2 ms·

It is an injection, but not a surjection. You need neither though. For example: y = x ^ 2 when even, (x+1) ^ 2 otherwise

• thaumasiotes 4y ago
  > It is an injection, but not a surjection. It's a bijection. (And therefore obviously also a surjection.) "One-to-one correspondence" is a correct, but incomplete, description.
  • quickthrower2 4y ago
    ∄ x ∈ ℕ: x² = 5 Unless you consider the domain of f the "renamed" range, in which case yes tautologically it is bijective.
    • mb7733 4y ago
      You're talking past one another. The other poster is discussing a correspondence between N and S, where S is the set of squares. So the function being discussed is f: N -> S, f(n) = n^2, which is obviously bijective. You seem to be talking about the function f: N -> N, which isn't surjective, but I don't see the relevance. Any surjective function can be made non-surjective by considering a larger range.
      • thaumasiotes 4y ago
        > You seem to be talking about the function f: N -> N, which isn't surjective, but I don't see the relevance. It's worse than just irrelevant. The claim is that we have two sets, ℕ and S, that are the same size. The argument that they are the same size is the observation that we've demonstrated a bijection between them. No one ever mentioned a function from ℕ to ℕ, because that would tell you nothing about S. It would be like saying "in order to show that 5 is odd, observe that 2 and 2 are both even". The idea is gibberish.