Question

The set $(A \cup B \cup C) \cap (A \cap B' \cap C')' \cap C'$ is equal to

• A. A $\cap$ B
• B. A $\cap$ C'
• C. B $\cap$ C'
• D. B' $\cap$ C'

Answer 1

Answer: (C)

B $\cap$ C'

Answer 2

1. Apply De Morgan's Law to $(A \cap B' \cap C')'$: $(A \cap B' \cap C')' = A' \cup B \cup C$.

2. Substitute this back into the original expression: $(A \cup B \cup C) \cap (A' \cup B \cup C) \cap C'$.

3. Simplify $(A \cup B \cup C) \cap (A' \cup B \cup C)$ using the identity $(P \cup Q) \cap (P' \cup Q) = Q$, where $P=A$ and $Q=B \cup C$. This yields $B \cup C$. The expression becomes $(B \cup C) \cap C'$.

4. Apply the distributive law: $(B \cup C) \cap C' = (B \cap C') \cup (C \cap C')$.

5. Since $C \cap C' = \emptyset$, the expression simplifies to $B \cap C'$.