Are you sure that by “one-to-one” Halmos means “bijective”? A more common usage is for it to mean “injective”. (But I don’t have NST and maybe he has an unusual idiom.)
There is a convention according to which a one-to-one function is injective, while a one-to-one correspondence is an injective function that is also surjective, ie, a bijection. (I don’t know whether Halmos uses this convention.)
Oh yes, for sure, but the context here was a statement that “onto” means surjective while “one-to-one” means bijective. Definitely talking functions. And I would be really surprised if Halmos were using “one-to-one” followed by anything other than “correspondence” to mean bijective.
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?
Are you sure that by “one-to-one” Halmos means “bijective”? A more common usage is for it to mean “injective”. (But I don’t have NST and maybe he has an unusual idiom.)
There is a convention according to which a one-to-one function is injective, while a one-to-one correspondence is an injective function that is also surjective, ie, a bijection. (I don’t know whether Halmos uses this convention.)
Oh yes, for sure, but the context here was a statement that “onto” means surjective while “one-to-one” means bijective. Definitely talking functions. And I would be really surprised if Halmos were using “one-to-one” followed by anything other than “correspondence” to mean bijective.
You guys must be right. And wikipedia corroborates. I’ll edit the post. Thanks.
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?