Combinatorics Through Guided Discovery, 2004a

10 1. What is Combinatorics?
If we have a function f from a set S to a set T, we draw a line of dots or circles,
called vertices to represent the elements of S and another (usually parallel) line of
circles or dots to represent the elements of T. We then draw an arrow from the
circle for x to the circle for y if f (x) =y. Sometimes, as in part (e) of the figure, if we
have a function from a set S to itself, we draw only one set of vertices representing
the elements of S, in which case we can have arrows both entering and leaving a
given vertex. As you see, the digraph can be more enlightening in this case if we
experiment with the function to find a nice placement of the vertices rather than
putting them in a row.
Notice that there is a simple test for whether a digraph whose vertices represent
the elements of the sets S and T is the digraph of a function from S to T. There
must be one and only one arrow leaving each vertex of the digraph representing an
element of S. The fact that there is one arrow means that f (x) is defined for each x
in S. The fact that there is only one arrow means that each x in S is related to exactly
one element of T. (Note that these remarks hold as well if we have a function from
S to S and draw only one set of vertices representing the elements of S.) For further
discussion of functions and digraphs see Sections A.1.1 and Subsection A.1.2 of
Appendix A.
◦ Problem 23. Draw the digraph of the function from the set {Alice, Bob,
Dawn, Bill} to the set {A, B, C, D, E} given by
f (X) = the first letter of the name X.
• Problem 24. A function f : S → T is called an onto function or surjection
if each element of T is f (x) for some x ∈ S. Choose a set S and a set T so
that you can draw the digraph of a function from S to T that is one-to-one
but not onto, and draw the digraph of such a function.
◦ Problem 25. Choose a set S and a set T so that you can draw the digraph of
a function from S to T that is onto but not one-to-one, and draw the digraph
of such a function.
• Problem 26. Digraphs of functions help us visualize the ideas of one-to-one
functions and onto functions.
(a) What does the digraph of a one-to-one function (injection) from a finite
set X to a finite set Y look like? (Look for a test somewhat similar to the
one we described for when a digraph is the digraph of a function.) (h)
(b) What does the digraph of an onto function look like?