DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

explanations in your earlier proofs. Your understanding of mathematical
induction will be relied on throughout the remainder of these notes. Work
through Appendix B if you’d like more practice with this proof technique.
◦67. Write down at least two ways that the Sum Principle can be expressed
as a sequence of statements indexed by the positive integers. Can you
think of a third way? Settle on a way that allows you to construct an
inductive process.
•68. Prove the Sum Principle by mathematical induction. (As commented
earlier in Problem 13, the base case of a partition into two blocks is
the definition of addition of two positive integers.)
Since in Problem 12 you proved the Product Principle follows logically from
the Sum Principle, you have now completed the proofs of both the Sum and
Product Principles.
2.2.1 Recurrences
In the last section you considered many situations that involved counting
items which are defined inductively. For instance, in Problem 57 you found
the relationship
sn = 2sn−1 . (2.3)
Also, when sn stands for the number of functions from [n] to [k], for a fixed
integer k, the inductive process you found in Problem 59 gave the equation
sn+1 = k sn . (2.4)
Equations (2.3) and (2.4) are examples of recurrence equations, which are
sometimes called recurrence relations, or simply recurrences. A recurrence
is an equation that expresses the n-th term of a sequence an in terms of
earlier values of ai. Other examples of recurrences are
an = an−1 + 7, (2.5)
an = 3an−1 + 2 n , (2.6)
an = an−3 + 3an−2 , (2.7)
an = a1an−1 + a2an−2 + . . . + an−1a1. (2.8)
A solution to a recurrence is any sequence that satisfies the recurrence. For
instance, the sequence given by sn = 2 n is a solution to recurrence (2.3).
Since sn = 17·2 n and sn = −13·2 n are two other solutions to recurrence (2.3),
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