An unusual issue involving contrapositives

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This was published on December 28th 2020 on WordPress:

https://explainingmaths.wordpress.com/2020/12/28/an-unusual-issue-involving-contrapositives/

I'm reposting it here because I am able to use MathJax on Blogger, which probably looks a bit better.

For the last question on my recent [as of 28/12/20] first-year pure mathematics assessed coursework (involving subsets of R), one of the possible solutions involved first proving the following true statement about real numbers x:

     If x is irrational, then 1/x is irrational.

But the contrapositive of this appears to say

      If 1/x is rational, then x is rational.

Now the first statement is surely true, but the second looks dodgy to me, because unless you know x is non-zero, 1/x may not even make sense!

If you interpret “If 1/x is rational” as “If 1/x makes sense and is rational” then you are OK. But I’m not convinced that is reasonable.

Perhaps the first “true” statement would be better stated as

    If x is irrational then 1/x makes sense and is irrational.

Then the contrapositive appears to be

If 1/x does not make sense or 1/x is rational then x is rational.

I suppose that one may be OK?


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