Sample MT 2 CS23022 Spring 2007
True/False
Indicate whether the sentence or statement is true or false.
____
1. The truth value of the statement, “If 2 is not an even integer, then 5 is not a positive integer.” is?
____
2. Assume it is true that: “Garfield likes lasagna.”, “Odi likes potatoes.”, and “Snoopy can fly a plane.”.
Then, determine the truth value of the statement below:
“Either Garfield likes lasagna and Snoopy can fly a plane, or Odi likes potatoes.”
____
3. The proposition,
, is equivalent to the proposition,
.
____
4. The proposition,
____
5. “If Debbie is elected class president, then Susan is elected vice president and Heather is elected treasurer.
Susan is not elected vice president. Therefore, Debbie is not elected class presdent.” Is this a valid
argument?
____
6. Let
of
, is a contradiction.
, and let the domain of x be the set of all real numbers. Then, the universal quantification
is:
,
____
7. Let
be a predicate and D be the domain of discourse. Then
true for just one value of x in the domain, D.
would be true provided that
is
____
8. To construct a direct proof of the implication,
, where D is the domain of discourse, we
select an arbitrary member, a, of the domain, D, and show that
is true.
____
9. To show that the implication,
show that the negation of q is true.
, is true, we can assume that the negation of p is true and use this to
____ 10. Proof by contradiction is based upon the fact that if r is any statement, then
____ 11. Let the Boolean expression
. Suppose
____ 12. Let the Boolean expression
must be a tautology.
and
. Then
. Suppose
.
. Then
.
____ 13. Let the Boolean expression
. Suppose
,
, and
. Then
.
____ 14. Let the Boolean expression
____ 15. Let
,
, and
and
be Boolean variables. Then
____ 16. For every Boolean expression
____ 17. The Boolean expression
. Then
.
, you can derive a dual expression by exchanging +’s for ’s and 0’s for 1’s.
is a minterm in the variables x, y, z, and w.
.
____ 18. Consider the following truth table for the Boolean function
. There will be 5 minterms used to
arrive at the DNF.
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
1
0
1
1
1
0
0
____ 19. The justification for the second step in the following partial proof is DeMorgan’s law.
____ 20. Let B be a Boolean algebra with 3 atoms. Then the number of elements of B is 9.
____ 21. A logic circuit consists of basic components called gates.
____ 22. An AND or OR gate can have more than two inputs.
____ 23. Two combinatorial circuits having inputs
and single output are equivalent if, whenever the
circuits receive the same inputs, they give the same output.
____ 24. If x and y are the inputs, then the output for a NOR gate is
.
____ 25. On method of arriving at the simplest circuit is to use a pictorial device called a Karnaugh map.
____ 26. The following is a correct K-map of the Boolean expression
____ 27. The circuit diagram above corresponds to the Boolean expression
.
.
____ 28. Given the circuit diagram above, if
,
, and
____ 29. Given the circuit diagram above, the output if
,
, then the output
, and
is
.
.
Multiple Choice
Identify the letter of the choice that best completes the statement or answers the question.
____ 30. Which of the following is correct if p and q are statements?
a.
is false if at least one p or q is false
b.
is false if p is false
c.
is false if both p and q are false
d.
is false if at least one p or q is false
____ 31. Which of the following is true?
a. 2 is a prime number and -5 is a nonnegative number.
b. 5 is a negative integer if and only if pigs can fly.
c. 12 is a prime number or -4 is not a negative number.
d. If 3 is not an even integer, then -3 is not a negative integer.
____ 32. The precedence order of logical connectives, listed in the order of being carried out first to last is correct for
which of the following?
a.
c.
b.
d.
____ 33. Let p, q, and r be propositions. What would make the following compound statement,
, false?
a. Make p true
b. Make r true
c. Make both q and r false
d. Either choices A or C will make it false
____ 34. Use a truth table to determine if the following statement is a tautology, contradiction, or neither:
a. tautology
c. neither
b. contradiction
____ 35. The following argument form is called _________________________.
a. modus tollens
c. disjunctive syllogism
b. modus ponens
d. hypothetical syllogism
____ 36. Supply the missing statement in the following argument:
a.
c.
b.
d.
____ 37. To determine whether the argument is valid, you would need to construct a truth table for which of the
following?
a.
b.
c.
d.
____ 38. Let
a.
b.
denote the sentence:
____ 39. An indirect proof of the statement, “if
_________________________.
a.
b.
. Which of the following is true in the domain of all integers?
c.
d.
, then
” , would require showing that
c.
d.
____ 40. A man is accused of robbing a bank. During the trial, the man being accused offers the following argument in
own defence:
“Suppose I actually did commit the robbery. Then it follows that at the time of the robbery, I would have had
to be present at the scene of the crime. In fact, at the time of the robbery, I was performing as a singer in front
of 2000 people, as any of them can testify. Therefore the assumption that I committed the robbery is false
since I could not have been in both places at the same time. So, I did not rob this bank.”
Which of the following applies?
a. method of direct proof
c. proof by contradiction
b. method of indirect proof
d. proof by induction
____ 41. To show that p, q, and r are equivalent statements, we need to show that _________________________.
a.
b.
c.
d. Either choice A or C would be correct
____ 42. Complete the last column of the truth table for the Boolean expression:
x
1
1
0
0
y
1
0
1
0
(x´ + y´) + x
a.
b.
c.
d.
____ 43. Complete the last column of the truth table for the Boolean expression:
x
1
1
0
0
y
1
0
1
0
(x · y) · x´
a.
b.
c.
d.
____ 44. Complete the last column of the truth table for the Boolean expression:
x
y
z
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1
0
a.
(x · y)´ + (z · x ´
)
b.
c.
d.
____ 45. Let and be Boolean variables. Then all the following assertions hold EXCEPT
_________________________.
a.
c.
b.
d.
____ 46. Let
a.
b.
,
, and
be Boolean variables. Which of the following is NOT true?
c.
d.
____ 47. Consider the following truth table for the Boolean expression
a.
b.
. Find the disjunctive normal form.
c.
d.
____ 48. The following graphic represents a(n) _________________________ gate.
a. AND
c. NAND
b. OR
d. NOT
____ 49. The following graphic represents a(n) _________________________ gate.
a. AND
c. NAND
b. OR
d. NOT
____ 50. The following graphic represents a(n) _________________________ gate.
a. AND
c. NAND
b. OR
d. NOT
____ 51. The circuit diagram below corresponds to which Boolean expression?
a.
b.
c.
d.
____ 52. The circuit diagram for the Boolean expression
a.
c.
is _________________________.
b.
d.
____ 53. The circuit diagram below corresponds to which Boolean expression?
a.
b.
c.
d.
____ 54. Given the circuit diagram below, all the following inputs for x, y, and z, respectively, will produce an output
of 1 EXCEPT _________________________.
a. 1 1 1
c. 0 1 0
b. 1 0 1
d. 0 0 0
____ 55. Complete the truth table for the circuit diagram below:
a. 0
1
1
0
1
0
0
1
b. 1
0
0
0
1
0
1
0
c. 0
1
1
1
0
1
0
1
d. 0
1
1
1
0
1
0
0
____ 56. The standard symbol corresponding to the truth table that follows is _________________________.
a. AND
c. NOR
b. NAND
d. OR
____ 57. Given the circuit diagram below, predict the output for s and c if
and
.
a.
c.
b.
d.
____ 58. Find the minimized sum-of-products Boolean expression corresponding to the following Karnaugh map:
a.
c.
b.
d.
____ 59. Find the minimized sum-of-products Boolean expression corresponding to the following Karnaugh map:
a.
b.
c.
d. Both A and B are correct
Short Answer
60. How many atoms are in the Boolean Algebra based on (4) ?
61. Consider two elements of (3), a= (x1,y1, z1) and b = (x2,y2,z2).
a) How is a • b defined ?
b) Evaluate c = a • b at a = ( 0,1,1 ) and b = (1,0,1)
.
62. How many functions are in the Boolean Algebra FUN(B(3)) ?,Why?
Problem
63. Given that
equivalent statements.
p
q
and that
, construct a truth table to show that A and B are
Sample MT 2 CS23022 Spring 2007
Answer Section
TRUE/FALSE
1.
2.
3.
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5.
6.
7.
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12.
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14.
15.
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18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
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T
T
F
T
T
F
T
T
F
T
T
F
T
T
F
T
F
T
F
F
T
T
T
F
T
F
F
T
F
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1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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29
30
37
35
46
56
56
64
66
67
772
772
772
773
773
775
776
779
789
792
795
796
801
804
803
814
798
798
798
PTS:
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1
1
1
1
1
1
1
1
1
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28
28
34
28
35
47
47
45
59
MULTIPLE CHOICE
30.
31.
32.
33.
34.
35.
36.
37.
38.
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
ANS:
D
B
C
D
A
D
C
B
B
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
ANS:
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D
C
D
C
D
B
A
D
D
D
A
B
C
B
A
B
C
B
D
C
D
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1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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66
67
69
770
770
779
773
773
777
795
795
795
798
798
798
798
798
803
806
815
815
SHORT ANSWER
60. ANS:
16
PTS: 1
61. ANS:
a) a • b = (x1• x2, y1• y2, z1• z2)
b) c = (0, 0, 1)
PTS: 1
62. ANS:
256
For each of the 8 atomic minterms, the function value can be either True or False.
This gives 2x2x2x2x2x2x2x2 = 256 different functions
PTS: 1
PROBLEM
63. ANS:
e
PTS: 1