Math 1030-5
Test 1
1. Identify the fallacy in the following statement and explain how it occurs. ”During the 2000 presidential
campaign, Al Gore proposed regulatory changes that would reduce the danger of global warming. The Bush
campaign countered with statements that began, ’Al Gore believes that government is the solution to all our
problems...’ ”
Just because Mr. Gore supported environmental regulations does not mean that he supports a larger government
in all cases, as the Bush campaign contended. They distorted his views in saying that he was soley pro-government.
This is the ”straw man” fallacy. This could also be seen as a diverson. Instead of talking about the environment
and the proposed regulations, the Bush camp talked about the size of government. Another possibility could be
a personal attack for the same reasons.
2. Identify the fallacy in the following statement and explain how it occurs. ”I believe in telepathy because no
one has ever proven that it doesn’t exist.”
This is an appeal to ignorance. Just because there is no proof that telepathy does not exist, does not mean that
it does.
3. Make a truth table to compare the statements ”if p, then q” and ’if not q, then not p”. Are they logically
equivalent?
p
T
T
F
F
q
T
F
T
F
not p
F
F
T
T
not q
F
T
F
T
If p, then q
T
F
T
T
If not q, then not p
T
F
T
T
Note that the thrid and fourth columns are not necessary, but are there for convenience. Since if p, then q has
the same truth values as if not q, then not p, we conclude that they are logically equivalent.
4. Given the conditional statement ”If its a penguin, then it can fly,” write its converse, inverse, and contrapositive.
Converse: If it can fly, then it is a penguin.
Inverse: If it is not a penguin, then it cannot fly.
Contrapositive: If it cannot fly, then it is not a penguin.
5. Draw a Venn diagram for the relationship between the set of all doctors and the set of all women. Be sure to
label enough so that I can understand your diagram.
Men who are not doctors
Women who are doctors
Women who are not doctors
Doctors who are not women
6. Draw a Venn diagram to represent the following information: Of the children on a school bus, 12 play soccer
only, 6 play softball only, 2 play soccer and softball, and 4 play netiher softball nor soccer.
1
Children on the bus that don’t play soccer or softball
4
Play softball,
not soccer
Play soccer
and
softball
Play soccer, not
softball
12
6
2
7. Using a Venn diagram, decide whether the following argument is valid.
P remise : It’s necessary for nurses to know CPR.
P remise : Tom is a nurse.
Conclusion : Tom knows CPR.
Nurses
X
Tom
People who know CPR
The Venn diagram above shows that the argument is valid.
8. Is the following argument inductive or deductive? Explain your answer.
”I see the same man leaving that office building every day at 5:00. He must work in the building.”
The argument is inductive, since it uses many specific cases (observations of the man leaving the building) to
support the more general conclusion that the man works there.
9. How many square inches are in 2 square yards?
2 yd2 ·
32 f t2 122 in2
·
1 yd2
1 f t2
2 · 9 · 144 = 2592,
2
2
There are 2592 in in 2 yd .
10. Although in reality 1 inch = 2.54 cm, for the purposes of this problem, let us say that 1 inch = 3 cm. Using
this assumption, convert 1 meter to yards.
1 m·
100 cm 1 in 1 f t 1 yd
·
·
·
1m
3 cm 12 in 3 f t
100
25
1 · 100
=
=
= .926
3 · 12 · 3
108
27
Therefore, 1 m is .926 yd.
2
11. If you sleep an average of 7.5 hours each night, how many hours do you sleep in a year?
7.5
hours
hours 365 nights
·
= 2737.5
.
night
1 year
year
12. Convert −15◦ C to Farenheit.
Recall the formula F = (9/5)C + 32. This gives us that
F =
9
(−15) + 32 = −27 + 32 = 5.
5
Thus, −15◦ C = 5◦ F .
13. Two cars, 150 kilometers apart, begin driving toward each other on a long, straight highway. One car travels
at 80 kilometers per hour and the other at 100 kilometers per hour. At the same time, a canary, starting on one
car, flies back and forth between the two cars as they approach each other. If the canary flies 120 kilometers per
hour and spends no time to turn around at each car, how far has it flown when the two cars collide?
We mentioned in class that there were two ways to approach this problem: the Calculus way and the easy way.
Of course, we’ll use the easy way. We know that distance = rate·time, so to find out how far the canary flew, we
only have to know how long he was flying, since his speed is given. The time that the canary flies is the same
amount of time that passes from the beginning of the scenario until the cars crash (and the canary is killed?).
Since the cars start 150 kilometers apart, we have
speed of car 1 · time + speed of car 2 · time = total distance
80 · t + 100 · t = 150
180t = 150
5
150
=
t=
180
6
Thus, the canary is in the air for 5/6 hours. We now have
distance canary travels = speed · time
d = 120 ·
5
= 100
6
The bird flies 100 kilometers.
14. Each of ten large barrels is filled with golf balls that all look alike. The balls in nine of the barrels weigh 1
ounce and the balls in one of the barrels weigh 2 ounces. You have a scale, but you are only allowed to use it
once. How can you determine which barrel has the heavier golf balls?
There may be more than one solution to this problem, but here is the one that I came up with. Let us number
the barrels 1 through 10, just to be able to tell them apart. I will take one ball from the first barrel, two from
the second, and so on, until the tenth. I will then weigh the balls I removed together. If the total weight is 11
ounces, then the first barrel held the heavy golf balls. If the weight is 12 ounces, then the second barrel was the
one, etc. The number of ounces above 10 is the number of heavy balls in the scale, and the number of heavy balls
is the same as the number of the barrel they came from.
3