Q.Let $U = \{1, 2, \dots, 10\}$, $A = \{1,3,5,7,9\}$ and $B = \{2,3,5,7\}$. Verify De Morgan's Law $(A \cap B)' = A' \cup B'$.

Compute the complement of the intersection directly, then compute the union of the two individual complements separately, and compare.

Working out each side independently within the universal set confirms the law.

Left-hand side: $(A \cap B)'$.

• $A \cap B$: elements common to $A=\{1,3,5,7,9\}$ and $B=\{2,3,5,7\}$ are $3, 5, 7$. So $A \cap B = \{3,5,7\}$.
• $(A\cap B)' = U - \{3,5,7\} = \{1,2,4,6,8,9,10\}$.

Right-hand side: $A' \cup B'$.

• $A' = U - A = \{1,\dots,10\} - \{1,3,5,7,9\} = \{2,4,6,8,10\}$.
• $B' = U - B = \{1,\dots,10\} - \{2,3,5,7\} = \{1,4,6,8,9,10\}$.
• $A' \cup B' = \{2,4,6,8,10\} \cup \{1,4,6,8,9,10\} = \{1,2,4,6,8,9,10\}$.

Compare. Both sides give $\{1,2,4,6,8,9,10\}$ — De Morgan's Law $(A\cap B)' = A'\cup B'$ is verified.

Intersecting $A'$ and $B'$ instead of taking their union on the right-hand side — that would test the OTHER De Morgan's Law instead. Also, arithmetic slips when listing $U - A$ or $U - B$, particularly forgetting one of the ten elements of $U$.