Question regarding elementary set proof I have some questions regarding my approach to a the following problem. I have given my solution and the answer from the solution manual.
The problem: Show that if $f:A \to B$ and $G,H$ are subsets of $B$ then $f^{-1}(G\cup H) =(f^{-1}(G) \cup f^{-1}(H)$. 
My method:
Let $x \in G \cup H$. This implies $x \in G$ or $x \in H$ and similarly $f^{-1}(x) \in f^{-1}(G)$ or $f^{-1}(x) \in f^{-1}(H)$. Therefore, $f^{-1}(x) \in f^{-1}(G) \cup f^{-1}(H)$ and finally $f^{-1}(G \cup H) \subseteq f^{-1}(G) \cup f^{-1}(H)$. 
Proving the other way, Let $y \in f^{-1}(G) \cup f^{-1}(H)$ which implies $f(y) \in G$ or $f(y) \in H$. This then leads us to $f(y) \in G \cup H$ and $y \in f^{-1}(G \cup H)$. This concludes this step with $f^{-1}(G) \cup f^{-1}(H)\subseteq f^{-1}(G\cup H)$.
Finally, we have shown $f^{-1}(G\cup H) =(f^{-1}(G) \cup f^{-1}(H)$. 
Solution manual method: 

I have opted to show that if $A \subseteq B$ and $B \subseteq A$ then $A = B$. I see the book uses a similar strategy except it uses iff to only require one direction. 
My questions:


*

*Is my solution correct?

*I see the solution manual starts out with $y \in f^{-1}(G \cup H)$? Am I correct starting out with $x \in G \subseteq H$?

*In this context how does one know that iff can be used rather than simply $\Rightarrow$?


Thanks!
 A: (1) Your solution is correct. There is almost always more than one solution to a question so don't get caught up too much in worrying about small differences between your solution and one you find in the solution manual. I think it is better to write out the solution in words as you have instead of what the solution manual has done; you don't usually see long lines of implications in a textbook and the solution manual is just trying to be brief. 
(2) You are correct in starting out with $x \in G \cup H$. Alternatively, you could have started out with $y \in f^{-1}(G \cup H)$ and then choose some $x \in G \cup H$ such that $y = f^{-1}(x)$ and continued with your proof as is. These are just small technicalities that don't need to be worried about. In the end you have still shown that $f^{-1}(G \cup H) \subseteq f^{-1}(G) \cup f^{-1}(H)$ and $f^{-1}(G) \cup f^{-1}(H) \subseteq f^{-1}(G \cup H)$ which is what was required.
(3) I don't think there is any general way to know when to use if and only if instead of proving both directions separately. As your mathematical maturity and problem-solving ability improve you will be able to pick up these things more easily. 
In this case, you can use $\iff$ since at each step, the preceding line implies the following line and each line also implies the preceding line. You can easily check that this is the case in each step.
