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 identityfor the operation *, if a *e = a = e * a,
a ∈A.
Given a binary operation * : A × A→A with the identity element ein 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 aand is denoted by a.
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