## Simplifying Expressions Involving Unions and Intersections of Sets

\[A, \:, B, \: C\]

:\[A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\]

\[A \cap (B \cup C)=(A \cap B) \cup (A \cap C)\]

We can use these laws to simplify many expressions involving unions and intersections. For example:

\[(A \cup B) \cap A' =A' \cap (A \cup B)=(A' \cap A) \cup (A' \cap B)= \emptyset \cup (A \cap B)=A \cap B \]

\[(A \cap B) \cup (A' \cap B)=(B \cap A) \cup (B \cap A')=B \cap (A \cup A')=B \cap \mathcal{U} =B\]