# 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
Onto means that it is surjective. – Janko Bracic Mar 14 '15 at 23:08

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

Onto means surjective: every element in the target space lies in the image of the function. Formally, if $f:X\to Y$ is onto, for all $y\in Y$ we can find at least one $x\in X$ such that $y=f(x)$.

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You do indeed hear these terms in relation to functions.

One-to-one means the same as injective. Onto means the same as surjective. One-to-one and onto means bijective.

A function can be just one of them or all three of them.

To answer your specific question, onto means each value of the codomain is mapped to by a member of the domain.

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Historically it's the opposite: surjective means onto. The foo-jective versions were introduced by Bourbaki and popularized in the New Math of the sixties. – Bill Dubuque Mar 14 '15 at 23:31
Interesting...it actually always just drove me crazy to have two vocabulary words to remember for the EXACT SAME idea! – fullyhip Mar 15 '15 at 2:56

First consider the definitions of what it means for a function to be one-to-one or onto (definitions adapted from John Durbin's Modern Algebra):

One-to-one: A mapping $\alpha\colon S\to T$ is said to be one-to-one if $$\alpha(x_1) = \alpha(x_2)\quad\text{implies}\quad x_1=x_2\quad (x_1,x_2\in S),$$ that is, if unequal elements in the domain have unequal images in the codomain.

Onto: If $\alpha\colon S\to T$ and $\alpha(S)=T$, then $\alpha$ is said to be onto. Thus $\alpha$ is onto if for each $y\in T$ there is at least one $x\in S$ such that $\alpha(x)=y$.

You can sort of visualize the above definition of onto with the following picture:

$\color{white}{\text{center it no}}$

Example: Let $S=\{x,y,z\}$ and $T=\{1,2,3\}$. Then a mapping $\alpha\colon S\to T$ may be defined by $\alpha(x)=2, \alpha(y)=1, \alpha(z)=3$. Another mapping, $\beta\colon S\to T$, is given by $\beta(x)=1,\beta(y)=3,\beta(z)=1$. The mapping $\alpha\colon S\to T$ looks like this:

$\color{white}{\text{center it now pleas keep goo}}$

And the mapping $\beta\colon S\to T$ looks like this:

$\color{white}{\text{center it now pleas keep goo}}$

For $\alpha\colon S\to T$, we can see this mapping is onto because each element in $T$ is being mapped to by some element in $S$:

• $y\mapsto 1$
• $x\mapsto 2$
• $z\mapsto 3$

But what about $\beta\colon S\to T$? Is this mapping onto? Can you see why not? Consider the following:

• $x\mapsto 1$
• $z\mapsto 1$
• $\color{red}{?\mapsto 2}$
• $y\mapsto 3$

For $\beta\colon S\to T$ to be onto, each element in $T$ must be mapped to by some element in $S$. Unfortunately, as we can see above by the part highlighted in red, no element in $S$ actually maps to $2$ which is in $T$. Thus, $\beta$ is not an onto mapping.

Similar reasoning will show that $\alpha$ is one-to-one but $\beta$ is not. Does it all make sense now?

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