Question

If A and B are sets:

i. Prove that (A ∩ B)' = (A' ∪ B)'
ii. Simplify as much as possible [(A ∩ B') ∪ (A' ∩ B')]'​

Answer 1

Applying De Morgan's laws to sets, (A ∩ B)' equals A' ∪ B'. Simplifying [(A ∩ B') ∪ (A' ∩ B')]' also involves De Morgan's laws, resulting in the simplified form A ∪ B.

Step-by-step explanation:

To address the student's question about sets, we begin by verifying the first claim which is the law of complementation: (A ∩ B)' = A' ∪ B'.

According to De Morgan's laws, the complement of the intersection of two sets is equal to the union of the complements of those sets. This means that if an element is not in both A and B, then it must be either not in A or not in B. Conversely, if an element is not in A or it is not in B, it cannot be in both A and B.

Now, for simplifying [(A ∩ B') ∪ (A' ∩ B')]', we apply De Morgan's laws again:

1. Complement the terms inside the bracket: [(A' ∪ B) ∩ (A ∪ B)].
2. Simplify the expression using the laws of sets: A' ∪ (A ∪ B) simplifies to A ∪ B since everything outside of A combined with A itself covers the entire space.

Therefore, the simplified form of [(A ∩ B') ∪ (A' ∩ B')]' is A ∪ B. This shows the power of De Morgan's laws in set theory and simplification of set expressions.