So, I can see the difference between something like:
A. A car is green if it is made in England.
B. A car is green if and only if it is made in England.
Then, if you had a Russian-made green car, it would be true for A. but not for B. So B is a stricter form of A. I'm trying to see how I can apply this logic to the statement
A function $f: A \to B$ is surjective if and only if for all $b \in B$, there exists an $a \in A$ such that $f(a) = b$.
I think what this means that if it were just normally implied (if x then y), you could have a surjection without the property: $\forall b \in B, \exists a \in A: f(a) = b$; i.e. there could be another property that allows a function to be surjective. But in saying if and only if, we are ensuring that a function can only be surjective if it has this property?