Plus homework points (max of
50) ________
1. Convert the following statements into propositional logic using ~, Ù, Ú, ®, and «. Let N
= “the network is down,” C = “classes
are cancelled,” and S = “the students
are happy.” (3 points each)
a. The network is down and the
students are not happy.
b. The network is down but
classes are not cancelled.
c. Classes
being cancelled is a sufficient condition for students to be happy.
d. Classes are cancelled if the
network is down.
e. If students are happy then
either classes are cancelled or the network is not down.
f. The students are happy if
and only if the network is not down.
2. Give the negation in English of each of the following: (4 points each)
a. The network is not down or
classes are cancelled.
b. Classes are cancelled and
the students are happy.
c. If classes are cancelled,
the students are happy.
d. If the network is down, then
classes are cancelled and the students are happy.
3. Use a truth table to determine if the following statements are logically equivalent. (8 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 (2 point)
Explain how
you can tell from the truth table? (3 points)
4. Test the following argument
for validity using a truth table. (8 points)
_{}
p 
q 

T 
T 

T 
F 

F 
T 

F 
F 

Is the
argument valid? YES or NO (2 point)
Explain how
you can tell from the truth table? (3 points)
5. For each of the following
valid arguments, identify the inference rule (found in Table 1.3.1 below) that
guarantees its validity. (12 points)
a. If the network is down, then
the students are happy.
The students are not happy.
\The network is not down.
Inference rule:
_____________________________________________
b. If the network is down, then
classes are cancelled.
If classes are cancelled, then the students are
happy.
\If the network is down, then
the students are happy.
Inference rule:
_____________________________________________
c. Either classes are cancelled
or the students are happy.
The students are not happy.
\Classes are cancelled.
Inference rule:
_____________________________________________
d. If classes are not
cancelled, then the students are not happy.
Classes are not cancelled.
\The students are not happy.
Inference rule:
_____________________________________________
Table 1.3.1
6. Consider the following
notation where the domain is the
student body at Lycoming:
B(x) is “x is a basketball player.”
T(x) is “x is tall.”
C(x) is “x is a cheerleader.”
K(x,y)
is “x knows y.”
Convert the following English statements to predicate logic form: (4 points
each)
a. All basketball players are
tall.
b. There is a basketball player
that is not tall.
c. Someone is a tall basketball
player.
d. Every basketball player
knows at least one cheerleader.
e. There is a cheerleader that
all basketball players know.
f. There is a basketball player
who knows all of the cheerleaders.
7. Consider the statement “There
are animals of every color.” Rewrite
this statement in the form “" colors c, $ ….” (3 points)
" colors
c, $
________________________________________
8. Consider the statement “There
is a book that everyone has read.”
Rewrite this statement in the form “$ a book b such that " ….” (3 points)
$ a book
b such that "
________________________________________
9. Give the contrapositive, converse, inverse,
and negation of the following
sentence. (12 points)
Original: " integers a, if a is odd, then a^{2} + a is even.
Contrapositive:
Converse:
Inverse:
Negation:
Which is/are logically
equivalent to the original statement? Circle all that apply. (4 points)
contrapositive converse inverse
negation
10. True or False:
For each statement, indicate whether it is true or false. (6 points)
a. " real
numbers x, $ a real number y such that x + y = 0.
b. $ a real number y such that " real numbers x, x
+ y = 0.
11. Disprove the following claim by giving a counterexample.
(4 points)
Claim: For all real number a and b, if a^{2} = b^{2}
then a = b.
Disproof:
12. Prove the following claim using the definitions of odd and even integers. (14 points)
Claim: If a is an odd integer, then a^{2} + a is even.
Proof:
13. Consider the following claim: (8 points)
Claim: For all integers a and b, if a  b then a^{2}  b^{2}.
a.
Write what you would suppose and what
you would need to show to
prove this statement by contraposition.
You do NOT need to write the
complete proof.
b.
Write what you would suppose and what
you would need to show to
prove this statement by contradiction.
You do NOT need to write the
complete proof.