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