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...

the common usage of sets in mathematics does not require the full power of Zermelo–Fraenkel set theory. In mathematical practice, sets can be manipulated...

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 (...

constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A...

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

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...

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

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...

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)...

(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...

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations...

{Z} ,n=2k\}} The set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation...

2^{-2}} is a quarter. Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2n members...

which is the set of all possible dependent set variables y resulting from satisfaction of the conditions of membership for x, is the power set of U such...

general result (now called Cantor's Theorem) that the power set of any set is strictly larger than the set itself. Cantor introduced the notion cardinal numbers...

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...

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...

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...

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,...

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...

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...

built up from the empty set by transfinitely iterating the power set operation. It is thus now possible again to reason about sets in a non-axiomatic fashion...

which is a subset of the power set of L α {\displaystyle L_{\alpha }} . Consequently, this is a tower of nested transitive sets. But L {\displaystyle L}...

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...

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

of families of sets satisfying certain restrictions. The set of all subsets of a given set S {\displaystyle S} is called the power set of S {\displaystyle...

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...

})} denotes the power set of A α {\displaystyle A_{\alpha }} (i.e., the set of all subsets of A α {\displaystyle A_{\alpha }} ), the set 2 A α ⊂ P ( A α...

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