Simplifying logical expressions

Statement

Simplify each statement.

(i) $p \lor (p \land \lnot p)$ (ii) $(p \land q) \lor \lnot p$ (iii) $(p \land \lnot q) \lor p$

Solution

(i) $p \land \lnot p$ is a contradiction ($F$), so $p \lor F = \boxed{p}$. (ii) By distributivity $(p \land q) \lor \lnot p = (\lnot p \lor p) \land (\lnot p \lor q) = T \land (\lnot p \lor q) = \boxed{\lnot p \lor q}$. (iii) By absorption $(p \land \lnot q) \lor p = \boxed{p}$, since every row where $p \land \lnot q$ is true already has $p$ true.