Question

For any two sets A and B, $(A-B)\cup (B-A)=$

• A. $(A-B)\cup B$
• B. $(B-A)\cup B$
• C. $(A\cup B)-(A\cap B)$
• D. $(A\cup B)\cap (A\cap B)$

Answer 1

The correct option is C

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

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

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

[Using distribution law]

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

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

[$A\cup A^{\prime }=U=B^{\prime }\cup B$]

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

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

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

= [$A\cup B$] $\cap$ [$(A\cup B)-(A\cap B)$]

= [$(A\cup B)-(A\cap B)$]