1. Tardigrade
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  3. Mathematics
  4. Let A, B and C be sets such that Φ ≠ A ∩ B ⊆ C,. Then which of the following statements is not true?

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Q. 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 MainJEE Main 2019Sets

Solution:

For $A = C, A - C = \Phi$
$\Rightarrow \ \Phi \ \subseteq \ B$
But $A \nsubseteq B$
$\Rightarrow \ $ option 1 is NOT true
$Let \ x \ \in \ (C x \in (C \cup A) \cap(C \cup B))$
$\Rightarrow \ \ x \in (C \ \cup A) and \ x \ \in (C \cup \ B)$
$\Rightarrow \ (x \in C \ or \ x \ \in A ) \ and \ (x \in C \ or \ x \ \in B)$
$\Rightarrow \ \ x \ \in \ C \ or \ x \ \in \ (A \cap B)$
$\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)$
$\Rightarrow \ \ C \ \subseteq (C \cup A) \cap (C \cup B) \ \ \ \ \ \ \ \ \ \ (2)$
$\Rightarrow \ \ from (1) and (2)$
$ \ \ \ \ \ \ \ \ \ C = (C \cup A) \cap (C \cup B)$
$\Rightarrow \ \ $ option 2 is true
Let x $\in $A and $x \notin B$
$ \ \ \ \ \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 3 is true
as $ \ \ \ \ \ \ \ \ C \supseteq (A \cap B)$
$ \ \ \ \ \ \Rightarrow \ \ B \cap C \supseteq (A \cap B)$
as $ \ \ \ \ \ \ \ \ \ \ \ \ A \cap B \le \Phi$
$ \ \ \ \ \ \Rightarrow \ \ \ \ B \cap C \le \Phi$
$\Rightarrow \ \ $ Option 4 is true

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