# Set Theory, Calendars, Clocks and Binomial Theorem

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## Set Theory - De Morgan's Laws

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. ${n(A \cup B)' =n(A' \cap B')}$
ii. If ${A \cap B=0}$, then ${A' \cup B'}$ is equal to the universal set
iii. If ${A \cup B}$ = universal set, then ${A' \cap B'}$ should be the null set.
iv. If ${A \subset B}$ then ${A' \cup B'=(A \cap B)'}$
1. 1
2. 2
3. 3
4. 4

Choice D. all four statements are true.

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## Detailed Solution

Let's look at these statements one at a time:

i. ${n(A \cup B)' =n(A' \cap B')}$
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. ${(A \cup B)' =(A' \cap B')}$
2. ${(A \cap B)' =(A' \cup B')}$

ii. ${A \cap B=0}$, then ${A' \cup B'}$ is equal to the universal set
${(A \cap B)' =(A' \cup B')}$

If A and B are disjoint sets, ${A \cap B=0}$ and ${(A \cap B)'}$ is the universal set. This statement is correct.

iii. If ${A \cup B}$ = Universal Set, then ${A' \cap B'}$ should be the null set. This means ${(A \cup B)'}$ is the null set.

iv. If ${A \subset B}$, then ${A' \cup B'= (A \cap B)'}$

If ${A \subset B}$ then ${A \cap B=A}$.
This implies that ${(A \cap B)'}=A'$

If ${A \subset B}$, then ${B' \subset A'}$ and ${A' \cup B'=A'}$

Thus ${A' \cup B'= (A \cap B)'}$. This statement is true.

Correct Answer: All four statements are true.

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