1. a) Show that if A and B are sets, then A ? B = B ? A
b) Show that if A and B are sets, then (A ? B) ? B = A.
where ? is symmetric difference
Solution
A> LHS=A ? B=A U B - AB=(A+B-AB)-AB=A+B-2AB .......1 (where AB is the intersection of
AB)
NOW,
RHS=B ? A=B U A +AB=(B+A-AB)-AB=A+B-2AB=LHS(proved)
b> LHS=(A?B)?B=(A+B-2AB)?B (Put the expression of A?B)
=A+B-2AB+B-2(A+B-2AB)B .....(IN EQN 1 PUT A= A+B-2AB)
=A+2B-2AB-2AB-2B+4AB=A (PROVED)