Q.Using the complement laws, simplify $A \cup A'$ and $A \cap A'$ for any subset $A$ of a universal set $U$, and state which laws these illustrate.

The complement of a set A, denoted A' (or A^c), is defined relative to a fixed universal set U as everything in U that is NOT in A: A' = {x / x∈U, x∉A}. For example, if U = {0,...,8} and A = {2,4,6,8}, then A' = {0,1,3,5,7}. On a Venn diagram, A' is shaded as everything inside the outer universal-set rectangle but outside the circle for A. Three simple but important properties: complementing twice returns the original set, (A')' = A; the complement of the empty set is the whole universe, φ' = U (since everything is 'outside' nothing); and the complement of the universal set is empty, U' = φ (nothing is left outside everything). Complement connects to union and intersection through DE MORGAN'S LA …