Binary operation

Let $S$ be a set . A binary operation on $S$ is a function

where $S\times S$ is the Cartesian product . Thus, for each pair $(a,b)\in S\times S$, the value $a*b$ is an element of $S$.

Binary operations are the basic input for algebraic structures (groups, rings, etc.), where one adds axioms such as associativity or commutativity.

Examples:

• Addition $+\colon\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is a binary operation.
• Matrix multiplication on the set of $n\times n$ real matrices is a binary operation.
• Subtraction on $\mathbb{N}$ is not a binary operation on $\mathbb{N}$ if $\mathbb{N}$ does not contain negatives (not closed).