Explanation
Analysis of Statements
(a) A−(B∩C)=(A∩B)−(A∩C)
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LHS: This represents elements in A that are not in the intersection of B and C.
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RHS: Elements in both A and B, but not in the intersection of A and C.
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Verdict: Incorrect. According to De Morgan's Law for sets:
(b) A−(B∪C)=(A−B)∩(A−C)
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LHS: Elements in A that are neither in B nor in C.
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RHS: Elements that are (in A but not in B) AND (in A but not in C).
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Verdict: Correct. This is a standard De Morgan's Law property:
(c) n(A−B)=n(A)−n(A∩B)
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This is the standard formula for the cardinality of the difference of two sets.
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The set (A−B) consists of elements belonging only to A. To find this, we subtract the common elements (the intersection) from the total count of A.
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Verdict: Correct.
(d) A∩(B−C)=(A∩B)∩(A−C)
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LHS: A∩(B∩Cc)=A∩B∩Cc
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RHS: (A∩B)∩(A∩Cc)=A∩B∩Cc
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Both sides simplify to the same expression.
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Verdict: Correct. (While mathematically correct, often in exams, identities like (b) and (c) are the primary focus).
Conclusion
Based on standard set theory identities, statements (b) and (c) are the most universally recognized fundamental properties. If you have to choose a combination, (b), (c), and (d) are technically all true.
