nyuuzyou/wikis huggingface dataset:
Power Series Expansion for Real Area Hyperbolic Sine/Mistake

== Source Work ==
:$8$. Taylor Series and Power Series:
::Appendix: Table $8.2$: Power Series of Important Functions
== Mistake ==
:$\sinh^{-1} x = \paren {\ln 2} x + \dfrac 1 {\paren {2 \times 2} x^2} - \dfrac {1 \times 3} {\paren {2 \times 4 \times 4} x^4} + \dfrac {1 \times 3 \times 5} {\paren {2 \times 4 \times 6 \times 6} x^6} + \cdots \quad \paren {\size x \ge 1}$
== Correction ==
There are two mistakes here:
:$(1): \quad$ The first term should read $\map \ln {2 x}$
:$(2): \quad$ The formula for $x \le -1$ should be:
::::$-\map \ln {-2 x} - \paren {\dfrac 1 {\paren {2 \times 2} x^2} - \dfrac {1 \times 3} {\paren {2 \times 4 \times 4} x^4} + \dfrac {1 \times 3 \times 5} {\paren {2 \times 4 \times 6 \times 6} x^6} + \cdots}$
== Sources ==
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
Category:Mistakes/Mathematics for Engineers and Scientists
Power Series Expansion for Real Area Hyperbolic Tangent

== Theorem ==
The (real) area hyperbolic tangent function has a Taylor series expansion:
for $\size x < 1$.
== Proof ==
From Sum of Infinite Geometric Sequence:
:$(1): \quad \ds \frac 1 {1 - x^2} = \sum_{n \mathop = 0}^\infty \paren {x^2}^n$
for $-1 < x < 1$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
For $-1 < x < 1$, the sequence $\sequence {\dfrac {x^{2 n + 1} } {2 n + 1} }$ is decreasing and converges to zero.
Hence the result.
== Also see ==
* Power Series Expansion for Real Area Hyperbolic Sine
* Power Series Expansion for Real Area Hyperbolic Cosine
* Power Series Expansion for Real Area Hyperbolic Cotangent
* Power Series Expansion for Real Area Hyperbolic Secant
* Power Series Expansion for Real Area Hyperbolic Cosecant
== Sources ==
* : $\S 20$: Series for Hyperbolic Functions: $20.41$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Hyperbolic Functions: $22.41.$
Category:Examples of Power Series
Category:Inverse Hyperbolic Tangent
Power Series Expansion for Reciprocal of 1-z to the m+1 by Logarithm of Reciprocal of 1-z

== Theorem ==
:$\dfrac 1 {\paren {1 - z}^{m + 1} } \map \ln {\dfrac 1 {1 - z} } = \ds \sum_{k \mathop \ge 1} \paren {H_{m + k} - H_m} \dbinom {m + k} k z^k$
where:
:$\dbinom {m + k} k$ denotes a binomial coefficient
:$H_m$ denotes the $m$th harmonic number.
== Proof ==
== Sources ==
* 
* 
* : $\S 1.2.9$: Generating Functions: $(25)$
Category:Examples of Power Series
Category:Logarithms
Power Series Expansion for Reciprocal of 1 + x/Proof 2

== Theorem ==
== Proof ==
== Sources ==
* : $\text I$. Algebra: The Binomial Theorem: Negative and fractional indices
* : $\S 20$: Binomial Series: $20.8$
* : $\S 22$: Taylor Series: Binomial Series: $22.8.$
Category:Power Series Expansion for Reciprocal of 1 + x
Power Series Expansion for Reciprocal of Cube Root of 1 + x

== Theorem ==
Let $x \in \R$ such that $-1 < x \le 1$.
Then:
:$\dfrac 1 {\sqrt [3] {1 + x} } = 1 - \dfrac 1 3 x + \dfrac {1 \times 4} {3 \times 6} x^2 - \dfrac {1 \times 4 \times 7} {3 \times 6 \times 9} x^3 + \cdots$
== Proof ==
== Sources ==
* : $\S 20$: Binomial Series: $20.13$
* : $\S 22$: Taylor Series: Binomial Series: $22.13.$
Category:Examples of Power Series
Power Series Expansion for Reciprocal of Cube of 1 + x/Proof 2

== Theorem ==
== Proof ==
== Sources ==
* : $\S 20$: Binomial Series: $20.10$
* : $\S 22$: Taylor Series: Binomial Series: $22.10.$
Category:Power Series Expansion for Reciprocal of Cube of 1 + x
Power Series Expansion for Reciprocal of Square Root of 1 + x

== Theorem ==
Let $x \in \R$ such that $-1 < x \le 1$.
Then:
== Proof ==
== Sources ==
* : $\S 20$: Binomial Series: $20.11$
* : $\S 22$: Taylor Series: Binomial Series: $22.11.$
Category:Examples of Power Series
Power Series Expansion for Reciprocal of Square of 1 + x/Corollary

== Corollary to Power Series Expansion for $\frac 1 {\paren {1 + x} }$ ==
Let $x \in \R$ such that $-1 < x < 1$.
Then:
== Proof ==
== Sources ==
* : $\text I$. Algebra: The Binomial Theorem: Negative and fractional indices
Category:Power Series Expansion for Reciprocal of Square of 1 + x
Power Series Expansion for Secant Function

== Theorem ==
The (real) secant function has a Taylor series expansion:
where $E_{2 n}$ denotes the Euler numbers.
This converges for $\size x < \dfrac \pi 2$.
== Proof ==
== Also presented as ==
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.25$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.25.$
Category:Power Series Expansion for Secant Function
Category:Examples of Power Series
Category:Euler Numbers
Category:Secant Function
Power Series Expansion for Secant Function/Also presented as

== Power Series Expansion for Secant Function: Also presented as == 
The Power Series Expansion for Secant Function can also be presented in the form:
where ${E_n}^*$ denotes the alternative form of the Euler numbers.
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.25$
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.25.$
Category:Power Series Expansion for Secant Function
Power Series Expansion for Sine Function

== Theorem ==
The sine function has the power series expansion:
valid for all $x \in \R$.
== Proof ==
From Derivative of Sine Function:
:$\dfrac \d {\d x} \sin x = \cos x$
From Derivative of Cosine Function:
:$\dfrac \d {\d x} \cos x = -\sin x$
Hence:
and so for all $m \in \N$:
where $k \in \Z$.
This leads to the Maclaurin series expansion:
From Series of Power over Factorial Converges, it follows that this series is convergent for all $x$.
== Also see ==
* Power Series Expansion for Cosine Function
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.21$
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : $\S 1.3.2$: Power series: $(1.46)$
* 
* 
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.21.$
Category:Sine Function
Category:Examples of Power Series
Category:Taylor Series
Power Series Expansion for Sine Integral Function

== Theorem ==
:$\ds \map \Si x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1} \times \paren {2 n + 1}!}$
where:
:$\Si$ denotes the sine integral function
:$x$ is a real number.
== Proof ==
== Sources ==
* : $\S 35$: Sine Integral $\ds \map \Si x = \int_0^x \frac {\sin u} u \rd u$: $35.11$
Category:Examples of Power Series
Category:Sine Integral Function
Power Series Expansion for Spence's Function

== Theorem ==
Spence's function has a power series expansion: 
:$\ds \map {\Li_2} z = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}$
for $z \in \C$.
This converges for $\size z \le 1$.
== Proof ==
Category:Spence's Function
Category:Examples of Power Series
Power Series Expansion for Square Root of 1 + x

== Theorem ==
Let $x \in \R$ such that $-1 < x \le 1$.
Then:
== Proof ==
== Sources ==
* : $\S 20$: Binomial Series: $20.12$
* : $\S 22$: Taylor Series: Binomial Series: $22.12.$
Category:Examples of Power Series
Power Series Expansion for Square Root of 1 - x

== Theorem ==
Let $x \in \R$ such that $-1 < x \le 1$.
Then:
== Proof ==
== Sources ==
* : $8$. Taylor Series and Power Series: Exercises: $8.2$ Expansion of a Function in a Power Series: $1. \ \text {(a)}$
Category:Examples of Power Series
Power Series Expansion for Tangent Function

== Theorem ==
The tangent function has a Taylor series expansion:
where $B_{2 n}$ denotes the Bernoulli numbers.
This converges for $\size x < \dfrac \pi 2$.
== Proof 1 ==
== Proof 2 ==
== Proof of Convergence ==
== Sequence of Terms ==
== Also presented as ==
== Also known as ==
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.23$
* : $4.$ Mathematics: $4.2$ Series
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* 
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.23.$
Category:Power Series Expansion for Tangent Function
Category:Examples of Power Series
Category:Bernoulli Numbers
Category:Tangent Function
Power Series Expansion for Tangent Function/Also presented as

== Power Series Expansion for Tangent Function: Also presented as == 
The Power Series Expansion for Tangent Function can also be presented in the form:
where ${B_n}^*$ denotes the archaic form of the Bernoulli numbers.
== Sources ==
* : $\S 20$: Series for Trigonometric Functions: $20.23$
* : $\S 22$: Taylor Series: Series for Trigonometric Functions: $22.23.$
Category:Power Series Expansion for Tangent Function
Power Series Expansion for Tangent Function/Mistake

== Source Work ==
:
: Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler
:: The Power Series for the Tangent
This mistake can be seen in the $1992$ edition as published by McGraw-Hill: ISBN 0-07-057566-5
== Mistake ==
:Based on our knowledge of the Bernoulli numbers, the first few terms of the [ Power Series Expansion for Tangent Function ] are easy to calculate explicitly,
::$\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {67} {2835} x^9 + \cdots$
The numerator of the coefficient for $x^9$ is in fact $62$, not $67$.
The correct calculation can be found in Power Series Expansion for Tangent Function/Sequence.
== Also see ==
* Power Series Expansion for Tangent Function
== Sources ==
* : Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler
Category:Mistakes/Calculus Gems
Power Series Expansion for Tangent Function/Proof 1

== Theorem ==
== Proof ==
From Power Series Expansion for Cotangent Function:
:$(1): \quad \cot x = \ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}$
Then:
=== Proof of Convergence ===
Category:Power Series Expansion for Tangent Function
Power Series Expansion for Tangent Function/Proof 2

== Theorem ==
== Proof ==
We have:
Thus:
Replacing $x$ with $2 i x$ in the of $(1)$:
Replacing $x$ with $2 i x$ in the of $(1)$:
Then from Cotangent Minus Tangent:
:$\tan x = \cot x - 2 \cot 2 x$
from which:
=== Proof of Convergence ===
== Sources ==
* : Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler
Category:Power Series Expansion for Tangent Function
Power Series Expansion for Tangent Function/Proof of Convergence

== Theorem ==
The radius of convergence of the Power Series Expansion for Tangent Function:
:$\ds \tan x = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}$
where $B_{2 n}$ denotes the Bernoulli numbers, is given as:
:$\size x < \dfrac \pi 2$
== Proof ==
By Combination Theorem for Limits of Real Functions we can deduce the following.
This is less than $1$ :
:$\size x < \dfrac \pi 2$
Hence by the Ratio Test, the series converges for $\size x < \dfrac \pi 2$.
Category:Power Series Expansion for Tangent Function
Power Series Expansion for Tangent Function/Sequence

== Theorem ==
The Power Series Expansion for Tangent Function begins:
:$\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {62} {2835} x^9 + \cdots$
== Proof ==
From Power Series Expansion for Tangent Function:
Enumerating the Bernoulli numbers:
Thus the appropriate arithmetic is performed on each coefficient:
Hence the result.
== Sources ==
* : $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
* : Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler
:::Beware: there is a mistake in the $5$th term as reported here.
Category:Power Series Expansion for Tangent Function
Power Series Expansion for nth Power of Logarithm of Reciprocal of 1-z

== Theorem ==
where $\ds \left[{k \atop n}\right]$ denotes an unsigned Stirling number of the first kind.
== Proof ==
== Sources ==
* : $\S 1.2.9$: Generating Functions: $(26)$
Category:Examples of Power Series
Category:Logarithms
Power Series is Differentiable on Interval of Convergence

== Theorem ==
Let $\xi \in \R$ be a real number.
Let $\ds \map f x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a (real) power series about $\xi$.
Let $\map f x$ have an interval of convergence $I$.
Then $\map f x$ is continuous on $I$, and differentiable on $I$ except possibly at its endpoints.
Also:
:$\ds \map {D_x} {\map f x} = \sum_{n \mathop = 1}^\infty n a_n \paren {x - \xi}^{n - 1}$
== Proof ==
Let the radius of convergence of $\map f x$ be $R$.
Suppose $x \in I$ such that $x$ is not an endpoint of $I$.
Then there exists $x_0 \in I$ such that $x$ lies between $x_0$ and $\xi$.
Thus:
:$\size {x - \xi} < \size {x_0 - \xi} < R$
Consider the series:
:$\ds \sum_{n \mathop = 2}^\infty \size {\frac {n \paren {n - 1} } 2 a_n X_0^{n - 2} }$ where $X_0 = x_0 - \xi$
From Radius of Convergence from Limit of Sequence‎, we have:
:$\ds \frac 1 R = \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = \limsup_{n \mathop \to \infty} \paren {\size {\frac {n \paren {n - 1} } 2} a_n}^{1/n}$
Thus we may deduce that:
:$\ds \sum_{n \mathop = 2}^\infty \size {\frac {n \paren {n - 1} } 2 a_n X_0^{n - 2} }$
converges.
Put $\delta = \size {x - x_0}$.
For values of $y$ such that $0 < \size {x - y} < \delta$, we consider:
:$\ds \Delta = \frac {\map f y - \map f x} {y - x} - \sum_{n \mathop = 1}^\infty n a_n \paren {x - \xi}^{n - 1} = \sum_{n \mathop = 1}^\infty a_n \paren {\frac {Y^n - X^n} {Y - X} - n X^{n - 1} }$
where $X = x - \xi, Y = y - \xi$.
We want to show that $\Delta \to 0$ as $Y \to X$.
From Difference of Two Powers, we have:
It follows that from all of these terms we can extract $\left({Y - X}\right)$ as a factor.
So the reduces to a product of $\paren {Y - X}$ with the sum of $n \paren {n - 1} / 2$ terms of the form $X^r Y^s$ where $r + s = n - 2$.
The number $n \paren {n - 1} / 2$ arises from Closed Form for Triangular Numbers:
:$1 + 2 + \cdots + \paren {n - 1} = \dfrac {n \paren {n - 1} } 2$
Since:
:$\size X = \size {x - \xi} < \size {x_0 - \xi} = \size {X_0}$
:$\size Y = \size {y - \xi} < \size {x - \xi} + \delta = \size {X_0}$
we have:
:$\size {\dfrac {Y^n - X^n} {Y - X} - n X^{n - 1} } < n \paren {n - 1} / 2 \size {X_0}^{n - 2} \size {Y - X}$
So:
:$\ds \size \Delta \le \size {Y - X} \sum_{n \mathop = 2}^\infty \frac {n \paren {n - 1} } 2 \size {a_n X_0^{n - 2} } \to 0$ as $Y \to X$
So $f$ is differentiable at all $x \in I$ which is not an endpoint, and that:
:$\ds \map {D_x} {\map f x} = \sum_{n \mathop = 1}^\infty n a_n \paren {x - \xi}^{n - 1}$
Now we need to investigate the question of left hand continuity and right hand continuity at the endpoints of $I$.
Abel's Theorem tells us that if $\ds \sum_{k \mathop = 0}^\infty a_k$ is convergent, then:
:$\ds \lim_{x \mathop \to 1^-} \paren {\sum_{k \mathop = 0}^\infty a_k x^k} = \sum_{k \mathop = 0}^\infty a_k$
== Sources ==
* : $\S 15.8$
* : Appendix: $\S 18.9$: Continuity and differentiation of power series
Category:Real Power Series
Category:Convergence
Category:Differential Calculus
Power Series is Taylor Series

== Theorem ==
Let $\ds \map f z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a complex power series about $\xi \in \C$.
Let $R$ be the radius of convergence of $f$.
Then, $f$ is of differentiability class $C^\infty$.
For all $n \in \N$:
:$a_n = \dfrac {\map {f^{\paren n} } \xi} {n!}$
Hence, $f$ is equal to its Taylor series expansion about $\xi$:
:$\ds \forall z \in \C, \size {z - \xi} < R: \quad \map f z = \sum_{n \mathop = 0}^\infty \dfrac {\paren {z - \xi}^n} {n!} \map {f^{\paren n} } \xi$
== Proof ==
First, we prove by induction over $k \in \N_{\ge 1}$ that:
:$\ds \map {f^{\paren k} } z = \sum_{n \mathop = k}^\infty a_n \paren {z - \xi}^{n - k} n^{\underline k}$
where $n^{\underline k}$ denotes the falling factorial.
=== Basis for the Induction ===
For $k = 1$, it follows from Derivative of Complex Power Series that 
:$\ds \map {f^{\paren k} } z = \sum_{n \mathop = 1}^\infty n a_n \paren {z - \xi}^{n - 1}$
As $n = n^{\underline 1}$, this proves the hypothesis.
From Radius of Convergence of Derivative of Complex Power Series, it follows that the equation holds for all $z \in \C$ with $\size {z - \xi} < R$.
=== Induction Hypothesis ===
For fixed $k \in \N_{\ge 1}$, the hypothesis is that:
:$\ds \map {f^{\paren k} } z = \sum_{n \mathop = k}^\infty a_n \paren {z - \xi}^{n - k} n^{\underline k}$
and $f^{\paren k}$ has radius of convergence $R$.
=== Induction Step ===
Note that $f^{\paren {k + 1} }$ is the derivative of $f^{\paren k}$.
From Radius of Convergence of Derivative of Complex Power Series, it follows that $f^{\paren {k + 1} }$ has radius of convergence $R$.
For $z \in \C$ with $\size {z - \xi} < R$, we have:
With $k \in \N$, we have:
Hence, it follows that $a_n = \dfrac {\map {f^{\paren n} } \xi} {n!}$.
By definition of Taylor series, it follows that $f$ is equal to its Taylor series expansion about $\xi$.
== Sources ==
* : $\S 1.4$
Category:Complex Power Series
Category:Taylor Series
Power Series is Termwise Differentiable within Radius of Convergence

== Theorem ==
Let $\ds \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$.
Let $R$ be the radius of convergence of the series.
Then:
:$\ds \frac \d {\d x} \map f x = \sum_{n \mathop = 0}^\infty \frac \d {\d x} a_n \paren {x - \xi}^n = \sum_{n \mathop = 1}^\infty n a_n \paren {x - \xi}^{n - 1}$
for $\size {x - \xi} < R$.
== Proof ==
Let $\rho \in \R$ such that $0 \le \rho < R$.
From Power Series Converges Uniformly within Radius of Convergence, $\map f x$ is uniformly convergent on $\set {x: \size {x - \xi} \le \rho}$.
From Real Polynomial Function is Continuous, each of $\map {f_n} x = a_n \paren {x - \xi}^n$ is a continuous function of $x$.
From Power Rule for Derivatives:
:$\dfrac \d {\d x} \paren {x - \xi}^n = n \paren {x - \xi}^{n - 1}$
and again from Real Polynomial Function is Continuous, each of $\dfrac \d {\d x} \map {f_n} x = n a_n \paren {x - \xi}^{n - 1}$ is a continuous function of $x$.
Then from Derivative of Uniformly Convergent Series of Continuously Differentiable Functions:
== Sources ==
* : $\S 1.3.2$: Power series
* 
* 
Category:Power Series
Power Series is Termwise Integrable within Radius of Convergence

== Theorem ==
Let $\ds \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$.
Let $R$ be the radius of convergence of $S$.
Then:
:$\ds \int_a^b \map f x \rd x = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \rd x = \sum_{n \mathop = 0}^\infty a_n \frac {x^{n + 1} } {n + 1}$
== Proof ==
Let $\rho \in \R$ such that $0 \le \rho < R$.
From Power Series Converges Uniformly within Radius of Convergence, $\map f x$ is uniformly convergent on $\set {x: \size {x - \xi} \le \rho}$.
From Real Polynomial Function is Continuous, each of $\map {f_n} x = a_n x^n$ is a continuous function of $x$.
Then from Definite Integral of Uniformly Convergent Series of Continuous Functions:
:$\ds \int_a^b \map f x \rd x = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \rd x$
The final result follows from Integral of Power.
== Sources ==
* : $\S 1.3.2$: Power series
* 
* 
Category:Power Series
Power Series of Sine of Odd Theta

== Theorem ==
Let $r \in \R$ such that $\size r < 1$.
Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$.
Then:
== Proof ==
From Euler's Formula:
:$\map \exp {i \theta} = \cos \theta + i \sin \theta$
Hence:
== Sources ==
* : $\S 4$. Elementary Functions of a Complex Variable: Exercise $10$
Category:Sine Function
Category:Trigonometric Series
Power Set/Examples/Arbitrary Example 1

== Example of Power Set ==
Let $A = \set {1, 2}$.
Then:
:$\powerset A = \set {\O, \set 1, \set 2, \set {1, 2} }$
== Sources ==
* : Chapter $1$: Preliminaries: $1.4$ Set Notation: Example $1.2$
Category:Examples of Power Sets
Power Set/Examples/Axiomatic Definition of 2

== Example of Power Set ==
Let $\O$ denote the empty set.
Let $S$ be the set defined as the $2$nd element of the von Neumann construction of the natural numbers:
:$S = \set {\O, \set \O}$
Then the power set of $S$ is:
:$\powerset S = \set {\O, \set \O, \set {\set \O}, \set {\O, \set \O} }$
and so has $2^2 = 4$ elements.
== Sources ==
* : $\S 2$. Sets of sets: Exercise $2$
Category:Examples of Power Sets
Power Set/Examples/Nested Sets of Empty Sets

== Example of Power Set ==
Let $\O$ denote the empty set.
Let $S$ be the set defined as:
:$S = \set {\O, \set \O, \set {\set \O} }$
Then the power set of $S$ is:
:$\powerset S = \set {\O, \set \O, \set {\set \O}, \set {\set {\set \O} }, \set {\O, \set \O}, \set {\O, \set {\set \O} }, \set {\set \O, \set {\set \O} }, \set {\O, \set \O, \set {\set \O} } }$
and so has $2^3 = 8$ elements.
== Sources ==
* : $\S 2$. Sets of sets: Exercise $2$
Category:Examples of Power Sets
Power Set/Examples/Set containing Empty Set

== Example of Power Set ==
Let $\O$ denote the empty set.
Let $S$ be the set defined as:
:$S = \set \O$
Then the power set of $S$ is:
:$\powerset S = \set {\O, \set \O}$
and so has $2^1 = 2$ elements.
== Sources ==
* : $\S 2$. Sets of sets: Exercise $2$
Category:Examples of Power Sets
Power Set/Examples/Set of 2 Elements

== Example of Power Set ==
Let $S = \set {1, 2}$.
Then the power set of $S$ is:
:$\powerset S = \set {\O, \set 1, \set 2, \set {1, 2} }$
and so has $2^2 = 4$ elements.
== Sources ==
* 
* 
Category:Examples of Power Sets
Power Set/Examples/Set of 3 Elements

== Example of Power Set ==
Let $S = \set {a, b, c}$.
Then the power set of $S$ is:
:$\powerset S = \set {\O, \set a, \set b, \set c, \set {a, b}, \set {b, c}, \set {a, c}, S}$
and so has $2^3 = 8$ elements.
Note that while $\set a \in \powerset S$, $a \notin \powerset S$.
== Sources ==
* : $\S 1.8$. Sets of sets: Example $25$
Category:Examples of Power Sets
Power Set Exists and is Unique

== Theorem ==
Let $V$ be a basic universe.
Let $x \in V$ be a set.
Let $\powerset x$ denote the power set of $x$.
Then $\powerset x$ is guaranteed to exist and is unique.
== Proof ==
By the an arbitrary subclass of $x$ can be created.
Hence we can create the class of all such subclasses.
Hence $\powerset x$ exists.
Let $\powerset x$, $\map \QQ x$ both be power sets of $x$.
From definition of power sets:
:$\forall T$:
::$T \in \powerset x \iff T \subseteq x$
::$T \in \map \QQ x \iff T \subseteq x$
From Biconditional is Commutative and Biconditional is Transitive:
:$T \in \powerset x \iff T \in \map \QQ x$
By the :
:$\powerset x = \map \QQ x$
Hence the power set is unique.
== Sources ==
* : Chapter $2$: Some Basics of Class-Set Theory: $\S 6$ The power axiom
Category:Power Set
Power Set and Two-Valued Functions are Isomorphic Boolean Rings

== Theorem ==
Let $S$ be a set.
Let $\mathbf 2$ be the Boolean ring two.
Let $\powerset S$ be the power set of $S$; by Power Set is Boolean Ring, it is a Boolean ring.
Let $\mathbf 2^S$ be the set of all mappings $f: S \to \mathbf 2$; by Two-Valued Functions form Boolean Ring, it is also a Boolean ring
Let $\chi_{\paren \cdot}: \powerset S \to \mathbf 2^S$ be the characteristic function operation.
Then $\chi_{\paren \cdot}$ is a ring isomorphism.
== Proof ==
From Support Operation Inverse to Characteristic Function Operation, $\chi_{\paren \cdot}$ is a bijection.
It therefore suffices to establish it is a ring homomorphism.
By Characteristic Function of Symmetric Difference:
:$\chi_{A * B} = \chi_A + \chi_B - 2 \chi_A \chi_B$
Since $\mathbf 2^S$ is a Boolean ring, by Idempotent Ring has Characteristic Two, the reduces to:
:$\chi_{A * B} = \chi_A + \chi_B$
showing that $\chi_{\paren \cdot}$ preserves ring addition.
By Characteristic Function of Intersection: Variant 1:
:$\chi_{A \cap B} = \chi_A \cdot \chi_B$
showing that $\chi_{\paren \cdot}$ preserves the ring product.
Hence $\chi_{\paren \cdot}$ is a ring homomorphism.
The result follows.
== Sources ==
* : $\S 3$
Category:Power Set
Category:Boolean Rings