How to prove infinitely many functions exist from one set to another

[C0nfused]

Hi everybody,
I guess most of you know how a function is defined from a set A to a set B. How do we prove that many different functions exist (usually, if sets A and B are for example R) from A to B? Of course we can come up with many different functions, real ones for example, but is there any other way of proving that the set of real functions for example has infinite elements?
Thanks

 

[hypermorphism]

If A and B are sets, then by the definition of "function", a function from A into B can be represented as a set of ordered pairs (elements of the Cartesian product AxB), and thus as a subset of AxB. I'm sure you can see where to go from there. :)

 

[C0nfused]

If A and B are sets, then by the definition of "function", a function from A into B can be represented as a set of ordered pairs (elements of the Cartesian product AxB), and thus as a subset of AxB. I'm sure you can see where to go from there. :)

So for each subset one function exists? For example RxR has infinite subsets so infinite different real functions can be defined?

 

[hypermorphism]

So for each subset one function exists?

Not necessarily. The set {(a,b), (a,c)} is not a representation of any function, for example. Rather, you're looking for subsets of the form {a}xB for all a in the domain of f, where the image of f is the set consisting of exactly one element from each such set. Since each of those sets is infinite for your nontrivial real intervals, the amount of such functions is unbounded for even the trivial domain of one real number.