Math 216  Exam 1  (150 points)                             Name _____________________

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 a2 + 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 a2 = b2 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 a2 + a is even.

 

Proof:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


13.  Consider the following claim:  (8 points)

 

Claim:  For all integers a and b, if a | b then a2 | b2.

 

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.