# What is the difference between a function and a map? [duplicate]

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Is there any difference between mapping and function?

I am an aspiring mathematician who just started out. What is the difference between a function and a map? Or are these notions equivalent?

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## marked as duplicate by Micah, Hagen von Eitzen, Chris Eagle, MJD, ThomasDec 10 '12 at 22:04

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

See this question. It is identical. – glebovg Dec 10 '12 at 21:04
I doubt there is even a precise definition of a function, because any definition has a counterexample. – glebovg Dec 10 '12 at 21:14
@glebovg, "function" has a precise definition: A function from a set $A$ to a set $B$ is a subset $f$ of $A \times B$ such that for any $a \in A$ there exists a unique $b \in B$ such that $(a,b) \in f$. In this notation, you would denote $b$ by $f(a)$ and the point is that a function is completely determined by its graph. – Santiago Canez Dec 10 '12 at 21:39

## 1 Answer

In my experience the two have always been interchangeable. Occasionally one is chosen for emphasis. For instance, sometimes one will use the term "function" to refer specifically to a map of sets with no structure, rather than a continuous map of topological spaces or a homomorphisms of some algebraic objects. Alternatively, the term "map" is sometimes reserved for geometric kinds of things (spaces, manifolds, etc.).

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