if u have A1={1,2,3} A2={4} A3={5,6,7} A4={8,9}
NOW r={A1XA1,A2XA2,A3XA3,A4XA4}
USE CARTESIAN MULTIPLICATION OF SETS YOU GET
R={((1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2 ),(3,3))U(4,4)U((5,5),(5,6),(5,7),(6,5),(6,6),(6,7 ),(7,5),(7,6),(7,7))U((8,8),(8,9),(9,8),(9,9))
NOW TAKE UNION AND U WILL FIND
R(a,b) has (a,b) belonging to A set
so A can be partionitioned in above way and the relation is defined
hence proved