Binary operation

A binary operation on a set $S$ is a function $*:S\times S\to S$, where $S\times S$ is the Cartesian product of $S$ with itself. The value of $*(x,y)$ is usually written $x*y$.

Binary operations are functions with two inputs that are “closed” in the sense that combining two elements of $S$ produces an element of $S$ again. Many algebraic structures begin with a set equipped with a binary operation.

Examples:

• Addition $+:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is a binary operation on the integers .
• For a fixed set $A$, the union map $(S,T)\mapsto S\cup T$ is a binary operation on the power set $\mathcal{P}(A)$, using union of subsets.