Let n be a fixed positive integer. Define a relation R on the set Z of integers by thenR isa)reflexiveb)symmetricc)transitived)equivalenceCorrect answer is option 'D'. Can you explain this answer?

Question

Let n be a fixed positive integer. Define a relation R on the set Z of integers by  thenR isa)reflexiveb)symmetricc)transitived)equivalenceCorrect answer is option 'D'. Can you explain this answer?

Answer

Correct Answer :- d

Explanation : Reflexive: For any a∈N, we have a−a=0=0×n⇒−a is divisible by n

⇒(a,a)∈R

Thus (a,a)∈R for all a∈Z. So, R is reflexive.

Symmetry: Let (a,b)∈R. Then,

⇒(a,b)∈R

⇒(a−b) is divisible by n.

⇒(a−b)=np for some p∈Z

⇒b−a=n(−p)

⇒b−a is divisible by n

Thus, (a,b)∈R

⇒(b,a)∈R for all a,b∈Z

So, R is symmetric on Z.

Transitive: Let a,b,c∈Z such that (a,b)∈R and (b,c)∈R. Then (a,b)∈R

⇒(a−b) is divisible by n.

⇒a−b=np for some p∈Z

(b,c)∈R

⇒(b−c) is divisible by n.

⇒b−c=nq for some q∈Z

therefore,(a,b)∈R and b−c∈R

⇒a−b=npb−c=nq

⇒(a−b)+(b−c)=np+nq

a−c=n(p+q)

⇒−c is divisible by n.

⇒(a−c)∈R

Thus, (a,b)∈R and (b,c)∈R

⇒(a,c)∈R for all a,b,c∈Z

So, R is a transitive relation on Z.

Thus, R being reflexive, symmetric and transitive, is an equivalence relation.

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