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-4600 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\dfrac x a > 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
| r... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\dfrac x a > 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
... | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Cotangent_Form/Proof | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Hyperbolic Cotangent",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Integral of Constant"
] |
proofwiki-4601 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x < a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tanh^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Tangent}}, which is defined where $\size {\dfrac x a} < 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \tanh u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x < a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tanh^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Tangent}}, which is defined where $\size {\dfrac x a} < 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \tanh u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
... | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Tangent_Form/Proof | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Hyperbolic Tangent",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Sum of Squares of Hyperbolic Secant and Tangent",
"Integral of Constant"
] |
proofwiki-4602 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth \dfrac x a$ in Logarit... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfr... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x greater than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_greater_than_a/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form"
] |
proofwiki-4603 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} } } + C
| c = $\artanh \dfrac x a$ in Logarith... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfrac... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form",
"Real Area Hyperbolic Tangent of x over a in Logarithm Form"
] |
proofwiki-4604 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = Partial Fraction Expansion
}}... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = [[Primitive of Reciproca... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Difference of Two Squares",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Partial Fraction Expansion",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Difference of Logarithms"
] |
proofwiki-4605 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = Primitive of $\dfrac 1 {x^2 - a^2}$ for $\size x < a$
}}
{{eq... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = [[Primitive of Reciprocal of x squared minus a squared/L... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_3 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a",
"Logarithm of Reciprocal"
] |
proofwiki-4606 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\rd x} {\paren {a x + b}^3}
| r = \frac 1 a \int \frac {\rd u} {u^3}
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \frac {-1} {2 u^2} + C
| c = Primitive of Power
}}
{{eqn | r = \frac {-1} {2 a \paren {a x + b}^2} + C
... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\rd x} {\paren {a x + b}^3}
| r = \frac 1 a \int \frac {\rd u} {u^3}
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \frac {-1} {2 u^2} + C
| c = [[Primitive of Power]]
}}
{{eqn | r ... | Primitive of Reciprocal of a x + b cubed/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_cubed/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Function of a x + b",
"Primitive of Power"
] |
proofwiki-4607 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From Primitive of Power of $a x + b$:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -3$.
{{qed}} | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From [[Primitive of Power of a x + b|Primitive of Power of $a x + b$]]:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -3$.
{{qed}} | Primitive of Reciprocal of a x + b cubed/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_cubed/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Power of a x + b"
] |
proofwiki-4608 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x + b}^2}
| r = \frac 1 a \int \frac {\d u} {u^2}
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \frac {-1} u + C
| c = Primitive of Power
}}
{{eqn | r = -\frac 1 {a \paren {a x + b} } + C
| c = su... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {a x + b}^2}
| r = \frac 1 a \int \frac {\d u} {u^2}
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \frac {-1} u + C
| c = [[Primitive of Power]]
}}
{{eqn | r = -\frac... | Primitive of Reciprocal of a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Function of a x + b",
"Primitive of Power"
] |
proofwiki-4609 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From Primitive of Power of $a x + b$:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -2$.
{{qed}} | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From [[Primitive of Power of a x + b|Primitive of Power of $a x + b$]]:
:$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$
where $n \ne 1$.
The result follows by setting $n = -2$.
{{qed}} | Primitive of Reciprocal of a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_+_b_squared/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Power of a x + b"
] |
proofwiki-4610 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | We aim to use Primitive of $\dfrac 1 {a x^2 + b x + c}$ with:
{{begin-eqn}}
{{eqn | l = a
| r = 3
}}
{{eqn | l = b
| r = 4
}}
{{eqn | l = c
| r = 2
}}
{{end-eqn}}
We note that:
{{begin-eqn}}
{{eqn | l = b^2 - 4 a c
| r = 4^2 - 4 \times 3 \times 2
}}
{{eqn | r = 16 - 24
}}
{{eqn | r = -8
}}
{{end... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | We aim to use [[Primitive of Reciprocal of a x squared plus b x plus c|Primitive of $\dfrac 1 {a x^2 + b x + c}$]] with:
{{begin-eqn}}
{{eqn | l = a
| r = 3
}}
{{eqn | l = b
| r = 4
}}
{{eqn | l = c
| r = 2
}}
{{end-eqn}}
We note that:
{{begin-eqn}}
{{eqn | l = b^2 - 4 a c
| r = 4^2 - 4 \times... | Primitive of Reciprocal of a x squared plus b x plus c/Examples/3 x^2 + 4 x + 2/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Examples/3_x^2_+_4_x_+_2/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Primitive of Reciprocal of a x squared plus b x plus c"
] |
proofwiki-4611 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {3 x^2 + 4 x + 2}
| r = \dfrac 1 3 \int \frac {\d x} {x^2 + \frac 4 3 x + \frac 2 3}
| c =
}}
{{eqn | r = \dfrac 1 3 \int \frac {\d x} {\paren {x + \frac 2 3}^2 + \paren {\frac 2 3 - \frac 4 9} }
| c =
}}
{{eqn | r = \dfrac 1 3 \int \frac {\d x} {\paren {x... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {3 x^2 + 4 x + 2}
| r = \dfrac 1 3 \int \frac {\d x} {x^2 + \frac 4 3 x + \frac 2 3}
| c =
}}
{{eqn | r = \dfrac 1 3 \int \frac {\d x} {\paren {x + \frac 2 3}^2 + \paren {\frac 2 3 - \frac 4 9} }
| c =
}}
{{eqn | r = \dfrac 1 3 \int \frac {\d x} {\paren {x... | Primitive of Reciprocal of a x squared plus b x plus c/Examples/3 x^2 + 4 x + 2/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/Examples/3_x^2_+_4_x_+_2/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-4612 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {a x^2 + c}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {\d x} {x^2 + \frac c a}
| c = Primitive of Constant Multiple of Function
}}
{{end-eqn}}
Let $a c > 0$.
Then $\dfrac c a > 0$ and:
{{begin-eqn}}
{{eqn | l... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {a x^2 + c}
| c =
}}
{{eqn | r = \frac 1 a \int \frac {\d x} {x^2 + \frac c a}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{end-eqn}}
Let $a c > 0$.
Then $\dfrac c a > 0$ and:
{{begin-eqn}}
{... | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/b_equal_to_0/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Constant Multiple of Function",
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form",
"Primitive of Power"
] |
proofwiki-4613 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $b = 0$.
From Primitive of Reciprocal of a x squared plus b x plus c, we have:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \\
\dfrac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\dfrac ... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $b = 0$.
From [[Primitive of Reciprocal of a x squared plus b x plus c]], we have:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \\
\dfrac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\d... | Primitive of Reciprocal of a x squared plus b x plus c/b equal to 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/b_equal_to_0/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of a x squared plus b x plus c"
] |
proofwiki-4614 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First:
{{begin-eqn}}
{{eqn | l = c
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {a x^2 + b x}
| c =
}}
{{eqn | r = \int \frac {\d x} {x \paren {a x + b} }
| c =
}}
{{eqn | r = \frac 1 b \ln \size {\frac x {a x + b} } + C
... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First:
{{begin-eqn}}
{{eqn | l = c
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {a x^2 + b x + c}
| r = \int \frac {\d x} {a x^2 + b x}
| c =
}}
{{eqn | r = \int \frac {\d x} {x \paren {a x + b} }
| c =
}}
{{eqn | r = \frac 1 b \ln \size {\frac x {a x + b} } + C
... | Primitive of Reciprocal of a x squared plus b x plus c/c equal to 0/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/c_equal_to_0/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by a x + b"
] |
proofwiki-4615 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $c = 0$.
From Primitive of $\dfrac 1 {a x^2 + b x + c}$, we have:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \\
\dfrac 1 {\sqrt {b^2 - 4 a c} } \ln \size {\dfrac {2 a x + b - ... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $c = 0$.
From [[Primitive of Reciprocal of a x squared plus b x plus c|Primitive of $\dfrac 1 {a x^2 + b x + c}$]], we have:
:<nowiki>$\ds \int \frac {\d x} {a x^2 + b x + c} = \begin {cases}
\dfrac 2 {\sqrt {4 a c - b^2} } \map \arctan {\dfrac {2 a x + b} {\sqrt {4 a c - b^2} } } + C & : b^2 - 4 a c < 0 \\ \\
\df... | Primitive of Reciprocal of a x squared plus b x plus c/c equal to 0/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_plus_b_x_plus_c/c_equal_to_0/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Definition:Primitive (Calculus)/Constant of Integration"
] |
proofwiki-4616 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \cos a x}
| r = \frac 2 {a \paren {p - q} } \int \frac {\d u} {u^2 + \dfrac {p + q} {p - q} }
| c = Weierstrass Substitution: $u = \tan \dfrac {a x} 2$
}}
{{end-eqn}}
Let $p^2 > q^2$.
Then, by Sign of Quotient of Factors of Difference of Squares:
:$\dfrac {... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \cos a x}
| r = \frac 2 {a \paren {p - q} } \int \frac {\d u} {u^2 + \dfrac {p + q} {p - q} }
| c = [[Primitive of Reciprocal of p plus q by Cosine of a x/Weierstrass Substitution|Weierstrass Substitution]]: $u = \tan \dfrac {a x} 2$
}}
{{end-eqn}}
Let $p... | Primitive of Reciprocal of p plus q by Cosine of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Cosine_of_a_x/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of p plus q by Cosine of a x/Weierstrass Substitution",
"Sign of Quotient of Factors of Difference of Squares",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Sign of Quotient of Factors of Difference of Squares",
"Primitive of Reciprocal of x squared minus ... |
proofwiki-4617 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First, let $\arctan \dfrac p q = \phi$.
Let $z = a x + \phi$.
{{begin-eqn}}
{{eqn | l = z
| r = \map \sin {a x + \phi}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \map \cos {a x + \phi}
| c = Derivative of $\sin a x$ etc.
}}
{{eqn | r = a \cos z
| c =
}}
{{end-e... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First, let $\arctan \dfrac p q = \phi$.
Let $z = a x + \phi$.
{{begin-eqn}}
{{eqn | l = z
| r = \map \sin {a x + \phi}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d z} {\d x}
| r = a \map \cos {a x + \phi}
| c = [[Derivative of Sine of a x|Derivative of $\sin a x$]] etc.
}}
{{eqn | r = ... | Primitive of Reciprocal of p plus q by Tangent of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Tangent_of_a_x/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Sine Function/Corollary",
"Tangent is Sine divided by Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Multiple of Sine plus Multiple of Cosine/Sine Form",
"Primitive of Constant Multiple of Function",
"Primitive of Cosine of a x over Sine of a x plus phi"
] |
proofwiki-4618 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | We have:
:$\dfrac \d {\d x} \paren {q \sin a x + p \cos a x} = a q \cos a x - a p \sin a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \tan a x}
| r = \int \frac {\d x} {p + q \dfrac {\sin a x} {\cos a x} }
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \int \frac {\cos a x \rd x} {p \... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | We have:
:$\dfrac \d {\d x} \paren {q \sin a x + p \cos a x} = a q \cos a x - a p \sin a x$
Thus:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p + q \tan a x}
| r = \int \frac {\d x} {p + q \dfrac {\sin a x} {\cos a x} }
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \int \frac {\cos a x \rd x... | Primitive of Reciprocal of p plus q by Tangent of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_plus_q_by_Tangent_of_a_x/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Tangent is Sine divided by Cosine",
"Primitive of Constant",
"Primitive of Reciprocal"
] |
proofwiki-4619 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $u = p^2 + q^2$ and $v = q^2 - p^2$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = u + v
| r = 2 q^2
}}
{{eqn | n = 2
| l = u - v
| r = 2 p ^2
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = u^2 - v^2
| r = \paren {u + v} \paren {u - v}
}}
{{eqn | l = u^2 - v^2
| r = \paren {2 q^2} \par... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $u = p^2 + q^2$ and $v = q^2 - p^2$.
Then:
{{begin-eqn}}
{{eqn | n = 1
| l = u + v
| r = 2 q^2
}}
{{eqn | n = 2
| l = u - v
| r = 2 p ^2
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = u^2 - v^2
| r = \paren {u + v} \paren {u - v}
}}
{{eqn | l = u^2 - v^2
| r = \paren {2 q^2... | Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_by_square_of_Sine_of_a_x_plus_q_squared_by_square_of_Cosine_of_a_x/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Power Reduction Formulas/Sine Squared",
"Power Reduction Formulas/Cosine Squared",
"Primitive of Reciprocal of p plus q by Cosine of a x"
] |
proofwiki-4620 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x}
| r = \int \frac {\sec^2 a x \d x} {p^2 \tan^2 a x + q^2}
| c = multiplying by $\dfrac {\sec^2 a x} {\sec^2 a x}$
}}
{{eqn | r = \frac 1 a \int \frac {\d t} {p^2 t^2 + q^2}
| c = substituting $t = \tan a x$
}}
{{eqn | r = \frac 1 {a p^2} \... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 \sin^2 a x + q^2 \cos^2 a x}
| r = \int \frac {\sec^2 a x \d x} {p^2 \tan^2 a x + q^2}
| c = multiplying by $\dfrac {\sec^2 a x} {\sec^2 a x}$
}}
{{eqn | r = \frac 1 a \int \frac {\d t} {p^2 t^2 + q^2}
| c = [[Integration by Substitution|substituting]] $t = \tan a x$
... | Primitive of Reciprocal of p squared by square of Sine of a x plus q squared by square of Cosine of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_p_squared_by_square_of_Sine_of_a_x_plus_q_squared_by_square_of_Cosine_of_a_x/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-4621 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First a pair of lemmata:
=== Lemma ===
{{:Primitive of Reciprocal of square of p plus q by Sine of a x/Lemma}}{{qed|lemma}}
=== Weierstrass Substitution ===
{{:Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution|Weierstrass Substitution}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \int \fr... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First a pair of [[Definition:Lemma|lemmata]]:
=== [[Primitive of Reciprocal of square of p plus q by Sine of a x/Lemma|Lemma]] ===
{{:Primitive of Reciprocal of square of p plus q by Sine of a x/Lemma}}{{qed|lemma}}
=== [[Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution|Weierstra... | Primitive of Reciprocal of square of p plus q by Sine of a x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_square_of_p_plus_q_by_Sine_of_a_x/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Definition:Lemma",
"Primitive of Reciprocal of square of p plus q by Sine of a x/Lemma",
"Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution",
"Primitive of Reciprocal of square of p plus q by Sine of a x/Weierstrass Substitution",
"Linear Combination of Integrals/Indefin... |
proofwiki-4622 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | === Lemma ===
{{:Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma}}{{qed|lemma}}
Let $x > 0$, and so $u > 0$.
Then we have:
:$\ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = -\int \frac {\d u} {\sqrt {a + b u + c u^2} }$
We consider the two cases where $c > 0$ and $c < 0$.
First we take $c... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | === [[Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma|Lemma]] ===
{{:Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma}}{{qed|lemma}}
Let $x > 0$, and so $u > 0$.
Then we have:
:$\ds \int \frac {\d x} {x \sqrt {a x^2 + b x + c} } = -\int \frac {\d u} {\sqrt {a + b u... | Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Lemma",
"Primitive of Reciprocal of Root of a x squared plus b x plus c/a greater than 0",
"Primitive of Reciprocal of Root of a x squared plus b x plus c/a less than 0",
"Primitive of Reciprocal of Root of a x squared plus b x plus c/a grea... |
proofwiki-4623 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = x \sqrt {a x^2 + b x + c}
| r = \frac x {\paren {a x^2 + b x + c}^{-\frac 1 2} }
| c =
}}
{{eqn | r = \frac{x \paren {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} } {\paren {a x^2 + b x + c}^{-\frac 1 2} \paren {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} }
| c =
}}
{{... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = x \sqrt {a x^2 + b x + c}
| r = \frac x {\paren {a x^2 + b x + c}^{-\frac 1 2} }
| c =
}}
{{eqn | r = \frac{x \paren {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} } {\paren {a x^2 + b x + c}^{-\frac 1 2} \paren {2 \sqrt c \sqrt {a x^2 + b x + c} + b x + 2 c} }
| c =
}}
{{... | Primitive of Reciprocal of x by Root of a x squared plus b x plus c/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_Root_of_a_x_squared_plus_b_x_plus_c/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Definition:Real Number",
"Primitive of Function under its Derivative"
] |
proofwiki-4624 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \paren {\frac 1 {b^2 x} - \frac a {b^2 \paren {a x + b} } - \frac a {b \paren {a x + b}^2} } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {b^2} \int \frac {\d x} x - \frac a {b^2} \int \frac {\d x} {a x + b} - \fr... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \paren {\frac 1 {b^2 x} - \frac a {b^2 \paren {a x + b} } - \frac a {b \paren {a x + b}^2} } \rd x
| c = [[Primitive of Reciprocal of x by a x + b squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \fra... | Primitive of Reciprocal of x by a x + b squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_squared/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by a x + b squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Primitive of Reciprocal of a x + b squared",
"Difference of Logarithms"
] |
proofwiki-4625 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \frac {b \rd x} {b x \paren {a x + b}^2}
| c = multiplying top and bottom by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b}^2}
| c = adding and subtracting $a x$
}}
{{eqn | r = \frac 1 b \... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^2}
| r = \int \frac {b \rd x} {b x \paren {a x + b}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b}^2}
| c = add... | Primitive of Reciprocal of x by a x + b squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b_squared/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal of x by a x + b",
"Primitive of Reciprocal of a x + b squared"
] |
proofwiki-4626 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b} }
| r = \int \paren {\dfrac 1 {b x} - \dfrac a {b \paren {a x + b} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 b \int \frac {\d x} x - \frac a b \int \frac {\d x} {a x + b}
| c = Linear Combination of Primitives
}}
{... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b} }
| r = \int \paren {\dfrac 1 {b x} - \dfrac a {b \paren {a x + b} } } \rd x
| c = [[Primitive of Reciprocal of x by a x + b/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac 1 b \int \frac {\d x} x - \frac a b \int \... | Primitive of Reciprocal of x by a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by a x + b/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Difference of Logarithms"
] |
proofwiki-4627 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b} }
| r = \int \frac {b \rd x} {b x \paren {a x + b} }
| c = multiplying top and bottom by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b} }
| c = adding and subtracting $a x$
}}
{{eqn | r = \frac 1 b \int... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b} }
| r = \int \frac {b \rd x} {b x \paren {a x + b} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x \paren {a x + b} }
| c = adding... | Primitive of Reciprocal of x by a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_a_x_+_b/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Difference of Logarithms"
] |
proofwiki-4628 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \int \paren {\frac 1 {a^2 x} - \frac x {a^2 \paren {x^2 + a^2} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {a^2} \int \frac {\d x} x - \frac 1 {a^2} \int \frac {x \rd x} {x^2 + a^2}
| c = Linear Combination... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \int \paren {\frac 1 {a^2 x} - \frac x {a^2 \paren {x^2 + a^2} } } \rd x
| c = [[Primitive of Reciprocal of x by x squared plus a squared/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = \frac 1 {a^2} \int \frac ... | Primitive of Reciprocal of x by x squared plus a squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by x squared plus a squared/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over x squared plus a squared",
"Difference of Logarithms"
] |
proofwiki-4629 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \int \frac {a^2 \rd x} {a^2 x \paren {x^2 + a^2} }
| c = multiplying top and bottom by $a^2$
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x \paren {x^2 + a^2} }
| c = adding and subtracting $x^2$
}}
{{eqn | r ... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \int \frac {a^2 \rd x} {a^2 x \paren {x^2 + a^2} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $a^2$
}}
{{eqn | r = \int \frac {\paren {x^2 + a^2 - x^2} \rd x} {a^2 x \paren {x^2 + a^2} }
... | Primitive of Reciprocal of x by x squared plus a squared/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Reciprocal",
"Primitive of x over x squared plus a squared",
"Difference of Logarithms"
] |
proofwiki-4630 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From Primitive of $\dfrac 1 {x \paren {x^n + a^n} }$:
:$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \frac 1 {2 a^2} \ln \size {\frac {x^2} {x^2 + a^2} } + C
| c = Primi... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From [[Primitive of Reciprocal of x by Power of x plus Power of a|Primitive of $\dfrac 1 {x \paren {x^n + a^n} }$]]:
:$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$
So:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^2 + a^2} }
| r = \frac 1 {... | Primitive of Reciprocal of x by x squared plus a squared/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_by_x_squared_plus_a_squared/Proof_3 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x by Power of x plus Power of a",
"Primitive of Reciprocal of x by Power of x plus Power of a",
"Absolute Value of Even Power"
] |
proofwiki-4631 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b} }
| r = \int \paren {\frac {a^2} {b^3 x} + \frac {-a} {b^2 x^2} + \frac 1 {b x^3} + \frac {-a^3} {b^3 \paren {a x + b} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac {a^2} {b^3} \int \frac {\d x} x + \frac {-a} {b^2} \int \fr... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b} }
| r = \int \paren {\frac {a^2} {b^3 x} + \frac {-a} {b^2 x^2} + \frac 1 {b x^3} + \frac {-a^3} {b^3 \paren {a x + b} } } \rd x
| c = [[Primitive of Reciprocal of x cubed by a x + b/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
... | Primitive of Reciprocal of x cubed by a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x cubed by a x + b/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Difference of Logarithms"
] |
proofwiki-4632 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b} }
| r = \int \frac {b \rd x} {b x^3 \paren {a x + b} }
| c = multiplying top and bottom by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x^3 \paren {a x + b} }
| c = adding and subtracting $a x$
}}
{{eqn | r = \frac 1 ... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b} }
| r = \int \frac {b \rd x} {b x^3 \paren {a x + b} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x^3 \paren {a x + b} }
| c = ... | Primitive of Reciprocal of x cubed by a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_cubed_by_a_x_+_b/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal of x squared by a x + b",
"Difference of Logarithms"
] |
proofwiki-4633 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = Derivative of Hyperbolic Cosine
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = \sqrt {x^2 - a^2... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a \sinh \theta
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = x
| r = a \cosh \theta
}}
{{eqn | ll= \leadsto
| l = \sqrt {x^... | Primitive of Reciprocal of x squared by Root of x squared minus a squared/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_Root_of_x_squared_minus_a_squared/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Integration by Substitution",
"Primitive of Reciprocal of Square of Hyperbolic Cosine of a x"
] |
proofwiki-4634 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b} }
| r = \int \paren {-\frac a {b^2 x} + \frac 1 {b x^2} + \frac {a^2} {b^2 \paren {a x + b} } } \rd x
| c = Partial Fraction Expansion
}}
{{eqn | r = -\frac a {b^2} \int \frac {\d x} x + \frac 1 b \int \frac {\d x} {x^2} + \frac {a^2} {b^2} \... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b} }
| r = \int \paren {-\frac a {b^2 x} + \frac 1 {b x^2} + \frac {a^2} {b^2 \paren {a x + b} } } \rd x
| c = [[Primitive of Reciprocal of x squared by a x + b/Partial Fraction Expansion|Partial Fraction Expansion]]
}}
{{eqn | r = -\frac a {b^2... | Primitive of Reciprocal of x squared by a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x squared by a x + b/Partial Fraction Expansion",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal",
"Primitive of Reciprocal of a x + b",
"Difference of Logarithms"
] |
proofwiki-4635 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b} }
| r = \int \frac {b \rd x} {b x^2 \paren {a x + b} }
| c = multiplying top and bottom by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x^2 \paren {a x + b} }
| c = adding and subtracting $a x$
}}
{{eqn | r = \frac 1 ... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 \paren {a x + b} }
| r = \int \frac {b \rd x} {b x^2 \paren {a x + b} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $b$
}}
{{eqn | r = \int \frac {\paren {a x + b - a x} \rd x} {b x^2 \paren {a x + b} }
| c = ... | Primitive of Reciprocal of x squared by a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_a_x_+_b/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Primitive of Reciprocal of x by a x + b",
"Logarithm of Reciprocal"
] |
proofwiki-4636 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}
| r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} }
}}
{{eqn | o =
| ro= -
| r = \frac {\paren {m + 2 n - 3} a} {... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From [[Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c]]:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {a x^2 + b x + c}^n}
| r = \frac {-1} {\paren {m - 1} c x^{m - 1} \paren {a x^2 + b x + c}^{n - 1} }
}}
{{eqn | o =
| ro= -
| r = \frac {\paren {m + 2 n - 3} ... | Primitive of Reciprocal of x squared by square of a x squared plus b x plus c/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_square_of_a_x_squared_plus_b_x_plus_c/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c"
] |
proofwiki-4637 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2}
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x^2 \paren {a x^2 + b x + c}^2}
| c = multiplying top and bottom by $c$
}}
{{eqn | r = \frac 1 c \int \frac {c \rd x} {x^2 \paren {a x^2 + b x + c}^2}
| c = Pr... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | First:
{{begin-eqn}}
{{eqn | o =
| r = \int \frac {\d x} {x^2 \paren {a x^2 + b x + c}^2}
| c =
}}
{{eqn | r = \int \frac {c \rd x} {c x^2 \paren {a x^2 + b x + c}^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $c$
}}
{{eqn | r = \frac 1 c \int \frac {c \... | Primitive of Reciprocal of x squared by square of a x squared plus b x plus c/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_by_square_of_a_x_squared_plus_b_x_plus_c/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Constant Multiple of Function",
"Linear Combination of Integrals/Indefinite",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Primitive of Power",
"Integration by Parts",
"Linear Combination of Integ... |
proofwiki-4638 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\size {\dfrac x a} > 1$
}}
{{eqn | ll= \leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d u}... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\size {\dfrac x a} > 1$
}}
{{eqn | ll= \leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d ... | Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Cotangent_Form/Proof | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Hyperbolic Cotangent",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Integral of Constant"
] |
proofwiki-4639 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth {\frac x a} + C
| c = Primitive of Reciprocal of $x^2 - a^2$ in $\arcoth$ form
}}
{{eqn | r = -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth {\dfrac x a}$ in Logar... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let $x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth {\frac x a} + C
| c = [[Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form|Primitive of Reciprocal of $x^2 - a^2$ in $\arcoth$ form]]
}}
{{eqn | r = -\frac 1 a \paren {\df... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form",
"Logarithm of Reciprocal",
"Integration by Substitution",
"Logarithm of Reciprocal"
] |
proofwiki-4640 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = \int \frac {\d x} {\paren {x - a} \paren {x + a} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {2 a... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = \int \frac {\d x} {\paren {x - a} \paren {x + a} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }
| c = [[Primitive of Reciprocal of x squared minus a squ... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Difference of Two Squares",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2/Partial Fraction Expansion",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Difference of Logarithms"
] |
proofwiki-4641 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From the $1$st logarithm form:
$\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$
From Primitive of Reciproca... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | From the [[Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1|$1$st logarithm form]]:
$\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_3 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form/Lemma"
] |
proofwiki-4642 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let:
:$a \tan \theta = x$
for $\theta \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
From Shape of Tangent Function, this substitution is valid for all real $x$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
| c = from above
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d \theta}
| r = a ... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Let:
:$a \tan \theta = x$
for $\theta \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
From [[Shape of Tangent Function]], this substitution is valid for all [[Definition:Real Number|real]] $x$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = a \tan \theta
| c = from above
}}
{{eqn | ll= \leadsto
| l = \fra... | Primitive of Reciprocal of x squared plus a squared/Arctangent Form/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_plus_a_squared/Arctangent_Form/Proof_1 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Shape of Tangent Function",
"Definition:Real Number",
"Derivative of Tangent Function",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Integral of Constant",
"Definition:Real Interval/Open",
"Definition:Primitive (Calcu... |
proofwiki-4643 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | We have that $x^2 + a^2$ is in the form $a x^2 + b x + c$, where $b^2 - 4 a c < 0$.
Thus from Primitive of $\dfrac 1 {a x^2 + b x + c}$ for $b^2 - 4 a c > 0$:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 2 {\sqrt {4 a c - b^2} } \map \arctan {\frac {2 a x + b} {\sqrt {4 a c - b^2} } } + C$
setting $a := 1, b := 0,... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | We have that $x^2 + a^2$ is in the form $a x^2 + b x + c$, where $b^2 - 4 a c < 0$.
Thus from [[Primitive of Reciprocal of a x squared plus b x plus c|Primitive of $\dfrac 1 {a x^2 + b x + c}$]] for $b^2 - 4 a c > 0$:
:$\ds \int \frac {\d x} {a x^2 + b x + c} = \frac 2 {\sqrt {4 a c - b^2} } \map \arctan {\frac {2 a x... | Primitive of Reciprocal of x squared plus a squared/Arctangent Form/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_plus_a_squared/Arctangent_Form/Proof_2 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Primitive of Reciprocal of a x squared plus b x plus c",
"Primitive of Reciprocal of a x squared plus b x plus c"
] |
proofwiki-4644 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 + a^2}
| r = \frac 1 a \int \frac {\d t} {t^2 + 1}
| c = Substitution of $x \to a t$
}}
{{eqn | r = \frac 1 a \int \frac {\d t} {\paren {1 + i t} \paren {1 - i t} }
| c = factoring
}}
{{eqn | r = \frac 1 {2 a} \paren {\int \frac {\d t} {1 + i t} + \int ... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 + a^2}
| r = \frac 1 a \int \frac {\d t} {t^2 + 1}
| c = [[Integration by Substitution|Substitution of $x \to a t$]]
}}
{{eqn | r = \frac 1 a \int \frac {\d t} {\paren {1 + i t} \paren {1 - i t} }
| c = factoring
}}
{{eqn | r = \frac 1 {2 a} \paren {\in... | Primitive of Reciprocal of x squared plus a squared/Arctangent Form/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_plus_a_squared/Arctangent_Form/Proof_3 | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Integration by Substitution",
"Primitive of Reciprocal",
"Sum of Logarithms",
"Arctangent Logarithmic Formulation"
] |
proofwiki-4645 | Primitive of Reciprocal | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Suppose $x > 0$.
Then:
:$\ln \size x = \ln x$
The result follows from Derivative of Natural Logarithm Function and the definition of primitive.
Suppose $x < 0$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \ln \size x
| r = \dfrac \d {\d x} \map \ln {-x}
| c = {{Defof|Absolute Value}}
}}
{{eqn | r = \fr... | :$\ds \int \frac {\d x} x = \ln \size x + C$
for $x \ne 0$. | Suppose $x > 0$.
Then:
:$\ln \size x = \ln x$
The result follows from [[Derivative of Natural Logarithm Function]] and the definition of [[Definition:Primitive (Calculus)|primitive]].
Suppose $x < 0$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \ln \size x
| r = \dfrac \d {\d x} \map \ln {-x}
| c... | Primitive of Reciprocal/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal | https://proofwiki.org/wiki/Primitive_of_Reciprocal/Proof | [
"Primitive of Reciprocal",
"Primitives involving Reciprocals",
"Logarithms",
"Reciprocals"
] | [] | [
"Derivative of Natural Logarithm Function",
"Definition:Primitive (Calculus)",
"Derivative of Composite Function",
"Derivative of Natural Logarithm Function",
"Definition:Primitive (Calculus)"
] |
proofwiki-4646 | Powers of Commuting Elements of Semigroup Commute | :$\forall m, n \in \N_{>0}: \paren {\circ^m a} \circ \paren {\circ^n b} = \paren {\circ^n b} \circ \paren {\circ^m a}$ | The proof proceeds by the Principle of Mathematical Induction:
Let $\map P n$ be the proposition:
:$\paren {\circ^n a} \circ b = b \circ \paren {\circ^n a}$ | :$\forall m, n \in \N_{>0}: \paren {\circ^m a} \circ \paren {\circ^n b} = \paren {\circ^n b} \circ \paren {\circ^m a}$ | The proof proceeds by the [[Principle of Mathematical Induction]]:
Let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\paren {\circ^n a} \circ b = b \circ \paren {\circ^n a}$ | Powers of Commuting Elements of Semigroup Commute | https://proofwiki.org/wiki/Powers_of_Commuting_Elements_of_Semigroup_Commute | https://proofwiki.org/wiki/Powers_of_Commuting_Elements_of_Semigroup_Commute | [
"Semigroups",
"Commutativity",
"Powers (Abstract Algebra)"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-4647 | All Elements of Right Operation are Right Zeroes | Let $\struct {S, \to}$ be an algebraic structure in which the operation $\to$ is the right operation.
Then no matter what $S$ is, $\struct {S, \to}$ is a semigroup all of whose elements are right zeroes.
Thus it can be seen that any right zero in a semigroup is not necessarily unique. | It is established in Structure under Right Operation is Semigroup that $\struct {S, \to}$ is a semigroup.
From the definition of right operation:
:$\forall x, y \in S: x \to y = y$
from which it can immediately be seen that all elements of $S$ are indeed right zeroes.
{{qed}}
From More than One Right Zero then No Left ... | Let $\struct {S, \to}$ be an [[Definition:Algebraic Structure|algebraic structure]] in which the [[Definition:Binary Operation|operation]] $\to$ is the [[Definition:Right Operation|right operation]].
Then no matter what $S$ is, $\struct {S, \to}$ is a [[Definition:Semigroup|semigroup]] all of whose elements are [[Def... | It is established in [[Structure under Right Operation is Semigroup]] that $\struct {S, \to}$ is a [[Definition:Semigroup|semigroup]].
From the definition of [[Definition:Right Operation|right operation]]:
:$\forall x, y \in S: x \to y = y$
from which it can immediately be seen that all elements of $S$ are indeed [[D... | All Elements of Right Operation are Right Zeroes | https://proofwiki.org/wiki/All_Elements_of_Right_Operation_are_Right_Zeroes | https://proofwiki.org/wiki/All_Elements_of_Right_Operation_are_Right_Zeroes | [
"Right Operation",
"Zero Elements"
] | [
"Definition:Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Right Operation",
"Definition:Semigroup",
"Definition:Right Zero",
"Definition:Right Zero",
"Definition:Semigroup"
] | [
"Structure under Right Operation is Semigroup",
"Definition:Semigroup",
"Definition:Right Operation",
"Definition:Right Zero",
"More than One Right Zero then No Left Zero",
"Definition:Left Zero"
] |
proofwiki-4648 | All Elements of Left Operation are Left Zeroes | Let $\struct {S, \leftarrow}$ be an algebraic structure in which the operation $\leftarrow$ is the left operation.
Then no matter what $S$ is, $\struct {S, \leftarrow}$ is a semigroup all of whose elements are left zeroes.
Thus it can be seen that any left zero in a semigroup is not necessarily unique. | It is established in Structure under Left Operation is Semigroup that $\struct {S, \leftarrow}$ is a semigroup.
From the definition of left operation:
:$\forall x, y \in S: x \leftarrow y = x$
from which it can immediately be seen that all elements of $S$ are indeed left zeroes.
{{qed}}
From More than One Right Zero th... | Let $\struct {S, \leftarrow}$ be an [[Definition:Algebraic Structure|algebraic structure]] in which the [[Definition:Binary Operation|operation]] $\leftarrow$ is the [[Definition:Left Operation|left operation]].
Then no matter what $S$ is, $\struct {S, \leftarrow}$ is a [[Definition:Semigroup|semigroup]] all of whose... | It is established in [[Structure under Left Operation is Semigroup]] that $\struct {S, \leftarrow}$ is a [[Definition:Semigroup|semigroup]].
From the definition of [[Definition:Left Operation|left operation]]:
:$\forall x, y \in S: x \leftarrow y = x$
from which it can immediately be seen that all elements of $S$ are... | All Elements of Left Operation are Left Zeroes | https://proofwiki.org/wiki/All_Elements_of_Left_Operation_are_Left_Zeroes | https://proofwiki.org/wiki/All_Elements_of_Left_Operation_are_Left_Zeroes | [
"Left Operation",
"Zero Elements"
] | [
"Definition:Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Left Operation",
"Definition:Semigroup",
"Definition:Left Zero",
"Definition:Left Zero",
"Definition:Semigroup"
] | [
"Structure under Left Operation is Semigroup",
"Definition:Semigroup",
"Definition:Left Operation",
"Definition:Left Zero",
"More than One Right Zero then No Left Zero",
"Definition:Right Zero"
] |
proofwiki-4649 | More than One Right Zero then No Left Zero | Let $\struct {S, \circ}$ be an algebraic structure.
If $\struct {S, \circ}$ has more than one left zero, then it has no right zero.
Likewise, if $\struct {S, \circ}$ has more than one right zero, then it has no left zero. | Let $\struct {S, \circ}$ be an algebraic structure with more than one left zero.
Take any two of these, and call them $z_{L_1}$ and $z_{L_2}$, where $z_{L_1} \ne z_{L_2}$.
Suppose $\struct {S, \circ}$ has a right zero.
Call it $z_R$.
Then, by the behaviour of $z_R$, $z_{L_1}$ and $z_{L_2}$:
:$z_{L_1} = z_{L_1} \circ z_... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
If $\struct {S, \circ}$ has more than one [[Definition:Left Zero|left zero]], then it has no [[Definition:Right Zero|right zero]].
Likewise, if $\struct {S, \circ}$ has more than one [[Definition:Right Zero|right zero]], then it ha... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]] with more than one [[Definition:Left Zero|left zero]].
Take any two of these, and call them $z_{L_1}$ and $z_{L_2}$, where $z_{L_1} \ne z_{L_2}$.
Suppose $\struct {S, \circ}$ has a [[Definition:Right Zero|right zero]].
Call it $z_R... | More than One Right Zero then No Left Zero | https://proofwiki.org/wiki/More_than_One_Right_Zero_then_No_Left_Zero | https://proofwiki.org/wiki/More_than_One_Right_Zero_then_No_Left_Zero | [
"Zero Elements"
] | [
"Definition:Algebraic Structure",
"Definition:Left Zero",
"Definition:Right Zero",
"Definition:Right Zero",
"Definition:Left Zero"
] | [
"Definition:Algebraic Structure",
"Definition:Left Zero",
"Definition:Right Zero",
"Category:Zero Elements"
] |
proofwiki-4650 | Left and Right Zero are the Same | Let $\struct {S, \circ}$ be an algebraic structure.
Let $z_L \in S$ be a left zero, and $z_R \in S$ be a right zero.
Then $z_L = z_R$, that is, both the left and right zero are the same, and are therefore a zero $z$.
Furthermore, $z$ is the ''only'' left and right zero for $\circ$. | Let $\struct {S, \circ}$ be an algebraic structure such that:
:$\exists z_L \in S: \forall x \in S: z_L \circ x = z_L$
:$\exists z_R \in S: \forall x \in S: x \circ z_R = z_R$
Then $z_L = z_L \circ z_R = z_R$ by both the above, hence the result.
The uniqueness of the left and right zero is a direct result of Zero Eleme... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]].
Let $z_L \in S$ be a [[Definition:Left Zero|left zero]], and $z_R \in S$ be a [[Definition:Right Zero|right zero]].
Then $z_L = z_R$, that is, both the left and right zero are the same, and are therefore a [[Definition:Zero Elemen... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure|algebraic structure]] such that:
:$\exists z_L \in S: \forall x \in S: z_L \circ x = z_L$
:$\exists z_R \in S: \forall x \in S: x \circ z_R = z_R$
Then $z_L = z_L \circ z_R = z_R$ by both the above, hence the result.
The uniqueness of the [[Definition:... | Left and Right Zero are the Same | https://proofwiki.org/wiki/Left_and_Right_Zero_are_the_Same | https://proofwiki.org/wiki/Left_and_Right_Zero_are_the_Same | [
"Zero Elements"
] | [
"Definition:Algebraic Structure",
"Definition:Left Zero",
"Definition:Right Zero",
"Definition:Zero Element",
"Definition:Left Zero",
"Definition:Right Zero"
] | [
"Definition:Algebraic Structure",
"Definition:Left Zero",
"Definition:Right Zero",
"Zero Element is Unique"
] |
proofwiki-4651 | Primitive of Cotangent Function | :$\ds \int \cot x \rd x = \ln \size {\sin x} + C$
where $\sin x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \cot x \rd x
| r = \int \frac {\cos x} {\sin x} \rd x
| c = {{Defof|Real Cotangent Function}}
}}
{{eqn | r = \int \frac {\paren {\sin x}'} {\sin x} \rd x
| c = Derivative of Sine Function
}}
{{eqn | r = \ln \size {\sin x} + C
| c = Primitive of Function under its D... | :$\ds \int \cot x \rd x = \ln \size {\sin x} + C$
where $\sin x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \cot x \rd x
| r = \int \frac {\cos x} {\sin x} \rd x
| c = {{Defof|Real Cotangent Function}}
}}
{{eqn | r = \int \frac {\paren {\sin x}'} {\sin x} \rd x
| c = [[Derivative of Sine Function]]
}}
{{eqn | r = \ln \size {\sin x} + C
| c = [[Primitive of Function under... | Primitive of Cotangent Function/Proof | https://proofwiki.org/wiki/Primitive_of_Cotangent_Function | https://proofwiki.org/wiki/Primitive_of_Cotangent_Function/Proof | [
"Primitive of Cotangent Function",
"Primitives of Trigonometric Functions",
"Primitives involving Cotangent Function",
"Cotangent Function"
] | [] | [
"Derivative of Sine Function",
"Primitive of Function under its Derivative"
] |
proofwiki-4652 | Inner Limit in Hausdorff Space by Open Neighborhoods | Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in a Hausdorff topological space $\struct {\XX, \tau}$.
Let $x \in \XX$.
Let $\map \mho x := \set {V \in \tau:\ x \in V}$ denote the set of open neighborhoods of $x$.
Let $\NN_\infty$ denote the set of cofinite subsets of $\N$:
:$\NN_\infty := \set {N \subs... | If $x \in \liminf_n C_n$ then there exist a sequence $\sequence {x_k}_{n \mathop \in \N}$ such that $x_k \to x$ while:
:$x_k \in C_{n_k}$
and
:$\sequence {n_k}_{k \mathop \in \N} \subseteq \N$ is a strictly increasing sequence of indices.
For any $V \in \map \mho x$ there exists $N_0 \in\N$ such that for all $i \ge N... | Let $\sequence {C_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]] in a [[Definition:Hausdorff Space|Hausdorff topological space]] $\struct {\XX, \tau}$.
Let $x \in \XX$.
Let $\map \mho x := \set {V \in \tau:\ x \in V}$ denote the [[Definition:Set|set]] of [[Definition:Open Neig... | If $x \in \liminf_n C_n$ then there exist a sequence $\sequence {x_k}_{n \mathop \in \N}$ such that $x_k \to x$ while:
:$x_k \in C_{n_k}$
and
:$\sequence {n_k}_{k \mathop \in \N} \subseteq \N$ is a strictly [[Definition:Increasing Sequence|increasing sequence]] of indices.
For any $V \in \map \mho x$ there exists $N... | Inner Limit in Hausdorff Space by Open Neighborhoods | https://proofwiki.org/wiki/Inner_Limit_in_Hausdorff_Space_by_Open_Neighborhoods | https://proofwiki.org/wiki/Inner_Limit_in_Hausdorff_Space_by_Open_Neighborhoods | [
"Set Theory",
"Hausdorff Spaces",
"Measure Theory",
"Limits of Sequence of Sets"
] | [
"Definition:Sequence",
"Definition:Set",
"Definition:T2 Space",
"Definition:Set",
"Definition:Open Neighborhood/Point",
"Definition:Set",
"Definition:Cofinite Subset",
"Definition:Inner Limit"
] | [
"Definition:Increasing/Sequence",
"Definition:Increasing/Sequence"
] |
proofwiki-4653 | Lagrange Polynomial Approximation | Let $f: D \to \R$ be $n + 1$ times differentiable in an interval $I \subseteq \R$.
Let $x_0, \dotsc, x_n \in I$ be pairwise distinct points.
Let $P$ be the Lagrange Interpolation Formula of degree at most $n$ such that:
:$\forall i \in \set {0, \dotsc, n}: \map P {x_i} = \map f {x_i}$
Let $\map R x = \map f x - \map P ... | This proof is similar to the proof of Taylor's theorem with the remainder in the Lagrange form, and is also based on Rolle's Theorem.
Observe that:
:$\map R {x_i} = 0$ for $i = 0, \dotsc, n$
and that:
:$R^{\paren {n + 1} } = f^{\paren {n + 1} }$
{{WLOG}}, assume that $x$ is different from all $x_i$ for $i = 0, \dotsc, ... | Let $f: D \to \R$ be $n + 1$ times [[Definition:Differentiable Real Function|differentiable]] in an [[Definition:Real Interval|interval]] $I \subseteq \R$.
Let $x_0, \dotsc, x_n \in I$ be [[Definition:Pairwise Distinct|pairwise distinct]] points.
Let $P$ be the [[Lagrange Interpolation Formula]] of [[Definition:Degre... | This proof is similar to the proof of [[Taylor's Theorem/One Variable/Proof by Rolle's Theorem|Taylor's theorem with the remainder in the Lagrange form]], and is also based on [[Rolle's Theorem]].
Observe that:
:$\map R {x_i} = 0$ for $i = 0, \dotsc, n$
and that:
:$R^{\paren {n + 1} } = f^{\paren {n + 1} }$
{{WLOG}},... | Lagrange Polynomial Approximation | https://proofwiki.org/wiki/Lagrange_Polynomial_Approximation | https://proofwiki.org/wiki/Lagrange_Polynomial_Approximation | [
"Real Analysis",
"Approximation Theory"
] | [
"Definition:Differentiable Mapping/Real Function",
"Definition:Real Interval",
"Definition:Distinct/Plural/Pairwise Distinct",
"Lagrange Interpolation Formula",
"Definition:Degree",
"Definition:Interval/Ordered_Set/Closed"
] | [
"Taylor's Theorem/One Variable/Proof by Rolle's Theorem",
"Rolle's Theorem",
"Extended Rolle's Theorem"
] |
proofwiki-4654 | Stabilizer of Subspace stabilizes Orthogonal Complement | Let $H$ be a finite-dimensional real or complex Hilbert space (that is, inner product space).
Let $t: H \to H$ be a normal operator on $H$.
Let $t$ stabilize a subspace $V$.
Then $t$ also stabilizes its orthogonal complement $V^\perp$. | Let $p: H \to V$ be the orthogonal projection of $H$ onto $V$.
Then the orthogonal projection of $H$ onto $V^\perp$ is $\mathbf 1 - p$, where $\mathbf 1$ is the identity map of $H$.
The fact that $t$ stabilizes $V$ can be expressed as:
:$\paren {\mathbf 1 - p} t p = 0$
or:
:$p t p = t p$
The goal is to show that:
:$p t... | Let $H$ be a [[Definition:Finite-Dimensional Hilbert Space|finite-dimensional]] [[Definition:Real Hilbert Space|real]] or [[Definition:Complex Hilbert Space|complex]] [[Definition:Hilbert Space|Hilbert space]] (that is, [[Definition:Inner Product Space|inner product space]]).
Let $t: H \to H$ be a [[Definition:Normal ... | Let $p: H \to V$ be the orthogonal projection of $H$ onto $V$.
Then the orthogonal projection of $H$ onto $V^\perp$ is $\mathbf 1 - p$, where $\mathbf 1$ is the [[Definition:Identity_Mapping|identity map]] of $H$.
The fact that $t$ stabilizes $V$ can be expressed as:
:$\paren {\mathbf 1 - p} t p = 0$
or:
:$p t p = t ... | Stabilizer of Subspace stabilizes Orthogonal Complement | https://proofwiki.org/wiki/Stabilizer_of_Subspace_stabilizes_Orthogonal_Complement | https://proofwiki.org/wiki/Stabilizer_of_Subspace_stabilizes_Orthogonal_Complement | [
"Hilbert Spaces"
] | [
"Definition:Hilbert Space/Finite-Dimensional",
"Definition:Hilbert Space/Real",
"Definition:Hilbert Space/Complex",
"Definition:Hilbert Space",
"Definition:Inner Product Space",
"Definition:Normal Operator",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement"
] | [
"Definition:Identity_Mapping",
"Definition:Inner Product",
"Definition:Adjoint_Operator",
"Definition:Trace (Linear Algebra)",
"Category:Hilbert Spaces"
] |
proofwiki-4655 | Primitive of Secant Function/Secant plus Tangent Form | :$\ds \int \sec x \rd x = \ln \size {\sec x + \tan x} + C$
where $\sec x + \tan x \ne 0$. | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tan x + \sec x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac \d {\d x} \tan x + \frac \d {\d x} \sec x
| c = Linear Combination of Derivatives
}}
{{eqn | r = \sec^2 x + \frac \d {\d x} \sec x
| c = Derivative of Tangent... | :$\ds \int \sec x \rd x = \ln \size {\sec x + \tan x} + C$
where $\sec x + \tan x \ne 0$. | Let:
{{begin-eqn}}
{{eqn | l = u
| r = \tan x + \sec x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac \d {\d x} \tan x + \frac \d {\d x} \sec x
| c = [[Linear Combination of Derivatives]]
}}
{{eqn | r = \sec^2 x + \frac \d {\d x} \sec x
| c = [[Derivative of T... | Primitive of Secant Function/Secant plus Tangent Form/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Secant_Function/Secant_plus_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Secant_Function/Secant_plus_Tangent_Form/Proof_1 | [
"Primitive of Secant Function"
] | [] | [
"Linear Combination of Derivatives",
"Derivative of Tangent Function",
"Derivative of Secant Function",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Primitive of Function under its Derivative"
] |
proofwiki-4656 | Primitive of Secant Function/Secant plus Tangent Form | :$\ds \int \sec x \rd x = \ln \size {\sec x + \tan x} + C$
where $\sec x + \tan x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \sec x \rd x
| r = \int \frac 1 {\cos x} \rd x
| c = Secant is Reciprocal of Cosine
}}
{{end-eqn}}
We make the Weierstrass Substitution:
{{begin-eqn}}
{{eqn | l = u
| r = \tan \frac x 2
}}
{{eqn | ll= \leadsto
| l = \cos x
| r = \frac {1 - u^2} {1 + u^2}
}}
... | :$\ds \int \sec x \rd x = \ln \size {\sec x + \tan x} + C$
where $\sec x + \tan x \ne 0$. | {{begin-eqn}}
{{eqn | l = \int \sec x \rd x
| r = \int \frac 1 {\cos x} \rd x
| c = [[Secant is Reciprocal of Cosine]]
}}
{{end-eqn}}
We make the [[Weierstrass Substitution]]:
{{begin-eqn}}
{{eqn | l = u
| r = \tan \frac x 2
}}
{{eqn | ll= \leadsto
| l = \cos x
| r = \frac {1 - u^2} {1 ... | Primitive of Secant Function/Secant plus Tangent Form/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Secant_Function/Secant_plus_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Secant_Function/Secant_plus_Tangent_Form/Proof_2 | [
"Primitive of Secant Function"
] | [] | [
"Secant is Reciprocal of Cosine",
"Weierstrass Substitution",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form",
"One Plus Tangent Half Angle over One Minus Tangent Half Angle"
] |
proofwiki-4657 | Subset of Natural Numbers is Cofinal iff Infinite | Consider the ordered set $\struct {\N, \le}$, where $\le$ is the usual ordering on the natural numbers.
Let $S \subseteq \N$.
Then $S$ is cofinal {{iff}} it is infinite. | From Rule of Transposition, we may replace the ''only if'' statement by its contrapositive.
Therefore, the following suffices: | Consider the [[Definition:Ordered Set|ordered set]] $\struct {\N, \le}$, where $\le$ is the [[Definition:Usual Ordering|usual ordering]] on the [[Definition:Natural Numbers|natural numbers]].
Let $S \subseteq \N$.
Then $S$ is [[Definition:Cofinal Subset|cofinal]] {{iff}} it is [[Definition:Infinite Set|infinite]]. | From [[Rule of Transposition]], we may replace the ''only if'' statement by its [[Definition:Contrapositive Statement|contrapositive]].
Therefore, the following suffices: | Subset of Natural Numbers is Cofinal iff Infinite | https://proofwiki.org/wiki/Subset_of_Natural_Numbers_is_Cofinal_iff_Infinite | https://proofwiki.org/wiki/Subset_of_Natural_Numbers_is_Cofinal_iff_Infinite | [
"Order Theory",
"Natural Numbers"
] | [
"Definition:Ordered Set",
"Definition:Usual Ordering",
"Definition:Natural Numbers",
"Definition:Cofinal Subset",
"Definition:Infinite Set"
] | [
"Rule of Transposition",
"Definition:Contrapositive Statement"
] |
proofwiki-4658 | Inner Limit in Hausdorff Space by Set Closures | Let $\struct {\XX, \tau}$ be a Hausdorff space.
Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $\XX$.
Then:
:$\ds \liminf_n C_n = \bigcap_{N \mathop \in \NN_\infty^\#} \map \cl {\bigcup_{n \mathop \in N} C_n}$
where:
:$\cl$ denotes set closure
:$\NN_\infty^\#$ denotes the set of cofinal subsets of $\... | $(1)$: Let:
:$\ds x \in \liminf_n \ C_n$
Let:
:$\Sigma \in \NN_\infty^\#$
Let $W$ be an open neighborhood of $x$.
Then there exists $N_0 \in \N$ such that for all $n \ge N_0$ such that $n \in \Sigma$:
:$W \cap C_n \ne \O$
Thus:
:$\ds x \in \map \cl {\bigcup_{n \mathop \in \Sigma} C_n}$
$(2)$: Let:
:$\ds x \notin \limi... | Let $\struct {\XX, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $\sequence {C_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]] in $\XX$.
Then:
:$\ds \liminf_n C_n = \bigcap_{N \mathop \in \NN_\infty^\#} \map \cl {\bigcup_{n \mathop \in N} C_n}$
where:
:$\cl$ ... | $(1)$: Let:
:$\ds x \in \liminf_n \ C_n$
Let:
:$\Sigma \in \NN_\infty^\#$
Let $W$ be an [[Definition:Open Neighborhood of Point|open neighborhood]] of $x$.
Then there exists $N_0 \in \N$ such that for all $n \ge N_0$ such that $n \in \Sigma$:
:$W \cap C_n \ne \O$
Thus:
:$\ds x \in \map \cl {\bigcup_{n \mathop... | Inner Limit in Hausdorff Space by Set Closures | https://proofwiki.org/wiki/Inner_Limit_in_Hausdorff_Space_by_Set_Closures | https://proofwiki.org/wiki/Inner_Limit_in_Hausdorff_Space_by_Set_Closures | [
"Set Theory",
"Hausdorff Spaces",
"Measure Theory",
"Limits of Sequence of Sets"
] | [
"Definition:T2 Space",
"Definition:Sequence",
"Definition:Set",
"Definition:Closure (Topology)",
"Definition:Set",
"Definition:Cofinal Subset"
] | [
"Definition:Open Neighborhood/Point",
"Definition:Open Neighborhood/Point",
"Definition:Set",
"Definition:Open Neighborhood/Point",
"Definition:Cofinal Subset"
] |
proofwiki-4659 | At Most Two Horizontal Asymptotes | The graph of a real function has at most two horizontal asymptotes. | Follows directly from the definition of a horizontal asymptote.
{{qed}} | The [[Definition:Graph of Mapping|graph]] of a [[Definition:Real Function|real function]] has at most two [[Definition:Horizontal Asymptote|horizontal asymptotes]]. | Follows directly from the definition of a [[Definition:Horizontal Asymptote|horizontal asymptote]].
{{qed}} | At Most Two Horizontal Asymptotes | https://proofwiki.org/wiki/At_Most_Two_Horizontal_Asymptotes | https://proofwiki.org/wiki/At_Most_Two_Horizontal_Asymptotes | [
"Asymptotes",
"Limits of Real Functions",
"Analytic Geometry"
] | [
"Definition:Graph of Mapping",
"Definition:Real Function",
"Definition:Horizontal Asymptote"
] | [
"Definition:Horizontal Asymptote"
] |
proofwiki-4660 | Ordinal is Transitive | Every ordinal is a transitive set. | Let $\alpha$ be an ordinal by Definition 1:
{{:Definition:Ordinal/Definition 1}}
Thus $\alpha$ is {{apriori}} transitive.
{{qed}} | Every [[Definition:Ordinal|ordinal]] is a [[Definition:Transitive Set|transitive set]]. | Let $\alpha$ be an [[Definition:Ordinal|ordinal]] by [[Definition:Ordinal/Definition 1|Definition 1]]:
{{:Definition:Ordinal/Definition 1}}
Thus $\alpha$ is {{apriori}} [[Definition:Transitive Set|transitive]].
{{qed}} | Ordinal is Transitive/Proof 1 | https://proofwiki.org/wiki/Ordinal_is_Transitive | https://proofwiki.org/wiki/Ordinal_is_Transitive/Proof_1 | [
"Ordinal is Transitive",
"Ordinals",
"Transitive Classes"
] | [
"Definition:Ordinal",
"Definition:Transitive Class"
] | [
"Definition:Ordinal",
"Definition:Ordinal/Definition 1",
"Definition:Transitive Class"
] |
proofwiki-4661 | Ordinal is Transitive | Every ordinal is a transitive set. | Let $\alpha$ be an ordinal by Definition 2:
{{:Definition:Ordinal/Definition 2}}
Thus $\alpha$ is {{apriori}} transitive.
{{qed}} | Every [[Definition:Ordinal|ordinal]] is a [[Definition:Transitive Set|transitive set]]. | Let $\alpha$ be an [[Definition:Ordinal|ordinal]] by [[Definition:Ordinal/Definition 2|Definition 2]]:
{{:Definition:Ordinal/Definition 2}}
Thus $\alpha$ is {{apriori}} [[Definition:Transitive Set|transitive]].
{{qed}} | Ordinal is Transitive/Proof 2 | https://proofwiki.org/wiki/Ordinal_is_Transitive | https://proofwiki.org/wiki/Ordinal_is_Transitive/Proof_2 | [
"Ordinal is Transitive",
"Ordinals",
"Transitive Classes"
] | [
"Definition:Ordinal",
"Definition:Transitive Class"
] | [
"Definition:Ordinal",
"Definition:Ordinal/Definition 2",
"Definition:Transitive Class"
] |
proofwiki-4662 | Ordinal is Transitive | Every ordinal is a transitive set. | Let $\alpha$ be an ordinal by Definition 3.
{{:Definition:Ordinal/Definition 3}}
That is, $\alpha$ is a transitive set.
{{explain|Determine exactly what is being proved here}}
{{qed}} | Every [[Definition:Ordinal|ordinal]] is a [[Definition:Transitive Set|transitive set]]. | Let $\alpha$ be an [[Definition:Ordinal|ordinal]] by [[Definition:Ordinal/Definition 3|Definition 3]].
{{:Definition:Ordinal/Definition 3}}
That is, $\alpha$ is a [[Definition:Transitive Set|transitive set]].
{{explain|Determine exactly what is being proved here}}
{{qed}} | Ordinal is Transitive/Proof 3 | https://proofwiki.org/wiki/Ordinal_is_Transitive | https://proofwiki.org/wiki/Ordinal_is_Transitive/Proof_3 | [
"Ordinal is Transitive",
"Ordinals",
"Transitive Classes"
] | [
"Definition:Ordinal",
"Definition:Transitive Class"
] | [
"Definition:Ordinal",
"Definition:Ordinal/Definition 3",
"Definition:Transitive Class"
] |
proofwiki-4663 | Ordinal is Transitive | Every ordinal is a transitive set. | Let $\alpha$ be an ordinal by Definition 4.
{{:Definition:Ordinal/Definition 4}}
The proof proceeds by the Principle of Superinduction.
From Empty Class is Transitive we start with the fact that $0$ is transitive.
{{qed|lemma}}
Let $x$ be transitive.
From Successor Set of Transitive Set is Transitive:
:$x^+$ is transit... | Every [[Definition:Ordinal|ordinal]] is a [[Definition:Transitive Set|transitive set]]. | Let $\alpha$ be an [[Definition:Ordinal|ordinal]] by [[Definition:Ordinal/Definition 4|Definition 4]].
{{:Definition:Ordinal/Definition 4}}
The proof proceeds by the [[Principle of Superinduction]].
From [[Empty Class is Transitive]] we start with the fact that $0$ is [[Definition:Transitive Set|transitive]].
{{qed|... | Ordinal is Transitive/Proof 4 | https://proofwiki.org/wiki/Ordinal_is_Transitive | https://proofwiki.org/wiki/Ordinal_is_Transitive/Proof_4 | [
"Ordinal is Transitive",
"Ordinals",
"Transitive Classes"
] | [
"Definition:Ordinal",
"Definition:Transitive Class"
] | [
"Definition:Ordinal",
"Definition:Ordinal/Definition 4",
"Principle of Superinduction",
"Empty Class is Transitive",
"Definition:Transitive Class",
"Definition:Transitive Class",
"Successor Set of Transitive Set is Transitive",
"Definition:Transitive Class",
"Class is Transitive iff Union is Subclas... |
proofwiki-4664 | Strictly Well-Founded Relation has no Relational Loops | Let $\prec$ be a strictly well-founded relation on $A$ and let $x_1, x_2, \ldots, x_n \in A$.
Then:
:$\neg \paren {x_1 \prec x_2 \land x_3 \prec x_4 \cdots \land x_n \prec x_1}$
That is, there are no relational loops within $A$. | Since $x_1, x_2, \ldots, x_n \in A$, there exists a $y$ such that $y = \set {x_1, x_2, \ldots, x_n}$.
Then $y$ is a non-empty subset of $A$.
So, by the definition of a strictly well-founded relation:
:$\exists w \in y: \forall z \in y: \neg w \prec z$
Now, suppose $x_1 \prec x_2 \land x_2 \prec x_3 \cdots \land x_n \pr... | Let $\prec$ be a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] on $A$ and let $x_1, x_2, \ldots, x_n \in A$.
Then:
:$\neg \paren {x_1 \prec x_2 \land x_3 \prec x_4 \cdots \land x_n \prec x_1}$
That is, there are no [[Definition:Relational Loop|relational loops]] within $A$. | Since $x_1, x_2, \ldots, x_n \in A$, there exists a $y$ such that $y = \set {x_1, x_2, \ldots, x_n}$.
Then $y$ is a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $A$.
So, by the definition of a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]]:
:$\exists w \in y: \f... | Strictly Well-Founded Relation has no Relational Loops | https://proofwiki.org/wiki/Strictly_Well-Founded_Relation_has_no_Relational_Loops | https://proofwiki.org/wiki/Strictly_Well-Founded_Relation_has_no_Relational_Loops | [
"Set Theory"
] | [
"Definition:Strictly Well-Founded Relation",
"Definition:Relational Loop"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Strictly Well-Founded Relation",
"Definition:Relational Loop",
"Definition:Strictly Well-Founded Relation"
] |
proofwiki-4665 | Epsilon Relation is Strictly Well-Founded | Let $\Epsilon$ denote the epsilon relation.
Then $\Epsilon$ is a strictly well-founded relation on every class $A$. | {{NotZFC}}
By the {{Axiom-link|Foundation}}:
:$\forall S: \paren {\exists x: x \in S \implies \exists y \in S: \forall x \in S: \neg x \in y}$
That is, by Nonempty Class has Members:
:$\forall S: \paren {S \ne \O \implies \exists y \in S: \forall x \in S: \neg x \in y}$
This holds for all sets $S$ whose construction i... | Let $\Epsilon$ denote the [[Definition:Epsilon Relation|epsilon relation]].
Then $\Epsilon$ is a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] on every [[Definition:Class (Class Theory)|class]] $A$. | {{NotZFC}}
By the {{Axiom-link|Foundation}}:
:$\forall S: \paren {\exists x: x \in S \implies \exists y \in S: \forall x \in S: \neg x \in y}$
That is, by [[Nonempty Class has Members]]:
:$\forall S: \paren {S \ne \O \implies \exists y \in S: \forall x \in S: \neg x \in y}$
This holds for all [[Definition:Set|se... | Epsilon Relation is Strictly Well-Founded | https://proofwiki.org/wiki/Epsilon_Relation_is_Strictly_Well-Founded | https://proofwiki.org/wiki/Epsilon_Relation_is_Strictly_Well-Founded | [
"Well-Founded Relations",
"Axiom of Foundation",
"Class Theory"
] | [
"Definition:Epsilon Relation",
"Definition:Strictly Well-Founded Relation",
"Definition:Class (Class Theory)"
] | [
"Nonempty Class has Members",
"Definition:Set",
"Axiom:Zermelo-Fraenkel Axioms",
"Definition:Set",
"Definition:Class (Class Theory)",
"Definition:Class (Class Theory)/Proper Class",
"Definition:Strictly Well-Founded Relation",
"Category:Well-Founded Relations",
"Category:Axiom of Foundation",
"Cat... |
proofwiki-4666 | Inner Limit in Normed Spaces by Open Balls | Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in a normed vector space $\struct {\XX, \norm {\, \cdot \,} }$.
Then the inner limit of $\sequence {C_n}_{n \mathop \in \N}$ is:
:$\ds \liminf_n C_n = \set {x: \forall \epsilon > 0: \exists N \in \NN_\infty: \forall n \in N: x \in C_n + B_\epsilon}$
where $... | The proof is an immediate result of Inner Limit in Hausdorff Space by Open Neighborhoods since the arbitrary open sets can be here replaced by open balls.
{{qed}} | Let $\sequence {C_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Set|sets]] in a [[Definition:Normed Vector Space|normed vector space]] $\struct {\XX, \norm {\, \cdot \,} }$.
Then the [[Definition:Inner Limit|inner limit]] of $\sequence {C_n}_{n \mathop \in \N}$ is:
:$\ds \liminf_n C_n ... | The proof is an immediate result of [[Inner Limit in Hausdorff Space by Open Neighborhoods]] since the arbitrary [[Definition:Open Set (Topology)|open sets]] can be here replaced by [[Definition:Open Ball|open balls]].
{{qed}} | Inner Limit in Normed Spaces by Open Balls | https://proofwiki.org/wiki/Inner_Limit_in_Normed_Spaces_by_Open_Balls | https://proofwiki.org/wiki/Inner_Limit_in_Normed_Spaces_by_Open_Balls | [
"Limits of Sequence of Sets"
] | [
"Definition:Sequence",
"Definition:Set",
"Definition:Normed Vector Space",
"Definition:Inner Limit",
"Definition:Open Ball"
] | [
"Inner Limit in Hausdorff Space by Open Neighborhoods",
"Definition:Open Set/Topology",
"Definition:Open Ball"
] |
proofwiki-4667 | Inner Limit is Closed Set | Let $\struct {S, \tau}$ be a Hausdorff topological space.
Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $S$.
Then the inner limit $\liminf_n C_n$ is a closed set. | According to Inner Limit in Hausdorff Space by Set Closures, the inner limit is given by an arbitrary intersection of closed sets which is closed in the topology $\tau$.
{{qed}}
Category:Limits of Sequence of Sets
olfc9bqbfl4j70ofagsfnhu9jetdors | Let $\struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff]] [[Definition:Topological Space|topological space]].
Let $\sequence {C_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of sets in $S$.
Then the [[Definition:Inner Limit|inner limit]] $\liminf_n C_n$ is a [[Definition:Closed Set (Topolo... | According to [[Inner Limit in Hausdorff Space by Set Closures]], the [[Definition:Inner Limit|inner limit]] is given by an arbitrary [[Definition:Set Intersection|intersection]] of [[Definition:Closed Set (Topology)|closed sets]] which is [[Definition:Closed Set (Topology)|closed]] in the [[Definition:Topology|topology... | Inner Limit is Closed Set | https://proofwiki.org/wiki/Inner_Limit_is_Closed_Set | https://proofwiki.org/wiki/Inner_Limit_is_Closed_Set | [
"Limits of Sequence of Sets"
] | [
"Definition:T2 Space",
"Definition:Topological Space",
"Definition:Sequence",
"Definition:Inner Limit",
"Definition:Closed Set/Topology"
] | [
"Inner Limit in Hausdorff Space by Set Closures",
"Definition:Inner Limit",
"Definition:Set Intersection",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Topology",
"Category:Limits of Sequence of Sets"
] |
proofwiki-4668 | Local Basis of Topological Vector Space | Let $\struct {\XX, \tau}$ be a topological vector space.
Let $0_\XX$ denote the zero vector of $\XX$.
Then there exists a local basis $\BB$ of $0_\XX$ with the following properties:
:$(1): \quad \forall W \in \BB: \exists V \in \BB$ such that $V + V \subseteq W$ (where the addition $V + V$ is meant in the sense of the ... | The proof will be carried out in various steps.
We will construct a collection of star-shaped neighborhoods of $0_\XX$.
Then we will show that it is indeed a local basis with the required properties.
Firstly we define the following set:
:$\BB_0 := \set {W \in \tau: 0 \in W, W \text{ is star-shaped} }$ | Let $\struct {\XX, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Let $0_\XX$ denote the [[Definition:Zero Vector|zero vector]] of $\XX$.
Then there exists a [[Definition:Local Basis|local basis]] $\BB$ of $0_\XX$ with the following properties:
:$(1): \quad \forall W \in \BB: \exists ... | The proof will be carried out in various steps.
We will construct a collection of [[Definition:Star Shaped_Set|star-shaped]] neighborhoods of $0_\XX$.
Then we will show that it is indeed a local basis with the required properties.
Firstly we define the following [[Definition:Set|set]]:
:$\BB_0 := \set {W \in \tau:... | Local Basis of Topological Vector Space | https://proofwiki.org/wiki/Local_Basis_of_Topological_Vector_Space | https://proofwiki.org/wiki/Local_Basis_of_Topological_Vector_Space | [] | [
"Definition:Topological Vector Space",
"Definition:Zero Vector",
"Definition:Local Basis",
"Definition:Minkowski Sum",
"Definition:Star Shaped_Set",
"Definition:Absorbent Set"
] | [
"Definition:Star Shaped_Set",
"Definition:Set"
] |
proofwiki-4669 | No Membership Loops | For any proper classes or sets $A_1, A_2, \ldots, A_n$:
:$\neg \paren {A_1 \in A_2 \land A_2 \in A_3 \land \cdots \land A_n \in A_1}$ | {{NotZFC}}
Either $A_1, A_2, \ldots, A_n$ are all sets, or there exists a proper class $A_m$ such that $1 \le m \le n$.
Suppose there exists a proper class $A_m$.
Then, by the definition of a proper class, $\neg A_m \in A_{m+1}$, since it is not a member of any class.
The result then follows directly.
Otherwise it foll... | For any [[Definition:Proper Class|proper classes]] or [[Definition:Set|sets]] $A_1, A_2, \ldots, A_n$:
:$\neg \paren {A_1 \in A_2 \land A_2 \in A_3 \land \cdots \land A_n \in A_1}$ | {{NotZFC}}
Either $A_1, A_2, \ldots, A_n$ are all [[Definition:Set|sets]], or there exists a [[Definition:Proper Class|proper class]] $A_m$ such that $1 \le m \le n$.
Suppose there exists a [[Definition:Proper Class|proper class]] $A_m$.
Then, by the definition of a [[Definition:Proper Class|proper class]], $\neg A... | No Membership Loops | https://proofwiki.org/wiki/No_Membership_Loops | https://proofwiki.org/wiki/No_Membership_Loops | [
"Class Theory",
"Axiom of Foundation"
] | [
"Definition:Class (Class Theory)/Proper Class",
"Definition:Set"
] | [
"Definition:Set",
"Definition:Class (Class Theory)/Proper Class",
"Definition:Class (Class Theory)/Proper Class",
"Definition:Class (Class Theory)/Proper Class",
"Definition:Class (Class Theory)",
"Definition:Set",
"Epsilon Relation is Strictly Well-Founded",
"Strictly Well-Founded Relation has no Rel... |
proofwiki-4670 | Composite of Isomorphisms is Isomorphism/Algebraic Structure | Let:
:$\struct {S_1, \odot_1, \odot_2, \ldots, \odot_n}$
:$\struct {S_2, *_1, *_2, \ldots, *_n}$
:$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be algebraic structures.
Let:
:$\phi: \struct {S_1, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
:$\psi: \struct {S_2, *_1, *_2, \ldots,... | If $\phi$ and $\psi$ are both isomorphisms, then they are by definition:
:homomorphisms
:bijections.
From Composite of Homomorphisms on Algebraic Structure is Homomorphism:
:$\phi \circ \psi$ and $\psi \circ \phi$ are both homomorphisms.
From Composite of Bijections is Bijection:
:$\phi \circ \psi$ and $\psi \circ \phi... | Let:
:$\struct {S_1, \odot_1, \odot_2, \ldots, \odot_n}$
:$\struct {S_2, *_1, *_2, \ldots, *_n}$
:$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be [[Definition:Algebraic Structure|algebraic structures]].
Let:
:$\phi: \struct {S_1, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
:$\... | If $\phi$ and $\psi$ are both [[Definition:Isomorphism (Abstract Algebra)|isomorphisms]], then they are by definition:
:[[Definition:Homomorphism (Abstract Algebra)|homomorphisms]]
:[[Definition:Bijection|bijections]].
From [[Composite of Homomorphisms on Algebraic Structure is Homomorphism]]:
:$\phi \circ \psi$ and $... | Composite of Isomorphisms is Isomorphism/Algebraic Structure | https://proofwiki.org/wiki/Composite_of_Isomorphisms_is_Isomorphism/Algebraic_Structure | https://proofwiki.org/wiki/Composite_of_Isomorphisms_is_Isomorphism/Algebraic_Structure | [
"Composite of Isomorphisms is Isomorphism",
"Isomorphisms (Abstract Algebra)"
] | [
"Definition:Algebraic Structure",
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Composition of Mappings",
"Definition:Isomorphism (Abstract Algebra)"
] | [
"Definition:Isomorphism (Abstract Algebra)",
"Definition:Homomorphism (Abstract Algebra)",
"Definition:Bijection",
"Composite of Homomorphisms is Homomorphism/Algebraic Structure",
"Definition:Homomorphism (Abstract Algebra)",
"Composite of Bijections is Bijection",
"Definition:Bijection",
"Definition... |
proofwiki-4671 | Composite of Isomorphisms is Isomorphism/R-Algebraic Structure | Let:
:$\struct {S_1, \ast_1}_R$
:$\struct {S_2, \ast_2}_R$
:$\struct {S_3, \ast_3}_R$
be $R$-algebraic structures with the same number of operations.
Let:
:$\phi: \struct {S_1, \ast_1}_R \to \struct {S_2, \ast_2}_R$
:$\psi: \struct {S_2, \ast_2}_R \to \struct {S_3, \ast_3}_R$
be isomorphisms.
Then the composite of $\ph... | If $\phi$ and $\psi$ are both isomorphisms, then they are by definition:
:homomorphisms
;bijections.
So:
:From Composite of Homomorphisms for R-Algebraic Structures is Homomorphism we have that $\phi \circ \psi$ and $\psi \circ \phi$ are both homomorphisms
:From Composite of Bijections is Bijection we have that $\phi \... | Let:
:$\struct {S_1, \ast_1}_R$
:$\struct {S_2, \ast_2}_R$
:$\struct {S_3, \ast_3}_R$
be [[Definition:R-Algebraic Structure|$R$-algebraic structures]] with the same number of operations.
Let:
:$\phi: \struct {S_1, \ast_1}_R \to \struct {S_2, \ast_2}_R$
:$\psi: \struct {S_2, \ast_2}_R \to \struct {S_3, \ast_3}_R$
be [[... | If $\phi$ and $\psi$ are both [[Definition:R-Algebraic Structure Isomorphism|isomorphisms]], then they are by definition:
:[[Definition:R-Algebraic Structure Homomorphism|homomorphisms]]
;[[Definition:Bijection|bijections]].
So:
:From [[Composite of Homomorphisms is Homomorphism/R-Algebraic Structure|Composite of Homo... | Composite of Isomorphisms is Isomorphism/R-Algebraic Structure | https://proofwiki.org/wiki/Composite_of_Isomorphisms_is_Isomorphism/R-Algebraic_Structure | https://proofwiki.org/wiki/Composite_of_Isomorphisms_is_Isomorphism/R-Algebraic_Structure | [
"Composite of Isomorphisms is Isomorphism",
"Isomorphisms (Abstract Algebra)"
] | [
"Definition:R-Algebraic Structure",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism",
"Definition:Composition of Mappings",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism"
] | [
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism",
"Definition:R-Algebraic Structure Homomorphism",
"Definition:Bijection",
"Composite of Homomorphisms is Homomorphism/R-Algebraic Structure",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism",
"Compos... |
proofwiki-4672 | Element of Transitive Class | Let $B$ be a transitive class.
Then:
:$A \in B \implies A \subsetneq B$
where $\subsetneq$ denotes a proper subset). | {{NotZFC}}
By the definition of a transitive class:
:$A \in B \implies A \subseteq B$
But $A \ne B$ because $\paren {A = B \land A \in B} \implies A \in A$, which by No Membership Loops is a contradiction.
Therefore $A \subsetneq B$.
{{qed}} | Let $B$ be a [[Definition:Transitive Class|transitive class]].
Then:
:$A \in B \implies A \subsetneq B$
where $\subsetneq$ denotes a [[Definition:Proper Subset|proper subset]]). | {{NotZFC}}
By the definition of a [[Definition:Transitive Class|transitive class]]:
:$A \in B \implies A \subseteq B$
But $A \ne B$ because $\paren {A = B \land A \in B} \implies A \in A$, which by [[No Membership Loops]] is a contradiction.
Therefore $A \subsetneq B$.
{{qed}} | Element of Transitive Class | https://proofwiki.org/wiki/Element_of_Transitive_Class | https://proofwiki.org/wiki/Element_of_Transitive_Class | [
"Class Theory",
"Transitive Classes"
] | [
"Definition:Transitive Class",
"Definition:Proper Subset"
] | [
"Definition:Transitive Class",
"No Membership Loops"
] |
proofwiki-4673 | Tangent Line to Convex Graph | Let $f$ be a real function that is:
:continuous on some closed interval $\closedint a b$
:differentiable and convex on the open interval $\openint a b$.
Then all the tangent lines to $f$ are below the graph of $f$.
{{explain|"below"}} | :500px
Let $\TT$ be the tangent line to $f$ at some point $\tuple {c, \map f c}$, $c \in \openint a b$.
Let the gradient of $\TT$ be $m$.
Let $\tuple {x_1, y_1}$ be an arbitrary point on $\TT$.
From the point-slope form of a straight line:
{{begin-eqn}}
{{eqn | l = y - y_1
| r = m \paren {x - x_1}
| c =
}}... | Let $f$ be a [[Definition:Real Function|real function]] that is:
:[[Definition:Continuous on Interval|continuous]] on some [[Definition:Closed Real Interval|closed interval]] $\closedint a b$
:[[Definition:Differentiable on Interval|differentiable]] and [[Definition:Convex Real Function|convex]] on the [[Definition:Ope... | :[[File:Concaveup.png|500px]]
Let $\TT$ be the [[Definition:Tangent Line|tangent line]] to $f$ at some [[Definition:Point|point]] $\tuple {c, \map f c}$, $c \in \openint a b$.
Let the [[Definition:Gradient|gradient]] of $\TT$ be $m$.
Let $\tuple {x_1, y_1}$ be an arbitrary [[Definition:Point|point]] on $\TT$.
From ... | Tangent Line to Convex Graph | https://proofwiki.org/wiki/Tangent_Line_to_Convex_Graph | https://proofwiki.org/wiki/Tangent_Line_to_Convex_Graph | [
"Convex Real Functions",
"Tangents",
"Differential Calculus",
"Analytic Geometry"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Convex Real Function",
"Definition:Real Interval/Open",
"Definition:Tangent Line",
"Definition:Graph of Mapping"
] | [
"File:Concaveup.png",
"Definition:Tangent Line",
"Definition:Point",
"Definition:Gradient",
"Definition:Point",
"Equation of Straight Line in Plane/Point-Slope Form",
"Definition:Line/Straight Line",
"Definition:Graph of Mapping",
"Definition:Graph of Mapping",
"Mean Value Theorem",
"Definition:... |
proofwiki-4674 | Characterization of Lower Semicontinuity | Let $f: S \to \overline \R$ be an extended real valued function.
Let $S$ be endowed with a topology $\tau$.
The following are equivalent:
:$(1): \quad$ $f$ is lower semicontinuous (LSC) on $S$.
:$(2): \quad$ All lower level sets of $f$ are closed in $S$.
:$(3): \quad$ The epigraph $\map {\operatorname{epi}} f$ of $f$ i... | For each $x_0 \in S$, let $\map \mho {x_0}$ be the set of open neighborhoods of $x_0$ in $\struct {X, \tau}$. | Let $f: S \to \overline \R$ be an [[Definition:Extended Real-Valued Function|extended real valued function]].
Let $S$ be endowed with a [[Definition:Topology|topology]] $\tau$.
The following are equivalent:
:$(1): \quad$ $f$ is [[Definition:Lower Semicontinuous on Subset|lower semicontinuous (LSC)]] on $S$.
:$(2): ... | For each $x_0 \in S$, let $\map \mho {x_0}$ be the [[Definition:Set|set]] of [[Definition:Open Neighborhood|open neighborhoods]] of $x_0$ in $\struct {X, \tau}$. | Characterization of Lower Semicontinuity | https://proofwiki.org/wiki/Characterization_of_Lower_Semicontinuity | https://proofwiki.org/wiki/Characterization_of_Lower_Semicontinuity | [
"Limits of Mappings",
"Characterization of Lower Semicontinuity",
"Lower Semicontinuity"
] | [
"Definition:Extended Real-Valued Function",
"Definition:Topology",
"Definition:Lower Semicontinuous/Subset",
"Definition:Lower Level Set",
"Definition:Closed Set/Topology",
"Definition:Epigraph",
"Definition:Closed Set/Topology",
"Definition:Product Topology"
] | [
"Definition:Set",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Set"
] |
proofwiki-4675 | Restriction of Strictly Well-Founded Relation is Strictly Well-Founded | Let $\struct {S, \RR}$ be a relational structure.
Let $\RR \subseteq S \times S$ be a strictly well-founded relation on $S$.
Let $T \subseteq S$ be a subset of $S$.
Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.
Then $\RR {\restriction_T}$ is a strictly well-founded relation on $T$. | By definition of strictly well-founded relation, every non-empty subset of $S$ has a minimal element.
By Subset Relation is Transitive, every subset of $T$ is also a subset of $S$.
Therefore every non-empty subset of $T$ has a minimal element.
Thus by definition, $\RR$ is a strictly well-founded relation on $T$.
{{qed}... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $\RR \subseteq S \times S$ be a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] on $S$.
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $\RR {\restriction_T} \subseteq T \times T$... | By definition of [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]], every [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$ has a [[Definition:Minimal Element|minimal element]].
By [[Subset Relation is Transitive]], every [[Definition:Subset|subset]] of $T$ is also a... | Restriction of Strictly Well-Founded Relation is Strictly Well-Founded | https://proofwiki.org/wiki/Restriction_of_Strictly_Well-Founded_Relation_is_Strictly_Well-Founded | https://proofwiki.org/wiki/Restriction_of_Strictly_Well-Founded_Relation_is_Strictly_Well-Founded | [
"Well-Founded Relations"
] | [
"Definition:Relational Structure",
"Definition:Strictly Well-Founded Relation",
"Definition:Subset",
"Definition:Restriction/Relation",
"Definition:Strictly Well-Founded Relation"
] | [
"Definition:Strictly Well-Founded Relation",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Minimal/Element",
"Subset Relation is Transitive",
"Definition:Subset",
"Definition:Subset",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Minimal/Element",
"Definition:Strict... |
proofwiki-4676 | Restriction of Strict Well-Ordering is Strict Well-Ordering | Let $R$ be a strict well-ordering of $A$.
Let $B \subseteq A$.
Then $R$ is a strict well-ordering of $B$. | By Restriction of Strictly Well-Founded Relation is Strictly Well-Founded, $R$ is a strictly well-founded relation on $B$.
By Restriction of Total Ordering is Total Ordering, $R$ is a total ordering on $B$.
By the above two statements, $R$ is a strict well-ordering of $B$.
{{explain|Indicate the specific definition whi... | Let $R$ be a [[Definition:Strict Well-Ordering|strict well-ordering]] of $A$.
Let $B \subseteq A$.
Then $R$ is a [[Definition:Strict Well-Ordering|strict well-ordering]] of $B$. | By [[Restriction of Strictly Well-Founded Relation is Strictly Well-Founded]], $R$ is a [[Definition:Strictly Well-Founded Relation|strictly well-founded relation]] on $B$.
By [[Restriction of Total Ordering is Total Ordering]], $R$ is a [[Definition:Total Ordering|total ordering]] on $B$.
By the above two statements... | Restriction of Strict Well-Ordering is Strict Well-Ordering | https://proofwiki.org/wiki/Restriction_of_Strict_Well-Ordering_is_Strict_Well-Ordering | https://proofwiki.org/wiki/Restriction_of_Strict_Well-Ordering_is_Strict_Well-Ordering | [
"Well-Orderings"
] | [
"Definition:Strict Well-Ordering",
"Definition:Strict Well-Ordering"
] | [
"Restriction of Strictly Well-Founded Relation is Strictly Well-Founded",
"Definition:Strictly Well-Founded Relation",
"Restriction of Total Ordering is Total Ordering",
"Definition:Total Ordering",
"Definition:Strict Well-Ordering"
] |
proofwiki-4677 | Alternative Definition of Ordinal | A set $S$ is an ordinal {{iff}} $S$ is transitive and is strictly well-ordered by the $\in$-relation. | === Necessary Condition ===
Suppose that $S$ is an ordinal.
Then $S$ is transitive.
By definition, the strict well-ordering on $S$ is given by the $\in$-relation.
Hence, the necessary condition is satisfied.
{{qed|lemma}} | A [[Definition:Set|set]] $S$ is an [[Definition:Ordinal|ordinal]] {{iff}} $S$ is [[Definition:Transitive Set|transitive]] and is [[Definition:Strict Well-Ordering|strictly well-ordered]] by the [[Definition:Epsilon Relation|$\in$-relation]]. | === Necessary Condition ===
Suppose that $S$ is an [[Definition:Ordinal|ordinal]].
Then [[Ordinal is Transitive|$S$ is transitive]].
By definition, the [[Definition:Strict Well-Ordering|strict well-ordering]] on $S$ is given by the [[Definition:Epsilon Relation|$\in$-relation]].
Hence, the [[Definition:Necessary C... | Alternative Definition of Ordinal | https://proofwiki.org/wiki/Alternative_Definition_of_Ordinal | https://proofwiki.org/wiki/Alternative_Definition_of_Ordinal | [
"Ordinals"
] | [
"Definition:Set",
"Definition:Ordinal",
"Definition:Transitive Class",
"Definition:Strict Well-Ordering",
"Definition:Epsilon Relation"
] | [
"Definition:Ordinal",
"Ordinal is Transitive",
"Definition:Strict Well-Ordering",
"Definition:Epsilon Relation",
"Definition:Conditional/Necessary Condition",
"Definition:Strict Well-Ordering",
"Definition:Epsilon Relation",
"Definition:Ordinal"
] |
proofwiki-4678 | Class of All Ordinals is Ordinal | The class of all ordinals $\On$ is an ordinal.
{{explain|Does this not contradict Class of All Ordinals is Proper Class? Sorry, I now read the discussion page. But the current Definition:Ordinal requires an ordinal to be a set, so $\On$ isn't an ordinal. We need a scond definition of ordinal for this theorem.}} | {{questionable|This whole proof probably needs to be rewritten from scratch.}}
{{NotZFC}}
The epsilon relation is equivalent to the strict subset relation when restricted to ordinals by Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.
It follows that:
:$\forall x \in \On: x \subset \On$
{{explain|We h... | The [[Definition:Class of All Ordinals|class of all ordinals]] $\On$ is an [[Definition:Ordinal|ordinal]].
{{explain|Does this not contradict [[Class of All Ordinals is Proper Class]]? Sorry, I now read the discussion page. But the current [[Definition:Ordinal]] requires an ordinal to be a set, so $\On$ isn't an ordin... | {{questionable|This whole proof probably needs to be rewritten from scratch.}}
{{NotZFC}}
The [[Definition:Epsilon Relation|epsilon relation]] is equivalent to the strict subset relation when [[Definition:Restriction of Relation|restricted]] to [[Definition:Ordinal|ordinals]] by [[Transitive Set is Proper Subset of O... | Class of All Ordinals is Ordinal | https://proofwiki.org/wiki/Class_of_All_Ordinals_is_Ordinal | https://proofwiki.org/wiki/Class_of_All_Ordinals_is_Ordinal | [
"Class of All Ordinals"
] | [
"Definition:Class of All Ordinals",
"Definition:Ordinal",
"Class of All Ordinals is Proper Class",
"Definition:Ordinal"
] | [
"Definition:Epsilon Relation",
"Definition:Restriction/Relation",
"Definition:Ordinal",
"Transitive Set is Proper Subset of Ordinal iff Element of Ordinal",
"Definition:Initial Segment",
"Definition:Class of All Ordinals",
"Definition:Class (Class Theory)",
"Definition:Class Equality",
"Definition:O... |
proofwiki-4679 | Transitive Set is Proper Subset of Ordinal iff Element of Ordinal | Let $A$ be an ordinal.
Let $B$ be a transitive set.
Then:
:$B \subsetneq A \iff B \in A$ | === Necessary Condition ===
Suppose that $B \in A$.
From Ordinal is Transitive, it follows that $B \subseteq A$.
Also, $B \ne A$ by Ordinal is not Element of Itself.
Therefore, $B \subsetneq A$, as desired.
{{qed|lemma}} | Let $A$ be an [[Definition:Ordinal|ordinal]].
Let $B$ be a [[Definition:Transitive Set|transitive set]].
Then:
:$B \subsetneq A \iff B \in A$ | === Necessary Condition ===
Suppose that $B \in A$.
From [[Ordinal is Transitive]], it follows that $B \subseteq A$.
Also, $B \ne A$ by [[Ordinal is not Element of Itself]].
Therefore, $B \subsetneq A$, as desired.
{{qed|lemma}} | Transitive Set is Proper Subset of Ordinal iff Element of Ordinal | https://proofwiki.org/wiki/Transitive_Set_is_Proper_Subset_of_Ordinal_iff_Element_of_Ordinal | https://proofwiki.org/wiki/Transitive_Set_is_Proper_Subset_of_Ordinal_iff_Element_of_Ordinal | [
"Ordinals",
"Transitive Classes"
] | [
"Definition:Ordinal",
"Definition:Transitive Class"
] | [
"Ordinal is Transitive",
"Ordinal is not Element of Itself",
"Ordinal is Transitive"
] |
proofwiki-4680 | Integral of Arcsine Function | :$\ds \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$
for $x \in \closedint {-1} 1$. | {{begin-eqn}}
{{eqn | l = \int \arcsin x \rd x
| r = \int 1 \cdot \arcsin x \rd x
}}
{{eqn | r = x \arcsin x - \int x \paren {\frac \rd {\rd x} \arcsin x} \rd x
| c = Integration by Parts
}}
{{eqn | r = x \arcsin x - \int \frac x {\sqrt {1 - x^2} } \rd x
| c = Derivative of Arcsine Function
}}
{{end-e... | :$\ds \int \arcsin x \rd x = x \arcsin x + \sqrt {1 - x^2} + C$
for $x \in \closedint {-1} 1$. | {{begin-eqn}}
{{eqn | l = \int \arcsin x \rd x
| r = \int 1 \cdot \arcsin x \rd x
}}
{{eqn | r = x \arcsin x - \int x \paren {\frac \rd {\rd x} \arcsin x} \rd x
| c = [[Integration by Parts]]
}}
{{eqn | r = x \arcsin x - \int \frac x {\sqrt {1 - x^2} } \rd x
| c = [[Derivative of Arcsine Function]]
}}... | Integral of Arcsine Function | https://proofwiki.org/wiki/Integral_of_Arcsine_Function | https://proofwiki.org/wiki/Integral_of_Arcsine_Function | [
"Integral Calculus"
] | [] | [
"Integration by Parts",
"Derivative of Arcsine Function",
"Integration by Substitution",
"Integration by Substitution",
"Integral of Power"
] |
proofwiki-4681 | Second Derivative of Convex Real Function is Non-Negative | Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.
Then $f$ is convex on $\openint a b$ {{iff}} its second derivative $f' ' \ge 0$ on $\openint a b$. | From Real Function is Convex iff Derivative is Increasing, $f$ is convex {{iff}} $f'$ is increasing.
From Derivative of Monotone Function, $f'$ is increasing {{iff}} its second derivative $f' ' \ge 0$.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then $f$ is [[Definition:Convex Real Function|convex]] on $\openint a b$ {{iff}} its [[Definition:Second Derivative|second deriv... | From [[Real Function is Convex iff Derivative is Increasing]], $f$ is [[Definition:Convex Real Function|convex]] {{iff}} $f'$ is [[Definition:Increasing Real Function|increasing]].
From [[Derivative of Monotone Function]], $f'$ is [[Definition:Increasing Real Function|increasing]] {{iff}} its [[Definition:Second Deriv... | Second Derivative of Convex Real Function is Non-Negative | https://proofwiki.org/wiki/Second_Derivative_of_Convex_Real_Function_is_Non-Negative | https://proofwiki.org/wiki/Second_Derivative_of_Convex_Real_Function_is_Non-Negative | [
"Differential Calculus",
"Convex Real Functions"
] | [
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Real Interval/Open",
"Definition:Convex Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative"
] | [
"Real Function is Convex iff Derivative is Increasing",
"Definition:Convex Real Function",
"Definition:Increasing/Real Function",
"Derivative of Monotone Function",
"Definition:Increasing/Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative"
] |
proofwiki-4682 | Ordinal Membership is Trichotomy | Let $\alpha$ and $\beta$ be ordinals.
Then:
:$\paren {\alpha = \beta} \lor \paren {\alpha \in \beta} \lor \paren {\beta \in \alpha}$
where $\lor$ denotes logical or. | From Class of All Ordinals is Well-Ordered by Subset Relation, $\On$ is a nest.
Hence:
:$\forall \alpha, \beta \in \On: \paren {\alpha \subsetneqq \beta} \lor \paren {\beta \subsetneqq \alpha} \lor \paren {\alpha = \beta}$
From Transitive Set is Proper Subset of Ordinal iff Element of Ordinal, this is equivalent to:
:$... | Let $\alpha$ and $\beta$ be [[Definition:Ordinal|ordinals]].
Then:
:$\paren {\alpha = \beta} \lor \paren {\alpha \in \beta} \lor \paren {\beta \in \alpha}$
where $\lor$ denotes [[Definition:Disjunction|logical or]]. | From [[Class of All Ordinals is Well-Ordered by Subset Relation]], $\On$ is a [[Definition:Nest (Class Theory)|nest]].
Hence:
:$\forall \alpha, \beta \in \On: \paren {\alpha \subsetneqq \beta} \lor \paren {\beta \subsetneqq \alpha} \lor \paren {\alpha = \beta}$
From [[Transitive Set is Proper Subset of Ordinal iff E... | Ordinal Membership is Trichotomy/Proof 1 | https://proofwiki.org/wiki/Ordinal_Membership_is_Trichotomy | https://proofwiki.org/wiki/Ordinal_Membership_is_Trichotomy/Proof_1 | [
"Ordinal Membership is Trichotomy",
"Ordinals"
] | [
"Definition:Ordinal",
"Definition:Disjunction"
] | [
"Class of All Ordinals is Well-Ordered by Subset Relation",
"Definition:Nest/Class Theory",
"Transitive Set is Proper Subset of Ordinal iff Element of Ordinal"
] |
proofwiki-4683 | Ordinal Membership is Trichotomy | Let $\alpha$ and $\beta$ be ordinals.
Then:
:$\paren {\alpha = \beta} \lor \paren {\alpha \in \beta} \lor \paren {\beta \in \alpha}$
where $\lor$ denotes logical or. | By Relation between Two Ordinals, it follows that:
:$\paren {\alpha = \beta} \lor \paren {\alpha \subset \beta} \lor \paren {\beta \subset \alpha}$
By Transitive Set is Proper Subset of Ordinal iff Element of Ordinal, the result follows.
{{qed}} | Let $\alpha$ and $\beta$ be [[Definition:Ordinal|ordinals]].
Then:
:$\paren {\alpha = \beta} \lor \paren {\alpha \in \beta} \lor \paren {\beta \in \alpha}$
where $\lor$ denotes [[Definition:Disjunction|logical or]]. | By [[Relation between Two Ordinals]], it follows that:
:$\paren {\alpha = \beta} \lor \paren {\alpha \subset \beta} \lor \paren {\beta \subset \alpha}$
By [[Transitive Set is Proper Subset of Ordinal iff Element of Ordinal]], the result follows.
{{qed}} | Ordinal Membership is Trichotomy/Proof 2 | https://proofwiki.org/wiki/Ordinal_Membership_is_Trichotomy | https://proofwiki.org/wiki/Ordinal_Membership_is_Trichotomy/Proof_2 | [
"Ordinal Membership is Trichotomy",
"Ordinals"
] | [
"Definition:Ordinal",
"Definition:Disjunction"
] | [
"Relation between Two Ordinals",
"Transitive Set is Proper Subset of Ordinal iff Element of Ordinal"
] |
proofwiki-4684 | Second Derivative of Concave Real Function is Non-Positive | Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.
Then $f$ is concave on $\openint a b$ {{iff}} its second derivative $f'' \le 0$ on $\openint a b$. | From Real Function is Concave iff Derivative is Decreasing, $f$ is concave {{iff}} $f'$ is decreasing.
From Derivative of Monotone Function, $f'$ is decreasing {{iff}} its second derivative $f'' \le 0$.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then $f$ is [[Definition:Concave Real Function|concave]] on $\openint a b$ {{iff}} its [[Definition:Second Derivative|second der... | From [[Real Function is Concave iff Derivative is Decreasing]], $f$ is [[Definition:Concave Real Function|concave]] {{iff}} $f'$ is [[Definition:Decreasing Real Function|decreasing]].
From [[Derivative of Monotone Function]], $f'$ is [[Definition:Decreasing Real Function|decreasing]] {{iff}} its [[Definition:Second De... | Second Derivative of Concave Real Function is Non-Positive | https://proofwiki.org/wiki/Second_Derivative_of_Concave_Real_Function_is_Non-Positive | https://proofwiki.org/wiki/Second_Derivative_of_Concave_Real_Function_is_Non-Positive | [
"Differential Calculus",
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Real Interval/Open",
"Definition:Concave Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative"
] | [
"Real Function is Concave iff Derivative is Decreasing",
"Definition:Concave Real Function",
"Definition:Decreasing/Real Function",
"Derivative of Monotone Function",
"Definition:Decreasing/Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative"
] |
proofwiki-4685 | Ordinal is Member of Class of All Ordinals | Let $A$ be an ordinal.
Then:
:$A \in \On \lor A = \On$
where $\On$ denotes the class of all ordinals. | We have {{hypothesis}} that $A$ is an ordinal
From Class of All Ordinals is Ordinal and Ordinal Membership is Trichotomy:
:$A \in \On \lor A = \On \lor \On \in A$
But by the Burali-Forti Paradox $\On$ is a proper class.
Therefore:
:$A \in \On \lor A = \On$
{{qed}}
{{explain|please say or link to why being a proper clas... | Let $A$ be an [[Definition:Ordinal|ordinal]].
Then:
:$A \in \On \lor A = \On$
where $\On$ denotes the [[Definition:Class of All Ordinals|class of all ordinals]]. | We have {{hypothesis}} that $A$ is an [[Definition:Ordinal|ordinal]]
From [[Class of All Ordinals is Ordinal]] and [[Ordinal Membership is Trichotomy]]:
:$A \in \On \lor A = \On \lor \On \in A$
But by the [[Burali-Forti Paradox]] $\On$ is a [[Definition:Proper Class|proper class]].
Therefore:
:$A \in \On \lor A = \O... | Ordinal is Member of Class of All Ordinals | https://proofwiki.org/wiki/Ordinal_is_Member_of_Class_of_All_Ordinals | https://proofwiki.org/wiki/Ordinal_is_Member_of_Class_of_All_Ordinals | [
"Class of All Ordinals"
] | [
"Definition:Ordinal",
"Definition:Class of All Ordinals"
] | [
"Definition:Ordinal",
"Class of All Ordinals is Ordinal",
"Ordinal Membership is Trichotomy",
"Burali-Forti Paradox",
"Definition:Class (Class Theory)/Proper Class"
] |
proofwiki-4686 | Ordinal is Subset of Class of All Ordinals | Suppose $A$ is an ordinal.
Then:
:$A \subseteq \On$
where $\On$ represents the class of all ordinals. | By Ordinal is Member of Class of All Ordinals:
:$A \in \On \lor A = \On$
In either case:
:$A \subseteq \On$
since $\On$ is transitive.
{{explain|Why is $\On$ necessarily transitive? This follows smoothly if it is assumed that $\On$ is a subclass of a Basic Universe, or otherwise from an axiomatic framework. Hence we ne... | Suppose $A$ is an [[Definition:Ordinal|ordinal]].
Then:
:$A \subseteq \On$
where $\On$ represents the [[Definition:Class of All Ordinals|class of all ordinals]]. | By [[Ordinal is Member of Class of All Ordinals]]:
:$A \in \On \lor A = \On$
In either case:
:$A \subseteq \On$
since $\On$ is [[Definition:Transitive Class|transitive]].
{{explain|Why is $\On$ necessarily transitive? This follows smoothly if it is assumed that $\On$ is a subclass of a Basic Universe, or otherwise fr... | Ordinal is Subset of Class of All Ordinals | https://proofwiki.org/wiki/Ordinal_is_Subset_of_Class_of_All_Ordinals | https://proofwiki.org/wiki/Ordinal_is_Subset_of_Class_of_All_Ordinals | [
"Class of All Ordinals"
] | [
"Definition:Ordinal",
"Definition:Class of All Ordinals"
] | [
"Ordinal is Member of Class of All Ordinals",
"Definition:Transitive Class"
] |
proofwiki-4687 | De Morgan's Laws (Set Theory)/Set Difference | {{:De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection}} | Suppose:
:$\ds x \in S \setminus \bigcap \mathbb T$
Note that by Set Difference is Subset we have that $x \in S$ (we need this later).
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S \setminus \bigcap \mathbb T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \notin
| r = \big... | {{:De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection}} | Suppose:
:$\ds x \in S \setminus \bigcap \mathbb T$
Note that by [[Set Difference is Subset]] we have that $x \in S$ (we need this later).
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S \setminus \bigcap \mathbb T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \notin
| r ... | De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection/Proof | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Difference | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Difference/General_Case/Difference_with_Intersection/Proof | [
"De Morgan's Laws"
] | [] | [
"Set Difference is Subset",
"De Morgan's Laws (Predicate Logic)/Denial of Universality"
] |
proofwiki-4688 | Classical Probability is Probability Measure | The classical probability model is a probability measure. | We check all the Kolmogorov axioms in turn: | The [[Definition:Classical Probability Model|classical probability model]] is a [[Definition:Probability Measure|probability measure]]. | We check all the [[Axiom:Kolmogorov Axioms|Kolmogorov axioms]] in turn: | Classical Probability is Probability Measure | https://proofwiki.org/wiki/Classical_Probability_is_Probability_Measure | https://proofwiki.org/wiki/Classical_Probability_is_Probability_Measure | [
"Probability Theory"
] | [
"Definition:Classical Probability Model",
"Definition:Probability Measure"
] | [
"Axiom:Kolmogorov Axioms"
] |
proofwiki-4689 | De Morgan's Laws (Set Theory)/Set Difference/General Case | Let $S$ and $T$ be sets.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T \subseteq \powerset T$.
Then:
==== Difference with Intersection ====
{{:De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection}}
==== Difference with Union ====
{{:De Morgan's Laws (Set Theory)/Set Differe... | Suppose:
:$\ds x \in S \setminus \bigcap \mathbb T$
Note that by Set Difference is Subset we have that $x \in S$ (we need this later).
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S \setminus \bigcap \mathbb T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \notin
| r = \big... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\powerset T$ be the [[Definition:Power Set|power set]] of $T$.
Let $\mathbb T \subseteq \powerset T$.
Then:
==== [[De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection|Difference with Intersection]] ====
{{:De Morgan's Laws (Set The... | Suppose:
:$\ds x \in S \setminus \bigcap \mathbb T$
Note that by [[Set Difference is Subset]] we have that $x \in S$ (we need this later).
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S \setminus \bigcap \mathbb T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \notin
| r ... | De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection/Proof | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Difference/General_Case | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Difference/General_Case/Difference_with_Intersection/Proof | [
"De Morgan's Laws"
] | [
"Definition:Set",
"Definition:Power Set",
"De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection",
"De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union"
] | [
"Set Difference is Subset",
"De Morgan's Laws (Predicate Logic)/Denial of Universality"
] |
proofwiki-4690 | De Morgan's Laws (Set Theory)/Relative Complement | {{:De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection}} | Let $x \in S$ througout.
{{begin-eqn}}
{{eqn | o =
| r = x \in \relcomp S {T_1 \cup T_2}
}}
{{eqn | o = \leadsto
| r = x \notin \paren {T_1 \cup T_2}
| c = {{Defof|Relative Complement}}
}}
{{eqn | o = \leadsto
| r = \neg \paren {x \in T_1 \lor x \in T_2}
| c = {{Defof|Set Union}}
}}
{{eqn... | {{:De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection}} | Let $x \in S$ througout.
{{begin-eqn}}
{{eqn | o =
| r = x \in \relcomp S {T_1 \cup T_2}
}}
{{eqn | o = \leadsto
| r = x \notin \paren {T_1 \cup T_2}
| c = {{Defof|Relative Complement}}
}}
{{eqn | o = \leadsto
| r = \neg \paren {x \in T_1 \lor x \in T_2}
| c = {{Defof|Set Union}}
}}
{{eq... | De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union/Proof 2 | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Relative_Complement | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Relative_Complement/Complement_of_Union/Proof_2 | [
"De Morgan's Laws"
] | [] | [
"De Morgan's Laws (Logic)/Conjunction of Negations",
"De Morgan's Laws (Logic)/Conjunction of Negations",
"Definition:Set Equality/Definition 1"
] |
proofwiki-4691 | De Morgan's Laws (Set Theory)/Set Complement | {{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}} | {{begin-eqn}}
{{eqn | l = \overline {T_1 \cap T_2}
| r = \mathbb U \setminus \paren {T_1 \cap T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | r = \paren {\mathbb U \setminus T_1} \cup \paren {\mathbb U \setminus T_2}
| c = De Morgan's Laws: Difference with Intersection
}}
{{eqn | r = \overline {T_1} \c... | {{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}} | {{begin-eqn}}
{{eqn | l = \overline {T_1 \cap T_2}
| r = \mathbb U \setminus \paren {T_1 \cap T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | r = \paren {\mathbb U \setminus T_1} \cup \paren {\mathbb U \setminus T_2}
| c = [[De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection|De M... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Proof 1 | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Proof_1 | [
"De Morgan's Laws"
] | [] | [
"De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection"
] |
proofwiki-4692 | De Morgan's Laws (Set Theory)/Set Complement | {{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}} | {{begin-eqn}}
{{eqn | o =
| r = x \in \overline {T_1 \cap T_2}
}}
{{eqn | o = \leadstoandfrom
| r = x \notin \paren {T_1 \cap T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {x \in T_1 \land x \in T_2}
| c = {{Defof|Set Intersection}}
}}
{{eqn | o = \l... | {{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}} | {{begin-eqn}}
{{eqn | o =
| r = x \in \overline {T_1 \cap T_2}
}}
{{eqn | o = \leadstoandfrom
| r = x \notin \paren {T_1 \cap T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {x \in T_1 \land x \in T_2}
| c = {{Defof|Set Intersection}}
}}
{{eqn | o = \l... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Proof 2 | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Proof_2 | [
"De Morgan's Laws"
] | [] | [
"De Morgan's Laws (Logic)/Disjunction of Negations",
"Definition:Set Equality/Definition 1"
] |
proofwiki-4693 | De Morgan's Laws (Set Theory)/Set Complement | {{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}} | {{begin-eqn}}
{{eqn | l = \map \complement {\map \complement A \cup \map \complement B}
| r = \map \complement {\map \complement A} \cap \map \complement {\map \complement B}
| c = De Morgan's Laws: Complement of Union
}}
{{eqn | r = A \cap B
| c = Complement of Complement
}}
{{eqn | ll= \leadstoandfr... | {{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}} | {{begin-eqn}}
{{eqn | l = \map \complement {\map \complement A \cup \map \complement B}
| r = \map \complement {\map \complement A} \cap \map \complement {\map \complement B}
| c = [[De Morgan's Laws (Set Theory)/Set Complement/Complement of Union|De Morgan's Laws: Complement of Union]]
}}
{{eqn | r = A \ca... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Proof 3 | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Proof_3 | [
"De Morgan's Laws"
] | [] | [
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Union",
"Complement of Complement",
"Definition:Set Complement",
"Complement of Complement"
] |
proofwiki-4694 | De Morgan's Laws (Set Theory)/Set Complement | {{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}} | {{begin-eqn}}
{{eqn | o =
| r = x \in \overline {T_1 \cup T_2}
}}
{{eqn | o = \leadstoandfrom
| r = x \notin \paren {T_1 \cup T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {x \in T_1 \lor x \in T_2}
| c = {{Defof|Set Union}}
}}
{{eqn | o = \leadstoan... | {{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}} | {{begin-eqn}}
{{eqn | o =
| r = x \in \overline {T_1 \cup T_2}
}}
{{eqn | o = \leadstoandfrom
| r = x \notin \paren {T_1 \cup T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {x \in T_1 \lor x \in T_2}
| c = {{Defof|Set Union}}
}}
{{eqn | o = \leadstoan... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Proof 2 | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Union/Proof_2 | [
"De Morgan's Laws"
] | [] | [
"De Morgan's Laws (Logic)/Conjunction of Negations",
"Definition:Set Equality/Definition 1"
] |
proofwiki-4695 | Image of Set Difference under Mapping/Corollary 1 | Let $f: S \to T$ be a mapping.
Let $S_1 \subseteq S_2 \subseteq S$.
Then:
:$\relcomp {f \sqbrk {S_2} } {f \sqbrk {S_1} } \subseteq f \sqbrk {\relcomp {S_2} {S_1} }$
where $\complement$ (in this context) denotes relative complement. | From {{Corollary|Image of Set Difference under Relation|1}}:
:$\relcomp {\RR \sqbrk {S_2} } {\RR \sqbrk {S_1} } \subseteq \RR \sqbrk {\relcomp {S_2} {S_1} }$
where $\RR \subseteq S \times T$ is a relation on $S \times T$.
As $f$, being a mapping, is also a relation, it follows directly that:
:$\relcomp {f \sqbrk {S_2} ... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $S_1 \subseteq S_2 \subseteq S$.
Then:
:$\relcomp {f \sqbrk {S_2} } {f \sqbrk {S_1} } \subseteq f \sqbrk {\relcomp {S_2} {S_1} }$
where $\complement$ (in this context) denotes [[Definition:Relative Complement|relative complement]]. | From {{Corollary|Image of Set Difference under Relation|1}}:
:$\relcomp {\RR \sqbrk {S_2} } {\RR \sqbrk {S_1} } \subseteq \RR \sqbrk {\relcomp {S_2} {S_1} }$
where $\RR \subseteq S \times T$ is a [[Definition:Relation|relation]] on $S \times T$.
As $f$, being a [[Definition:Mapping|mapping]], is also a [[Definition:Re... | Image of Set Difference under Mapping/Corollary 1 | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping/Corollary_1 | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping/Corollary_1 | [
"Image of Set Difference under Mapping"
] | [
"Definition:Mapping",
"Definition:Relative Complement"
] | [
"Definition:Relation",
"Definition:Mapping",
"Definition:Relation",
"Category:Image of Set Difference under Mapping"
] |
proofwiki-4696 | Image of Set Difference under Mapping/Corollary 2 | Let $f: S \to T$ be a mapping.
Let $X$ be a subset of $S$.
Then:
:$\relcomp {\Img f} {f \sqbrk X} \subseteq f \sqbrk {\relcomp S X}$
where:
:$\Img f$ denotes the image of $f$
:$\complement_{\Img f}$ denotes the complement relative to $\Img f$.
This can be expressed in the language and notation of direct image mappings ... | From {{Corollary|Image of Set Difference under Relation|2}}:
:$\relcomp {\Img \RR} {\RR \sqbrk X} \subseteq \RR \sqbrk {\relcomp S X}$
where $\RR \subseteq S \times T$ is a relation on $S \times T$.
As $f$, being a mapping, is also a relation, it follows directly that:
:$\relcomp {\Img f} {f \sqbrk X} \subseteq f \sqbr... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then:
:$\relcomp {\Img f} {f \sqbrk X} \subseteq f \sqbrk {\relcomp S X}$
where:
:$\Img f$ denotes the [[Definition:Image of Mapping|image]] of $f$
:$\complement_{\Img f}$ denotes the [[Definition:Relative Complem... | From {{Corollary|Image of Set Difference under Relation|2}}:
:$\relcomp {\Img \RR} {\RR \sqbrk X} \subseteq \RR \sqbrk {\relcomp S X}$
where $\RR \subseteq S \times T$ is a [[Definition:Relation|relation]] on $S \times T$.
As $f$, being a [[Definition:Mapping|mapping]], is also a [[Definition:Relation|relation]], it ... | Image of Set Difference under Mapping/Corollary 2 | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping/Corollary_2 | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping/Corollary_2 | [
"Image of Set Difference under Mapping"
] | [
"Definition:Mapping",
"Definition:Subset",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Relative Complement",
"Definition:Direct Image Mapping",
"Definition:Composition of Mappings"
] | [
"Definition:Relation",
"Definition:Mapping",
"Definition:Relation"
] |
proofwiki-4697 | Preimage of Set Difference under Mapping/Corollary 1 | Let $f: S \to T$ be a mapping.
Let $T_1 \subseteq T_2 \subseteq T$.
Then:
:$\relcomp {f^{-1} \sqbrk {T_2} } {f^{-1} \sqbrk {T_1} } = f^{-1} \sqbrk {\relcomp {T_2} {T_1} }$
where:
:$\complement$ (in this context) denotes relative complement
:$f^{-1} \sqbrk {T_1}$ denotes preimage. | From One-to-Many Image of Set Difference: Corollary 1 we have:
:$\relcomp {\RR \sqbrk {T_2} } {\RR \sqbrk {T_1} } = \RR \sqbrk {\relcomp {T_2} {T_1} }$
where $\RR \subseteq T \times S$ is a one-to-many relation on $T \times S$.
Hence as $f^{-1}: T \to S$ is a one-to-many relation:
:$\relcomp {f^{-1} \sqbrk {T_2} } {f^{... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $T_1 \subseteq T_2 \subseteq T$.
Then:
:$\relcomp {f^{-1} \sqbrk {T_2} } {f^{-1} \sqbrk {T_1} } = f^{-1} \sqbrk {\relcomp {T_2} {T_1} }$
where:
:$\complement$ (in this context) denotes [[Definition:Relative Complement|relative complement]]
:$f^{-1} \sqbrk {T_... | From [[One-to-Many Image of Set Difference#Corollary 1|One-to-Many Image of Set Difference: Corollary 1]] we have:
:$\relcomp {\RR \sqbrk {T_2} } {\RR \sqbrk {T_1} } = \RR \sqbrk {\relcomp {T_2} {T_1} }$
where $\RR \subseteq T \times S$ is a [[Definition:One-to-Many Relation|one-to-many relation]] on $T \times S$.
He... | Preimage of Set Difference under Mapping/Corollary 1 | https://proofwiki.org/wiki/Preimage_of_Set_Difference_under_Mapping/Corollary_1 | https://proofwiki.org/wiki/Preimage_of_Set_Difference_under_Mapping/Corollary_1 | [
"Preimages under Mappings",
"Set Difference",
"Set Complement"
] | [
"Definition:Mapping",
"Definition:Relative Complement",
"Definition:Preimage/Mapping/Subset"
] | [
"One-to-Many Image of Set Difference",
"Definition:One-to-Many Relation",
"Inverse of Mapping is One-to-Many Relation",
"Category:Preimages under Mappings",
"Category:Set Difference",
"Category:Set Complement"
] |
proofwiki-4698 | Complement of Preimage equals Preimage of Complement | Let $f: S \to T$ be a mapping.
Let $T_1$ be a subset of $T$.
Then:
:$\relcomp S {f^{-1} \sqbrk {T_1} } = f^{-1} \sqbrk {\relcomp T {T_1} }$
where:
:$\complement_S$ (in this context) denotes relative complement
:$f^{-1} \sqbrk {T_1}$ denotes preimage. | From {{Corollary|One-to-Many Image of Set Difference|2}} we have:
:$\relcomp {\Img \RR} {\RR \sqbrk {S_1} } = \RR \sqbrk {\relcomp S {S_1} }$
where:
:$S_1 \subseteq S$
:$\RR \subseteq T \times S$ is a one-to-many relation on $T \times S$.
Hence as $f^{-1}: T \to S$ is a one-to-many relation:
:$\relcomp {\Preimg f} {f^{... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $T_1$ be a [[Definition:Subset|subset]] of $T$.
Then:
:$\relcomp S {f^{-1} \sqbrk {T_1} } = f^{-1} \sqbrk {\relcomp T {T_1} }$
where:
:$\complement_S$ (in this context) denotes [[Definition:Relative Complement|relative complement]]
:$f^{-1} \sqbrk {T_1}$ den... | From {{Corollary|One-to-Many Image of Set Difference|2}} we have:
:$\relcomp {\Img \RR} {\RR \sqbrk {S_1} } = \RR \sqbrk {\relcomp S {S_1} }$
where:
:$S_1 \subseteq S$
:$\RR \subseteq T \times S$ is a [[Definition:One-to-Many Relation|one-to-many relation]] on $T \times S$.
Hence as [[Inverse of Mapping is One-to-Man... | Complement of Preimage equals Preimage of Complement | https://proofwiki.org/wiki/Complement_of_Preimage_equals_Preimage_of_Complement | https://proofwiki.org/wiki/Complement_of_Preimage_equals_Preimage_of_Complement | [
"Mapping Theory",
"Relative Complement"
] | [
"Definition:Mapping",
"Definition:Subset",
"Definition:Relative Complement",
"Definition:Preimage/Mapping/Subset"
] | [
"Definition:One-to-Many Relation",
"Inverse of Mapping is One-to-Many Relation",
"Preimage of Mapping equals Domain"
] |
proofwiki-4699 | One-to-Many Image of Set Difference/Corollary 1 | Let $\RR \subseteq S \times T$ be a relation which is one-to-many.
Let $A \subseteq B \subseteq S$.
Then:
:$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk {\relcomp B A}$
where $\complement$ (in this context) denotes relative complement. | We have that $A \subseteq B$.
Then by definition of relative complement:
:$\relcomp B A = B \setminus A$
:$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk B \setminus \RR \sqbrk A$
Hence, when $A \subseteq B$:
:$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk {\relcomp B A}$
means exactly the same thing as:
:$\... | Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]] which is [[Definition:One-to-Many Relation|one-to-many]].
Let $A \subseteq B \subseteq S$.
Then:
:$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk {\relcomp B A}$
where $\complement$ (in this context) denotes [[Definition:Relative Complement|re... | We have that $A \subseteq B$.
Then by definition of [[Definition:Relative Complement|relative complement]]:
:$\relcomp B A = B \setminus A$
:$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk B \setminus \RR \sqbrk A$
Hence, when $A \subseteq B$:
:$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk {\relcomp B A}... | One-to-Many Image of Set Difference/Corollary 1 | https://proofwiki.org/wiki/One-to-Many_Image_of_Set_Difference/Corollary_1 | https://proofwiki.org/wiki/One-to-Many_Image_of_Set_Difference/Corollary_1 | [
"Relative Complement",
"One-to-Many Image of Set Difference"
] | [
"Definition:Relation",
"Definition:One-to-Many Relation",
"Definition:Relative Complement"
] | [
"Definition:Relative Complement",
"One-to-Many Image of Set Difference",
"Category:Relative Complement",
"Category:One-to-Many Image of Set Difference"
] |
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