Combinatorics Through Guided Discovery, 2004a

Chapter 2
Applications of Induction and
Recursion in Combinatorics
and Graph Theory
2.1 Some Examples of Mathematical Induction
If you are unfamiliar with the Principle of Mathematical Induction, you should
read Appendix B (a portion of which is repeated here).
2.1.1 Mathematical induction
The principle of mathematical induction states that
In order to prove a statement about an integer n,ifwecan
1. Prove the statement when n = b, for some fixed integer b
2. Show that the truth of the statement for n = k − 1 implies the truth
of the statement for n = k whenever k > b,
then we can conclude the statement is true for all integers n ≥ b.
As an example, let us give yet another proof that a set with n elements has 2 n
subsets. This proof uses essentially the same bijections we used in proving the
Pascal Equation. The statement we wish to prove is the statement that “A set of
size n has 2 n subsets.”
Our statement is true when n =0, because a set of size 0 is the empty set
and the empty set has 1=2 0 subsets. (This step of our proof is called a
base step.) Now suppose that k > 0 and every set with k − 1 elements
has 2 k−1 subsets. Suppose S = {a 1 , a 2 ,...a k } is a set with k elements.
We partition the subsets of S into two blocks. Block B 1 consists of the
subsets that do not contain a n and block B 2 consists of the subsets that
do contain a n . Each set in B 1 is a subset of {a 1 , a 2 ,...a k−1 }, and each
subset of {a 1 , a 2 ,...a k−1 } is in B 1 . Thus B 1 is the set of all subsets of
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