Let A x Z 0 x 10 0 1 2 3 4 5 6 7 8 9 10 Define the relati

Let A = {x Z : 0 x 10} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define the relation C on A by C = {(x, y) A × A : 3|(x y)}. That is, xCy if and only if x y is divisible by 3. This is an equivalence relation. (You do not need to prove this.) Identify all of the equivalence classes of this relation.

Solution

x-y is divisible by 3 means x and y give same remainder modulo 3

So one equivalence is set of all numbers giving same remainder modulo 3

THere are 3 possible remainders modulo 3:0,1 and 2

So we have 3 equivalence classes

Equivalence class corresponding to remainder 0 has elements

{0,3,6,9}

Equivalence class corresponding to remainder 1 has elements

{1,4,7,10}

Equivalence class corresponding to remainder 2 has elements

{2,5,8}