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If $A, B$ and $C$ be sets. Then show that $A \cap(B \cup C)$ $=(A \cap B) \cup(A \cap C)$.
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Verified Answer
Let $x \in A \cap(B \cup C)$
$\Rightarrow x \in A$ and $x \in(B \cup C)$
$\Rightarrow x \in A$ and $(x \in B$ or $x \in C)$
$\Rightarrow(x \in A$ and $x \in B)$ or $(x \in A$ and $x \in C)$
$\Rightarrow x \in A \cap B$ or $x \in A \cap C$ $x \in(A \cap B) \cup(A \cap C)$
$\Rightarrow A \cap(B \cup C) \subset(A \cap B) \cup(A \cap C) \ldots(\mathrm{i})$
Again, let $y \in(A \cap B) \cup(A \cap C)$
$\Rightarrow y \in(A \cap B)$ or $y \in(A \cap C)$
$\Rightarrow(y \in A$ and $y \in B) \operatorname{or}(y \in A$ and $y \in C)$
$\Rightarrow y \in A$ and $(y \in B$ or $y \in C)$
$\Rightarrow y \in A$ and $y \in(B \cup C)$
$\Rightarrow y \in A \cap(B \cap C)$
$\Rightarrow(A \cap B) \cup(A \cap C) \subset A \cap(B \cup C) \quad \ldots$ (ii)
From Eqs. (i) and (ii),
$A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$.
$\Rightarrow x \in A$ and $x \in(B \cup C)$
$\Rightarrow x \in A$ and $(x \in B$ or $x \in C)$
$\Rightarrow(x \in A$ and $x \in B)$ or $(x \in A$ and $x \in C)$
$\Rightarrow x \in A \cap B$ or $x \in A \cap C$ $x \in(A \cap B) \cup(A \cap C)$
$\Rightarrow A \cap(B \cup C) \subset(A \cap B) \cup(A \cap C) \ldots(\mathrm{i})$
Again, let $y \in(A \cap B) \cup(A \cap C)$
$\Rightarrow y \in(A \cap B)$ or $y \in(A \cap C)$
$\Rightarrow(y \in A$ and $y \in B) \operatorname{or}(y \in A$ and $y \in C)$
$\Rightarrow y \in A$ and $(y \in B$ or $y \in C)$
$\Rightarrow y \in A$ and $y \in(B \cup C)$
$\Rightarrow y \in A \cap(B \cap C)$
$\Rightarrow(A \cap B) \cup(A \cap C) \subset A \cap(B \cup C) \quad \ldots$ (ii)
From Eqs. (i) and (ii),
$A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$.
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