Answer to Question #302378 in Discrete Mathematics for Justin

Question

Simplify (A ∩ B) U (A ∩ B’) U (A’ ∩ B) U (A’ ∩ B’)

Answer 1

Let $\bf{U}$ be a universal set. Then, the distributive law for any three sets $A,B,C$ is given by, $A\cap (B \cup C) = (A\cap B)\cup(A \cap C)$.

$\begin{aligned} (A ∩ B) \cup (A ∩ B') ~\cup \\ (A' ∩ B) \cup (A' ∩ B') &= (A \cap (B\cup B'))\cup (A'\cap (B\cup B')) \quad(\text{Since $\cup$ is associative})\\ &=(A \cap \textbf{U})\cup (A' \cap \textbf{U})\\&\quad\qquad(\text{Union of a set and its complement is the universal set})\\ & = A \cup A' = \textbf{U} \end{aligned}$