Sets can themselves be treated as mathematical objects and combined into new sets.

Power sets

The power set of $A$, written $\mathcal P(A)$, is the set of all subsets of $A$.

If

$$A=\lbrace a,b\rbrace,$$

then

$$\mathcal P(A)=\lbrace\varnothing,\lbrace a\rbrace,\lbrace b\rbrace,\lbrace a,b\rbrace\rbrace.$$

A finite set with $n$ elements has

$$2^n$$

subsets, because each element can independently be included or excluded.

Ordered pairs

An ordered pair $(a,b)$ remembers which element is first and which is second, so in general

$$(a,b)\ne(b,a).$$

Cartesian products

For sets $A$ and $B$, their Cartesian product is

$$A\times B=\lbrace(a,b):a\in A,\ b\in B\rbrace.$$

Cartesian products turn two sets into a space of possible pairings. They provide the natural setting for relations, functions, coordinate systems and multidimensional data.