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 = {1, 2, 3, 4, 5} and R = {(a, b): | a – b | is even}

R = {(1, 3), (1, 5), (3, 5), (2, 4), (3, 1), (5, 1), (5, 3), (4, 2)}

(a) (i) Let us take any element a of set A.

Then, | a – a | = 0 is even,

⇒ R is reflexive.

(ii) If | a – b | is even, then | b – a | is also even, where R = {{a, b): | a – b | is even}

⇒ R is symmetric.

(iii) Further a – c = a – b + b – c

If | a – b | and | b – c | are even, then their sum | a – b + b – c | is also even.

⇒ | a – c | is even, R is transitive.

Hence, R is an equivalence relation.

(b) Elements of {1, 3, 5} are related to each other.

Since | 1 – 3 | = 2, | 3 – 5| = 2, | 1 – 5 | = 4. All are even numbers.

⇒ Elements of {1, 3, 5} are related to each other. Similarly, elements of {2, 4} are related to each other, since | 2 – 4 | = 2 is an even number. No element of set {1, 3, 5} is related to any element of {2,4}, because | 2 – 1 | is not even, | 2 – 3 | is not even, etc.