Page 20, Problem 9a

The set of all functions from a finite set to a finite set.

Proof. To be begin, consider the example of a function f from a set of2 elements, {1, 2} to a set of 35 elements:

f: {1,2} -> { 1, 2, .., 35}.

The function f is defined by the value it takes at 1, f(1), and thevalue it takes at 2, f(2).

The value f(1) can be one of 35 possible values. Similarly, the valuef(2) can be one of 35 possible values. Thus there number of functionsin this case is 35*35.

Generalizing, the number of functions

f: {1,2, .. ,m} -> { 1, 2, .., n}

from a set of m elements to a set of n elements is nm.