Venn Diagrams and the Equality of Set Expressions Two sets are equal if and only if they each represent the same region(s) on a Venn diagratm. Venn diagrams can be used to verify each of the following properties. CHAPTER 2 Reviea Exerses For all sets A, B, and C: 101 See Example 6 and Check Youue Progress 6 on page 71, and then try Exercises 45 to 48 on page 102. Commutative Properties ANB=BNA AUB=BU A Associative Properties (AN B) N C = AN (B n C) An(B U C) = (A N B) U (A n C) (AU B) U C = A U (B U C) AU (Bnc)= (AU B)n (AU C) Distributive Properties 2.4 Applications of Sets Applications Many counting problems that arise in applications involving surveys can be solved by using sets and Venn diagrams. The Inclusion-Exclusion Formula For all finite sets A and B, See Examples 1 and 2 on page 81, and then try Exercises 53 and 54 on page 102. n(A U B) = n(A) + n(B) – n(A N B) See Examples 3 and 4 on page 83, and then 2.5 Infinite Sets Exercises 55 and 56 on page 103. try One-to-One Correspondence and Equivalent Sets Two sets A and B re equivalent, denoted by A ~ B, if and only if A and B can be placed See Examples 1 and 3 on pages 90 and 92, and then try Exercises 57 to 60 on page 103. in a one-to-one correspondence. Infinite Set A set is an infinite set if it can be placed in a one-to-one correspondence with a proper subset of itself. See Example 2 on page 91, and then try Exercises 61 and 62 on page 103. CHAPTER 2 REVIEW EXERCISES In Exercises 11 and 12, state whether each of the fol- lowing sets are equal, equivalent, both, or neither. 11. {2, 4, 6, 8}, {x\x E N and x < 5} - In Exercises 1 to 6, use the roster method to represent each set. 1. The set of months of the year with a name that starts 12. {8, 9}, the set of single digit whole numbers greater than 7 with the letter J 2. The set of states in the United States that do not share a common border with another state I In Exercises 13 to 16, determine whether the statement is A.A2 3. The set of whole numbers less than 8 64 true or false. 13. {3} E {1,2, 3, 4} 4. The set of integers that satisfy x 14. -11 EI 15. {a, b, c} ~ {1, 5, 9} 5. The set of natural numbers that satisfy x + 3 <7 6. The set of counting numbers larger than -3 and less than or equal to 6 In Exercises 17 to 24, let U = {2, 6, 8, 10, 12, 14, 16, 18}, A = {2, 6, 10}, B = {6, 10, 16, 18}, and C = {14, 16}. Find each of the following. 16. The set of small numbers is a well-defined set %3D - In Exercises 7 to 10, use set-builder notation to write 18. AUB each set. 17. ANB 20. BUC' 7. The set of integers greater than -6 19. A' N C 22. (A U C)' N B' 8. {April, June, September, November} 21. A U (B N C) 24. (AUBUC)' 9. {Kansas, Kentucky} 23. (A N B')' 10. {1, 8, 27, 64, 125} froper given storlemen

Algebra and Trigonometry (6th Edition)
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ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Venn Diagrams and the Equality of Set Expressions Two sets are equal
if and only if they each represent the same region(s) on a Venn diagratm.
Venn diagrams can be used to verify each of the following properties.
CHAPTER 2
Reviea Exerses
For all sets A, B, and C:
101
See Example 6 and Check Youue Progress
6 on page 71, and then try Exercises 45
to 48 on page 102.
Commutative Properties
ANB=BNA
AUB=BU A
Associative Properties
(AN B) N C = AN (B n C) An(B U C) = (A N B) U (A n C)
(AU B) U C = A U (B U C) AU (Bnc)= (AU B)n (AU C)
Distributive Properties
2.4 Applications of Sets
Applications Many counting problems that arise in applications
involving surveys can be solved by using sets and Venn diagrams.
The Inclusion-Exclusion Formula For all finite sets A and B,
See Examples 1 and 2 on page 81, and
then try Exercises 53 and 54 on page 102.
n(A U B) = n(A) + n(B) – n(A N B)
See Examples 3 and 4 on page 83, and
then
2.5 Infinite Sets
Exercises 55 and 56 on page 103.
try
One-to-One Correspondence and Equivalent Sets Two sets A and B
re equivalent, denoted by A ~ B, if and only if A and B can be placed
See Examples 1 and 3 on pages 90
and 92, and then try Exercises 57 to 60 on
page 103.
in a one-to-one correspondence.
Infinite Set A set is an infinite set if it can be placed in a one-to-one
correspondence with a proper subset of itself.
See Example 2 on page 91, and then try
Exercises 61 and 62 on page 103.
CHAPTER 2 REVIEW EXERCISES
In Exercises 11 and 12, state whether each of the fol-
lowing sets are equal, equivalent, both, or neither.
11. {2, 4, 6, 8}, {x\x E N and x < 5}
- In Exercises 1 to 6, use the roster method to represent
each set.
1. The set of months of the year with a name that starts
12. {8, 9}, the set of single digit whole numbers greater
than 7
with the letter J
2. The set of states in the United States that do not share a
common border with another state
I In Exercises 13 to 16, determine whether the statement is
A.A2
3. The set of whole numbers less than 8
64
true or false.
13. {3} E {1,2, 3, 4}
4. The set of integers that satisfy x
14. -11 EI
15. {a, b, c} ~ {1, 5, 9}
5. The set of natural numbers that satisfy x + 3 <7
6. The set of counting numbers larger than -3 and less
than or equal to 6
In Exercises 17 to 24, let U = {2, 6, 8, 10, 12, 14, 16, 18},
A = {2, 6, 10}, B = {6, 10, 16, 18}, and C = {14, 16}.
Find each of the following.
16. The set of small numbers is a well-defined set
%3D
- In Exercises 7 to 10, use set-builder notation to write
18. AUB
each set.
17. ANB
20. BUC'
7. The set of integers greater than -6
19. A' N C
22. (A U C)' N B'
8. {April, June, September, November}
21. A U (B N C)
24. (AUBUC)'
9. {Kansas, Kentucky}
23. (A N B')'
10. {1, 8, 27, 64, 125}
froper
given storlemen
Transcribed Image Text:Venn Diagrams and the Equality of Set Expressions Two sets are equal if and only if they each represent the same region(s) on a Venn diagratm. Venn diagrams can be used to verify each of the following properties. CHAPTER 2 Reviea Exerses For all sets A, B, and C: 101 See Example 6 and Check Youue Progress 6 on page 71, and then try Exercises 45 to 48 on page 102. Commutative Properties ANB=BNA AUB=BU A Associative Properties (AN B) N C = AN (B n C) An(B U C) = (A N B) U (A n C) (AU B) U C = A U (B U C) AU (Bnc)= (AU B)n (AU C) Distributive Properties 2.4 Applications of Sets Applications Many counting problems that arise in applications involving surveys can be solved by using sets and Venn diagrams. The Inclusion-Exclusion Formula For all finite sets A and B, See Examples 1 and 2 on page 81, and then try Exercises 53 and 54 on page 102. n(A U B) = n(A) + n(B) – n(A N B) See Examples 3 and 4 on page 83, and then 2.5 Infinite Sets Exercises 55 and 56 on page 103. try One-to-One Correspondence and Equivalent Sets Two sets A and B re equivalent, denoted by A ~ B, if and only if A and B can be placed See Examples 1 and 3 on pages 90 and 92, and then try Exercises 57 to 60 on page 103. in a one-to-one correspondence. Infinite Set A set is an infinite set if it can be placed in a one-to-one correspondence with a proper subset of itself. See Example 2 on page 91, and then try Exercises 61 and 62 on page 103. CHAPTER 2 REVIEW EXERCISES In Exercises 11 and 12, state whether each of the fol- lowing sets are equal, equivalent, both, or neither. 11. {2, 4, 6, 8}, {x\x E N and x < 5} - In Exercises 1 to 6, use the roster method to represent each set. 1. The set of months of the year with a name that starts 12. {8, 9}, the set of single digit whole numbers greater than 7 with the letter J 2. The set of states in the United States that do not share a common border with another state I In Exercises 13 to 16, determine whether the statement is A.A2 3. The set of whole numbers less than 8 64 true or false. 13. {3} E {1,2, 3, 4} 4. The set of integers that satisfy x 14. -11 EI 15. {a, b, c} ~ {1, 5, 9} 5. The set of natural numbers that satisfy x + 3 <7 6. The set of counting numbers larger than -3 and less than or equal to 6 In Exercises 17 to 24, let U = {2, 6, 8, 10, 12, 14, 16, 18}, A = {2, 6, 10}, B = {6, 10, 16, 18}, and C = {14, 16}. Find each of the following. 16. The set of small numbers is a well-defined set %3D - In Exercises 7 to 10, use set-builder notation to write 18. AUB each set. 17. ANB 20. BUC' 7. The set of integers greater than -6 19. A' N C 22. (A U C)' N B' 8. {April, June, September, November} 21. A U (B N C) 24. (AUBUC)' 9. {Kansas, Kentucky} 23. (A N B')' 10. {1, 8, 27, 64, 125} froper given storlemen
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