Math 310/520
3–9–2009
Problem Set 11
Problem Set 11 is due by the end of class on Monday, March 16. If you use a free makeup, you may
turn it in as late as the end of class on Friday, March 20.
•Problems marked “[MATH 520]” are to be done by Math 520 students; Math 310 students will not get
credit for doing them. Other problems are to be done by everyone.
1. Give a counterexample to each of the following statements.
(a) “|x+y|=|x|+|y|for all x, y ∈R.”
(b) “For all x,y ∈R, if cos x= cos y, then x=y.”
(c) “For all x, y, z ∈R, if x
y≤ −1 and y
z≤ −1, then x
z≤ −1.”
2. Suppose that the universe is {1,2,3,4,5,6,7,8,9,10}, and that
A={1,3,5,7,9}and B={4,5,6,7,8,9,10}.
List the elements of the following sets.
(a) A.
(b) A−B.
(c) A∩B.
(d) A∪B.
page 28: 1.4, 1.12
page 103: 4.36, 4.38
•Your proofs should be written using the definitions of the set constructions, with each step justified.
page 151: 6.18
[MATH 520]
page 151: 6.16
•You might find the following inequality useful:
n2+n+ 1
n(n+ 1)2>n2+n
n(n+ 1)2=n(n+ 1)
n(n+ 1)2=1
n+ 1.
Many are stubborn in pursuit of the path they have chosen, few in pursuit of the goal. -Friedrich
Nietzsche
c
2009 by Bruce Ikenaga 1