If A and B are sets, then prove that A−B,


If $A$ and $B$ are sets, then prove that $A-B, \mid A \cap B$ and $B-A$ are pair wise disjoint.


(i) $(A-B)$ and $(A \cap B)$

Let $a \in A-B$

$\Rightarrow a \in \mathrm{A}$ and $a \notin B$

$\Rightarrow a \notin A \cap B$

Hence, $(A-B)$ and $A \cap B$ are disjoint sets.

(ii) $(B-A)$ and $(A \cap B)$

Let $a \in B-A$

$\Rightarrow a \in B$ and $a \notin A$

$\Rightarrow a \notin \mathrm{A} \cap \mathrm{B}$

Hence, $(B-A)$ and $A \cap B$ a re disjoint sets.

(iii) $(A-B)$ and $(B-A)$

$(A-B)=\{x: x \in A$ and $x \notin B\}$

$(B-A)=\{x: x \in B$ and $x \notin A\}$

Hence, $(A-B)$ and $(B-A)$ are disjoint sets.

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