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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 ($\div:\R\times\R^*\to\R$) 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.

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