Let A, B and C be sets such that phi&nbs | vedclass

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Question

For $A\, = \,C,\,A - C\, = \,\phi $

$ \Rightarrow \phi \, \subseteq \,B$

But $A\, \subseteq \,B$

$ \Rightarrow \,$ option $A$ is NO

Vedclass · Accepted Answer

If $\left( {A - C} \right) \subseteq B$ then $A \subseteq B$

Vedclass · Answer

For $A\, = \,C,\,A - C\, = \,\phi $

$ \Rightarrow \phi \, \subseteq \,B$

But $A\, \subseteq \,B$

$ \Rightarrow \,$ option $A$ is NOT true

Let $x\, \in \,\,(C\,x\, \in \,(C\, \cup \,A)\,\, \cap (C\, \cup \,B)\,)$

$ \Rightarrow \,x\,(C\, \cup \,A)$ 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 $x\, \in \,\,(A\, \cap \,B)$

$ \Rightarrow \,(x\, \in \,C$   or  $x\, \in \,C$  (as $A\, \cup \,B \subseteq \,C\,$ )

$ \Rightarrow \,x\, \in \,C$

$ \Rightarrow (C\, \cup \,A)\,\, \cap (C\, \cup \,B)\, \subseteq \,C\,\,\,(1)$

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)\, \subseteq \,C\,\,\,(2)$

$ \Rightarrow $ from $(1)$ and $(2)$

$C\, = \,(C\, \cup \,A)\,\, \cap (C\, \cup \,B)$

$ \Rightarrow $ option $B$ is true

Let $x\, \in \,A$ and $x\, 
otin \,B$

$\Rightarrow \,x\, \in \,(A - B)$

$ \Rightarrow \,x\, \in \,C$ (as $A - B \subseteq \,C$ )

Let $x\, \in \,A$ and $x\, \in \,B$

$ \Rightarrow \,x\, \in \,(A \cap B)$

$ \Rightarrow \,x\, \in \,C$ (as $A \cap B \subseteq \,C$ )

Hence $x\, \in \,A\,\, \Rightarrow \,x\, \in \,C$

$ \Rightarrow A\, \subseteq \,C$

$ \Rightarrow $ option $C$ is true

As $C \supseteq \,(A \cap B)$

$ \Rightarrow B \cap C \supseteq \,(A \cap B)$

As $A \cap B 
e \phi $

$ \Rightarrow B \cap C 
e \phi $

Hence the correct answer is option $(A)$

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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