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


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.


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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