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)