Subset
Power set

The set of all subsets of S {\displaystyle S} is called its power set, and is denoted by P (S) {\displaystyle {\mathcal {P}}(S)}.

The inclusion relation ⊆ {\displaystyle \subseteq } is a partial order on the set P (S) {\displaystyle {\mathcal {P}}(S)} defined by A ≤ B ⟺ A ⊆ B {\displaystyle A\leq B\iff A\subseteq B}. We may also partially order P (S) {\displaystyle {\mathcal {P}}(S)} by reverse set inclusion by defining A ≤ B if and only if B ⊆ A. {\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.}

For the power set P (S) {\displaystyle \operatorname {\mathcal {P}} (S)} of a set S, the inclusion partial order is—up to an order isomorphism —the Cartesian product of k = | S | {\displaystyle k=|S|} (the cardinality of S) copies of the partial order on { 0, 1 } {\displaystyle \{0,1\}} for which 0 < 1. {\displaystyle 0<1.} This can be illustrated by enumerating S = { s 1, s 2, …, s k }, {\displaystyle S=\left\{s_{1},s_{2},\ldots,s_{k}\right\},}, and associating with each subset T ⊆ S {\displaystyle T\subseteq S} (i.e., each element of 2 S {\displaystyle 2^{S}}) the k -tuple from { 0, 1 } k, {\displaystyle \{0,1\}^{k},} of which the i th coordinate is 1 if and only if s i {\displaystyle s_{i}} is a member of T.

The set of all k {\displaystyle k} -subsets of A {\displaystyle A} is denoted by (A k) {\displaystyle {\tbinom {A}{k}}}, in analogue with the notation for binomial coefficients, which count the number of k {\displaystyle k} -subsets of an n {\displaystyle n} -element set. In set theory, the notation k {\displaystyle ^{k}} is also common, especially when k {\displaystyle k} is a transfinite cardinal number.
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