Numbers and Polynomials
Test 3
Answer FOUR questions. Be sure to give reasons for each step.
1. Where possible, in each case give (with justiﬁcation) an example of a
nonempty set of real numbers that is bounded above and has:
(i) a greatest element;
(ii) a least upper bound but no greatest element;
(iii) no least upper bound.
2. Let
A
be a nonempty subset of
N
that is bounded above (by a real number).
Show that its least upper bound lub
A
is an element of
A
.
3. Let
P
: =
R
\
Q
denote the set of all irrational real numbers. For
A, B
⊂
R
let
A
+
B
: =
{
a
+
b
:
a
∈
A, b
∈
B
}
. Identify each of the following subsets
of
R
:
(i)
Q
+
Q
;
(ii)
Q
+
P
;
(iii)
P
+
P
.
4. Let
p
be a prime number and let
a, b
∈
Q
. Show explicitly (and carefully:
attend to the denominator) that if
a
+
b
√
p
is nonzero then there exist
x, y
∈
Q
such that 1
/
(
a
+
b
√
p
) =
x
+
y
√
p
.
5. Let
p
be a prime number and
n
a positive integer. Show that if
√
n
∈
Q
[
√
p
]
and
n
is not the square of an integer then
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 Fall '08
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 upper bound

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