Do cryptographic hash functions reach each possible values, i.e., are they surjective?

Question

Take a commonly used binary hash function - for example, SHA-256. As the name implies, it outputs a 256 bit value.

Let A be the set of all possible 256 bit bina

Answer 1

Not necessarily. That would depend on the hash function.

It would probably be ideal if the hash function was surjective, but there are things that're usually more important, such as a low likelihood of collisions.