CS 23022 Practice Question for Final Eaxm Spring 2007
True/False
Indicate whether the statement is true or false.
____
1. A proof by the first principle of mathematical induction assumes that the inductive hypothesis is true.
____
2. To show that
is divisible by 3 for every positive integer n, the basis step in an inductive proof would be
to show that
____
is divisible by 3 when n equals 1.
3. In the second principle of mathematical induction, to show that proposition P(n) is true for all positive
integers n, after showing the basis step for to be true, we assume P(k) is true, where k is an integer
and use it to prove that
is true.
,
____
4. Let A = {grape, banana, apple} and B = {red, blue, green, yellow}. Let R = {(grape, blue), (banana, yellow),
(banana, yellow), (green, apple)}. Then R is a relation from A to B.
____
5. Let Z be the set of all integers and let R be a relation on
Then it is true that
____
____
7. Let
.
.
6. Let Z be the set of all integers and let R be a relation on
true that
defined by
defined by
. Then it is
.
be the set of all positive integers and R be a ternary relation on
defined by
. Then it is true that
____
8. Let
____
9. The digraph shown above has a total of four loops.
.
. Then the digraph shown above represents the relation
____ 10. Let R be a relation on set
.
defined by the rule
____ 11. Let P be the set of people. Then the relation R on
if
, defined by
.
. Then the range of R,
has
the domain, D(R): is the set of all people who have money owed to them.
____ 12. Let R be a relation from set X into set Y. The inverse of R , denoted by
by
.
is the relation from Y to X defined
____ 13. Let R be a relation from set A into set B, and let S be a relation from set B into set C. If there exists some
such that
and
, then
.
____ 14. Let R be a relation on a nonempty set . Then
.
____ 15. In a local election to select one board member, there are 3 Democrats, 5 Republicans, and 2 Independents
running for the same office. There are 30 possible outcomes to this election.
____ 16. If task can be performed in
ways and task
performing tasks and in succession is
can be performed in
.
ways, then the number of ways of
____ 17. A set A contains 5 distinct elements. The number of possible subsets that can be constructed from A is 120.
____ 18. Of 300 patients surveyed, 110 had high blood pressure, 73 had high cholesterol, and 42 suffered from both.
The number of patients that had neither high blood pressure nor high cholesterol was 159.
____ 19. On January 3, 100 shoppers were surveyed at the mall. It was determined that 67 came to get a book signed
by the author, 42 came to meet friends for lunch, and 26 came to do both. The number in the survey who did
not come for the book signing nor to have lunch was 17.
____ 20. The Pigeonhole Principle states that if
in the same hole.
or more pigeons are put into n holes, at least two pigeons are put
____ 21. Say you have five pairs of mittens, all pairs different colors, jumbled together in a drawer. You go into the
room without turning on the light. If you want to be sure to get the red pair, you must draw out 6 individual
mittens.
____ 22. The permutation of 7 distinct items, taken 3 at a time, is 343.
____ 23. A list of items implies a permutation whereas a set of items implies a combination.
____ 24. The number of combinations of 5 different objects taken 3 at a time is 10.
____ 25. The number of combinations of 5 different objects taken 5 at a time is equal to 120.
____ 26. A child has 3 pennies, 5 nickels, 2 dimes, and 1 quarter. The number of groups of 3 coins that include none
of the dimes is 84.
____ 27. Five candidates for mayor are to participate in a debate. Candidates are lined up on stage behind podiums
facing the audience. If two of the candidates refuse to be positioned next to each other, the candidates can be
arranged in 60 different ways.
____ 28. The coefficient of
____ 29. The coefficient of
in
is 15.
in the expansion of
____ 30. In Pascal’s triangle, if rows are labeled
numbered
, then the value
is 126.
and positions (from left to right) in a row are
would be in row 5, position 4.
____ 31. The 6th term of the Fibonacci sequence is 13.
____ 32. The initial conditions for the Fibonacci sequence are
and
Multiple Choice
Identify the choice that best completes the statement or answers the question.
.
____ 33. To use the first principle of mathematical induction to prove that one can climb a ladder, the inductive
hypothesis would be _________________________.
a. to show that the first rung of the ladder can be climbed
b. to show that the first k rungs of the ladder can be climbed
c. to assume that the first k rungs of the ladder (
) can be climbed
st
d. to show that the
rung of the ladder can be climbed given that the first k rungs can be
climbed
____ 34. Using the first principle of mathematical induction to prove that
be _________________________.
a. to show that
b.
c.
to show that
, where k is an integer and
to show that
for all
, implies
, the inductive step would
is true
for all
d.
to show that
for all
____ 35. Let
be the statement that every positive integer greater than 1 can be factored into primes. Proving this
fact using the second principle of mathematical induction, the inductive step would be to show
_________________________.
a. if
is true where n is an integer with
, then
is true
b. if
is true where n is an integer with
, then
is true
c. that
is true for some
d. if all
for all
are true, then
is true
____ 36. Let
and
. If relation
, then which of the following
is not correct?
a.
c.
b.
d. Both B and C
____ 37. For a relation R from set A into set B, which of the following is true?
a.
c.
b.
d. Both A and C
____ 38. Let Z be the set of all integers and R a relation on
defined by
following ordered pairs belongs to R?
a.
c.
b.
d. Both A and B
____ 39. Let
be the set of all rational numbers and R a relation on
defined by
which one of the following does not satisfy the relation?
a.
c.
b. 1 R 2
d.
____ 40. Match the arrow diagram below with the correct binary relation.
. Which of the
. Then
a.
c.
b.
d.
____ 41. Which of the relations below matches the following digraph?
a.
b.
c.
d.
____ 42. Let
and
_________________________.
a.
b.
. Let
if and only if
. the domain,
is
c.
d.
____ 43. Let A be the set of all real numbers and relation R defined on
by
. Which
of the following is true?
a. Domain of R ,
b. Range of R ,
c. Domain of R ,
d. Both A and B
____ 44. Let set
. Let R be a relation defined on
the set _________________________.
a.
b.
c.
by
. Then
is
d.
____ 45. Let
,
defined as follows:
____ 46.
____ 47.
____ 48.
____ 49.
____ 50.
____ 51.
____ 52.
____ 53.
____ 54.
and
. Let the relations R from
and S from
be
Then all of the following are true EXCEPT _________________________.
a.
c.
b.
d.
You have 3 trees, 6 bushes, and 5 shrubs to plant. You must decide which item to plant first. If all the plants
are unique in some way, how many choices are there?
a. 14
c. 3
b. 90
d. 846
A binary digit (bit) is either a 0 or a 1. If 8 bits make up a byte, how many different bytes are possible?
a. 16
c. 256
b. 8
d. 64
A computer password is to consist of 4 digits followed by a letter. The first digit must not be zero. How
many passwords can be created?
a. 26,000
c. 21,060
b. 23,400
d. 16,848
For sets A and B,
. Find
.
a. 68
c. 22
b. 15
d. 8
A survey of 500 college students indicated that 320 drank alcohol, 150 smoked, and 70 neither drank nor
smoked. The number of students who both drank and smoked was _________________________.
a. 30
c. 110
b. 40
d. 80
Count Dracula has five pairs of shoes, all different, jumbled together in his closet. He goes in to get a pair
and because vampires don’t dare get in the sunlight, he fumbles in the dark. How many single shoes must he
take with him to be certain to have a matching pair among them?
a. 3
c. 5
b. 6
d. 10
Suppose there are 70 people in a club. Then at least _____ of them must have their birthday in the same
month.
a. 1
c. 5
b. 11
d. 6
Which of the following represents a permutation?
a. The 5 oldest coins in your collection.
b. The Democrats in congress.
c. The 5 most recent inductees into the Music Hall of Fame.
d. An arrangement of paintings on your wall.
An algebra class has 30 students. Four students will be selected to go to the board and solve 4 problems. If
each student solves only one problem, how many ways can this be done?
a. 27,405
c.
b. 657,720
____ 55. Evaluate
a. 21
b. 84
d.
.
c. 504
d. 720
____ 56. A bag contains three red marbles, three blue ones, five green ones, and two yellow ones. How many sets of
four marbles are possible?
a. 715
c. 330
b. 17,160
d. 90
____ 57. From a group of 9 children playing soldiers, how many ways can a general, 2 lieutenants, a sergeant, and a
corporal be selected?
a. 7,650
c. 181,440
b. 15,120
d. 362,880
____ 58. Jason has won a contest that entitled him to select 5 computer games from a list of 12 or 2 reference books
from a list of 20. How many ways can he make his selection?
a. 10
c. 782
b. 150,480
d. 95,420
____ 59. The coefficient of x in
a. 4
b. 32
is _________________________.
c. 16
d. 64
____ 60. The coefficient of
in
is _________________________.
a. 1024
c. 4320
b. 5760
d. 1620
____ 61. Pascal’s Identity can be stated as follows, where n and r are integers such that
_________________________.
a.
c.
b.
d.
____ 62. Assume rows are numbered starting with 1. The fifth tow of Pascal’s triangle contains the values
_________________________.
a. 1 3 5 3 1
c. 1 5 10 10 5 1
b. 1 4 6 4 1
d. 1 2 3 2 1
Problem
63. Prove by induction: For all positive integers n, 5n-1 is divisible by 4.
.
CS 23022 Practice Question for Final Eaxm Spring 2007
Answer Section
TRUE/FALSE
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135
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175
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183
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417
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135
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175
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MULTIPLE CHOICE
33.
34.
35.
36.
37.
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C
B
D
C
D
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D
A
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D
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A
C
B
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B
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457
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PROBLEM
63. ANS:
Let P(n) be : 4|(5n - 1)
Prove P(n) is true for all integers n>0
Base Case: P(0) is 4|(51 - 1) = 4|(4) is true, therefore P(0) is true
I.H. : P(k) is true for some integer k=n>0
I.S. : Prove P(k+1) is true , i.e 4|(5k+1 - 1)
(5k+1 - 1) = 5 x (5k - 1) + 4
By I.H. 4|(5 x (5k - 1))
also 4|4
then 4|(5 x (5k - 1) + 4)
therefore P(k+1) is true and P(n) is true for all integers n>0.
PTS: 1