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Stream: learning: questions

Topic: help with lemma

Matteo Capucci (he/him) (Aug 26 2024 at 16:00):

Smooth brain moment: is the following true for a function $f$?

Suppose $f$ is invertible iff $f$ is surjective. Then $f$ is injective.

Todd Trimble (Aug 26 2024 at 16:33):

It would help to put quantifiers in as appropriate. I can say that if $A$ is a finite set, then it is unconditionally true that for all $f: A \to A$, $f$ is surjective iff $f$ is invertible. That doesn't mean that $f: A \to A$ is always injective. But perhaps I misunderstood what is being asked.

Damiano Mazza (Aug 26 2024 at 16:37):

I am probably missing something, but the way it's written I understand it as just a formula of the form $((A\land B)\Leftrightarrow B)\Rightarrow A$. This is not provable because it is falsified by taking both $A$ and $B$ false. For functions, you take $f$ to be neither surjective nor injective, so the "iff" hypothesis holds, but not the conclusion. But, again, I may be misunderstanding the statement.

Matteo Capucci (he/him) (Aug 26 2024 at 16:41):

Todd Trimble said:

It would help to put quantifiers in as appropriate. I can say that if $A$ is a finite set, then it is unconditionally true that for all $f: A \to A$, $f$ is surjective iff $f$ is invertible. That doesn't mean that $f: A \to A$ is always injective. But perhaps I misunderstood what is being asked.

That's a perfect counterexample.

Patrick Nicodemus (Aug 29 2024 at 04:25):

smooth-brains should check out the differential geometry zulip, this is more of an arrow-brain server

Matteo Capucci (he/him) (Aug 30 2024 at 09:51):

too many arrows smoothed my brain :laughing: