Recall that De Morgan's laws relate the intersection and union of sets to their complements. Start by writing down the definitions of set union, intersection, and complement.

To prove the first law, consider an arbitrary element in the left-hand side (LHS) and show it must belong to the right-hand side (RHS), and vice versa. Use the logical equivalence of set operations with Boolean algebra.

For the second law, construct a Venn diagram to visualize the sets and their complements. Then, use the diagram to argue why the LHS and RHS must represent the same set, ensuring you cover all possible regions of the diagram.