Question

Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b) : a, b ∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]

Answer 1

A={0,1,2,3,4,5,6,7,8,9,10,11,12}
R={(a,b):a,b ∈ Z, |a−b| is divisible by 4}
For reflexive,
for every a ∈ A
|a−a| = 0 which is divisible by 4
then (a,a) ∈ R
Hence, it is reflexive.
For symmetric

If (a,b) ∈ R then (b,a) ∈ R
|a−b| = |b−a|
Hence, it is symmetric.
For transitive

If (a,b) ∈ R ⇒ |a−b| is divisible by 4 (Say |a−b|=4k₁ ⇒ a−b = ±4k₁)

and (b,c) ∈ R ⇒|b−c| is divisible by 4 (Say |b−c| = 4k₂  ⇒ b−c = ±4k₂)

∴|a−c|=|±4k₁ ±  4k₂| which is divisible by 4

then (a,c) ∈ R

Hence, it is transitive.
Also, the relation is the equivalence.

Set of elements related to 1 is {(1,1),(1,5),(1,9),(5,1),(9,1)}

Let (x,2) ∈ R; (x ∈ A)

|x−2|= 4k (k is whole number, k≤3)

∴ x=2,6,10

Equivalence class [2] is {2,6,10}