question_answer

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? [JEE Main 12-4-2019 Afternoon]

A) If\[(A-C)\subseteq B,\]then\[A\subseteq B\]

B) \[\left( C\cup A \right)\cap \left( C\cup B \right)=C\]

C) If\[\left( A-B \right)\subseteq C,\]then\[A\subseteq C\]

D) \[B\cap C\ne \phi \]

Correct Answer: A

Solution :

for\[A=C,A-C=\phi \]\[\Rightarrow \phi \subseteq B\] But\[AB\] \[\Rightarrow \]option is NOT true Let\[x\in (Cx\in \left( C\cup A \right)\cap \left( C\cup B \right)\] \[\Rightarrow x\in \left( C\cup A \right)\,\,\,\,and\,\,\,\,x\in \left( C\cup B \right)\] \[\Rightarrow \left( x\in C\,\text{or}\,x\in A \right)\]and\[\left( x\in C\,\text{or}\,x\in B \right)\] \[x\in C\,\,\,\,or\,\,\,\,x\in (A\cap B)\] \[\Rightarrow x\in C\,or\,x\in C\]\[(as\,A\cup B\subseteq C)\] \[\Rightarrow x\in C\] \[\Rightarrow \left( C\cup A \right)\cap \left( C\cup B \right)\subseteq C\]                    (1) Now\[x\in C\Rightarrow x\in \left( C\cup A \right)\]and\[x\in \left( C\cup B \right)\] \[\Rightarrow x\in \left( C\cup A \right)\cap \left( C\cup B \right)\] \[\Rightarrow C\subseteq \left( C\cup A \right)\cap \left( C\cup B \right)\]                    (2) \[\Rightarrow \]from (1) and (2)             \[C=\left( C\cup A \right)\cap \left( C\cup B \right)\] \[\Rightarrow \]option is true Let \[x\in A\]and\[xB\] \[\Rightarrow \]\[x\in (A-B)\]\[\Rightarrow \]\[x\in C\]\[(as\,A-B\subseteq C)\] Let\[x\in A\]and\[x\in B\] \[\Rightarrow \]\[x\in \left( A\cap B \right)\]\[\Rightarrow \]\[x\in C\]\[(as\,A\cap B\subseteq C)\] Hence \[x\in A\Rightarrow x\in C\]\[\Rightarrow A\subseteq C\] \[\Rightarrow \] Option is true as         \[C\supseteq \left( A\cap B \right)\] \[\Rightarrow B\cap C\supseteq \left( A\cap B \right)\] as\[A\cap B\ne \phi \] \[\Rightarrow B\cap C\ne \phi \] \[\Rightarrow \]Option [d] is true.