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If I have the statement. "I can ride my bike only if tires aren't broken"

and I have P(X) = "I can ride my bike" and I have Q(X) = "My tires are broken"

Would the above statement be P -> Not(Q)

Also what would the contrapositive of this be?

in English and predicate..? Thank you.

EidA5
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  • Not sure why you have labelled the statements P(X) and Q(X) rather than just P and Q, since they have no dependence on a variable X. – Matt Dickau Jun 03 '16 at 15:52
  • Oh is that wrong to use P(X) or Q(X)? – EidA5 Jun 03 '16 at 15:56
  • Not wrong per se, but misleading. P(X) would normally be taken to be an assertion about a variable object, X, such as "X can ride his/her bike" or "X is a prime number". (This is called a predicate.) There is no need to include such a variable for the statement "I can ride my bike" since it is a specific assertion. (A proposition). – Matt Dickau Jun 03 '16 at 18:32

3 Answers3

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We have $P\Rightarrow \neg Q.$ The contrapositive is $Q\Rightarrow \neg P.$

"If your tires are broken, then you cannot ride your bike."

$P$ "only if" $Q$ is really just $P\Rightarrow Q.$ If you can ride your bike, then you know that your tires are not broken. So your initial intuition is correct.

jdods
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That's right.

The contrapositive would be $Q\to\lnot P$ ("If my tires are broken then I can't ride my bike").

MPW
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Given $P$ and $Q$ as you define them, then "I can ride my bike only if my tires aren't broken" is indeed $P\to\neg Q$. The contrapositive $Q\to\neg P$ would be most naturally read as "If my tires are broken, then I can't ride my bike".

Matt Dickau
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