Question

If $A$ and $B$ are any two sets, then state the following statement is true/false

$A-(A-B)=A\cap B$

• A. True
• B. False

Answer 1

The correct option is A True

$L.H.S=A-(A-B)$

$=A\cap (A-B{)}^{\prime }$ [ Set difference ]

$=A\cap (A\cap B^{\prime }{)}^{\prime }$ [ Set difference ]

$=A\cap (A\cup (B^{\prime }{)}^{\prime })$ [ De-Morgan's Law ]

$=A\cap (A^{\prime }\cup B)$ [ Double complement ]

$=(A\cap A^{\prime })\cup (A\cap B)$ [ Distribution law ]

$=\varphi \cup (A\cap B)$ [ Since, A set and its complement are disjoint ]

$=A\cap B$

$=R.H.S$

$\therefore$ Given statement is true.