How is "one to one" a synonym of "injective"? (Terminology question)

Question

Am I the only one to find these alternative names stupid?

• one-to-one = injective
• one-to-one correspondance = bijective
• onto = surjective

Why don't they simply do:

• one to one = bijective
• into = injective
• onto = surjective

Edit: To be clear, I am asking about the "one to one", "onto" etc. I am used to injective, surjective, bijective

I proposed "into" for injective for the same reason people use "onto" for surjective. Onto means there are more elements in the domain than in the range, the function $f: D\mapsto A$ is covering all the elements in the set of arrival. Into would mean the elements $f(x)~\text{such that}~x\in\text{domain }$ (the range) is a strict/proper subset of the arrival set.

Also it is important to realize, and it's something I didn't always know, when people write $g: X\mapsto Y$ is a function, they mean that $\forall x\in X$ $x$ has an image by $g$ in $Y$. But $Y$ is just a set, it's not the range, and not all elements of $Y$ have a pre-image by $g$.

Answer 1

(Too long to really include as comment since this doesn't really include much of an answer.)

I agree that they're kind of dumb.

• "One-to-one" kind of makes sense when you think of the definition, but I could see it being easily misinterpreted. I could see it being envisioned as a statement of "one inputs goes to one output," but then that is just more descriptive of a function which takes on single values (as opposed to, say, multivalued functions). I can also see your idea somewhat for calling "one-to-one" bijective instead.

• "Onto" ... this one I had to think about a lot. I kind of look at the domain of the function mapping "onto" and wholly covering the whole codomain, sort of? I don't know, it's still kind of underwhelming.

• "One-to-one correspondence" is easily the dumbest of the three because of "one-to-one." A function could also be considered a sort of "correspondence" so it's like ... why, oh why, would you deliberately choose something so easily confused?

That said, at least we have the alternative names (injective, surjective, bijective) respectively. I always use those unless I'm just not thinking about it. I don't know inherently where these three terms come from, though, but I'd rather have terminologies that I can't crack, as opposed to those that I can see people confusing with different ideas.

Good luck trying to find an alternate terminology that would suddenly upset the current paradigm, though. Personally your alternate/suggested terminology honestly isn't much better.

Part of the problem probably lies with someone coming up with the former terminologies and them simply sticking. Really, this could be reframed into an interesting math history discussion -- one focusing on these questions:

• Where the did terms "one-to-one", "onto," and "one-to-one correspondence" come from?
• What about "injective," "surjective," "bijective?"
• Which came first, and why did the second terminology become introduced? (Why rock the boat, in other words?)
• Is there any motivating reason for one over the other, i.e. is one more intuitive in some context of mathematics than the other? (Some notations and terminologies are more easily understood in some frameworks, and then get dragged carelessly elsewhere where they're harder to understand.)

Answer 2

No,there names are very meaningful.the root of all these names were "projection" and according to their projection type they are named of those special cases.like, $$projection\implies injection/surjection \implies injective/surjective$$

Now,

$1)$how injective/injection came from???

$\implies$ look..what does basically "one to one" mean is that one is pushing something to other..or one is injecting to other..one member of the domain only points to one element of the codomain.all members of codomain must have to be pointed is not necessary.

$2)$What is "surjection/surjective"?

$\implies$the prefix " sur" means in addition with or something extra..like surplus...say a mapping is given like $a \to 1$,$b \to 2$, $c \to 3$,$d\to 3$. Now see,$3$ is not only pointed by only $c$ but also by $d$.it is a suplus..type.... The elements of the domain are mapped onto all the elements of the codomain.now here..one is only pointed by one element is not necessary, what is necessary is all the members of codomain must be mapped by.so we are using the word "onto" codomain.
$3)$How "bijective"?

$\implies$ holds both of the projection.

Look,I haven't described all the terms elaborately..because what they mean I think is very nicely described in your textbook..I have just given the intuitive keypoints.

Answer 3

"Onto" makes sense in the following way: a function $f \colon X \rightarrow Y$ is surjective if and only if the image of $f$ covers the whole set $Y$. The image of $f$ is a subset of $Y$ by definition, and if $f$ is surjective then then there is no point in $Y$ that is not covered by the image (think of a table cloth covering a table).

For injective, consider the following: take an element in the image of the function $f \colon X \rightarrow Y$ (note that the image is a subset of $Y$ by definition). Suppose you connect each point $y$ in the image of $f$ with the points in $f^{-1}[\{y\}] := \{ x \in X \, | \, f(x) = y \}$ (this is also called the pre-image of the set $\{ y \}$) with a needle. If $f$ is injective then, just like a syringe, the connecting needle pierces the domain at exactly one point; i.e. there is only one needle for each point in the image (and no needle for the points that are not in the image). (I concede, this is a somewhat clumsy visual aide, but it might help).

Another comment:

Into would mean the elements $f(x)$ such that $x \in$ domain (the range) is a strict/proper subset of the arrival set.

But that's not what injective means. Using your definition any bijective $f \colon X \rightarrow Y$ would imply that $image(f) \subset Y$; but any bijection is also a surjection, so $f$ is a surjection iff $image(f) = Y$, which contradicts your definition of injection.

(Also, not meaning to be pedantic, the elements $f(x)$ are not a set. They form a set which is called the image of $f$.)

If you struggle with the nomenclature, understandably, just abandon it and use the formal injective/surjective/bijective. As @MathematicsStudent1122 already wrote in a comment, it's unambiguous and well-understand in the mathematical community.