The problem asks to show that if $A$, $B$, and $C$ are sets, then the following equality holds:

$A \cap (B \cap C)' = A' \cup (B' \cup C')$

Part (a): Prove this by showing that each side is a subset of the other.

We'll do this in two parts: first, prove that $A \cap (B \cap C)' \subseteq A' \cup (B' \cup C')$, and then prove that $A' \cup (B' \cup C') \subseteq A \cap (B \cap C)'$.

Proof 1: $A \cap (B \cap C)' \subseteq A' \cup (B' \cup C')$

1. Let $x \in A \cap (B \cap C)'$. This means:

   • $x \in A$, and
   • $x \in (B \cap C)'$, i.e., $x \notin B \cap C$.

2. If $x \notin B \cap C$, then $x \notin B$ or $x \notin C$, meaning:

   • $x \in B' \cup C'$.

3. Thus, $x \in A$ and $x \in (B' \cup C')$. Therefore, we must show that $x \in A' \cup (B' \cup C')$.

4. Since $x \in A$, this implies $x \notin A'$, so $x$ belongs to $(B' \cup C')$.

5. Therefore, $x \in A' \cup (B' \cup C')$.

Thus, $A \cap (B \cap C)' \subseteq A' \cup (B' \cup C')$.

Proof 2: $A' \cup (B' \cup C') \subseteq A \cap (B \cap C)'$

1. Let $x \in A' \cup (B' \cup C')$. This means:

   • Either $x \in A'$, or
   • $x \in (B' \cup C')$.

2. If $x \in A'$, then $x \notin A$, so $x \notin A \cap (B \cap C)$. Hence, $x \in (B \cap C)'$, which means $x \in A \cap (B \cap C)'$.

3. If $x \in (B' \cup C')$, this implies $x \notin B$ or $x \notin C$, so $x \notin B \cap C$, which implies $x \in (B \cap C)'$.

4. Thus, $x \in A \cap (B \cap C)'$.

Therefore, $A' \cup (B' \cup C') \subseteq A \cap (B \cap C)'$.

Conclusion

Since we have shown both $A \cap (B \cap C)' \subseteq A' \cup (B' \cup C')$ and $A' \cup (B' \cup C') \subseteq A \cap (B \cap C)'$, we conclude that:

$A \cap (B \cap C)' = A' \cup (B' \cup C')$

Would you like more details or have any questions on this proof?

Here are 5 related questions to expand on this:

1. How do you prove other set identities, like De Morgan's laws, using subset arguments?
2. Can you provide an example with specific sets to illustrate the equality?
3. How do you create a membership table for this proof?
4. What other techniques are useful in proving set theory identities?
5. How would the proof change if $A$, $B$, or $C$ were empty?

Tip: When proving set equalities, breaking the problem into two parts — showing mutual subset relations — is a reliable approach to establish equality.