Combinatorics Through Guided Discovery, 2004a

136 A. Relations
Problem 333. Draw the digraph of the relation from the set {A, M, P, S} to
the set {Sam, Mary, Pat, Ann, Polly, Sarah} given by “is the first letter of.”
Problem 334. Draw the digraph of the relation from the set {Sam, Mary,
Pat, Ann, Polly, Sarah} to the set {A, M, P, S} given by “has as its first letter.”
Problem 335. Draw the digraph of the relation on the set {Sam, Mary, Pat,
Ann, Polly, Sarah} given by “has the same first letter as.”
A.1.3
Digraphs of Functions
Problem 336. When we draw the digraph of a function f , we draw an arrow
from the vertex representing x to the vertex representing f (x). One of the
relations you considered in Problems 333 and Problem 334 is the relation of
a function.
(a) Which relation is the relation of a function?
(b) How does the digraph help you visualize that one relation is a function
and the other is not?
Problem 337. Digraphs of functions help us to visualize whether or not
they are onto or one-to-one. For example, let both S and T be the set
{−2, −1, 0, 1, 2} and let S ′ and T ′ be the set {0, 1, 2}. Let f (x) =2−|x|.
(a) Draw the digraph of the function f assuming its domain is S and its
range is T. Use the digraph to explain why or why not this function
maps S onto T.
(b) Use the digraph of the previous part to explain whether or not the
function is one-to one.
(c) Draw the digraph of the function f assuming its domain is S and its
range is T ′ . Use the digraph to explain whether or not the function is
onto.
(d) Use the digraph of the previous part to explain whether or not the
function is one-to-one.
(e) Draw the digraph of the function f assuming its domain is S ′ and its
range is T ′ . Use the digraph to explain whether the function is onto.