(adsbygoogle = window.adsbygoogle || []).push({}); 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 [itex]x\mapsto x^2[/itex] ". 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 [itex]g\subset X\times Y[/itex] and that [itex]f=(X,Y,g)[/itex]. f is said to be a function from X into Y if

(a) [itex]x\in X\Rightarrow \exists y\in Y\ (x,y)\in g[/itex]

(b) [itex](x,y)\in g\ \land\ (x,z)\in g \Rightarrow y=z[/itex].

Option 2:

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

(a) [itex]x\in X\Rightarrow \exists y\in Y\ (x,y)\in g[/itex]

(b) [itex](x,y)\in g\ \land\ (x,z)\in g \Rightarrow y=z[/itex].

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

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# The terms function and map .

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