24 March, 2014
Exam 2
Math 2200
UID:
Name:
Instructions:
• Answer each question in the space provided. Scratch paper will be pro
vided and should be attached to your test when turned in, but your final
answer should be presented as neatly as possible in the space provided.
• No calculators are allowed.
• You may refer to a note card no larger than 3 x 5 inches.
• Answers should be justified as appropriate.
Question
Points
1
10
2
10
3
10
4
10
5
10
6
5
7
5
8
0
Total:
60
Score
Math 2200
Exam 2, Page 2 of 8
1. (10 points) Let
A= {1,2,3}
B = {3,4,5}
C= {1,4}
List all elements of the following sets:
(a) AU(BflC)
L1
A
(b) (A—C)uB
7g5f
3L,
C)
(c) A—(CUB)
1,
Cui3
A—
.s:
1
I31
1z
(d) A x C
(
i), (i,L
Czi),
(?,i) (3,1), (vi)
(e) P(C) (the power set of C)
I
,
qj
j,
q
24 March, 2014
Math 2200
Exam 2, Page 3 of 8
2. (10 points) Consider the function
(a) Show that
f
f
Z x Z
L)
1
ç(
f
Z given by
f(x, y)
=
x
—
y.
is not one-to-one.
5tv1c_
(b) Show that
—+
24 March, 2014
is onto.
LeéZ.
3
i-
_—-€.
3. (10 points) Consider the relation
a c(mod2) and 6 d(mod3).
‘-‘-i
on the set Z x Z given by (a, b)
r (c,
d) if
is reflexive.
(a) Show that
f-
24 March, 2014
Exam 2, Page 4 of 8
Math 2200
)
(
k
1
EZl.
ci’
b
7)
ct
(‘
3)
()‘4)
t4
is symmetric.
(b) Show that
(c,d
(c) Show that
54A
1>
r’
of
tS
.
yV1k’L(1.
-ad
0
(3)
(3)-’
(oo)
6 CO,o)
#(?,3)
H3y
is transitive.
(cd,)
(d)?
(e)
Zx
5C’
‘—(g.
(d
2)
(-i )
O
dI,(d
is not antisymmetric.
[oo)d
(d) Show that
(-odi).
cc
)’- (k,)
Cc
d
1
S
d
Iiec
•
p
1
d(d 3)
[y
Lb
()
)E
(, “) ‘---(rE)
cj
A
Ce(’ i),
d€(vd 3%),
t’’
(Q)’-(c,d)
Exam 2, Page 5 of 8
Math 2200
24 March, 2014
011 the set Z x Z given by
4. (10 points) Consider the relation
d (mod 3) as in the previous problem. Problem 3
(a. b) r (c. d) if a
c (mod 2) and b
shows that this is an equivalence relation.
(a) List four elements of Z x Z in the same equivaleice class as (0, 0).
(0,)
(-;-3)
(, )
1 L
(L
(b) List all equivalence classes of the relation.
L(oo) )
J
iJ]
1
C (o
[Ri)
Math 2200
Exam 2, Page 6 of 8
24 March, 2014
5. (10 points) Prove or disprove each of the following statements:
(a) If ac and bc, then (ab)c.
Fcthe
21’, qjq
44iLj
(b) If xy and xz, then xyz.
‘
cLwLe
yz S”-Ex
svzL.<
iv1frL1
x\yz
3
.id
Math 2200
Exam 2, Page 7 of 8
24 March, 2014
6. (5 points) Find the greatest common divisor of 222 and 629.
(LC1 2222-’- tS
zz
J8
V6-37
537f
cd(3ü)
,o
3_i
7. (5 points) Find integers s and t such that 17s + 12t
I7
L2 2s
5::
2(I22.s)
L
5.5-2
s(7
n)
—
I ‘)
& -7
2•
1
=
1.
24 March, 2014
Exam 2, Page 8 of 8
Math 2200
8. (5 points (bomis)) Let a, n
(i) a
e
Z and
ii
>
o.
Show that the following are eqiivalent:
0(modn)
(ii) a mod ri
—
0
(iii) nla
(Ij)
rc
( ‘)
vck
SO-
f
‘
(‘;
)
c nc
.i.
O
0.
—
TP
c
‘t
sc
=
SoVr-C
+O_
&;)— ()
1Q
h-
(-o).
The€
n).