“power set”的概念、定义、翻译、参考文献-科学参考

power set

Mathematics

The set of all subsets of a set S is the power set of S, denoted by ℘(S). Suppose that S has n elements a₁, a₂,…, a_n, and let A be a subset of S. For each element a_i of S, there are two possibilities: either a_i ε‎ A or not. Considering all n elements leads to 2ⁿ possibilities in all. Hence, S has 2ⁿ subsets; that is, ℘(S) has 2ⁿ elements. If S = {a,b,c}, the 8 ( = 2³) elements of ℘(S) are

$Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, and {a, b, c} .$

Cantor’s Diagonal Theorem states that the power set of a set always has greater cardinality than the cardinality of the set.

Computer

(of a set S) The set of all subsets of S, typically denoted by 2^S. It can be described as
${A | A \subseteq S}$
The number of elements in the power set of S is 2^N, where N is the number of elements in S.

Philosophy

The power set of a set S is the set of all subsets of S. See also Cantor’s theorem.

单词	power set
释义	power set Mathematics The set of all subsets of a set S is the power set of S, denoted by ℘(S). Suppose that S has n elements a₁, a₂,…, a_n, and let A be a subset of S. For each element a_i of S, there are two possibilities: either a_i ε‎ A or not. Considering all n elements leads to 2ⁿ possibilities in all. Hence, S has 2ⁿ subsets; that is, ℘(S) has 2ⁿ elements. If S = {a,b,c}, the 8 ( = 2³) elements of ℘(S) are $Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, and {a, b, c} .$ Cantor’s Diagonal Theorem states that the power set of a set always has greater cardinality than the cardinality of the set. Computer (of a set S) The set of all subsets of S, typically denoted by 2^S. It can be described as ${A \| A \subseteq S}$ The number of elements in the power set of S is 2^N, where N is the number of elements in S. Philosophy The power set of a set S is the set of all subsets of S. See also Cantor’s theorem.
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