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.
