The question is from CAT Set theory. This is about De Morgan's Laws. We need to verify the given statements. CAT exam is known to test on basics rather than high funda ideas. CAT also tests multiple ideas in the same question, and 2IIMs CAT question bank provides you with CAT questions that can help you gear for CAT Exam CAT 2020. Set Theory (especially constructing venn diagrams) is a frequently tested topic. Make sure you know the basics from this chapter.
Question 2: 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)'
Let's look at these statements one at a time:
i. 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:
1. (A ∪ B)' =(A' ∩ B')
2. (A ∩ B)' =(A' ∪ B')
ii. A ∩ B=0, then A' ∪ B' is equal to the universal set
(A ∩ B)' =(A' ∪ B')
If A and B are disjoint sets, A ∩ B=0 and (A ∩ B)' is the universal set. This statement is correct.
iii. If A ∪ B = Universal Set, then A' ∩ B' should be the null set. This means (A ∪ B)' is the null set.
iv. If A ⊂ B, then A' ∪ B'= (A ∩ B)'
If A ⊂ B then A ∩ B=A.
This implies that (A ∩ B)'=A'
If A ⊂ B, then B' ⊂ A' and A' ∪ B'=A'
Thus A' ∪ B'= (A ∩ B)'. This statement is true.
The question is "How many of the following have to be true?"
Choice D is the correct answer.
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