Binary operation on a set

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Suppose G is a non-empty set.
The G X G = {(a,b) : a E G, b E G)}. If f : G X G → G then
f is called a binary operation on a set G.
The image of the ordered pair (a,b) under the function f is denoted by afb.

A binary operation on asset G is sometimes also said to be
the binary composition in the set G.
If * is a binary composition in G then, a * b E G, a, b E G.
Therefore g is closed with respect to the composition denoted by *.


https://www.includehelp.com/basics/a...structure.aspx

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A binary operation on a set is an operation whose two domains and the codomain
are the same set.

Examples include the familiar arithmetic operations of addition, subtraction, multiplication.

Other examples are readily found in different areas of mathematics,
such as vector addition, matrix multiplication and conjugation in groups.


Binary operations are the keystone of most algebraic structures,
that are studied in algebra, in particular in semigroups, monoids,
groups, rings, fields, and vector spaces.

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