Tautology contradiction or neither

Statement

Decide whether each statement is a tautology (always true), a contradiction (always false), or neither.

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

Solution

(i) Tautology: law of excluded middle, always true. (ii) Contradiction: law of non-contradiction, always false. (iii) Tautology: $\lnot q \to \lnot p$ is the contrapositive of $p \to q$, logically equivalent to it, so the biconditional is always true. (iv) Neither: with $p=T,\,q=F$ we get $T \to F = F$; with $p=F$ we get $F \to \dots = T$. It is satisfiable but not always true.