Q. 45.0( 1 Vote )

# Let n be a fixed positive integer. Define a relation R on Z as follows :(a, b) ∈ R ⇔ a – b is divisible by n.Show that R is an equivalence relation on Z.

R = {(a, b) : a – b is divisible by n} on Z.

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) R

Let a Z

a – a = 0 × n

a – a is divisible by n

(a, a) R

R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) R, then (b, a) R

Let (a, b) R

a – b = np For some p Z

b – a = n(–p)

b – a is divisible by n

(b, a) R

R is symmetric

Transitive : : For Transitivity, we need to prove that-

If (a, b) R and (b, c) R, then (a, c) R

Let (a, b) R and (b, c) R

a – b = np and b – c = nq For some p, q Z

a – c = n (p + q)

a – c = is divisible by n

(a, c) R

R is transitive

R being reflexive, symmetric and transitive on Z.

R is an equivalence relation on Z

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