What I mean is, given two sets $A$ and $B$ (which are infinite or finite) is it true that if there does not exist a function $f:A\to B$ such that $f$ is onto, then must there exist a function $g:A\to B$ such that $g$ is one to one?
To me this feels intuitive since it seems like if no matter how the elements of $A$ are laid out over the elements of $B$ not every element of $B$ is covered, then it should mean that I should be able to take every input of a mapping that maps to a non-unique output and redirect them to a unique element of $B$.
To make what I mean clearer, I'll give an example of a finite set where this is easiest to demonstrate (though this can be demonstrated with infinite sets). Suppose $A=\{1,2,3\}$ and $B=\{1,2,3,4\}$, then clearly no function from $A$ to $B$ is surjective, because no matter how I map $A$ to $B$, I cannot map to all 4 elements of $B$ with only the 3 elements of $A$; however, I can in fact make an injection, namely $f=\{(1,1),(2,2),(3,3)\}$.
Another example is with mapping the naturals to its power set. Notice that although there does not exist a surjection from $\mathbb{N}$ to $\mathcal{P}(\mathbb{N})$, there does exist an injection from $\mathbb{N}$ to $\mathcal{P}(\mathbb{N})$: $$ f(n\in\mathbb{N})=\{n\}. $$
My question is whether I've overlooked a pair of sets (excluding $A=\emptyset$ or $B=\emptyset$) such that this is not true, and if not how I would prove this. One of my more recent attempts go like this:
Let $A$ and $B$ be sets and let there not exist a $f:A\to B$ such that $f$ is surjective. Also assume a particular $f$ is non-injective; therefore, $\exists x_1,x_2\in A\ni f(x_1)=f(x_2)$. Additionally, since $f$ is non surjective, I know there exists a $y\in B$ such that $\neg \exists x\in A\ni f(x)=y$. From this I know that I can always create a new function $$ f'(x)=\begin{cases} f(x) & \text{ if } x\neq x_1 \\ y & \text{ if } x=x_1 \end{cases} $$ If $A$ where finite I could clearly do this a limited number of times to produce an injection, but I don't quite see how I could use this to generalize this to functions with infinite domains.
Thank you for any help I am extremely new to topology.
