Q.Let $U = \{1, 2, \dots, 10\}$, $A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$. Verify De Morgan's Law $(A \cup B)' = A' \cap B'$.

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

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

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

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

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

Computing $A'$ and $B'$ correctly but then taking their UNION instead of their intersection on the right-hand side (that would instead verify the OTHER De Morgan's Law, for $(A\cap B)'$, not this one). Also, forgetting the universal set $U$ entirely and trying …