Power set

The power set of a given set $S$ is the set $\mathcal{P}(S)$ of subsets of that set.

The empty set has only one subset, itself. Thus $\mathcal{P}(\emptyset) = \{\emptyset\}$ .

A set $\{a\}$ with a single element has two subsets, the empty set and the entire set. Thus $\mathcal{P}(\{a\}) = \{\emptyset, \{a\}\}$ .

A set $\{a, b\}$ with two elements has four subsets, and $\mathcal{P}(\{a, b\}) = \{\emptyset, \{a\}, \{b\}, \{a, b\}\}$ .

Similarly, for any finite set with $n$ elements, the power set has $2^n$ elements.

Note that for any set $\displaystyle S$ such that $\displaystyle a \in S$ , $\displaystyle \{ a \} \subseteq S$ , so the power set of any set $\displaystyle S$ has a cardinality at least as large as that of $\displaystyle S$ itself. Specifically, sets of cardinality 1 or 0 are the only sets that have power sets of the same cardinality, since if $\displaystyle S$ is a finite set with cardinality at least 2, then $\displaystyle \mathcal{P}( S )$ clearly has cardinality greater than 2. A similar result holds for infinite sets: for no infinite set $\displaystyle S$ is there a bijection between $\displaystyle S$ and $\displaystyle \mathcal{P} (S)$ .

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Power set

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