Binary operation

A binary operation is an operation with arity two, involving two operands.

A binary operation on a set $S$ is a function that maps elements of the Cartesian product:

$f:S\times S\to S$

Examples

In the set of real numbers, and in any field for that matter:

• Addition ($+$);
• Subtraction ($-$);
• Multiplication ($\times$);
• While not a binary operation in the strictest sense, as division by zero is undefined, division ($\div :\mathbb {R} \times \mathbb {R} ^{*}\to \mathbb {R}$) is commonly thought of as an operation.

Notation

Because a binary operation $+$ on a set $S$ is also a function from $S \times S$ to $S$ , and therefore a relation and a subset of the cartesian product $(S\times S)\times S$ , the following notations are valid:

• ${\bigl (}(x,y),z{\bigr )}\in +$ , when viewing $+$ as a set;
• $(x,y)+z$, when viewing $+$ as a relation;
• $+(x,y)=z$, when viewing $+$ as a function;

However, we will adopt the preferred notation $x + y$ as an alternative to the function notation $+(x,y)$ . One should not confuse this preferred notation to the relation notation; the preferred notation for binary operations is an expression for a value in the codomain, while the relation notation is an expression of a statement.