Map, injection or both? We are given the relation $R$ between $\mathbb{N}$ and $\mathbb{N}$:
$$
R\colon \mathbb{N}\rightarrow \mathbb{N}\colon n \mapsto n^3- 3n^2 - n
$$
and we are asked: is $R$ a map, bijection, injection or surjection?
First of all, it is clear that not every natural number $n$ has a corresponding mapping, e.g. 1 would map to -3, which is $\notin \mathbb{N}$, similarly for 2 (which would map to -6) and 3 (would map to -2). In conclusion: not every element in the source set $\mathbb{N}$ has an associated target element, and also not every element in the target set $\mathbb{N}$ has an associated source element.
This rules out bijection and surjection. It should be an injection, since every target element has a unique source element. My colleague thinks that $R$ is both a map and an injection, since "all injections are maps". However, I believe that $R$ is not a map, since not every element in the source set has a target element. Various sources on the internet (and books) seem to define maps and injections as being restricted to the domain of $R$, in which case my colleague is correct that injections are maps.
We are stuck. Which one of us is right?
EDIT
We are following the (non-standard?) definitions from a course on Discrete Mathematics, which are as follows:

*

*A relation between two sets is called a mapping if from every element there departs exactly one arrow.

*A relation is called an injection if from every element at most one arrow departs and in every element at most one arrow arrives.

 A: A (binary) relation between sets $X$ and $Y$ is nothing else than a subset of the Cartesian product $X × Y$. In your question we have $R \subset \mathbb N \times \mathbb  N$. In my opinion it is misleading to write it the form $n \mapsto n^3- 3n^2 - n
$, I would prefer to write
$$R = \{ (n,m) \in \mathbb  N \times \mathbb  N \mid m = n^3- 3n^2 - n \} .$$
I recommend to have a look at https://en.wikipedia.org/wiki/Binary_relation.

*

*$R$ is not a function $\mathbb  N \to \mathbb  N$. As you have shown, for $n = 1, 2,3$ we do not have $m \in \mathbb  N$ such that $(n,m) \in R$. However, we may regard it is as a partial function which means that for each $n \in \mathbb N$ we have at most one $m \in \mathbb N$ such that $(n,m) \in R$. The restriction  $\mathbb N \setminus \{1, 2, 3\} \to \mathbb N$ is a function.


*$R$ is an injective relation. This means that if $(n, m) \in R$ and $(n', m) \in R$, then $n = n'$. In fact, consider the function $\phi : \mathbb  R \to \mathbb R,\phi(x ) = x^3 -3x^2-x$. Its derivative $\phi'(x) = 3x^2 - 6x -1$ is positive for $x > \xi = 1 + \sqrt{4/3}$, thus $\phi$ is strictly increasing on $(\xi,\infty)$. Since $ 4 > \xi$, we get injectivity.


*$R$ is not a surjective relation which means that there exists $m \in \mathbb N$ such for all $n \in \mathbb N$ we have $(n,m) \notin R$. In fact, we have $\phi(4)= 12, \phi(5) = 45$. Since $\phi$ is strictly increasing on $(\xi,\infty)$, we may take $m = 13$.
A: I would say it is a (not well-defined) map from $\mathbb{N}$ to $\mathbb{N}$ (since it takes inputs from $\mathbb{N}$ and outputs into the target space $\mathbb{N}$). It is not an injection since it is not even well-defined.
But it is a well-defined map from $\mathbb{N}\setminus\{1,2,3\}$ to $\mathbb{N}$, in fact it is an injection
A: In the definition you give at the end of your post we have that every injective function is a injective relation but every injective relation is not a injective function (injective function here means a relation which is a function and a injective relation).
For example, let $A=\{a,b,c\}$, $B=\{b,c\}$ and $R=\{(b,b),(c,c)\}\subset A\times B$ then $R$ is not a function from $A$ to $B$ as there is no arrow departing from $a$ (the requirement of your definition of mapping would be that there is an arrow departing from $a$).
On the other hand, from every element of $A$ there is at most one arrow departing (The wording "at-most" suggests that no arrow departing is allowed) and also at most one arrow arrives to each element of $B$ making $R$ an injective relation.
Summary: Injections (in terms of relations) are not always mappings.
