Combinatorics Through Guided Discovery, 2004a

1.2. Basic Counting Principles 5
1.2.1 The sum and product principles
These problems contain among them the kernels of many of the fundamental
ideas of combinatorics. For example, with luck, you just stated the sum principle
(illustrated in Figure 1.2.1), and product principle (illustrated in Figure 1.2.2) in
Problems 9 and Problem 10. These two counting principles are the basis on which
we will develop many other counting principles.
Figure 1.2.1: The union of these two disjoint sets has size 17.
Figure 1.2.2: The union of four disjoint sets of size five.
You may have noticed some standard mathematical words and phrases such
as set, ordered pair, function and so on creeping into the problems. One of our
goals in these notes is to show how most counting problems can be recognized as
counting all or some of the elements of a set of standard mathematical objects. For
example Problem 4 is meant to suggest that the question we asked in Problem 3
was really a problem of counting all the ordered pairs consisting of a bread choice
and a filling choice. We use A × B to stand for the set of all ordered pairs whose first
element is in A and whose second element is in B and we call A × B the Cartesian
product of A and B, so you can think of Problem 4 as asking you for the size of the
Cartesian product of M and N, that is, asking you to count the number of elements
of this Cartesian product.
When a set S is a union of disjoint sets B 1 , B 2 ,...,B m we say that the sets
B 1 , B 2 ,...,B m are a partition of the set S. Thus a partition of S is a (special kind
of) set of sets. So that we don’t find ourselves getting confused between the set S