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

Answer:

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

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

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

Step-by-step 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:

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

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

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