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प्रश्न
For any two sets A and B, prove the following:
\[A - \left( A - B \right) = A \cap B\]
उत्तर
\[LHS = A - \left( A - B \right)\]
\[ = A - \left( A \cap B' \right)\]
\[ = A \cap \left( A \cap B' \right)'\]
\[ = A \cap \left\{ A' \cup \left( B' \right)' \right\}\]
\[ = A \cap \left( A' \cup B \right)\]
\[ = \left( A \cap A' \right) \cup \left( A \cup B \right)\]
\[ = \left( \varnothing \right) \cup \left( A \cup B \right)\]
\[ = \left( A \cup B \right) = RHS\]
Hence proved.
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