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I tried to find a more intuitive way of explaining to myself how injective and surjective functions work. Does the following make sense? I'm assuming you have a function defined in the form $f(x)=y$.

  1. Injective functions, for every unique $y$-value, have at most one corresponding $x$.

  2. Surjective functions, for every unique $y$-value, have at least one corresponding $x$.

  3. Bijective functions are both injective and surjective, so for every unique $y$-value, they have exactly one corresponding $x$.

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  • $\begingroup$ I saw your title and I wanted to comment exactly your three points. So yeah, I think that's a pretty good explanation. $\endgroup$ – M. Van Oct 20 '17 at 22:58
  • $\begingroup$ Yes, your statements make sense. This answer may help. It uses words with suffix "morphism" that you won't have encountered yet but you should be able to translate it into words about functions. math.stackexchange.com/questions/2039702/… $\endgroup$ – Ethan Bolker Oct 20 '17 at 23:20
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For the function $$ f(x) = 2 $$ what is your $y$? The $2$ or else?

That is one reason why one provides the sets the function maps: $$ f : X \to Y $$ One characterization of a surjective function would be $f(X) = Y$.

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