### a. The smallest value of n(z) and the corresponding value of n(x).

We need to find out what is the least number of elements of z in the venn diagram. As no elements are given ,use the size of the area a a proxy to represent the number of elements. So in this case, the smallest value of n(z) is when its area is the smallest. Do note that area z is the region in the venn diagram that is complement (outside) of P and of Q. In this case, when the area of P and Q does not intersect ( x=0), and when both the area occupies the biggest region in the venn diagram. the size of n(z) is the smallest.

n(ε) - n(P) -n(Q) = 130-37-50 = 43

n(z) = 43 and n(x) = 0.

### b. The largest value of n(z) and the corresponding value of n(x).

The largest number of elements in z occurs when area z in the venn diagram is the largest (please remember that z is the complement of P and Q), and the area of P and Q is at its smallest. This can happen if the area of P and Q overlap each other and P becomes a proper subset of Q. All the elements in P is also the elements of Q (x=w) but some elements of Q are not in P.

n(ε) -n(Q) = 130-50 = 80

n(z) = 80 and n(x) = 37.