For two propositions \(x,y\), the implication \(x\Rightarrow y\) is logically equivalent to its contrapositive \(\neg y\Rightarrow \neg x\), formally, the compound proposition. \[(x\Rightarrow y)\Leftrightarrow(\neg y\Rightarrow \neg x)\]
is a tautology (always valid).
\(\text{"if Socrates is a philosopher, then Socrates is a person"}\) is logically equivalent to \(\text{"if Socrates is not a person, then Socrates is not a philosopher"}\).
\((a > 2)\Rightarrow (2 < a)\) is logically equivalent to \((2 \ge a) \Rightarrow (a \le 2)\).
Proofs: 1