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I suppose my question is of more philosophical than mathematical interest. Take the natural logarithm as an example. We can say that $$y = \ln{x}$$ implies $$e^{y} = x$$ due to being inverse functions. I guess we take "implies" to mean biconditionality, which is met when we're looking at inverse functions. But are there any examples where we'd say one function implies another, but they're not inverses? I think not, but looking for confirmation.

I know this is a silly question, but I hope it can be useful or interesting to some people.

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    $\begingroup$ We don't say "one function implies another". One statement can imply another statement, as for example the two equations you've written. A function is not a statement. $\endgroup$
    – hardmath
    Feb 5, 2021 at 2:28
  • $\begingroup$ @hardmath that's fair $\endgroup$
    – 1ijk
    Feb 5, 2021 at 2:32
  • $\begingroup$ To add a bit more rigor, those equations would only become statements if the values of $x$ and $y$ are quantified (in the logical sense, or "qualified" if we want to speak more informally). $\endgroup$
    – hardmath
    Feb 5, 2021 at 3:22

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If $y=|x|$ then $y^2=|x|^2$.

We can't conclude the reverse direction. It is possible that $y=-|x|$.

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  • $\begingroup$ hm, but that is an implication. I think the problem in my question might be that I'm taking "implies" to be a biconditional, when what I was reading did mean something weaker, and just a casual observation at that. $\endgroup$
    – 1ijk
    Feb 5, 2021 at 2:38

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