Binary Operations,commutative,associative,identity,inverse

Binary Operations: A binary operation *  on a set A is a function * :  A × A → A. We denote * (a, b) by a * b.

A binary operation * on the set X is called commutative, if a * b = b * a, for every a, b ∈ X.

A binary operation * : A × A →  A is said to be associative if (a * b) * c = a * (b * c), a, b, c, ∈ A.

Given a binary operation * : A × A→ A, an element e ∈ A, if it exists,

is called identity for the operation *, if a *e = a = e * a,

 a ∈ A.

Given a binary operation * : A × A→A with the identity element e in A,

an element a ∈ A is said to be invertible with respect to the operation *, if there exists an element b in A such that a * b = e = b * a and b is called the inverse of a and is denoted by a.