HOMEWORK 2
DUE: Fri., Oct. 7
NAME:
DIRECTIONS:
•
Turn in your homework as
SINGLESIDED
typed or handwritten pages.
•
STAPLE
your homework together. Do not use paper clips, folds, etc.
•
STAPLE
this page to the front of your homework.
•
Be sure to write your name on your homework.
•
Show all work,
clearly and in order
.
You will lose point 0.5 points for each instruction not followed.
Questions
Points
Score
1
1
2
1
3
1
4
2
5
3
6
1
7
1
Total
10
1
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HOMEWORK 2
DUE: Fri., Oct. 7
NAME:
Problem 1:
(1 point) Suppose
A
negationslash
=
∅
and
B
negationslash
=
∅
. Show that
A
×
B
=
B
×
A
iff
A
=
B
.
Problem 2:
(1 point) If
A
,
B
, and
C
are finite sets, show that
#
(
A
∪
B
∪
C
) =
#
A
+
#
B
+
#
C
−
#
(
A
∩
B
)
−
#
(
A
∩
C
)
−
#
(
B
∩
C
) +
#
(
A
∩
B
∩
C
)
.
Problem 3:
(1 point) If
a
,
b
∈
Z
, show (
−
a
)(
−
b
) =
ab
.
Problem 4:
(2 points) If
a
,
b
∈
Z
,
(a)
(1 point) Suppose 0
<a
and 0
<b
. Show that
a<b
iff
a
2
<b
2
.
(b)
(1 point) Suppose
a<
0 and
b<
0. Show that
a<b
iff
b
2
<a
2
.
Problem 5:
(3 points) If
n
,
k
are nonnegative integers, we define the binomial coefficient,
(
n
k
)
, by
parenleftbigg
n
k
parenrightbigg
=
n
!
k
!(
n
−
k
)!
,
where
n
! =
n
·
(
n
−
1)
· · ·
2
·
1, and we set 0! = 1.
(a)
(2 points) Prove that
parenleftbigg
n
r
parenrightbigg
+
parenleftbigg
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 Fall '07
 SCHMIDT
 Mathematical Induction, Negative and nonnegative numbers, Natural number, positive integer

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