discrete math Assignment
Mohsin
mat_280_exam_1__09-26-161.pdf
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
2. Write the truth table for the proposition (r q) (p → 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 proposition using only pq and the connective V with the given truth table.
p
q ?
T T F
T F F
F T F
F F T
4. Determine whether p (q r) is equivalent to q (p 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: ________________________
5. Write a proposition equivalent to ( p q) using only pq and the connective . . Support your
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
6. Prove that q p and its contrapositive are logically 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?
(q r ) → (p s)
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
…”)
If the air temperature is above 75 and it is raining, we will not 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
In the questions below P(xy) means “x and y are real numbers such that x 4y 30”. Determine the truth
value of the statement. Explain your answer.
16. xyP(xy).
17. xyP(xy).
18. In the questions below suppose that Q(x) is “x 4 3x”, where x is a real number. Find the truth
value of the statement.
a. Q(2). b. xQ(x). c. xQ(x).
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.
a. xyP(xy). b. yx (x ≤ y (P(x,y)).
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.
Write the statement using these predicates and any 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. Write the statement using these predicates and any 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.
a. x(C(x) F(x)). Circle 1, 2, or 3
b. x(C(x) F(x)). Circle 1, 2, or 3
c. x(F(x) C(x)) 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.
