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In 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. [1] In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. [2] The powerset of S is variously denoted as P(S), 𝒫 (S ...
Power (statistics) In frequentist statistics, power is a measure of the ability of an experimental design and hypothesis testing setup to detect a particular effect if it is truly present. In typical use, it is a function of the test used (including the desired level of statistical significance), the assumed distribution of the test (for ...
In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set , the set of all subsets of known as the power set of has a strictly greater cardinality than itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets.
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
Powerset construction. In the theory of computation and automata theory, the powerset construction or subset construction is a standard method for converting a nondeterministic finite automaton (NFA) into a deterministic finite automaton (DFA) which recognizes the same formal language. It is important in theory because it establishes that NFAs ...
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. [1] In terms of set-builder notation, that is [2][3] A table can be created by taking the Cartesian product of a set of rows and a set of columns.
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 2 n members. Integer powers of 2 are important in computer science. The positive integer powers 2 n give the number of possible values for an n-bit integer binary number; for example, a byte may take 2 8 = 256 different values.
Axiom of power set. The elements of the power set of the set {x, y, z} ordered with respect to inclusion. In mathematics, the axiom of power set[1] is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set the existence of a set , the power set of , consisting precisely of the subsets of .