Introductory Discrete Mathematics

Introductory Discrete Mathematics
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1. List the (distinct) elements in each of the following sets:
{x Z | xy = 15 for y Z}Solution: C = { 1, 3, 5, 15, -1, -3, -5, -15} (both positive and negative of 15)
e. {a N | a <-4 and a >a}
Solution:It is impossible to list the elements since they are infinite in number. “N” refers to natural numbers and therefore “-4” is irrelevant.
2. List five elements in each of the following sets:
d. {nN|n2 + n is a multiple of 3}
Solution:The numbers are in the form n2+n=3p for some integer P hence:
n={3,5,6,8,9}
9a. List all the subsets of the set {a, b, c, d} that contain
Solution
i.) four elements – {a,b,c,d}ii.) three elements – {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}iii.) two elements – {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}iv.) one element – {a}, {b}, {c}, {d}v.) no elements – {0}
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3. Let A = {(-1,2), (4, 5), (0, 0), (6, -5), (5, 1), (4, 3)}.
a. {a + b | (a, b) A}
Solution: A = {1, 9, 0, 6, 7}.
b. {a | a >0 and (a, b) A for some b}
Solution:{4,6,5,4}
4. List the elements in the sets A = {(a, b) N x N|a < b, b < 3 } and B = {a/b | a, b {-1, 1, 2} }.
Solution:A = {(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)}
Solution:B = {(1,2), (2,1),(1,-1)}
10. (Explain your answers) The universal set for this problem is the set of students attending Miskatomic University. Using only the set theoretical notation we have introduced in this chapter, rewrite each of the following assertions.
Solutions
Computer science majors had a test on Friday.

CS⊆TNo math major ate pizza last Thursday.
M∩P=∅ Since M∩P ate Pizza last Thursday.
Some math majors did not eat pizza last Thursday.
is M∩Pc≠∅ Since M∩Pc did not eat Pizza last Thursday.
Those computer science majors who ate pizza on Thursday did not have a test on Friday ate pizza on Thursday.
is (CS∩TC)⊆P Since CS∩TC represents Computer Science Majors who did not have test on Friday.
Math or computer science majors who ate pizza on Thursday did not have a test on Friday.
is ((M∪CS)∩P)⊆TC Since ((M∪CS)∩P) are Math or Computer Science Majors who ate Pizza last Thursday.
15. Let A = {1, 2, 4, 5, 6, 9}, B = {1, 2, 3, 4}, and C = {5, 6, 7, 8}.
Draw a Venn diagram showing the relationship between these sets. Show which elements are in which region.
Solution
∅∅5,6
1,2,4

b. What are the elements in each of the following sets?
i. (A U B) ∩CSolution:
A U B= {1, 2, 3,4, 5, 6, 9}A U B∩C={5,6}ii. A B A={1, 2,4, 5, 6, 9}
iii. A U B A ∩C=A U B∩A ∩Cc={1, 2,3,4, 9}
iv. A⨁ CA U C-A∩C={5,6}
v. (A ∩C) x (A ∩B)={5,1,5,2,5,4,6,1,6,2,6,4}Page 56
Let B denote the set of books in a college library and S denote the set of students attending that college. Interpret the Cartesian product S x B. Give a sensible example of a binary relation from S to B.
f:S×B⟼0,1. Suppose, (x,y) is a set of students and books respectively, so that xϵS and yϵB, then f(x,y) is a binary function since one student can only read a book at a time.
Let A denote the set of names of streets in St. John’s, Newfoundland, and B denote the names of the residents of St. John’s. Interpret the Cartesian product A x B. Give a sensible example of a binary relation from A to B.
f:S×B⟼{0,1}. Suppose, (a,b) is a set of streets and names respectively, so that aϵA and bϵB, then f(a,b) is a binary function since an individual can only reside in one street at any given time.
Book Price Length
U $10 100 pages
W $25 125 pages
X $20 150 pages
Y $10 200 pages
Z $5 100 pages
12. Let A be the set of books for sale in a certain university bookstore and assume that among these are books with the following properties.
a. Suppose (a, b) R if and only if the price of book a is greater than or equal to the price of book b and the lengthof a is greater than or equal to the length of b. Is R reflexive? Symmetric? Antisymmetric? Transitive?
Reflexive
Workbook Page 12
1. List the members of the following set.
a. {x; x Z+, x<7}
{1,2,3,4,5,6}
Use a single Venn Diagram to illustrate the relation A B and B C. Is is true that A C?

Its true that A C
5. Mark each of the following as TRUE or FALSE.
b. A U B A
Solution. FALSE
9. Consider the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} together with subsets A = {1, 4, 7, 10}, B = {1,
2,3, 4, 5}, C ={2, 4, 6, 8}. Determine each of the following sets.
__
A∩C
Solution
A =2,3,5,6,8,9, C={2,4,6,8}A ∩C={2,6,8}______
c. A B
Solution
A⨁B=A U B-A∩B={2,3,5,7,10}A⨁B={1,4,6,8,9} 10. Consider the universal set U = {a, e, i, o, u, m, s, t, h} together with subjects A = {m, a, t, h}, B = {s,e,t},
and C = { a, e, i}. Determine each of the following sets.
__
B U C
Solution:
B∪C={e,o,u,m,h}