Binary operation

A binary operation is a function that maps two elements of a given set to one element of that set.

In general form:

$$ a*b=c,\, a,b,c \in X $$

where $a,b,c$ do not have to be distinct.

Formal definition

Let $X \neq \emptyset$ be a set.

Any function $*:X \times X \to X$ is called a binary operation. $X$ is said to be closed under $*$.