Contrapositive
Theorem 1.3 (Contrapositive). For any statements \(P\) and \(Q\), the conditional \(P \implies Q\) and its contrapositive \(\lnot Q \implies \lnot P\) are logically equivalent: they are true in exactly the same cases.
Proof. Compare the two in every case:
\[ \begin {array}{cccc|c|c} P & Q & \lnot P & \lnot Q & P \implies Q & \lnot Q \implies \lnot P \\ \hline T & T & F & F & T & T \\ T & F & F & T & F & F \\ F & T & T & F & T & T \\ F & F & T & T & T & T \end {array} \]
The last two columns agree in every row.