CMSC-203 Discrete Math: Special Problem 2 (fall 99)
Graphing Functions
Consider the following two functions that map
Z_55 to Z_55:
f(x) = x^2 mod 55
g(x) = x^3 mod 55
for all x in Z_55. Here, x^3 = expt(x,3)
and Z_55 = {0, 1, 2, ..., 54}.
- Draw a separate directed graph for each function. The
vertices are the natural numbers Z_55. For each
x,y in Z_55, draw an edge from x to y if and only
if f(x) = y. Lay out your graphs in a neat and symmetrical
fashion with no edges crossing. See Chapter 11 for the
definition of a directed graph.
- For each function, determine which of the
following properties it has:
injection, surjection, bijection, endomorphism, permutation.
Explain your answers intuitively using your graphs.