DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

most of the techniques are developed in problems, you might not realize how
much you have learned and most likely it will help to write a summary of
important facts. Every once in a while you should do that—go through the
notes with an eye for what you have learned and how you might approach
older problems differently. You should review at the end of each chapter,
and some of you might prefer more frequent reviews, done either on your
own or in your group.
+ 45. As a group, identify four or five important principles of counting developed
in this chapter. Also, identify at least four techniques used. (For
this, I would call the Bijection Principle a counting principle, whereas
I consider the idea of using a function to count to be a technique.
There is not a clear line of separation between them.)
+ 46. When you originally solved Problems 1 to 9, you probably did not
think explicitly in terms of any basic principles, although you probably
used some principles implicitly. Revisit those problems from the
point of view of categorizing the counting principle used in your solution.
For this, compose a 9×3 table whose rows are labeled Problem 1,
Problem 2,..., Problem 9 and columns by Sum Principle, Product Principle,
Bijection Principle. In each row, indicate with an X each principle
that is natural to use to solve the problem which names the row.
In some problems, several principles might be used.
+ 47. As a group, discuss the function interpretation of Problems 18 and 19
and compose a similar problem of your own which can be solved using
this technique. Share the problems with the other groups and evaluate
their solutions. Be sure the solutions include a clear explanation of
why the constructed function can be used to solve the problem. In
addition, solutions must correctly identify and count the size of the
domain and co-domain of the function.
+ 48. Evaluate this solution to Problem 26(b):
Define the function f : S −→ T by f(a1 a2 · · · an) equals
the set of all subscripts such that ai = 1. This is a bijection
since S and T both have 2 n elements.
1.6.1 An overview of problem solving
You should think about how your approach to counting problems has matured
over the course of this chapter. There are some fairly general tech-
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