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 nonnegative integers, $2^n > n$ so the power set of any set has a cardinality at least as large as the set itself. An analogous result holds for infinite sets: for any set $S$ , there is no bijection between $S$ and $\mathcal{P}(S)$ (or equivalently, there is no injection from $\mathcal{P}(S)$ to $S$ ).

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Power set

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