Question

Let $U=\{1,2,3,4,5,6\}$, $A=\{2,3\}$ and $B=\{3,4,5\}$.
Which of the following is true?

• A. $(A\cup B{)}^{\prime }=A^{\prime }\cup B^{\prime }$
• B. $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$
• C. $(A\cap B{)}^{\prime }=A^{\prime }\cap B^{\prime }$
• D. None of the above

Answer 1

Answer: (B)

Clearly, $A^{\prime }=\{1,4,5,6\},B^{\prime }=\{1,2,6\}$.
Hence, $A^{\prime }\cap B^{\prime }=\{1,6\}$.
Also, $A\cup B=\{2,3,4,5\}$, so that $(A\cup B)=\{1,6\}$
$(A\cup B{)}^{\prime }=\{1,6\}=A^{\prime }\cap B^{\prime }$
It can be shown that the above result is true in general. If $A$ and $B$ are any two subsets of the universal set $U$, then
$(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$
Similarly, $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$