Let R = {(a, b): a, b ∈ Z and (a – b) is divisible by 5}. Show tha

1.	Let R = {(a, b): a, b ∈ Z and (a – b) is divisible by 5}. Show that R is an equivalence relation on Z.
Answer» It is given that R = {(a, b): a, b ∈ Z and (a – b) is divisible by 5} We know that (a, b) ∈ R where a – b is divisible by 5. (i) Reflexive- We know that (a, a) ∈ R as a – a = 0 divisible by 5. Therefore, R is reflexive. (ii) Symmetric- We know that (a, b) ∈ R as a – b is divisible by 5. So we get a – b = 5k and b – a = – 5k Hence, b – a is also divisible by 5 and (b, a) ∈ R Therefore, R is symmetric. (iii) Transitive- We know that if (a, b) and (b, c) ∈ R as a – b is divisible by 5 So we get a – b = 5m and b – c is divisible by 5 where b – c = 5n By adding we get a – c = 5 (m + n) Here, a – c is divisible by 5 and (a, c) ∈ R which is transitive. R is reflexive, symmetric and transitive and therefore is an equivalence relation.

Let R = {(a, b): a, b ∈ Z and (a – b) is divisible by 5}. Show that R is an equivalence relation on Z.

Answer»

It is given that

R = {(a, b): a, b ∈ Z and (a – b) is divisible by 5}

We know that

(a, b) ∈ R where a – b is divisible by 5.

(i) Reflexive-

We know that (a, a) ∈ R as a – a = 0 divisible by 5.

Therefore, R is reflexive.

(ii) Symmetric-

We know that (a, b) ∈ R as a – b is divisible by 5.

So we get

a – b = 5k and b – a = – 5k

Hence, b – a is also divisible by 5 and (b, a) ∈ R

Therefore, R is symmetric.

(iii) Transitive-

We know that if (a, b) and (b, c) ∈ R as a – b is divisible by 5

So we get

a – b = 5m and b – c is divisible by 5 where b – c = 5n

By adding we get

a – c = 5 (m + n)

Here, a – c is divisible by 5 and (a, c) ∈ R which is transitive.

R is reflexive, symmetric and transitive and therefore is an equivalence relation.

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