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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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