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.