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To prove that two functions $f$ and $g$ are equal, we've to prove three statements independently:

First: $f$ and $g$ have the same domain.

Second: $f$ and $g$ have the same co-domain.

Third: $f(x)=g(x)$ for every x belonging to their domain.

My doubt is, if we prove the third statement, doesn't that imply the other two? I think the first two statements are a direct result of the third statement.

khaxan
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    Third statement does not have unambiguous meaning without the first (what is "their domain", if you do not know they have the same domain?). As for the second... are $f\colon\mathbb{R}\to\mathbb{R}$ with $f(x)=x^2$ the same function as $g\colon\mathbb{R}\to [0,\infty)$, with $g(x)=x^2$ the same function? $f$ is not surjective, but $g$ is. – Arturo Magidin May 14 '23 at 05:41
  • "you put an $x$ such that $f(x)$ gives an output but $g(x)$ is not defined at"--is that what you mean when you say (i) statement is required? If that's the case, i think considering only (iii) statement gives the same result as considering both (i) and (iii). – khaxan May 14 '23 at 06:49
  • I mean that it doesn't make sense to talk about "their domain" unless you already know they have their same domain. Just like you cannot talk about two people and "their father", unless you already knwo they have the same father. – Arturo Magidin May 14 '23 at 17:27
  • I meant this. So shouldn't (iii) imply (i) – khaxan May 15 '23 at 15:05
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    I don't know how many times I need to say this, but here it is for the third time: (iii)doesn't make sense without (i). It is not an intelligible assertion, because you cannot talk about "their" (implied: common) domain unless you first establish that they have the same domain. So, as written (iii) cannot imlly (i) because (iii) is nonsense until you prove (i). – Arturo Magidin May 15 '23 at 17:23
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    First: "They have a common graph" is not the same as "they take the same value at each point of their domain". Second, in Set Theory, sometimes you don't care about the codomain (which is about (ii), not (i) and (iii)), and the question you are pointing to takes that view. See also here – Arturo Magidin May 15 '23 at 17:25
  • @ArturoMagidin, thanks for the clarification !!! forgive my naivety – khaxan May 16 '23 at 14:26

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