Let $f\colon A\to B$ be a function

Question

The following sentence is often used in mathematical articles and writings:

Let $f\colon A\to B$ be a function.

I wonder, why this senctence is used often, since it is redundant. It is like I would say

Let $x < 0$ be a real number.

In both sentences we first convey the condition which should be satisfied by the considered variable, and after that we declare the type of the variable.

Why not write

Let $f$ be a function from $A$ to $B$ (which can be displayed as follows using symbolic notation: $f\colon A\to B$)

Let $x$ be a real number less then $0$.

?

Answer 1

When writing $f:A\to B$ one often mean that $f$ is a function, unless the specific context implies otherwise. As some comments mention, in certain fields of mathematics, the notation actually mean something else. However also stating that $f$ is a function implicitly gives us that $f$ may be any function between $A$ and $B$. Thus if we are talking about vector spaces, stating that $f$ is a function, without stating that it is linear, means either that it is not linear or that we have to show this in a (probably) non-trivial way.

Regarding

Let $x<0$ be a real number.

Unless you have stated earlier that you will somehow exclusively be talking about real numbers, stating $x<0$ may refer to an integer $x$ which is less than zero, thus adding that $x$ is a real number is certainly not redundant.

Answer 2

I've thought about this too. The problem, in short, is that algebraic structures tend to be identified with their underlying sets. For example, you may wish to define $X \rightarrow Y$ to be the set of all functions with source $X$ and target $Y$. In this case, its certainly redundant to say that $f$ is a function. However, you may also wish to define that if $R$ and $S$ are rings, then $R \rightarrow S$ is the set of all ring homomorphisms with source $R$ and target $S$. Unfortunately, rings are often identified with their underlying set, so all of a sudden the notation $R \rightarrow S$ has become ambiguous.

One way of disambiguating here is to write $\mathbf{Set}(R,S)$ for the set of functions $R \rightarrow S$, and $\mathbf{Ring}(R,S)$ for the set of ring homomorphisms $R \rightarrow S$. But, its usually easier to just write the arrow and say what you mean in natural language.