Question

A universal set U consists of elements. If sets​ A, B, and C are proper subsets of U and ​n(U)​, ​n(A ​B)​n(A ​C)​n(B ​C)​, ​n(A B ​C)​, and​ n(A B ​C)​, determine each of the following.

a. n(AUB)
b. n(A'UC)
c. n(AnB)'

Answer 1

`n(A\ u\ B) = 10`

`n(A'\ u\ C) = 10`

`n(A\ n\ B)' = 6`

Explanation:

Given

`n(U) = 12`

`n(A\ n\ B) =n(A\ n\ C) = n(B\ n\ C) =6`

`n(A\ n\ B\ n\ C)=4`

`n(A\ u\ B\ u\ C)=10`

Required

Solve a, b and c

There are several ways to solve this; the best is by using Venn diagram (see attachment for diagram)

Solving (a):

`n(A\ u\ B)`

This is calculated as:

`n(A\ u\ B) = n(A) + n(B) - n(A\ n\ B)`

From the attachment

`n(A) = 0+2+4+2 = 8`

`n(B) = 0+2+4+2 = 8`

`n(A\ n\ B) = 4 +2 = 6`

So:

`n(A\ u\ B) = 8 + 8 - 6`

`n(A\ u\ B) = 10`

Solving (b):

`n(A'\ u\ C)`

This is calculated as:

`n(A'\ u\ C) = n(A') + n(C) -n(A'\ n\ C)`

From the attachment

`n(A) = n(U) - n(A) = 12 - 8 = 4`

`n(C) = 0+2+4+2 = 8`

`n(A'\ n\ C) = 2`

So:

`n(A'\ u\ C) = 4 + 8 - 2`

`n(A'\ u\ C) = 10`

Solving (c):

`n(A\ n\ B)'`

This is calculated as:

`n(A\ n\ B)' = n(U) - n(A\ n\ B)`

`n(A\ n\ B)' = 12- 6`

`n(A\ n\ B)' = 6`