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-8800 | Vector Cross Product is Anticommutative/Complex | :$\forall z_1, z_2 \in \C: z_1 \times z_2 = -\paren {z_2 \times z_1}$ | Let:
: $z_1 := x_1 + i y_1, z_2 = x_2 + i y_2$
Then:
{{begin-eqn}}
{{eqn | l = z_1 \times z_2
| r = x_1 y_2 - y_1 x_2
| c = {{Defof|Vector Cross Product|subdef = Complex|index = 1|Complex Cross Product}}
}}
{{eqn | r = -\paren {x_2 y_1 - y_2 x_1}
| c = Real Addition is Commutative and Real Multiplicat... | :$\forall z_1, z_2 \in \C: z_1 \times z_2 = -\paren {z_2 \times z_1}$ | Let:
: $z_1 := x_1 + i y_1, z_2 = x_2 + i y_2$
Then:
{{begin-eqn}}
{{eqn | l = z_1 \times z_2
| r = x_1 y_2 - y_1 x_2
| c = {{Defof|Vector Cross Product|subdef = Complex|index = 1|Complex Cross Product}}
}}
{{eqn | r = -\paren {x_2 y_1 - y_2 x_1}
| c = [[Real Addition is Commutative]] and [[Real Mult... | Vector Cross Product is Anticommutative/Complex | https://proofwiki.org/wiki/Vector_Cross_Product_is_Anticommutative/Complex | https://proofwiki.org/wiki/Vector_Cross_Product_is_Anticommutative/Complex | [
"Vector Cross Product is Anticommutative",
"Complex Cross Product"
] | [] | [
"Real Addition is Commutative",
"Real Multiplication is Commutative"
] |
proofwiki-8801 | Polar Form of Complex Conjugate | Let $z := r \paren {\cos \theta + i \sin \theta} \in \C$ be a complex number expressed in polar form.
Then:
:$\overline z = r \paren {\cos \theta - i \sin \theta}$
where $\overline z$ denotes the complex conjugate of $z$. | {{begin-eqn}}
{{eqn | l = z
| r = r \paren {\cos \theta + i \sin \theta}
| c =
}}
{{eqn | r = \paren {r \cos \theta} + i \paren {r \sin \theta}
| c =
}}
{{eqn | ll= \leadsto
| l = \overline z
| r = \paren {r \cos \theta} - i \paren {r \sin \theta}
| c = {{Defof|Complex Conjugate}}
... | Let $z := r \paren {\cos \theta + i \sin \theta} \in \C$ be a [[Definition:Polar Form of Complex Number|complex number expressed in polar form]].
Then:
:$\overline z = r \paren {\cos \theta - i \sin \theta}$
where $\overline z$ denotes the [[Definition:Complex Conjugate|complex conjugate]] of $z$. | {{begin-eqn}}
{{eqn | l = z
| r = r \paren {\cos \theta + i \sin \theta}
| c =
}}
{{eqn | r = \paren {r \cos \theta} + i \paren {r \sin \theta}
| c =
}}
{{eqn | ll= \leadsto
| l = \overline z
| r = \paren {r \cos \theta} - i \paren {r \sin \theta}
| c = {{Defof|Complex Conjugate}}
... | Polar Form of Complex Conjugate | https://proofwiki.org/wiki/Polar_Form_of_Complex_Conjugate | https://proofwiki.org/wiki/Polar_Form_of_Complex_Conjugate | [
"Complex Conjugates",
"Polar Form of Complex Number"
] | [
"Definition:Complex Number/Polar Form",
"Definition:Complex Conjugate"
] | [] |
proofwiki-8802 | Exponential Form of Complex Conjugate | Let $z := r e^{i \theta} \in \C$ be a complex number expressed in exponential form.
Then:
:$\overline z = r e^{-i \theta}$
where $\overline z$ denotes the complex conjugate of $z$. | {{begin-eqn}}
{{eqn | l = z
| r = r e^{i \theta}
| c =
}}
{{eqn | r = r \paren {\cos \theta + i \sin \theta}
| c = Euler's Formula
}}
{{eqn | ll= \leadsto
| l = \overline z
| r = r \paren {\cos \theta - i \sin \theta}
| c = Polar Form of Complex Conjugate
}}
{{eqn | r = r e^{-i \the... | Let $z := r e^{i \theta} \in \C$ be a [[Definition:Exponential Form of Complex Number|complex number expressed in exponential form]].
Then:
:$\overline z = r e^{-i \theta}$
where $\overline z$ denotes the [[Definition:Complex Conjugate|complex conjugate]] of $z$. | {{begin-eqn}}
{{eqn | l = z
| r = r e^{i \theta}
| c =
}}
{{eqn | r = r \paren {\cos \theta + i \sin \theta}
| c = [[Euler's Formula]]
}}
{{eqn | ll= \leadsto
| l = \overline z
| r = r \paren {\cos \theta - i \sin \theta}
| c = [[Polar Form of Complex Conjugate]]
}}
{{eqn | r = r e^... | Exponential Form of Complex Conjugate | https://proofwiki.org/wiki/Exponential_Form_of_Complex_Conjugate | https://proofwiki.org/wiki/Exponential_Form_of_Complex_Conjugate | [
"Complex Conjugates"
] | [
"Definition:Complex Number/Polar Form/Exponential Form",
"Definition:Complex Conjugate"
] | [
"Euler's Formula",
"Polar Form of Complex Conjugate"
] |
proofwiki-8803 | Complex Dot Product in Exponential Form | Let $z_1 := r_1 e^{i \theta_1}, z_2 := r_2 e^{i \theta_2} \in \C$ be complex numbers expressed in exponential form.
Then:
:$z_1 \circ z_2 = r_1 r_2 \, \map \cos {\theta_2 - \theta_1}$
where $z_1 \circ z_2$ denotes the dot product of $z_1$ and $z_2$. | {{begin-eqn}}
{{eqn | l = z_1 \circ z_2
| r = \map \Re {\overline {z_1} z_2}
| c = {{Defof|Dot Product|subdef = Complex|index = 3}}
}}
{{eqn | r = \map \Re {r_1 e^{-i \theta_1} r_2 e^{i \theta_2} }
| c = Exponential Form of Complex Conjugate
}}
{{eqn | r = \map \Re {r_1 r_2 e^{i \paren {\theta_2 - \th... | Let $z_1 := r_1 e^{i \theta_1}, z_2 := r_2 e^{i \theta_2} \in \C$ be [[Definition:Exponential Form of Complex Number|complex numbers expressed in exponential form]].
Then:
:$z_1 \circ z_2 = r_1 r_2 \, \map \cos {\theta_2 - \theta_1}$
where $z_1 \circ z_2$ denotes the [[Definition:Complex Dot Product|dot product]] of ... | {{begin-eqn}}
{{eqn | l = z_1 \circ z_2
| r = \map \Re {\overline {z_1} z_2}
| c = {{Defof|Dot Product|subdef = Complex|index = 3}}
}}
{{eqn | r = \map \Re {r_1 e^{-i \theta_1} r_2 e^{i \theta_2} }
| c = [[Exponential Form of Complex Conjugate]]
}}
{{eqn | r = \map \Re {r_1 r_2 e^{i \paren {\theta_2 -... | Complex Dot Product in Exponential Form | https://proofwiki.org/wiki/Complex_Dot_Product_in_Exponential_Form | https://proofwiki.org/wiki/Complex_Dot_Product_in_Exponential_Form | [
"Complex Dot Product"
] | [
"Definition:Complex Number/Polar Form/Exponential Form",
"Definition:Dot Product/Complex"
] | [
"Exponential Form of Complex Conjugate",
"Product of Complex Numbers in Exponential Form"
] |
proofwiki-8804 | Complex Cross Product in Exponential Form | Let $z_1 := r_1 e^{i \theta_1}, z_2 := r_2 e^{i \theta_2} \in \C$ be complex numbers expressed in exponential form.
Then:
:$z_1 \times z_2 = r_1 r_2 \map \sin {\theta_2 - \theta_1}$
where $z_1 \times z_2$ denotes the dot product of $z_1$ and $z_2$. | {{begin-eqn}}
{{eqn | l = z_1 \times z_2
| r = \map \Im {\overline {z_1} z_2}
| c = {{Defof|Vector Cross Product|subdef = Complex|index = 3|Complex Cross Product}}
}}
{{eqn | r = \map \Im {r_1 e^{-i \theta_1} r_2 e^{i \theta_2} }
| c = Exponential Form of Complex Conjugate
}}
{{eqn | r = \map \Im {r_1... | Let $z_1 := r_1 e^{i \theta_1}, z_2 := r_2 e^{i \theta_2} \in \C$ be [[Definition:Exponential Form of Complex Number|complex numbers expressed in exponential form]].
Then:
:$z_1 \times z_2 = r_1 r_2 \map \sin {\theta_2 - \theta_1}$
where $z_1 \times z_2$ denotes the [[Definition:Complex Dot Product|dot product]] of $... | {{begin-eqn}}
{{eqn | l = z_1 \times z_2
| r = \map \Im {\overline {z_1} z_2}
| c = {{Defof|Vector Cross Product|subdef = Complex|index = 3|Complex Cross Product}}
}}
{{eqn | r = \map \Im {r_1 e^{-i \theta_1} r_2 e^{i \theta_2} }
| c = [[Exponential Form of Complex Conjugate]]
}}
{{eqn | r = \map \Im ... | Complex Cross Product in Exponential Form | https://proofwiki.org/wiki/Complex_Cross_Product_in_Exponential_Form | https://proofwiki.org/wiki/Complex_Cross_Product_in_Exponential_Form | [
"Complex Cross Product"
] | [
"Definition:Complex Number/Polar Form/Exponential Form",
"Definition:Dot Product/Complex"
] | [
"Exponential Form of Complex Conjugate",
"Product of Complex Numbers in Exponential Form"
] |
proofwiki-8805 | Dot Product Distributes over Addition | :$\paren {\mathbf u + \mathbf v} \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$ | {{begin-eqn}}
{{eqn | l = \left({\mathbf u + \mathbf v}\right) \cdot \mathbf w
| r = \sum_{i \mathop = 1}^n \left({u_i + v_i}\right) w_i
| c = {{Defof|Vector Sum}} and {{Defof|Dot Product}}
}}
{{eqn | r = \sum_{i \mathop = 1}^n \left({u_i w_i + v_i w_i}\right)
| c = Real Multiplication Distributes ove... | :$\paren {\mathbf u + \mathbf v} \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$ | {{begin-eqn}}
{{eqn | l = \left({\mathbf u + \mathbf v}\right) \cdot \mathbf w
| r = \sum_{i \mathop = 1}^n \left({u_i + v_i}\right) w_i
| c = {{Defof|Vector Sum}} and {{Defof|Dot Product}}
}}
{{eqn | r = \sum_{i \mathop = 1}^n \left({u_i w_i + v_i w_i}\right)
| c = [[Real Multiplication Distributes o... | Dot Product Distributes over Addition/Proof 1 | https://proofwiki.org/wiki/Dot_Product_Distributes_over_Addition | https://proofwiki.org/wiki/Dot_Product_Distributes_over_Addition/Proof_1 | [
"Dot Product Distributes over Addition",
"Dot Product",
"Vector Addition",
"Examples of Distributive Operations"
] | [] | [
"Real Multiplication Distributes over Addition"
] |
proofwiki-8806 | Dot Product Distributes over Addition | :$\paren {\mathbf u + \mathbf v} \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$ | Let the vectors $\mathbf u$, $\mathbf v$ and $\mathbf w$ be embedded in a Cartesian $3$-space.
It is noted that $\mathbf u$, $\mathbf v$ and $\mathbf w$ are not necessarily coplanar.
:520px
Let instances of $\mathbf u$ and $\mathbf w$ be selected so their initial points are at some point $O$.
Let an instance of $\mathb... | :$\paren {\mathbf u + \mathbf v} \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$ | Let the [[Definition:Vector|vectors]] $\mathbf u$, $\mathbf v$ and $\mathbf w$ be embedded in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
It is noted that $\mathbf u$, $\mathbf v$ and $\mathbf w$ are not necessarily [[Definition:Coplanar Vectors|coplanar]].
:[[File:Dot-product-distributes-over-addition.pn... | Dot Product Distributes over Addition/Proof 2 | https://proofwiki.org/wiki/Dot_Product_Distributes_over_Addition | https://proofwiki.org/wiki/Dot_Product_Distributes_over_Addition/Proof_2 | [
"Dot Product Distributes over Addition",
"Dot Product",
"Vector Addition",
"Examples of Distributive Operations"
] | [] | [
"Definition:Vector",
"Definition:Cartesian 3-Space",
"Definition:Coplanar Vectors",
"File:Dot-product-distributes-over-addition.png",
"Definition:Initial Point of Vector",
"Definition:Initial Point of Vector",
"Definition:Terminal Point of Vector",
"Definition:Terminal Point of Vector",
"Definition:... |
proofwiki-8807 | Dot Product Distributes over Addition | :$\paren {\mathbf u + \mathbf v} \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$ | From Dot Product Operator is Bilinear:
:$\paren {c \mathbf u + \mathbf v} \cdot \mathbf w = c \paren {\mathbf u \cdot \mathbf w} + \paren {\mathbf v \cdot \mathbf w}$
Setting $c = 1$ yields the result.
{{qed}} | :$\paren {\mathbf u + \mathbf v} \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$ | From [[Dot Product Operator is Bilinear]]:
:$\paren {c \mathbf u + \mathbf v} \cdot \mathbf w = c \paren {\mathbf u \cdot \mathbf w} + \paren {\mathbf v \cdot \mathbf w}$
Setting $c = 1$ yields the result.
{{qed}} | Dot Product Distributes over Addition/Proof 3 | https://proofwiki.org/wiki/Dot_Product_Distributes_over_Addition | https://proofwiki.org/wiki/Dot_Product_Distributes_over_Addition/Proof_3 | [
"Dot Product Distributes over Addition",
"Dot Product",
"Vector Addition",
"Examples of Distributive Operations"
] | [] | [
"Dot Product Operator is Bilinear"
] |
proofwiki-8808 | Dot Product Operator is Bilinear | :$\paren {c \mathbf u + \mathbf v} \cdot \mathbf w = c \paren {\mathbf u \cdot \mathbf w} + \paren {\mathbf v \cdot \mathbf w}$
:$\mathbf u \cdot \paren {c \mathbf v + \mathbf w} = c \paren {\mathbf u \cdot \mathbf v} + \paren {\mathbf u \cdot \mathbf w}$ | Let $V$ have $n$ dimensions. | :$\paren {c \mathbf u + \mathbf v} \cdot \mathbf w = c \paren {\mathbf u \cdot \mathbf w} + \paren {\mathbf v \cdot \mathbf w}$
:$\mathbf u \cdot \paren {c \mathbf v + \mathbf w} = c \paren {\mathbf u \cdot \mathbf v} + \paren {\mathbf u \cdot \mathbf w}$ | Let $V$ have [[Definition:Dimension of Vector Space|$n$ dimensions]]. | Dot Product Operator is Bilinear | https://proofwiki.org/wiki/Dot_Product_Operator_is_Bilinear | https://proofwiki.org/wiki/Dot_Product_Operator_is_Bilinear | [
"Dot Product",
"Bilinear Mappings"
] | [] | [
"Definition:Dimension of Vector Space"
] |
proofwiki-8809 | Dot Product Associates with Scalar Multiplication | :$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$ | {{begin-eqn}}
{{eqn | l = \left({c \mathbf u}\right) \cdot \mathbf v
| r = \sum_{i \mathop = 1}^n \left({c u_i}\right) v_i
| c = {{Defof|Dot Product}}
}}
{{eqn | r = \sum_{i \mathop = 1}^n c \left({ u_i v_i }\right)
| c = Real Multiplication is Associative
}}
{{eqn | r = c \sum_{i \mathop = 1}^n u_i v... | :$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$ | {{begin-eqn}}
{{eqn | l = \left({c \mathbf u}\right) \cdot \mathbf v
| r = \sum_{i \mathop = 1}^n \left({c u_i}\right) v_i
| c = {{Defof|Dot Product}}
}}
{{eqn | r = \sum_{i \mathop = 1}^n c \left({ u_i v_i }\right)
| c = [[Real Multiplication is Associative]]
}}
{{eqn | r = c \sum_{i \mathop = 1}^n u... | Dot Product Associates with Scalar Multiplication/Proof 1 | https://proofwiki.org/wiki/Dot_Product_Associates_with_Scalar_Multiplication | https://proofwiki.org/wiki/Dot_Product_Associates_with_Scalar_Multiplication/Proof_1 | [
"Dot Product Associates with Scalar Multiplication",
"Dot Product"
] | [] | [
"Real Multiplication is Associative",
"Real Multiplication Distributes over Addition"
] |
proofwiki-8810 | Dot Product Associates with Scalar Multiplication | :$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$ | {{begin-eqn}}
{{eqn | l = \paren {c \mathbf u} \cdot \mathbf v
| r = \norm {c \mathbf u} \norm {\mathbf v} \cos \angle c \mathbf u, \mathbf v
| c = {{Defof|Dot Product}}
}}
{{eqn | r = \sqrt {\sum_{i \mathop = 1}^n \paren {c u_i}^2} \norm {\mathbf v} \cos \angle c \mathbf u, \mathbf v
| c = {{Defof|Ve... | :$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$ | {{begin-eqn}}
{{eqn | l = \paren {c \mathbf u} \cdot \mathbf v
| r = \norm {c \mathbf u} \norm {\mathbf v} \cos \angle c \mathbf u, \mathbf v
| c = {{Defof|Dot Product}}
}}
{{eqn | r = \sqrt {\sum_{i \mathop = 1}^n \paren {c u_i}^2} \norm {\mathbf v} \cos \angle c \mathbf u, \mathbf v
| c = {{Defof|Ve... | Dot Product Associates with Scalar Multiplication/Proof 2 | https://proofwiki.org/wiki/Dot_Product_Associates_with_Scalar_Multiplication | https://proofwiki.org/wiki/Dot_Product_Associates_with_Scalar_Multiplication/Proof_2 | [
"Dot Product Associates with Scalar Multiplication",
"Dot Product"
] | [] | [] |
proofwiki-8811 | Dot Product Associates with Scalar Multiplication | :$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$ | From Dot Product Operator is Bilinear:
:$\left({c \mathbf u + \mathbf v}\right) \cdot \mathbf w = c \left({\mathbf u \cdot \mathbf w}\right) + \left({\mathbf v \cdot \mathbf w}\right)$
Setting $\mathbf v = 0$ and renaming $\mathbf w$ yields the result.
{{qed}} | :$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$ | From [[Dot Product Operator is Bilinear]]:
:$\left({c \mathbf u + \mathbf v}\right) \cdot \mathbf w = c \left({\mathbf u \cdot \mathbf w}\right) + \left({\mathbf v \cdot \mathbf w}\right)$
Setting $\mathbf v = 0$ and renaming $\mathbf w$ yields the result.
{{qed}} | Dot Product Associates with Scalar Multiplication/Proof 3 | https://proofwiki.org/wiki/Dot_Product_Associates_with_Scalar_Multiplication | https://proofwiki.org/wiki/Dot_Product_Associates_with_Scalar_Multiplication/Proof_3 | [
"Dot Product Associates with Scalar Multiplication",
"Dot Product"
] | [] | [
"Dot Product Operator is Bilinear"
] |
proofwiki-8812 | Semantic Tableau Algorithm Terminates | Let $\mathbf A$ be a WFF of propositional logic.
Then the Semantic Tableau Algorithm for $\mathbf A$ terminates.
Each leaf node of the resulting semantic tableau is marked. | Let $t$ be an unmarked leaf of the semantic tableau $T$ being constructed.
Let $\map b t$ be the number of binary logical connectives occurring in its label $\map U t$.
Let $\map n t$ be the number of negations occurring in $\map U t$.
Let $\map i t$ be the number of biconditionals and exclusive ors occurring in $\map ... | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Then the [[Semantic Tableau Algorithm]] for $\mathbf A$ terminates.
Each [[Definition:Leaf Node|leaf node]] of the resulting [[Definition:Semantic Tableau|semantic tableau]] is [[Definition:Marked Leaf|marked]]. | Let $t$ be an [[Definition:Unmarked Leaf|unmarked leaf]] of the [[Definition:Semantic Tableau|semantic tableau]] $T$ being constructed.
Let $\map b t$ be the number of [[Definition:Binary Logical Connective|binary logical connectives]] occurring in its label $\map U t$.
Let $\map n t$ be the number of [[Definition:Lo... | Semantic Tableau Algorithm Terminates | https://proofwiki.org/wiki/Semantic_Tableau_Algorithm_Terminates | https://proofwiki.org/wiki/Semantic_Tableau_Algorithm_Terminates | [
"Propositional Logic"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Semantic Tableau Algorithm",
"Definition:Tree (Graph Theory)/Leaf Node",
"Definition:Semantic Tableau",
"Definition:Marked Leaf"
] | [
"Definition:Marked Leaf",
"Definition:Semantic Tableau",
"Definition:Logical Connective/Binary",
"Definition:Logical Not",
"Definition:Biconditional",
"Definition:Exclusive Or",
"Definition:Tree (Graph Theory)/Leaf Node",
"Semantic Tableau Algorithm",
"Definition:Alpha-Formula",
"Definition:Alpha-... |
proofwiki-8813 | Complex Cross Product Distributes over Addition | Let $z_1, z_2, z_3 \in \C$ be complex numbers.
Then:
:$z_1 \times \paren {z_2 + z_3} = z_1 \times z_2 + z_1 \times z_3$
where $\times$ denotes cross product. | Let:
:$z_1 = x_1 + i y_1$
:$z_2 = x_2 + i y_2$
:$z_3 = x_3 + i y_3$
Then:
{{begin-eqn}}
{{eqn | l = z_1 \times \paren {z_2 + z_3}
| r = \paren {x_1 + i y_1} \times \paren {\paren {x_2 + i y_2} + \paren {x_3 + i y_3} }
| c =
}}
{{eqn | r = \paren {x_1 + i y_1} \times \paren {\paren {x_2 + x_3} + i \paren {y... | Let $z_1, z_2, z_3 \in \C$ be [[Definition:Complex Number|complex numbers]].
Then:
:$z_1 \times \paren {z_2 + z_3} = z_1 \times z_2 + z_1 \times z_3$
where $\times$ denotes [[Definition:Complex Cross Product|cross product]]. | Let:
:$z_1 = x_1 + i y_1$
:$z_2 = x_2 + i y_2$
:$z_3 = x_3 + i y_3$
Then:
{{begin-eqn}}
{{eqn | l = z_1 \times \paren {z_2 + z_3}
| r = \paren {x_1 + i y_1} \times \paren {\paren {x_2 + i y_2} + \paren {x_3 + i y_3} }
| c =
}}
{{eqn | r = \paren {x_1 + i y_1} \times \paren {\paren {x_2 + x_3} + i \paren {... | Complex Cross Product Distributes over Addition | https://proofwiki.org/wiki/Complex_Cross_Product_Distributes_over_Addition | https://proofwiki.org/wiki/Complex_Cross_Product_Distributes_over_Addition | [
"Complex Cross Product",
"Complex Addition"
] | [
"Definition:Complex Number",
"Definition:Vector Cross Product/Complex"
] | [
"Real Multiplication Distributes over Addition",
"Real Addition is Commutative"
] |
proofwiki-8814 | Complement of Closed Set in Complex Plane is Open | Let $S \subseteq \C$ be a closed subset of the complex plane $\C$.
Then the complement of $S$ in $\C$ is open. | {{begin-eqn}}
{{eqn | o =
| c = $S$ is closed
}}
{{eqn | o = \leadsto
| c = $\forall z \in \C: z$ is a limit point of $S \implies z \in S$
| cc= {{Defof|Closed Set (Complex Analysis)|Closed Set}}
}}
{{eqn | o = \leadsto
| c = $\forall z \in \C: z \notin S \implies z$ is not a limit point of $S$... | Let $S \subseteq \C$ be a [[Definition:Closed Set (Complex Analysis)|closed subset]] of the [[Definition:Complex Plane|complex plane]] $\C$.
Then the [[Definition:Relative Complement|complement]] of $S$ in $\C$ is [[Definition:Open Set (Complex Analysis)|open]]. | {{begin-eqn}}
{{eqn | o =
| c = $S$ is [[Definition:Closed Set (Complex Analysis)|closed]]
}}
{{eqn | o = \leadsto
| c = $\forall z \in \C: z$ is a [[Definition:Limit Point (Complex Analysis)|limit point]] of $S \implies z \in S$
| cc= {{Defof|Closed Set (Complex Analysis)|Closed Set}}
}}
{{eqn | o =... | Complement of Closed Set in Complex Plane is Open | https://proofwiki.org/wiki/Complement_of_Closed_Set_in_Complex_Plane_is_Open | https://proofwiki.org/wiki/Complement_of_Closed_Set_in_Complex_Plane_is_Open | [
"Complex Analysis"
] | [
"Definition:Closed Set/Complex Analysis",
"Definition:Complex Number/Complex Plane",
"Definition:Relative Complement",
"Definition:Open Set/Complex Analysis"
] | [
"Definition:Closed Set/Complex Analysis",
"Definition:Limit Point/Complex Analysis",
"Definition:Limit Point/Complex Analysis",
"Rule of Transposition",
"Definition:Limit Point/Complex Analysis",
"Definition:Deleted Neighborhood/Complex Analysis",
"Definition:Open Set/Complex Analysis"
] |
proofwiki-8815 | Complement of Open Set in Complex Plane is Closed | Let $S \subseteq \C$ be an open subset of the complex plane $\C$.
Then the complement of $S$ in $\C$ is closed. | {{begin-eqn}}
{{eqn | o =
| c = $S$ is open in $\C$
| cc=
}}
{{eqn | o = \leadsto
| c = $\forall z \in S$: there exists a deleted $\epsilon$-neighborhood $\map {N_\epsilon} z \setminus \set z$ of $z$ entirely in $S$
| cc= {{Defof|Open Set (Complex Analysis)|Open Set}}
}}
{{eqn | o = \leadsto
... | Let $S \subseteq \C$ be an [[Definition:Open Set (Complex Analysis)|open subset]] of the [[Definition:Complex Plane|complex plane]] $\C$.
Then the [[Definition:Relative Complement|complement]] of $S$ in $\C$ is [[Definition:Closed Set (Complex Analysis)|closed]]. | {{begin-eqn}}
{{eqn | o =
| c = $S$ is [[Definition:Open Set (Complex Analysis)|open]] in $\C$
| cc=
}}
{{eqn | o = \leadsto
| c = $\forall z \in S$: there exists a [[Definition:Deleted Neighborhood (Complex Analysis)|deleted $\epsilon$-neighborhood]] $\map {N_\epsilon} z \setminus \set z$ of $z$ enti... | Complement of Open Set in Complex Plane is Closed | https://proofwiki.org/wiki/Complement_of_Open_Set_in_Complex_Plane_is_Closed | https://proofwiki.org/wiki/Complement_of_Open_Set_in_Complex_Plane_is_Closed | [
"Complex Analysis"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Complex Number/Complex Plane",
"Definition:Relative Complement",
"Definition:Closed Set/Complex Analysis"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Deleted Neighborhood/Complex Analysis",
"Definition:Limit Point/Complex Analysis",
"Definition:Limit Point/Complex Analysis",
"Rule of Transposition",
"Definition:Limit Point/Complex Analysis",
"Definition:Closed Set/Complex Analysis"
] |
proofwiki-8816 | Finite Union of Open Sets in Complex Plane is Open | Let $S_1, S_2, \ldots, S_n$ be open sets of $\C$.
Then $\ds \bigcup_{k \mathop = 1}^n S_k$ is an open set of $\C$. | Let $\ds z \in \bigcup_{k \mathop = 1}^n S_k$.
Then by definition of finite union:
:$\exists k \in \set {1, 2, \ldots, n}: z \in S_k$
By definition of open set:
:$\exists \epsilon \in \R_{>0}: \map {N_\epsilon} z \subseteq S_k$
where $\map {N_\epsilon} z$ is the $\epsilon$-neighborhood of $z$ for $\epsilon$.
By Set is ... | Let $S_1, S_2, \ldots, S_n$ be [[Definition:Open Set (Complex Analysis)|open sets]] of $\C$.
Then $\ds \bigcup_{k \mathop = 1}^n S_k$ is an [[Definition:Open Set (Complex Analysis)|open set]] of $\C$. | Let $\ds z \in \bigcup_{k \mathop = 1}^n S_k$.
Then by definition of [[Definition:Finite Union|finite union]]:
:$\exists k \in \set {1, 2, \ldots, n}: z \in S_k$
By definition of [[Definition:Open Set (Complex Analysis)|open set]]:
:$\exists \epsilon \in \R_{>0}: \map {N_\epsilon} z \subseteq S_k$
where $\map {N_\eps... | Finite Union of Open Sets in Complex Plane is Open | https://proofwiki.org/wiki/Finite_Union_of_Open_Sets_in_Complex_Plane_is_Open | https://proofwiki.org/wiki/Finite_Union_of_Open_Sets_in_Complex_Plane_is_Open | [
"Complex Plane",
"Set Union"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Open Set/Complex Analysis"
] | [
"Definition:Set Union/Finite Union",
"Definition:Open Set/Complex Analysis",
"Definition:Neighborhood (Complex Analysis)",
"Set is Subset of Union",
"Subset Relation is Transitive",
"Definition:Open Set/Complex Analysis"
] |
proofwiki-8817 | Limit Point of Set in Complex Plane not Element is Boundary Point | Let $S \subseteq \C$ be a subset of the complex plane.
Let $z \in \C$ be a limit point of $S$ such that $z \notin S$.
Then $z$ is a boundary point of $S$. | Let $z \in \C$ be a limit point of $S$ such that $z \notin S$.
Suppose $z$ is not a boundary point of $S$.
Then there exists an $\epsilon$-neighborhood $\map {N_\epsilon} z$ of $z$ such that either:
:$(1): \quad$ All elements of $\map {N_\epsilon} z$ are in $S$
or
:$(2): \quad$ All elements of $\map {N_\epsilon} z$ are... | Let $S \subseteq \C$ be a [[Definition:Subset|subset]] of the [[Definition:Complex Plane|complex plane]].
Let $z \in \C$ be a [[Definition:Limit Point (Complex Analysis)|limit point]] of $S$ such that $z \notin S$.
Then $z$ is a [[Definition:Boundary Point (Complex Analysis)|boundary point]] of $S$. | Let $z \in \C$ be a [[Definition:Limit Point (Complex Analysis)|limit point]] of $S$ such that $z \notin S$.
Suppose $z$ is not a [[Definition:Boundary Point (Complex Analysis)|boundary point]] of $S$.
Then there exists an [[Definition:Neighborhood (Complex Analysis)|$\epsilon$-neighborhood]] $\map {N_\epsilon} z$ of... | Limit Point of Set in Complex Plane not Element is Boundary Point | https://proofwiki.org/wiki/Limit_Point_of_Set_in_Complex_Plane_not_Element_is_Boundary_Point | https://proofwiki.org/wiki/Limit_Point_of_Set_in_Complex_Plane_not_Element_is_Boundary_Point | [
"Complex Analysis"
] | [
"Definition:Subset",
"Definition:Complex Number/Complex Plane",
"Definition:Limit Point/Complex Analysis",
"Definition:Boundary Point (Complex Analysis)"
] | [
"Definition:Limit Point/Complex Analysis",
"Definition:Boundary Point (Complex Analysis)",
"Definition:Neighborhood (Complex Analysis)",
"Definition:Element",
"Definition:Element",
"Proof by Contradiction",
"Definition:Limit Point/Complex Analysis",
"Proof by Contradiction"
] |
proofwiki-8818 | Condition for Quartic with Real Coefficients to have Wholly Imaginary Root | Let $Q$ be the quartic equation:
:$(1): \quad z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$
such that all of $a_1, a_2, a_3, a_4$ are real numbers.
Then $Q$ has a root which is wholly imaginary {{iff}}:
:$\text {(a)}: \quad a_3^2 + a_1^2 a_4 = a_1 a_2 a_3$
:$\text {(b)}: \quad a_1 a_3 > 0$ | === Necessary Condition ===
We have:
{{begin-eqn}}
{{eqn | l = a_3^2 + a_1^2 a_4
| r = a_1 a_2 a_3
| c =
}}
{{eqn | ll= \leadsto
| l = a_4
| r = \frac {a_3} {a_1} \paren {a_2 - \frac {a_3} {a_1} }
| c =
}}
{{end-eqn}}
This leads to the factorisation of $(1)$:
{{begin-eqn}}
{{eqn | l = z^... | Let $Q$ be the [[Definition:Quartic Equation|quartic equation]]:
:$(1): \quad z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$
such that all of $a_1, a_2, a_3, a_4$ are [[Definition:Real Number|real numbers]].
Then $Q$ has a [[Definition:Root of Polynomial|root]] which is [[Definition:Wholly Imaginary|wholly imaginary]] {{... | === Necessary Condition ===
We have:
{{begin-eqn}}
{{eqn | l = a_3^2 + a_1^2 a_4
| r = a_1 a_2 a_3
| c =
}}
{{eqn | ll= \leadsto
| l = a_4
| r = \frac {a_3} {a_1} \paren {a_2 - \frac {a_3} {a_1} }
| c =
}}
{{end-eqn}}
This leads to the factorisation of $(1)$:
{{begin-eqn}}
{{eqn | l ... | Condition for Quartic with Real Coefficients to have Wholly Imaginary Root | https://proofwiki.org/wiki/Condition_for_Quartic_with_Real_Coefficients_to_have_Wholly_Imaginary_Root | https://proofwiki.org/wiki/Condition_for_Quartic_with_Real_Coefficients_to_have_Wholly_Imaginary_Root | [
"Quartic Equations"
] | [
"Definition:Quartic Equation",
"Definition:Real Number",
"Definition:Root of Polynomial",
"Definition:Complex Number/Wholly Imaginary"
] | [
"Solution to Quadratic Equation",
"Definition:Complex Number/Real Part",
"Definition:Root of Polynomial",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Root of Polynomial",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Root of Polyn... |
proofwiki-8819 | Soundness and Completeness of Semantic Tableaux | Let $\mathbf A$ be a WFF of propositional logic.
Let $T$ be a completed semantic tableau for $\mathbf A$.
Then $\mathbf A$ is unsatisfiable {{iff}} $T$ is closed. | The two directions of this theorem are respectively addressed on:
:Soundness Theorem for Semantic Tableaux
:Completeness Theorem for Semantic Tableaux
{{qed}} | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Let $T$ be a [[Definition:Completed Tableau|completed semantic tableau]] for $\mathbf A$.
Then $\mathbf A$ is [[Definition:Unsatisfiable (Boolean Interpretations)|unsatisfiable]] {{iff}} $T$ is [[Definition:Closed Tableau|clos... | The two directions of this theorem are respectively addressed on:
:[[Soundness Theorem for Semantic Tableaux]]
:[[Completeness Theorem for Semantic Tableaux]]
{{qed}} | Soundness and Completeness of Semantic Tableaux | https://proofwiki.org/wiki/Soundness_and_Completeness_of_Semantic_Tableaux | https://proofwiki.org/wiki/Soundness_and_Completeness_of_Semantic_Tableaux | [
"Propositional Logic"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Semantic Tableau/Completed",
"Definition:Unsatisfiable/Boolean Interpretations",
"Definition:Semantic Tableau/Closed"
] | [
"Soundness Theorem for Semantic Tableaux",
"Completeness Theorem for Semantic Tableaux"
] |
proofwiki-8820 | Semantic Tableau Algorithm is Decision Procedure for Tautologies | The Semantic Tableau Algorithm is a decision procedure for tautologies. | Let $\mathbf A$ be a WFF of propositional logic.
The Semantic Tableau Algorithm applied to $\neg \mathbf A$ yields a completed tableau for $\neg \mathbf A$.
By {{Corollary|Soundness and Completeness of Semantic Tableaux|2}}, this completed tableau decides if $\mathbf A$ is a tautology.
{{qed}} | The [[Semantic Tableau Algorithm]] is a [[Definition:Decision Procedure for Tautologies|decision procedure for tautologies]]. | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
The [[Semantic Tableau Algorithm]] applied to $\neg \mathbf A$ yields a [[Definition:Completed Tableau|completed tableau]] for $\neg \mathbf A$.
By {{Corollary|Soundness and Completeness of Semantic Tableaux|2}}, this [[Definit... | Semantic Tableau Algorithm is Decision Procedure for Tautologies | https://proofwiki.org/wiki/Semantic_Tableau_Algorithm_is_Decision_Procedure_for_Tautologies | https://proofwiki.org/wiki/Semantic_Tableau_Algorithm_is_Decision_Procedure_for_Tautologies | [
"Propositional Logic"
] | [
"Semantic Tableau Algorithm",
"Definition:Decision Procedure/Tautologies"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Semantic Tableau Algorithm",
"Definition:Semantic Tableau/Completed",
"Definition:Semantic Tableau/Completed",
"Definition:Tautology/Formal Semantics/Boolean Interpretations"
] |
proofwiki-8821 | Soundness Theorem for Semantic Tableaux | Let $\mathbf A$ be a WFF of propositional logic.
Let $T$ be a completed tableau for $\mathbf A$.
Suppose that $T$ is closed.
Then $\mathbf A$ is unsatisfiable for boolean interpretations. | We will prove inductively the following claim for every node $t$ of $T$:
:If all leaves that are descendants of $t$ are marked closed, then $\map U t$ is unsatisfiable.
By the Semantic Tableau Algorithm, we know this statement to hold for the leaf nodes themselves.
For, a leaf $t$ is marked closed {{iff}} $\map U t$ co... | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Let $T$ be a [[Definition:Completed Tableau|completed tableau]] for $\mathbf A$.
Suppose that $T$ is [[Definition:Closed Tableau|closed]].
Then $\mathbf A$ is [[Definition:Unsatisfiable (Boolean Interpretations)|unsatisfiable... | We will prove inductively the following claim for every [[Definition:Node of Tree|node]] $t$ of $T$:
:If all [[Definition:Leaf Node|leaves]] that are [[Definition:Descendant Node|descendants]] of $t$ are [[Definition:Marked Closed Leaf|marked closed]], then $\map U t$ is [[Definition:Unsatisfiable (Boolean Interpretat... | Soundness Theorem for Semantic Tableaux | https://proofwiki.org/wiki/Soundness_Theorem_for_Semantic_Tableaux | https://proofwiki.org/wiki/Soundness_Theorem_for_Semantic_Tableaux | [
"Propositional Logic",
"Named Theorems"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Semantic Tableau/Completed",
"Definition:Semantic Tableau/Closed",
"Definition:Unsatisfiable/Boolean Interpretations"
] | [
"Definition:Tree (Graph Theory)/Node",
"Definition:Tree (Graph Theory)/Leaf Node",
"Definition:Rooted Tree/Descendant",
"Definition:Marked Leaf/Closed",
"Definition:Unsatisfiable/Boolean Interpretations",
"Semantic Tableau Algorithm",
"Definition:Tree (Graph Theory)/Leaf Node",
"Definition:Tree (Graph... |
proofwiki-8822 | Completeness Theorem for Semantic Tableaux | Let $\mathbf A$ be a WFF of propositional logic.
Let $\mathbf A$ be unsatisfiable for boolean interpretations.
Then every completed tableau for $\mathbf A$ is closed. | {{AimForCont}} some open tableau $T$ for $\mathbf A$ exists.
We will prove that $\mathbf A$ must be satisfiable.
This claim is proved by inductively establishing the following claim for all nodes $t$ of $T$:
:If a descendant leaf node of $t$ is marked open, then $\map U t$ is satisfiable.
By the Semantic Tableau Algori... | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Let $\mathbf A$ be [[Definition:Unsatisfiable (Boolean Interpretations)|unsatisfiable for boolean interpretations]].
Then every [[Definition:Completed Tableau|completed tableau]] for $\mathbf A$ is [[Definition:Closed Tableau|... | {{AimForCont}} some [[Definition:Open Tableau|open tableau]] $T$ for $\mathbf A$ exists.
We will prove that $\mathbf A$ must be [[Definition:Satisfiable (Boolean Interpretations)|satisfiable]].
This claim is proved by inductively establishing the following claim for all [[Definition:Node of Tree|nodes]] $t$ of $T$:
... | Completeness Theorem for Semantic Tableaux | https://proofwiki.org/wiki/Completeness_Theorem_for_Semantic_Tableaux | https://proofwiki.org/wiki/Completeness_Theorem_for_Semantic_Tableaux | [
"Completeness Theorem",
"Propositional Logic",
"Named Theorems"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Unsatisfiable/Boolean Interpretations",
"Definition:Semantic Tableau/Completed",
"Definition:Semantic Tableau/Closed"
] | [
"Definition:Semantic Tableau/Open",
"Definition:Satisfiable/Boolean Interpretations",
"Definition:Tree (Graph Theory)/Node",
"Definition:Rooted Tree/Descendant",
"Definition:Tree (Graph Theory)/Leaf Node",
"Definition:Marked Leaf/Open",
"Definition:Satisfiable/Boolean Interpretations",
"Semantic Table... |
proofwiki-8823 | Cosine to Power of Even Integer/Proof 2 | Let $n \in \Z$ be an even integer.
Then:
:$\ds \cos^n \theta = \frac 1 {2^{n - 1} } \sum_{k \mathop = 0}^{n / 2} \paren {\binom n k \cos \paren {n - 2 k} \theta}$
That is:
:$\cos^n \theta = \dfrac 1 {2^{n - 1} } \paren {\cos n \theta + n \cos \paren {n - 2} \theta + \dfrac {n \paren {n - 1} } 2 \cos \paren {n - 4} \the... | {{begin-eqn}}
{{eqn | l = \cos^n \theta
| r = \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^n
| c = De Moivre's Theorem
}}
{{eqn | r = \frac {\paren {e^{i \theta} + e^{-i \theta} }^n} {2^n}
| c =
}}
{{eqn | r = \frac 1 {2^n} \sum_{k \mathop = 0}^n \binom n k e^{\paren {n - k} i \theta} e^{-k i \th... | Let $n \in \Z$ be an [[Definition:Even Integer|even integer]].
Then:
:$\ds \cos^n \theta = \frac 1 {2^{n - 1} } \sum_{k \mathop = 0}^{n / 2} \paren {\binom n k \cos \paren {n - 2 k} \theta}$
That is:
:$\cos^n \theta = \dfrac 1 {2^{n - 1} } \paren {\cos n \theta + n \cos \paren {n - 2} \theta + \dfrac {n \paren {n - 1... | {{begin-eqn}}
{{eqn | l = \cos^n \theta
| r = \paren {\frac {e^{i \theta} + e^{-i \theta} } 2}^n
| c = [[De Moivre's Theorem]]
}}
{{eqn | r = \frac {\paren {e^{i \theta} + e^{-i \theta} }^n} {2^n}
| c =
}}
{{eqn | r = \frac 1 {2^n} \sum_{k \mathop = 0}^n \binom n k e^{\paren {n - k} i \theta} e^{-k i... | Cosine to Power of Even Integer/Proof 2 | https://proofwiki.org/wiki/Cosine_to_Power_of_Even_Integer/Proof_2 | https://proofwiki.org/wiki/Cosine_to_Power_of_Even_Integer/Proof_2 | [
"Cosine to Power of Even Integer"
] | [
"Definition:Even Integer"
] | [
"De Moivre's Formula",
"Binomial Theorem"
] |
proofwiki-8824 | Conjugate of Real Polynomial is Polynomial in Conjugate | Let $\map P z$ be a polynomial in a complex number $z$.
Let the coefficients of $P$ all be real.
Then:
:$\overline {\map P z} = \map P {\overline z}$
where $\overline z$ denotes the complex conjugate of $z$. | Let $\map P z$ be expressed as:
:$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$
Then:
{{begin-eqn}}
{{eqn | l = \overline {\map P z}
| r = \overline {a_n z^n + a_{n-1} z^{n - 1} + \cdots + a_1 z + a_0}
| c =
}}
{{eqn | r = \overline {a_n z^n} + \overline {a_{n - 1} z^{n - 1} } + \cdots + \overline ... | Let $\map P z$ be a [[Definition:Polynomial|polynomial]] in a [[Definition:Complex Number|complex number]] $z$.
Let the [[Definition:Polynomial Coefficient|coefficients]] of $P$ all be [[Definition:Real Number|real]].
Then:
:$\overline {\map P z} = \map P {\overline z}$
where $\overline z$ denotes the [[Definition:C... | Let $\map P z$ be expressed as:
:$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$
Then:
{{begin-eqn}}
{{eqn | l = \overline {\map P z}
| r = \overline {a_n z^n + a_{n-1} z^{n - 1} + \cdots + a_1 z + a_0}
| c =
}}
{{eqn | r = \overline {a_n z^n} + \overline {a_{n - 1} z^{n - 1} } + \cdots + \overline... | Conjugate of Real Polynomial is Polynomial in Conjugate | https://proofwiki.org/wiki/Conjugate_of_Real_Polynomial_is_Polynomial_in_Conjugate | https://proofwiki.org/wiki/Conjugate_of_Real_Polynomial_is_Polynomial_in_Conjugate | [
"Complex Conjugates"
] | [
"Definition:Polynomial",
"Definition:Complex Number",
"Definition:Coefficient of Polynomial",
"Definition:Real Number",
"Definition:Complex Conjugate"
] | [
"Sum of Complex Conjugates",
"Product of Complex Conjugates",
"Complex Number equals Conjugate iff Wholly Real",
"Product of Complex Conjugates/General Result"
] |
proofwiki-8825 | Absolute Value of Components of Complex Number no greater than Root 2 of Modulus | Let $z = x + i y \in \C$ be a complex number.
Then:
:$\size x + \size y \le \sqrt 2 \cmod z$
where:
:$\size x$ and $\size y$ denote the absolute value of $x$ and $y$
:$\cmod z$ denotes the complex modulus of $z$. | Let $z = x + i y \in \C$ be an arbitrary complex number.
{{AimForCont}} the contrary:
{{begin-eqn}}
{{eqn | l = \size x + \size y
| o = >
| r = \sqrt 2 \cmod z
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\size x + \size y}^2
| o = >
| r = 2 \cmod z^2
| c = squaring both sides
... | Let $z = x + i y \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\size x + \size y \le \sqrt 2 \cmod z$
where:
:$\size x$ and $\size y$ denote the [[Definition:Absolute Value|absolute value]] of $x$ and $y$
:$\cmod z$ denotes the [[Definition:Complex Modulus|complex modulus]] of $z$. | Let $z = x + i y \in \C$ be an arbitrary [[Definition:Complex Number|complex number]].
{{AimForCont}} the contrary:
{{begin-eqn}}
{{eqn | l = \size x + \size y
| o = >
| r = \sqrt 2 \cmod z
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\size x + \size y}^2
| o = >
| r = 2 \cmod z^2
... | Absolute Value of Components of Complex Number no greater than Root 2 of Modulus | https://proofwiki.org/wiki/Absolute_Value_of_Components_of_Complex_Number_no_greater_than_Root_2_of_Modulus | https://proofwiki.org/wiki/Absolute_Value_of_Components_of_Complex_Number_no_greater_than_Root_2_of_Modulus | [
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Absolute Value",
"Definition:Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Real Number"
] |
proofwiki-8826 | Exclusive Or with Factor of Exclusive Or | :$\paren {p \oplus q} \oplus q \dashv \vdash p$ | {{begin-eqn}}
{{eqn | l = \paren {p \oplus q} \oplus q
| o = \dashv \vdash
| r = p \oplus \paren {q \oplus q}
| c = Exclusive Or is Associative
}}
{{eqn | o = \dashv \vdash
| r = p \oplus \bot
| c = Exclusive Or with Itself
}}
{{eqn | o = \dashv \vdash
| r = p
| c = Exclusive O... | :$\paren {p \oplus q} \oplus q \dashv \vdash p$ | {{begin-eqn}}
{{eqn | l = \paren {p \oplus q} \oplus q
| o = \dashv \vdash
| r = p \oplus \paren {q \oplus q}
| c = [[Exclusive Or is Associative]]
}}
{{eqn | o = \dashv \vdash
| r = p \oplus \bot
| c = [[Exclusive Or with Itself]]
}}
{{eqn | o = \dashv \vdash
| r = p
| c = [[E... | Exclusive Or with Factor of Exclusive Or | https://proofwiki.org/wiki/Exclusive_Or_with_Factor_of_Exclusive_Or | https://proofwiki.org/wiki/Exclusive_Or_with_Factor_of_Exclusive_Or | [
"Exclusive Or"
] | [] | [
"Exclusive Or is Associative",
"Exclusive Or with Itself",
"Exclusive Or with Contradiction"
] |
proofwiki-8827 | Biconditional with Factor of Biconditional | :$\paren {p \iff q} \iff q \dashv \vdash p$ | {{begin-eqn}}
{{eqn | l = \paren {p \iff q} \iff q
| o = \dashv \vdash
| r = p \iff \paren {q \iff q}
| c = Biconditional is Associative
}}
{{eqn | o = \dashv \vdash
| r = p \iff \top
| c = Biconditional with Itself
}}
{{eqn | o = \dashv \vdash
| r = p
| c = Biconditional with ... | :$\paren {p \iff q} \iff q \dashv \vdash p$ | {{begin-eqn}}
{{eqn | l = \paren {p \iff q} \iff q
| o = \dashv \vdash
| r = p \iff \paren {q \iff q}
| c = [[Biconditional is Associative]]
}}
{{eqn | o = \dashv \vdash
| r = p \iff \top
| c = [[Biconditional with Itself]]
}}
{{eqn | o = \dashv \vdash
| r = p
| c = [[Biconditi... | Biconditional with Factor of Biconditional | https://proofwiki.org/wiki/Biconditional_with_Factor_of_Biconditional | https://proofwiki.org/wiki/Biconditional_with_Factor_of_Biconditional | [
"Biconditional"
] | [] | [
"Biconditional is Associative",
"Biconditional is Reflexive",
"Biconditional with Tautology"
] |
proofwiki-8828 | Conditional iff Biconditional of Antecedent with Conjunction | :$p \implies q \dashv \vdash p \iff \paren {p \land q}$ | We apply the Method of Truth Tables.
As can be seen by inspection, the appropriate truth values match for all boolean interpretations.
$\begin{array}{|ccc||ccccc|} \hline
p & \implies & q & p & \iff & (p & \land & q) \\
\hline
\F & \T & \F & \F & \T & \F & \F & \F \\
\F & \T & \T & \F & \T & \F & \F & \T \\
\T & \F & \... | :$p \implies q \dashv \vdash p \iff \paren {p \land q}$ | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the appropriate [[Definition:Truth Value|truth values]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|ccc||ccccc|} \hline
p & \implies & q & p & \iff & (p & \land & q) \\
\hline
\F & \T & \F & \F & \T... | Conditional iff Biconditional of Antecedent with Conjunction | https://proofwiki.org/wiki/Conditional_iff_Biconditional_of_Antecedent_with_Conjunction | https://proofwiki.org/wiki/Conditional_iff_Biconditional_of_Antecedent_with_Conjunction | [
"Conditional",
"Biconditional",
"Conjunction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8829 | Conditional iff Biconditional of Consequent with Disjunction | :$p \implies q \dashv \vdash q \iff \paren {p \lor q}$ | We apply the Method of Truth Tables.
As can be seen by inspection, the appropriate truth values match for all boolean interpretations.
$\begin{array}{|ccc||ccccc|} \hline
p & \implies & q & q & \iff & (p & \lor & q) \\
\hline
\F & \T & \F & \F & \T & \F & \F & \F \\
\F & \T & \T & \T & \T & \F & \T & \T \\
\T & \F & \F... | :$p \implies q \dashv \vdash q \iff \paren {p \lor q}$ | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the appropriate [[Definition:Truth Value|truth values]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|ccc||ccccc|} \hline
p & \implies & q & q & \iff & (p & \lor & q) \\
\hline
\F & \T & \F & \F & \T ... | Conditional iff Biconditional of Consequent with Disjunction | https://proofwiki.org/wiki/Conditional_iff_Biconditional_of_Consequent_with_Disjunction | https://proofwiki.org/wiki/Conditional_iff_Biconditional_of_Consequent_with_Disjunction | [
"Conditional",
"Biconditional",
"Disjunction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8830 | Conjunction iff Biconditional of Biconditional with Disjunction | :$p \land q \dashv \vdash \paren {p \iff q} \iff \paren {p \lor q}$ | We apply the Method of Truth Tables.
As can be seen by inspection, the appropriate truth values match for all boolean interpretations.
$\begin{array}{|ccc||ccccccc|} \hline
p & \land & q & (p & \iff & q) & \iff & (p & \lor & q) \\
\hline
\F & \F & \F & \F & \T & \F & \F & \F & \F & \F \\
\F & \F & \T & \F & \F & \T & \... | :$p \land q \dashv \vdash \paren {p \iff q} \iff \paren {p \lor q}$ | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the appropriate [[Definition:Truth Value|truth values]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|ccc||ccccccc|} \hline
p & \land & q & (p & \iff & q) & \iff & (p & \lor & q) \\
\hline
\F & \F & \... | Conjunction iff Biconditional of Biconditional with Disjunction | https://proofwiki.org/wiki/Conjunction_iff_Biconditional_of_Biconditional_with_Disjunction | https://proofwiki.org/wiki/Conjunction_iff_Biconditional_of_Biconditional_with_Disjunction | [
"Biconditional",
"Conjunction",
"Disjunction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Boolean Interpretation"
] |
proofwiki-8831 | Functionally Incomplete Logical Connectives/Conjunction and Disjunction | :$\set {\land, \lor}$: And and Or | Let $v_T$ be the boolean interpretation that assigns $T$ to each propositional symbol.
Then it follows by the nature of the truth functions for $\land$ and $\lor$ that:
:$\map {v_T} {\mathbf A} = T$
for each WFF $\mathbf A$ comprising only $\land$ and $\lor$.
On the other hand:
:$\map {v_T} {\neg p} = F$
Therefore, $\n... | :$\set {\land, \lor}$: [[Definition:Conjunction|And]] and [[Definition:Disjunction|Or]] | Let $v_T$ be the [[Definition:Boolean Interpretation|boolean interpretation]] that assigns $T$ to each [[Definition:Propositional Symbol|propositional symbol]].
Then it follows by the nature of the [[Definition:Truth Function|truth functions]] for $\land$ and $\lor$ that:
:$\map {v_T} {\mathbf A} = T$
for each [[Def... | Functionally Incomplete Logical Connectives/Conjunction and Disjunction | https://proofwiki.org/wiki/Functionally_Incomplete_Logical_Connectives/Conjunction_and_Disjunction | https://proofwiki.org/wiki/Functionally_Incomplete_Logical_Connectives/Conjunction_and_Disjunction | [
"Functional Completeness"
] | [
"Definition:Conjunction",
"Definition:Disjunction"
] | [
"Definition:Boolean Interpretation",
"Definition:Language of Propositional Logic/Alphabet/Letter",
"Definition:Truth Function",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Functionally Complete"
] |
proofwiki-8832 | Functionally Incomplete Logical Connectives/Negation and Biconditional | :$\set {\neg, \iff}$: Not and Iff | Let $\mathbf A$ be a WFF comprising only the logical connectives $\neg$ and $\iff$.
The central claim to this proof is that:
:The number of models for $\mathbf A$ is even.
In this claim, we disregard the obvious exceptions of the form $\neg \cdots \neg p$.
Firstly, note that including any propositional symbols not occu... | :$\set {\neg, \iff}$: [[Definition:Logical Not|Not]] and [[Definition:Biconditional|Iff]] | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF]] comprising only the [[Definition:Logical Connective|logical connectives]] $\neg$ and $\iff$.
The central claim to this proof is that:
:The number of [[Definition:Model (Boolean Interpretations)|models]] for $\mathbf A$ is even.
In this claim, we disr... | Functionally Incomplete Logical Connectives/Negation and Biconditional | https://proofwiki.org/wiki/Functionally_Incomplete_Logical_Connectives/Negation_and_Biconditional | https://proofwiki.org/wiki/Functionally_Incomplete_Logical_Connectives/Negation_and_Biconditional | [
"Functional Completeness"
] | [
"Definition:Logical Not",
"Definition:Biconditional"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Logical Connective",
"Definition:Model (Boolean Interpretations)",
"Definition:Language of Propositional Logic/Alphabet/Letter",
"Definition:Truth Table",
"Definition:Boolean Interpretation",
"Definition:Model (Boolean Interpre... |
proofwiki-8833 | Theory of Set of Formulas is Theory | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF$ be a set of $\LL$-formulas.
Let $\map T \FF$ be the $\LL$-theory of $\FF$.
Then $\map T \FF$ is a theory. | Let $\phi$ be a $\LL$-formula.
Suppose that $\map T \FF \models_{\mathscr M} \phi$.
By definition of $\map T \FF$:
:$\FF \models_{\mathscr M} \psi$
for every $\psi \in \map T \FF$.
Hence by definition of semantic consequence:
:$\FF \models_{\mathscr M} \map T \FF$
By Semantic Consequence is Transitive, it follows that:... | 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|$\LL$-formulas]].
Let $\map T \FF$ be the [[Definition:Theory of Set of Formulas|$\LL$-theory]] of $... | Let $\phi$ be a [[Definition:Logical Formula|$\LL$-formula]].
Suppose that $\map T \FF \models_{\mathscr M} \phi$.
By definition of $\map T \FF$:
:$\FF \models_{\mathscr M} \psi$
for every $\psi \in \map T \FF$.
Hence by definition of [[Definition:Semantic Consequence|semantic consequence]]:
:$\FF \models_{\math... | Theory of Set of Formulas is Theory | https://proofwiki.org/wiki/Theory_of_Set_of_Formulas_is_Theory | https://proofwiki.org/wiki/Theory_of_Set_of_Formulas_is_Theory | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Set",
"Definition:Logical Formula",
"Definition:Theory/Set of Formulas",
"Definition:Theory"
] | [
"Definition:Logical Formula",
"Definition:Semantic Consequence",
"Semantic Consequence is Transitive"
] |
proofwiki-8834 | Semantic Consequence is Transitive | Let $\LL$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\LL$.
Let $\FF, \GG$ and $\HH$ be sets of $\LL$-formulas.
Suppose that:
{{begin-eqn}}
{{eqn | l = \FF
| o = \models_{\mathscr M}
| r = \GG
}}
{{eqn | l = \GG
| o = \models_{\mathscr M}
| r = \HH
}}
{{end-eqn}}
Then $\FF... | Let $\MM$ be an $\mathscr M$-structure.
By assumption, if $\MM$ is a model of $\FF$, it is one of $\GG$ as well.
But any model of $\GG$ is also a model of $\HH$.
In conclusion, any model of $\FF$ is also a model of $\HH$.
Hence the result, by definition of semantic consequence.
{{qed}}
Category:Formal Semantics
ekmhdnp... | Let $\LL$ be a [[Definition:Logical Language|logical language]].
Let $\mathscr M$ be a [[Definition:Formal Semantics|formal semantics]] for $\LL$.
Let $\FF, \GG$ and $\HH$ be [[Definition:Set|sets]] of [[Definition:Logical Formula|$\LL$-formulas]].
Suppose that:
{{begin-eqn}}
{{eqn | l = \FF
| o = \models_{\m... | Let $\MM$ be an [[Definition:Structure (Formal Systems)|$\mathscr M$-structure]].
By assumption, if $\MM$ is a [[Definition:Model of Set of Formulas|model]] of $\FF$, it is one of $\GG$ as well.
But any [[Definition:Model of Set of Formulas|model]] of $\GG$ is also a [[Definition:Model of Set of Formulas|model]] of $... | Semantic Consequence is Transitive | https://proofwiki.org/wiki/Semantic_Consequence_is_Transitive | https://proofwiki.org/wiki/Semantic_Consequence_is_Transitive | [
"Formal Semantics"
] | [
"Definition:Logical Language",
"Definition:Formal Semantics",
"Definition:Set",
"Definition:Logical Formula"
] | [
"Definition:Formal Semantics/Structure",
"Definition:Model (Logic)/Set of Logical Formulas",
"Definition:Model (Logic)/Set of Logical Formulas",
"Definition:Model (Logic)/Set of Logical Formulas",
"Definition:Model (Logic)/Set of Logical Formulas",
"Definition:Model (Logic)/Set of Logical Formulas",
"De... |
proofwiki-8835 | Lemniscate of Bernoulli as Locus in Complex Plane | The locus of $z$ on the complex plane such that:
:$\cmod {z - a} \cmod {z + a} = a^2$
is a lemniscate of Bernoulli. | {{begin-eqn}}
{{eqn | l = \cmod {z - a} \cmod {z + a}
| r = a^2
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\paren {x - a}^2 + y^2} \paren {\paren {x + a}^2 + y^2}
| r = a^4
| c = {{Defof|Complex Modulus}}
}}
{{eqn | ll= \leadsto
| l = \paren {x^2 - 2 a x + a^2 + y^2} \paren {x^2 + ... | The [[Definition:Locus|locus]] of $z$ on the [[Definition:Complex Plane|complex plane]] such that:
:$\cmod {z - a} \cmod {z + a} = a^2$
is a [[Definition:Lemniscate of Bernoulli|lemniscate of Bernoulli]]. | {{begin-eqn}}
{{eqn | l = \cmod {z - a} \cmod {z + a}
| r = a^2
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {\paren {x - a}^2 + y^2} \paren {\paren {x + a}^2 + y^2}
| r = a^4
| c = {{Defof|Complex Modulus}}
}}
{{eqn | ll= \leadsto
| l = \paren {x^2 - 2 a x + a^2 + y^2} \paren {x^2 + ... | Lemniscate of Bernoulli as Locus in Complex Plane | https://proofwiki.org/wiki/Lemniscate_of_Bernoulli_as_Locus_in_Complex_Plane | https://proofwiki.org/wiki/Lemniscate_of_Bernoulli_as_Locus_in_Complex_Plane | [
"Analytic Geometry",
"Geometry of Complex Plane"
] | [
"Definition:Locus",
"Definition:Complex Number/Complex Plane",
"Definition:Lemniscate of Bernoulli"
] | [
"Difference of Two Squares",
"Definition:Lemniscate of Bernoulli",
"Definition:Lemniscate of Bernoulli/Major Semiaxis"
] |
proofwiki-8836 | Product of Tangent and Cotangent | :$\tan \theta \cot \theta = 1$ | {{begin-eqn}}
{{eqn | l = \tan \theta \cot \theta
| r = \frac {\sin \theta} {\cos \theta} \cot \theta
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {\sin \theta} {\cos \theta} \frac {\cos \theta} {\sin \theta}
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {\sin \theta} ... | :$\tan \theta \cot \theta = 1$ | {{begin-eqn}}
{{eqn | l = \tan \theta \cot \theta
| r = \frac {\sin \theta} {\cos \theta} \cot \theta
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {\sin \theta} {\cos \theta} \frac {\cos \theta} {\sin \theta}
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {\sin ... | Product of Tangent and Cotangent | https://proofwiki.org/wiki/Product_of_Tangent_and_Cotangent | https://proofwiki.org/wiki/Product_of_Tangent_and_Cotangent | [
"Tangent Function",
"Cotangent Function"
] | [] | [
"Tangent is Sine divided by Cosine",
"Cotangent is Cosine divided by Sine",
"Category:Tangent Function",
"Category:Cotangent Function"
] |
proofwiki-8837 | Product of Cotangents of Fractions of Pi | Let $m \in \Z$ such that $m > 1$.
Then:
:$\ds \prod_{k \mathop = 1}^{m - 1} \cot \frac {k \pi} {2 m} = 1$ | We have:
{{begin-eqn}}
{{eqn | l = \frac {k \pi} {2 m} + \frac {\paren {m - k} \pi} {2 m}
| r = \frac {\paren {k + m - k} \pi} {2 m}
| c =
}}
{{eqn | r = \frac \pi 2
| c =
}}
{{end-eqn}}
That means:
:$(1): \quad \dfrac \pi 2 - \dfrac {k \pi} {2 m} = \dfrac {\paren {m - k} \pi} {2 m}$
Let $m$ be odd.... | Let $m \in \Z$ such that $m > 1$.
Then:
:$\ds \prod_{k \mathop = 1}^{m - 1} \cot \frac {k \pi} {2 m} = 1$ | We have:
{{begin-eqn}}
{{eqn | l = \frac {k \pi} {2 m} + \frac {\paren {m - k} \pi} {2 m}
| r = \frac {\paren {k + m - k} \pi} {2 m}
| c =
}}
{{eqn | r = \frac \pi 2
| c =
}}
{{end-eqn}}
That means:
:$(1): \quad \dfrac \pi 2 - \dfrac {k \pi} {2 m} = \dfrac {\paren {m - k} \pi} {2 m}$
Let $m$ be... | Product of Cotangents of Fractions of Pi | https://proofwiki.org/wiki/Product_of_Cotangents_of_Fractions_of_Pi | https://proofwiki.org/wiki/Product_of_Cotangents_of_Fractions_of_Pi | [
"Cotangent Function"
] | [] | [
"Definition:Odd Integer",
"Definition:Even Integer",
"Cotangent of Complement equals Tangent",
"Product of Tangent and Cotangent",
"Definition:Odd Integer",
"Cotangent of Complement equals Tangent",
"Product of Tangent and Cotangent",
"Cotangent of 45 Degrees"
] |
proofwiki-8838 | Uncountable Set less Countable Set is Uncountable | Let $S$ be an uncountable set.
Let $T \subseteq S$ be a countable subset of $S$.
Then:
:$S \setminus T$ is uncountable
where $\setminus$ denotes set difference. | {{AimForCont}} $S \setminus T$ were countable.
By definition of relative complement:
:$S \setminus T = \relcomp S T$
Thus from Union with Relative Complement:
:$\paren {S \setminus T} \cup T = S$
But from Finite Union of Countable Sets is Countable it follows that $S$ is countable.
From this contradiction it follows th... | Let $S$ be an [[Definition:Uncountable Set|uncountable set]].
Let $T \subseteq S$ be a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] of $S$.
Then:
:$S \setminus T$ is [[Definition:Uncountable Set|uncountable]]
where $\setminus$ denotes [[Definition:Set Difference|set difference]]. | {{AimForCont}} $S \setminus T$ were [[Definition:Countable Set|countable]].
By definition of [[Definition:Relative Complement|relative complement]]:
:$S \setminus T = \relcomp S T$
Thus from [[Union with Relative Complement]]:
:$\paren {S \setminus T} \cup T = S$
But from [[Finite Union of Countable Sets is Countabl... | Uncountable Set less Countable Set is Uncountable | https://proofwiki.org/wiki/Uncountable_Set_less_Countable_Set_is_Uncountable | https://proofwiki.org/wiki/Uncountable_Set_less_Countable_Set_is_Uncountable | [
"Uncountable Sets",
"Countable Sets"
] | [
"Definition:Uncountable/Set",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Uncountable/Set",
"Definition:Set Difference"
] | [
"Definition:Countable Set",
"Definition:Relative Complement",
"Union with Relative Complement",
"Finite Union of Countable Sets is Countable",
"Definition:Countable Set",
"Proof by Contradiction",
"Definition:Uncountable/Set",
"Category:Uncountable Sets",
"Category:Countable Sets"
] |
proofwiki-8839 | Irrational Numbers are Uncountably Infinite | The set $\R \setminus \Q$ of irrational numbers is uncountable. | From Real Numbers are Uncountable, $\R$ is an uncountable set.
From Rational Numbers are Countably Infinite $\Q$ is countable.
The result follows from Uncountable Set less Countable Set is Uncountable.
{{qed}}
{{AoC|Uncountable Set less Countable Set is Uncountable}} | The [[Definition:Set|set]] $\R \setminus \Q$ of [[Definition:Irrational Number|irrational numbers]] is [[Definition:Uncountable Set|uncountable]]. | From [[Real Numbers are Uncountable]], $\R$ is an [[Definition:Uncountable Set|uncountable set]].
From [[Rational Numbers are Countably Infinite]] $\Q$ is [[Definition:Countable Set|countable]].
The result follows from [[Uncountable Set less Countable Set is Uncountable]].
{{qed}}
{{AoC|Uncountable Set less Countabl... | Irrational Numbers are Uncountably Infinite | https://proofwiki.org/wiki/Irrational_Numbers_are_Uncountably_Infinite | https://proofwiki.org/wiki/Irrational_Numbers_are_Uncountably_Infinite | [
"Irrational Numbers",
"Uncountable Sets"
] | [
"Definition:Set",
"Definition:Irrational Number",
"Definition:Uncountable/Set"
] | [
"Real Numbers are Uncountably Infinite",
"Definition:Uncountable/Set",
"Rational Numbers are Countably Infinite",
"Definition:Countable Set",
"Uncountable Set less Countable Set is Uncountable"
] |
proofwiki-8840 | Product of Diagonals from Point of Regular Polygon | Let $A_0, A_1, \ldots, A_{n - 1}$ be the vertices of a regular $n$-gon $P = A_0 A_1 \cdots A_{n - 1}$ which is circumscribed by a unit circle.
Then:
:$\ds \prod_{k \mathop = 2}^{n - 2} A_0 A_k = \frac {n \csc^2 \frac \pi n} 4$
where $A_0 A_k$ is the length of the line joining $A_0$ to $A_k$. | First it is worth examining the degenerate case $n = 3$, where there are no such lines.
In this case:
{{begin-eqn}}
{{eqn | l = \frac {n \csc^2 \frac \pi n} 4
| r = \frac 4 3 \csc^2 \frac \pi 3
| c =
}}
{{eqn | r = \frac 3 4 \paren {\frac {2 \sqrt 3} 3}^2
| c = Cosecant of $60 \degrees$
}}
{{eqn | r ... | Let $A_0, A_1, \ldots, A_{n - 1}$ be the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Regular Polygon|regular $n$-gon]] $P = A_0 A_1 \cdots A_{n - 1}$ which is [[Definition:Circle Circumscribed around Polygon|circumscribed]] by a [[Definition:Unit Circle|unit circle]].
Then:
:$\ds \prod_{k \mathop = 2}... | First it is worth examining the [[Definition:Degenerate Case|degenerate case]] $n = 3$, where there are no such lines.
In this case:
{{begin-eqn}}
{{eqn | l = \frac {n \csc^2 \frac \pi n} 4
| r = \frac 4 3 \csc^2 \frac \pi 3
| c =
}}
{{eqn | r = \frac 3 4 \paren {\frac {2 \sqrt 3} 3}^2
| c = [[Cose... | Product of Diagonals from Point of Regular Polygon | https://proofwiki.org/wiki/Product_of_Diagonals_from_Point_of_Regular_Polygon | https://proofwiki.org/wiki/Product_of_Diagonals_from_Point_of_Regular_Polygon | [
"Polygons"
] | [
"Definition:Polygon/Vertex",
"Definition:Polygon/Regular",
"Definition:Circumscribe/Circle around Polygon",
"Definition:Unit Circle",
"Definition:Linear Measure/Length",
"Definition:Line/Segment"
] | [
"Definition:Degenerate Case",
"Cosecant of 60 Degrees",
"Definition:Continued Product",
"Definition:Continued Product/Vacuous Product",
"Definition:Continued Product/Vacuous Product",
"File:RegularPolygonWithDiagonals.png",
"Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle",
... |
proofwiki-8841 | Multiple Angle Formula for Sine | :$\ds \frac {\map \sin {n z} } {\sin z} = 2^{n - 1} \prod_{k \mathop = 1}^{n - 1} \paren {\cos z - \cos \frac {k \pi} n}$
for $\sin z \ne 0$. | We have:
{{begin-eqn}}
{{eqn | l = \frac {\map \sin {n z} } {\sin z}
| r = \frac {e^{i n z} - e^{-i n z} } {e^{i z} - e^{-i z} }
| c = Euler's Sine Identity
}}
{{eqn | r = \frac {e^{i n z} - e^{-i n z} } {e^{i z} - e^{-i z} } \times \frac {e^{i n z} } {e^{i n z} } \times \frac {2^{n - 1} } {2^{n - 1} } \tim... | :$\ds \frac {\map \sin {n z} } {\sin z} = 2^{n - 1} \prod_{k \mathop = 1}^{n - 1} \paren {\cos z - \cos \frac {k \pi} n}$
for $\sin z \ne 0$. | We have:
{{begin-eqn}}
{{eqn | l = \frac {\map \sin {n z} } {\sin z}
| r = \frac {e^{i n z} - e^{-i n z} } {e^{i z} - e^{-i z} }
| c = [[Euler's Sine Identity]]
}}
{{eqn | r = \frac {e^{i n z} - e^{-i n z} } {e^{i z} - e^{-i z} } \times \frac {e^{i n z} } {e^{i n z} } \times \frac {2^{n - 1} } {2^{n - 1} } ... | Multiple Angle Formula for Sine | https://proofwiki.org/wiki/Multiple_Angle_Formula_for_Sine | https://proofwiki.org/wiki/Multiple_Angle_Formula_for_Sine | [
"Sine Function",
"Complex Roots of Unity"
] | [] | [
"Euler's Sine Identity",
"Definition:Polynomial Equation",
"Definition:Root of Unity/Complex",
"Difference of Two Squares",
"Euler's Formula",
"Exponent Combination Laws/Product of Powers",
"Euler's Identity",
"Exponent Combination Laws/Product of Powers",
"Exponent Combination Laws/Product of Power... |
proofwiki-8842 | Condition on Conjugate from Real Product of Complex Numbers | Let $z_1, z_2 \in \C$ be complex numbers such that $z_1 z_2 \in \R_{\ne 0}$.
Then:
:$\exists p \in \R: z_1 = p \overline {z_2}$
where $\overline {z_2}$ denotes the complex conjugate of $z_2$. | Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.
As $z_1 z_2$ is real:
:$(1): \quad z_1 z_2 = x_1 x_2 - y_1 y_2$
and:
:$(2): \quad x_1 y_2 + y_1 x_2 = 0$
So:
{{begin-eqn}}
{{eqn | l = \frac {\paren {z_1} } {\paren {\overline {z_2} } }
| r = \frac {x_1 + i y_1} {x_2 - i y_2}
| c = {{Defof|Complex Conjugate}}
}}
{... | Let $z_1, z_2 \in \C$ be [[Definition:Complex Number|complex numbers]] such that $z_1 z_2 \in \R_{\ne 0}$.
Then:
:$\exists p \in \R: z_1 = p \overline {z_2}$
where $\overline {z_2}$ denotes the [[Definition:Complex Conjugate|complex conjugate]] of $z_2$. | Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.
As $z_1 z_2$ is [[Definition:Real Number|real]]:
:$(1): \quad z_1 z_2 = x_1 x_2 - y_1 y_2$
and:
:$(2): \quad x_1 y_2 + y_1 x_2 = 0$
So:
{{begin-eqn}}
{{eqn | l = \frac {\paren {z_1} } {\paren {\overline {z_2} } }
| r = \frac {x_1 + i y_1} {x_2 - i y_2}
| c = {... | Condition on Conjugate from Real Product of Complex Numbers | https://proofwiki.org/wiki/Condition_on_Conjugate_from_Real_Product_of_Complex_Numbers | https://proofwiki.org/wiki/Condition_on_Conjugate_from_Real_Product_of_Complex_Numbers | [
"Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate"
] | [
"Definition:Real Number",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Definition:Real Number"
] |
proofwiki-8843 | Real Natural Logarithm is Restriction of Complex Natural Logarithm | Let $\ln: \C_{\ne 0} \to \C$ be the complex natural logarithm.
Let $\ln': \R_{>0} \to \R$ be the real natural logarithm.
Then:
:$\ln' = \ln \restriction_{\R_{>0} \times \R}$
That is, the real natural logarithm is the restriction of the complex natural logarithm. | From Domain of Real Natural Logarithm:
: $\operatorname{Dom} \left({\ln'}\right) = \R_{>0}$
From Image of Real Natural Logarithm:
: $\operatorname{Im} \left({\ln'}\right) = \R$
Let $z \in \C$ such that $z = x + i y$.
Let $z$ be expressed in exponential form as $z = r e^{i \theta}$.
Let $x > 0$ and $y = 0$.
Thus $z \in ... | Let $\ln: \C_{\ne 0} \to \C$ be the [[Definition:Complex Natural Logarithm|complex natural logarithm]].
Let $\ln': \R_{>0} \to \R$ be the [[Definition:Real Natural Logarithm|real natural logarithm]].
Then:
:$\ln' = \ln \restriction_{\R_{>0} \times \R}$
That is, the [[Definition:Real Natural Logarithm|real natural l... | From [[Domain of Real Natural Logarithm]]:
: $\operatorname{Dom} \left({\ln'}\right) = \R_{>0}$
From [[Image of Real Natural Logarithm]]:
: $\operatorname{Im} \left({\ln'}\right) = \R$
Let $z \in \C$ such that $z = x + i y$.
Let $z$ be expressed in [[Definition:Exponential Form of Complex Number|exponential form]] ... | Real Natural Logarithm is Restriction of Complex Natural Logarithm | https://proofwiki.org/wiki/Real_Natural_Logarithm_is_Restriction_of_Complex_Natural_Logarithm | https://proofwiki.org/wiki/Real_Natural_Logarithm_is_Restriction_of_Complex_Natural_Logarithm | [
"Natural Logarithms"
] | [
"Definition:Natural Logarithm/Complex",
"Definition:Natural Logarithm/Positive Real",
"Definition:Natural Logarithm/Positive Real",
"Definition:Restriction/Relation",
"Definition:Natural Logarithm/Complex"
] | [
"Domain of Real Natural Logarithm",
"Image of Real Natural Logarithm",
"Definition:Complex Number/Polar Form/Exponential Form",
"Category:Natural Logarithms"
] |
proofwiki-8844 | Rule of Commutation/Disjunction/Formulation 2/Forward Implication | :$\vdash \paren {p \lor q} \implies \paren {q \lor p}$ | Using a tableau proof for instance 1 of a Gentzen proof system:
{| border="1"
|+$\vdash \paren {p \lor q} \implies \paren {q \lor p}$
|-
! Line !!
! Pool
! Formula
! Rule
! Depends upon
! Notes
|-
| 1 || ||
| $\neg p, q, p$
| Axiom || ||
|-
| 2 || ||
| $\neg q, q, p$
| Axiom || ||
|-
| 3 || ||
| $\neg \paren {p \lor q}... | :$\vdash \paren {p \lor q} \implies \paren {q \lor p}$ | Using a [[Definition:Tableau Proof (Natural Deduction)|tableau proof]] for [[Definition:Gentzen Proof System/Instance 1|instance 1 of a Gentzen proof system]]:
{| border="1"
|+$\vdash \paren {p \lor q} \implies \paren {q \lor p}$
|-
! Line !!
! Pool
! Formula
! Rule
! Depends upon
! Notes
|-
| 1 || ||
| $\neg p, q, p... | Rule of Commutation/Disjunction/Formulation 2/Forward Implication | https://proofwiki.org/wiki/Rule_of_Commutation/Disjunction/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/Rule_of_Commutation/Disjunction/Formulation_2/Forward_Implication | [
"Rule of Commutation"
] | [] | [
"Definition:Tableau Proof (Natural Deduction)",
"Definition:Gentzen Proof System/Instance 1",
"Definition:Gentzen Proof System/Instance 1/Alpha-Rule",
"Definition:Gentzen Proof System/Instance 1/Beta-Rule",
"Definition:Gentzen Proof System/Instance 1/Beta-Rule"
] |
proofwiki-8845 | Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2/Forward Implication | :$\vdash \paren {p \lor \paren {q \land r} } \implies \paren {\paren {p \lor q} \land \paren {p \lor r} }$ | {{BeginTableau|\vdash \paren {p \lor \paren {q \land r} } \implies \paren {\paren {p \lor q} \land \paren {p \lor r} }|instance 1 of a Gentzen proof system}}
{{TableauLine | n = 1
| f = \neg p, p, q
| rlnk = Definition:Gentzen Proof System/Instance 1#Axioms
| rtxt = Axiom... | :$\vdash \paren {p \lor \paren {q \land r} } \implies \paren {\paren {p \lor q} \land \paren {p \lor r} }$ | {{BeginTableau|\vdash \paren {p \lor \paren {q \land r} } \implies \paren {\paren {p \lor q} \land \paren {p \lor r} }|[[Definition:Gentzen Proof System/Instance 1|instance 1 of a Gentzen proof system]]}}
{{TableauLine | n = 1
| f = \neg p, p, q
| rlnk = Definition:Gentzen Proof System... | Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2/Forward Implication/Proof 1 | https://proofwiki.org/wiki/Rule_of_Distribution/Disjunction_Distributes_over_Conjunction/Left_Distributive/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/Rule_of_Distribution/Disjunction_Distributes_over_Conjunction/Left_Distributive/Formulation_2/Forward_Implication/Proof_1 | [
"Rule of Distribution"
] | [] | [
"Definition:Gentzen Proof System/Instance 1"
] |
proofwiki-8846 | Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2/Forward Implication | :$\vdash \paren {p \lor \paren {q \land r} } \implies \paren {\paren {p \lor q} \land \paren {p \lor r} }$ | {{BeginTableau|\vdash \paren {p \lor \paren {q \land r} } \implies \paren {\paren {p \lor q} \land \paren {p \lor r} }|Instance 2 of the Hilbert-style systems}}
{{TableauLine
| n = 1
| f = \paren {q \land r} \implies q
| rlnk = Rule of Simplification/Sequent Form/Formulation 2/Form 1
| rtxt = Rule of Simplification... | :$\vdash \paren {p \lor \paren {q \land r} } \implies \paren {\paren {p \lor q} \land \paren {p \lor r} }$ | {{BeginTableau|\vdash \paren {p \lor \paren {q \land r} } \implies \paren {\paren {p \lor q} \land \paren {p \lor r} }|[[Definition:Hilbert Proof System/Instance 2|Instance 2 of the Hilbert-style systems]]}}
{{TableauLine
| n = 1
| f = \paren {q \land r} \implies q
| rlnk = Rule of Simplification/Sequent Form/Formul... | Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 2/Forward Implication/Proof 2 | https://proofwiki.org/wiki/Rule_of_Distribution/Disjunction_Distributes_over_Conjunction/Left_Distributive/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/Rule_of_Distribution/Disjunction_Distributes_over_Conjunction/Left_Distributive/Formulation_2/Forward_Implication/Proof_2 | [
"Rule of Distribution"
] | [] | [
"Definition:Hilbert Proof System/Instance 2"
] |
proofwiki-8847 | Provable by Gentzen Proof System iff Negation has Closed Tableau/Formula | Let $\mathbf A$ be a WFF of propositional logic.
Then $\mathbf A$ is a $\mathscr G$-theorem {{iff}}:
:$\neg \mathbf A$ has a closed semantic tableau
where $\neg \mathbf A$ is the negation of $\mathbf A$. | This is a specific instance of Provable by Gentzen Proof System iff Negation has Closed Tableau: Set of Formulas.
{{qed}} | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Then $\mathbf A$ is a [[Definition:Theorem (Formal Systems)|$\mathscr G$-theorem]] {{iff}}:
:$\neg \mathbf A$ has a [[Definition:Closed Tableau|closed semantic tableau]]
where $\neg \mathbf A$ is the [[Definition:Logical Not|... | This is a specific instance of [[Provable by Gentzen Proof System iff Negation has Closed Tableau/Set of Formulas|Provable by Gentzen Proof System iff Negation has Closed Tableau: Set of Formulas]].
{{qed}} | Provable by Gentzen Proof System iff Negation has Closed Tableau/Formula | https://proofwiki.org/wiki/Provable_by_Gentzen_Proof_System_iff_Negation_has_Closed_Tableau/Formula | https://proofwiki.org/wiki/Provable_by_Gentzen_Proof_System_iff_Negation_has_Closed_Tableau/Formula | [
"Propositional Logic"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Theorem/Formal System",
"Definition:Semantic Tableau/Closed",
"Definition:Logical Not"
] | [
"Provable by Gentzen Proof System iff Negation has Closed Tableau/Set of Formulas"
] |
proofwiki-8848 | Provable by Gentzen Proof System iff Negation has Closed Tableau/Set of Formulas | Let $U$ be a set of WFFs of propositional logic.
Then:
:$U$ is a $\mathscr G$-theorem
{{iff}}:
:$\bar U$ has a closed semantic tableau
where $\bar U$ is the set comprising the logical complements of all WFFs in $U$. | Denote with $\bar{\mathbf A}$ and $\bar U$ the logical complement of a WFF $\mathbf A$ and set $U$, respectively. | Let $U$ be a [[Definition:Set|set]] of [[Definition:WFF of Propositional Logic|WFFs of propositional logic]].
Then:
:$U$ is a [[Definition:Theorem (Formal Systems)|$\mathscr G$-theorem]]
{{iff}}:
:$\bar U$ has a [[Definition:Closed Tableau|closed semantic tableau]]
where $\bar U$ is the [[Definition:Set|set]] compris... | Denote with $\bar{\mathbf A}$ and $\bar U$ the [[Definition:Logical Complement|logical complement]] of a [[Definition:WFF of Propositional Logic|WFF]] $\mathbf A$ and [[Definition:Set|set]] $U$, respectively. | Provable by Gentzen Proof System iff Negation has Closed Tableau/Set of Formulas | https://proofwiki.org/wiki/Provable_by_Gentzen_Proof_System_iff_Negation_has_Closed_Tableau/Set_of_Formulas | https://proofwiki.org/wiki/Provable_by_Gentzen_Proof_System_iff_Negation_has_Closed_Tableau/Set_of_Formulas | [
"Propositional Logic"
] | [
"Definition:Set",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Theorem/Formal System",
"Definition:Semantic Tableau/Closed",
"Definition:Set",
"Definition:Logical Complement",
"Definition:Language of Propositional Logic/Formal Grammar/WFF"
] | [
"Definition:Logical Complement",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Set",
"Definition:Logical Complement",
"Definition:Logical Complement",
"Definition:Logical Complement"
] |
proofwiki-8849 | Domain of Real Natural Logarithm | Let $\ln$ be the natural logarithm function on the real numbers.
Then the domain of $\ln$ is the set of strictly positive real numbers:
:$\Dom \ln = \R_{>0}$ | By definition of natural logarithm:
:$\ln^{-1} = \exp$
From Exponential Tends to Zero and Infinity:
:$\Dom \exp = \openint {-\infty} {+\infty}$
:$\Img \exp = \openint 0 {+\infty}$
From Exponential is Strictly Increasing:
:$\exp$ is strictly increasing.
From Strictly Monotone Real Function is Bijective, $\exp: \R \to \R... | Let $\ln$ be the [[Definition:Real Natural Logarithm|natural logarithm function on the real numbers]].
Then the [[Definition:Domain of Mapping|domain]] of $\ln$ is the set of [[Definition:Strictly Positive Real Number|strictly positive real numbers]]:
:$\Dom \ln = \R_{>0}$ | By definition of [[Definition:Natural Logarithm/Positive Real/Definition 2|natural logarithm]]:
:$\ln^{-1} = \exp$
From [[Exponential Tends to Zero and Infinity]]:
:$\Dom \exp = \openint {-\infty} {+\infty}$
:$\Img \exp = \openint 0 {+\infty}$
From [[Exponential is Strictly Increasing]]:
:$\exp$ is [[Definition:Stri... | Domain of Real Natural Logarithm | https://proofwiki.org/wiki/Domain_of_Real_Natural_Logarithm | https://proofwiki.org/wiki/Domain_of_Real_Natural_Logarithm | [
"Natural Logarithms"
] | [
"Definition:Natural Logarithm/Positive Real",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Strictly Positive/Real Number"
] | [
"Definition:Natural Logarithm/Positive Real/Definition 2",
"Exponential Tends to Zero and Infinity",
"Exponential is Strictly Increasing",
"Definition:Strictly Increasing/Real Function",
"Strictly Monotone Real Function is Bijective",
"Definition:Bijection",
"Category:Natural Logarithms"
] |
proofwiki-8850 | Image of Real Natural Logarithm | Let $\ln$ be the natural logarithm function on the real numbers.
Then the image of $\ln$ is the set of real numbers:
:$\Img \ln = \R$ | By definition of natural logarithm:
:$\ln^{-1} = \exp$
From Exponential Tends to Zero and Infinity:
:$\Dom \exp = \openint {-\infty} {+\infty}$
:$\Img \exp = \openint 0 {+\infty}$
From Exponential is Strictly Increasing:
:$\exp$ is strictly increasing.
From Strictly Monotone Real Function is Bijective, $\exp: \R \to \R... | Let $\ln$ be the [[Definition:Real Natural Logarithm|natural logarithm function on the real numbers]].
Then the [[Definition:Image of Mapping|image]] of $\ln$ is the set of [[Definition:Real Number|real numbers]]:
:$\Img \ln = \R$ | By definition of [[Definition:Natural Logarithm/Positive Real/Definition 2|natural logarithm]]:
:$\ln^{-1} = \exp$
From [[Exponential Tends to Zero and Infinity]]:
:$\Dom \exp = \openint {-\infty} {+\infty}$
:$\Img \exp = \openint 0 {+\infty}$
From [[Exponential is Strictly Increasing]]:
:$\exp$ is [[Definition:Stri... | Image of Real Natural Logarithm | https://proofwiki.org/wiki/Image_of_Real_Natural_Logarithm | https://proofwiki.org/wiki/Image_of_Real_Natural_Logarithm | [
"Natural Logarithms"
] | [
"Definition:Natural Logarithm/Positive Real",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Real Number"
] | [
"Definition:Natural Logarithm/Positive Real/Definition 2",
"Exponential Tends to Zero and Infinity",
"Exponential is Strictly Increasing",
"Definition:Strictly Increasing/Real Function",
"Strictly Monotone Real Function is Bijective",
"Definition:Bijection",
"Category:Natural Logarithms"
] |
proofwiki-8851 | Soundness and Completeness of Gentzen Proof System | Let $\mathscr G$ be instance 1 of a Gentzen proof system.
Let $\mathrm{BI}$ be the formal semantics of boolean interpretations.
Then $\mathscr G$ is a sound and complete proof system for $\mathrm{BI}$. | This is an immediate consequence of:
* Provable by Gentzen Proof System iff Negation has Closed Tableau
* Soundness and Completeness of Semantic Tableaux
{{qed}} | Let $\mathscr G$ be [[Definition:Gentzen Proof System/Instance 1|instance 1 of a Gentzen proof system]].
Let $\mathrm{BI}$ be the [[Definition:Formal Semantics of Boolean Interpretations|formal semantics of boolean interpretations]].
Then $\mathscr G$ is a [[Definition:Sound Proof System|sound]] and [[Definition:Com... | This is an immediate consequence of:
* [[Provable by Gentzen Proof System iff Negation has Closed Tableau]]
* [[Soundness and Completeness of Semantic Tableaux]]
{{qed}} | Soundness and Completeness of Gentzen Proof System | https://proofwiki.org/wiki/Soundness_and_Completeness_of_Gentzen_Proof_System | https://proofwiki.org/wiki/Soundness_and_Completeness_of_Gentzen_Proof_System | [
"Propositional Logic"
] | [
"Definition:Gentzen Proof System/Instance 1",
"Definition:Boolean Interpretation/Formal Semantics",
"Definition:Sound Proof System",
"Definition:Complete Proof System"
] | [
"Provable by Gentzen Proof System iff Negation has Closed Tableau",
"Soundness and Completeness of Semantic Tableaux"
] |
proofwiki-8852 | Deduction Theorem | Let $\mathscr H$ be instance 1 of a Hilbert proof system.
Then the deduction rule:
::$\dfrac{U,\mathbf A \vdash \mathbf B}{U \vdash \mathbf A \implies \mathbf B}$
is a derived rule for $\mathscr H$. | For any proof of $U, \mathbf A \vdash \mathbf B$, we indicate how to transform it into a proof of $U \vdash \mathbf A \implies \mathbf B$ without using the deduction rule.
This is done by applying the Second Principle of Mathematical Induction to the length $n$ of the proof of $U,\mathbf A \vdash \mathbf B$.
If $n = 1$... | Let $\mathscr H$ be [[Definition:Hilbert Proof System/Instance 1|instance 1 of a Hilbert proof system]].
Then the [[Definition:Deduction Rule|deduction rule]]:
::$\dfrac{U,\mathbf A \vdash \mathbf B}{U \vdash \mathbf A \implies \mathbf B}$
is a [[Definition:Derived Rule|derived rule]] for $\mathscr H$. | For any [[Definition:Formal Proof|proof]] of $U, \mathbf A \vdash \mathbf B$, we indicate how to transform it into a proof of $U \vdash \mathbf A \implies \mathbf B$ without using the [[Definition:Deduction Rule|deduction rule]].
This is done by applying the [[Second Principle of Mathematical Induction]] to the length... | Deduction Theorem | https://proofwiki.org/wiki/Deduction_Theorem | https://proofwiki.org/wiki/Deduction_Theorem | [
"Deduction Theorem",
"Propositional Logic",
"Named Theorems"
] | [
"Definition:Hilbert Proof System/Instance 1",
"Definition:Deduction Rule",
"Definition:Derived Rule"
] | [
"Definition:Proof System/Formal Proof",
"Definition:Deduction Rule",
"Second Principle of Mathematical Induction",
"Definition:Proof System/Formal Proof",
"Definition:Axiom/Formal Systems",
"Definition:Theorem/Formal System",
"Modus Ponendo Ponens",
"Law of Identity/Formulation 2/Proof 2",
"Modus Po... |
proofwiki-8853 | Soundness Theorem for Hilbert Proof System | Let $\mathscr H$ be instance 1 of a Hilbert proof system.
Let $\mathrm{BI}$ be the formal semantics of boolean interpretations.
Then $\mathscr H$ is a sound proof system for $\mathrm{BI}$:
:Every $\mathscr H$-theorem is a tautology. | Recall the axioms of $\mathscr H$:
{{begin-axiom}}
{{axiom | lc = '''Axiom $1$:'''
| m = \mathbf A \implies \paren {\mathbf B \implies \mathbf A}
}}
{{axiom | lc = '''Axiom $2$:'''
| m = \paren {\mathbf A \implies \paren {\mathbf B \implies \mathbf C} } \implies \paren {\paren {\mathbf A \implies \mat... | Let $\mathscr H$ be [[Definition:Hilbert Proof System/Instance 1|instance 1 of a Hilbert proof system]].
Let $\mathrm{BI}$ be the [[Definition:Formal Semantics of Boolean Interpretations|formal semantics of boolean interpretations]].
Then $\mathscr H$ is a [[Definition:Sound Proof System|sound proof system]] for $\m... | Recall the [[Definition:Axiom (Formal Systems)|axioms]] of $\mathscr H$:
{{begin-axiom}}
{{axiom | lc = '''Axiom $1$:'''
| m = \mathbf A \implies \paren {\mathbf B \implies \mathbf A}
}}
{{axiom | lc = '''Axiom $2$:'''
| m = \paren {\mathbf A \implies \paren {\mathbf B \implies \mathbf C} } \implies ... | Soundness Theorem for Hilbert Proof System | https://proofwiki.org/wiki/Soundness_Theorem_for_Hilbert_Proof_System | https://proofwiki.org/wiki/Soundness_Theorem_for_Hilbert_Proof_System | [
"Hilbert Proof System Instance 1"
] | [
"Definition:Hilbert Proof System/Instance 1",
"Definition:Boolean Interpretation/Formal Semantics",
"Definition:Sound Proof System",
"Definition:Theorem/Formal System",
"Definition:Tautology/Formal Semantics/Boolean Interpretations"
] | [
"Definition:Axiom/Formal Systems",
"Definition:Tautology/Formal Semantics/Boolean Interpretations",
"True Statement is implied by Every Statement/Formulation 2/Proof by Truth Table",
"Self-Distributive Law for Conditional/Formulation 1/Proof by Truth Table",
"Talk:Self-Distributive Law for Conditional",
"... |
proofwiki-8854 | Equivalence of Definitions of Dominate (Set Theory) | Let $S, T$ be sets.
{{TFAE|def = Dominate (Set Theory)|view = Dominate|context = Set Theory}} | === Definition 1 implies Definition 2 ===
Let $f: S \to T$ be an injection.
By Injection to Image is Bijection, $f$ is a bijection from $S$ to the image of $f$.
{{qed|lemma}} | Let $S, T$ be [[Definition:Set|sets]].
{{TFAE|def = Dominate (Set Theory)|view = Dominate|context = Set Theory}} | === Definition 1 implies Definition 2 ===
Let $f: S \to T$ be an [[Definition:Injection|injection]].
By [[Injection to Image is Bijection]], $f$ is a [[Definition:Bijection|bijection]] from $S$ to the [[Definition:Image of Mapping|image of $f$]].
{{qed|lemma}} | Equivalence of Definitions of Dominate (Set Theory) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Dominate_(Set_Theory) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Dominate_(Set_Theory) | [
"Set Theory"
] | [
"Definition:Set"
] | [
"Definition:Injection",
"Injection to Image is Bijection",
"Definition:Bijection",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Bijection",
"Definition:Injection"
] |
proofwiki-8855 | Immediate Successor under Total Ordering is Unique | Let $\preceq$ be a total ordering.
Let $b$ be an immediate successor to $a$.
Then $b$ is unique.
That is, if $b$ and $b'$ are both immediate successor to $a$, then $b = b'$. | Let $b$ and $b'$ both be immediate successors to $a$.
We have that $\preceq$ is a total ordering.
{{WLOG}}:
:$b \preceq b'$
By virtue of $b'$ being a immediate successor of $a$:
:$\neg \exists c \in S: a \prec c \prec b'$
However, since $b$ is also an immediate successor:
:$a \prec b$
Hence, it cannot be the case that ... | Let $\preceq$ be a [[Definition:Total Ordering|total ordering]].
Let $b$ be an [[Definition:Immediate Successor Element|immediate successor]] to $a$.
Then $b$ is [[Definition:Unique|unique]].
That is, if $b$ and $b'$ are both [[Definition:Immediate Successor Element|immediate successor]] to $a$, then $b = b'$. | Let $b$ and $b'$ both be [[Definition:Immediate Successor Element|immediate successors]] to $a$.
We have that $\preceq$ is a [[Definition:Total Ordering|total ordering]].
{{WLOG}}:
:$b \preceq b'$
By virtue of $b'$ being a [[Definition:Immediate Successor Element|immediate successor]] of $a$:
:$\neg \exists c \in ... | Immediate Successor under Total Ordering is Unique | https://proofwiki.org/wiki/Immediate_Successor_under_Total_Ordering_is_Unique | https://proofwiki.org/wiki/Immediate_Successor_under_Total_Ordering_is_Unique | [
"Total Orderings",
"Successor Elements"
] | [
"Definition:Total Ordering",
"Definition:Immediate Successor Element",
"Definition:Unique",
"Definition:Immediate Successor Element"
] | [
"Definition:Immediate Successor Element",
"Definition:Total Ordering",
"Definition:Immediate Successor Element",
"Definition:Immediate Successor Element"
] |
proofwiki-8856 | Immediate Predecessor under Total Ordering is Unique | Let $\preceq$ be a total ordering.
Let $a$ be an immediate predecessor to $b$.
Then $a$ is unique.
That is, if $a$ and $a'$ are both immediate predecessors to $b$, then $a = a'$. | Let $a$ and $a'$ both be immediate predecessors of $b$.
We have that $\preceq$ is a total ordering.
{{WLOG}}, suppose:
:$a \preceq a'$
By virtue of $a$ being a immediate predecessor of $b$:
:$\neg \exists c \in S: a \prec c \prec b$
However, since $a'$ is also an immediate predecessor:
:$a' \prec b$
Hence, it cannot be... | Let $\preceq$ be a [[Definition:Total Ordering|total ordering]].
Let $a$ be an [[Definition:Immediate Predecessor Element|immediate predecessor]] to $b$.
Then $a$ is [[Definition:Unique|unique]].
That is, if $a$ and $a'$ are both [[Definition:Immediate Predecessor Element|immediate predecessors]] to $b$, then $a = ... | Let $a$ and $a'$ both be [[Definition:Immediate Predecessor Element|immediate predecessors]] of $b$.
We have that $\preceq$ is a [[Definition:Total Ordering|total ordering]].
{{WLOG}}, suppose:
:$a \preceq a'$
By virtue of $a$ being a [[Definition:Immediate Predecessor Element|immediate predecessor]] of $b$:
:$\ne... | Immediate Predecessor under Total Ordering is Unique | https://proofwiki.org/wiki/Immediate_Predecessor_under_Total_Ordering_is_Unique | https://proofwiki.org/wiki/Immediate_Predecessor_under_Total_Ordering_is_Unique | [
"Total Orderings",
"Predecessor Elements"
] | [
"Definition:Total Ordering",
"Definition:Immediate Predecessor Element",
"Definition:Unique",
"Definition:Immediate Predecessor Element"
] | [
"Definition:Immediate Predecessor Element",
"Definition:Total Ordering",
"Definition:Immediate Predecessor Element",
"Definition:Immediate Predecessor Element"
] |
proofwiki-8857 | Equivalence of Definitions of Complex Inverse Tangent Function | {{TFAE|def = Complex Inverse Tangent}}
Let $S$ be the subset of the complex plane:
:$S = \C \setminus \set {0 + i, 0 - i}$ | The proof strategy is to how that for all $z \in \C$:
:$\set {w \in \C: \tan w = z} = \set {\dfrac 1 {2 i} \map \ln {\dfrac {i - z} {i + z} } + k \pi: k \in \Z}$
Note that when $z = 0 + i$:
{{begin-eqn}}
{{eqn | l = i - z
| r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {i - z} {i + z}
|... | {{TFAE|def = Complex Inverse Tangent}}
Let $S$ be the [[Definition:Subset|subset]] of the [[Definition:Complex Plane|complex plane]]:
:$S = \C \setminus \set {0 + i, 0 - i}$ | The proof strategy is to how that for all $z \in \C$:
:$\set {w \in \C: \tan w = z} = \set {\dfrac 1 {2 i} \map \ln {\dfrac {i - z} {i + z} } + k \pi: k \in \Z}$
Note that when $z = 0 + i$:
{{begin-eqn}}
{{eqn | l = i - z
| r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {i - z} {i + z}
... | Equivalence of Definitions of Complex Inverse Tangent Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Inverse_Tangent_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Inverse_Tangent_Function | [
"Inverse Tangent"
] | [
"Definition:Subset",
"Definition:Complex Number/Complex Plane"
] | [] |
proofwiki-8858 | Equivalence of Definitions of Complex Inverse Cotangent Function | {{TFAE|def = Complex Inverse Cotangent}}
Let $S$ be the subset of the complex plane:
:$S = \C \setminus \set {0 + i, 0 - i}$ | The proof strategy is to how that for all $z \in S$:
:$\set {w \in \C: \cot w = z} = \set {\dfrac 1 {2 i} \map \ln {\dfrac {z + i} {z - i} } + k \pi: k \in \Z}$
Note that when $z = 0 - i$:
{{begin-eqn}}
{{eqn | l = z + i
| r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {z + i} {z - i}
| ... | {{TFAE|def = Complex Inverse Cotangent}}
Let $S$ be the [[Definition:Subset|subset]] of the [[Definition:Complex Plane|complex plane]]:
:$S = \C \setminus \set {0 + i, 0 - i}$ | The proof strategy is to how that for all $z \in S$:
:$\set {w \in \C: \cot w = z} = \set {\dfrac 1 {2 i} \map \ln {\dfrac {z + i} {z - i} } + k \pi: k \in \Z}$
Note that when $z = 0 - i$:
{{begin-eqn}}
{{eqn | l = z + i
| r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {z + i} {z - i}
... | Equivalence of Definitions of Complex Inverse Cotangent Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Inverse_Cotangent_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Inverse_Cotangent_Function | [
"Inverse Cotangent"
] | [
"Definition:Subset",
"Definition:Complex Number/Complex Plane"
] | [] |
proofwiki-8859 | Equivalence of Definitions of Complex Inverse Hyperbolic Tangent | {{TFAE|def = Complex Inverse Hyperbolic Tangent}}
Let $S$ be the subset of the complex plane:
:$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$ | The proof strategy is to how that for all $z \in S$:
:$\set {w \in \C: z = \tanh w} = \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$
Note that when $z = -1 + 0 i$:
{{begin-eqn}}
{{eqn | l = 1 + z
| r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {1 + z} {1 - z}
... | {{TFAE|def = Complex Inverse Hyperbolic Tangent}}
Let $S$ be the [[Definition:Subset|subset]] of the [[Definition:Complex Plane|complex plane]]:
:$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$ | The proof strategy is to how that for all $z \in S$:
:$\set {w \in \C: z = \tanh w} = \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$
Note that when $z = -1 + 0 i$:
{{begin-eqn}}
{{eqn | l = 1 + z
| r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {1 + z} {1 - z}
... | Equivalence of Definitions of Complex Inverse Hyperbolic Tangent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Inverse_Hyperbolic_Tangent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Inverse_Hyperbolic_Tangent | [
"Inverse Hyperbolic Tangent"
] | [
"Definition:Subset",
"Definition:Complex Number/Complex Plane"
] | [] |
proofwiki-8860 | Equivalence of Definitions of Complex Inverse Hyperbolic Cotangent | {{TFAE|def = Complex Inverse Hyperbolic Cotangent}}
Let $S$ be the subset of the complex plane:
:$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$ | The proof strategy is to how that for all $z \in S$:
:$\set {w \in \C: z = \coth w} = \set {\dfrac 1 2 \map \ln {\dfrac {z + 1} {z - 1} } + k \pi i: k \in \Z}$
Note that when $z = -1 + 0 i$:
{{begin-eqn}}
{{eqn | l = z + 1
| r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {z + 1} {z - 1}
... | {{TFAE|def = Complex Inverse Hyperbolic Cotangent}}
Let $S$ be the [[Definition:Subset|subset]] of the [[Definition:Complex Plane|complex plane]]:
:$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$ | The proof strategy is to how that for all $z \in S$:
:$\set {w \in \C: z = \coth w} = \set {\dfrac 1 2 \map \ln {\dfrac {z + 1} {z - 1} } + k \pi i: k \in \Z}$
Note that when $z = -1 + 0 i$:
{{begin-eqn}}
{{eqn | l = z + 1
| r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {z + 1} {z - 1}
... | Equivalence of Definitions of Complex Inverse Hyperbolic Cotangent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Inverse_Hyperbolic_Cotangent | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complex_Inverse_Hyperbolic_Cotangent | [
"Inverse Hyperbolic Cotangent"
] | [
"Definition:Subset",
"Definition:Complex Number/Complex Plane"
] | [] |
proofwiki-8861 | Natural Numbers form Inductive Set | Let $\N$ denote the natural numbers as subset of the real numbers $\R$.
Then $\N$ is an inductive set. | By definition of the natural numbers:
:$\N = \ds \bigcap \II$
where $\II$ is the collection of all inductive sets.
Suppose that $n \in \N$.
Then by definition of intersection:
:$\forall I \in \II: n \in I$
Because all these $I$ are inductive:
:$\forall I \in \II: n + 1 \in I$
Again by definition of intersection:
:$n + ... | Let $\N$ denote the [[Definition:Natural Numbers in Real Numbers|natural numbers]] as [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] $\R$.
Then $\N$ is an [[Definition:Inductive Set as Subset of Real Numbers|inductive set]]. | By definition of the [[Definition:Natural Numbers in Real Numbers|natural numbers]]:
:$\N = \ds \bigcap \II$
where $\II$ is the collection of all [[Definition:Inductive Set as Subset of Real Numbers|inductive sets]].
Suppose that $n \in \N$.
Then by definition of [[Definition:Intersection of Set of Sets|intersecti... | Natural Numbers form Inductive Set/Proof 1 | https://proofwiki.org/wiki/Natural_Numbers_form_Inductive_Set | https://proofwiki.org/wiki/Natural_Numbers_form_Inductive_Set/Proof_1 | [
"Natural Numbers",
"Inductive Sets",
"Natural Numbers form Inductive Set"
] | [
"Definition:Natural Numbers/Inductive Sets in Real Numbers",
"Definition:Subset",
"Definition:Real Number",
"Definition:Inductive Set/Subset of Real Numbers"
] | [
"Definition:Natural Numbers/Inductive Sets in Real Numbers",
"Definition:Inductive Set/Subset of Real Numbers",
"Definition:Set Intersection/Set of Sets",
"Definition:Inductive Set/Subset of Real Numbers",
"Definition:Set Intersection/Set of Sets"
] |
proofwiki-8862 | Natural Numbers form Inductive Set | Let $\N$ denote the natural numbers as subset of the real numbers $\R$.
Then $\N$ is an inductive set. | By the given definition of the natural numbers:
:$\N = \bigcap \II$
where $\II$ is the collection of all inductive sets.
The result is a direct application of Intersection of Inductive Set as Subset of Real Numbers is Inductive Set.
{{qed}} | Let $\N$ denote the [[Definition:Natural Numbers in Real Numbers|natural numbers]] as [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] $\R$.
Then $\N$ is an [[Definition:Inductive Set as Subset of Real Numbers|inductive set]]. | By the given definition of the [[Definition:Natural Numbers in Real Numbers|natural numbers]]:
:$\N = \bigcap \II$
where $\II$ is the collection of all [[Definition:Inductive Set as Subset of Real Numbers|inductive sets]].
The result is a direct application of [[Intersection of Inductive Set as Subset of Real Numbe... | Natural Numbers form Inductive Set/Proof 2 | https://proofwiki.org/wiki/Natural_Numbers_form_Inductive_Set | https://proofwiki.org/wiki/Natural_Numbers_form_Inductive_Set/Proof_2 | [
"Natural Numbers",
"Inductive Sets",
"Natural Numbers form Inductive Set"
] | [
"Definition:Natural Numbers/Inductive Sets in Real Numbers",
"Definition:Subset",
"Definition:Real Number",
"Definition:Inductive Set/Subset of Real Numbers"
] | [
"Definition:Natural Numbers/Inductive Sets in Real Numbers",
"Definition:Inductive Set/Subset of Real Numbers",
"Intersection of Inductive Set as Subset of Real Numbers is Inductive Set"
] |
proofwiki-8863 | Square Root of Complex Number in Cartesian Form | Let $z \in \C$ be a complex number.
Let $z = x + i y$ where $x, y \in \R$ are real numbers.
Let $z$ not be wholly real, that is, such that $y \ne 0$.
Then the square root of $z$ is given by:
:$z^{1/2} = \pm \paren {a + i b}$
where:
{{begin-eqn}}
{{eqn | l = a
| r = \sqrt {\frac {x + \sqrt {x^2 + y^2} } 2}
|... | Let $a + i b \in z^{1/2}$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {a + i b}^2
| r = x + i y
| c = {{Defof|Square Root|subdef = Complex Number|index = 4|Square Root of Complex Number}}
}}
{{eqn | n = 1
| ll= \leadsto
| l = a^2 + 2 i a b - b^2
| r = x + i y
| c = Square of Sum and $i^2... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $z = x + i y$ where $x, y \in \R$ are [[Definition:Real Number|real numbers]].
Let $z$ not be [[Definition:Wholly Real|wholly real]], that is, such that $y \ne 0$.
Then the [[Definition:Complex Square Root|square root]] of $z$ is given by:
:$z^{1... | Let $a + i b \in z^{1/2}$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {a + i b}^2
| r = x + i y
| c = {{Defof|Square Root|subdef = Complex Number|index = 4|Square Root of Complex Number}}
}}
{{eqn | n = 1
| ll= \leadsto
| l = a^2 + 2 i a b - b^2
| r = x + i y
| c = [[Square of Sum]] an... | Square Root of Complex Number in Cartesian Form | https://proofwiki.org/wiki/Square_Root_of_Complex_Number_in_Cartesian_Form | https://proofwiki.org/wiki/Square_Root_of_Complex_Number_in_Cartesian_Form | [
"Square Root of Complex Number in Cartesian Form",
"Complex Analysis",
"Square Roots",
"Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Real Number",
"Definition:Complex Number/Wholly Real",
"Definition:Square Root/Complex Number"
] | [
"Square of Sum",
"Definition:Complex Number/Imaginary Part",
"Definition:Complex Number/Real Part",
"Solution to Quadratic Equation",
"Definition:Square Root",
"Definition:Square Root/Positive",
"Definition:Discriminant of Polynomial/Quadratic Equation",
"Definition:Square Root/Negative",
"Differenc... |
proofwiki-8864 | Square Root of Complex Number in Cartesian Form | Let $z \in \C$ be a complex number.
Let $z = x + i y$ where $x, y \in \R$ are real numbers.
Let $z$ not be wholly real, that is, such that $y \ne 0$.
Then the square root of $z$ is given by:
:$z^{1/2} = \pm \paren {a + i b}$
where:
{{begin-eqn}}
{{eqn | l = a
| r = \sqrt {\frac {x + \sqrt {x^2 + y^2} } 2}
|... | We have that:
:$-15 - 8 i = 17 \map \cis {\theta + 2 k \pi}$
where:
:$\cos \theta = -\dfrac {15} {17}$
:$\sin \theta = -\dfrac 8 {17}$
Then the square roots of $-15 - 8 i$ are:
{{begin-eqn}}
{{eqn | n = 1
| l = z_1
| r = \sqrt {17} \cis \dfrac \theta 2
| c =
}}
{{eqn | n = 2
| l = z_2
| r... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $z = x + i y$ where $x, y \in \R$ are [[Definition:Real Number|real numbers]].
Let $z$ not be [[Definition:Wholly Real|wholly real]], that is, such that $y \ne 0$.
Then the [[Definition:Complex Square Root|square root]] of $z$ is given by:
:$z^{1... | We have that:
:$-15 - 8 i = 17 \map \cis {\theta + 2 k \pi}$
where:
:$\cos \theta = -\dfrac {15} {17}$
:$\sin \theta = -\dfrac 8 {17}$
Then the [[Definition:Square Root|square roots]] of $-15 - 8 i$ are:
{{begin-eqn}}
{{eqn | n = 1
| l = z_1
| r = \sqrt {17} \cis \dfrac \theta 2
| c =
}}
{{eqn | n... | Square Root of Complex Number in Cartesian Form/Examples/-15-8i/Proof 1 | https://proofwiki.org/wiki/Square_Root_of_Complex_Number_in_Cartesian_Form | https://proofwiki.org/wiki/Square_Root_of_Complex_Number_in_Cartesian_Form/Examples/-15-8i/Proof_1 | [
"Square Root of Complex Number in Cartesian Form",
"Complex Analysis",
"Square Roots",
"Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Real Number",
"Definition:Complex Number/Wholly Real",
"Definition:Square Root/Complex Number"
] | [
"Definition:Square Root",
"Definition:Cartesian Plane/Quadrants/Third",
"Definition:Cartesian Plane/Quadrants/Second"
] |
proofwiki-8865 | Zero is Identity in Naturally Ordered Semigroup | Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $0$ be the zero of $\struct {S, \circ, \preceq}$.
Then $0$ is the identity for $\circ$.
That is:
:$\forall n \in S: n \circ 0 = n = 0 \circ n$ | By definition of an ordering:
:$0 \preceq 0$
Thus from {{NOSAxiom|3}}:
:$\exists p \in S: 0 \circ p = 0$
By the definition of zero:
:$0 \preceq 0 \circ 0$ and $0 \preceq p$
Thus since $\preceq$ is compatible with $\circ$:
:$0 \circ 0 \preceq 0 \circ p = 0$
Thus:
:$0 \circ 0 \preceq 0$ and $0 \preceq 0 \circ 0$
Hence, ... | Let $\struct {S, \circ, \preceq}$ be a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
Let $0$ be the [[Definition:Zero of Naturally Ordered Semigroup|zero]] of $\struct {S, \circ, \preceq}$.
Then $0$ is the [[Definition:Identity Element|identity]] for $\circ$.
That is:
:$\forall n \in S: n... | By definition of an [[Definition:Ordering|ordering]]:
:$0 \preceq 0$
Thus from {{NOSAxiom|3}}:
:$\exists p \in S: 0 \circ p = 0$
By the definition of [[Definition:Zero of Naturally Ordered Semigroup|zero]]:
:$0 \preceq 0 \circ 0$ and $0 \preceq p$
Thus since $\preceq$ is [[Definition:Relation Compatible with O... | Zero is Identity in Naturally Ordered Semigroup | https://proofwiki.org/wiki/Zero_is_Identity_in_Naturally_Ordered_Semigroup | https://proofwiki.org/wiki/Zero_is_Identity_in_Naturally_Ordered_Semigroup | [
"Naturally Ordered Semigroup"
] | [
"Definition:Naturally Ordered Semigroup",
"Definition:Zero (Number)/Naturally Ordered Semigroup",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Ordering",
"Definition:Zero (Number)/Naturally Ordered Semigroup",
"Definition:Relation Compatible with Operation",
"Definition:Antisymmetric Relation",
"Definition:Semigroup",
"Definition:Associative Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-8866 | Reduction Formula for Integral of Power of Sine | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Let:
:$I_n := \ds \int \sin^n x \rd x$
Then:
:$I_n = \dfrac {n - 1} n I_{n - 2} - \dfrac {\sin^{n - 1} x \cos x} n$
is a reduction formula for $\ds \int \sin^n x \rd x$. | Let $n \ge 2$.
Let:
:$\ds I_n := \int \sin^n x \rd x$
Then:
{{begin-eqn}}
{{eqn | l = I_n
| r = \int \sin^n x \rd x
| c =
}}
{{eqn | r = \int \sin^{n - 1} x \sin x \rd x
| c =
}}
{{eqn | r = \int \sin^{n - 1} x \map \rd {-\cos x}
| c = Derivative of Cosine Function
}}
{{eqn | r = - \sin^{n - 1... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let:
:$I_n := \ds \int \sin^n x \rd x$
Then:
:$I_n = \dfrac {n - 1} n I_{n - 2} - \dfrac {\sin^{n - 1} x \cos x} n$
is a [[Definition:Reduction Formula (Calculus)|reduction formula]] for $\ds \int \sin^n x \rd x$. | Let $n \ge 2$.
Let:
:$\ds I_n := \int \sin^n x \rd x$
Then:
{{begin-eqn}}
{{eqn | l = I_n
| r = \int \sin^n x \rd x
| c =
}}
{{eqn | r = \int \sin^{n - 1} x \sin x \rd x
| c =
}}
{{eqn | r = \int \sin^{n - 1} x \map \rd {-\cos x}
| c = [[Derivative of Cosine Function]]
}}
{{eqn | r = - \sin... | Reduction Formula for Integral of Power of Sine/Proof 1 | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Sine | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Sine/Proof_1 | [
"Reduction Formula for Integral of Power of Sine",
"Reduction Formulae (Calculus)",
"Primitives involving Sine Function"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Reduction Formula (Calculus)"
] | [
"Derivative of Cosine Function",
"Integration by Parts",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Linear Combination of Integrals/Indefinite",
"Sum of Squares of Sine and Cosine",
"Primitive of Sine Function"
] |
proofwiki-8867 | Reduction Formula for Integral of Power of Sine | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
Let:
:$I_n := \ds \int \sin^n x \rd x$
Then:
:$I_n = \dfrac {n - 1} n I_{n - 2} - \dfrac {\sin^{n - 1} x \cos x} n$
is a reduction formula for $\ds \int \sin^n x \rd x$. | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin^{n - 1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {n - 1} \sin ^{n - 2} x \cos x
| c... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let:
:$I_n := \ds \int \sin^n x \rd x$
Then:
:$I_n = \dfrac {n - 1} n I_{n - 2} - \dfrac {\sin^{n - 1} x \cos x} n$
is a [[Definition:Reduction Formula (Calculus)|reduction formula]] for $\ds \int \sin^n x \rd x$. | With a view to expressing the problem in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \sin^{n - 1} x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \paren {n - 1} \sin ^{n - 2} x \cos x
| ... | Reduction Formula for Integral of Power of Sine/Proof 2 | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Sine | https://proofwiki.org/wiki/Reduction_Formula_for_Integral_of_Power_of_Sine/Proof_2 | [
"Reduction Formula for Integral of Power of Sine",
"Reduction Formulae (Calculus)",
"Primitives involving Sine Function"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Reduction Formula (Calculus)"
] | [
"Derivative of Composite Function",
"Derivative of Sine Function",
"Power Rule for Derivatives",
"Primitive of Sine Function",
"Integration by Parts",
"Sum of Squares of Sine and Cosine",
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-8868 | Reduction Formula for Definite Integral of Power of Sine | Let $n \in \Z_{> 0}$ be a positive integer.
Let $I_n$ be defined as:
:$\ds I_n = \int_0^{\frac \pi 2} \sin^n x \rd x$
Then $\sequence {I_n}$ is a decreasing sequence of real numbers which satisfies:
:$n I_n = \paren {n - 1} I_{n - 2}$
Thus:
:$I_n = \dfrac {n - 1} n I_{n - 2}$
is a reduction formula for $I_n$. | From Shape of Sine Function:
:$\forall x \in \closedint 0 {\dfrac \pi 2}: 0 \le \sin x \le 1$
and so on the same interval:
:$0 \le \sin^{n + 1} x \le \sin^n x$
therefore:
:$\forall n \in \N: 0 < I_{n + 1} < I_n$
From Reduction Formula for Integral of Power of Sine:
:$\ds \int \sin^n x \rd x = \dfrac {n - 1} n \int \sin... | Let $n \in \Z_{> 0}$ be a [[Definition:Positive Integer|positive integer]].
Let $I_n$ be defined as:
:$\ds I_n = \int_0^{\frac \pi 2} \sin^n x \rd x$
Then $\sequence {I_n}$ is a [[Definition:Decreasing Real Sequence|decreasing sequence of real numbers]] which satisfies:
:$n I_n = \paren {n - 1} I_{n - 2}$
Thus:
:$... | From [[Shape of Sine Function]]:
:$\forall x \in \closedint 0 {\dfrac \pi 2}: 0 \le \sin x \le 1$
and so on the same [[Definition:Open Real Interval|interval]]:
:$0 \le \sin^{n + 1} x \le \sin^n x$
therefore:
:$\forall n \in \N: 0 < I_{n + 1} < I_n$
From [[Reduction Formula for Integral of Power of Sine]]:
:$\ds \in... | Reduction Formula for Definite Integral of Power of Sine | https://proofwiki.org/wiki/Reduction_Formula_for_Definite_Integral_of_Power_of_Sine | https://proofwiki.org/wiki/Reduction_Formula_for_Definite_Integral_of_Power_of_Sine | [
"Reduction Formula for Definite Integral of Power of Sine",
"Reduction Formulae (Calculus)",
"Definite Integrals involving Sine Function"
] | [
"Definition:Positive/Integer",
"Definition:Decreasing/Sequence/Real Sequence",
"Definition:Reduction Formula (Calculus)"
] | [
"Shape of Sine Function",
"Definition:Real Interval/Open",
"Reduction Formula for Integral of Power of Sine",
"Cosine of Right Angle",
"Sine of Zero is Zero"
] |
proofwiki-8869 | Stirling's Formula/Proof 2/Lemma 4 | Let $I_n$ be defined as:
:$\ds I_n = \int_0^{\frac \pi 2} \sin^n x \rd x$
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {I_{2 n} } {I_{2 n + 1} } = 1$ | {{begin-eqn}}
{{eqn | l = n I_n
| r = \paren {n - 1} I_{n - 2}
| cc= Reduction Formula for Definite Integral of Power of Sine
}}
{{eqn | ll= \leadsto
| l = 1
| o = <
| r = \frac {I_{ 2n} } {I_{2 n + 1} }
| cc=
}}
{{eqn | o = <
| r = \frac {I_{2 n - 1} } {I_{2 n + 1} }
| ... | Let $I_n$ be defined as:
:$\ds I_n = \int_0^{\frac \pi 2} \sin^n x \rd x$
Then:
:$\ds \lim_{n \mathop \to \infty} \frac {I_{2 n} } {I_{2 n + 1} } = 1$ | {{begin-eqn}}
{{eqn | l = n I_n
| r = \paren {n - 1} I_{n - 2}
| cc= [[Reduction Formula for Definite Integral of Power of Sine]]
}}
{{eqn | ll= \leadsto
| l = 1
| o = <
| r = \frac {I_{ 2n} } {I_{2 n + 1} }
| cc=
}}
{{eqn | o = <
| r = \frac {I_{2 n - 1} } {I_{2 n + 1} }
... | Stirling's Formula/Proof 2/Lemma 4 | https://proofwiki.org/wiki/Stirling's_Formula/Proof_2/Lemma_4 | https://proofwiki.org/wiki/Stirling's_Formula/Proof_2/Lemma_4 | [
"Sine Function",
"Stirling's Formula"
] | [] | [
"Reduction Formula for Definite Integral of Power of Sine",
"Reduction Formula for Definite Integral of Power of Sine",
"Reduction Formula for Definite Integral of Power of Sine"
] |
proofwiki-8870 | Definite Integral from 0 to Half Pi of Even Power of Sine x | {{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \sin^{2 n} x \rd x
| r = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2
| c =
}}
{{eqn | r = \dfrac {1 \cdot 3 \cdot 5 \cdots \paren {2 n - 1} } {2 \cdot 4 \cdot 6 \cdots 2 n} \dfrac \pi 2
| c =
}}
{{end-eqn}}
for $n \in \Z_{>0}$. | {{improve|This would be more rigorous if implemented as a formal induction proof.}}
Let $I_n = \ds \int_0^{\frac \pi 2} \sin^n x \rd x$.
Then:
{{begin-eqn}}
{{eqn | l = I_{2 n}
| r = \frac {2 n - 1} {2 n} I_{2 n - 2}
| c = Reduction Formula for Definite Integral of Power of Sine
}}
{{eqn | r = \frac {\paren... | {{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \sin^{2 n} x \rd x
| r = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2
| c =
}}
{{eqn | r = \dfrac {1 \cdot 3 \cdot 5 \cdots \paren {2 n - 1} } {2 \cdot 4 \cdot 6 \cdots 2 n} \dfrac \pi 2
| c =
}}
{{end-eqn}}
for $n \in \Z_{>0}$. | {{improve|This would be more rigorous if implemented as a formal induction proof.}}
Let $I_n = \ds \int_0^{\frac \pi 2} \sin^n x \rd x$.
Then:
{{begin-eqn}}
{{eqn | l = I_{2 n}
| r = \frac {2 n - 1} {2 n} I_{2 n - 2}
| c = [[Reduction Formula for Definite Integral of Power of Sine]]
}}
{{eqn | r = \frac... | Definite Integral from 0 to Half Pi of Even Power of Sine x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Even_Power_of_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Even_Power_of_Sine_x/Proof_1 | [
"Definite Integral from 0 to Half Pi of Even Power of Sine x",
"Reduction Formula for Definite Integral of Power of Sine",
"Definite Integrals involving Sine Function"
] | [] | [
"Reduction Formula for Definite Integral of Power of Sine",
"Reduction Formula for Definite Integral of Power of Sine",
"Reduction Formula for Definite Integral of Power of Sine",
"Integral of Constant",
"Integral of Constant",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Defin... |
proofwiki-8871 | Definite Integral from 0 to Half Pi of Odd Power of Sine x | {{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \sin^{2 n + 1} x \rd x
| r = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = \dfrac {2 \cdot 4 \cdot 6 \cdots 2 n} {3 \cdot 5 \cdot 7 \cdots \paren {2 n + 1} }
| c =
}}
{{end-eqn}}
for $n \in \Z_{>0}$. | {{improve|This would be more rigorous if implemented as a formal induction proof.}}
Let $I_n = \ds \int_0^{\frac \pi 2} \sin^n x \rd x$.
Then:
{{begin-eqn}}
{{eqn | l = I_{2 n + 1}
| r = \frac {2 n} {2 n + 1} I_{2 n - 1}
| c = Reduction Formula for Definite Integral of Power of Sine
}}
{{eqn | r = \frac {2 ... | {{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \sin^{2 n + 1} x \rd x
| r = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = \dfrac {2 \cdot 4 \cdot 6 \cdots 2 n} {3 \cdot 5 \cdot 7 \cdots \paren {2 n + 1} }
| c =
}}
{{end-eqn}}
for $n \in \Z_{>0}$. | {{improve|This would be more rigorous if implemented as a formal induction proof.}}
Let $I_n = \ds \int_0^{\frac \pi 2} \sin^n x \rd x$.
Then:
{{begin-eqn}}
{{eqn | l = I_{2 n + 1}
| r = \frac {2 n} {2 n + 1} I_{2 n - 1}
| c = [[Reduction Formula for Definite Integral of Power of Sine]]
}}
{{eqn | r = \f... | Definite Integral from 0 to Half Pi of Odd Power of Sine x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Odd_Power_of_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Odd_Power_of_Sine_x/Proof_1 | [
"Definite Integral from 0 to Half Pi of Odd Power of Sine x",
"Reduction Formula for Definite Integral of Power of Sine",
"Definite Integrals involving Sine Function"
] | [] | [
"Reduction Formula for Definite Integral of Power of Sine",
"Reduction Formula for Definite Integral of Power of Sine",
"Reduction Formula for Definite Integral of Power of Sine",
"Primitive of Sine Function",
"Cosine of Right Angle",
"Cosine of Zero is One",
"Definition:Fraction/Numerator",
"Definiti... |
proofwiki-8872 | Zero Complement is Not Empty | Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $S^*$ be the zero complement of $S$.
Then $S^*$ is not empty. | From {{NOSAxiom|4}}, we have:
:$\exists m, n \in S: m \ne n$
That is, there are at least two distinct elements in $S$.
Therefore, there must be at least one element in $S^* = S \setminus \set 0$.
So:
:$S^* = S \setminus \set 0 \ne \O$
{{qed}} | Let $\struct {S, \circ, \preceq}$ be a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
Let $S^*$ be the [[Definition:Zero Complement|zero complement]] of $S$.
Then $S^*$ is not [[Definition:Empty Set|empty]]. | From {{NOSAxiom|4}}, we have:
:$\exists m, n \in S: m \ne n$
That is, there are at least two [[Definition:Distinct Elements|distinct]] [[Definition:Element|elements]] in $S$.
Therefore, there must be at least one [[Definition:Element|element]] in $S^* = S \setminus \set 0$.
So:
:$S^* = S \setminus \set 0 \ne \O$
{... | Zero Complement is Not Empty | https://proofwiki.org/wiki/Zero_Complement_is_Not_Empty | https://proofwiki.org/wiki/Zero_Complement_is_Not_Empty | [
"Naturally Ordered Semigroup"
] | [
"Definition:Naturally Ordered Semigroup",
"Definition:Zero Complement",
"Definition:Empty Set"
] | [
"Definition:Distinct/Plural",
"Definition:Element",
"Definition:Element"
] |
proofwiki-8873 | Stirling's Formula/Proof 2/Lemma 5 | :$\dfrac {n!} {n^n \sqrt n e^{-n} } \to \sqrt {2 \pi}$ as $n \to \infty$ | By previous work done in Stirling's Formula: Proof 2 it is noted that $\sequence {\dfrac {n!} {n^n \sqrt n e^{-n} } }_{n \mathop \in \N}$ is a convergent sequence.
Let $\dfrac {n!} {n^n \sqrt n e^{-n} } \to C$ as $n \to \infty$.
Let $\ds I_n = \int_0^{\frac \pi 2} \sin^n x \rd x$.
Then:
{{begin-eqn}}
{{eqn | l = \frac ... | :$\dfrac {n!} {n^n \sqrt n e^{-n} } \to \sqrt {2 \pi}$ as $n \to \infty$ | By previous work done in [[Stirling's Formula/Proof 2|Stirling's Formula: Proof 2]] it is noted that $\sequence {\dfrac {n!} {n^n \sqrt n e^{-n} } }_{n \mathop \in \N}$ is a [[Definition:Convergent Sequence (Analysis)|convergent sequence]].
Let $\dfrac {n!} {n^n \sqrt n e^{-n} } \to C$ as $n \to \infty$.
Let $\ds I_n... | Stirling's Formula/Proof 2/Lemma 5 | https://proofwiki.org/wiki/Stirling's_Formula/Proof_2/Lemma_5 | https://proofwiki.org/wiki/Stirling's_Formula/Proof_2/Lemma_5 | [
"Stirling's Formula"
] | [] | [
"Stirling's Formula/Proof 2",
"Definition:Convergent Sequence/Analysis",
"Definite Integral from 0 to Half Pi of Even Power of Sine x",
"Definite Integral from 0 to Half Pi of Odd Power of Sine x",
"Stirling's Formula/Proof 2/Lemma 4"
] |
proofwiki-8874 | Limit of Error in Stirling's Formula | Consider Stirling's Formula:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
The ratio of $n!$ to its approximation $\sqrt {2 \pi n} \paren {\dfrac n e}^n$ is bounded as follows:
:$e^{1 / \paren {12 n + 1} } < \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } < e^{1 / 12 n}$ | Taking the natural logarithm of the ratio of $n!$ to its approximation $\sqrt {2 \pi n} \paren {\dfrac n e}^n$, we obtain:
{{begin-eqn}}
{{eqn | l = \map \ln {\dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } }
| r = \map \ln {n!} - \map \ln { {\sqrt {2 \pi n} n^n e^{-n} } }
| c = Difference of Logarithms
}}
{{eqn ... | Consider [[Stirling's Formula]]:
:$n! \sim \sqrt {2 \pi n} \paren {\dfrac n e}^n$
The ratio of $n!$ to its approximation $\sqrt {2 \pi n} \paren {\dfrac n e}^n$ is [[Definition:Bounded Real Sequence|bounded]] as follows:
:$e^{1 / \paren {12 n + 1} } < \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } < e^{1 / 12 n}$ | Taking the [[Definition:Natural Logarithm/Complex|natural logarithm]] of the ratio of $n!$ to its approximation $\sqrt {2 \pi n} \paren {\dfrac n e}^n$, we obtain:
{{begin-eqn}}
{{eqn | l = \map \ln {\dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } }
| r = \map \ln {n!} - \map \ln { {\sqrt {2 \pi n} n^n e^{-n} } }
... | Limit of Error in Stirling's Formula | https://proofwiki.org/wiki/Limit_of_Error_in_Stirling's_Formula | https://proofwiki.org/wiki/Limit_of_Error_in_Stirling's_Formula | [
"Stirling's Formula"
] | [
"Stirling's Formula",
"Definition:Bounded Sequence/Real"
] | [
"Definition:Natural Logarithm/Complex",
"Difference of Logarithms",
"Sum of Logarithms",
"Logarithm of Power/Natural Logarithm",
"Natural Logarithm of e is 1",
"Stirling's Formula/Proof 2/Lemma 3",
"Definition:Increasing/Sequence/Real Sequence",
"Difference of Logarithms",
"Definition:Fraction/Numer... |
proofwiki-8875 | Zero Strictly Precedes One | Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $0$ be the zero of $S$.
Let $1$ be the one of $S$.
Then:
:$0 \prec 1$ | This follows directly from the definition of $\prec$.
First note that:
:$\forall n \in S: 0 \preceq n$
from the definition of zero.
Next, from the definition of one:
:$0 \ne 1$
Thus:
{{begin-eqn}}
{{eqn | o =
| r = 0 \preceq 1 \land 0 \ne 1
}}
{{eqn | o = \leadsto
| r = 0 \prec 1
| c = {{Defof|Strict... | Let $\struct {S, \circ, \preceq}$ be a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
Let $0$ be the [[Definition:Zero of Naturally Ordered Semigroup|zero]] of $S$.
Let $1$ be the [[Definition:One of Naturally Ordered Semigroup|one]] of $S$.
Then:
:$0 \prec 1$ | This follows directly from the [[Definition:Strictly Precede|definition of $\prec$]].
First note that:
:$\forall n \in S: 0 \preceq n$
from the definition of [[Definition:Zero of Naturally Ordered Semigroup|zero]].
Next, from the definition of [[Definition:One of Naturally Ordered Semigroup|one]]:
:$0 \ne 1$
T... | Zero Strictly Precedes One | https://proofwiki.org/wiki/Zero_Strictly_Precedes_One | https://proofwiki.org/wiki/Zero_Strictly_Precedes_One | [
"Naturally Ordered Semigroup"
] | [
"Definition:Naturally Ordered Semigroup",
"Definition:Zero (Number)/Naturally Ordered Semigroup",
"Definition:Unit (One)/Naturally Ordered Semigroup"
] | [
"Definition:Strictly Precede",
"Definition:Zero (Number)/Naturally Ordered Semigroup",
"Definition:Unit (One)/Naturally Ordered Semigroup"
] |
proofwiki-8876 | Stirling's Formula/Proof 2/Lemma 2 | The sequence $\sequence {d_n}$ defined as:
:$d_n = \map \ln {n!} - \paren {n + \dfrac 1 2} \ln n + n$
is decreasing. | The proof strategy is to demonstrate that the sign of $d_n - d_{n + 1}$ is positive.
{{begin-eqn}}
{{eqn | l = d_n - d_{n + 1}
| r = \map \ln {n!} - \paren {n + \frac 1 2} \ln n + n
| c =
}}
{{eqn | o =
| ro= -
| r = \paren {\map \ln {\paren {n + 1}!} - \paren {n + 1 + \frac 1 2} \map \ln {n +... | The [[Definition:Real Sequence|sequence]] $\sequence {d_n}$ defined as:
:$d_n = \map \ln {n!} - \paren {n + \dfrac 1 2} \ln n + n$
is [[Definition:Decreasing Real Sequence|decreasing]]. | The proof strategy is to demonstrate that the sign of $d_n - d_{n + 1}$ is [[Definition:Positive Real Number|positive]].
{{begin-eqn}}
{{eqn | l = d_n - d_{n + 1}
| r = \map \ln {n!} - \paren {n + \frac 1 2} \ln n + n
| c =
}}
{{eqn | o =
| ro= -
| r = \paren {\map \ln {\paren {n + 1}!} - \pa... | Stirling's Formula/Proof 2/Lemma 2 | https://proofwiki.org/wiki/Stirling's_Formula/Proof_2/Lemma_2 | https://proofwiki.org/wiki/Stirling's_Formula/Proof_2/Lemma_2 | [
"Stirling's Formula"
] | [
"Definition:Real Sequence",
"Definition:Decreasing/Sequence/Real Sequence"
] | [
"Definition:Positive/Real Number",
"Stirling's Formula/Proof 2/Lemma 1",
"Definition:Decreasing/Sequence/Real Sequence"
] |
proofwiki-8877 | Power Series Expansion for Real Arctangent Function | The arctangent function has a Taylor series expansion:
:<nowiki>$\arctan x = \begin {cases} \ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \\
\ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \ge 1 \\ \\
\ds -\fr... | From Sum of Infinite Geometric Sequence:
:$(1): \quad \ds \sum_{n \mathop = 0}^\infty \paren {-x^2}^n = \frac 1 {1 + x^2}$
for $-1 < x < 1$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
{{begin-eqn}}
{{eqn | l = \int_0^x \frac 1 {1 + t^2} \rd t
| r =... | The [[Definition:Inverse Tangent|arctangent function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
:<nowiki>$\arctan x = \begin {cases} \ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \\
\ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac... | From [[Sum of Infinite Geometric Sequence]]:
:$(1): \quad \ds \sum_{n \mathop = 0}^\infty \paren {-x^2}^n = \frac 1 {1 + x^2}$
for $-1 < x < 1$.
From [[Power Series is Termwise Integrable within Radius of Convergence]], $(1)$ can be [[Definition:Integration|integrated]] term by term:
{{begin-eqn}}
{{eqn | l = \int_0^... | Power Series Expansion for Real Arctangent Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arctangent_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arctangent_Function | [
"Examples of Power Series",
"Arctangent Function"
] | [
"Definition:Inverse Tangent",
"Definition:Taylor Series"
] | [
"Sum of Infinite Geometric Sequence",
"Power Series is Termwise Integrable within Radius of Convergence",
"Definition:Primitive (Calculus)/Integration",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Integral of Power",
"Definition:Real Sequence",
"Definition:Decreasing/Sequence... |
proofwiki-8878 | Ordering in terms of Addition | Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Then $\forall m, n \in S$:
:$m \preceq n \iff \exists p \in S: m \circ p = n$ | === Necessary Condition ===
From {{NOSAxiom|3}}, we have:
:$\forall m, n \in S: m \preceq n \implies \exists p \in S: m \circ p = n$
{{qed|lemma}} | Let $\struct {S, \circ, \preceq}$ be a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
Then $\forall m, n \in S$:
:$m \preceq n \iff \exists p \in S: m \circ p = n$ | === Necessary Condition ===
From {{NOSAxiom|3}}, we have:
:$\forall m, n \in S: m \preceq n \implies \exists p \in S: m \circ p = n$
{{qed|lemma}} | Ordering in terms of Addition | https://proofwiki.org/wiki/Ordering_in_terms_of_Addition | https://proofwiki.org/wiki/Ordering_in_terms_of_Addition | [
"Naturally Ordered Semigroup"
] | [
"Definition:Naturally Ordered Semigroup"
] | [] |
proofwiki-8879 | Difference in Naturally Ordered Semigroup is Unique | Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $n, m \in S$ such that $m \preceq n$.
Then there exists a unique difference $n - m$ of $m$ and $n$. | Since $m \preceq n$, by {{NOSAxiom|3}}:
:$\exists p \in S: m \circ p = n$
Now suppose that $p, q \in S$ are such that:
:$m \circ p = m \circ q = n$
Then it follows from {{NOSAxiom|2}} that:
:$p = q$
Hence the result.
{{qed}} | Let $\struct {S, \circ, \preceq}$ be a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
Let $n, m \in S$ such that $m \preceq n$.
Then there exists a [[Definition:Unique|unique]] [[Definition:Difference (Naturally Ordered Semigroup)|difference]] $n - m$ of $m$ and $n$. | Since $m \preceq n$, by {{NOSAxiom|3}}:
:$\exists p \in S: m \circ p = n$
Now suppose that $p, q \in S$ are such that:
:$m \circ p = m \circ q = n$
Then it follows from {{NOSAxiom|2}} that:
:$p = q$
Hence the result.
{{qed}} | Difference in Naturally Ordered Semigroup is Unique | https://proofwiki.org/wiki/Difference_in_Naturally_Ordered_Semigroup_is_Unique | https://proofwiki.org/wiki/Difference_in_Naturally_Ordered_Semigroup_is_Unique | [
"Naturally Ordered Semigroup"
] | [
"Definition:Naturally Ordered Semigroup",
"Definition:Unique",
"Definition:Subtraction/Naturally Ordered Semigroup"
] | [] |
proofwiki-8880 | Equivalence of Definitions of Beta Function | {{TFAE|def = Beta Function}}
For $\map \Re x, \map \Re y > 0$: | === Definition 1 is equivalent to Definition 2 ===
{{begin-eqn}}
{{eqn | l = \map \Beta {x, y}
| r = \int_0^1 t^{x - 1} \paren {1 - t}^{y - 1} \rd t
| c = {{Defof|Beta Function|index = 1}}
}}
{{eqn | r = \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 2} \paren {\cos \theta}^{2 y - 2} 2 \sin \theta \cos \theta... | {{TFAE|def = Beta Function}}
For $\map \Re x, \map \Re y > 0$: | === [[Definition:Beta Function/Definition 1|Definition 1]] is equivalent to [[Definition:Beta Function/Definition 2|Definition 2]] ===
{{begin-eqn}}
{{eqn | l = \map \Beta {x, y}
| r = \int_0^1 t^{x - 1} \paren {1 - t}^{y - 1} \rd t
| c = {{Defof|Beta Function|index = 1}}
}}
{{eqn | r = \int_0^{\pi / 2} \p... | Equivalence of Definitions of Beta Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Beta_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Beta_Function | [
"Beta Function"
] | [] | [
"Definition:Beta Function/Definition 1",
"Definition:Beta Function/Definition 2",
"Integration by Substitution",
"Definition:Beta Function/Definition 2"
] |
proofwiki-8881 | Ordering of Naturally Ordered Semigroup is Strongly Compatible | :$\forall m, n, p \in S: m \preceq n \iff m \circ p \preceq n \circ p$ | The forward implication is immediate from $\preceq$ being compatible with $\circ$:
:$\forall m, n, p \in S: m \preceq n \implies m \circ p \preceq n \circ p$
Conversely, suppose that $m \circ p \preceq n \circ p$.
Suppose that $n \prec m$.
Then as $\preceq$ is compatible with $\circ$:
:$n \circ p \preceq m \circ p$
Sin... | :$\forall m, n, p \in S: m \preceq n \iff m \circ p \preceq n \circ p$ | The forward implication is immediate from $\preceq$ being [[Definition:Relation Compatible with Operation|compatible]] with $\circ$:
:$\forall m, n, p \in S: m \preceq n \implies m \circ p \preceq n \circ p$
Conversely, suppose that $m \circ p \preceq n \circ p$.
Suppose that $n \prec m$.
Then as $\preceq$ is [[De... | Ordering of Naturally Ordered Semigroup is Strongly Compatible | https://proofwiki.org/wiki/Ordering_of_Naturally_Ordered_Semigroup_is_Strongly_Compatible | https://proofwiki.org/wiki/Ordering_of_Naturally_Ordered_Semigroup_is_Strongly_Compatible | [
"Naturally Ordered Semigroup"
] | [] | [
"Definition:Relation Compatible with Operation",
"Definition:Relation Compatible with Operation",
"Definition:Ordering",
"Definition:Total Ordering"
] |
proofwiki-8882 | Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible | :$\forall m, n, p \in S: m \prec n \iff m \circ p \prec n \circ p$ | By {{NOSAxiom|2}}, all $n \in S$ are cancellable.
Hence from Strict Ordering Preserved under Product with Cancellable Element:
:$\forall m, n, p \in S: m \prec n \implies m \circ p \prec n \circ p$
By {{NOSAxiom|1}}, $\preceq$ is a total ordering.
Therefore, the contrapositive of:
:$\forall m, n, p \in S: m \circ p \pr... | :$\forall m, n, p \in S: m \prec n \iff m \circ p \prec n \circ p$ | By {{NOSAxiom|2}}, all $n \in S$ are [[Definition:Cancellable Element|cancellable]].
Hence from [[Strict Ordering Preserved under Product with Cancellable Element]]:
:$\forall m, n, p \in S: m \prec n \implies m \circ p \prec n \circ p$
By {{NOSAxiom|1}}, $\preceq$ is a [[Definition:Total Ordering|total ordering]].... | Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible | https://proofwiki.org/wiki/Strict_Ordering_of_Naturally_Ordered_Semigroup_is_Strongly_Compatible | https://proofwiki.org/wiki/Strict_Ordering_of_Naturally_Ordered_Semigroup_is_Strongly_Compatible | [
"Naturally Ordered Semigroup"
] | [] | [
"Definition:Cancellable Element",
"Strict Ordering Preserved under Product with Cancellable Element",
"Definition:Total Ordering",
"Definition:Contrapositive Statement"
] |
proofwiki-8883 | Sum with One is Immediate Successor in Naturally Ordered Semigroup | Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $1$ be the one of $S$.
Let $n \in S$.
Then $n \circ 1$ is the immediate successor of $n$.
That is, for all $m \in S$:
:$n \prec m \iff n \circ 1 \preceq m$ | By Zero Strictly Precedes One, $0 \prec 1$, where $0$ is the zero of $S$.
Hence from Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible:
:$n \circ 0 \prec n \circ 1$
and by Zero is Identity in Naturally Ordered Semigroup, $n \circ 0 = n$.
Now suppose that $n \prec m$.
Then by {{NOSAxiom|3}}, there ex... | Let $\struct {S, \circ, \preceq}$ be a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
Let $1$ be the [[Definition:One of Naturally Ordered Semigroup|one]] of $S$.
Let $n \in S$.
Then $n \circ 1$ is the [[Definition:Immediate Successor Element|immediate successor]] of $n$.
That is, for all ... | By [[Zero Strictly Precedes One]], $0 \prec 1$, where $0$ is the [[Definition:Zero of Naturally Ordered Semigroup|zero]] of $S$.
Hence from [[Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible]]:
:$n \circ 0 \prec n \circ 1$
and by [[Zero is Identity in Naturally Ordered Semigroup]], $n \circ 0 = ... | Sum with One is Immediate Successor in Naturally Ordered Semigroup | https://proofwiki.org/wiki/Sum_with_One_is_Immediate_Successor_in_Naturally_Ordered_Semigroup | https://proofwiki.org/wiki/Sum_with_One_is_Immediate_Successor_in_Naturally_Ordered_Semigroup | [
"Naturally Ordered Semigroup"
] | [
"Definition:Naturally Ordered Semigroup",
"Definition:Unit (One)/Naturally Ordered Semigroup",
"Definition:Immediate Successor Element"
] | [
"Zero Strictly Precedes One",
"Definition:Zero (Number)/Naturally Ordered Semigroup",
"Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible",
"Zero is Identity in Naturally Ordered Semigroup",
"Definition:Zero (Number)/Naturally Ordered Semigroup",
"Definition:Unit (One)/Naturally Ordered... |
proofwiki-8884 | Riemann Zeta Function at Even Integers | The Riemann $\zeta$ function can be calculated for even integers as follows:
{{begin-eqn}}
{{eqn | l = \map \zeta {2 n}
| r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 {1^{2 n} } + \frac 1 {2^{2 n} } + \frac 1 {3^{2 n} } + \frac 1 {4^{2 n} } + ... | {{begin-eqn}}
{{eqn | l = \frac {\sin \pi x} {\pi x}
| r = \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k^2} }
| c = Euler Formula for Sine Function
}}
{{eqn | ll= \leadsto
| l = \map \ln {\frac {\sin \pi x} {\pi x} }
| r = \ln \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k^2} ... | The [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]] can be calculated for [[Definition:Even Integer|even integers]] as follows:
{{begin-eqn}}
{{eqn | l = \map \zeta {2 n}
| r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 {1^{2 n} }... | {{begin-eqn}}
{{eqn | l = \frac {\sin \pi x} {\pi x}
| r = \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k^2} }
| c = [[Euler Formula for Sine Function]]
}}
{{eqn | ll= \leadsto
| l = \map \ln {\frac {\sin \pi x} {\pi x} }
| r = \ln \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k... | Riemann Zeta Function at Even Integers/Lemma/Proof 1 | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Even_Integers | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Even_Integers/Lemma/Proof_1 | [
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Bernoulli Numbers",
"Analytic Number Theory"
] | [
"Definition:Riemann Zeta Function",
"Definition:Even Integer",
"Definition:Bernoulli Numbers",
"Definition:Positive/Integer"
] | [
"Euler Formula for Sine Function",
"Laws of Logarithms",
"Definition:Differentiation",
"Sum of Infinite Geometric Sequence",
"Tonelli's Theorem"
] |
proofwiki-8885 | Riemann Zeta Function at Even Integers | The Riemann $\zeta$ function can be calculated for even integers as follows:
{{begin-eqn}}
{{eqn | l = \map \zeta {2 n}
| r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 {1^{2 n} } + \frac 1 {2^{2 n} } + \frac 1 {3^{2 n} } + \frac 1 {4^{2 n} } + ... | From Laurent Series Expansion for Cotangent Function:
:$\ds \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$
where:
:$z \in \C$ such that $\cmod z < 1$
:$\zeta$ is the Riemann Zeta function.
Letting $x \in \R$ replace $z$, and multiplying through by $x$:
{{begin-eqn}}
{{eqn | l =... | The [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]] can be calculated for [[Definition:Even Integer|even integers]] as follows:
{{begin-eqn}}
{{eqn | l = \map \zeta {2 n}
| r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 {1^{2 n} }... | From [[Laurent Series Expansion for Cotangent Function]]:
:$\ds \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$
where:
:$z \in \C$ such that $\cmod z < 1$
:$\zeta$ is the [[Definition:Riemann Zeta Function|Riemann Zeta function]].
Letting $x \in \R$ replace $z$, and multiplyi... | Riemann Zeta Function at Even Integers/Lemma/Proof 2 | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Even_Integers | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Even_Integers/Lemma/Proof_2 | [
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Bernoulli Numbers",
"Analytic Number Theory"
] | [
"Definition:Riemann Zeta Function",
"Definition:Even Integer",
"Definition:Bernoulli Numbers",
"Definition:Positive/Integer"
] | [
"Laurent Series Expansion for Cotangent Function",
"Definition:Riemann Zeta Function"
] |
proofwiki-8886 | Riemann Zeta Function at Even Integers | The Riemann $\zeta$ function can be calculated for even integers as follows:
{{begin-eqn}}
{{eqn | l = \map \zeta {2 n}
| r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 {1^{2 n} } + \frac 1 {2^{2 n} } + \frac 1 {3^{2 n} } + \frac 1 {4^{2 n} } + ... | === Lemma ===
{{:Riemann Zeta Function at Even Integers/Lemma}}
We also have:
{{begin-eqn}}
{{eqn | l = \pi x \cot {\pi x}
| r = i \pi x \frac {e^{i \pi x} + e^{- i \pi x} } {e^{i \pi x} - e^{- i \pi x} }
| c = Euler's Cotangent Identity
}}
{{eqn | r = i \pi x \frac {e^{2 i \pi x} + 1} {e^{2 i \pi x} - 1}
... | The [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]] can be calculated for [[Definition:Even Integer|even integers]] as follows:
{{begin-eqn}}
{{eqn | l = \map \zeta {2 n}
| r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 {1^{2 n} }... | === [[Riemann Zeta Function at Even Integers/Lemma|Lemma]] ===
{{:Riemann Zeta Function at Even Integers/Lemma}}
We also have:
{{begin-eqn}}
{{eqn | l = \pi x \cot {\pi x}
| r = i \pi x \frac {e^{i \pi x} + e^{- i \pi x} } {e^{i \pi x} - e^{- i \pi x} }
| c = [[Euler's Cotangent Identity]]
}}
{{eqn | r = ... | Riemann Zeta Function at Even Integers/Proof 1 | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Even_Integers | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Even_Integers/Proof_1 | [
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Bernoulli Numbers",
"Analytic Number Theory"
] | [
"Definition:Riemann Zeta Function",
"Definition:Even Integer",
"Definition:Bernoulli Numbers",
"Definition:Positive/Integer"
] | [
"Riemann Zeta Function at Even Integers/Lemma",
"Euler's Cotangent Identity",
"Odd Bernoulli Numbers Vanish"
] |
proofwiki-8887 | Riemann Zeta Function at Even Integers | The Riemann $\zeta$ function can be calculated for even integers as follows:
{{begin-eqn}}
{{eqn | l = \map \zeta {2 n}
| r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 {1^{2 n} } + \frac 1 {2^{2 n} } + \frac 1 {3^{2 n} } + \frac 1 {4^{2 n} } + ... | Let $k \in \N$.
Let $\map S x$ be equal to $x^{2 k}$ on $\closedint {-\pi} \pi$ and be periodic with period $2 \pi$.
Let $\ds \map I {2 m, n} = \int_0^\pi x^{2 m} \cos n x \rd x$.
Let $\map A {2 m, n} = \dfrac {\pi^{2 m - 1} \paren {-1}^n 2 m} {n^2}$.
Let $\map B {2 m, n} = -\dfrac {2 m \paren {2 m - 1} } {n^2}$.
By Fo... | The [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]] can be calculated for [[Definition:Even Integer|even integers]] as follows:
{{begin-eqn}}
{{eqn | l = \map \zeta {2 n}
| r = \paren {-1}^{n + 1} \dfrac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 {1^{2 n} }... | Let $k \in \N$.
Let $\map S x$ be equal to $x^{2 k}$ on $\closedint {-\pi} \pi$ and be [[Definition:Periodic Real Function|periodic]] with [[Definition:Period of Periodic Real Function|period]] $2 \pi$.
Let $\ds \map I {2 m, n} = \int_0^\pi x^{2 m} \cos n x \rd x$.
Let $\map A {2 m, n} = \dfrac {\pi^{2 m - 1} \paren... | Riemann Zeta Function at Even Integers/Proof 2 | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Even_Integers | https://proofwiki.org/wiki/Riemann_Zeta_Function_at_Even_Integers/Proof_2 | [
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Bernoulli Numbers",
"Analytic Number Theory"
] | [
"Definition:Riemann Zeta Function",
"Definition:Even Integer",
"Definition:Bernoulli Numbers",
"Definition:Positive/Integer"
] | [
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Fourier Series for Even Function over Symmetric Range",
"Integration by Parts",
"Integration by Parts",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Sum of Bernoulli Numbers by Power of Two a... |
proofwiki-8888 | Tangent Inequality | :$x < \tan x$
for all $x$ in the interval $\openint 0 {\dfrac \pi 2}$. | Let $\map f x = \tan x - x$.
By Derivative of Tangent Function, $\map {f'} x = \sec^2 x - 1$.
By Shape of Secant Function, $\sec^2 x > 1$ for $x \in \openint 0 {\dfrac \pi 2}$.
Hence $\map {f'} x > 0$.
From Derivative of Monotone Function, $\map f x$ is strictly increasing in this interval.
Since $\map f 0 = 0$, it fol... | :$x < \tan x$
for all $x$ in the [[Definition:Open Real Interval|interval]] $\openint 0 {\dfrac \pi 2}$. | Let $\map f x = \tan x - x$.
By [[Derivative of Tangent Function]], $\map {f'} x = \sec^2 x - 1$.
By [[Shape of Secant Function]], $\sec^2 x > 1$ for $x \in \openint 0 {\dfrac \pi 2}$.
Hence $\map {f'} x > 0$.
From [[Derivative of Monotone Function]], $\map f x$ is [[Definition:Strictly Increasing|strictly increasi... | Tangent Inequality | https://proofwiki.org/wiki/Tangent_Inequality | https://proofwiki.org/wiki/Tangent_Inequality | [
"Tangent Function",
"Inequalities"
] | [
"Definition:Real Interval/Open"
] | [
"Derivative of Tangent Function",
"Shape of Secant Function",
"Derivative of Monotone Function",
"Definition:Strictly Increasing",
"Definition:Real Interval/Open",
"Category:Tangent Function",
"Category:Inequalities"
] |
proofwiki-8889 | Integral Representation of Riemann Zeta Function in terms of Gamma Function | For $\Re \paren s > 1$, the Riemann Zeta function is given by:
:$\ds \map \zeta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {t^{s - 1}} {e^t - 1} \rd t$
where $\Gamma$ is the Gamma function. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \rd t
| r = \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \times \dfrac {e^{-t} } {e^{-t} }\rd t
| c = multiplying by 1
}}
{{eqn | r = \int_0^\infty \frac {t^{s - 1} e^{-t} } {1 - e^{-t} } \rd t
| c =
}}
{{eqn | r = \int_0^\infty t^{s ... | For $\Re \paren s > 1$, the [[Definition:Riemann Zeta Function|Riemann Zeta function]] is given by:
:$\ds \map \zeta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {t^{s - 1}} {e^t - 1} \rd t$
where $\Gamma$ is the [[Definition:Gamma Function|Gamma function]]. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \rd t
| r = \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \times \dfrac {e^{-t} } {e^{-t} }\rd t
| c = multiplying by 1
}}
{{eqn | r = \int_0^\infty \frac {t^{s - 1} e^{-t} } {1 - e^{-t} } \rd t
| c =
}}
{{eqn | r = \int_0^\infty t^{s ... | Integral Representation of Riemann Zeta Function in terms of Gamma Function | https://proofwiki.org/wiki/Integral_Representation_of_Riemann_Zeta_Function_in_terms_of_Gamma_Function | https://proofwiki.org/wiki/Integral_Representation_of_Riemann_Zeta_Function_in_terms_of_Gamma_Function | [
"Integral Representation of Riemann Zeta Function in terms of Gamma Function",
"Definite Integrals involving Exponential Function",
"Gamma Function",
"Riemann Zeta Function",
"Analytic Number Theory"
] | [
"Definition:Riemann Zeta Function",
"Definition:Gamma Function"
] | [
"Sum of Infinite Geometric Sequence",
"Exponent Combination Laws/Product of Powers",
"Fubini's Theorem",
"Integration by Substitution"
] |
proofwiki-8890 | Bohr-Mollerup Theorem | Let $f: \R_{>0} \to \R_{>0}$ be a real function which is positive on $\openint 0 \to$.
Let $\ln \mathop \circ f$ be convex on $\R_{>0}$.
Let $f$ satisfy the conditions:
:<nowiki>$\map f {x + 1} = \begin{cases}
1 & : x = 0 \\
x \map f x & : x > 0 \end{cases}$</nowiki>
Then:
:$\forall x \in \R_{>0}: \map f x = \map \Gamm... | Let $s, t \in \R_{>0}$ such that $s \le t \le s + 1$.
Let $t = \alpha s + \beta \paren {s + 1}$ where $\alpha, \beta \in \R_{>0}, \alpha + \beta = 1$.
Then:
:$t = \paren {\alpha + \beta} s + \beta = s + \beta$
and so:
:$\beta = t - s$
Thus:
{{begin-eqn}}
{{eqn | l = \map \ln {\map f t}
| o = \le
| r = \alph... | Let $f: \R_{>0} \to \R_{>0}$ be a [[Definition:Real Function|real function]] which is [[Definition:Strictly Positive Real Number|positive]] on $\openint 0 \to$.
Let $\ln \mathop \circ f$ be [[Definition:Convex Real Function|convex]] on $\R_{>0}$.
Let $f$ satisfy the conditions:
:<nowiki>$\map f {x + 1} = \begin{cases... | Let $s, t \in \R_{>0}$ such that $s \le t \le s + 1$.
Let $t = \alpha s + \beta \paren {s + 1}$ where $\alpha, \beta \in \R_{>0}, \alpha + \beta = 1$.
Then:
:$t = \paren {\alpha + \beta} s + \beta = s + \beta$
and so:
:$\beta = t - s$
Thus:
{{begin-eqn}}
{{eqn | l = \map \ln {\map f t}
| o = \le
| r = \... | Bohr-Mollerup Theorem | https://proofwiki.org/wiki/Bohr-Mollerup_Theorem | https://proofwiki.org/wiki/Bohr-Mollerup_Theorem | [
"Gamma Function"
] | [
"Definition:Real Function",
"Definition:Strictly Positive/Real Number",
"Definition:Convex Real Function",
"Definition:Gamma Function",
"Definition:Gamma Function/Euler Form"
] | [
"Definition:Convex Real Function",
"Squeeze Theorem/Sequences/Real Numbers",
"Definition:Gamma Function/Euler Form",
"Gamma Difference Equation"
] |
proofwiki-8891 | Equivalence of Definitions of Convex Real Function | Let $f$ be a real function which is defined on a real interval $I$.
{{TFAE|def = Convex Real Function}} | Let $f$ be convex real function on $I$ according to definition 1.
That is:
:$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$
{{WLOG}}, assume $x \le y$.
Make the substitutions $x_1 = x, x_2 = \alpha x + \beta y, x_3 = y$.
As $\... | Let $f$ be a [[Definition:Real Function|real function]] which is defined on a [[Definition:Real Interval|real interval]] $I$.
{{TFAE|def = Convex Real Function}} | Let $f$ be [[Definition:Convex Real Function/Definition 1|convex real function on $I$ according to definition 1]].
That is:
:$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \le \alpha \map f x + \beta \map f y$
{{WLOG}}, assume $x \le y$.
Make the substitutions... | Equivalence of Definitions of Convex Real Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convex_Real_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convex_Real_Function | [
"Convex Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval"
] | [
"Definition:Convex Real Function/Definition 1",
"Definition:Convex Real Function/Definition 3",
"Definition:Convex Real Function/Definition 2"
] |
proofwiki-8892 | Equivalence of Definitions of Concave Real Function | Let $f$ be a real function which is defined on a real interval $I$.
{{TFAE|def = Concave Real Function}} | Let $f$ be concave real function on $I$ according to definition 1.
That is:
:$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \ge \alpha \map f x + \beta \map f y$
Make the substitutions $x_1 = x, x_2 = \alpha x + \beta y, x_3 = y$.
As $\alpha + \beta = 1$, we have... | Let $f$ be a [[Definition:Real Function|real function]] which is defined on a [[Definition:Real Interval|real interval]] $I$.
{{TFAE|def = Concave Real Function}} | Let $f$ be [[Definition:Concave Real Function/Definition 1|concave real function on $I$ according to definition 1]].
That is:
:$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \ge \alpha \map f x + \beta \map f y$
Make the substitutions $x_1 = x, x_2 = \alpha x +... | Equivalence of Definitions of Concave Real Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Concave_Real_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Concave_Real_Function | [
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval"
] | [
"Definition:Concave Real Function/Definition 1",
"Definition:Concave Real Function/Definition 3",
"Definition:Concave Real Function/Definition 2",
"Category:Concave Real Functions"
] |
proofwiki-8893 | Power Series Expansion for Real Arcsine Function | The (real) arcsine function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \arcsin x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}
| c =
}}
{{eqn | r = x + \frac {x^3} {2 \times 3} + \frac {\paren {1 \times 3} x^5} {2 \times 4 \time... | From the General Binomial Theorem:
{{begin-eqn}}
{{eqn | l = \paren {1 - x^2}^{-1/2}
| r = 1 + \frac 1 2 x^2 + \frac {1 \times 3} {2 \times 4} x^4 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^6 + \cdots
| c =
}}
{{eqn | n = 1
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{... | The [[Definition:Real Arcsine|(real) arcsine]] function has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \arcsin x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}
| c =
}}
{{eqn | r = x + \frac {x^3} {2 \ti... | From the [[General Binomial Theorem]]:
{{begin-eqn}}
{{eqn | l = \paren {1 - x^2}^{-1/2}
| r = 1 + \frac 1 2 x^2 + \frac {1 \times 3} {2 \times 4} x^4 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^6 + \cdots
| c =
}}
{{eqn | n = 1
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!}... | Power Series Expansion for Real Arcsine Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arcsine_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arcsine_Function | [
"Examples of Power Series",
"Arcsine Function"
] | [
"Definition:Inverse Sine/Real/Arcsine",
"Definition:Taylor Series",
"Definition:Convergent Series"
] | [
"Binomial Theorem/General Binomial Theorem",
"Power Series is Termwise Integrable within Radius of Convergence",
"Definition:Primitive (Calculus)/Integration",
"Derivative of Arcsine Function",
"Definition:Convergent Series",
"Stirling's Formula",
"Convergence of P-Series",
"Definition:Convergent Seri... |
proofwiki-8894 | Power Series Expansion for Real Arccosine Function | The arccosine function has a Taylor Series expansion:
{{begin-eqn}}
{{eqn | l = \arccos x
| r = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}
| c =
}}
{{eqn | r = \frac \pi 2 - \paren {x + \frac {x^3} {2 \times 3} + \frac {\paren {1 \... | {{begin-eqn}}
{{eqn | l = \arccos x
| r = \frac {\pi} 2 - \arcsin x
| c = Sum of Arcsine and Arccosine
}}
{{eqn | r = \frac {\pi} 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}
| c = Power Series Expansion for Real Arcsine Function
}}
{{en... | The [[Definition:Real Arccosine|arccosine]] function has a [[Definition:Taylor Series|Taylor Series]] expansion:
{{begin-eqn}}
{{eqn | l = \arccos x
| r = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}
| c =
}}
{{eqn | r = \frac \pi 2... | {{begin-eqn}}
{{eqn | l = \arccos x
| r = \frac {\pi} 2 - \arcsin x
| c = [[Sum of Arcsine and Arccosine]]
}}
{{eqn | r = \frac {\pi} 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}
| c = [[Power Series Expansion for Real Arcsine Function]]... | Power Series Expansion for Real Arccosine Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arccosine_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arccosine_Function | [
"Examples of Power Series",
"Arccosine Function"
] | [
"Definition:Inverse Cosine/Real/Arccosine",
"Definition:Taylor Series",
"Definition:Convergent Series"
] | [
"Sum of Arcsine and Arccosine",
"Power Series Expansion for Real Arcsine Function",
"Power Series Expansion for Real Arcsine Function"
] |
proofwiki-8895 | Real Function is Concave iff its Negative is Convex | Let $f$ be a real function.
Let $I \subseteq \R$ be an interval of $\R$.
Then $f$ is concave on $I$ {{iff}} $-f$ is convex on $\R$. | === Necessary Condition ===
Let $f$ be concave on $I$.
Let $\alpha, \beta \in \R_{>0}$ such that $\alpha + \beta = 1$.
Then:
{{begin-eqn}}
{{eqn | l = \map f {\alpha x + \beta y}
| o = \ge
| r = \alpha \map f x + \beta \map f y
| c = {{Defof|Concave Real Function}}
}}
{{eqn | ll= \leadsto
| l = ... | Let $f$ be a [[Definition:Real Function|real function]].
Let $I \subseteq \R$ be an [[Definition:Real Interval|interval of $\R$]].
Then $f$ is [[Definition:Concave Real Function|concave on $I$]] {{iff}} $-f$ is [[Definition:Convex Real Function|convex]] on $\R$. | === Necessary Condition ===
Let $f$ be [[Definition:Concave Real Function|concave on $I$]].
Let $\alpha, \beta \in \R_{>0}$ such that $\alpha + \beta = 1$.
Then:
{{begin-eqn}}
{{eqn | l = \map f {\alpha x + \beta y}
| o = \ge
| r = \alpha \map f x + \beta \map f y
| c = {{Defof|Concave Real Functio... | Real Function is Concave iff its Negative is Convex | https://proofwiki.org/wiki/Real_Function_is_Concave_iff_its_Negative_is_Convex | https://proofwiki.org/wiki/Real_Function_is_Concave_iff_its_Negative_is_Convex | [
"Convex Real Functions",
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Concave Real Function",
"Definition:Convex Real Function"
] | [
"Definition:Concave Real Function",
"Definition:Convex Real Function",
"Definition:Convex Real Function",
"Definition:Concave Real Function"
] |
proofwiki-8896 | Equivalence of Definitions of Strictly Concave Real Function | Let $f$ be a real function which is defined on a real interval $I$.
{{TFAE|def = Strictly Concave Real Function}} | Let $f$ be strictly concave real function on $I$ according to definition 1.
That is:
:$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} > \alpha \map f x + \beta \map f y$
Make the substitutions $x_1 = x, x_2 = \alpha x + \beta y, x_3 = y$.
As $\alpha + \beta = 1$, ... | Let $f$ be a [[Definition:Real Function|real function]] which is defined on a [[Definition:Real Interval|real interval]] $I$.
{{TFAE|def = Strictly Concave Real Function}} | Let $f$ be [[Definition:Concave Real Function/Definition 1/Strictly|strictly concave real function on $I$ according to definition 1]].
That is:
:$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} > \alpha \map f x + \beta \map f y$
Make the substitutions $x_1 = x, ... | Equivalence of Definitions of Strictly Concave Real Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Strictly_Concave_Real_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Strictly_Concave_Real_Function | [
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval"
] | [
"Definition:Concave Real Function/Definition 1/Strictly",
"Definition:Concave Real Function/Definition 3/Strictly",
"Definition:Concave Real Function/Definition 2/Strictly",
"Category:Concave Real Functions"
] |
proofwiki-8897 | Equivalence of Definitions of Strictly Convex Real Function | Let $f$ be a real function which is defined on a real interval $I$.
{{TFAE|def = Strictly Convex Real Function}} | Let $f$ be strictly convex real function on $I$ according to definition 1.
That is:
:$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} < \alpha \, \map f x + \beta \, \map f y$
Make the substitutions $x_1 = x, x_2 = \alpha x + \beta y, x_3 = y$.
As $\alpha + \beta =... | Let $f$ be a [[Definition:Real Function|real function]] which is defined on a [[Definition:Real Interval|real interval]] $I$.
{{TFAE|def = Strictly Convex Real Function}} | Let $f$ be [[Definition:Convex Real Function/Definition 1/Strictly|strictly convex real function on $I$ according to definition 1]].
That is:
:$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} < \alpha \, \map f x + \beta \, \map f y$
Make the substitutions $x_1 =... | Equivalence of Definitions of Strictly Convex Real Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Strictly_Convex_Real_Function | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Strictly_Convex_Real_Function | [
"Convex Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval"
] | [
"Definition:Convex Real Function/Definition 1/Strictly",
"Definition:Convex Real Function/Definition 3/Strictly",
"Definition:Convex Real Function/Definition 2/Strictly",
"Category:Convex Real Functions"
] |
proofwiki-8898 | Concave Real Function is Left-Hand and Right-Hand Differentiable | Let $f$ be a real function which is concave on the open interval $\openint a b$.
Then the left-hand derivative $\map {f'_-} x$ and right-hand derivative $\map {f'_+} x$ both exist for all $x \in \openint a b$. | Let $f$ be concave on $\openint a b$.
Then by definition of concavity:
:$\forall x_1, x_2, x_3 \in \openint a b: x_1 < x_2 < x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$
Let $0 < h_1 < h_2$.
Substitute $x_1 = x$, $x_2 = x + h_1$, $x_3 = x + h_2$. Then:
:$... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Concave Real Function|concave]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then the [[Definition:Left-Hand Derivative|left-hand derivative]] $\map {f'_-} x$ and [[Definition:Right-Hand Derivative|right-hand deriv... | Let $f$ be [[Definition:Concave Real Function|concave]] on $\openint a b$.
Then by definition of [[Definition:Concave Real Function|concavity]]:
:$\forall x_1, x_2, x_3 \in \openint a b: x_1 < x_2 < x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$
Let $0 ... | Concave Real Function is Left-Hand and Right-Hand Differentiable | https://proofwiki.org/wiki/Concave_Real_Function_is_Left-Hand_and_Right-Hand_Differentiable | https://proofwiki.org/wiki/Concave_Real_Function_is_Left-Hand_and_Right-Hand_Differentiable | [
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Concave Real Function",
"Definition:Real Interval/Open",
"Definition:Left-Hand Derivative",
"Definition:Right-Hand Derivative"
] | [
"Definition:Concave Real Function",
"Definition:Concave Real Function",
"Definition:Decreasing/Real Function",
"Limit of Monotone Real Function/Decreasing"
] |
proofwiki-8899 | Concave Real Function is Continuous | Let $f$ be a real function which is concave on the open interval $\openint a b$.
Then $f$ is continuous on $\openint a b$. | From Concave Real Function is Left-Hand and Right-Hand Differentiable, $f$ is left-hand and right-hand differentiable on $\openint a b$.
From Left-Hand and Right-Hand Differentiable Function is Continuous, $f$ is continuous on $\openint a b$.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Concave Real Function|concave]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then $f$ is [[Definition:Continuous on Interval|continuous]] on $\openint a b$. | From [[Concave Real Function is Left-Hand and Right-Hand Differentiable]], $f$ is [[Definition:Left-Hand Derivative|left-hand]] and [[Definition:Right-Hand Derivative|right-hand differentiable]] on $\openint a b$.
From [[Left-Hand and Right-Hand Differentiable Function is Continuous]], $f$ is [[Definition:Continuous o... | Concave Real Function is Continuous | https://proofwiki.org/wiki/Concave_Real_Function_is_Continuous | https://proofwiki.org/wiki/Concave_Real_Function_is_Continuous | [
"Concave Real Functions",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Concave Real Function",
"Definition:Real Interval/Open",
"Definition:Continuous Real Function/Interval"
] | [
"Concave Real Function is Left-Hand and Right-Hand Differentiable",
"Definition:Left-Hand Derivative",
"Definition:Right-Hand Derivative",
"Left-Hand and Right-Hand Differentiable Function is Continuous",
"Definition:Continuous Real Function/Interval"
] |
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