Combinatorics Through Guided Discovery, 2004a

1.2. Basic Counting Principles 9
1.2.2 Functions and directed graphs
As another example how standard mathematical language relates to counting problems,
Problem 7 explicitly asked you to relate the idea of counting functions to the
question of Problem 6. You have probably learned in algebra or calculus how to
draw graphs in the Cartesian plane of functions from a set of numbers to a set of
numbers. You may recall how we can determine whether a graph is a graph of a
function by examining whether each vertical straight line crosses the graph at most
one time. You might also recall how we can determine whether such a function is
one-to-one by examining whether each horizontal straight line crosses the graph at
most one time. The functions we deal with will often involve objects which are not
numbers, and will often be functions from one finite set to another. Thus graphs
in the cartesian plane will often not be available to us for visualizing functions.
However, there is another kind of graph called a directed graph or digraph
that is especially useful when dealing with functions between finite sets. We
take up this topic in more detail in Appendix A, particularly Subsection A.1.2 and
Subsection A.1.3. InFigure 1.2.3 we show several examples of digraphs of functions.
Figure 1.2.3: What is a digraph of a function?