Math 216  Exam 1  (150 points)                                 Name _____________________

Homework points (max of 50) ________

 

1.      Convert the following statements into propositional logic using ~, Ù, Ú, ®, and «.  Let h = “the cats are happy”, f = “the cats have been fed”, and d = “the dogs is around.”  (2 points each)

 

a.       The cats have been fed and they are happy.

 

 

b.      If the dogs are around, then the cats are not happy.

 

 

c.       Neither the cats have been fed nor the dogs are around.

 

 

d.      If either the cats have not been fed or the dog is around, then the cats are not happy.

 

 

e.       The cats are happy but they have not been fed.

 

 

f.        The cats are happy only if they’ve been fed.

 

 

g.       The cats being fed is a necessary condition for them to be happy.

 

 

 

2.      Give the negation in English of each of the following:  (3 points each)

 

a.       The cats have been fed and the dogs are not around.

 

 

 

 

 

b.      If the dogs are around, then the cats are not 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 points)

 

 

4.      Test the following argument for validity using a truth table.  (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

 

 

Is the argument valid?  YES or NO  (2 points)


5.      Consider the following notation where the domain is the whole world:

 

C(x) is “x is a cat.”

D(x) is “x is a dog.”

F(x) is “x is fearless.”

L(x,y) is “x likes y.”

 

Convert the following English statements to predicate logic form: (16 points)

 

a.       All cats are fearless.

 

 

 

 

 

 

b.      There exists a fearless dog.

 

 

 

 

 

 

c.       Some cats are not fearless.

 

 

 

 

 

 

 

d.      There is a cat that is liked by all dogs.

 

 

 

 

 


6.      Each of the following statements refers to the Tarski World given.  For each, give its truth value and the negation of the sentence in English.  (8 points)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a.       There is a triangle x such that for all squares y, x is above y.

 

Truth value:  True or False

Negation:

 

 

 

 

 

 

 

b.      For all circles x, there is a triangle y such that x is to the right of y.

 

Truth value:  True or False

Negation:

 

 

 

 


7.      Give the contrapositive, converse, and inverse of the following sentence in English.  (6 points)

 

Original:  If an integer is divisible by 2, then it is even.

 

Contrapositive:

 

 

 

 

Converse:

 

 

 

 

Inverse:

 

 

 

 

 

Which is logically equivalent to the original statement?  (3 points)

 

contrapositive, converse, or inverse

 

 

 

 

8.      Select the negation of the statement “All integers are positive.” There is only one correct answer.  (3 points)

 

a.       All integers are not positive.

b.      Some integers are not positive.

c.       Some integers are positive.

d.      I’m positive there are some integers.

 

 


9.       Fill in the blanks in the following proof that for all integers a and b, if a | b then a | b2.  (6 points)

 

Proof:  Suppose a and b are any integers such that ____________.  By the definition of divisibility, b = __________ for some __________ k.  Observe that b2 = b×b = _______ = a(ak2).  But since a and k are integers, then so is _______.  Hence, by the definition of divisibility, __________________, as was to be shown.

 

10.  Prove the following claim using the definitions of odd and even integers.  (10 points)

 

 

Claim:  The sum of two odd integers is even.

 

Proof:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


11.  Consider the following claim:  (8 points)

 

Claim:  For all integers n, if n2 is even then n is even.

 

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.