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Question
If A and B are sets, then prove that \[A - B, A \cap B \text{ and } B - A\] are pair wise disjoint.
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Solution
\[\left( i \right) \left( A - B \right) \text{ 0and } \left( A \cap B \right)\]
\[\text{ Let } a \in A - B\]
\[ \Rightarrow a \in A \text{ and } a \not\in B\]
\[ \Rightarrow a \not\in A \cap B\]
\[\text{ Hence }, \left( A - B \right) \text{ and } A \cap B \text{ are disjoint sets } . \]
\[\left( ii \right) \left( B - A \right) and \left( A \cap B \right)\]
\[\text{ Let } a \in B - A\]
\[ \Rightarrow a \in B \text{ and } a \not\in A\]
\[ \Rightarrow a \not\in A \cap B\]
\[\text{ Hence }, \left( B - A \right) \text{ and } A \cap B \text{ are disjoint sets } . \]
\[\left( iii \right) \left( A - B \right) \text{ and } \left( B - A \right)\]
\[\left( A - B \right) = \left\{ x: x \in A \text{ and }x \not\in B \right\}\]
\[\left( B - A \right) = \left\{ x: x \in B \text{ and } x \not\in A \right\}\]
\[Hence, \left( A - B \right) \text{ and } \left( B - A \right) \text{ are disjoint sets } . \]
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