Looks to me like Halmos does intend “one-to-one” to mean “injective”. What he writes is “A function that always maps distinct elements onto distinct elements is called one-to-one (usually a one-to-one correspondence).” Then he mentions inclusion maps as examples of one-to-one functions.
My two main sources of confusion in that sentence are:
He says “distinct elements onto distinct elements”, which suggests both injection and surjection.
He says “is called one-to-one (usually a one-to-one correspondence)”, which might suggest that “one-to-one” and “one-to-one correspondence” are synonyms—since that is what he usually uses the parantheses for when naming concepts.
I find Halmos somewhat contradictory here.
But I’m convinced you’re right. I’ve edited the post. Thanks.
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?
Looks to me like Halmos does intend “one-to-one” to mean “injective”. What he writes is “A function that always maps distinct elements onto distinct elements is called one-to-one (usually a one-to-one correspondence).” Then he mentions inclusion maps as examples of one-to-one functions.
My two main sources of confusion in that sentence are:
He says “distinct elements onto distinct elements”, which suggests both injection and surjection.
He says “is called one-to-one (usually a one-to-one correspondence)”, which might suggest that “one-to-one” and “one-to-one correspondence” are synonyms—since that is what he usually uses the parantheses for when naming concepts.
I find Halmos somewhat contradictory here.
But I’m convinced you’re right. I’ve edited the post. Thanks.
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?