Math 3333
Homework 1 Solutions
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Show your work. If a problem requires a proof, explain and justify your steps carefully.
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Homework should be submitted
in class
on the indicated due date. Submissions by
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1. Recall that the absolute value of a real number
x
is defined as follows:

x

=
x
if
x
≥
0
,

x
if
x <
0
.
Give a mathematical proof for each of the statements below. In all cases, you
may use basic facts about inequalities and the algebra of real numbers.
In
particular, you may assume that the product of nonnegative integers is nonneg
ative, and that the product of a nonnegative number and a nonpositive number
is nonpositive.
(a)

x

2
=
x
2
for all real numbers
x
. (Hint: Use a proof by cases.)
Assume
x
∈
R
. We will divide the proof into two cases.
Case 1: Assume
x
≥
0. Then,

x

=
x
. Therefore,

x

2
=
x
2
.
Case 2: Assume
x <
0. Then,

x

=

x
. Therefore,

x

2
= (

x
)
2
=
x
2
.
In both cases, we have

x

2
=
x
2
. Therefore,

x

2
=
x
2
for all
x
∈
R
.
(b)
x
≤ 
x

for all real numbers
x
. (Hint: Use a proof by cases.)
Assume
x
∈
R
. We will divide the proof into two cases.
Case 1: Assume
x
≥
0. Then,

x

=
x
. Therefore,
x
≤ 
x

.
Case 2: Assume
x <
0. Then,

x

=

x
. Therefore,
x <
0
<

x
=

x

.
Therefore,
x
≤ 
x

.
In both cases, we have
x
≤ 
x

. Therefore,
x
≤ 
x

for all
x
∈
R
.
(c)

xy

=

x

y

for all real numbers
x
and
y
.
Assume
x, y
∈
R
. By part (a) we have

xy

2
= (
xy
)
2
=
x
2
y
2
=

x

2

y

2
= (

x

y

)
2
That is,

xy

2
= (

x

y

)
2
. Therefore,

xy

=

x

y

or

xy

=

x

y

. Since

xy
 ≥
0 and

x

y
 ≥
0, the equation

xy

=

x

y

is satisfied if and
only if
x
= 0 or
y
= 0. In either case, the former equation

xy

=

x

y

is satisfied as well.
We conclude that

xy

=

x

y

for all real numbers
x, y
∈
R
.
Alternate proof (by cases):
Assume
x, y
∈
R
. We will divide the proof into four cases.
Case 1: Assume
x
≥
0 and
y
≥
0.
Then,
xy
≥
0.
Therefore,

x

=
x
,

y

=
y
, and

xy

=
xy
. It follows that

xy

=
xy
=

x

y

.
Case 2: Assume
x
≥
0 and
y <
0.
Then,
xy
≤
0.
Therefore,

x

=
x
,

y

=

y
, and

xy

=

xy
. It follows that

xy

=

xy
=
x
(

y
) =

x

y

.
Case 3: Assume
x <
0 and
y
≥
0. Then,
xy <
0. Therefore,

x

=

x
,

y

=
y
, and

xy

=

xy
. It follows that

xy

=

xy
= (

x
)
y
=

x

y

.
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