CS173: Discrete Mathematical Structures
Spring 2006
Homework #4
Due Sun 02/12/06, 8AM.
1) Let S be the set of all UIUC CS students and C the set of all courses offered in
Department of Computer Science for spring 2006.
a) For
x
โSand yโC, we say y
=
f(x)if and only if student x has registered course
y. Does f:SโCdefine a function? Why or why not?
b) For
x
โSand yโP
C (P
Cis the power set of C), we say y
=
f(x) if and only if
student x has registered every course in set y. Does f:SโP
Cdefine a function?
Why or why not?
c) For
x
โSand yโP
C (
P
Cis the power set of C), we say y
=
f(x) if and only if
student x has registered for every course in set y and for no course in set
Cโy.
Does f:SโP
Cdefine a function? Why or why not?
2) Which of the followings are functions from the domain to the codomain given?
Which functions are one-to-one? Which functions are onto? If it is a bijection,
describe its inverse function.
a) f:ZโZwhere f is defined by f(x)=x2
b) f:ZโRwhere f is defined by f(x)=x
c) f:R+โR+where f is defined by f(x)
=
log(x
+
1)
d) f:ZโNwhere f is defined by f(x)=x
x
e) f:RรRรRโR+โช{0}where f is defined by f(x,y,z)=x2+y2+z2
f) f:RรRโRรRwhere f is defined by f(x,y)
=
(x
+
1, y
+
1)
3) Let f:SโTand
g
:
T
โ
U
be functions.
a) Prove that if
g๎fis one-to-one, so is f.
b) Find an example where g๎fis one-to-one but g is not one-to-one.
4) The Growth of Functions
a) Show that
x
3is
O(x4)but that
x
4is not O(x3).
b) Give as good a big-O estimate as possible for each of the following (A formal
proof is not required, but give your reasoning):
Docsity.com
