Consider two subsets P and Q of a universal set U. Given that n(U) = 104, n(P) = 63, n(P'∩Q) = 28 and n(P∩Q) = 16, find: (a) n(Q) (b) n(P∩Q') (c) n(P'∩Q') (d) n((P' ∩Q) ∪(P ∩ Q')) (e) n((P∩Q)')

Answer 1

This kind of problem is best approached using a Venn diagram, where the universal set U contains sets P and Q, with P and Q overlapping.

Such a Venn diagram will have four regions: (P∪Q)', P-Q, Q-P, and P∩Q. The number of items in each region can be deduced from the quantities given. For instance:

n(P-Q) = n(P) - n(P∩Q) = 63 - 16 = 47

n(Q-P) = n(P'∩Q) = 28

n((P∪Q)') = n(U) - (n(P-Q) + n(P∩Q) + n(Q-P)) = 104 - (47 + 16 + 28) = 13

From the quantities of these four regions, the size of any other region can be constructed by combining them, for example:

n(Q) = n(P∩Q) + n(Q-P) = 16 + 28 = 44