I just began learning about mathematical proofs (the questions I have in homework are mainly about proving bijections between sets) and I found this subject extremely puzzling and confusing and it seems like I can't understand this damn subject...
I understand that if you show that if $f(a)=f(b)$, then $a=b$ and the function is injective. So if I have a function from set $A$ (which is the set of odd natural numbers (without zero)) to set $B$ (even natural numbers (without zero)) $f(a)=a+1$ then you show that it is injective by saying that $a=b$ if $f(a)=f(b)$ because we can take $f(a)=f(b) = a+1=b+1$ and remove $1$ from both sides and turn it to $a=b$. This is a "proof" that the function is injective? Why? Because I could use some operations to turn $a+1=b+1$ to $a=b$?. And how do you prove that a function is surjective?
Yes that a correct proof for injectivity.
For sujectivity we need to prove that for every $n$ even exists $a$ odd such that $f(a)=a+1=n$ which is true.
Therefore $f$ is injective and surjective and thus invertible.