From "Mathematical Reasoning for Elementary Teachers 7th edition"
From page 79:
Definition: One-to-One correspondence:
A one-to-one correspondence between sets A and B is an assigment, for each element of A, of exactly one element of B in such a way that all the elements of B are used. It can also be thought of as a pairing between elements of A and B such that each element of A is matched with one and only one element of B and every element of B has an element of A assigned to it.
So, in my mind, the definition above describes a bijective correspondence between sets and a one-to-one corrrespondence would be the standard $r(x)=r(y) \rightarrow x=y$ definition.
That being said, I feel like throughout my time of reading maths, sometimes it seems that the phrase "one-to-one" is meant to mean bijective.
Can somebody enlighten me to what's going on here? Am I not understanding something?
Best Answer
First, here are the definitions in a more "notched" way, if you will.
Injection
Let $f$ be a function, and Dom$(f) = X$ and Rng$(f) = Y$ (remember that Rng$(f) \subseteq Cod(f)$) such that $f : X \to Y$. We call $f(x)$ the image of $x$ under $f$. We say that $f$ is an injective function if for all $x_1,x_2 \in X$ there does not exist an image of either $x_1$ or $x_2$ such that $f(x_1) = f(x_2)$ without implying that $x_1 = x_2$. We denote that as $X \leq Y$ and it may be referred to as a one-to-one mapping. It could also be defined such that for all $x \in X$, there exists a unique $y \in Y$, where $y = f(x)$.
Surjection
Let $f$ be a function, and Dom$(f) = X$ and Rng$(f) = Y$, where $f : X \to Y$. We say $f$ is a surjective function if for all $y \in X$, there exists an $x \in X$ such that $y = f(x)$. This is denoted by $X \geq Y$. It may also be referred to as an onto mapping.
Bijection
By CBS$^{1}$ we say that if a function $f$ is both surjective & injective that it is a bijective function. More formally, let Dom$(f) = X$ and Rng$(f) = Y$, where $f : X \to Y$ such that for all $y \in Y$ there exists a unique $x \in X$ where $y = f(x)$. It is denoted by $X \sim Y$, and is often referred to as a one-to-one correspondence, but it is better for clarity to refer to it as a one-to-one & onto mapping.
Commentary:
The "one-to-one" label is more-so referring to the intuitive idea of the diagram where it looks like it is taking one element from the domain and injecting it into one distinct element in the range (or what will eventually become so) in a "one-to-one" fashion. Think about it like you exchanging pens with your friend; you only have blue pens, and he only has green pens, and both of you are the function here, so you'll only give him exactly one blue pen (from your domain) if he gives exactly one green pen (which will become your range, but so far it is his codomain, and so the correct thing to say is that he will give you from his codomain), and you keep doing that until you run out of blue pens, and each blue pen that you now have is your range. You both may have ran out of pens to exchange or you ran out first (as denoted by $X \leq Y$, but this is only notation, remember).
For the "onto" label, it is also referring to the more intuitive idea of stretching your domain onto the range (or what will eventually become so) to takeover all of the codomain. If we circle back to the pen exchange function, first you make sure that your friend doesn't have more green pens than blues pens that you have, and then you give him all your blue pens (from your domain), and he gives you all his green pens (from his codomain, and after you have the green pens, its now your range) even if he has a lesser amount of green pens.
Lastly, for the "one-to-one & onto" label it is simply a combination of the aforementioned. It intuitively looks like it is taking over the whole codomain and also injecting every element in the domain into distinct elements in the range. In the pen exchange function, it will be an exact fair trade. You give your friend exactly one blue pen (from your domain), and he gives you exactly one green pen (from his codomain), and you keep repeating that process until his whole codomain is now your range (meaning you took all his green pens), and both of you run out of pens at the same time (meaning he also took all your blue pens, with both being the same amount).
All of these ideas are at the heart of the study of infinity in set theory, and so they're special. Also, fun fact... if $X \leq Y$ then $Y \geq X$ (meaning if $X$ is injective on $Y$ then $Y$ is surjective on $X$... if this seems obvious, thank the notation because the proof is not)
Footnote:
1: Cantor–Schröder–Bernstein theorem that showed for a given function that is both surjective & injective that it must be bijective. (More accurately, that if the cardinality [size] of two sets are less than or equal to and greater than or equal each other, then they're equal.)