must... | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF$ be an [[Definition:Unsatisfiable Set of Formulas|$\mathscr M$-unsatisfiable set of formulas]] from $\LL$.
Let $\phi \in \FF$ be a [[Definition:Tautology (Formal... | Suppose $\FF \setminus \set {\phi}$ were [[Definition:Satisfiable Set of Formulas|satisfiable]].
Then by [[Satisfiable Set Union Tautology is Satisfiable]], so would $\FF$ be, because:
:$\FF = \paren {\FF \setminus \set {\phi} } \cup \set {\phi}$
by [[Set Difference Union Intersection]] and [[Intersection with Subse... | Unsatisfiable Set minus Tautology is Unsatisfiable | https://proofwiki.org/wiki/Unsatisfiable_Set_minus_Tautology_is_Unsatisfiable | https://proofwiki.org/wiki/Unsatisfiable_Set_minus_Tautology_is_Unsatisfiable | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Unsatisfiable/Set of Formulas",
"Definition:Tautology/Formal Semantics",
"Definition:Unsatisfiable/Set of Formulas"
] | [
"Definition:Satisfiable/Set of Formulas",
"Satisfiable Set Union Tautology is Satisfiable",
"Set Difference Union Intersection",
"Intersection with Subset is Subset",
"Definition:Unsatisfiable/Set of Formulas"
] |
proofwiki-8749 | Semantic Consequence as Tautological Conditional | Let $\FF$ be a finite set of WFFs of propositional logic.
Let $\mathbf A$ be another WFF.
Then the following are equivalent:
{{begin-eqn}}
{{eqn | l = \FF
| o = \models_{\mathrm {BI} }
| r = \mathbf A
}}
{{eqn | o = \models_{\mathrm {BI} }
| r = \bigwedge \FF \implies \mathbf A
}}
{{end-eqn}}
that is... | === Necessary Condition ===
let:
:$\FF \models_{\mathrm {BI} } \mathbf A$
{{AimForCont}} $\ds \bigwedge \FF \implies \mathbf A$ were not a tautology.
Then there exists a boolean interpretation $v$ such that:
:$\map v {\ds \bigwedge \FF} = \T$
:$\map v {\mathbf A} = \F$
by definition of the boolean interpretation of $\i... | Let $\FF$ be a [[Definition:Finite Set|finite set]] of [[Definition:WFF of Propositional Logic|WFFs of propositional logic]].
Let $\mathbf A$ be another [[Definition:WFF of Propositional Logic|WFF]].
Then the following are [[Definition:Logical Equivalence|equivalent]]:
{{begin-eqn}}
{{eqn | l = \FF
| o = \mod... | === Necessary Condition ===
let:
:$\FF \models_{\mathrm {BI} } \mathbf A$
{{AimForCont}} $\ds \bigwedge \FF \implies \mathbf A$ were not a [[Definition:Tautology (Boolean Interpretations)|tautology]].
Then there exists a [[Definition:Boolean Interpretation|boolean interpretation]] $v$ such that:
:$\map v {\ds \bigw... | Semantic Consequence as Tautological Conditional | https://proofwiki.org/wiki/Semantic_Consequence_as_Tautological_Conditional | https://proofwiki.org/wiki/Semantic_Consequence_as_Tautological_Conditional | [
"Boolean Interpretations"
] | [
"Definition:Finite Set",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Logical Equivalence",
"Definition:Semantic Consequence/Boolean Interpretations",
"Definition:Tautology/Formal Semantics/Boolean Interpretat... | [
"Definition:Tautology/Formal Semantics/Boolean Interpretations",
"Definition:Boolean Interpretation",
"Definition:Conditional/Boolean Interpretation",
"Definition:Conjunction/Boolean Interpretation",
"Definition:Model (Boolean Interpretations)",
"Definition:Tautology/Formal Semantics/Boolean Interpretatio... |
proofwiki-8750 | Semantic Consequence of Set Union Formula | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be a set of logical formulas from $\LL$.
Let $\phi$ be an $\mathscr M$-semantic consequence of $\FF$.
Let $\psi$ be another logical formula.
Then:
:$\FF \cup \set \psi \models_{\mathscr M} \phi$
that is, $\phi$ is also a semant... | This is an immediate consequence of Semantic Consequence of Superset.
{{qed}} | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF$ be a [[Definition:Set|set]] of [[Definition:Logical Formula|logical formulas]] from $\LL$.
Let $\phi$ be an [[Definition:Semantic Consequence|$\mathscr M$-seman... | This is an immediate consequence of [[Semantic Consequence of Superset]].
{{qed}} | Semantic Consequence of Set Union Formula | https://proofwiki.org/wiki/Semantic_Consequence_of_Set_Union_Formula | https://proofwiki.org/wiki/Semantic_Consequence_of_Set_Union_Formula | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Set",
"Definition:Logical Formula",
"Definition:Semantic Consequence",
"Definition:Logical Formula",
"Definition:Semantic Consequence"
] | [
"Semantic Consequence of Superset"
] |
proofwiki-8751 | Semantic Consequence of Superset | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be a set of logical formulas from $\LL$.
Let $\phi$ be an $\mathscr M$-semantic consequence of $\FF$.
Let $\FF'$ be another set of logical formulas.
Then:
:$\FF \cup \FF' \models_{\mathscr M} \phi$
that is, $\phi$ is also a sem... | Any model of $\FF \cup \FF'$ is a fortiori also a model of $\FF$.
By definition of semantic consequence all models of $\FF$ are models of $\phi$.
Therefore all models of $\FF \cup \FF'$ are also models of $\phi$.
Hence:
:$\FF \cup \FF' \models_{\mathscr M} \phi$
as desired.
{{qed}}
Category:Formal Semantics
cowaaykdur0... | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF$ be a [[Definition:Set|set]] of [[Definition:Logical Formula|logical formulas]] from $\LL$.
Let $\phi$ be an [[Definition:Semantic Consequence|$\mathscr M$-seman... | Any [[Definition:Model (Logic)|model]] of $\FF \cup \FF'$ is [[Definition:A Fortiori|a fortiori]] also a [[Definition:Model (Logic)|model]] of $\FF$.
By definition of [[Definition:Semantic Consequence|semantic consequence]] all [[Definition:Model (Logic)|models]] of $\FF$ are [[Definition:Model (Logic)|models]] of $\p... | Semantic Consequence of Superset | https://proofwiki.org/wiki/Semantic_Consequence_of_Superset | https://proofwiki.org/wiki/Semantic_Consequence_of_Superset | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Set",
"Definition:Logical Formula",
"Definition:Semantic Consequence",
"Definition:Set",
"Definition:Logical Formula",
"Definition:Semantic Consequence"
] | [
"Definition:Model (Logic)",
"Definition:A Fortiori",
"Definition:Model (Logic)",
"Definition:Semantic Consequence",
"Definition:Model (Logic)",
"Definition:Model (Logic)",
"Definition:Model (Logic)",
"Definition:Model (Logic)",
"Category:Formal Semantics"
] |
proofwiki-8752 | Semantic Consequence of Set minus Tautology | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be a set of logical formulas from $\LL$.
Let $\phi$ be an $\mathscr M$-semantic consequence of $\FF$.
Let $\psi \in \FF$ be a tautology.
Then:
:$\FF \setminus \set \psi \models_{\mathscr M} \phi$
that is, $\phi$ is also a seman... | Let $\MM$ be a model of $\FF \setminus \set \psi$.
Since $\psi$ is a tautology, it follows that:
:$\MM \models_{\mathscr M} \psi$
Hence:
:$\MM \models \FF$
which, {{hypothesis}}, entails:
:$\MM \models \phi$
Since $\MM$ was arbitrary, it follows by definition of semantic consequence that:
:$\FF \setminus \set \psi \mod... | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF$ be a [[Definition:Set|set]] of [[Definition:Logical Formula|logical formulas]] from $\LL$.
Let $\phi$ be an [[Definition:Semantic Consequence|$\mathscr M$-seman... | Let $\MM$ be a [[Definition:Model (Logic)|model]] of $\FF \setminus \set \psi$.
Since $\psi$ is a [[Definition:Tautology|tautology]], it follows that:
:$\MM \models_{\mathscr M} \psi$
Hence:
:$\MM \models \FF$
which, {{hypothesis}}, entails:
:$\MM \models \phi$
Since $\MM$ was arbitrary, it follows by definiti... | Semantic Consequence of Set minus Tautology | https://proofwiki.org/wiki/Semantic_Consequence_of_Set_minus_Tautology | https://proofwiki.org/wiki/Semantic_Consequence_of_Set_minus_Tautology | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Set",
"Definition:Logical Formula",
"Definition:Semantic Consequence",
"Definition:Tautology",
"Definition:Semantic Consequence"
] | [
"Definition:Model (Logic)",
"Definition:Tautology",
"Definition:Semantic Consequence"
] |
proofwiki-8753 | Equivalence of Definitions of Closed Set in Metric Space | {{TFAE|def = Closed Set (Metric Space)|view = Closed Set|context = Metric Space|contextview = Metric Spaces}}
Let $M = \struct {A, d}$ be a metric space.
Let $H \subseteq A$. | Let $H'$ denote the set of limit points of $H$. | {{TFAE|def = Closed Set (Metric Space)|view = Closed Set|context = Metric Space|contextview = Metric Spaces}}
Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $H \subseteq A$. | Let $H'$ denote the set of [[Definition:Limit Point (Metric Space)|limit points]] of $H$. | Equivalence of Definitions of Closed Set in Metric Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Closed_Set_in_Metric_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Closed_Set_in_Metric_Space | [
"Closed Sets"
] | [
"Definition:Metric Space"
] | [
"Definition:Limit Point/Metric Space",
"Definition:Limit Point/Metric Space",
"Definition:Limit Point/Metric Space",
"Definition:Limit Point/Metric Space"
] |
proofwiki-8754 | Set of Literals Satisfiable iff No Complementary Pairs | Let $S$ be a set of literals.
Then $S$ is satisfiable {{iff}} it contains no complementary pairs. | === Necessary Condition ===
Suppose $S$ contained the complementary pair $\set {p, \neg p}$.
By the Principle of Non-Contradiction, $\set {p, \neg p}$ is unsatisfiable.
By Superset of Unsatisfiable Set is Unsatisfiable, so is $S$.
Hence if $S$ is satisfiable, it cannot contain any complementary pairs.
{{qed|lemma}} | Let $S$ be a [[Definition:Set|set]] of [[Definition:Literal|literals]].
Then $S$ is [[Definition:Satisfiable (Boolean Interpretations)|satisfiable]] {{iff}} it contains no [[Definition:Complementary Pair|complementary pairs]]. | === Necessary Condition ===
Suppose $S$ contained the [[Definition:Complementary Pair|complementary pair]] $\set {p, \neg p}$.
By the [[Principle of Non-Contradiction]], $\set {p, \neg p}$ is [[Definition:Unsatisfiable (Boolean Interpretations)|unsatisfiable]].
By [[Superset of Unsatisfiable Set is Unsatisfiable]], ... | Set of Literals Satisfiable iff No Complementary Pairs | https://proofwiki.org/wiki/Set_of_Literals_Satisfiable_iff_No_Complementary_Pairs | https://proofwiki.org/wiki/Set_of_Literals_Satisfiable_iff_No_Complementary_Pairs | [
"Boolean Interpretations",
"Propositional Logic"
] | [
"Definition:Set",
"Definition:Literal",
"Definition:Satisfiable/Boolean Interpretations",
"Definition:Logical Complement/Complementary Pair"
] | [
"Definition:Logical Complement/Complementary Pair",
"Principle of Non-Contradiction",
"Definition:Unsatisfiable/Boolean Interpretations",
"Superset of Unsatisfiable Set is Unsatisfiable",
"Definition:Satisfiable/Boolean Interpretations",
"Definition:Logical Complement/Complementary Pair",
"Definition:Lo... |
proofwiki-8755 | Equivalence of Definitions of Exterior Point (Complex Analysis) | {{TFAE|def = Exterior Point (Complex Analysis)|view = exterior point|context = Complex Analysis}}
Let $S \subseteq \C$ be a subset of the complex plane.
Let $z_0 \in \C$. | Let $S \subseteq \C$. | {{TFAE|def = Exterior Point (Complex Analysis)|view = exterior point|context = Complex Analysis}}
Let $S \subseteq \C$ be a [[Definition:Subset|subset]] of the [[Definition:Complex Plane|complex plane]].
Let $z_0 \in \C$. | Let $S \subseteq \C$. | Equivalence of Definitions of Exterior Point (Complex Analysis) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Exterior_Point_(Complex_Analysis) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Exterior_Point_(Complex_Analysis) | [
"Complex Analysis"
] | [
"Definition:Subset",
"Definition:Complex Number/Complex Plane"
] | [] |
proofwiki-8756 | Bolzano-Weierstrass Theorem/General Form | Every infinite bounded space in a real Euclidean space has at least one limit point. | The proof of this theorem will be given as a series of lemmata that culminate in the actual theorem in the end.
Unless otherwise stated, all real spaces occurring in the proofs are equipped with the Euclidean metric and its related Euclidean topology. | Every infinite [[Definition:Bounded Metric Space|bounded space]] in a [[Definition:Real Euclidean Space|real Euclidean space]] has at least one [[Definition:Limit Point (Metric Space)|limit point]]. | The proof of this theorem will be given as a series of [[Definition:Lemma|lemmata]] that culminate in the actual theorem in the end.
Unless otherwise stated, all [[Definition:Real Euclidean Space|real spaces]] occurring in the proofs are equipped with the [[Definition:Euclidean Metric|Euclidean metric]] and its relate... | Bolzano-Weierstrass Theorem/General Form | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem/General_Form | https://proofwiki.org/wiki/Bolzano-Weierstrass_Theorem/General_Form | [
"Limits of Sequences",
"Bolzano-Weierstrass Theorem"
] | [
"Definition:Bounded Metric Space",
"Definition:Euclidean Space/Real",
"Definition:Limit Point/Metric Space"
] | [
"Definition:Lemma",
"Definition:Euclidean Space/Real",
"Definition:Euclidean Metric",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:Euclidean Metric"
] |
proofwiki-8757 | Quintuple Angle Formulas/Sine/Corollary | For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
:$\dfrac {\sin 5 \theta} {\sin \theta} = 16 \cos^4 \theta - 12 \cos^2 \theta + 1$
where $\sin$ denotes sine and $\cos$ denotes cosine. | First note that when $\theta = 0, \pm \pi, \pm 2 \pi \ldots$:
:$\sin \theta = 0$
so $\dfrac {\sin 5 \theta} {\sin \theta}$ is undefined.
Therefore for the rest of the proof it is assumed that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
{{begin-eqn}}
{{eqn | l = \sin 5 \theta
| r = 5 \sin \theta - 20 \sin^3 \theta +... | For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
:$\dfrac {\sin 5 \theta} {\sin \theta} = 16 \cos^4 \theta - 12 \cos^2 \theta + 1$
where $\sin$ denotes [[Definition:Sine|sine]] and $\cos$ denotes [[Definition:Cosine|cosine]]. | First note that when $\theta = 0, \pm \pi, \pm 2 \pi \ldots$:
:$\sin \theta = 0$
so $\dfrac {\sin 5 \theta} {\sin \theta}$ is undefined.
Therefore for the rest of the proof it is assumed that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
{{begin-eqn}}
{{eqn | l = \sin 5 \theta
| r = 5 \sin \theta - 20 \sin^3 \the... | Quintuple Angle Formulas/Sine/Corollary | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Sine/Corollary | https://proofwiki.org/wiki/Quintuple_Angle_Formulas/Sine/Corollary | [
"Quintuple Angle Formula for Sine",
"Sine Function"
] | [
"Definition:Sine",
"Definition:Cosine"
] | [
"Quintuple Angle Formulas/Sine",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-8758 | Complex Multiplication as Geometrical Transformation | Let $z_1 = \left\langle{r_1, \theta_1}\right\rangle$ and $z_2 = \left\langle{r_2, \theta_2}\right\rangle$ be complex numbers expressed in polar form.
Let $z_1$ and $z_2$ be represented on the complex plane $\C$ in vector form.
Let $z = z_1 z_2$ be the product of $z_1$ and $z_2$.
Then $z$ can be interpreted as the resul... | :500px
Let $z = r e^{i \alpha}$.
By Product of Complex Numbers in Exponential Form:
:$z = r_1 r_2 e^{i \left({\theta_1 + \theta_2}\right)}$
Adding $\theta_2$ to $\theta_1$ is equivalent to rotation about the origin of $\C$ by $\theta_2$ in the positive direction.
Similarly, the modulus of $z$ is obtained by multiplying... | Let $z_1 = \left\langle{r_1, \theta_1}\right\rangle$ and $z_2 = \left\langle{r_2, \theta_2}\right\rangle$ be [[Definition:Polar Form of Complex Number|complex numbers expressed in polar form]].
Let $z_1$ and $z_2$ be represented on the [[Definition:Complex Plane|complex plane]] $\C$ in [[Definition:Complex Number as V... | :[[File:Complex-Multiplication-as-Rotation.png|500px]]
Let $z = r e^{i \alpha}$.
By [[Product of Complex Numbers in Exponential Form]]:
:$z = r_1 r_2 e^{i \left({\theta_1 + \theta_2}\right)}$
Adding $\theta_2$ to $\theta_1$ is equivalent to rotation about the [[Definition:Origin|origin]] of $\C$ by $\theta_2$ in th... | Complex Multiplication as Geometrical Transformation | https://proofwiki.org/wiki/Complex_Multiplication_as_Geometrical_Transformation | https://proofwiki.org/wiki/Complex_Multiplication_as_Geometrical_Transformation | [
"Complex Multiplication",
"Polar Form of Complex Number",
"Geometry of Complex Plane",
"Complex Multiplication as Geometrical Transformation"
] | [
"Definition:Complex Number/Polar Form",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Number as Vector",
"Definition:Multiplication/Complex Numbers",
"Definition:Coordinate System/Origin",
"Definition:Axis/Positive Direction",
"Definition:Complex Modulus"
] | [
"File:Complex-Multiplication-as-Rotation.png",
"Product of Complex Numbers in Exponential Form",
"Definition:Coordinate System/Origin",
"Definition:Axis/Positive Direction",
"Definition:Complex Modulus",
"Definition:Complex Modulus"
] |
proofwiki-8759 | Argument of Quotient equals Difference of Arguments | Let $z_1$ and $z_2$ be complex numbers.
Then:
:$\map \arg {\dfrac {z_1} {z_2} } = \map \arg {z_1} - \map \arg {z_1} + 2 k \pi$
where:
:$\arg$ denotes the argument of a complex number
:$k$ can be $0$, $1$ or $-1$. | Let $z_1$ and $z_2$ be expressed in polar form.
:$z_1 = \polar {r_1, \theta_1}$
:$z_2 = \polar {r_2, \theta_2}$
From Division of Complex Numbers in Polar Form:
:$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \paren {\map \cos {\theta_1 - \theta_2} + i \, \map \sin {\theta_1 - \theta_2} }$
By the definition of argument:
:$\ma... | Let $z_1$ and $z_2$ be [[Definition:Complex Number|complex numbers]].
Then:
:$\map \arg {\dfrac {z_1} {z_2} } = \map \arg {z_1} - \map \arg {z_1} + 2 k \pi$
where:
:$\arg$ denotes the [[Definition:Argument of Complex Number|argument]] of a [[Definition:Complex Number|complex number]]
:$k$ can be $0$, $1$ or $-1$. | Let $z_1$ and $z_2$ be [[Definition:Polar Form of Complex Number|expressed in polar form]].
:$z_1 = \polar {r_1, \theta_1}$
:$z_2 = \polar {r_2, \theta_2}$
From [[Division of Complex Numbers in Polar Form]]:
:$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \paren {\map \cos {\theta_1 - \theta_2} + i \, \map \sin {\theta_1 - ... | Argument of Quotient equals Difference of Arguments | https://proofwiki.org/wiki/Argument_of_Quotient_equals_Difference_of_Arguments | https://proofwiki.org/wiki/Argument_of_Quotient_equals_Difference_of_Arguments | [
"Complex Division",
"Argument of Complex Number"
] | [
"Definition:Complex Number",
"Definition:Argument of Complex Number",
"Definition:Complex Number"
] | [
"Definition:Complex Number/Polar Form",
"Division of Complex Numbers in Polar Form",
"Definition:Argument of Complex Number",
"Cosine of Angle plus Full Angle",
"Sine of Angle plus Full Angle",
"Definition:Argument of Complex Number",
"Definition:Argument of Complex Number/Principal Range",
"Cosine of... |
proofwiki-8760 | Conditions on Rational Solution to Polynomial Equation | Let $P$ be the polynomial equation:
:$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
where $a_0, \ldots, a_n$ are integers.
Let $\dfrac p q$ be a root of $P$ expressed in canonical form.
Then $p$ is a divisor of $a_0$ and $q$ is a divisor of $a_n$. | By definition of the canonical form of a rational number, $p$ and $q$ are coprime.
Substitute $\dfrac p q$ for $z$ in $P$ and multiply by $q^n$:
:$(1): \quad a_n p^n + a_{n - 1} p^{n - 1} q + \cdots + a_1 p q^{n - 1} + a_0 q^n = 0$
Dividing $(1)$ by $p$ gives:
:$(2): \quad a_n p^{n - 1} + a_{n - 1} p^{n - 2} q + \cdots... | Let $P$ be the [[Definition:Polynomial Equation|polynomial equation]]:
:$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
where $a_0, \ldots, a_n$ are [[Definition:Integer|integers]].
Let $\dfrac p q$ be a [[Definition:Root of Polynomial|root]] of $P$ expressed in [[Definition:Canonical Form of Rational Numbe... | By definition of the [[Definition:Canonical Form of Rational Number|canonical form of a rational number]], $p$ and $q$ are [[Definition:Coprime Integers|coprime]].
Substitute $\dfrac p q$ for $z$ in $P$ and multiply by $q^n$:
:$(1): \quad a_n p^n + a_{n - 1} p^{n - 1} q + \cdots + a_1 p q^{n - 1} + a_0 q^n = 0$
Divi... | Conditions on Rational Solution to Polynomial Equation | https://proofwiki.org/wiki/Conditions_on_Rational_Solution_to_Polynomial_Equation | https://proofwiki.org/wiki/Conditions_on_Rational_Solution_to_Polynomial_Equation | [
"Polynomial Equations"
] | [
"Definition:Polynomial Equation",
"Definition:Integer",
"Definition:Root of Polynomial",
"Definition:Rational Number/Canonical Form",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Rational Number/Canonical Form",
"Definition:Coprime/Integers",
"Definition:Integer",
"Definition:Coprime/Integers",
"Euclid's Lemma",
"Definition:Divisor (Algebra)/Integer",
"Euclid's Lemma",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-8761 | Cardinality of Finite Vector Space | Let $K$ be a Galois field.
Let $V$ be a $K$-vector space.
Let the dimension of $V$ be finite.
Then:
:$\size V = \size K^{\map \dim V}$ | By Isomorphism from $\R^n$ via $n$-Term Sequence, $V$ is isomorphic to the $K$-vector space $K^{\map \dim V}$.
Thus:
:$\size V = \size {K^{\map \dim V} }$
By Cardinality of Finite Cartesian Space:
:$\size {K^{\map \dim V} } = \size K^{\map \dim V}$
Thus:
:$\size V = \size K^{\map \dim V}$
{{qed}}
Category:Vector Spaces... | Let $K$ be a [[Definition:Galois Field|Galois field]].
Let $V$ be a [[Definition:Vector Space|$K$-vector space]].
Let the [[Definition:Dimension of Vector Space|dimension]] of $V$ be [[Definition:Finite Set|finite]].
Then:
:$\size V = \size K^{\map \dim V}$ | By [[Isomorphism from R^n via n-Term Sequence|Isomorphism from $\R^n$ via $n$-Term Sequence]], $V$ is [[Definition:R-Algebraic Structure Isomorphism|isomorphic]] to the [[Definition:Vector Space on Cartesian Product|$K$-vector space $K^{\map \dim V}$]].
Thus:
:$\size V = \size {K^{\map \dim V} }$
By [[Cardinality o... | Cardinality of Finite Vector Space | https://proofwiki.org/wiki/Cardinality_of_Finite_Vector_Space | https://proofwiki.org/wiki/Cardinality_of_Finite_Vector_Space | [
"Vector Spaces"
] | [
"Definition:Galois Field",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Finite Set"
] | [
"Isomorphism from R^n via n-Term Sequence",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism",
"Definition:Vector Space on Cartesian Product",
"Cardinality of Cartesian Product of Finite Sets/General Result/Corollary",
"Category:Vector Spaces"
] |
proofwiki-8762 | Sum of Roots of Polynomial | Let $P$ be the polynomial equation:
: $a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The sum of the roots of $P$ is $-\dfrac {a_{n - 1} } {a_n}$. | Let the roots of $P$ be $z_1, z_2, \ldots, z_n$.
Then $P$ can be written in factored form as:
:$\ds a_n \prod_{k \mathop = 1}^n \paren {z - z_k} = a_n \paren {z - z_1} \paren {z - z_2} \cdots \paren {z - z_n}$
Using Viète's Formulas, $P$ can be expressed as:
:$a_n \paren {z^n - \paren {z_1 + z_2 + \cdots + z_n} z^{n - ... | Let $P$ be the [[Definition:Polynomial Equation|polynomial equation]]:
: $a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The [[Definition:Summation|sum]] of the [[Definition:Root of Polynomial|roots]] of $P$ is $-\dfrac {a_{n - 1} } {a_n}$. | Let the [[Definition:Root of Polynomial|roots]] of $P$ be $z_1, z_2, \ldots, z_n$.
Then $P$ can be written in [[Polynomial Factor Theorem|factored form]] as:
:$\ds a_n \prod_{k \mathop = 1}^n \paren {z - z_k} = a_n \paren {z - z_1} \paren {z - z_2} \cdots \paren {z - z_n}$
Using [[Viète's Formulas]], $P$ can be expr... | Sum of Roots of Polynomial/Proof 1 | https://proofwiki.org/wiki/Sum_of_Roots_of_Polynomial | https://proofwiki.org/wiki/Sum_of_Roots_of_Polynomial/Proof_1 | [
"Polynomial Equations",
"Sum of Roots of Polynomial"
] | [
"Definition:Polynomial Equation",
"Definition:Summation",
"Definition:Root of Polynomial"
] | [
"Definition:Root of Polynomial",
"Polynomial Factor Theorem",
"Viète's Formulas",
"Definition:Coefficient of Polynomial",
"Definition:Power (Algebra)/Exponent"
] |
proofwiki-8763 | Sum of Roots of Polynomial | Let $P$ be the polynomial equation:
: $a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The sum of the roots of $P$ is $-\dfrac {a_{n - 1} } {a_n}$. | From Viète's Formulas:
:$\ds a_{n - k} = \paren {-1}^k a_n \sum_{1 \mathop \le i_1 \mathop < \dotsb \mathop < i_k \mathop \le n} z_{i_1} \dotsm z_{i_k}$
for $k = 1, 2, \ldots, n$.
The result follows for $k = 1$.
{{qed}} | Let $P$ be the [[Definition:Polynomial Equation|polynomial equation]]:
: $a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The [[Definition:Summation|sum]] of the [[Definition:Root of Polynomial|roots]] of $P$ is $-\dfrac {a_{n - 1} } {a_n}$. | From [[Viète's Formulas]]:
:$\ds a_{n - k} = \paren {-1}^k a_n \sum_{1 \mathop \le i_1 \mathop < \dotsb \mathop < i_k \mathop \le n} z_{i_1} \dotsm z_{i_k}$
for $k = 1, 2, \ldots, n$.
The result follows for $k = 1$.
{{qed}} | Sum of Roots of Polynomial/Proof 2 | https://proofwiki.org/wiki/Sum_of_Roots_of_Polynomial | https://proofwiki.org/wiki/Sum_of_Roots_of_Polynomial/Proof_2 | [
"Polynomial Equations",
"Sum of Roots of Polynomial"
] | [
"Definition:Polynomial Equation",
"Definition:Summation",
"Definition:Root of Polynomial"
] | [
"Viète's Formulas"
] |
proofwiki-8764 | Sum of Roots of Polynomial | Let $P$ be the polynomial equation:
: $a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The sum of the roots of $P$ is $-\dfrac {a_{n - 1} } {a_n}$. | Proof by induction.
For $n \in \N_{> 0}$, let $\map S n$ be the statement:
:For polynomial $P$ of degree $n$, the sum of the roots of $P$ is $-\dfrac {a_{n - 1} } {a_n}$.
=== Basis of Induction ===
Let $P$ be the polynomial of degree $1$:
:$\map P x = a_0 + a_1 z$
The single root of $P$ is:
:$a_0 + a_1 z = 0 \implies z... | Let $P$ be the [[Definition:Polynomial Equation|polynomial equation]]:
: $a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The [[Definition:Summation|sum]] of the [[Definition:Root of Polynomial|roots]] of $P$ is $-\dfrac {a_{n - 1} } {a_n}$. | Proof by [[Definition:Principle of Mathematical Induction|induction]].
For $n \in \N_{> 0}$, let $\map S n$ be the [[Definition:Statement|statement]]:
:For [[Definition:Polynomial|polynomial]] $P$ of [[Definition:Degree of Polynomial|degree]] $n$, the [[Definition:Summation|sum]] of the [[Definition:Root of Polynomia... | Sum of Roots of Polynomial/Proof 3 | https://proofwiki.org/wiki/Sum_of_Roots_of_Polynomial | https://proofwiki.org/wiki/Sum_of_Roots_of_Polynomial/Proof_3 | [
"Polynomial Equations",
"Sum of Roots of Polynomial"
] | [
"Definition:Polynomial Equation",
"Definition:Summation",
"Definition:Root of Polynomial"
] | [
"Principle of Mathematical Induction",
"Definition:Statement",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Definition:Summation",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Definition:Root of Polynomial",
"Definition:Induction Hyp... |
proofwiki-8765 | Product of Roots of Polynomial | Let $P$ be the polynomial equation:
:$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The product of the roots of $P$ is $\dfrac {\paren {-1}^n a_0} {a_n}$. | Let the roots of $P$ be $z_1, z_2, \ldots, z_n$.
Then $P$ can be written in factored form as:
:$\ds a_n \prod_{k \mathop = 1}^n \paren {z - z_k} = a_0 \paren {z - z_1} \paren {z - z_2} \dotsm \paren {z - z_n}$
From Viète's Formulas, $P$ can be expressed as:
:$a_n \paren {z^n - \paren {z_1 + z_2 + \dotsb + z_n} z^{n - 1... | Let $P$ be the [[Definition:Polynomial Equation|polynomial equation]]:
:$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The [[Definition:Continued Product|product]] of the [[Definition:Root of Polynomial|roots]] of $P$ is $\dfrac {\paren {-1}^n a_0} {a_n}$. | Let the [[Definition:Root of Polynomial|roots]] of $P$ be $z_1, z_2, \ldots, z_n$.
Then $P$ can be written in [[Polynomial Factor Theorem|factored form]] as:
:$\ds a_n \prod_{k \mathop = 1}^n \paren {z - z_k} = a_0 \paren {z - z_1} \paren {z - z_2} \dotsm \paren {z - z_n}$
From [[Viète's Formulas]], $P$ can be expres... | Product of Roots of Polynomial/Proof 1 | https://proofwiki.org/wiki/Product_of_Roots_of_Polynomial | https://proofwiki.org/wiki/Product_of_Roots_of_Polynomial/Proof_1 | [
"Product of Roots of Polynomial",
"Polynomial Equations"
] | [
"Definition:Polynomial Equation",
"Definition:Continued Product",
"Definition:Root of Polynomial"
] | [
"Definition:Root of Polynomial",
"Polynomial Factor Theorem",
"Viète's Formulas",
"Definition:Coefficient of Polynomial",
"Definition:Power (Algebra)/Exponent"
] |
proofwiki-8766 | Product of Roots of Polynomial | Let $P$ be the polynomial equation:
:$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The product of the roots of $P$ is $\dfrac {\paren {-1}^n a_0} {a_n}$. | Proof by induction.
For $n \in \N_{> 0}$, let $\map S n$ be the statement:
:For polynomial $P$ of degree $n$, the product of the roots of $P$ is $\dfrac {\paren {-1}^n a_0} {a_n}$.
=== Basis of Induction ===
Let $P$ be the polynomial of degree $1$:
:$\map P x = a_0 + a_1 z$
The single root of $P$ is:
:$a_0 + a_1 z = 0 ... | Let $P$ be the [[Definition:Polynomial Equation|polynomial equation]]:
:$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$
such that $a_n \ne 0$.
The [[Definition:Continued Product|product]] of the [[Definition:Root of Polynomial|roots]] of $P$ is $\dfrac {\paren {-1}^n a_0} {a_n}$. | Proof by [[Definition:Principle of Mathematical Induction|induction]].
For $n \in \N_{> 0}$, let $\map S n$ be the [[Definition:Statement|statement]]:
:For [[Definition:Polynomial|polynomial]] $P$ of [[Definition:Degree of Polynomial|degree]] $n$, the [[Definition:Continued Product|product]] of the [[Definition:Root ... | Product of Roots of Polynomial/Proof 2 | https://proofwiki.org/wiki/Product_of_Roots_of_Polynomial | https://proofwiki.org/wiki/Product_of_Roots_of_Polynomial/Proof_2 | [
"Product of Roots of Polynomial",
"Polynomial Equations"
] | [
"Definition:Polynomial Equation",
"Definition:Continued Product",
"Definition:Root of Polynomial"
] | [
"Principle of Mathematical Induction",
"Definition:Statement",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Definition:Continued Product",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Definition:Root of Polynomial",
"Definition:Induc... |
proofwiki-8767 | Sum of Cosines of Fractions of Pi | Let $n \in \Z$ such that $n > 1$.
Then:
:$\ds \sum_{k \mathop = 1}^{n - 1} \cos \frac {2 k \pi} n = -1$ | Consider the equation:
:$z^n - 1 = 0$
whose solutions are the complex roots of unity:
:$1, e^{2 \pi i / n}, e^{4 \pi i / n}, e^{6 \pi i / n}, \ldots, e^{2 \paren {n - 1} \pi i / n}$
By Sum of Roots of Polynomial:
:$1 + e^{2 \pi i / n} + e^{4 \pi i / n} + e^{6 \pi i / n} + \cdots + e^{2 \paren {n - 1} \pi i / n} = 0$
Fr... | Let $n \in \Z$ such that $n > 1$.
Then:
:$\ds \sum_{k \mathop = 1}^{n - 1} \cos \frac {2 k \pi} n = -1$ | Consider the [[Definition:Polynomial Equation|equation]]:
:$z^n - 1 = 0$
whose solutions are the [[Definition:Complex Roots of Unity|complex roots of unity]]:
:$1, e^{2 \pi i / n}, e^{4 \pi i / n}, e^{6 \pi i / n}, \ldots, e^{2 \paren {n - 1} \pi i / n}$
By [[Sum of Roots of Polynomial]]:
:$1 + e^{2 \pi i / n} + e^{4 ... | Sum of Cosines of Fractions of Pi | https://proofwiki.org/wiki/Sum_of_Cosines_of_Fractions_of_Pi | https://proofwiki.org/wiki/Sum_of_Cosines_of_Fractions_of_Pi | [
"Cosine Function"
] | [] | [
"Definition:Polynomial Equation",
"Definition:Root of Unity/Complex",
"Sum of Roots of Polynomial",
"Euler's Formula",
"Definition:Complex Number/Real Part"
] |
proofwiki-8768 | Sum of Sines of Fractions of Pi | :$\forall n \in \Z_{>1}: \ds \sum_{k \mathop = 1}^{n - 1} \sin \frac {2 k \pi} n = 0$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \sin \frac {2 k \pi} n
| r = \sin \frac {2 \paren 1 \pi} n + \sin \frac {2 \paren 2 \pi} n + \sin \frac {2 \paren 3 \pi} n + \cdots + \sin \frac {2 \paren {n - 3} \pi} n + \sin \frac {2 \paren {n - 2} \pi} n + \sin \frac {2 \paren {n - 1} \pi} n
| c =... | :$\forall n \in \Z_{>1}: \ds \sum_{k \mathop = 1}^{n - 1} \sin \frac {2 k \pi} n = 0$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \sin \frac {2 k \pi} n
| r = \sin \frac {2 \paren 1 \pi} n + \sin \frac {2 \paren 2 \pi} n + \sin \frac {2 \paren 3 \pi} n + \cdots + \sin \frac {2 \paren {n - 3} \pi} n + \sin \frac {2 \paren {n - 2} \pi} n + \sin \frac {2 \paren {n - 1} \pi} n
| c =... | Sum of Sines of Fractions of Pi/Proof 2 | https://proofwiki.org/wiki/Sum_of_Sines_of_Fractions_of_Pi | https://proofwiki.org/wiki/Sum_of_Sines_of_Fractions_of_Pi/Proof_2 | [
"Sum of Sines of Fractions of Pi",
"Sine Function"
] | [] | [
"Sine of Angle plus Full Angle",
"Sine Function is Odd",
"Definition:Odd Integer",
"Definition:Term of Expression",
"Definition:Even Integer",
"Definition:Even Integer",
"Definition:Term of Expression",
"Definition:Odd Integer",
"Definition:Term of Expression",
"Definition:Additive Inverse",
"De... |
proofwiki-8769 | Area of Parallelogram in Complex Plane | Let $z_1$ and $z_2$ be complex numbers expressed as vectors.
Let $ABCD$ be the parallelogram formed by letting $AD = z_1$ and $AB = z_2$.
Then the area $\AA$ of $ABCD$ is given by:
:$\AA = z_1 \times z_2$
where $z_1 \times z_2$ denotes the cross product of $z_1$ and $z_2$. | :400px
From Area of Parallelogram:
:$\AA = \text{base} \times \text{height}$
In this context:
:$\text {base} = \cmod {z_2}$
and:
:$\text {height} = \cmod {z_1} \sin \theta$
The result follows by definition of complex cross product.
{{qed}} | Let $z_1$ and $z_2$ be [[Definition:Complex Number as Vector|complex numbers expressed as vectors]].
Let $ABCD$ be the [[Definition:Parallelogram|parallelogram]] formed by letting $AD = z_1$ and $AB = z_2$.
Then the [[Definition:Area|area]] $\AA$ of $ABCD$ is given by:
:$\AA = z_1 \times z_2$
where $z_1 \times z_2$ ... | :[[File:AreaOfParallelogramComplex.png|400px]]
From [[Area of Parallelogram]]:
:$\AA = \text{base} \times \text{height}$
In this context:
:$\text {base} = \cmod {z_2}$
and:
:$\text {height} = \cmod {z_1} \sin \theta$
The result follows by definition of [[Definition:Vector Cross Product/Complex/Definition 2|complex c... | Area of Parallelogram in Complex Plane | https://proofwiki.org/wiki/Area_of_Parallelogram_in_Complex_Plane | https://proofwiki.org/wiki/Area_of_Parallelogram_in_Complex_Plane | [
"Area of Parallelogram",
"Complex Cross Product",
"Geometry of Complex Plane"
] | [
"Definition:Complex Number as Vector",
"Definition:Quadrilateral/Parallelogram",
"Definition:Area",
"Definition:Vector Cross Product/Complex"
] | [
"File:AreaOfParallelogramComplex.png",
"Area of Parallelogram",
"Definition:Vector Cross Product/Complex/Definition 2"
] |
proofwiki-8770 | Area of Triangle in Determinant Form | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be points in the Cartesian plane.
The area $\AA$ of the triangle whose vertices are at $A$, $B$ and $C$ is given by:
:$\AA = \dfrac 1 2 \size {\paren {\begin {vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end {vmatrix} } }$ | {{begin-eqn}}
{{eqn | l = \map \Area T
| r = \dfrac 1 2 \size {\paren {\begin {vmatrix} 0 & 0 & 1 \\ b & a & 1 \\ x & y & 1 \end {vmatrix} } }
| c = Area of Triangle in Determinant Form
}}
{{eqn | r = \dfrac 1 2 \size {b y - a x}
| c = Determinant of Order 3
}}
{{end-eqn}}
{{qed}} | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be [[Definition:Point|points]] in the [[Definition:Cartesian Plane|Cartesian plane]].
The [[Definition:Area|area]] $\AA$ of the [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Vertex of Polygon|vertices]] are at $A$, $B$ and $C$ i... | {{begin-eqn}}
{{eqn | l = \map \Area T
| r = \dfrac 1 2 \size {\paren {\begin {vmatrix} 0 & 0 & 1 \\ b & a & 1 \\ x & y & 1 \end {vmatrix} } }
| c = [[Area of Triangle in Determinant Form]]
}}
{{eqn | r = \dfrac 1 2 \size {b y - a x}
| c = [[Determinant of Order 3]]
}}
{{end-eqn}}
{{qed}} | Area of Triangle in Determinant Form with Vertex at Origin/Proof 1 | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form_with_Vertex_at_Origin/Proof_1 | [
"Area of Triangle in Determinant Form",
"Areas of Triangles",
"Determinants"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Vertex"
] | [
"Area of Triangle in Determinant Form",
"Determinant/Examples/Order 3"
] |
proofwiki-8771 | Area of Triangle in Determinant Form | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be points in the Cartesian plane.
The area $\AA$ of the triangle whose vertices are at $A$, $B$ and $C$ is given by:
:$\AA = \dfrac 1 2 \size {\paren {\begin {vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end {vmatrix} } }$ | Let the polar coordinates of $B$ and $C$ be:
{{begin-eqn}}
{{eqn | l = B
| r = \polar {r_1, \theta_1}
}}
{{eqn | l = C
| r = \polar {r_2, \theta_2}
}}
{{end-eqn}}
Let $\theta$ be the angle between $AB$ and $AC$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map \Area {\triangle ABC}
| r = \dfrac 1 2 AB \cd... | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be [[Definition:Point|points]] in the [[Definition:Cartesian Plane|Cartesian plane]].
The [[Definition:Area|area]] $\AA$ of the [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Vertex of Polygon|vertices]] are at $A$, $B$ and $C$ i... | Let the [[Definition:Polar Coordinates|polar coordinates]] of $B$ and $C$ be:
{{begin-eqn}}
{{eqn | l = B
| r = \polar {r_1, \theta_1}
}}
{{eqn | l = C
| r = \polar {r_2, \theta_2}
}}
{{end-eqn}}
Let $\theta$ be the [[Definition:Angle|angle]] between $AB$ and $AC$.
Then we have:
{{begin-eqn}}
{{eqn | l ... | Area of Triangle in Determinant Form with Vertex at Origin/Proof 2 | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form_with_Vertex_at_Origin/Proof_2 | [
"Area of Triangle in Determinant Form",
"Areas of Triangles",
"Determinants"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Vertex"
] | [
"Definition:Polar Coordinates",
"Definition:Angle",
"Area of Triangle in Terms of Two Sides and Angle",
"Sine of Difference",
"Definition:Area",
"Definition:Positive/Real Number",
"Definition:Negative/Real Number",
"Definition:Sign of Number",
"Definition:Absolute Value"
] |
proofwiki-8772 | Area of Triangle in Determinant Form | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be points in the Cartesian plane.
The area $\AA$ of the triangle whose vertices are at $A$, $B$ and $C$ is given by:
:$\AA = \dfrac 1 2 \size {\paren {\begin {vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end {vmatrix} } }$ | :400px
Let $A$, $B$ and $C$ be defined as complex numbers in the complex plane.
The vectors from $C$ to $A$ and from $C$ to $B$ are given by:
:$z_1 = \paren {x_1 - x_3} + i \paren {y_1 - y_3}$
:$z_2 = \paren {x_2 - x_3} + i \paren {y_2 - y_3}$
From Area of Triangle in Terms of Side and Altitude, $\AA$ is half that of a... | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be [[Definition:Point|points]] in the [[Definition:Cartesian Plane|Cartesian plane]].
The [[Definition:Area|area]] $\AA$ of the [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Vertex of Polygon|vertices]] are at $A$, $B$ and $C$ i... | :[[File:AreaOfTriangleComplex.png|400px]]
Let $A$, $B$ and $C$ be defined as [[Definition:Complex Number|complex numbers]] in the [[Definition:Complex Plane|complex plane]].
The [[Definition:Complex Number as Vector|vectors]] from $C$ to $A$ and from $C$ to $B$ are given by:
:$z_1 = \paren {x_1 - x_3} + i \paren {y_... | Area of Triangle in Determinant Form/Proof 1 | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form/Proof_1 | [
"Area of Triangle in Determinant Form",
"Areas of Triangles",
"Determinants"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Vertex"
] | [
"File:AreaOfTriangleComplex.png",
"Definition:Complex Number",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Number as Vector",
"Area of Triangle in Terms of Side and Altitude",
"Definition:Quadrilateral/Parallelogram",
"Area of Parallelogram in Complex Plane",
"Determinant/Examples/O... |
proofwiki-8773 | Area of Triangle in Determinant Form | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be points in the Cartesian plane.
The area $\AA$ of the triangle whose vertices are at $A$, $B$ and $C$ is given by:
:$\AA = \dfrac 1 2 \size {\paren {\begin {vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end {vmatrix} } }$ | :400px
Let $A$, $B$ and $C$ be as defined..
Let $O$ be the origin of the Cartesian plane in which $\triangle ABC$ is embedded.
Taking into account the signs of the areas of the various triangles involved:
:$\triangle ABC = \triangle OAB + \triangle OBC + \triangle OCA$
as it is seen that $\triangle OBC$ and $\triangle ... | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be [[Definition:Point|points]] in the [[Definition:Cartesian Plane|Cartesian plane]].
The [[Definition:Area|area]] $\AA$ of the [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Vertex of Polygon|vertices]] are at $A$, $B$ and $C$ i... | :[[File:Area-of-Triangle-Determinant.png|400px]]
Let $A$, $B$ and $C$ be as defined..
Let $O$ be the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian plane]] in which $\triangle ABC$ is embedded.
Taking into account the [[Definition:Sign of Area of Triangle|signs]] of the [[Definition:Area... | Area of Triangle in Determinant Form/Proof 2 | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form/Proof_2 | [
"Area of Triangle in Determinant Form",
"Areas of Triangles",
"Determinants"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Vertex"
] | [
"File:Area-of-Triangle-Determinant.png",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Sign of Area of Triangle",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Clockwise",
"Area of Triangle in Determinant Form with Vertex at Origin/Proof 2",
"Det... |
proofwiki-8774 | Area of Triangle in Determinant Form | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be points in the Cartesian plane.
The area $\AA$ of the triangle whose vertices are at $A$, $B$ and $C$ is given by:
:$\AA = \dfrac 1 2 \size {\paren {\begin {vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end {vmatrix} } }$ | :400px
Let $A$, $B$ and $C$ be defined as $\tuple {x_1, y_1}$, $\tuple {x_2, y_2}$ and $\tuple {x_3, y_3}$ respectively.
From the figure, we see that:
{{begin-eqn}}
{{eqn | l = \map \Area {ABC}
| r = \map \Area {PACR} + \map \Area {RCBQ} - \map \Area {PABQ}
| c =
}}
{{eqn | r = \dfrac {\paren {x_3 - x_1} \... | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be [[Definition:Point|points]] in the [[Definition:Cartesian Plane|Cartesian plane]].
The [[Definition:Area|area]] $\AA$ of the [[Definition:Triangle (Geometry)|triangle]] whose [[Definition:Vertex of Polygon|vertices]] are at $A$, $B$ and $C$ i... | :[[File:Area-of-Triangle-Determinant-Proof-3.png|400px]]
Let $A$, $B$ and $C$ be defined as $\tuple {x_1, y_1}$, $\tuple {x_2, y_2}$ and $\tuple {x_3, y_3}$ respectively.
From the figure, we see that:
{{begin-eqn}}
{{eqn | l = \map \Area {ABC}
| r = \map \Area {PACR} + \map \Area {RCBQ} - \map \Area {PABQ}
... | Area of Triangle in Determinant Form/Proof 3 | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form | https://proofwiki.org/wiki/Area_of_Triangle_in_Determinant_Form/Proof_3 | [
"Area of Triangle in Determinant Form",
"Areas of Triangles",
"Determinants"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Vertex"
] | [
"File:Area-of-Triangle-Determinant-Proof-3.png",
"Area of Trapezium",
"Determinant/Examples/Order 3"
] |
proofwiki-8775 | Equation of Circle in Complex Plane/Formulation 2 | Let $\C$ be the complex plane.
Let $C$ be a circle in $\C$.
Then $C$ may be written as:
:$\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$
where:
:$\alpha \in \R_{\ne 0}$ is real and non-zero
:$\gamma \in \R$ is real
:$\beta \in \C$ is complex such that $\cmod \beta^2 > \alpha \gamma$.
The cur... | {{begin-eqn}}
{{eqn | l = \cmod {z - a}
| r = r
| c = Equation of Circle in Complex Plane: Formulation 1
}}
{{eqn | ll= \leadsto
| l = \cmod {z - a}^2
| r = r^2
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {z - a} \overline {\paren {z - a} }
| r = r^2
| c = Modulus in Term... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $C$ be a [[Definition:Circle|circle]] in $\C$.
Then $C$ may be written as:
:$\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$
where:
:$\alpha \in \R_{\ne 0}$ is [[Definition:Real Number|real]] and non-zero
:$\gamma \in \R$ is [... | {{begin-eqn}}
{{eqn | l = \cmod {z - a}
| r = r
| c = [[Equation of Circle in Complex Plane/Formulation 1|Equation of Circle in Complex Plane: Formulation 1]]
}}
{{eqn | ll= \leadsto
| l = \cmod {z - a}^2
| r = r^2
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {z - a} \overline {\paren... | Equation of Circle in Complex Plane/Formulation 2/Proof 1 | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_2 | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_2/Proof_1 | [
"Equation of Circle in Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Circle",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Line/Straight Line"
] | [
"Equation of Circle in Complex Plane/Formulation 1",
"Modulus in Terms of Conjugate",
"Sum of Complex Conjugates",
"Modulus in Terms of Conjugate",
"Equation of Line in Complex Plane/Formulation 1",
"Definition:Line/Straight Line"
] |
proofwiki-8776 | Equation of Circle in Complex Plane/Formulation 2 | Let $\C$ be the complex plane.
Let $C$ be a circle in $\C$.
Then $C$ may be written as:
:$\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$
where:
:$\alpha \in \R_{\ne 0}$ is real and non-zero
:$\gamma \in \R$ is real
:$\beta \in \C$ is complex such that $\cmod \beta^2 > \alpha \gamma$.
The cur... | From Equation of Circle in Cartesian Plane: Formulation 2, the equation for a circle is:
:$A \paren {x^2 + y^2} + B x + C y + D = 0$
provided that:
:$B^2 + C^2 \ge 4 A D$
:$A > 0$.
Thus:
{{begin-eqn}}
{{eqn | l = A \paren {x^2 + y^2} + B x + C y + D
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = A z \ov... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $C$ be a [[Definition:Circle|circle]] in $\C$.
Then $C$ may be written as:
:$\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$
where:
:$\alpha \in \R_{\ne 0}$ is [[Definition:Real Number|real]] and non-zero
:$\gamma \in \R$ is [... | From [[Equation of Circle/Cartesian/Formulation 2|Equation of Circle in Cartesian Plane: Formulation 2]], the equation for a [[Definition:Circle|circle]] is:
:$A \paren {x^2 + y^2} + B x + C y + D = 0$
provided that:
:$B^2 + C^2 \ge 4 A D$
:$A > 0$.
Thus:
{{begin-eqn}}
{{eqn | l = A \paren {x^2 + y^2} + B x + C y + ... | Equation of Circle in Complex Plane/Formulation 2/Proof 2 | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_2 | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_2/Proof_2 | [
"Equation of Circle in Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Circle",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Line/Straight Line"
] | [
"Equation of Circle/Cartesian/Formulation 2",
"Definition:Circle",
"Product of Complex Number with Conjugate",
"Sum of Complex Number with Conjugate",
"Difference of Complex Number with Conjugate",
"Definition:Real Number",
"Equation of Line in Complex Plane/Formulation 1",
"Definition:Line/Straight L... |
proofwiki-8777 | Equation of Line in Complex Plane/Formulation 1 | Let $\C$ be the complex plane.
Let $L$ be a straight line in $\C$.
Then $L$ may be written as:
:$\beta z + \overline \beta \overline z + \gamma = 0$
where $\gamma$ is real and $\beta$ may be complex. | From Equation of Straight Line in Plane, the equation for a straight line is:
:$A x + B y + C = 0$
Thus:
{{begin-eqn}}
{{eqn | l = A x + B y + C
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \frac A 2 \paren {z + \overline z} + B y + C
| r = 0
| c = Sum of Complex Number with Conjugate
}}
{... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $L$ be a [[Definition:Straight Line|straight line]] in $\C$.
Then $L$ may be written as:
:$\beta z + \overline \beta \overline z + \gamma = 0$
where $\gamma$ is [[Definition:Real Number|real]] and $\beta$ may be [[Definition:Complex Number|complex]]. | From [[Equation of Straight Line in Plane]], the equation for a [[Definition:Straight Line|straight line]] is:
:$A x + B y + C = 0$
Thus:
{{begin-eqn}}
{{eqn | l = A x + B y + C
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \frac A 2 \paren {z + \overline z} + B y + C
| r = 0
| c = [[Sum ... | Equation of Line in Complex Plane/Formulation 1 | https://proofwiki.org/wiki/Equation_of_Line_in_Complex_Plane/Formulation_1 | https://proofwiki.org/wiki/Equation_of_Line_in_Complex_Plane/Formulation_1 | [
"Equation of Line in Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Line/Straight Line",
"Definition:Real Number",
"Definition:Complex Number"
] | [
"Equation of Straight Line in Plane",
"Definition:Line/Straight Line",
"Sum of Complex Number with Conjugate",
"Difference of Complex Number with Conjugate"
] |
proofwiki-8778 | Vertices of Equilateral Triangle in Complex Plane | Let $z_1$, $z_2$ and $z_3$ be complex numbers.
Then:
: $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle
{{iff}}:
:${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$ | :400px | Let $z_1$, $z_2$ and $z_3$ be [[Definition:Complex Number|complex numbers]].
Then:
: $z_1$, $z_2$ and $z_3$ represent on the [[Definition:Complex Plane|complex plane]] the [[Definition:Vertex of Polygon|vertices]] of an [[Definition:Equilateral Triangle|equilateral triangle]]
{{iff}}:
:${z_1}^2 + {z_2}^2 + {z_3}^2 ... | :[[File:EquilateralTriangleInComplexPlane.png|400px]] | Vertices of Equilateral Triangle in Complex Plane | https://proofwiki.org/wiki/Vertices_of_Equilateral_Triangle_in_Complex_Plane | https://proofwiki.org/wiki/Vertices_of_Equilateral_Triangle_in_Complex_Plane | [
"Equilateral Triangles",
"Geometry of Complex Plane",
"Vertices of Equilateral Triangle in Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Complex Plane",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)/Equilateral"
] | [
"File:EquilateralTriangleInComplexPlane.png"
] |
proofwiki-8779 | Product of Sines of Fractions of Pi | Let $m \in \Z$ such that $m > 1$.
Then:
:$\ds \prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} m = \frac m {2^{m - 1} }$ | From Product Formula for Sine, we have:
:$\ds \map \sin {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}$
Therefore:
{{begin-eqn}}
{{eqn | l = \map \sin {n z}
| r = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}
| c =
}}
{{eqn | r = 2^{n - 1} \map \si... | Let $m \in \Z$ such that $m > 1$.
Then:
:$\ds \prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} m = \frac m {2^{m - 1} }$ | From [[Product Formula for Sine]], we have:
:$\ds \map \sin {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}$
Therefore:
{{begin-eqn}}
{{eqn | l = \map \sin {n z}
| r = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}
| c =
}}
{{eqn | r = 2^{n - 1} \m... | Product of Sines of Fractions of Pi/Proof 2 | https://proofwiki.org/wiki/Product_of_Sines_of_Fractions_of_Pi | https://proofwiki.org/wiki/Product_of_Sines_of_Fractions_of_Pi/Proof_2 | [
"Product of Sines of Fractions of Pi",
"Product Formula for Sine",
"Sine Function"
] | [] | [
"Product Formula for Sine",
"Definition:Limit of Complex Function",
"Limit of Sinc Function at Zero"
] |
proofwiki-8780 | Product of Complex Conjugates/General Result | Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers.
Let $\overline z$ be the complex conjugate of the complex number $z$.
Then:
:$\ds \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$ | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$
$\map P 1$ is trivially true, as this just says $\overline {z_1} = \overline {z_1}$. | Let $z_1, z_2, \ldots, z_n \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\overline z$ be the [[Definition:Complex Conjugate|complex conjugate]] of the [[Definition:Complex Number|complex number]] $z$.
Then:
:$\ds \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$
$\map P 1$ is trivially true, as this just says $\overline {z_1} = \overline {z_... | Product of Complex Conjugates/General Result | https://proofwiki.org/wiki/Product_of_Complex_Conjugates/General_Result | https://proofwiki.org/wiki/Product_of_Complex_Conjugates/General_Result | [
"Product of Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-8781 | Quotient of Complex Conjugates | Let $z_1, z_2 \in \C$ be complex numbers.
Let $\overline z$ be the complex conjugate of the complex number $z$.
Then:
:$\overline {\paren {\dfrac {z_1} {z_2} } } = \dfrac {\paren {\overline {z_1} } } {\paren {\overline {z_2} } }$
for $z_2 \ne 0$. | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \overline {\paren {\frac {z_1} {z_2} } }
| r = \overline {\paren {\frac {x_1 x_2 + y_1 y_2} { {x_2}^2 + {y_2}^2} + i \frac {x_2 y_1 - x_1 y_2} { {x_2}^2 + {y_2}^2} } }
| c = Division of Compl... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\overline z$ be the [[Definition:Complex Conjugate|complex conjugate]] of the [[Definition:Complex Number|complex number]] $z$.
Then:
:$\overline {\paren {\dfrac {z_1} {z_2} } } = \dfrac {\paren {\overline {z_1} } } {\paren {\overline {z_2}... | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \overline {\paren {\frac {z_1} {z_2} } }
| r = \overline {\paren {\frac {x_1 x_2 + y_1 y_2} { {x_2}^2 + {y_2}^2} + i \frac {x_2 y_1 - x_1 y_2} { {x_2}^2 + {y_2}^2} } }
| c = [[Division of C... | Quotient of Complex Conjugates | https://proofwiki.org/wiki/Quotient_of_Complex_Conjugates | https://proofwiki.org/wiki/Quotient_of_Complex_Conjugates | [
"Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number"
] | [
"Division of Complex Numbers",
"Division of Complex Numbers"
] |
proofwiki-8782 | Condition for Points in Complex Plane to form Parallelogram | Let $A = z_1$, $B = z_2$, $C = z_3$ and $D = z_4$ represent on the complex plane the vertices of a quadrilateral.
Then $ABCD$ is a parallelogram {{iff}}:
:$z_1 - z_2 + z_3 - z_4 = 0$ | :400px
$ABCD$ is a parallelogram {{iff}}:
:$\vec {AB} = \vec {DC}$
Then:
{{begin-eqn}}
{{eqn | l = \vec {AB}
| r = \vec {DC}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = z_2 - z_1
| r = z_3 - z_4
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = z_1 - z_2 + z_3 - z_4
| r = 0
| c... | Let $A = z_1$, $B = z_2$, $C = z_3$ and $D = z_4$ represent on the [[Definition:Complex Plane|complex plane]] the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Quadrilateral|quadrilateral]].
Then $ABCD$ is a [[Definition:Parallelogram|parallelogram]] {{iff}}:
:$z_1 - z_2 + z_3 - z_4 = 0$ | :[[File:ParallelogramInComplexPlane.png|400px]]
$ABCD$ is a [[Definition:Parallelogram|parallelogram]] {{iff}}:
:$\vec {AB} = \vec {DC}$
Then:
{{begin-eqn}}
{{eqn | l = \vec {AB}
| r = \vec {DC}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = z_2 - z_1
| r = z_3 - z_4
| c =
}}
{{eqn | ll= ... | Condition for Points in Complex Plane to form Parallelogram | https://proofwiki.org/wiki/Condition_for_Points_in_Complex_Plane_to_form_Parallelogram | https://proofwiki.org/wiki/Condition_for_Points_in_Complex_Plane_to_form_Parallelogram | [
"Parallelograms",
"Geometry of Complex Plane",
"Condition for Points in Complex Plane to form Parallelogram"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Polygon/Vertex",
"Definition:Quadrilateral",
"Definition:Quadrilateral/Parallelogram"
] | [
"File:ParallelogramInComplexPlane.png",
"Definition:Quadrilateral/Parallelogram"
] |
proofwiki-8783 | Quadrilateral with Bisecting Diagonals is Parallelogram | Let $ABCD$ be a quadrilateral.
Let the diagonals of $ABCD$ bisect each other.
Then $ABCD$ is a parallelogram. | The diagonals of $ABCD$ bisect each other if the position vectors of the midpoints of the diagonals are the same point.
Let $z_1, z_2, z_3, z_4$ be the position vectors of the vertices of $ABCD$.
Thus:
{{begin-eqn}}
{{eqn | l = z_1 + \frac {z_3 - z_1} 2
| r = z_2 + \frac {z_4 - z_2} 2
| c = condition for bi... | Let $ABCD$ be a [[Definition:Quadrilateral|quadrilateral]].
Let the [[Definition:Diagonal of Quadrilateral|diagonals]] of $ABCD$ [[Definition:Bisection|bisect]] each other.
Then $ABCD$ is a [[Definition:Parallelogram|parallelogram]]. | The [[Definition:Diagonal of Quadrilateral|diagonals]] of $ABCD$ [[Definition:Bisection|bisect]] each other if the [[Definition:Complex Number as Vector|position vectors]] of the [[Definition:Midpoint of Line|midpoints]] of the [[Definition:Diagonal of Quadrilateral|diagonals]] are the same [[Definition:Point|point]].
... | Quadrilateral with Bisecting Diagonals is Parallelogram | https://proofwiki.org/wiki/Quadrilateral_with_Bisecting_Diagonals_is_Parallelogram | https://proofwiki.org/wiki/Quadrilateral_with_Bisecting_Diagonals_is_Parallelogram | [
"Quadrilaterals",
"Parallelograms"
] | [
"Definition:Quadrilateral",
"Definition:Diameter of Quadrilateral",
"Definition:Bisection",
"Definition:Quadrilateral/Parallelogram"
] | [
"Definition:Diameter of Quadrilateral",
"Definition:Bisection",
"Definition:Complex Number as Vector",
"Definition:Line/Midpoint",
"Definition:Diameter of Quadrilateral",
"Definition:Point",
"Definition:Complex Number as Vector",
"Definition:Polygon/Vertex",
"Definition:Bisection",
"Condition for ... |
proofwiki-8784 | Medians of Triangle Meet at Centroid | Let $\triangle ABC$ be a triangle.
Then the medians of $\triangle ABC$ meet at a single point.
This point is called the centroid of $\triangle ABC$. | :360px
Let $A'$ be the midpoint of $BC$.
Let $B'$ be the midpoint of $AC$.
Let $C'$ be the midpoint of $AB$.
Hence $AA'$, $BB'$ and $CC'$ are the medians of $\triangle ABC$.
Let $AA'$ and $BB'$ intersect at $G$.
Hence $A'B'$ is a midline of $\triangle ABC$.
By the Midline Theorem, $A'B'$ is parallel to $AB$ and half th... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then the [[Definition:Median of Triangle|medians]] of $\triangle ABC$ meet at a single [[Definition:Point|point]].
This point is called the [[Definition:Centroid of Triangle|centroid]] of $\triangle ABC$. | :[[File:Medians-meet-at-Centroid.png|360px]]
Let $A'$ be the [[Definition:Midpoint of Line|midpoint]] of $BC$.
Let $B'$ be the [[Definition:Midpoint of Line|midpoint]] of $AC$.
Let $C'$ be the [[Definition:Midpoint of Line|midpoint]] of $AB$.
Hence $AA'$, $BB'$ and $CC'$ are the [[Definition:Median of Triangle|medi... | Medians of Triangle Meet at Centroid/Proof 1 | https://proofwiki.org/wiki/Medians_of_Triangle_Meet_at_Centroid | https://proofwiki.org/wiki/Medians_of_Triangle_Meet_at_Centroid/Proof_1 | [
"Medians of Triangle Meet at Centroid",
"Centroids of Triangles",
"Medians of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Median of Triangle",
"Definition:Point",
"Definition:Centroid/Triangle"
] | [
"File:Medians-meet-at-Centroid.png",
"Definition:Line/Midpoint",
"Definition:Line/Midpoint",
"Definition:Line/Midpoint",
"Definition:Median of Triangle",
"Definition:Intersection (Geometry)",
"Definition:Midline of Triangle",
"Midline Theorem",
"Definition:Parallel (Geometry)/Lines",
"Definition:H... |
proofwiki-8785 | Midpoints of Sides of Quadrilateral form Parallelogram | Let $\Box ABCD$ be a quadrilateral.
Let $E, F, G, H$ be the midpoints of $AB, BC, CD, DA$ respectively.
Then $\Box EFGH$ is a parallelogram. | Let $z_1, z_2, z_3, z_4$ be the position vectors of the vertices of $\Box ABCD$.
The midpoints of the sides of $\Box ABCD$ are, then:
{{begin-eqn}}
{{eqn | l = OE
| r = \frac {z_1 + z_2} 2
| c =
}}
{{eqn | l = OF
| r = \frac {z_2 + z_3} 2
| c =
}}
{{eqn | l = OG
| r = \frac {z_3 + z_4} 2... | Let $\Box ABCD$ be a [[Definition:Quadrilateral|quadrilateral]].
Let $E, F, G, H$ be the [[Definition:Midpoint of Line|midpoints]] of $AB, BC, CD, DA$ respectively.
Then $\Box EFGH$ is a [[Definition:Parallelogram|parallelogram]]. | Let $z_1, z_2, z_3, z_4$ be the [[Definition:Complex Number as Vector|position vectors]] of the [[Definition:Vertex of Polygon|vertices]] of $\Box ABCD$.
The [[Definition:Midpoint of Line|midpoints]] of the [[Definition:Side of Polygon|sides]] of $\Box ABCD$ are, then:
{{begin-eqn}}
{{eqn | l = OE
| r = \frac {... | Midpoints of Sides of Quadrilateral form Parallelogram | https://proofwiki.org/wiki/Midpoints_of_Sides_of_Quadrilateral_form_Parallelogram | https://proofwiki.org/wiki/Midpoints_of_Sides_of_Quadrilateral_form_Parallelogram | [
"Quadrilaterals",
"Parallelograms"
] | [
"Definition:Quadrilateral",
"Definition:Line/Midpoint",
"Definition:Quadrilateral/Parallelogram"
] | [
"Definition:Complex Number as Vector",
"Definition:Polygon/Vertex",
"Definition:Line/Midpoint",
"Definition:Polygon/Side",
"Definition:Polygon/Opposite",
"Definition:Complex Number as Vector",
"Definition:Polygon/Opposite",
"Definition:Quadrilateral/Parallelogram"
] |
proofwiki-8786 | Line from Bisector of Side of Parallelogram to Vertex Trisects Diagonal | Let $ABCD$ be a parallelogram.
Let $E$ be the midpoint of $AD$.
Then the point at which the line $BE$ meets $AC$ trisects $AC$. | :400px
Let the given intersection be at $F$.
We have that $E$ is the midpoint of $AD$.
Thus:
{{begin-eqn}}
{{eqn | l = \vec {AB} + \vec {BE}
| r = \frac {\vec {AD} } 2
| c =
}}
{{eqn | ll= \leadsto
| l = \vec {BE}
| r = \frac {\vec {AD} } 2 - \vec {AB}
| c =
}}
{{eqn | ll= \leadsto
... | Let $ABCD$ be a [[Definition:Parallelogram|parallelogram]].
Let $E$ be the [[Definition:Midpoint of Line|midpoint]] of $AD$.
Then the [[Definition:Point|point]] at which the [[Definition:Line Segment|line]] $BE$ meets $AC$ [[Definition:Trisection|trisects]] $AC$. | :[[File:TrisectorOfDiagonalOfParallelogram.png|400px]]
Let the given [[Definition:Intersection (Geometry)|intersection]] be at $F$.
We have that $E$ is the [[Definition:Midpoint of Line|midpoint]] of $AD$.
Thus:
{{begin-eqn}}
{{eqn | l = \vec {AB} + \vec {BE}
| r = \frac {\vec {AD} } 2
| c =
}}
{{eqn | ... | Line from Bisector of Side of Parallelogram to Vertex Trisects Diagonal | https://proofwiki.org/wiki/Line_from_Bisector_of_Side_of_Parallelogram_to_Vertex_Trisects_Diagonal | https://proofwiki.org/wiki/Line_from_Bisector_of_Side_of_Parallelogram_to_Vertex_Trisects_Diagonal | [
"Parallelograms"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Line/Midpoint",
"Definition:Point",
"Definition:Line/Segment",
"Definition:Trisection"
] | [
"File:TrisectorOfDiagonalOfParallelogram.png",
"Definition:Intersection (Geometry)",
"Definition:Line/Midpoint",
"Definition:Coincident Straight Lines",
"Definition:Parallel (Geometry)/Lines"
] |
proofwiki-8787 | Sum of Complex Numbers in Exponential Form | Let $z_1 = r_1 e^{i \theta_1}$ and $z_2 = r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form.
Let $z_3 = r_3 e^{i \theta_3} = z_1 + z_2$.
Then:
:$r_3 = \sqrt { {r_1}^2 + {r_2}^2 + 2 r_1 r_2 \map \cos {\theta_1 - \theta_2} }$
:$\theta_3 = \map \arctan {\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2} {r_... | We have:
{{begin-eqn}}
{{eqn | l = r_1 e^{i \theta_1} + r_2 e^{i \theta_2}
| r = r_1 \paren {\cos \theta_1 + i \sin \theta_1} + r_2 \paren {\cos \theta_2 + i \sin \theta_2}
| c = {{Defof|Polar Form of Complex Number}}
}}
{{eqn | r = \paren {r_1 \cos \theta_1 + r_2 \cos \theta_2} + i \paren {r_1 \sin \theta_... | Let $z_1 = r_1 e^{i \theta_1}$ and $z_2 = r_2 e^{i \theta_2}$ be [[Definition:Exponential Form of Complex Number|complex numbers expressed in exponential form]].
Let $z_3 = r_3 e^{i \theta_3} = z_1 + z_2$.
Then:
:$r_3 = \sqrt { {r_1}^2 + {r_2}^2 + 2 r_1 r_2 \map \cos {\theta_1 - \theta_2} }$
:$\theta_3 = \map \arctan... | We have:
{{begin-eqn}}
{{eqn | l = r_1 e^{i \theta_1} + r_2 e^{i \theta_2}
| r = r_1 \paren {\cos \theta_1 + i \sin \theta_1} + r_2 \paren {\cos \theta_2 + i \sin \theta_2}
| c = {{Defof|Polar Form of Complex Number}}
}}
{{eqn | r = \paren {r_1 \cos \theta_1 + r_2 \cos \theta_2} + i \paren {r_1 \sin \theta... | Sum of Complex Numbers in Exponential Form | https://proofwiki.org/wiki/Sum_of_Complex_Numbers_in_Exponential_Form | https://proofwiki.org/wiki/Sum_of_Complex_Numbers_in_Exponential_Form | [
"Complex Addition"
] | [
"Definition:Complex Number/Polar Form/Exponential Form"
] | [
"Complex Modulus of Sum of Complex Numbers"
] |
proofwiki-8788 | De Moivre's Formula/Positive Integer Index | Let $z \in \C$ be a complex number expressed in polar form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | $\cos x + i \sin x$ is a complex number expressed in polar form $\left\langle{r, \theta}\right\rangle$ whose complex modulus is $1$ and whose argument is $x$.
From De Moivre's Formula: Positive Integer Index:
:$\forall n \in \Z_{>0}: \left({r \left({\cos x + i \sin x}\right)}\right)^n = r^n \left({\cos \left({n x}\righ... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|polar form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | $\cos x + i \sin x$ is a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|polar form]] $\left\langle{r, \theta}\right\rangle$ whose [[Definition:Complex Modulus|complex modulus]] is $1$ and whose [[Definition:Argument of Complex Number|argument]] is $x$.
From [[De Moi... | De Moivre's Formula/Positive Integer Index/Corollary/Proof 1 | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index/Corollary/Proof_1 | [
"De Moivre's Formula"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form",
"Definition:Complex Modulus",
"Definition:Argument of Complex Number",
"De Moivre's Formula/Positive Integer Index"
] |
proofwiki-8789 | De Moivre's Formula/Positive Integer Index | Let $z \in \C$ be a complex number expressed in polar form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | {{begin-eqn}}
{{eqn | l = \paren {\cos \theta + i \sin \theta}^n
| r = \paren {e^{i \theta} }^n
| c = Euler's Formula
}}
{{eqn | r = e^{i n \theta}
| c = Exponential of Product
}}
{{eqn | r = \cos n \theta + i \sin n \theta
| c = Euler's Formula
}}
{{end-eqn}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|polar form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | {{begin-eqn}}
{{eqn | l = \paren {\cos \theta + i \sin \theta}^n
| r = \paren {e^{i \theta} }^n
| c = [[Euler's Formula]]
}}
{{eqn | r = e^{i n \theta}
| c = [[Exponential of Product]]
}}
{{eqn | r = \cos n \theta + i \sin n \theta
| c = [[Euler's Formula]]
}}
{{end-eqn}} | De Moivre's Formula/Positive Integer Index/Corollary/Proof 2 | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index/Corollary/Proof_2 | [
"De Moivre's Formula"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Euler's Formula",
"Exponential of Product",
"Euler's Formula"
] |
proofwiki-8790 | De Moivre's Formula/Positive Integer Index | Let $z \in \C$ be a complex number expressed in polar form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | Proof by induction:
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$\paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$
$\map P 1$ is the case:
:$\paren {r \paren {\cos x + i \sin x} }^1 = r^1 \paren {\map \cos {1 x} + i \, \map \sin {1 x} }$
which is triviall... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|polar form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$
$\map P 1$ is the case:
:$\paren {r \paren {\cos x + i \sin x} }^1 = ... | De Moivre's Formula/Positive Integer Index/Proof 1 | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index/Proof_1 | [
"De Moivre's Formula"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Product of Complex Numbers in Polar Form",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"De Moivre's Formula/Positive Integer Index/Proof 1",
"Product of Complex Nu... |
proofwiki-8791 | De Moivre's Formula/Positive Integer Index | Let $z \in \C$ be a complex number expressed in polar form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | From Product of Complex Numbers in Polar Form: General Result:
:$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \, \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$
Setting $z_1 = z_2 = \cdots = z_n = r \paren {\cos x + i \sin x}$ gives the result. | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|polar form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | From [[Product of Complex Numbers in Polar Form/General Result|Product of Complex Numbers in Polar Form: General Result]]:
:$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \, \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$
Setting $z_1 = z_2 = \cdots = z_... | De Moivre's Formula/Positive Integer Index/Proof 2 | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index/Proof_2 | [
"De Moivre's Formula"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Product of Complex Numbers in Polar Form/General Result"
] |
proofwiki-8792 | De Moivre's Formula/Negative Integer Index | Let $z \in \C$ be a complex number expressed in complex form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{\le 0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | Let $n = 0$.
Then:
{{begin-eqn}}
{{eqn | l = r^0 \paren {\map \cos {0 x} + i \map \sin {0 x} }
| r = 1 \times \paren {\cos 0 + i \sin 0}
| c = {{Defof|Power (Algebra)|Zeroth Power}}
}}
{{eqn | r = 1 \paren {1 + i 0}
| c = Cosine of Zero is One and Sine of Zero is Zero
}}
{{eqn | r = 1
| c =
}}
... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|complex form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall n \in \Z_{\le 0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$ | Let $n = 0$.
Then:
{{begin-eqn}}
{{eqn | l = r^0 \paren {\map \cos {0 x} + i \map \sin {0 x} }
| r = 1 \times \paren {\cos 0 + i \sin 0}
| c = {{Defof|Power (Algebra)|Zeroth Power}}
}}
{{eqn | r = 1 \paren {1 + i 0}
| c = [[Cosine of Zero is One]] and [[Sine of Zero is Zero]]
}}
{{eqn | r = 1
|... | De Moivre's Formula/Negative Integer Index | https://proofwiki.org/wiki/De_Moivre's_Formula/Negative_Integer_Index | https://proofwiki.org/wiki/De_Moivre's_Formula/Negative_Integer_Index | [
"De Moivre's Formula"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Cosine of Zero is One",
"Sine of Zero is Zero",
"De Moivre's Formula/Positive Integer Index"
] |
proofwiki-8793 | De Moivre's Formula/Rational Index | Let $z \in \C$ be a complex number expressed in complex form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall p \in \Q: \paren {r \paren {\cos x + i \sin x} }^p = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$ | Write $p = \dfrac a b$, where $a, b \in \Z$, $b \ne 0$.
Then:
{{begin-eqn}}
{{eqn | l = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }
| r = \paren {r^p \paren {\map \cos {p x} + i \, \map \sin {p x} } }^{\frac b b}
}}
{{eqn | r = \paren {r^{b p} \paren {\map \cos {b p x} + i \, \map \sin {b p x} } }^{\frac... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|complex form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall p \in \Q: \paren {r \paren {\cos x + i \sin x} }^p = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$ | Write $p = \dfrac a b$, where $a, b \in \Z$, $b \ne 0$.
Then:
{{begin-eqn}}
{{eqn | l = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }
| r = \paren {r^p \paren {\map \cos {p x} + i \, \map \sin {p x} } }^{\frac b b}
}}
{{eqn | r = \paren {r^{b p} \paren {\map \cos {b p x} + i \, \map \sin {b p x} } }^{\fra... | De Moivre's Formula/Rational Index | https://proofwiki.org/wiki/De_Moivre's_Formula/Rational_Index | https://proofwiki.org/wiki/De_Moivre's_Formula/Rational_Index | [
"De Moivre's Formula"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"De Moivre's Formula/Integer Index",
"De Moivre's Formula/Integer Index"
] |
proofwiki-8794 | Dot Product Operator is Commutative | :$\mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u$ | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf v
| r = \sum_{i \mathop = 1}^n u_i v_i
| c = {{Defof|Dot Product}}
}}
{{eqn | r = \sum_{i \mathop = 1}^n v_i u_i
| c = Real Multiplication is Commutative
}}
{{eqn | r = \mathbf v \cdot \mathbf u
| c = {{Defof|Dot Product}}
}}
{{end-eqn}}
{{qed}} | :$\mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u$ | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf v
| r = \sum_{i \mathop = 1}^n u_i v_i
| c = {{Defof|Dot Product}}
}}
{{eqn | r = \sum_{i \mathop = 1}^n v_i u_i
| c = [[Real Multiplication is Commutative]]
}}
{{eqn | r = \mathbf v \cdot \mathbf u
| c = {{Defof|Dot Product}}
}}
{{end-eqn}}
{{qe... | Dot Product Operator is Commutative/Proof 1 | https://proofwiki.org/wiki/Dot_Product_Operator_is_Commutative | https://proofwiki.org/wiki/Dot_Product_Operator_is_Commutative/Proof_1 | [
"Dot Product Operator is Commutative",
"Dot Product",
"Examples of Commutative Operations"
] | [] | [
"Real Multiplication is Commutative"
] |
proofwiki-8795 | Dot Product Operator is Commutative | :$\mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u$ | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf v
| r = \norm {\mathbf u} \norm {\mathbf v} \cos \angle \mathbf u, \mathbf v
| c = {{Defof|Dot Product|subdef = Real Euclidean Space}}
}}
{{eqn | r = \norm {\mathbf v} \norm {\mathbf u} \cos \angle \mathbf u, \mathbf v
| c = Real Multiplication is Comm... | :$\mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u$ | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf v
| r = \norm {\mathbf u} \norm {\mathbf v} \cos \angle \mathbf u, \mathbf v
| c = {{Defof|Dot Product|subdef = Real Euclidean Space}}
}}
{{eqn | r = \norm {\mathbf v} \norm {\mathbf u} \cos \angle \mathbf u, \mathbf v
| c = [[Real Multiplication is Co... | Dot Product Operator is Commutative/Proof 2 | https://proofwiki.org/wiki/Dot_Product_Operator_is_Commutative | https://proofwiki.org/wiki/Dot_Product_Operator_is_Commutative/Proof_2 | [
"Dot Product Operator is Commutative",
"Dot Product",
"Examples of Commutative Operations"
] | [] | [
"Real Multiplication is Commutative",
"Cosine Function is Even"
] |
proofwiki-8796 | Dot Product with Self is Non-Negative | Let $\mathbf u$ be a vector in the real Euclidean space $\R^n$.
Then:
:$\mathbf u \cdot \mathbf u \ge 0$
where $\cdot$ denotes the dot product operator. | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf u
| r = \sum_{i \mathop = 1}^n {u_i}^2
| c = {{Defof|Dot Product|subdef = General Context}}
}}
{{eqn | o = \ge
| r = 0
| c = as $u_i \in \R$ it follows that ${u_i}^2 \ge 0$
}}
{{end-eqn}}
{{qed}} | Let $\mathbf u$ be a [[Definition:Vector (Real Euclidean Space)|vector]] in the [[Definition:Real Euclidean Space|real Euclidean space]] $\R^n$.
Then:
:$\mathbf u \cdot \mathbf u \ge 0$
where $\cdot$ denotes the [[Definition:Dot Product|dot product operator]]. | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf u
| r = \sum_{i \mathop = 1}^n {u_i}^2
| c = {{Defof|Dot Product|subdef = General Context}}
}}
{{eqn | o = \ge
| r = 0
| c = as $u_i \in \R$ it follows that ${u_i}^2 \ge 0$
}}
{{end-eqn}}
{{qed}} | Dot Product with Self is Non-Negative/Proof 1 | https://proofwiki.org/wiki/Dot_Product_with_Self_is_Non-Negative | https://proofwiki.org/wiki/Dot_Product_with_Self_is_Non-Negative/Proof_1 | [
"Dot Product with Self is Non-Negative",
"Dot Product"
] | [
"Definition:Vector/Real Euclidean Space",
"Definition:Euclidean Space/Real",
"Definition:Dot Product"
] | [] |
proofwiki-8797 | Dot Product with Self is Non-Negative | Let $\mathbf u$ be a vector in the real Euclidean space $\R^n$.
Then:
:$\mathbf u \cdot \mathbf u \ge 0$
where $\cdot$ denotes the dot product operator. | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf u
| r = \norm {\mathbf u}^2
| c = Dot Product of Vector with Itself
}}
{{eqn | o = \ge
| r = 0
| c = Square of Real Number is Non-Negative
}}
{{end-eqn}}
{{qed}} | Let $\mathbf u$ be a [[Definition:Vector (Real Euclidean Space)|vector]] in the [[Definition:Real Euclidean Space|real Euclidean space]] $\R^n$.
Then:
:$\mathbf u \cdot \mathbf u \ge 0$
where $\cdot$ denotes the [[Definition:Dot Product|dot product operator]]. | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf u
| r = \norm {\mathbf u}^2
| c = [[Dot Product of Vector with Itself]]
}}
{{eqn | o = \ge
| r = 0
| c = [[Square of Real Number is Non-Negative]]
}}
{{end-eqn}}
{{qed}} | Dot Product with Self is Non-Negative/Proof 2 | https://proofwiki.org/wiki/Dot_Product_with_Self_is_Non-Negative | https://proofwiki.org/wiki/Dot_Product_with_Self_is_Non-Negative/Proof_2 | [
"Dot Product with Self is Non-Negative",
"Dot Product"
] | [
"Definition:Vector/Real Euclidean Space",
"Definition:Euclidean Space/Real",
"Definition:Dot Product"
] | [
"Dot Product of Vector with Itself",
"Square of Real Number is Non-Negative"
] |
proofwiki-8798 | Dot Product with Self is Zero iff Zero Vector | :$\mathbf u \cdot \mathbf u = 0 \iff \mathbf u = \mathbf 0$ | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf u
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{i \mathop = 1}^n u_i^2
| r = 0
| c = {{Defof|Dot Product|subdef = General Context}}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall i
| l = u_i
| r = 0
| c =
... | :$\mathbf u \cdot \mathbf u = 0 \iff \mathbf u = \mathbf 0$ | {{begin-eqn}}
{{eqn | l = \mathbf u \cdot \mathbf u
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{i \mathop = 1}^n u_i^2
| r = 0
| c = {{Defof|Dot Product|subdef = General Context}}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall i
| l = u_i
| r = 0
| c =
... | Dot Product with Self is Zero iff Zero Vector/Proof 1 | https://proofwiki.org/wiki/Dot_Product_with_Self_is_Zero_iff_Zero_Vector | https://proofwiki.org/wiki/Dot_Product_with_Self_is_Zero_iff_Zero_Vector/Proof_1 | [
"Dot Product with Self is Zero iff Zero Vector",
"Dot Product"
] | [] | [] |
proofwiki-8799 | Dot Product with Self is Zero iff Zero Vector | :$\mathbf u \cdot \mathbf u = 0 \iff \mathbf u = \mathbf 0$ | Let $\mathbf u \cdot \mathbf u = 0$.
Then:
{{begin-eqn}}
{{eqn | l = 0
| r = \norm {\mathbf u }^2 \cos \angle \mathbf u, \mathbf u
| c = {{Defof|Dot Product|subdef = Real Euclidean Space}}
}}
{{eqn | r = \norm {\mathbf u}^2 \cos 0
| c =
}}
{{eqn | r = \norm {\mathbf u}^2
| c =
}}
{{end-eqn}}
The... | :$\mathbf u \cdot \mathbf u = 0 \iff \mathbf u = \mathbf 0$ | Let $\mathbf u \cdot \mathbf u = 0$.
Then:
{{begin-eqn}}
{{eqn | l = 0
| r = \norm {\mathbf u }^2 \cos \angle \mathbf u, \mathbf u
| c = {{Defof|Dot Product|subdef = Real Euclidean Space}}
}}
{{eqn | r = \norm {\mathbf u}^2 \cos 0
| c =
}}
{{eqn | r = \norm {\mathbf u}^2
| c =
}}
{{end-eqn}}
T... | Dot Product with Self is Zero iff Zero Vector/Proof 2 | https://proofwiki.org/wiki/Dot_Product_with_Self_is_Zero_iff_Zero_Vector | https://proofwiki.org/wiki/Dot_Product_with_Self_is_Zero_iff_Zero_Vector/Proof_2 | [
"Dot Product with Self is Zero iff Zero Vector",
"Dot Product"
] | [] | [] |
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