The terms function and map .

[Fredrik]

The terms "function" and "map".

I have noticed that the term "map" is used more often than "function" when a map/function is defined using the "mapsto" arrow, as in "the map $x\mapsto x^2$ ". It has occurred to me that when a function is defined this way, it's usually not clear what the codomain is. So I'm wondering if the choice of the word "map" has something to do with this. Is it common to define "map" differently than "function"? (One way to do it would be to use the term "function" only for the first kind of function below, and "map" only for the second kind).





These are two standard definitions of "function".

Option 1:

Suppose that $g\subset X\times Y$ and that $f=(X,Y,g)$. f is said to be a function from X into Y if

(a) $x\in X\Rightarrow \exists y\in Y\ (x,y)\in g$
(b) $(x,y)\in g\ \land\ (x,z)\in g \Rightarrow y=z$.

Option 2:

Suppose that $g\subset X\times Y$ and that $f=(X,Y,g)$. g is said to be a function from X into Y if

(a) $x\in X\Rightarrow \exists y\in Y\ (x,y)\in g$
(b) $(x,y)\in g\ \land\ (x,z)\in g \Rightarrow y=z$.

Note that when the definitions are expressed this way, they only differ by one character.

 

[Office_Shredder]

The difference between the two definitions is whether f or g is called the function? That doesn't seem to be a very productive difference to me.

An example of a "proper" way to denote a function using the mapsto arrow$$f:X\to Y$$, $$x\mapsto f(x)$$ where f(x) is your formula of course. Usually your domain and codomain are suppressed because they're obvious from context; this would be no different from just saying "let f(x)=x²" and not saying what the domain/codomain are.

The word map itself means the same thing as function. It's probably used more as you read higher levels of mathematics, and the mapsto arrow is used more at the same time because the standard "f(x)=..." formula is no longer sufficient notation, so it's coincidence more than anything else that you notice the two together

 

[Fredrik]

The word map itself means the same thing as function. It's probably used more as you read higher levels of mathematics, and the mapsto arrow is used more at the same time because the standard "f(x)=..." formula is no longer sufficient notation, so it's coincidence more than anything else that you notice the two together

That's my impression too. The only source I've seen actually claim that "map" and "function" can have different definitions is Wikipedia, and they didn't have a reference for that claim.

 

[adriank]

The terms "map" and "function" are often synonymous, but sometimes "map" could mean a morphism in some concrete category, such as a group homomorphism (a "map of groups") or a continuous function (a "map of topological spaces"). I know Hatcher explicitly defines a map to be a continuous function. The term "function" alone usually means a map of sets, ignoring any other structure.

That's how I see it, anyway. Which term is used is generally the preference of the author.