Show that the relation R in the set A = {1, 2, 3, 4, 5} Given by R = {

1.	Show that the relation R in the set A = {1, 2, 3, 4, 5} Given by R = {(a, b) / \|a – b\| is even} is an equivalence relation.
Answer» (i) Since \|a – a\| is even, ∴ (a, a) ∈ R ∴ R is reflexive. (ii) Let (a, b) ∈ R Then \|a – b\| is even ∴ \|b – a\| is even ∴ (b, a) ∈ R ∴ R is symmetric. (iii) Let (a, b), (b, c) ∈ R Then a – b = ±2m, b – c = ±2n ∴ a – c = ±2(m + n), where m, n are integers. ∴ (a, c) ∈ R ∴ R is transitive Thus, R is an equivalence relation.

Show that the relation R in the set A = {1, 2, 3, 4, 5} Given by R = {(a, b) / |a – b| is even} is an equivalence relation.

Answer»

(i) Since |a – a| is even,

∴ (a, a) ∈ R ∴ R is reflexive.

(ii) Let (a, b) ∈ R Then |a – b| is even

∴ |b – a| is even

∴ (b, a) ∈ R ∴ R is symmetric.

(iii) Let (a, b), (b, c) ∈ R

Then a – b = ±2m, b – c = ±2n

∴ a – c = ±2(m + n), where m, n are integers.

∴ (a, c) ∈ R ∴ R is transitive

Thus, R is an equivalence relation.