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Binary operation

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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 S × S to S:

f \colon S \times S \rightarrow 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}^* \rightarrow \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 \left(S \times S\right) \times S, the following notations are valid:

  • \left(\left(x,y\right),z\right)\in +, when viewing + as a set;
  • \left(x,y\right)\,+\,z, when viewing + as a relation;
  • +\left(x,y\right)=z, when viewing + as a function;

However, we will adopt the preferred notation x+y as an alternative to the function notation +\left(x,y\right). 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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