Math 216 Exam 1 (150 points) Name _____________________
Homework points (max of 50)
________
1. Convert the following statements into propositional logic using ~, Ů, Ú, ®, and «. Let T = “the team won the game,” C = “the coach is happy,” and F = “the fans are happy.” (2 points each)
a. The team won the game and
the coach is happy.
b. The team won the game but
the coach is not happy.
c. If the team loses the game
then neither the coach nor the fans are happy.
d. The team winning the game is
a sufficient condition for the fans to be happy.
e. The fans are happy if and
only if the team wins the game.
2. Give the negation in English of each of the following: (3 points each)
a. The coach is happy and the
fans are happy.
b. The coach is happy but the
fans are not happy.
c. If the team wins then the
fans are happy.
3. Use a truth table to determine if the following statements are logically equivalent. (6 points)
|
p |
q |
r |
|
|
T |
T |
T |
|
|
T |
T |
F |
|
|
T |
F |
T |
|
|
T |
F |
F |
|
|
F |
T |
T |
|
|
F |
T |
F |
|
|
F |
F |
T |
|
|
F |
F |
F |
|
Are they
logically equivalent? YES or NO (1 point) How
can you tell from the truth table?
(2 points)
4. Test the following argument
for validity using a truth table. (5 points)
|
p |
q |
|
|
T |
T |
|
|
T |
F |
|
|
F |
T |
|
|
F |
F |
|
Is the
argument valid? YES or NO (1 point) How
can you tell from the truth table?
(2 points)
5. For each of the following
valid arguments, identify the inference rule (found in Table 1.3.1 below) that
guarantees its validity. (8 points)
a. If the cat is hungry then she
will meow.
The cat is hungry.
\The cat is meowing.
Inference rule:
_____________________________________________
b. Either the cat is hungry or
the cat is sleeping.
The cat is not sleeping.
\The cat is hungry.
Inference rule:
_____________________________________________
c. If the cat is hungry then
she will meow.
The cat is not meowing.
\The cat is not hungry.
Inference rule:
_____________________________________________
d. If the cat is hungry then
she is not sleeping.
If the cat is not sleeping, then she is meowing.
\If the cat is hungry then
she is meowing.
Inference rule:
_____________________________________________
Table 1.3.1
6. Consider the following
notation where the domain is the
student body at Lycoming:
E(x) is “x is energetic.”
C(x) is “x is a CS major.”
M(x) is “x is a Math major.”
K(x,y) is “x knows y.”
Convert the following English statements to predicate logic form: (3 points
each)
a. All CS majors are energetic.
b. There is a Math major who is
energetic.
c. There are students who are
both CS and Math majors.
d. This is a CS major who knows
every student on campus.
e. Every CS major knows at
least one Math major.
f. There is a Math major that
every CS major knows.
7. Consider the statement
“Everybody is older than somebody.”
Rewrite this statement in the form “" people x, $ ….” (3 points)
" people x, $
________________________________________
8. Consider the statement
“Somebody is older than everybody.”
Rewrite this statement in the form “$ a person x such that " ….” (3 points)
$ a person x such that "
________________________________________
9. Give the contrapositive, converse, inverse,
and negation of the following
sentence. (8 points)
Original: " real numbers x, if x2 ≥ 1, then x
> 0.
Contrapositive:
Converse:
Inverse:
Negation:
Which is/are logically
equivalent to the original statement? Circle all that apply. (2 points)
contrapositive converse inverse
negation
Which is/are true? Circle
all that apply. (2 points)
original contrapositive converse inverse
negation
10. Disprove the following claim by giving a counterexample.
(4 points)
Claim: For all integers n, if n is prime then
(-1)n = -1.
Disproof:
11. Prove the following claim using the definitions of rational numbers. (10 points)
Claim: The sum of any two rational numbers is a
rational number.
Proof:
12. Consider the following claim: (6 points)
Claim: For all integers n, if 5 divides n2
then 5 divides n.
a.
Write what you would suppose and what
you would need to show to
prove this statement by contradiction.
b.
Write what you would suppose and what
you would need to show to
prove this statement by contraposition.