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. The pigeonhole principle states that once one more hash beyond the size of A has been generated that there is a probability of collision of 1, but it does not state that every single element of A has been generated.