Do any one know about this Schroder-Bernstein theorem? Who is this person? Where was this theorem originally proved? The theorem says as long as there is a one-to-one map from X to Y and a one-to-one map from Y to X, there will be a one-to-one and onto map from X to Y. I see somethings wrong in this theorem. I have seen two different proofs; bothe of them embeded with logic errors. The theorem is published in many discreet math. book. I can't see how you can make a one-to-one and onto map from [-1,1] to (-1,1), but it's very easy to make a one-to-one map from one to another in this case.