If the hash length is greater than the input length, can we assume that the probability of a collision is strictly zero?

Question

Mathematics

Inspired by a neighboring question about the possibility of hashing collisions. Suppose we have some crypto-resistant function that generates a hash of a certain length from a sequence of bytes of arbitrary length. Is it true that if the length of the input of this function is less than the length of the output, then there will certainly be no collisions on the set of all admissible inputs of this function with a length less than the length of the output? At least the cardinality of the set of possible inputs is clearly less than the cardinality of the set of possible outputs, that is, this statement can be true.

Answer 1

In general, it's not true. For each specific case, it is necessary to consider the hash function and mathematically prove such a statement.
However, the function f(x) = x satisfies your condition.