Question

Show that if Rin A = {XEN/x < 10) given by R = {(a, b) a=b} is an equivalence
relation.​

Answer 1

A = { x ∈ Z : 0 ≤ x ≤ 12} = {0,1,2,3,4,5,6,7,8,9,10,11,12}

(i) R = { (a,b) : |a - b| is a multiple of 4}

For any element a ∈A, we have (a, a) ∈ R as |a - a = 0|is a multiple of 4.

∴R is reflexive.

Now, let (a, b) ∈ R ⇒ |a - b| is a multiple of 4.

⇒ |-(a - b)| = ⇒ |b - a| is a multiple of 4.

⇒ (b, a) ∈ R

∴R is symmetric.

Now, let (a, b), (b, c) ∈ R.

⇒ |(a - b)| is a multiple of 4 and |(b - c)| is a multiple of