Question:
Let A, B and C be sets such that \(\phi \ne A \cap B \subseteq C\). Then which of the following statements is not true ?
Let A, B and C be sets such that \(\phi \ne A \cap B \subseteq C\). Then which of the following statements is not true ?
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Check each statement using \(A = (A-B)\cup(A\cap B)\) and distributive laws; find a counterexample for the false one.
Updated On: Nov 11, 2026
- If \((A - B) \subseteq C\), then \(A \subseteq C\)
- \(B \cap C \ne \phi\)
- \((C \cup A) \cap (C \cup B) = C\)
- If \((A - C) \subseteq B\), then \(A \subseteq B\)
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The Correct Option is D
Solution and Explanation
(1) \(A = (A-B)\cup(A\cap B)\); both parts are in C, so \(A\subseteq C\). True.
(2) \(B\cap C \supseteq B\cap(A\cap B) = A\cap B \ne \phi\). True.
(3) \((C\cup A)\cap(C\cup B) = C\cup(A\cap B) = C\) since \(A\cap B\subseteq C\). True.
(4) Take \(A=\{1,2\}\), \(B=\{1\}\), \(C=\{1,2\}\). Then \(A\cap B = \{1\}\subseteq C\), \(A - C = \phi \subseteq B\), but \(A \not\subseteq B\). So statement (4) is not true.
Answer: (4) If \((A - C) \subseteq B\), then \(A \subseteq B\)
(2) \(B\cap C \supseteq B\cap(A\cap B) = A\cap B \ne \phi\). True.
(3) \((C\cup A)\cap(C\cup B) = C\cup(A\cap B) = C\) since \(A\cap B\subseteq C\). True.
(4) Take \(A=\{1,2\}\), \(B=\{1\}\), \(C=\{1,2\}\). Then \(A\cap B = \{1\}\subseteq C\), \(A - C = \phi \subseteq B\), but \(A \not\subseteq B\). So statement (4) is not true.
Answer: (4) If \((A - C) \subseteq B\), then \(A \subseteq B\)
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