Question

Is the following statement True or False?
$(A-B)\cup (B-A)=(A\cup B)\cap (A^{\prime }\cup B^{\prime })$
If True then write answer as $1$
If False then write answer as $0$

Answer 1

$(A-B)\cup (B-A)=(A\cap B^{\prime })\cup (B\cap A^{\prime })$ (Using $A-B=A\cap B^{\prime })$

$=(A\cup (B\cap A^{\prime }\left)\right)\cap (B^{\prime }\cup (B\cap A^{\prime }\left)\right)$

$=\left(\right(A\cup B)\cap (A\cup A^{\prime }\left)\right)\cap \left(\right(B^{\prime }\cup B)\cap (B^{\prime }\cup A^{\prime }\left)\right)$

$=\left(\right(A\cup B)\cap U)\cap (U\cap (B^{\prime }\cup A^{\prime }\left)\right)$

$=(A\cup B)\cap (B^{\prime }\cup A^{\prime })$

Hence statement is true. Answer is 1.