MATH 410: Homework 1 Solutions
Section 1.1
1. For each of the following statements determine whether it is true or false and justify (infor
mally) your answer.
(a) The sum of irrational numbers is irrational.
Solution:
False. For example
π
+ (

π
) = 0.
(b) The product of irrational numbers is irrational.
Solution:
False. For example (
π
)
(
1
π
)
= 1.
(c) If
n
is natural and
n
2
is odd then
n
is odd.
Solution:
True. If
n
were not odd then it would be even and then so would
n
2
.
2. For each of the following statements determine whether it is true or false and justify (infor
mally) your answer.
(a) Every nonempty set of real numbers that is bounded above has a largest member.
Solution:
False, for example [0
,
1).
(b) If
S
is a nonempty set of positive real numbers then 0
≤
inf
S
.
Solution:
True. If inf
S <
0 then
1
2
inf
S <
0 would be a greater lower bound.
(c) If
S
is a set of real numbers that is bounded above and
B
is a nonempty subset of
S
then
sup
B
≤
sup
S
.
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 Spring '08
 staff
 Math, Supremum, Order theory, upper bound, J2

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