--- aliases: [contrapositive, contrapositives] --- #logic ## Definition > [!tldr] Definition > The **contrapositive** of the [[Conditional statements|conditional statement]] $P \rightarrow Q$ is the conditional statement $(\neg Q) \rightarrow (\neg P)$. That is, we form the converse by replacing the [[Conditional statements|hypothesis]] of the original statement with the [[Negation|negation]] of the original conclusion, and replacing the [[Conditional statements|conclusion]] of the original statement the [[negation]] of the original hypothesis. Notes: > [!important] **The contrapositive of a conditional statement is always [[logically equivalent]] to the original statement**. > To see this, here are the [[Truth tables|truth tables]] for $P \rightarrow Q$ and its contrapositive $(\neg Q) \rightarrow (\neg P)$ side by side (with intermediate columns for the negations): | $P$ | $Q$ | $\stackrel{\Downarrow}{P \rightarrow Q}$ | $(\neg Q)$ | $(\neg P)$ | $\stackrel{\Downarrow}{(\neg Q) \rightarrow (\neg P)}$ | | ----- | ----- | ----------------- | ----------------- | ------- | ------ | | True | True | True | False | False | True | | True | False | False | True | False | False | False | True | True | False | True | True | False | False | True | True | True | True The original statement is in column 3 and the contrapositive in column 6 (indicated by vertical arrows). Notice the truth values are the same. ## Examples Here are some conditional statements and their contrapositives: | Statement | Converse | | -------------------------------- | -------------------------------- | | If it's snowing, then it's cold. | If it's not cold, then it's not snowing. | | If $x > 4$ then $x^2 > 16$. | If $x^2 \leq 16$ then $x \leq 4$. | | My coffee being hot is a consequence of it being freshly brewed. | My coffee not being freshly brewed is a consequence of it not being hot. | | | ## Resources <iframe src="https://player.vimeo.com/video/588861844?h=3596e8dbfd" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe> <p><a href="https://vimeo.com/588861844">Screencast 2.4: Converse, contrapositive, and inverse</a> from <a href="https://vimeo.com/user132700952">Robert Talbert</a> on <a href="https://vimeo.com">Vimeo</a>.</p> Other resources: - Tutorial: [Contrapositive](https://www.regentsprep.org/contrapositive/) - Tutoral: [Converse and contrapositive](https://www.cs.odu.edu/~toida/nerzic/content/logic/prop_logic/converse/converse_intro.html)