For any sets A and B, prove that:

(i) (A – B) B = ϕ


(ii) A (B – A) = A B


(iii) (A – B) (A B) = A


(iv) (A B) – B = A – B


(iv) A – (A B) = A – B


Two sets are shown with the following Venn Diagram


The yellow region is denoted by 1.


Blue region is denoted by 2.


The common region is denoted by 3.



(i) A - B denotes region 1


B denotes region (2+3)


So their intersection is a null set


(A - B)B = ø


(ii) B - A denotes region 2


A denotes region (1+3)


So their union denotes region (1+2+3) which is the union of A and B


A(B - A) = AB


(iii) A - B denotes region 1
A
B denotes region 3
Their union denotes region (1+3) which is set A
(A - B)(AB) = A


(iv) AB denotes region (1+2+3)
(AUB) - B denotes region (1+2+3) - (2+3) = 1
A - B denotes region 1
(AB) - B = A - B


(v) Wrong question


1