Binary operation - Imported Knowlpedia Content

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 $x*y$ .

Remarks

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.