Power set

2020 Mathematics Subject Classification: Primary: 03E [MSN][ZBL]

of a set $X$

The set of all subsets of $X$, denoted $\mathcal{P}(X)$. One has $A \in \mathcal{P}(X) \Leftrightarrow A \subseteq X$. The power set of a finite set of $n$ elements has $2^n$ elements. Cantor's theorem states that a set and its power set can never be put into one-to-one correspondence, hence cannot have the same cardinality.

The power set forms a Boolean algebra with the operations of union of sets, intersection of sets and relative complement.

References

[1]	P. R. Halmos, Naive Set Theory, Springer (1960) ISBN 0-387-90092-6