Question

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. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Answer 1

(a, a) ∈ R |a – a| =0 is even, which is true

R is reflexive,

a R b and b R c

⇒ |a – b| is even and |b – c| is even

⇒ (a- b) is even and b – c is even

⇒ (a- b) + (b – c) is even

⇒ (a- c) is even ⇒ |a – c| is even

⇒ a R c, hence R is transitive

since R is reflexive, symmetric and transitive

R is an equivalence relation.

|1 – 3| = 2, even, |1 – 5| = 4, even |3 – 5| = 2,

even hence all the elements of {1, 3, 5} are related to each other.

|2 – 4| = 2, even, hence all the elements {2,4} are related to each other.

|1-2| = 1, |3-2| = 1, |5-2| = 3, |1-4| = 3,

|3 – 4| = 1, |5 – 4| = 1, are all odd.

Therefore no elements of {1, 3, 5} are related to {2, 4}.