Q5
With the help of venn diagrams,verify the two distributive properties in the following cases w.r.t union and intersection.

(i) A⊆B,A∩C=Φ and B and c are overlapping

Solution:

Distributivity of union over intersection:

In Fig 1, the shaded regions represents A and B∩C, so the shaded region represents AU(B∩C).

In fig (ii) the horizontal lines represent AUB, the vertical lines represent AUC and the rectangles represent (AUB)∩(AUC).It is clear from both fig that the shaded region in fig (i) is equal to the region formed by the rectangles in fig (ii). So

AU(B∩C)=(AUB)∩(AUC)

Distibutivity of intersection over union:

In fig (i) the shaded regions represent A∩B.Since A∩C={ },so there is no region to represent A∩c in fig (i).

in fig (ii), the horizontal lines represent BUC, the vertical lines represents A and the rectangles represent A∩(BUC). it is clear from both figures that the shaded region in fig (i) is equal to the region formed by the rectangles in fig(ii), So

A∩(BUC)=(A∩B)U(A∩C)

(ii) A and b are overlapping , B and c are overlapping but A and C are disjoint.

Solution:

Distributivity of union over intersection:

In fig (i) the shaded regions represents A and B∩C,So; as a whole it represents AU(B∩C).

In Fig(ii) ,the horizontal lines represents AUB and vertical lines represent AUC.thus the regions of rectangles represents (AUB)∩(AUC).So

AU(B∩C)=(AUB)∩(AUC)

Distributivity of intersection over union:

in fig (i) horizontal lines represent BUC and area of rectangles represents A∩(BUC).

In fig (ii), there is no region to represent A∩C; and region or horizontal lines represent A∩B and alos; (A∩B)U(A∩C). So

A∩(BUC)=(A∩B)U(A∩C)