Using Zorn's Lemma to prove existence of a function

Question

Using Zorn's Lemma, I want to prove that for any two sets $A,B$ either there is an injective function $f:A\to B$ or there is an injective function $g:B\to A$. Any ideas on how to approach this proof?

Answer 1

Consider the set $\mathcal{F}$ consisting of all pairs $(X,f)$ where $X\subseteq A$ and $f\colon X\to B$ is an injective function. The set is not empty, because $(\emptyset,\emptyset)\in\mathcal{F}$. You can order it by declaring that $(X,f)\le(Y,g)$ if and only if $X\subseteq Y$ and the restriction of $g$ to $X$ equals $f$.

Suppose $\mathcal{C}$ is a chain in $\mathcal{F}$; define $$ Z=\bigcup\{X:(X,f)\in\mathcal{C}\} $$ and define $h\colon Z\to B$ in the obvious way, showing it is injective. Clearly $(Z,h)$ is an upper bound for $\mathcal{C}$.

By Zorn's lemma, there is a maximal element $(W,i)\in\mathcal{F}$. If $i$ is surjective, it has an inverse and we have found an injection $B\to A$.

So, assume $i$ is not surjective. If $W=A$ we are done. Otherwise, take $x\in A\setminus W$ and $b\in B\setminus i(W)$. Can you get a contradiction?

Answer 2

Let $\mathcal T$ be the set of all bijections $h\colon A'\to B'$ with $A'\subseteq A$ and $B'\subseteq B$.