A function on a set

Question

Recently in an exam question, they had used the wording: a function $f$ on set $[1,2,\ldots,n]$ , and so i assumed that this meant that the function had domain and codomain equal to $[1,2,\ldots,n]$ and I solved the question and got the correct answer but apparently we had to take the codomain as whole numbers which would give a different answer. (like I got that the answer was $p(8)$, the number of partitions of $8$, while the given answer is $p(7)$ )

I remember seeing/ reading in a lot of places that a function on set $S$ meant the function was $f: S \to S$ and that this is also pretty common notation, but now i can't remember where I read this,

So my question is that,

This is a common notation right? Like a function on set S means domain and codomain is $S$

And also, more importantly, could you please give me any example of any math resource where this notation is used, the reason I am asking is that the organisers are accepting answer key challenges, so I would like to ask them to accept both $p(7)$ and $p(8)$ as answers, and I would like to ideally attach some math books/ handouts whatever where this notation is used, so that they can see that this is indeed common notation.

I really need to get the question right to have a chance of qualifying for the next round, so I am asking for your help.

Looking forward to any possible help, please.

Thanks!

Answer 1

These are the conventions that I have seen:

• A relation on $S$ is a subset of $S \times S$
• A function on $S$ is a function whose domain is $S$. The codomain is not specified from this description. In some fields, there is an implied usual codomain. For instance, in manifolds, sometimes functions are assumed to have codomain $\mathbb R$, and functions between different manifolds are called mappings instead. But not always.

Despite the fact that functions are a special kind of relation, a function on $S$ is not a special case of a relation on $S$.