Question

A universal set U consists of 16 elements. If sets​ A, B, and C are proper subsets of U and ​n(U) = 16​, ​n(A ∩ ​B) = ​n(A ∩ ​C) = ​n(B ∩ ​C) = 6​, ​n(A ∩ B ∩ ​C) = 2​, and​ n(A ∪ B ∪ ​C) = 14​, determine each of the following. ​a)​ n(A ∪ ​B) ​b) n (A' ∪ C )​c) n (A ∩ B )'

Answer 1

a) 14

b) 12

c) 10

Explanation:

The easiest way to do this is with a Venn diagram. So we put a 2 in the intersection between A,B AND C (​n(A ∩ B ∩ ​C) = 2​). And we are going to called that region as V.

​n(A ∩ ​B) = 6​, so the sum of the region (that we can see in the picture) V and ii should be 6. V+ii= 6; ii= 4

The same happen with n(A ∩ ​C) = ​n(B ∩ ​C). Region iv and vi =4

n(A ∪ B ∪ ​C) = 14​ and the sum of all the other region is already 14. So the region i, iii and vii are going to be 0. But n(U)=16, so there will be another region, called viii=2 (16-14)

n(A U B)=n(A)+n(B)-n(A ∩ B) --> region (A)= i, ii, v, iv + region (B) = iii, ii, v, vi- region (A ∩ B)= v, ii= 4+2+4+4+2+4-6=14

n(A' U C)--> regions of n(A')= region vi, iii, vii, viii + regions of n(C)= region vii, vi, v, iv= region iii, iv, v, vi, vii, viii=0+4+2+4+0+2=12

n(A ∩ B)'=n(U)-n(A ∩ B)--> n(U)=region i, ii, iii, iv, v, vi, vii, viii-n(A ∩ B)=region v, ii=16-6=10