For any two sets A and B, prove that A-(A-B)=A - Vidyadip

Question

For any two sets A and B, prove that
$\text{A}-(\text{A}-\text{B})=\text{A}\cap\text{B}$

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Answer

Let $\text{x}\in \text{A}- \text{(A} - \text{B)}\Leftrightarrow\text{x}\in \text{A}$ and
$\Leftrightarrow\text{x}\in\text{A and x}\in \text{(A} ∩ \text{B)}$
$\Leftrightarrow\text{x}\in\text{A}\cap\text{(A} \cap \text{B})$
$\Leftrightarrow \text{x}\in\text{(A}\cap\text{B})$
$\therefore\text{A} - \text{(A – B) = (A}\cap\text{B})$

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