When talking about relations, injections, surjection, etc… we usually consider two different sets of input to the relation. For example, the input sets might be $\{1, 2\}$ and $\mathbb{N}$.
These two sets are like the range and co-domain, except that they are sets of input, not sets of output.
When I write math, usually refer to the two input sets the small domain and the big domain.
However, that is not standard.
I do not like to invent new terminology if there is already standard wording for such things.
Most math textbooks written in the last 50 years contain something like the following:
We define a relation to be any set of ordered pairs.
Let $R$ be the relation $\{ (1, 11), (2, 22) \}$
Let the domain of $R$ be the set $\{1, 2, 3, 4, 5\}$.Earlier in the book, we defined the domain of any relation to be the set of natural numbers which appear on the left of at least one ordered pair in the relation. For example, if $R$ = { (1, 11), (2, 22) } then the domain of $R$ is set containing 1 and 2 and nothing else.
We now have that the domain of $R$ is simultaneously both $\mathbb{N}$ and is $\\{1, 2\\}$ That is a contradiction, but we will ignore it.
Next, we ask the student to prove or disprove that every element of the domain of the relation $R$ is an element of the domain of relation $R$.
In other words, we ask the student to prove or disprove that:
$\forall x \in \{1, 2, 3, 4, 5\}, \exists y \in \mathbb{N}: (x, y) \in \{ (1, 11), (2, 22) \}$
The Wikipedia articles on relations, injections, and surjections usually have the same issues as the textbooks.
Mainly, the word “domain” used for 2 or 3 different things:
- The $\text{domain}_1$ of $R$ is $\{a \in \mathbb{N}: \exists b \in \mathbb{N}: (a, b) \in R \}$
- For any set $A$, if $A$ is a super-set of the $\text{domain}_1$ of $R$ then $A$ is a $\text{domain}_2$ of $R$. $R$ has lots and lots of $\text{domains}_2$
- The domain $_3$ of relation $R$ is a very specific super-set of the set of things appearing on the left side of at least one ordered pair. The domain$ _3$ of relation $R$ is unique.
What words do contemporary mathematicians use to describe each type of domain?
Given some cartesian product of two sets $A \times B$ then $R \subset A \times B$ is a relation on $A \times B$. Any element of $A$ is in the domain of that relation. If you restrict yourself to $R$ then $\{a : (a,b) \in R)\}$ is the domain of $R$. The context typically tells you if you're dealing with the superset $A$ or the restriction but the difference is rarely material.