From this link I understood when we can say that two functions are equal. My doubt is can I interpret the same as follows: Let $X$ , $Y$ and $Z$ be three sets, Let $f : Z \rightarrow X$ and $g : Z \rightarrow Y$, be two functions, then if f = g, we can conclude that $X=Y$.
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It depends on the details... In the usual def we have that $f$ is defined for every $z \in Z$ but not necessarily every $x \in X$ is in the image of $f$. If so, we may have that $X \ne Y$. – Mauro ALLEGRANZA Jan 19 '21 at 09:00
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No. If $f: [0,1]\to [0,1]$ and $g: [0,1]\to \mathbb R$ are defined by $f(x)=x$ and $g(x)=x$ for all $x$ then $f=g$.
PS What is involed here is a matter of conventions. Some authors may not consider my functions as equal.
Kavi Rama Murthy
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@azif00 No. For surjectivity you have to take the codomain into consideration. – Kavi Rama Murthy Jan 19 '21 at 09:40
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What I understood from the explained example is, even though the mapping function is same, the codomain seems to be different. But instead of considering the codomain, if we consider the range, then can we say X=Y. – user3555809 Jan 19 '21 at 13:52
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Thankyou for the insight, can you please provide any reference for the above statement, so that I can do a detailed reading. – user3555809 Jan 19 '21 at 14:13