Let U be the universal set and A, B, C, the subsets of U such that
U= {x|x £ Z; 0<x<=30}
A= {U{all primes}}, B= {all multiples of 3}, C= {1, 2, 5, 8, 9, 12, 15} . . .
Using the above information AU(B/C) and (B\C) n (AnA^c)
First, let's find A ∪ (B/C):
A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} (All prime numbers less than or equal to 30)
B = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30} (All multiples of 3 less than or equal to 30)
C = {1, 2, 5, 8, 9, 12, 15}
B/C = {3, 6, 18, 21, 24, 27, 30} (All multiples of 3 that are not in C)
A ∪ (B/C) = {2, 3, 5, 6, 7, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 27, 29, 30}
Next, let's find (B\C) ∩ (A ∩ A^c):
A^c = {0, 1, 4, 6, 8, 10, 14, 16, 20, 22, 25, 26, 28, 30} (Complement of A)
B\C = {6, 18, 21, 24, 27} (All multiples of 3 not in C)
(B\C) ∩ (A ∩ A^c) = {6, 18, 21, 24, 27} ∩ {6, 18, 21, 24, 27} = {6, 18, 21, 24, 27}
Therefore, AU(B/C) = {2, 3, 5, 6, 7, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 27, 29, 30}
And (B\C) ∩ (A ∩ A^c) = {6, 18, 21, 24, 27}.