It is somewhat confusing, but remember that srujectivity is defined with respect to a particular codomain; a function is surjective if its range is equal to its codomain, and thus whether it's surjective depends on what its codomain is considered to be; every function maps its domain onto its range. "f maps X onto Y" means that f is surjective with respect to Y". So, for instance, the exponential function maps the real numbers onto the positive real numbers. It's surjective *with respect to positive real numbers. Saying "the exponential function maps real numbers onto real numbers" would not be correct, because it's not surjective with respect to the entire set of real numbers. So saying that a one-to-one function maps distinct elements onto a set of distinct elements can be considered to be correct, albeit not as clear as saying "to" rather than "onto". It also suffer from a lack of clarity in that it's not clear what the "always" is supposed to range over; are there functions that sometimes do map distinct elements to distinct elements, but sometimes don't?