Question

If A, B and C are any three sets, then $A-(B \cup C)$ is equal to

• A. $(A-B) \cap C$
• B. $(A-B) \cup C$
• C. $(A-B) \cap (A-C)$
• D. $(A-B) \cup (A-C)$

Answer 1

Answer: (C)

$(A-B) \cap (A-C)$

Answer 2

The expression $A-(B \cup C)$ means elements that are in set A but not in the union of sets B and C. Using set identities:

1. $A-(B \cup C)$ is equivalent to $A \cap (B \cup C)'$ by the definition of set difference.
2. By De Morgan's Law, $(B \cup C)'$ is equivalent to $B' \cap C'$.
3. Substituting this into the expression, we get $A \cap (B' \cap C')$.
4. This can be rewritten as $(A \cap B') \cap (A \cap C')$, using the associative and idempotent properties of intersection ($A \cap A = A$).
5. Finally, converting back to set difference notation, $A \cap B'$ is $A-B$, and $A \cap C'$ is $A-C$.
6. Therefore, $(A \cap B') \cap (A \cap C')$ becomes $(A-B) \cap (A-C)$.