Expressing the exclusive or with basic connectives

Statement

Using only the connectives $\lnot$, $\land$, $\lor$, write a statement in the variables $p$ and $q$ that has the same truth values as the exclusive or $p \oplus q$.

Solution

$p \oplus q$ is true when exactly one of $p$, $q$ is true. These two cases are ”$p$ true and $q$ false” and ”$p$ false and $q$ true”, that is $p \oplus q \;\equiv\; (p \land \lnot q) \lor (\lnot p \land q).$ Check with the truth table:

$p$	$q$	$(p\land\lnot q)\lor(\lnot p\land q)$
T	T	F
T	F	T
F	T	T
F	F	F

The column matches that of $p \oplus q$.