Combinatorics Through Guided Discovery, 2004a

34 2. Applications of Induction and Recursion in Combinatorics and Graph Theory
(a) Use this definition to prove the rule of exponents a m+n = a m a n for
nonnegative m and n. (h)
(b) Use this definition to prove the rule of exponents a mn =(a m ) n . (h)
+ Problem 77. Suppose that f is a function on the nonnegative integers such
that f (0) = 0 and f (n) =n + f (n −1). Prove that f (n) =n(n +1)/2. Notice
that this gives a third proof that 1+2+···+ n = n(n +1)/2, because this
sum satisfies the two conditions for f . (The sum has no terms and is thus 0
when n =0.)
⇒
Problem 78. Give an inductive definition of the summation notation
∑ n
i=1 a i. Use it and the distributive law b(a + c) =ba + bc to prove the
distributive law
n∑ n∑
b a i = ba i .
i=1 i=1
2.1.4 Proving the general product principle (Optional)
We stated the sum principle as
If we have a partition of a set S, then the size of S is the sum of the sizes
of the blocks of the partition.
In fact, the simplest form of the sum principle says that the size of the sum of two
disjoint (finite) sets is the sum of their sizes.
Problem 79. Prove the sum principle we stated for partitions of a set from
the simplest form of the sum principle. (h)
We stated the simplest form of the product principle as
If we have a partition of a set S into m blocks, each of size n, then S has
size mn.
In Problem 14 we gave a more general form of the product principle which can be
stated as
Let S be a set of functions f from [n] to some set X. Suppose that
• there are k 1 choices for f (1), and
• suppose that for each choice of f (1), f (2),... f (i − 1), there are k i
choices for f (i).
Then the number of functions in the set S is k 1 k 2 ···k n .