Equivalence
Relations: A relation R on a set A is called an equivalence relation if and
only if

(i)
R is reflexive, i.e., for all a ∈ R, (a, a) ∈ R,

(ii)
R is symmetric, i.e., (a, b) ∈ R ⇒ (b, a) ∈ R, for all a, b ∈ A, and

(iii)
R is transitive, i.e., (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R, for all a, b, c
∈ A.

One
of the most trivial examples of an equivalence relation is that of ‘equality’.

For
any elements a,b,c in a set A,

(i)
a = a, i.e., reflexivity

(ii)
a = b⇒ b = a, i.e., symmetricity

(iii)
a = b and b = c ⇒ a = c , i.e., transitivity.

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