Let A and B two sets such that A∩X=B∩X=φ and A∪X=B∩X for some set X To show: A=B A=A∩(A∪X) =A∩(A∪X)(A∪X=B∪X) =(A∩B)∪(A∩X) (Distributive law) =(A∩B)∪φ(∵A∩X=φ) =A∩B.....(i) Now, B=B∩(B∪X) =B∩(A∪X)(∵A∪X=B∪X) =(B∩A)∪(B∩X)(Distributive law) =(B∩A)∪φ(∵B∩X=φ) =B∩A=A∩B...(ii) Hence, form (i) and (ii), we get, A=B.