Question

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

Answer 1

It is given that

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

Reflexivity-

If a ∈ A and |a – a| = 0 which is even

We get

(a, a) ∈ R for a ∈ A.

Hence, R is reflexive.

Symmetric-

If (a, b) ∈ R where |a – b| and |b – a| is even

We get (b, a) ∈ R

Here (a, b) ∈ R and (b, a) ∈ R

Hence, R is symmetric.

Transitivity-

If (a, b) ∈ R and (b, c) ∈ R

We know that |a – b| and |b – c| is even

Case I- If b is even we get (a, b) ∈ R and (b, c) ∈ R

Where |a – b| and |b – c| is even

Both a and c is even

We get |a – c| is even and (a, c) ∈ R

Case II – If b is odd we get (a, b) ∈ R and (b, c) ∈ R

Where |a – b| and |b – c| is even

Both a and c is odd

We get |a – c| is even and (a, c) ∈ R

Hence, R is transitive.

Therefore, R is an equivalence relation.