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gjm comments on Book Review: Naive Set Theory (MIRI research guide) - Less Wrong Discussion
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Comments (12)
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:
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?