📂Set Theory

Mathematical Logical Proof of the Law of Contraposition

Law ¹

$$ p \to q \iff \lnot q \to \lnot p $$

Explanation

If a proposition is true, then its contrapositive is also true; if a proposition is false, then its contrapositive is also false. Of course, if the converse holds, then by the law of contraposition the reverse of the original proposition also holds.

Such expressions can be too difficult for people unfamiliar with mathematics. Let’s understand through an intuitive example:

• $p$ : The weather is hot
• $q$ : I sweat
• $p \to q$ : If the weather is hot, I sweat

If it is true that when the weather is hot I sweat, then if I am not sweating, whatever else may be the case, I can tell that the weather is not hot.

Proof

$$ \begin{align*} p \to q \iff & \lnot p \lor q \\ \iff & \lnot p \lor \lnot (\lnot q) \\ \iff & \lnot (\lnot q) \lor \lnot p \\ \iff & \lnot q \to \lnot p \end{align*} $$

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