Elementary Math – Set Notation – Find the Largest and Smallest value from the venn diagram.

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.

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