Very many counting problems can be formulated in terms of counting the number of functions between two sets (possibly satisfying certain properties) or counting the number of subsets of a given set (possibly satisfying certain properties). In this chapter we give a brief introduction to these ideas. They naturally lead to the binomial coefficients, one of the most important families of numbers in all mathematics.
Counting sets of functions
Suppose that X and Y are finite sets. It is natural to ask how many different functions there are with domain X and codomain Y. For example, if X is a set of people and Y is a set of dishes on a menu, then each function X → Y represents a choice of (one) dish for each person, so that the number of functions represents the number of possible orders by the set of people. If S is a set of students and T is a set of tutors then a function S → T is an assignment of a tutor for each student.
There is another increased level of abstraction here. The idea of treating a set, for example the set of integers, as a single mathematical object is one stage of abstraction. A further stage is viewing a function from one set to another as a single mathematical object. But now we are forming a new set whose elements are the functions from one set to another and asking questions about this set.
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