Set power

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In mathematics, the power set of a given set is another set made up of all subsets of the given set. For example, given the set:

${displaystyle A={1,2,3}}$

the power set is:

${displaystyle {mathcal {P}}(A)={varnothing{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}}}$

The power set $A$ also called group of parties of $A$ or group of parties of $A$ and denotes ${displaystyle {mathcal {P}}(A)}$ Where ${displaystyle 2^{|A|}}$ is the cardinal of the parts of $A$ I mean, ${displaystyle |{mathcal {P}}(A)|=2^{|A|}}$ .

Definition

The power set of $A$ is the class or collection of the subsets of $A$ :

The power of $A$ (o) group of parties or group of parties) is the set ${displaystyle {mathcal {P}}(A)}$ formed by all subsets of ${displaystyle A:}$
${displaystyle bin {mathcal {P}}(A){text{ cuando }}bsubseteq A}$

Examples

The power set $A = a, 2, c!$ is:

{displaystyle {mathcal {P}}(A)={varnothing{a},{2},{c},{a,2},{a,c},{2,c},{a,2,c}}}

The power set $B = x!$ is:

{displaystyle {mathcal {P}}(B)={varnothing{x}}}

Properties

The power set of any set contains at least one subset. Also, it is not equipotent with the base.

The empty set is in the power set of any set:
${displaystyle varnothing in {mathcal {P}}(A){text{ para cualquier }}A}$
A whole is always an element of its power set:
${displaystyle Ain {mathcal {P}}(A){text{ para cualquier }}A}$

Cardinal

Whenever the empty set is not an element of a set, the following is true: The number of elements in the power set is precisely a power of the number of elements in the original set:

The cardinal of the whole power of a finite set $A$ is 2 elevated to the cardinal $A$ :
${displaystyle |{mathcal {P}}(A)|=2^{|A|}}$

This relation is the origin of the $2 A$ notation for the power set. One way to derive it is by using the binomial coefficients. If the set $A$ has $n$ elements, the number of subsets with $k$ elements is equal to the combinatorial number $C (n, k)$ . A subset of $A$ can have at least 0 elements, and $n$ at most, and therefore:

{displaystyle |{mathcal {P}}(A)|={n choose 0}+{n choose 1}+ldots +{n choose k}+ldots +{n choose n}=2^{n}=2^{|A|}}

This relationship can also be proved by noting that the power set of $A$ is equivalent to the set of functions with domain $A$ and codomain ${0, 1}$ , $f : A \to {0, 1}$ . Each function then corresponds to a subset, if the image of an element is interpreted as an indicator of whether said element belongs to the subset: 0 indicates "does not belong", 1 indicates "belongs". The number of these characteristic functions of $A$ is precisely $2 n$ , if $| A | = n$ .

In the case of an infinite set the identification between subsets and functions is equally valid, and the cardinal of the power set remains equal to $2 | A |$ , in terms of infinite cardinals and their arithmetic. The power set always has a greater cardinal than the original set, as established by Cantor's theorem, so there is never a bijective map between a set and its power set.

The minimum of the cardinals of power sets is 1, exactly that of the power set of the vacuum set

Boolean Algebra

Main article: Algebra de Boole

The power set of a given set has a Boolean algebra structure, considering the operations of union, intersection and complement, and is commonly used as an example of such a structure. In fact, a finite Boolean algebra is always isomorphic to the Boolean algebra of the power set of some finite set. In the general case—including infinite algebras—a Boolean algebra is always isomorphic to a subalgebra of a power set.

Axiom of the power set

Main article: Axiom of the power set

In axiomatic set theory, the existence of the power set in general cannot be proved from more basic properties, so it is postulated through an axiom. Without this axiom it is not possible to prove the existence of uncountable sets.

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