Truth value of logical expressions

Statement

Let $p$, $q$, $r$ be three true statements. Compute the truth value of each expression.

(i) $(p \land q) \lor \lnot r$ (ii) $\lnot p \to (q \land r)$ (iii) $\lnot(p \leftrightarrow \lnot q) \land r$ (iv) $\lnot[\,p \land (\lnot q \to r)\,]$

Solution

With $p=q=r=T$: (i) $(T \land T) \lor \lnot T = T \lor F = \mathbf{T}$. (ii) $\lnot T \to (T \land T) = F \to T = \mathbf{T}$ (the antecedent is false). (iii) $p \leftrightarrow \lnot q = T \leftrightarrow F = F$; hence $\lnot F \land r = T \land T = \mathbf{T}$. (iv) $\lnot q \to r = F \to T = T$; $p \land T = T$; $\lnot T = \mathbf{F}$.