Question

Asked in: JEE Main-2019

Let A , B and C be sets such that 

\phi \neq A\cap B\subseteq C.  Then which of the following 

statements is not true ? 

A.

B\cap C\neq \phi

B.

If\: \: (A-B)\subseteq C, then\: \: A\subseteq C

C.

(C\cup A)\cap(C\cup B)=C

D.

If\: \: (A-C)\subseteq B, then\: \: A\subseteq B

Answers (2)

 

A. For A = C, A - C = \Phi

\Rightarrow \Phi \subseteq B 
But A\nsubseteq B 
Therefore, option A. is not true 
Let 

\\ x \in (Cx\in(C\cupA)\cap(C\cupB)) \\ \Rightarrow x\in(C \cupA)and x \in(C\cup B) \\ \Rightarrow (x\in C \ or\ x \in A) and (x\in C \ or\ x \in B) \\ \Rightarrow x \in C \ or\ x \in (A\capB) \\ \Rightarrow x \in C \ or\ x \in C \\ \Rightarrow x\in C \\ \Rightarrow (C\cup A) \cap (C\cupB)\subseteq C
Now  

\\ x\in C \\ \Rightarrow x\in(C\cup A)\ and\ x \in(C\cup B) \\ \Rightarrow x\in(C\cup A)\cap (C\cup B) \\ \Rightarrow C \subseteq (C\cup A)\cap (C\cup B) 
Therefore,

C=(C\cup A)\cap (C\cup B) 
Therefore, option B. is true

 

A. For A = C, A - C = \Phi

\Rightarrow \Phi \subseteq B 
But A\nsubseteq B 
Therefore, option A. is not true 
B. Let 

\\ x \in (Cx\in(C\cupA)\cap(C\cupB)) \\ \Rightarrow x\in(C \cupA)and x \in(C\cup B) \\ \Rightarrow (x\in C \ or\ x \in A) and (x\in C \ or\ x \in B) \\ \Rightarrow x \in C \ or\ x \in (A\capB) \\ \Rightarrow x \in C \ or\ x \in C \\ \Rightarrow x\in C \\ \Rightarrow (C\cup A) \cap (C\cupB)\subseteq C
Now  

\\ x\in C \\ \Rightarrow x\in(C\cup A)\ and\ x \in(C\cup B) \\ \Rightarrow x\in(C\cup A)\cap (C\cup B) \\ \Rightarrow C \subseteq (C\cup A)\cap (C\cup B) 
Therefore,

C=(C\cup A)\cap (C\cup B) 
Therefore, option B. is true

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