To help you with this question, try constructing a Venn Diagram with two sets.

1. Set Theory

2. Set Theory
A´ is defined as the complement of A, as in, set of all elements that are part of the
universal set but not in A. How many of the following have to be true?
i. n(A  B)´ = n(A´  B´)
ii. If A  B = 0, then A´  B´ is equal to the universal set
iii. If A  B = Universal set, then A´  B´ should be the null set
iv. If A ⊂B, then A´  B´ = (A  B)´
(a) 1 (b) 2
(c) 3 (d) 4

3. Set Theory
De Morgan’s Laws:
IIII II
A B
IV
Region IV:
• (A U B)' = A' ∩ B‘
Regions I, III and IV:
• (A ∩ B)' = A' U B'

4. Set Theory
Let us look at these statements one at a time.
(a) n(A  B)´ = n(A´  B´) -- This is a direct statement of De Morgan´s law. This is
definitely true. Just to recap, De Morgan´s laws are as follows:
(A  B)´ = A´  B´
(A  B) ´ = A ´  B ´
(b) If A  B = 0, then A´  B´ is equal to the universal set.
(A  B)´ = A ´  B ´
If A and B are disjoint sets, AB = 0 and (A  B)´ = Universal set. So, Statement b is also
true.

5. Set Theory
(c) If A  B = Universal set, then A´  B´ should be the null set.
A  B = Universal set => (A  B )´ = Null set.
(d) If A ⊂ B, then A´  B´ = (A  B)´
If A ⊂ B, then A  B = A => (A  B)´ = A´
If A ⊂ B, then B’ ⊂ A’ and A´  B´ = A´
=> A´  B´ = (A  B)’
All 4 statements are true
Answer choice (d)

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