binary operation

A binary operation is an operation of two elements, called operands, to produce a third well-defined element.

Formally,
a binary operation on a set $S$ is a function of the elements of the Cartesian product $S\times S$ to $S$:

i.e. a total functional relation, so a set of ordered pairs

such that

• $\mathrm{\forall }a,b\in S:\mathrm{\exists }c\in S(c=\omega (a,b))$
• $\mathrm{\forall }a,b,c,d\in S:c=\omega (a,b)\mathtt{\text{ AND }}d=\omega (a,b)⟺c=d$

Examples include

• addition
• subtraction
• multiplication
• division
• exponentiation
• logical connectives
• composition of relations
• convolution
• set operations