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.
