Is there any difference between mapping and function?

Question

I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to $\mathbb R$. But I am not ok with this answer. I need a simple way to explain the differences between mapping and function to a lay man together with some illustration (if possible).

Thanks for any help.

Answer 1

I'm afraid the person who told you that was wrong. There is no difference between a mapping and a function, they are just different terms used for the same mathematical object. Generally, I say "mapping" when I want to emphasize that what I am talking about pairs elements in one set with elements in another, and "function" when I want to emphasize that the thing I am talking about takes input and returns output. But that's just a personal preference, and there is no convention I'm aware of.

Answer 2

Although in most cases the words function and mapping can be used interchangeably, in several parts of mathematics differences in emphasis, especially in analysis and differential geometry. I can think of two.

First, especially in differential geometry, "mapping" is the universal word, and the word "function" is used for mappings that map to $\mathbb{R}$ or $\mathbb{C}$. Thus a mapping which maps to $\mathbb{R}^n$ for instance would not be called a function. This convention is not always adhered to (you might occasionally read about "vector-valued functions"), but this is the usual interpretation.

Second, especially in analysis, it is not uncommon to call members of $L^p$ "functions", even though they are actually equivalence classes of mappings. Again the idea is that functions should assign numbers to some objects (e.g. points in some space) in a suitable sense. Thus functions are thought of being objects studied in analysis, whereas "mapping" is thought of being a term from set theory.

Answer 3

Not that much difference in the long run. When I use the word function I generally mean that a point maps to a single point. So, if a point might map to several points, I am not going to use that word, more likely mapping or transformation. In a recent article I had one of these, each point went to several points, and each point in the image probably had several pre-images, so I emphasized, in a traditional phrase, that the mapping was "many-to-many." Now, both primage and image were equivalence classes under a weaker equivalence, so the mapping did induce a function from "genus" to "genus," but was not well-defined on the level of isometry classes of quadratic forms.

Anyway, if a point goes to only a single point, you are allowed to call it a function.

EDIT: I see, you have finished college and are just asking about preferences. I've got to think about popularity in English... Function is used for $\mathbb C \mapsto \mathbb C,$ also maps from any smooth manifold to the reals. I might use function for almost any map into $\mathbb R^n$ from almost anything, but would be less likely to use function for a mapping between two other manifolds. Various kinds of mappings in algebra are unlikely to be called function.

Answer 4

By Nii: To my best understanding, mapping is just a process of matching elements of one set to elements of another set. Mapping is not a function unless some conditions are defined. Thus every mapping is a retation but not necessary a function.