1W
PROCTORS STATEMENT
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 p
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 verifi
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 verifi
5
5. Lesson 5
(1) Let A = cfw_a, b, c, d and R = cfw_(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 denition of digr
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 deman
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 deman
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 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 .
General instructions: Write up the solutions for the problems in each lesson. Scan the work into
a pdf le. No other le format is accepted. If there are multiple pages for an assignment, merge
the page
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 i
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 rela
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 w
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,
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
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
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 deman
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 .
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
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
(1) Write the following sets using the roster form:
(a) cfw_x Z|3 x2 < 100 (Careful, that is Z, not N!)
In layman's terms, list all the square numbers between 3 and 100 in the
domain of all integers (
(1) Write the following sets using the roster form:
(a) cfw_x Z|3 x2 < 100 (Careful, that is Z, not N!)
In layman's terms, list all the square numbers between 3 and
100 in the domain of all integers (
(1) Let A = cfw_a, b, c, d and R = cfw_(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!
a
b
c
d
2. Lesson 2
(1) Let P (x) : x2 4. The domain for x is all positive integers (1, 2, 3, . . .). Determine the
truth values of the following propositions.
(a) P (1)
p =1, 1^2 = 1 which is less then or eq
(1) Give a direct proof that the sum an even integer and an odd integer is odd.
Hint: Start by letting m be an even integer and letting n be an odd integer. That means
m = 2k for some integer k and n
Problems 1 through 5 are worth are worth 5 points each. Problem 6 is a bonus problem worth
3 points. The total score for a problem set cannot be greater than 25. Sample problems with
solutions are fou
2. Lesson 2
(1) Let P (x) : x2 4. The domain for x is all positive integers (1, 2, 3, . . .). Determine the
truth values of the following propositions.
(a) P (1)
p =1, 1^2 = 1 which is less then or eq
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
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 I