Math 163
Test 1 Spring 2007
Name______________________
Show your work. Give appropriate formula and indicate the answers.
2. If it is raining, then it is raining.
Ans: True
3. If 1  0, then 3  4.
Ans: True
4. If 2  1  3, then 2  3  1.
Ans: True
5. If 1  1  2 or 1  1  3, then 2  2  3 and 2  2  4.
Ans: False
6. .Write the truth table for the proposition (r  q)  (p  r).
q Ans:
r (r
p  q)  (p 
r)
T T T T
T T F T
T F T F
T F F T
F T T T
F T F F
F F T F
F F F F
.Write the truth table for the proposition (r  q)  (p  r).
q Ans:
r (r
p  q)  (p 
r)
T T T T
T T F T
T F T F
T F F T
F T T T
F T F F
F F T F
F F F F
8.Find a proposition with three variables p, q, and r that is true when p and r are true and
q is false, and false otherwise
Ans: (a) p  q  r.
10.Find a proposition with three variables p, q, and r that is never true
Ans: (p  p)  (q  q)  (r  r).
11. Find a proposition using only pq and the connective  with the given truth
table.
p q ?
T T F
T F T
F
T T
F
F F
Ans: (p  q)  (p  q).
13. Determine whether p  (q  r) is equivalent to (p  q)  r.
Ans: Not equivalent. Let p, q, and r be false.
14. Determine whether (p  q)  (p  q)  q.
Ans: Both truth tables are identical:
p q (p  q)  (p  q
q)
T T T
T
T F F
F
F T T
T
F F F
F
17.Prove that the proposition “if it is not hot, then it is hot” is equivalent to “it is hot”.
Ans: Both propositions are true when “it is hot” is true and both are false when
“it is hot” is false.
18. Write a proposition equivalent to p  q using only pq and the connective: .
Ans: p  q.
19. Write a proposition equivalent to p  q using only pq and the connective .
Ans: (p  q).
23. Determine whether the following two propositions are logically equivalent: p 
(q  r)p  (r  q).
Ans: Yes.
25. Determine whether this proposition is a tautology: ((p  q)  p)  q.
Ans: No.
26. Determine whether this proposition is a tautology: ((p  q)  q)  p.
Ans: Yes.
34.Write the contrapositive, converse, and inverse of the following: If you try hard, then
you will win.
Ans: Contrapositive: If you will not win, then you do not try hard. Converse: If
you will win, then you try hard. Inverse: If you do not try hard, then you
will not win.
35. Write the contrapositive, converse, and inverse of the following: You sleep late if
it is Saturday.
Ans: Contrapositive: If you do not sleep late, then it is not Saturday. Converse: If
you sleep late, then it is Saturday. Inverse: If it is not Saturday, then you do
not sleep late.
40.Using c for “it is cold” and d for “it is dry”, write “It is neither cold nor dry” in
symbols.
Ans: c  d.
41. Using c for “it is cold” and r for “it is rainy”, write “It is rainy if it is not cold” in
symbols.
Ans: c  r.
36.It is Thursday and it is cold.
Ans: It is not Thursday or it is not cold.
37. I will go to the play or read a book, but not both.
Ans: I will go to the play and read a book, or I will not go to the play and not
read a book.
38. If it is rainy, then we go to the movies.
Ans: It is rainy and we do not go to the movies.
45.On the island of knights and knaves you encounter two people, A and B. Person A
says, "B is a knave." Person B says, "We are both knights." Determine whether
each person is a knight or a knave.
Ans: A is a knight, B is a knave.
46.On the island of knights and knaves you encounter two people. A and B. Person A
says, "B is a knave." Person B says, "At least one of us is a knight." Determine
whether each person is a knight or a knave.
Ans: A is a knave, B is a knight.
Use the following to answer questions 58-59:
In the questions below P(xy) means “x and y are real numbers such that x  2y  5”.
Determine whether the statement is true.
58. xyP(xy).
Ans: True, since for every real number x we can find a real number y such that x
 2y  5, namely y  (5  x)2.
59. xyP(xy).
Ans: False, if it were true for some number x0, then x0 = 5 -2y for every y, which
is not possible.
Use the following to answer questions 60-62:
In the questions below P(mn) means “m ≤ n”, where the universe of discourse for m and
n is the set of nonnegative integers. What is the truth value of the statement?
60. nP(0n).
Ans: True
61. nmP(mn).
Ans: False
62. mnP(mn).
Ans: True
In the questions below suppose the variable x represents students and y represents
courses, and:
U(y): y is an upper-level course M(y): y is a math course F(x): x is a freshman
B(x): x is a full-time student T(xy): student x is taking course y.
Write the statement using these predicates and any needed quantifiers.
69. Eric is taking MTH 281.
Ans: T(Eric, MTH 281).
70. All students are freshmen.
Ans: xF(x).
71. Every freshman is a full-time student.
Ans: x(F(x)  B(x)).
72. No math course is upper-level.
Ans: y(M(y)  U(y)).
Use the following to answer questions 73-75:
In the questions below suppose the variable x represents students and y represents
courses, and:
U(y): y is an upper-level course M(y): y is a math 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.
73. Every student is taking at least one course.
Ans: xy T(xy).
74. There is a part-time student who is not taking any math course.
Ans: xy[A(x)  (M(y)  T(xy))].
75. Every part-time freshman is taking some upper-level course.
Ans: xy[(F(x)  A(x))  (U(y)  T(xy))].
In the questions below assume that the universe for x is all people and the universe for y
is the set of all movies. Write the statement in good English, using the predicates
S(xy): x saw y L(xy): x liked y.
Do not use variables in your answer.
111. yS(Margarety).
Ans: There is a movie that Margaret did not see.
112. yxL(xy).
Ans: There is a movie that everyone liked.
113. xyL(xy).
Ans: Everyone liked at least one movie.
9.Prove that A  (B  C)  (A  B)  (A  C) by giving a Venn diagram proof.
Ans:
.Prove that A  B  A  B by giving a Venn diagram proof.
Ans:
Use the following to answer questions 12-15:
Use a Venn diagram to determine which relationship, , =, , is true for the pair of sets.
12. A  B, A  (B  A).
Ans: .
13. A  (B  C), (A  B)  C.
Ans: .
14. (A  B)  (A  C), A  (B  C).
Ans: .
15. (A  C)  (B  C), A  B.
Ans: .
22.A  (B  C)  (A  B)  C.
Ans: False
26. A  B  A  A .
Ans: True
27. If A  C  B  C, then A  B.
Ans: False
29.If A  B  A  B, then A  B.
Ans: True
30. If A  B  A, then B  A.
Ans: False
31. There is a set A such that  P(A)   12.
Ans: False
32. A  A  A.
Ans: False
38. Suppose U = {1, 2, ..., 9}, A = all multiples of 2, B = all multiples of 3, and C =
{3, 4, 5, 6, 7}. Find C - (B - A).
Ans: {4,5,6,7}.
39. Suppose S = {1, 2, 3, 4, 5}. Find P( S ) .
Ans: 32.
In the questions below suppose A  xy and B  xx. Mark the statement TRUE or
FALSE
40. x  B.
Ans: False
41.   P(B).
Ans: True
42. x  A  B.
Ans: False
43.  P(A)   4.
Ans: True
In the questions below suppose A  abc. Mark the statement TRUE or FALSE.
44. bc  P(A).
Ans: True
45. a  P(A).
Ans: True
46.   A.
Ans: True
47.   P(A).
Ans: True
48.   A  A.
Ans: True
49. ac  A.
Ans: True
50. ab  A  A.
Ans: False
51. (cc)  A  A.
Ans: True
In the questions below suppose A  abc. Mark the statement TRUE or FALSE.
44. bc  P(A).
Ans: True
45. a  P(A).
Ans: True
46.   A.
Ans: True
47.   P(A).
Ans: True
48.   A  A.
Ans: True
49. ac  A.
Ans: True
50. ab  A  A.
Ans: False
51. (cc)  A  A.
Ans: True
73.x  x  Z and x2  10.
Ans: 7.
74. P(abcd), where P denotes the power set.
Ans: 16.
75. 1357….
Ans: Infinite.
76. A  B, where A  12345 and B  123.
Ans: 15.
77. x  x  N and 9x2  1  0.
Ans: 0.
78. P(A), where A is the power set of abc.
Ans: 256.
79. A  B, where A  abc and B  .
Ans: 0.
83.1101001000….
Ans: Infinite.
84. S  T, where S  abc and T  12345.
Ans: 15.
85. x  x  Z and x2  8.
Ans: 5.
83.1101001000….
Ans: Infinite.
84. S  T, where S  abc and T  12345.
Ans: 15.
85. x  x  Z and x2  8.
Ans: 5.
 x  2 if x  0
94.G  R  R where G ( x)  
 x  1 if x  4
Ans: Not a function; the cases overlap. For example, G(1) is equal to both 3 and
0.
 x 2 if x  2
95. f  R  R where f ( x)  
 x  1 if x  4
Ans: Not a function; f(3) not defined.
96. G  Q  Q where G(pq)  q.
Ans: Not a function; f(12)  2 and f(24)  4.
97. Give an example of a function f  Z  Z that is 1-1 and not onto Z.
Ans: f(n)  2n.
98. Give an example of a function f  Z  Z that is onto Z but not 1-1.
n
Ans: f (n)    .
2
99. Give an example of a function f  Z  N that is both 1-1 and onto N.
2n if n  0
Ans: f (n)  
 2n  1 if x  0
103.Suppose f  N  N has the rule f(n)  4n  1. Determine whether f is 1-1.
Ans: Yes.
104. Suppose f  N  N has the rule f(n)  4n  1. Determine whether f is onto N.
Ans: No.
105. Suppose f  Z  Z has the rule f(n)  3n2  1. Determine whether f is 1-1.
Ans: No.
106. Suppose f  Z  Z has the rule f(n)  3n  1. Determine whether f is onto Z.
Ans: No.
107. Suppose f  N  N has the rule f(n)  3n2  1. Determine whether f is 1-1.
Ans: Yes.
108. Suppose f  N  N has the rule f(n)  4n2  1. Determine whether f is onto N.
Ans: No.
109. Suppose f  R  R where f(x)  x2.
(a) Draw the graph of f.
(b) Is f 1-1?
(c) Is f onto R?
Ans: (a)
(b) No.
(c) No.
110. Suppose f  R  R where f(x)  x2.
(a) If S  x  1  x  6, find f(S).
(b) If T  345, find f1(T).
Ans: (a) 0123
(b) [612).
111. Determine whether f is a function from the set of all bit strings to the set of
integers if f(S) is the position of a 1 bit in the bit string S.
Ans: No; there may be no 1 bits or more than one 1 bit.
112. Determine whether f is a function from the set of all bit strings to the set of
integers if f(S) is the number of 0 bits in S.
Ans: Yes.
113. Determine whether f is a function from the set of all bit strings to the set of
integers if f(S) is the largest integer i such that the ith bit of S is 0 and f(S)  1
when S is the empty string (the string with no bits).
Ans: No; f not defined for the string of all 1's, for example S  11111.
114. Let f(x)  x33. Find f(S) if S is:
(a) 210123.
(b) 012345.
(c) 15711.
(d) 261014.
Ans: (a) 31029.
(b) 0292141.
(c) 041114443.
(d) 272333914.
109. Suppose f  R  R where f(x)  x2.
(a) Draw the graph of f.
(b) Is f 1-1?
(c) Is f onto R?
Ans: (a)
(b) No.
(c) No.
110. Suppose f  R  R where f(x)  x2.
(a) If S  x  1  x  6, find f(S).
(b) If T  345, find f1(T).
Ans: (a) 0123
(b) [612).
In the questions below suppose g  A  B and f  B  C where A  B  C  1234, g
 (14)(21)(31)(42) and f  (13)(22)(34)(42).
124. Find f  g.
Ans: (12)(23)(33)(42).
125. Find g  f.
Ans: (11)(21)(32)(41).
126. Find g  g.
Ans: (12)(24)(34)(41).