Answer:
1. n[(A U B) - C] = {23, 24, 27, 29, 33, 36, 37, 39, 41, 42, 43, 45, 47, 48, 51, 53, 54, 57, 59, 61, 63}
2. n[(A - B) U C] = n[A U C] = {6, 10, 12, 15, 20, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 60, 61}
3. D. I, II and III.
Explanation:
U = {21, 22, 23, ..., 64}
A prime number is a number that can be divided only by 1 and itself.
A = {23, 29, 31, 37, 41, 43, 47, 53, 59, 61}
B = {24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63}
C = {6, 10, 12, 15, 20, 30, 60}
1. Find n[(A U B) - C]
n(A U B) = {23, 24, 27, 29, 30, 33, 36, 37, 39, 41, 42, 43, 45, 47, 48, 51, 53, 54, 57, 59, 60, 61, 63}
n[(A U B) - C] = {23, 24, 27, 29, 30, 33, 36, 37, 39, 41, 42, 43, 45, 47, 48, 51, 53, 54, 57, 59, 60, 61, 63} - {6, 10, 12, 15, 20, 30, 60}
Since only 30 and 60 are common to n(A U B) and C, we therefore remove them it and have:
n[(A U B) - C] = {23, 24, 27, 29, 33, 36, 37, 39, 41, 42, 43, 45, 47, 48, 51, 53, 54, 57, 59, 61, 63}
2. Find n[(A - B) U C]
To get n(A - B) we remove all the elements of B in A. Since there are no common elements between A and B, we therefore have:
A - B = A = {23, 29, 31, 37, 41, 43, 47, 53, 59, 61}
C = {6, 10, 12, 15, 20, 30, 60}
Therefore, we have:
n[(A - B) U C] = n[A U C] = {6, 10, 12, 15, 20, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 60, 61}
3. Which of the following is/are true?
I. A ∩ B = A ∩ C
A ∩ B = ∅
A ∩ C = ∅
Therefore, A ∩ B = A ∩ C is true.
II. A - B = A - C
A - B = A = {23, 29, 31, 37, 41, 43, 47, 53, 59, 61}
A - C = A = {23, 29, 31, 37, 41, 43, 47, 53, 59, 61}
Therefore, A - B = A - C is true.
III. A ∩ (B ∪ C) = ∅
A = {23, 29, 31, 37, 41, 43, 47, 53, 59, 61}
B U C = {6, 10, 12, 15, 20, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63}
Therefore, A ∩ (B ∪ C) = ∅ is true.
Therefore, the correct option is D i.e. I, II and III are true.