# Transformation T is… “onto”?

I thought you have to say a mapping is onto something... like, you don't say, "the book is on the top of"...

Our book starts out by saying "a mapping is said to be onto R^m", but thereafter, it just says "the mapping is onto", without saying onto what. Is that simply the author's version of being too lazy to write the codomain (sorry for saying something negative, but that's what it looks like to me at the moment), or does it have a different meaning?

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The word "onto" is often used as a synonym for the word "surjective". –  Sebastian Jan 31 '11 at 9:37
The same meaning is intended. If the range is known, then there is no need to repeat it. While this could be confusing, it leads to a shorter and, ultimately, more comprehensible text. If you're the type that annotates textbooks, you can just add an annotation. –  Yuval Filmus Jan 31 '11 at 9:38
@Sebastian: OH that makes a lot more sense, thanks. Feel free to put that as the answer so I can mark it! :) @Yuval: Huh, okay... I beg to differ on the comprehensibility, though. :\ –  Mehrdad Jan 31 '11 at 9:42
@Mehrdad: You're absolutely right though that "onto" makes no sense unless the codomain is understood. So for instance, I can write "$f: X \rightarrow Y$ is onto", but writing "$f(x) = x^2$ is not onto" is sloppy: this is true if the domain and codomain are both $\mathbb{R}$, but false if the domain and codomain are both $\mathbb{C}$, for instance. (Note further that mathematically erudite people generally prefer "surjective" to "onto". For one thing it reads better since, as you suggest, "onto" is traditionally a preposition, with which one is not supposed to end a sentence!) –  Pete L. Clark Jan 31 '11 at 10:16
I guess I don't see how surjective is really any better, since it also should have the codomain specified. I've noticed that the further I get in math the more things are left ambiguous on their face, but everyone knows what you are talking about. For example: $f\in L^2$ isn't really a function or writing $z>0$ instead of $z$ real and $>0$. –  Brian Jan 31 '11 at 13:04

As I mentioned in my comment, the word "onto" is often used as a synonym for the word "surjective". In the same spirit, you can use "one-to-one" instead of "injective". See for example the corresponding Wikipedia article.

Edit: I agree with the comments by Qiaochu and Jonas that "one-to-one" is a little ambiguous and could refer to a bijection. So it is probably best to stick to the unambiguous terms "injective" and "surjective".

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I find "one-to-one" confusing. For the longest time I couldn't remember whether it meant injective or bijective (and I am not sure its usage is entirely consistent on this matter). I think it is best to stick to injective and surjective. –  Qiaochu Yuan Jan 31 '11 at 21:04
@Qiaochu: Coincidentally I was just looking in Lefschetz's Algebraic topology, where a map is defined to be "one-one" if it is both "univalent" and "onto" (i.e., bijective). The phrase "one-to-one correspondence" is sometimes used in place of "bijection", which can also cause confusion with "one-to-one". –  Jonas Meyer Jan 31 '11 at 21:06

This confused me in my first linear algebra class, too. The psychological difference between "onto" and "surjective" is that the latter is only ever introduced as an adjective, whereas prior experience makes us want to read "onto" as a preposition. I don't think this problem arises for "one-to-one", because again we first learn this phrase as an adjective, so there's nothing to confuse it with.

Oxford English dictionary has numerous definitions of the preposition "onto", but the only instance it gives for usage as an adjective is in mathematics.

Math. In form onto. Designating a mapping of one set on to another.

The following is the earliest quotation given there for this usage:

1942 S. Lefschetz Algebraic Topol. i. 7 If a transformation is ‘onto’, the inverse image of the complement of a set is the complement of the inverse image of that set.

I am confused by this quotation, as the result is true for maps that are not onto. However, a quick search of the book shows other uses of the adjective "onto" in the modern sense. The next is more apt:

1951 N. Jacobson Lect. Abstr. Algebra I. 4 If α is a mapping of S into T, and β is a mapping of T into S such that αβ = $1_S$ and βα = $1_T$, then α and β are 1−1, onto mappings and β = α$^{−1}$.

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+1 Ah, thanks for the great explanation! :) –  Mehrdad Jan 31 '11 at 21:28