Transformation T is... "onto"?

Question

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?

Answer 1

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".

Answer 2

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.

B. adj.

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$^{−1}$.