Q.If $U = \{1, 2, \dots, 12\}$ and $A = \{2, 4, 6, 8, 10, 12\}$, find $A'$ and verify that $(A')' = A$.

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 LAWS — (A∪B)' = A'∩B' and (A∩B)' = A'∪B' — which let you convert a complement of a compound expression into a compound expression of complements (and vice versa), a technique used constantly when simplifying or verifying set identities and when working with intervals like (a,b]' on the real number line.