Question:
Let A and B be subsets of X and \(C = (A \cap B’) \cup (A’ \cap B)\), where A’ and B’ are complements of A and B respectively in X. What is C equal to?
मान लीजिए A और B, X के उपसमुच्चय हैं तथा \(C = (A \cap B’) \cup (A’ \cap B)\), जहाँ A’ और B’ क्रमशः X में A और B के पूरक समुच्चय हैं। C किसके बराबर है?
Options:
- \((A \cup B)’ – (A \cap B)\)
- \((A’ \cup B) – (A \cap B)\)
- \((A \cup B) – (A \cap B)\)
- \((A’ \cup B’) – (A’ \cap B’)\)
- \((A \cup B)’ – (A \cap B)\)
- \((A’ \cup B) – (A \cap B)\)
- \((A \cup B) – (A \cap B)\)
- \((A’ \cup B’) – (A’ \cap B’)\)
Correct Answer: c)
Explanation:
By definition of set difference:
\(A \cap B’ = A – B\) and \(A’ \cap B = B – A\).
Their union is the symmetric difference:
\[C = (A – B) \cup (B – A) = (A \cup B) – (A \cap B)\]
Hence, the correct option is (c).
समुच्चय अंतर की परिभाषा से:
\(A \cap B’ = A – B\) तथा \(A’ \cap B = B – A\)।
इनका सम्मिलन (union) सममित अंतर कहलाता है:
\[C = (A – B) \cup (B – A) = (A \cup B) – (A \cap B)\]
अतः सही विकल्प (c) है।