Math 2200
Name:
Exam 1
Key
10 February, 2014
UID:
)()3II59
Instructions:
• Answer each cluestion 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
10
7
10
8
10
9
0
Total:
80
Score
Math 2200
Exam 1, Page 2 of 8
10 February, 2014
1. (10 points)
(a) Fill in the truth table below.
p
q
pq q
(pq)Aq p
F
TTTF
T
TF
F
T
FF1
T
7F
T
I
r
F
(b) Is the compound proposition ((p —+ q) A —‘q)
or a contingency?
(((
p-%)A -ici)
((pq)Aq)p
,
p
,c
—+
4.L
p a tautology, a contradiction,
r
0
c
all
oji
fyvz4n3
c2. (10 points) Rewrite each proposition below so that negation symbols appear immedi
ately before predicates.
(a) (P(x)AQ(x))
1
P(x)
V
‘
O1x
(OeM
kI%I)
(b) —‘VxyP(x,y)
—1
c1
)c’c/y -iP(
)
1
)
(c) x Ey P(x. y) V Va Q(a))
—1
Vx
1
3J
1]y
VA’7’y
P(x,y) A
P(x,y) A
‘P(i,y) ,
,\/c C?()
VCA O(c)
Voi 6H)
(JJ%(OV1’J
(idc’Aot(
law)
Math 2200
Exam 1, Page 3 of 8
10 February, 2014
2 = x + y where x and y are integers.
3. (10 points) Let P(x. y) be the statement x
Determine the truth value of each statement below. (Write something justifying your
answer— it need not be a complete argument, but show me your thought process.)
(a) P(3,6)
.,1.
-,
D
.
L
1
ffr
(b) rP(x,4)
3x
ci.
S’ vic-c
(c)
vi
xxf’1
I (f pcope4y
o
VyP(2,y)
Wy, 2
z
y
(d) VxyP(x,y)
‘Ii
—)
I-
A
(e) ExVyP(x,y)
1?
-((
1r
y
X
vcI
/:kd
iPjt
Math 2200
Exam 1, Page 4 of 8
10 February, 2014
4. (10 points) Determine whether each argument is valid. Justify your answer.
(a) If Gimli is a dwarf, then he has a beard. Gimli has a beard. Therefore, Gimli is a
dwarf.
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I
,,,
6ead
ciii
I
wa
(b) Either elves are tall or hobbits are tall. Hobbits are not tall. Therefore, elves are
tall.
pvc
(c) If Aragorn is a Hobbit, then he eats six meals a day. Aragorn does not eat six meals
a day. Therefore, Aragorn is not a hobbit.
1-oI(ers
Math 2200
Exam 1, Page 5 of 8
10 February, 2014
5. (10 points) Prove that if x and y are consecutive integers, then x + y is odd.
acw
fl-eu
y>X;
1)
‘
yx-I.
X+y x- (x.-g)
2+i
)(
is c,i
iifeeç
6. (10 points) Prove that if
SiIppc)$(
x is
2
.k4y
is
is irrational, then x is irrational.
honcf
1.
c,id
cs-f
b
vch
t’i-cAf
)(
_i
d
,,
62
frL.e
2
X
coi
i
Irposihoi.
OE’)
Math 2200
Exam 1. Page 6 of 8
10 February, 2014
7. (10 points) Prove that iy is odd if and only if x and y are odd.
odd
‘S
-
X r
SLfpoe
WLOt3-,
y
y
)( cid
,
X
5frCItl
Cç.-
1
4
t)i’
czdd
evl.
tiii.
htri
fL
’s
1
SOV
xZi.
(2*
Z (iy)
;
7•
eVen.
y
4
x’ic
s,ppi€
a
xy
add.
y
.f
)2j
7vi+I.
(2v’.I)(zY4-I)
nvi
7
s
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d
0
d
21f?v #1
t
yV1
c’yu-
7v’ivivitv,
a
oil
GEl)
Math 2200
Exam 1, Page 7 of 8
10 February, 2014
8. (10 points) Prove that any rn x ii checkerboard with an even number of squares can be
tiled by dominos.
You may use the previous questions in this proof if you find them helpful.
Ti
Cck(Y-’Q
5Lc-
iyi
1)tciJ-
rvO.y
‘ias
,
evil.
eih_r
11—k
of
•
Lit c,y)1 I bl 05
fl
h
jshovi
y
vvi
,i
vvicy
111 i 1
fY.
rc),v
ii
h5
whc,le ckeck-e’ tcycI.
Y’vv12
e’er
k,rzvil
qvc-s
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akl
r”qflr
.vi
7,
v-L
15
C1(rtC7O?7
hLea
‘1
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toy
e4c
h
I1llk1j
Math 2200
Exam 1, Page 8 of 8
10 February, 2014
9. (5 points (bonus)) Shown below are the four possible shapes for tetrominos. Can the
standard 8 x 8 checkerboard be tiled by each of these tetrominos?
Minimal partial credit will be given on this problem— if you just do the easy cases, you
won’t get much. Sarah brought graph paper if you need it.
Four
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