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 \(\mathbb{R}\)), 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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