I Is f(x) an Injective Function? Understanding Proof and Notation

[PeroK]

If $y \in f(f^{-1}(E))$ then there is some $x \in f^{-1}(E)$ such that $y = f(x)$. From this $x \in f^{-1}(E)$ implies that $f(x) \in E$ such that $y = f(x) \in E$.

If $y \in E$, by surjection there is some $x \in f^{-1}(E)$ such that $y = f(x)$. So by definition, $x =f(x) \in f(f^{-1}(E))$.

This looks good. However, you ought to structure it a bit better. Especially the second part you need to say up front you assume $f$ is surjective.

 

[CaptainAmerica17]

Let $y \in E$. Assume that $f$ is surjective. There is some $x \in f^{-1}(E)$ such that $y = f(x)$. So by definition, $x =f(x) \in f(f^{-1}(E))$.

At least I finally got something understandable. I didn't have nearly as much trouble proving things about inverse images themselves (i.e. $f^{-1}(G \cup H) = f^{-1}(G) \cup f^{-1}(H)$). This forced me to more properly understand what is actually being said by "injection" and "surjection".

As an aside, if you don't mind answering: I'm starting my first semester of college in the fall (for math, of course). This is one of the reasons, besides my own interest, that I've started working on proof-writing and real analysis on my own time. Would a proof like the one I've written above be passable in an actual course? Or do you think it would be docked credit for not being so well-written? The school I'm attending focuses a lot on research, and I would love to be prepared enough to get involved (even in a minimal capacity). I've been kind of nervous recently thinking about it.

 

[WWGD]

Wow, I really overcomplicated things XD

Superheroes tend to do that ;).

 

[member 587159]

Superheroes tend to do that ;).

I think you are the second one to make a superhero joke with this user :P

 

[WWGD]

I think you are the second one to make a superhero joke with this user :P

Us non-superheroes tend to do that ;). Thanks for the setup.

 

[CaptainAmerica17]

Us non-superheroes tend to do that ;). Thanks for the setup.

Lol