The University of Maine at Augusta Name ______.docx

The University of Maine at Augusta Name:
________________________
Mathematics Department Date:
_________________________
MAT 280 F16
Location:______________________
MAT 280
Exam 1 Chapter 1.1-1.6
Please answer the following questions. Part credit is possible if
the work indicates an understanding of
the objective under investigation. Students may use their
laptops, tablets, textbooks, notes, calculators
and scrap paper. If used, scrap paper should be turned in with
the exam. Students may not use smart
phones. If available homework should be turned in with the
exam.
Time: 2:45 If staff is available, extra time is permitted.

1. Find a proposition with the given truth table.
p q ?
T T F
T F T
F T T
F F T
→ r). Use as many columns as necessary.
Label each column.
p Ans: 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
Name: ________________________
3. Find a proposit
V with the given truth table.
p
q
?
T T F
T F F
F T F
F F T

r). Use as many columns as necessary.
Label each used column. Credit will only be when a completed
truth table accompanies the answer.
p Ans: 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
Name: ________________________

answer with a truth table. If necessary, insert columns in the
truth table below to support your
answer
p
q
T T
T F
F T
F F
equivalent. If they are not equivalent, explain
why. Do the same for its inverse.
p
q
p
q

T T T T
T F T F
F T F T
F F F F
7. In the questions below write the statement in the form “If
…, then ….”
a. Whenever the temperature drops below 35 degrees, children
should wear boots
during recess.
b. You have completed your program’s requirements only
if you are eligible to graduate.
Name: ________________________
8. Write the contrapositive, converse, and inverse of the
following:
If I have a valid passport, I will be
able to travel to Cuba.
.

a. Contrapositive:
b. Converse:
c. Inverse:
9. How many rows are required to show the truth table for the
following compound proposition?
10. Are the following system specifications consistent?
If the file system is not locked, the new messages will be
queued.
If the file system is not locked, the system is functioning
normally and conversely.
If new messages are not queued, then they will be sent to the
message buffer.
If the file system is not locked then new messages will be sent
to the message buffer.
New messages will not be sent to the message buffer.

11. In the questions below write the negation of the statement.
(Don't write “It is not true that
…”)
go swimming.
12. Explain why the negation of “ENG 101 and ENG 317 are
program requirements for CIS ” is
not “ENG 101 and ENG 317 are not program requirements for
CIS”.
13. Using C for “it is cold” and W for “it is Winter”, write “To
be Cold it is necessary that it is
Winter” in symbols
14. On the island of knights and knaves you encounter two
people, A and B. B says "A is a
knave." Person A says, "At least one of us is a knight."
Determine whether each person is a
knight or a knave. If it is not possible to determine, state why.
Name_____________________________
15. Find the output of the combinatorial circuits

value of the statement. Explain your answer.
18. In the questions below suppo
where x is a real number. Find the truth
value of the statement.
19. In the question below suppose P(x,y) is a predicate and
the universe for the variables x and
y is {1,2,3}. Suppose P(1,3), P(2,1), P(2,2), P(2,3), P(2,3),
P(3,1), P(3,2) are true, and P(x,y)

is false otherwise. Determine whether the following statements
are true.
Name:____________________________
20. In the questions below suppose the variable x represents
students and y represents courses,
and: U(y): y is an upper-level course E(y): y is a English
course F(x): x is a freshman
A(x): x is a part-time student T(x,y): student x is taking
course y.
Write the statement using these predicates and any
needed quantifiers.
a. There is a part-time student who is not taking any upper-
level courses.
b. Every freshman is taking at least one English course.

c. No freshman is taking is taking an upper level English
courses
21. Express the negations of these propositions using
quantifiers, and in English.
a. There is a student in this class who has taken every
mathematics course offered
at this school.
b. There is a student in this class who has been in at least one
room of every
building on campus.
22. In the questions below suppose the variables x and y
represent real numbers, and
L(x, y) : x < y Q(x, y) : x = y E(x) : x is
even I(x) : x is an integer.
needed quantifiers.
a. Every integer is odd
b. If x < y then x is not equal to y.
C. There is no smallest integer.

Name: ________________________
23. Suppose the variable x represents people, and
F(x): x is friendly T(x): x is tall A(x): x is
angry S(x): x is a student.
needed quantifiers.
a. All friendly students are tall
b. No friendly students are angry
24. In the questions below suppose the variable x represents
students, F(x) means “x is a
Freshman”, and C(x) means “x is an Computer major”. Match
the statement in symbols with one of
the English statements in this list:
1. Some freshmen are Computer majors. 2. At least one
Computer major is a freshman.
3. No Computer major is a freshman.

Circle 1, 2, or 3
25. Determine whether the following argument is valid:
p → r
q → r
q ∨ ¬r
... ¬p
Name__________________________
26. Show the premises and explain which rule of inference is
used to reach the
conclusion.
“Linda, a student in this class, owns a red convertible. Everyone
who owns a red
convertible has gotten at least one speeding ticket. Therefore,
someone in this class

has gotten a speeding ticket.” Make sure to use the correct
quantifiers.
27. Is the following a valid argument. Explain
Premise: If I publicly insult my mother-in-law, then my wife
will be angry at me.
Premise: I will not insult my mother-in-law.
Conclusion: Hence, my wife will never be angry at me.
The University of Maine at Augusta Name:
________________________
Mathematics Department Date:
________________________
MAT 280 F16
MAT 280
Exam 2 Proofs

Please answer the following questions. You my use your book
and other outside resources.
However, all work should represent the individual student’s
efforts. This exam covers
primarily sections 1.7 and 1.8, however there will be some
questions from 1.5 and 1.6.
1. Show that the premises, “ Arron is registered as an instate
student” and “All instate students have
permanent addresses in Maine” imply the conclusion “Arron’s
permanent address is in Maine”
2. Determine whether the following argument is valid.
Rainy days make gardens grow.
Gardens don't grow if it is not hot.
It always rains on a day that is not hot.
Therefore, if it is not hot, then it is hot.
3. What is the rule of inference used in the following:
If I work all week on this take home exam, then I can answer
the questions correctly. If I
answer the questions correctly, I will pass the exam. Therefore,
if I work all week on this take

home exam, then I will pass the exam.
4. Suppose you wish to prove a theorem of the form “if p then
q”.
a. If you give a direct proof, what do you assume and what do
you prove?
b. If you give an indirect proof, what do you assume and what
do you prove?
c. If you give a proof by contradiction, what do you assume and
what do you prove?
5. Suppose you are allowed to give either a direct proof or a
proof by contraposition of the
following: if 5n + 3 is even, then n is odd.
a. Which type of proof would be easier to give, direct or
indirect?
Explain why.

b. Prove: If 5n +3 is even then n is odd.
6. In the following proof, name the proof technique:
Prove that p→q
Proof continued…..
7. Prove: If x is odd, then 7 x - 4 is odd
Proof:
1. Suppose x is odd. [Show 7x - 4 is odd .]
2. If x is odd then by definition,
x = 2k + 1, where k is an integer
So…

⇒ 7x - 4 = 7( 2k + 1) - 4 (substitution)
= 14k + 7 - 4
= 14k + 3
=2(7k +1) + 1
= 2m +1 is odd for integer m =7k + 1
Therefore, 7 x - 4 is odd.
Which of the following applies?
a. method of direct proof d. proof by contrapositive
b. method of indirect proof
c. proof by contradiction
e. b and d
8. What is wrong with the following
"proof" that -2 = 2, using backward reasoning?
Assume that -2 = 2.
(-2)2 = 22 Squaring both sides
4 = 4
Therefore -2 = 2.

9. It is important to ask a key question when attempting to
prove a statement. For the key question,
“How can I show that 2 lines in a plane are perpendicular?”
which of the following answer(s) is/are
incorrect? Explain why.
a. Show that the product of the slopes
of the two lines equal -1.
c. Show that the lines are on
adjacent sides of a rectangle.
d. None
b. Show that each of the lines is
perpendicular to a third line.
10. In class we mentioned that Key questions help to give
oneself direction when starting a proof.
Suppose you are asked to prove, “If a, b, and c are integers and
a. State a possible Key question that would help you to work
through this proof.

11. Show that the statement ”The product of 2 irrational
numbers is irrational” is false by finding a
counterexample.
12. Given a class of 25 students prove that at least 4 of them
were born on the same day of the
week.

13. Give a direct proof of the following: “If m and n have
different parities, then the sum of the m
and n is odd”.
14. Give a proof by contradiction of the following:
“If b is an odd integer, then b2 is odd”.
15. Prove:
If a, b and c are even integers, then abc is divisible
by 8.
16. Prove:
If x and y are rational numbers then x divided by y
is a rational number.

17. Prove or disprove that if m is an integer then
12
is an integer.
18. Prove:
If a and b are integers with a ≠ 0 and x is a real number
such that
ax2 + bx + b – a = 0, then a│b.
Note: a│b means that a divides b or a

is a factor of b.
19. Given that a and b are integers and a ≠ 0. Prove that if
a|300 and a|(b+300), then a|b.
(see number 14 for definition of a|b.)

The University of Maine at Augusta Name ______.docx

Recommended

Recommended

More Related Content

Similar to The University of Maine at Augusta Name ______.docx

Similar to The University of Maine at Augusta Name ______.docx (20)

More from christalgrieg

More from christalgrieg (20)

Recently uploaded

Recently uploaded (20)

The University of Maine at Augusta Name ______.docx