Q.For $U = \{1,2,3,4,5,6,7,8\}$, $A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$, verify De Morgan's law $(A \cup B)' = A' \cap B'$.

Compute the left side $(A \cup B)'$ and the right side $A' \cap B'$ separately, then check the two sets are identical.

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

Its complement within $U = \{1,\dots,8\}$ is

Right-hand side $A' \cap B'$. First the individual complements:

Their intersection (elements in both):

Conclusion. Both sides equal $\{7, 8\}$, so …

The companion law $(A \cap B)' = A' \cup B'$ can be verified the same way: here $A \cap B = \{3,4\}$ so $(A \cap B)' = {1,2,5,6,7 …

A frequent error is computing $A' \cup B'$ instead of $A' \cap B'$ on the right side (mixing up which De Morgan law pairs union with intersection), or takin …