$A - (A - B)$ is equivalent to which expression

B

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A \cup B

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A \cap B

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B - A

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Solution

The correct option is D $A \cap B$
$A - (A - B)$
$= A - (A \cap B^{C})$
$= A \cap {(A \cap B^{C})}^{c}$
$= A \cap [A^{c} \cup {(B^{c})}^{c}]$
$= A \cap [A^{c} \cup B]$ since ${(B^{c})}^{c} = B$
$= (A \cap A^{c}) \cup (A \cap B)$
$= ϕ \cup (A \cap B)$ since $A \cap A^{c} = ϕ$
$= A \cap B$

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