Question

In the set Z of integers, define m Rn if m is divisible by 7. prove that R is an equivalence relation​

Answer 1

Answer:

The proof is explained step-wise below :

Step-by-step explanation:

To show : R is an equivalence relation.

Proof : In order to show R is an equivalence relation, We need to show R is reflexive, symmetric and transitive.

(i) Reflexive : Let m ∈ Z. Then m - m is divisible by 7. Therefore mRm holds for all m in Z and R is reflexive.

(ii) Let m, n ∈ Z and mRn hold. Then m – n is divisible by 7 and therefore n – m is divisible by 7.

Thus, mRn ⇒ nRm and therefore R is symmetric.

(iii) Let m, n, o ∈ Z and mRn, nRo both hold. Then m - n and n - o are both divisible by 7.

Therefore m - o = (m – n) + (n – o) is divisible by 7.

Thus, R is an equivalence relation.

Hence Proved.

Thus, aRb and bRc  ⇒ aRc and therefore R is transitive.

Since R is reflexive, symmetric and transitive so, R is an equivalence relation on Z.