mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed...

Zermelo–Fraenkel set theory with the axiom of choice. (ZFC is the most widely-studied version of axiomatic set theory.) The power set of a set S is the set of all...

of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set x {\displaystyle x} the existence of a set P (...

example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} }. Some basic sets of central importance are the set of natural numbers, the set of real numbers...

union is infinite. The power set of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely...

mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable...

that set, where P {\displaystyle {\mathcal {P}}} represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in...

itself; equivalently, the power set of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its cardinality)...

IBM POWER is a reduced instruction set computer (RISC) instruction set architecture (ISA) developed by IBM. The name is an acronym for Performance Optimization...

In set theory, the complement of a set A, often denoted by A ∁ {\displaystyle A^{\complement }} (or A′), is the set of elements not in A. When all elements...

(or power) than the former set. Another example in an Euler diagram: A is a proper subset of B. C is a subset but not a proper subset of B. The set of...

where Z 0 {\displaystyle Z_{0}} is any infinite set and P {\displaystyle {\mathcal {P}}} is the power set operation. Moreover, one of Zermelo's axioms invoked...

In mathematics, fuzzy sets (also known as uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently...

mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle...

In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing...

The set of all even integers, expressed in set-builder notation. In set theory and its applications to logic, mathematics, and computer science, set-builder...

addition modulo 2. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element...

mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related...

empty set are also members of any set A is vacuously true). The set of all subsets of a given set A is called the power set of A and is denoted by 2 A {\displaystyle...

follows. For a collection T of subsets of X (that is, for any subset of the power set P(X) of X), let T σ {\displaystyle T_{\sigma }} be all countable unions...

set concerns the power set of the set of all sets. Because this power set is a set of sets, it would necessarily be a subset of the set of all sets,...

) . {\displaystyle A\cup (B\cap C)=(A\cup B)\cap (A\cup C).} The power set of a set ⁠ U {\displaystyle U} ⁠, together with the operations given by union...

domain is called the universe denoted U. The range is the set of subsets of U called the power set of U and denoted P(U). Thus the relation ∈ {\displaystyle...

algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations...

set of subsets of a given set (its power set) ordered by inclusion (see Fig. 1). Similarly, the set of sequences ordered by subsequence, and the set of...

theorem). They include, for instance: the set of all subsets of R, i.e., the power set of R, written P(R) or 2R the set RR of all functions from R to R Both...

exponent or power. Exponentiation is written as bn, where b is the base and n is the power; this is pronounced as "b (raised) to the (power of) n". When...

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously...

Reliance Power Limited (R-Power), formerly Reliance Energy Generation Limited (REGL) is a part of the Reliance Anil Dhirubhai Ambani Group. It was established...

between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships...