(related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
Now, we want to reverse the question and ask for the number of different maps of a finite set, say with two elements \{a,b\}, into an arbitrary, non-empty set X, i.e. f:\{a,b\}\to X? How many possible maps of this kind are there? In order to answer this question, observe that a given f assigns uniquely a\to f(a) and b\to f(b), where x:=f(a) and y:=f(b) are some unique two elements of x,y\in X. In the set theory, this is called an ordered pair (x,y). Such an ordered pair is an element of the Cartesian product X\times X. Vice versa, any given ordered pair (x,y)\in X\times X uniquely determines a map f by defining f(a):=x and f(b):=y.
Therefore, there is a one-to-one correspondence between maps of finite sets with n elements into a given non-empty set X. This connection is summarized in the following proposition:
