nyuuzyou/wikis huggingface dataset:
Hyperbolic Sine Function is Odd

== Theorem ==
Let $\sinh: \C \to \C$ be the hyperbolic sine function on the set of complex numbers.
Then $\sinh$ is odd:
:$\map \sinh {-x} = -\sinh x$
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Hyperbolic Cosine Function is Even
* Hyperbolic Tangent Function is Odd
* Hyperbolic Cotangent Function is Odd
* Hyperbolic Secant Function is Even
* Hyperbolic Cosecant Function is Odd
* Sine Function is Odd
== Sources ==
* : $8.14$: Functions of Negative Arguments
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
* 
* 
* 
Category:Hyperbolic Sine Function
Category:Hyperbolic Sine Function is Odd
Category:Examples of Odd Functions
Hyperbolic Sine in terms of Sine

== Theorem ==
Let $z \in \C$ be a complex number.
Then:
:$i \sinh z = \map \sin {i z}$
where:
:$\sin$ denotes the complex sine
:$\sinh$ denotes the hyperbolic sine
:$i$ is the imaginary unit: $i^2 = -1$.
== Proof ==
== Also presented as ==
This identity is also seen in the form:
:$\sinh z = -i \map \sin {i z}$
which can be seen to follow from the other form by multiplication by $-i$.
== Also see ==
* Hyperbolic Cosine in terms of Cosine
* Hyperbolic Tangent in terms of Tangent
* Hyperbolic Cotangent in terms of Cotangent
* Hyperbolic Secant in terms of Secant
* Hyperbolic Cosecant in terms of Cosecant
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $(4.22)$
* : $\S 8$: Hyperbolic Functions: $8.74$: Relationship between Hyperbolic and Trigonometric Functions
* : $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
* 
Category:Sine Function
Category:Hyperbolic Sine Function
Hyperbolic Sine of Complex Number

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\sinh \paren {a + b i} = \sinh a \cos b + i \cosh a \sin b$
where:
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Hyperbolic Cosine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Sine Function
Category:Complex Numbers
Category:Hyperbolic Sine of Complex Number
Hyperbolic Sine of Complex Number/Proof 1

== Theorem ==
== Proof ==
== Also see ==
* Hyperbolic Cosine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Sine of Complex Number
Hyperbolic Sine of Complex Number/Proof 2

== Theorem ==
== Proof ==
== Also see ==
* Hyperbolic Cosine of Complex Number
* Hyperbolic Tangent of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Sine of Complex Number
Hyperbolic Sine of Sum

== Theorem ==
:$\map \sinh {a + b} = \sinh a \cosh b + \cosh a \sinh b$
where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine.
=== Corollary ===
== Proof ==
== Also see ==
* Hyperbolic Cosine of Sum
* Hyperbolic Tangent of Sum
* Hyperbolic Cotangent of Sum
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.20$: Addition Formulas
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
* 
Category:Hyperbolic Sine Function
Category:Addition Formulas for Hyperbolic Functions
Hyperbolic Sine of Sum/Corollary

== Corollary of Hyperbolic Sine of Sum ==
:$\map \sinh {a - b} = \sinh a \cosh b - \cosh a \sinh b$
where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine.
== Proof ==
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.20$: Addition Formulas
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
Category:Subtraction Formulas for Hyperbolic Functions
Category:Hyperbolic Sine Function
Hyperbolic Tangent Function is Odd

== Theorem ==
Let $\tanh: \C \to \C$ be the hyperbolic tangent function on the set of complex numbers.
Then $\tanh$ is odd:
:$\map \tanh {-x} = -\tanh x$
== Proof ==
== Also see ==
* Hyperbolic Sine Function is Odd
* Hyperbolic Cosine Function is Even
* Hyperbolic Cotangent Function is Odd
* Hyperbolic Secant Function is Even
* Hyperbolic Cosecant Function is Odd
* Tangent Function is Odd
== Sources ==
* : $8.16$: Functions of Negative Arguments
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
Category:Hyperbolic Tangent Function
Category:Examples of Odd Functions
Hyperbolic Tangent Half-Angle Substitution

== Theorem ==
:$\ds \int \map F {\sinh x, \cosh x} \rd x = 2 \int \map F {\frac {2 u} {1 - u^2}, \frac {1 + u^2} {1 - u^2} } \frac {\d u} {1 - u^2}$
where $u = \tanh \dfrac x 2$.
== Proof ==
The result follows from Integration by Substitution.
== Also see ==
* Weierstrass Substitution
Category:Integral Substitutions
Category:Hyperbolic Tangent Function
Category:Primitives involving Hyperbolic Sine Function
Category:Primitives involving Hyperbolic Cosine Function
Category:Hyperbolic Tangent Half-Angle Substitutions
Hyperbolic Tangent Half-Angle Substitution for Cosine

== Corollary to Double Angle Formula for Hyperbolic Cosine ==
:$\cosh 2 x = \dfrac {1 + \tanh^2 x}{1 - \tanh^2 x}$
where $\cosh$ and $\tanh$ denote hyperbolic cosine and hyperbolic tangent respectively.
== Proof ==
Category:Hyperbolic Cosine Function
Category:Hyperbolic Tangent Half-Angle Substitutions
Hyperbolic Tangent Half-Angle Substitution for Sine

== Corollary to Double Angle Formula for Hyperbolic Sine ==
:$\sinh 2 x = \dfrac {2 \tanh x} {1 - \tanh^2 x}$
where $\sin$ and $\tan$ denote hyperbolic sine and hyperbolic tangent respectively.
== Proof ==
Category:Hyperbolic Sine Function
Category:Hyperbolic Tangent Half-Angle Substitutions
Hyperbolic Tangent Less than X

== Theorem ==
:$\tanh x \le x$
for $x \ge 0$.
== Proof ==
Let $\map f x = x - \tanh x$.
By Derivative of Hyperbolic Tangent:
:$\map {f'} x = 1 - \sech^2 x$
Since $\cosh x \ge 1$ for all $x \in \R$, we can deduce that:
:$\map {f'} x \ge 0$
From Derivative of Monotone Function, $\map f x$ is increasing.
By definition of hyperbolic tangent:
:$\map f x = 0$
It follows that $\map f x \ge 0$ for $x \ge 0$.
Hence the result.
Category:Hyperbolic Tangent Function
Hyperbolic Tangent in terms of Tangent

== Theorem ==
Let $z \in \C$ be a complex number.
Then:
:$i \tanh z = \map \tan {i z}$
where:
:$\tan$ denotes the tangent function
:$\tanh$ denotes the hyperbolic tangent
:$i$ is the imaginary unit: $i^2 = -1$.
== Proof ==
 
== Also see ==
* Hyperbolic Sine in terms of Sine
* Hyperbolic Cosine in terms of Cosine
* Hyperbolic Cotangent in terms of Cotangent
* Hyperbolic Secant in terms of Secant
* Hyperbolic Cosecant in terms of Cosecant
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.76$: Relationship between Hyperbolic and Trigonometric Functions
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
Category:Tangent Function
Category:Hyperbolic Tangent Function
Hyperbolic Tangent of Complex Number

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
=== Formulation 1 ===
=== Formulation 2 ===
=== Formulation 3 ===
=== Formulation 4 ===
where:
:$\tan$ denotes the real tangent function
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function
:$\tanh$ denotes the hyperbolic tangent function.
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Tangent Function
Category:Complex Numbers
Category:Hyperbolic Tangent of Complex Number
Hyperbolic Tangent of Complex Number/Formulation 1

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\tanh \paren {a + b i} = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}$
where:
:$\tanh$ denotes the hyperbolic tangent function
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function.
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Tangent of Complex Number
Hyperbolic Tangent of Complex Number/Formulation 2

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\tanh \paren {a + b i} = \dfrac {\tanh a + i \tan b} {1 + i \tanh a \tan b}$
where:
:$\tan$ denotes the real tangent function
:$\tanh$ denotes the hyperbolic tangent function.
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Tangent of Complex Number
Hyperbolic Tangent of Complex Number/Formulation 3

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\tanh \paren {a + b i} = \dfrac {\tanh a + \tanh a \tan^2 b} {1 + \tanh^2 a \tan^2 b} + \dfrac {\tan b - \tanh^2 a \tan b} {1 + \tanh^2 a \tan^2 b} i$
where:
:$\tan$ denotes the real tangent function
:$\tanh$ denotes the hyperbolic tangent function.
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
Category:Hyperbolic Tangent of Complex Number
Hyperbolic Tangent of Complex Number/Formulation 4

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
:$\map \tanh {a + b i} = \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \cos 2 b}$
where:
:$\tanh$ denotes the hyperbolic tangent function
:$\sin$ denotes the real sine function
:$\cos$ denotes the real cosine function
:$\sinh$ denotes the hyperbolic sine function
:$\cosh$ denotes the hyperbolic cosine function.
== Proof ==
== Also see ==
* Hyperbolic Sine of Complex Number
* Hyperbolic Cosine of Complex Number
* Hyperbolic Cosecant of Complex Number
* Hyperbolic Secant of Complex Number
* Hyperbolic Cotangent of Complex Number
== Sources ==
* : $\S 4$. Elementary Functions of a Complex Variable: Exercise $9 \ \text{(i)}$
Category:Hyperbolic Tangent of Complex Number
Hyperbolic Tangent of Sum

== Theorem ==
:$\map \tanh {a + b} = \dfrac {\tanh a + \tanh b} {1 + \tanh a \tanh b}$
where $\tanh$ denotes the hyperbolic tangent.
=== Corollary ===
== Proof ==
== Also see ==
* Hyperbolic Sine of Sum
* Hyperbolic Cosine of Sum
* Hyperbolic Cotangent of Sum
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.22$: Addition Formulas
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
Category:Hyperbolic Tangent Function
Category:Addition Formulas for Hyperbolic Functions
Hyperbolic Tangent of Sum/Corollary

== Corollary to Hyperbolic Tangent of Sum ==
:$\map \tanh {a - b} = \dfrac {\tanh a - \tanh b} {1 - \tanh a \tanh b}$
where $\tanh$ denotes the hyperbolic tangent.
== Proof ==
== Sources ==
* : $\S 8$: Hyperbolic Functions: $8.22$: Addition Formulas
* : $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
Category:Subtraction Formulas for Hyperbolic Functions
Category:Hyperbolic Tangent Function
Hypergeometric Continued Fraction

== Theorem ==
Let $a, b \in \C$.
Let $a, b \notin \Z_{\le 0}$.
=== Formulation 1 ===
=== Formulation 2 ===
== Also see ==
* Hypergeometric Function Difference Equation
Category:Continued Fractions
Category:Hypergeometric Function Difference Equation
Category:Hypergeometric Functions
Category:Hypergeometric Series
Hypergeometric Continued Fraction/Formulation 1

== Theorem ==
Let $a \in \C$.
Let $a \notin \Z_{\le 0}$.
:$\ds \frac {\map { {}_0 \operatorname F_1} {a + 1; x} } {\map { {}_0 \operatorname F_1} {a; x} } = \cfrac a {a + \cfrac x {\paren {a + 1} + \cfrac x {\paren {a + 2} + \cfrac x {\paren {a + 3} + \cfrac x {\ddots + \cfrac {\ddots} {\ddots } } } } } }$
where:
:$\map { {}_0 \operatorname F_1} {a; x}$ is the generalized hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac 1 { a^{\overline k} } \dfrac {x^k} {k!}$
:$x^{\overline k}$ denotes the $k$th rising factorial power of $x$
== Proof ==
:First, we demonstate:
:$\ds \frac {\map { {}_0 \operatorname F_1} {a + 1; x} } {\map { {}_0 \operatorname F_1} {a; x} } = \frac 1 {1 + \dfrac x {\paren {a + 1} a } \dfrac {\map { {}_0 \operatorname F_1} {a + 2; x} } {\map { {}_0 \operatorname F_1} {a + 1; x} } }$
Therefore:
Category:Continued Fractions
Category:Hypergeometric Functions
Hypergeometric Continued Fraction/Formulation 2

== Theorem ==
Let $a, b \in \C$.
Let $b \notin \Z_{\le 0}$.
:$\ds \frac {\map { {}_1 \operatorname F_1} {a + 1; b; x} } {\map { {}_1 \operatorname F_1} {a; b; x} } = \cfrac 1 {1 - \cfrac x {b + \cfrac {\paren {a + 1} x} {\paren {b + 1} + \cfrac {\paren {a - b } x} {\paren {b + 2} + \cfrac {\paren {a + 2} x} {\paren {b + 3} + \cfrac {\paren {a - b - 1 } x} {\paren {b + 4} + \cfrac {\ddots} {\ddots} } } } } } }$
where:
:$\map { {}_1 \operatorname F_1} {a; b; x}$ is the generalized hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} } { b^{\overline k} } \dfrac {x^k} {k!}$
:$x^{\overline k}$ denotes the $k$th rising factorial power of $x$.
== Proof ==
:First, we demonstate:
:$\ds \frac {\map { {}_1 \operatorname F_1} {a + 1; b; x} } {\map { {}_1 \operatorname F_1} {a; b; x} } = \frac 1 {1 - \dfrac x b \dfrac {\map { {}_1 \operatorname F_1} {a + 1; b + 1; x} } {\map { {}_1 \operatorname F_1} {a + 1; b; x} } }$
:Next, we demonstate:
:$\ds \frac {\map { {}_1 \operatorname F_1} {a; b + 1; x} } {\map { {}_1 \operatorname F_1} {a; b; x} } = \frac 1 {1 + \dfrac {a x} {\paren {b + 1} b } \dfrac {\map { {}_1 \operatorname F_1} {a + 1; b + 2; x} } {\map { {}_1 \operatorname F_1} {a; b + 1; x} } }$
:Finally, we demonstate:
:$\ds \frac {\map { {}_1 \operatorname F_1} {a + 1; b + 1; x} } {\map { {}_1 \operatorname F_1} {a; b; x} } = \frac 1 {1 + \dfrac {\paren {a - b } x} {\paren {b + 1} b } \dfrac {\map { {}_1 \operatorname F_1} {a + 1; b + 2; x} } {\map { {}_1 \operatorname F_1} {a + 1; b + 1; x} } }$
Therefore:
Category:Continued Fractions
Category:Hypergeometric Functions
Hypergeometric Distribution approaches Binomial Distribution

== Theorem ==
Let $X$ be a discrete random variable which obeys a hypergeometric distribution over a population consisting of $M$ satisfactory units and $N$ unsatisfactory units:
:$\map \Pr {X = r} = \dfrac {\dbinom M r \dbinom N {n - r} } {\dbinom {M + N} r}$
where:
:$n$ is the number of trials
:$r$ is the number of satisfactory units drawn from the population without replacement.
When $M$ and $N$ are large compared to $n$, $X$ approaches the binomial distribution $\Binomial n p$ where:
:$p = \dfrac M {M + N}$
== Proof ==
== Sources ==
* 
* 
Category:Hypergeometric Distribution
Category:Binomial Distribution
Hypergeometric Function Difference Equation

== Theorem ==
Let $a, b \in \C$.
Let $b \notin \Z_{\le 0}$.
=== Formulation 1 ===
=== Formulation 2 ===
=== Formulation 3 ===
=== Formulation 4 ===
== Also see ==
* Gaussian Hypergeometric Function Difference Equation
Category:Hypergeometric Function Difference Equation
Category:Hypergeometric Functions
Category:Hypergeometric Series
Hypergeometric Function Difference Equation/Formulation 1

== Theorem ==
where:
:$\map { {}_0 \operatorname F_1} {a; x}$ is the generalized hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac 1 { a^{\overline k} } \dfrac {x^k} {k!}$
:$x^{\overline k}$ denotes the $k$th rising factorial power of $x$.
== Proof ==
== Also see ==
* Hypergeometric Function Difference Equation/Formulation 2
* Hypergeometric Function Difference Equation/Formulation 3
* Hypergeometric Function Difference Equation/Formulation 4
Category:Hypergeometric Function Difference Equation
Hypergeometric Function Difference Equation/Formulation 2

== Theorem ==
where:
:$\map { {}_1 \operatorname F_1} {a; b; x}$ is the generalized hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} } { b^{\overline k} } \dfrac {x^k} {k!}$
:$x^{\overline k}$ denotes the $k$th rising factorial power of $x$.
== Proof ==
== Also see ==
* Hypergeometric Function Difference Equation/Formulation 1
* Hypergeometric Function Difference Equation/Formulation 3
* Hypergeometric Function Difference Equation/Formulation 4
Category:Hypergeometric Function Difference Equation
Hypergeometric Function Difference Equation/Formulation 3

== Theorem ==
where:
:$\map { {}_1 \operatorname F_1} {a; b; x}$ is the generalized hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} } { b^{\overline k} } \dfrac {x^k} {k!}$
:$x^{\overline k}$ denotes the $k$th rising factorial power of $x$.
== Proof ==
== Also see ==
* Hypergeometric Function Difference Equation/Formulation 1
* Hypergeometric Function Difference Equation/Formulation 2
* Hypergeometric Function Difference Equation/Formulation 4
Category:Hypergeometric Function Difference Equation
Hypergeometric Function Difference Equation/Formulation 4

== Theorem ==
where:
:$\map { {}_1 \operatorname F_1} {a; b; x}$ is the generalized hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} } { b^{\overline k} } \dfrac {x^k} {k!}$
:$x^{\overline k}$ denotes the $k$th rising factorial power of $x$.
== Proof ==
== Also see ==
* Hypergeometric Function Difference Equation/Formulation 1
* Hypergeometric Function Difference Equation/Formulation 2
* Hypergeometric Function Difference Equation/Formulation 3
Category:Hypergeometric Function Difference Equation
Hyperplane in Normed Vector Space generated by Unbounded Linear Functional is Everywhere Dense

== Theorem ==
Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
Let $\phi : X \to \Bbb F$ be a linear functional that is not bounded.
Let $U$ be a hyperplane in $X$ given by: 
:$U = \map \ker \phi$
Then: 
:$U$ is everywhere dense in $X$.
== Proof ==
Since $\phi$ is not bounded, for each $n \in \N$ we can pick $v_n \ne 0$ such that:
:$\cmod {\map \phi {v_n} } \ge n \norm {v_n}$
Set:
:$\ds x_n = \frac {v_n} {\norm {v_n} }$
for each $n \in \N$.
Then for each $n \in \N$, we have:
:$\cmod {\map \phi {x_n} } \ge n$
from linearity with: 
:$\norm {x_n} = 1$
Fix $x \in X$ and $\epsilon > 0$.
We aim to show that there exists $y \in U$ such that: 
:$\norm {x - y} < \epsilon$
It suffices to find a sequence $\sequence {y_n}_{n \mathop \in \N}$ in $U$ such that: 
:$y_n \to x$ 
Then, from the definition of convergent sequence, we can find $N \in \N$ such that: 
:$\norm {x - y_n} < \epsilon$
For each $n \in \N$, let: 
:$\ds y_n = x - \frac {\map \phi x} {\map \phi {x_n} } x_n$
Then we have: 
So, we have: 
:$y_n \in \map \ker \phi$
and so:
:$y_n \in U$
We also have: 
Since: 
:$\cmod {\map \phi {x_n} } \ge n$
We have: 
:$\ds \frac {\cmod {\map \phi x} } {\cmod {\map \phi {x_n} } } \le \frac {\cmod {\map \phi x} } n$
So, we have: 
:$\ds \norm {x - y_n} \le \frac {\cmod {\map \phi x} } n$
We therefore have: 
:$\ds \norm {x - y_n} \to 0$ as $n \to \infty$.
From Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence, we have: 
:$\ds y_n \to x$
So $\sequence {y_n}_{n \mathop \in \N}$ is the desired sequence converging to $x$.
Since $x$ was arbitrary, we have that: 
:$U$ is everywhere dense in $X$.
Category:Hyperplanes
Category:Normed Vector Spaces
Hypothetical Syllogism

== Theorem ==
The (rule of the) hypothetical syllogism is a valid deduction sequent in propositional logic:
:If we can conclude that $p$ implies $q$, and if we can also conclude that $q$ implies $r$, then we may infer that $p$ implies $r$.
=== Formulation 1 ===
=== Formulation 2 ===
=== Formulation 3 ===
=== Formulation 4 ===
=== Formulation 5 ===
== Also known as ==
It is referred to by some authors as the principle of syllogism
It is also known as the transitivity law.
Its abbreviation in a tableau proof is $\textrm{HS}$.
== Examples ==
== Sources ==
* : $\S \text{II}.12$: Laws of sentential calculus
* : $\S 3.9$: Derivation by Substitution
* 
Category:Hypothetical Syllogism
Category:Conditional
Hypothetical Syllogism/Examples/Ancient Chinese Proverb

== Example of Use of Hypothetical Syllogism ==
:If there is light in the soul,
::then there will be beauty in the person.
:If there is beauty in the person,
::then there will be harmony in the house.
:If there is harmony in the house,
::then there will be order in the nation.
:If there is order in the nation,
::then there will be peace in the world.
The conclusion is:
:If there is light in the soul, then there will be peace in the world.
== Proof ==
Let:
:$p$ denote the statement:
::There is light in the soul.
:$q$ denote the statement:
::There is beauty in the person.
:$r$ denote the statement:
::There is harmony in the house.
:$s$ denote the statement:
::There is order in the nation.
:$t$ denote the statement:
::There is peace in the world.
The proverb can be expressed in symbols as follows:
Hence we have:
</onlyinclude>
== Sources ==
* : Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...: Ponderable $1.1.2$
Category:Examples of Hypothetical Syllogism
Hypothetical Syllogism/Formulation 1

== Theorem ==
== Proof 1 ==
== Proof 2 ==
== Proof by Truth Table ==
== Sources ==
* : Chapter $1$: The Propositional Calculus $1$: $2$ Conditionals and Negation: Exercise $1 \ \text{(i)}$
* : $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables
* : $3$: The Method of Deduction: $3.1$: Formal Proof of Validity: Rules of Inference: $3.$
* : $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
* : $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs: $6$
* : $\S 1.13$: Tableau Problems (TAB1): Example $2$
Category:Hypothetical Syllogism
Hypothetical Syllogism/Formulation 1/Proof 2

== Theorem ==
== Proof ==
This proof uses $\mathscr H_2$, Instance 2 of the Hilbert proof systems.
Recall the sequent form of the Hypothetical Syllogism:
:$\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$
Applying the Rule of Detachment $\text {RST} 3$ twice, we obtain:
:$\vdash \paren {p \implies q} \implies \paren {p \implies r}$
and subsequently:
:$\vdash p \implies r$
as desired.
== Sources ==
* : $\S 4.7$: The Derivation of Formulae: $DR \, 1$
Category:Hypothetical Syllogism
Category:Hilbert Proof System Instance 2
Hypothetical Syllogism/Formulation 1/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables to the propositions in turn.
As can be seen for all boolean interpretations by inspection, where the truth values under the main connective on the is $\T$, that under the one on the is also $\T$:
$\begin{array}{|ccccccc||ccc|} \hline
(p & \implies & q) & \land & (q & \implies & r) & p & \implies & r \\
\hline
\F & \T & \F & \T & \F & \T & \F & \F & \T & \F \\
\F & \T & \F & \T & \F & \T & \T & \F & \T & \T \\
\F & \T & \T & \F & \T & \F & \F & \F & \T & \F \\
\F & \T & \T & \T & \T & \T & \T & \F & \T & \T \\
\T & \F & \F & \F & \F & \T & \F & \T & \F & \F \\
\T & \F & \F & \F & \F & \T & \T & \T & \T & \T \\
\T & \T & \T & \F & \T & \F & \F & \T & \F & \F \\
\T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\
\hline
\end{array}$
Hence the result.
== Sources ==
* : $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables
Category:Truth Table Proofs
Category:Hypothetical Syllogism