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I'd like to know why, given two sets $X$ and $Y$, there always exists an injection from $X$ to $Y$ or a surjection from $X$ to $Y$. In other words, why we can always compare the cardinality of two sets.

I appreciate any help!

Andrés E. Caicedo
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Jiu
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  • What previous knowledge do you have in set theory? Do you know about the axiom of choice? Do you know about Zorn's lemma? The well-ordering theorem? Do you know Hartogs' lemma? Anything? Something? – Asaf Karagila Oct 04 '18 at 11:36
  • @AsafKaragila I know the axiom of choice and Zorn's lemma. I have just looked up the well-ordering theorem. – Jiu Oct 04 '18 at 11:38
  • Once you realize that an injection from $X$ into $Y$ is the same as a surjection from $Y$ onto $X$ (as long as $X$ is non-empty), the question has many answers. See the duplicate, as well as the linked questions there: https://math.stackexchange.com/questions/linked/421638 – Asaf Karagila Oct 04 '18 at 11:42
  • @AsafKaragila Thanks I'll check them out. – Jiu Oct 04 '18 at 11:44

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