CBSE BoardEnglish MediumSTD 11 ScienceMathsModel Paper 22 Marks
Question
If $A=\{a, b, c, d, e\}, B=\{a, c, e, g\}$ and $C=\{b, e, f, g\},$ verify that : $(A \cap B) \cap C=A \cap(B \cap C)$
✓
Answer
Suppose $x$ be any element of $(A \cap B) \cap C$
$\Rightarrow x \in(A \cap B) \text { } x \in C$
$\Rightarrow x \in A \text { } x \in B \text { } x \in C$
$\Rightarrow x \in A \text { } x \in(B \cap C)$
$\Rightarrow x \in A \cap(B \cap C)$
$\Rightarrow(A \cap B) \cap C \subset A \cap(B \cap C) \ldots( i )$
Now, suppose $x$ be an element of $A \cap(B \cap C)$
Then, $x \in A$ $(B \cap C)$
$\Rightarrow x \in A \text { } x \in B \text { } x \in C$
$\Rightarrow x \in(A \cap B) \text { } x \in C$
$\Rightarrow x \in(A \cap B) \cap C$
$\Rightarrow A \cap(B \cap C) \subset(A \cap B) \cap C \ldots \ldots(\text { ii) }$
Using $(i)$ $(ii),$ we have $(A \cap B) \cap C=A \cap(B \cap C)$
Hence, proved.
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