Chapter 1: Speaking Mathematically
Section 1.1
6.
a
.
s
is negative
b
. negative; the cube root of
s
is negative (
Or
:
3
√
s
is negative)
c
. is negative;
3
√
s
is negative (
Or
: the cube root of
s
is negative)
9.
a
. have at most two real solutions
b
. has at most two real solutions
c
. has at most two real
solutions
d
. is a quadratic equation; has at most two real solutions
e
.
E
has at most two
real solutions
Section 1.2
6.
T
2
and
T

3
each have two elements, and
T
0
and
T
1
each have one element.
Justification
:
T
2
=
{
2
,
2
2
}
=
{
2
,
4
}
,
T

3
=
{
3
,
(

3)
2
}
=
{
3
,
9
}
,
T
1
=
{
1
,
1
2
}
=
{
1
,
1
}
=
{
1
}
, and
T
0
=
{
0
,
0
2
}
=
{
0
,
0
}
=
{
0
}
.
9.
c
. No: The only elements in
{
1
,
2
}
are 1 and 2
,
and
{
2
}
is not equal to either of these.
d
. Yes:
{
3
}
is one of the elements listed in
{
1
,
{
2
}
,
{
3
}}
.
e
. Yes:
{
1
}
is the set whose only element is 1.
g
. Yes: The only element in
{
1
}
is 1, and 1 is an element in
{
1
,
2
}
.
h
. No: The only elements in
{{
1
}
,
2
}
are
{
1
}
and 2, and 1 is not equal to either of these.
j
. Yes: The only element in
{
1
}
is 1, which is is an element in
{
1
}
. So every element in
{
1
}
is
in
{
1
}
.
12. All four sets have nine elements.
a
.
S
×
T
=
{
(2
,
1)
,
(2
,
3)
,
(2
,
5)
,
(4
,
1)
,
(4
,
3)
,
(4
,
5)
,
(6
,
1)
,
(6
,
3)
,
(6
,
5)
}
b. T
×
S
=
{
(1
,
2)
,
(3
,
2)
,
(5
,
2)
,
(1
,
4)
,
(3
,
4)
,
(5
,
4)
,
(1
,
6)
,
(3
,
6)
,
(5
,
6)
}
c
.
S
×
S
=
{
(2
,
2)
,
(2
,
4)
,
(2
,
6)
,
(4
,
2)
,
(4
,
4)
,
(4
,
6)
,
(6
,
2)
,
(6
,
4)
,
(6
,
6)
}
d. T
×
T
=
{
(1
,
1)
,
(1
,
3)
,
(1
,
5)
,
(3
,
1)
,
(3
,
3)
,
(3
,
5)
,
(5
,
1)
,
(5
,
3)
,
(5
,
5)
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 Spring '11
 Shirley
 Math, real solutions

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