MATH 208 DISCRETE MATHEMATICS: LESSON 2 PROBLEMS
Problems are worth 20 points each.
(1) Let P (x) : x2 4. Determine the truth values of the following propositions. Assume the
domain for the variable is all positive integers: 1, 2, 3, 4, 5, and so on.
(a)
MATH 208 DISCRETE MATHEMATICS: LESSON 6 PROBLEMS
Problems are worth 20 points each.
(1) Let A be the set of people alive on earth. For each relation dened below, determine
if it is an equivalence relation on A. If it is, describe the equivalence classes.
MATH 208 DISCRETE MATHEMATICS: LESSON 10 PROBLEMS
Problems are worth 25 points each.
Warning: Its not unusual to nd these problems really tough. One diculty is that these problems
will make some demands on your algebra skills. Another diculty is just gett
MATH 208 DISCRETE MATHEMATICS: LESSON 11 PROBLEMS
Problems are worth 25 points each.
(1) Suppose there are two algorithms to solve a certain problem. Algorithm one has a
worst case scenario function w1 (n) = n2 , and algorithm two has worst case scenario
MATH 208 DISCRETE MATHEMATICS: LESSON 19 PROBLEMS
Problems are worth 20 points each.
(1) (a) Show that in any groups of eight people, at least two were born on the same
day of the week.
(b) Show that in any group of 80 people, at least 12 were born on the
MATH 208 DISCRETE MATHEMATICS: LESSON 3 PROBLEMS
Problems are worth 20 points each.
(1) List the members of the following sets.
(a) cfw_x Z x2 < 100 (think about negative numbers too).
Z|3
(b) cfw_x IN|x 14
(2) Use set-builder notation to give a descripti
MATH 208 DISCRETE MATHEMATICS: LESSON 2 PROBLEMS
Problems are worth 20 points each.
(1) Let P (x) : x2 4. Determine the truth values of the following propositions. Assume the
domain for the variable is all positive integers: 1, 2, 3, 4, 5, and so on.
(a)
MATH 208 DISCRETE MATHEMATICS: LESSON 9 PROBLEMS
Problems are worth 20 points each.
(1) List the rst ve terms of the sequence dened recursively by a0 = 2, and, for
n 1, an = a2 1.
n1
(2) List the rst ve terms of the sequence with initial terms u1 = 1 and
MATH 208 DISCRETE MATHEMATICS: LESSON 4 PROBLEMS
Problems are worth 20 points each.
Warning: These problems may be a challenge. Constructing proofs is not an easy skill to learn.
For these exercises, feel free to use familiar facts about integers. For exa
MATH 208 DISCRETE MATHEMATICS: LESSON 10 PROBLEMS
Problems are worth 25 points each.
Warning: Its not unusual to nd these problems really tough. One diculty is that these problems
will make some demands on your algebra skills. Another diculty is just gett
]oshua Klopfenstein
MATH 208 DISCRETE MATHEMATICS: LESSON 5 PROBLEMS
(1) Let A = {a, b, c, d} and R = {(a, a),(a, c),(b, d),(c, a),(c, c),(d, b)} be a relation on A.
Draw a digraph which represents R. You might want to review the deﬁnition of
digraph! Q C
M A T H 208 D IS C R E T E M A T H E M A T IC S : L E S SO N 1 P R O B L E M S
P ro b le m s are w orth 20 p oin ts each.
(1) Determine which of the following sentences are propositions.
(a) Today is Tuesday.
(b) W hy are yon whining?
(c) The Vikings are
I. 0 Yes H is pfoeogiiwn Bernou"-t
kc: or Palm I
b) 3 bemoan H is A $043 How V
0) (do in Comte Mn 7'3 no can), b proati,
a! A's: .
A) 1) be most (3,);an arm
3 '1 (q p\ q (-19 \I Q Dayad'i" vax
:3 P \I (19 J 7,) Dough MIXRON
E (F \I '1?) \ q) A5o30kh LVS
i
MATH 208 DISCRETE MATHEMATICS: LESSON 8 PROBLEMS
Problems are worth 20 points each.
(1) In words, x is the largest integer less than or equal to x. Complete the sentence:
In words, x is the smallest . . . . . .
Draw a (college algebra) graph of f (x) = x
MATH 208 DISCRETE MATHEMATICS: LESSON 9 PROBLEMS
Problems are worth 20 points each.
(1) List the rst ve terms of the sequence dened recursively by a0 = 2, and, for
n 1, an = a2 1.
n1
(2) List the rst ve terms of the sequence with initial terms u1 = 1 and
(1)
(a) a H b a and b are the same height.
Yes . Transitive relation: a H b => b H c =>a H c (a and bare ofsame height and band care ofsame
height, then a will have same height with c)
Yes. Symmetric relation: a H b =>b Ha (if a will same height with b, t
PROCTOR’S STATEMENT
This is to certify that wrote an
examination in the course _Math 208— under my personal supervision
and received no outside aid from any source whatsoever. The student was verified
through a picture 1D prior to taking the examination
PROCTORS STATEMENT
This is to certify th a t_ wrote an
examination in the course_ Math 208_ under my personal supervision
and received no outside aid from any source whatsoever. The student was verified
through a picture ID prior to taking the examination
MATH 208 DISCRETE MATHEMATICS: LESSON 12 PROBLEMS
Problems are worth 20 points each.
For the proofs requested below, use facts and theorem given in this lesson as justications.
Assume all letters represent integers.
(1) Mimic the proof given in the sample
MATH 208 DISCRETE MATHEMATICS: LESSON 11 PROBLEMS
Problems are worth 25 points each.
(1) Suppose there are two algorithms to solve a certain problem. Algorithm one has a
worst case scenario function w1 (n) = n2 , and algorithm two has worst case scenario
MATH 208 DISCRETE MATHEMATICS: LESSON 6 PROBLEMS
Problems are worth 20 points each.
(1) Let A be the set of people alive on earth. For each relation dened below, determine
if it is an equivalence relation on A. If it is, describe the equivalence classes.
MATH 208 DISCRETE MATHEMATICS: LESSON 13 PROBLEMS
Problems are worth 20 points each.
(1) Use
(a)
(b)
(c)
(d)
the Euclidean algorithm to compute gcd(a, b) in each case.
a = 233, b = 89
a = 1001, b = 11
a = 2457, b = 1452
a = 924, b = 780
(2) Compute gcd(98
MATH 208 DISCRETE MATHEMATICS: LESSON 10 PROBLEMS
Problems are worth 25 points each.
Warning: Its not unusual to nd these problems really tough. One diculty is that these problems
will make some demands on your algebra skills. Another diculty is just gett
MATH 208 DISCRETE MATHEMATICS: LESSON 8 PROBLEMS
Problems are worth 20 points each.
(1) In words, x is the largest integer less than or equal to x. Complete the sentence:
In words, x is the smallest . . . . . .
Draw a (college algebra) graph of f (x) = x
1.
2.
3.
4.
5.
6.
Lesson 11
a. Suppose the integers and are less than 0.
Since < 0, multiplying both sides of 0 > by will give 0 < according to
one of the rules of inequalities given in the lesson. A theorem provided in the
lesson tells us that 0
Lesson 6
1. Let A be the set of all living people
a. This is an equivalence relation on A. It is reflexive as every object is equivalent to
itself because all people are the same height as themselves. It is symmetric as
any person a has a height equiv
Lesson 4
1. Assume that m is even and n is odd. Show that m + n is odd. By definition of an even
number = 2 for some integer . By definition of an odd number = 2 + 1 for
some integer . Summing them gives + = 2 + 2 + 1 = 2 + 2 + 1 =
2 + +