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With motivation from this post, a certain question came to my mind. Is it "easy" to prove 2 functions are the same?

While I searched on the internet for that matter, I only found out about ways to prove or disprove it for certain functions. I found here that finding the equivalence of two functions is undecidable (but for "programming" functions).

Is there anything similar for "math" functions?

The only thing that comes to my mind is, if they have the same domain, codomain and we can transform them into an "irreducible" form.

(I'm talking about functions which domain is infinite, else we can just check all the inputs).

Alex Them
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Apr 16 '22 at 22:26
  • What I'm asking is, if proving the equivalence between two functions is an undecidable problem, with undecidable problem meaning that (https://en.m.wikipedia.org/wiki/Undecidable_problem) – Alex Them Apr 16 '22 at 22:32

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Equivalence of two functions is simply equivalence of the graph of the function, i.e. $$ (x,f(x)) = (x, g(x)) $$.

Problem with this approach is that the graph contain infinite amount of data, and comparing each point in the graph by enumerating the points runs into problems with the infinities. So you need to find another way how to decide if the graphs are the same in some range $[a,b]$. But once you solve that, equivalence of the functions follow.

tp1
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  • So there is a way to determine that? I mean, I'm not asking for simple functions that you just compute the compositions and check if the closed type is the same. My question is more general, not for solving an exercise etc – Alex Them Apr 16 '22 at 22:47
  • @Alex Them: The actual principle is that $ { x \in R | f(x)=g(x) } $, i.e. set builder notation for equalizer is creating a subset of $R$ by filtering out those values of $R$ where $f(x)\neq g(x)$, thus the result is smaller output set which still can contain large part of $R$. – tp1 Jun 03 '22 at 21:33