Perfect hash function

Is a perfect hash function always bijective ? In Wiki "In mathematical terms, it is a total injective function" but it does not mean, that it is bijective.

• If the input domain is larger than the output, how could it be bijective? – mikeazo Oct 4 '15 at 23:38
• Please note, that what you refer as a perfect hash is not a perfect hash in a cryptographic sense. A perfect cryptographic hash is random oracle and that is almost never injective. The Wikipedia article refers to hashing for non cryptographic purposes, e.g. database lookup. And that is a complete different story. – user27950 Oct 5 '15 at 7:48

So if you define the function as mapping a particular set to e.g. $\mathbb{N}$ or $[1, n]$ where $n$ is larger than the size of the set, it is not a bijection.