Mapping plane/set into/onto itself (What exactly does this mean?)

[vantroff]

I've seen in books things like "G is mapping of plane into itself", "map of a set into itself" or "map of set/plane onto itself".
What exactly to map into/onto itself means? Do this means that when G maps into itself we get G as a result or we can also associate points on G to other points as long as they are on G?
If we have set S={1,2,3} what will mean to map it into itself?
The flowing thing?
1->1 S→S
2->2 S₂→S₂
3->3 S₃→S₃

Will "f:S→S where the image is S itself (i.e f(S)=S)" will be the correct notation(is there difference between the two, if they are correct at all?)

Some simple examples will be helpful.

Most of things I wrote probably make no sense, but I'm totally confused and google don't want to assist when I search about "mapping" and "maps". Giving me the right thing to search for or where to read about these things will be highly appreciated.
I also suppose that onto and into have different meaning, but I don't know what.

Thanks in advance to anyone who reply!

 

[andrewkirk]

For a set S, and a function ('map') f whose domain is S, we say f 'maps S into itself' if f(x) is in S for every x in S.
We say f 'maps S onto itself' if the above applies and the additional condition applies that for every y in S there is some x in S such that y=f(x). This can also be written as f(S)=S.

There are maps from a set onto itself that do not map each element to itself. For instance, with your three-element set, the map f such that
##f(S_1)=S_2##
##f(S_2)=S_3##
##f(S_3)=S_1##
maps S onto itself.

Maps of a set onto itself are sometimes called 'permutations'.

 

[vantroff]

Thank you for the answer! I think that I understand it now. IDK why in all the books where I checked it wasn't explained that simple (it wasnt explained at all)...