A,B Sets injective map A into B or bijection subset A onto B First, the statement of the question: Let A,B be two non-empty sets. Show that there exists an injective map of A into B, or there exists a bijective map of a subset of A onto B. My idea is to use Zorn's Lemna (which is covered in the chapter this question is in, next chapter is on cardinality) on the family of injective maps of subsets of A into B, thus determining that there is a maximal element, but I'm not quite sure how this would imply the existence of a bijection. If S is the set of injective maps of subsets of A into B, my idea would be to define a partial ordering $f<g$ iff $f(A')\subset{g(A'')}$, where A' and A'' are the subsets for f and g respectively. It is clear that S is thus inductively ordered so there must therefore exists a maximal element of S by Zorn's Lemna however Im not sure as stated above what this would mean in this context. All help and feedback appreciated. Thank you

Let $\mathcal S$ be the set of maps $f\colon A'\to B$ where $A'\subseteq A$ and $f$ is injective. If $f\colon A'\to B$ and $g\colon A''\to B$ are elements of $\mathcal S$, say $fe A$ and let $a\in A\setminus M$. Assume there exists $b\in B\setminus m(M)$. Then we can extend $m$ to $M\cup\\{a\\}$ by mapping $a\mapsto b$, contradicting maximality of $M$. Therefore, no such $b$ exists, i.e., $m$ is onto.