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Encyclopedia > Power set

In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X âŠ† Y; Y is a superset of (or includes) X; Y âŠ‡ X... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory. ...

Any subset F of is called a family of sets over S. A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X âŠ† Y; Y is a superset of (or includes) X; Y âŠ‡ X... In mathematics, the concept of hypergraph generalizes the notion of a graph. ...

For example, if S is the set {A, B, C} then the complete list of subsets of S is as follows:

• {} (the empty set)
• {A}
• {B}
• {C}
• {A, B}
• {A, C}
• {B, C}
• {A, B, C}

and hence the power set of S is In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...

{}, {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C}

If S is a finite set with |S|=n elements, then the power set of S contains elements. (One can - and computers actually do - represent the elements of 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.) This article is about the unit of information. ...

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 set of natural numbers for instance can be put in a one-to-one correspondence with the set of real numbers (by identifying an infinite 0-1 sequence with the set of indices where the ones occur). Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In 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. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

The power set of a set S, together with the operations of union, intersection and complement forms the prototypical example of a boolean algebra. 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. 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. ... In 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. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... For the use of binary numbers in computer systems, please see the article binary arithmetic. ...

The power set of a set S forms an Abelian group 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. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, a semigroup is a set with an associative binary operation on it. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...

In set theory, XY is the set of all functions from Y to X. As 2 can be defined as {0,1} (see natural number), 2S is the set of all functions from S to {0,1}. By identifying a function in 2S with the corresponding preimage of 1, we see that there is a bijection between 2S and , where each function is the characteristic function of the subset in with which it is identified. Hence 2S and could be considered identical set-theoretically. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In 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. ... In the mathematical subfield of set theory, the indicator function is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... Results from FactBites:

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 Power set - Wikipedia, the free encyclopedia (448 words) The power set of the set of natural numbers for instance can be put in a one-to-one correspondence with the set of real numbers (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 union, intersection and complement forms the prototypical example of a boolean algebra. The power set of a set S forms an Abelian group 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.
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