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In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set.
In this case S is usually called the universal set and any subset F of P(S) is called a family of sets over S.
For example, if S is the set {A, B, C} then the complete list of subsets of S is as follows:
and hence the power set of S is
If n = |S| is the number of elements of S, then the respective power set contains |P(S)| = 2n elements. (One can - and computers actually do - represent the elements of P(S) as n-bit numbers; the n-th bit refers to presence or absence of the n-th element of S. There are 2n such numbers.)
One can also consider the power set of infinite sets. Cantor's diagonal argument shows that the power set of a set (infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be 'greater' than the original set). The power set of the natural numbers for instance can be put in a one-to-one correspondenceIn mathematics, a bijection bijective function or one-to-one correspondence is a function that is both injective ("one-to-one") and surjective ("onto"), and therefore bijections are also called one-to-one and onto . Intuitively, a bijective function creat with the set of real numberIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers mays (by identifying an infinite 0-1 sequence with the set of indices where the ones occur).
The power set of a set S, together with the operations of unionAbstract algebra Algebra In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. This article uses mathematical symbols. Basic definition If, intersectionIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. This article uses mathematical symbols. The intersecti and complementIn set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement . Relative complement If A and B are sets, then the relative complement of a A in B also known as the set theoretic forms the prototypical example of a boolean algebraIn mathematics and computer science, Boolean algebras or Boolean lattices are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and compleme. In fact, one can show that any finite boolean algebra is isomorphic to the boolean algebra of the power set of a finite set. For infinite boolean algebras this is no longer true, but every infinite boolean algebra is a subalgebra of a power set boolean algebra.
The power set of a set S forms an Abelian groupAbstract algebra Algebra Group theory In mathematics, an abelian group is a commutative group, i. a group G ) such that a b b a for all a and b in G''. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions fo when considered with the operation of symmetric difference (with the empty set as its unit and each set being its own inverse) and a commutative semigroup when considered with the operation of intersection. It can hence be shown (by proving the distributive laws) that the power set considered together with both of these operations forms a commutative ring.
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