1. Welcome to MM150 – Unit 2 Seminar
Instructor: Larry Musolino
2.1 – Set Concepts
2.2 – Subsets
2.3 –Venn Diagrams and Set Operations
2.4 – More Venn Diagrams

2. Reminder on Seminars Reminder on Seminars:
Seminars Available (attend any one):
Wednesday
Thursday
Saturday
Unit 1
Anne Poe Weds June 6th 12noon
Charlie Johnson Thursday June 7th 10 PM
Unit 2
Dawn Thomas Weds June 13th 12noon
Charlie Johnson Thursday June 14th 10 PM
Larry Musolino Saturday June 16th 11am
Unit 3
Anne Poe Weds June 20th 12noon
Unit 4
Anne Poe Weds June 27th 12noon
Larry Musolino Saturday June 30th 11am
Unit 5
Charlie Johnson Thursday July 5th 10 PM
Unit 6
Dawn Thomas Weds July 11th 12noon
Larry Musolino Saturday July 14th 11am
Unit 7
Dawn Thomas Weds July 18th 12noon
Unit 8
Dawn Thomas Weds July 25th 12noon
Larry Musolino Saturday July 28th 11am
Unit 9
Anne Poe Weds Aug 1st 12noon
Charlie Johnson Thursday Aug 2nd 10 PM
Times are Eastern Time Zone

3. Weekly Responsibilities
Readings and Video Lectures
Attend Weekly Seminar
Optional, attendance is not graded.
Participate in weekly discussion topic, note you must post initial response and then at least two follow-up responses to other students
Complete the MyMathLab (MML) Graded Practice (20 problems).
Reminder on MML Graded Practice:
Must be completed by Tues 11:59ET for each weekly Unit.
If you have questions or encounter problems with MML, post your questions in Discussion Board, or contact Math Center

4. Summary of Student Responsibilities
Due Date
For weekly Units 1 to 9 (DISCUSSION BOARD):
Provide initial response to the Discussion Board Question
Post at least 2 follow-up responses to fellow students
By the end of the weekly unit, Tuesdays 11:59pm ET
For weekly Units 1 to 9 (SEMINAR):
Attend one of the Live Seminars as shown on previous Slide
Attending the live seminar is OPTIONAL and you are not graded on SEMINAR attendance
Live Seminar times as shown on previous slide.
For weekly Units 1 to (MML GRADED PRACTICE):
Complete the MyMathLab set of 20 problems
For Unit (FINAL PROJECT):
(a) Post the Final Project in the Course
Final Project due by end of week 9

5. Accessing Kaplan Math Center
Live, one-on-one tutoring.
To access the Math Center:
For live tutoring: go to “My Studies” on the KU Campus
choose “Academic Support Center”
select “Math Tutoring
In the past students have indicated this is a great resource, please take advantage of this !!!!

6. 2.1
Set Concepts

7. Set
A collection of objects, which are called elements or members of the set.
Listing the elements of a set inside a pair of braces, { }, is called roster form.
The symbol read “is an element of,” is used to indicate membership in a set.
The symbol means “is not an element of.”

8. Sets - Applications
Many applications of set and set operations in real-life:
Biology – taxonomy is science of classifying living things in sets,
Example: Living organisms classified into six sets called: animalia, plantae, archaea, eubacteria, fungi, protista.
Business Application – Insurance companies pool together applicants in sets based on various risk factors such as age, gender, smoker vs. non-smoker, etc. for actuarial analysis.
Scientific studies – During clinical trials, patients are arranged in various sets for evaluation:
E.g. placebo vs. non-placebo
Chemistry – Elements are classified into certain sets in the Periodic Table with common properties, e.g metals, non-metals, noble gases, etc.

9. Well-defined Set Well-Defined Set:
A set which has no question about what elements should be included.
Its elements can be clearly determined.
No opinion is associated with the members.
Examples:
Set of Integers between 4 and 9 inclusive:
{4, 5, 6, 7, 8, 9}
Not Well-Defined Set:
Set of Three Best NFL Teams of All Time.
Subjective and Set Elements will vary depending on opinions.

10. Roster Form
This is the form of the set where the elements are all listed within curly braces { } , each separated by commas.
Example:
Set N is the set of all natural numbers less than or equal to 25.
Solution: N = {1, 2, 3, 4, 5,…, 25}
The 25 after the ellipsis indicates that the elements continue up to and including the number 25.

11. Set-Builder (or Set-Generator) Notation
A formal statement that describes the members of a set is written between the braces.
A variable may represent any one of the members of the set.
Example: Write set B = {2, 4, 6, 8, 10} in set-builder notation.
Solution:

12. You Try It #1
Write the following set in both set-builder notation and roster notation:
Set A is the set of natural numbers between 4 and 10, inclusive.

13. You Try It #1 - Solution
Write the following set in both set-builder notation and roster notation:
Set A is the set of natural numbers between 4 and 10, inclusive.
Solution:
Roster Notation: A = { 4, 5, 6, 7, 8, 9, 10 }
Set-Builder Notation:

14. You Try It #2
Write the following set in both set-builder notation and roster notation:
Set B is the set of natural numbers greater than 12.

15. You Try It #2 - Solution
Write the following set in both set-builder notation and roster notation:
Set B is the set of natural numbers greater than 12.
Solution:
Roster Notation: B = { 13, 14, 15, 16, … }
Set-Builder Notation:

16. Finite Set
A set that contains no elements or the number of elements in the set is a natural number.
Example:
Set S = {2, 3, 4, 5, 6, 7} is a finite set because the number of elements in the set is 6, and 6 is a natural number.

17. Infinite Set
An infinite set contains an indefinite (uncountable) number of elements.
The set of natural numbers is an example of an infinite set because it continues to increase forever without stopping, making it impossible to count its members.
N = {1, 2, 3, 4, 5, …}

18. Cardinal Number
The number of elements in set A is its cardinal number.
Symbol: n(A)
Example:
Set A = {Toyota, Ford, Chevy, Honda}
n(A) = 4

19. Equal Sets
Equal sets have the exact same elements in them, regardless of their order.
Symbol: A = B
Example of Equal Sets:
Set A = {Toyota, Ford, Chevy, Honda}
Set B = {Chevy, Honda, Ford, Toyota}

20. Equivalent Sets
Equivalent sets have the same number of elements in them.
Symbol: n(A) = n(B)
Example of Equivalent Sets:
Set A = {Toyota, Ford, Chevy, Honda}
Set B = {Dolphins, Jets, Packers, Eagles}
Note that if Two Sets are Equal then they must also be equivalent.
Note that if Two Sets are Equivalent they are not necessarily equal.

21. Empty (or Null) Set
A null set (or empty set ) contains absolutely NO elements.
Symbol:

22. Universal Set
The universal set contains all of the possible elements which could be discussed in a particular problem.
Symbol: U

23. You Try It #3
Given that:
Find n(C), n(D), n(E)

24. You Try It #3 - Solution Given that: Find n(C), n(D), n(E)
n(C) = 6 since C = { 2, 3, 4, 5, 6, 7 }
n(D) = 1 since D = { 4 }
n(E) = 0

25. You Try It #4
State whether the two Sets are Equal, Equivalent, Both, or Neither:
Page 76 Problem 79
Set A = {algebra, geometry, trigonometry}
Set B = {geometry, trigonometry, algebra}
Page 76 Problem 81
Set A = {grapes, apples, oranges}
Set B = {grapes, peaches, apples, oranges}
Page 76 Problem 83
Set A is the set of letters in word tap
Set B is the set of letters in word ant

26. You Try It #4 - Solution
State whether the two Sets are Equal, Equivalent, Both, or Neither:
Page 76 Problem 79
Set A = {algebra, geometry, trigonometry}
Set B = {geometry, trigonometry, algebra} Answer: Both
Page 76 Problem 81
Set A = {grapes, apples, oranges}
Set B = {grapes, peaches, apples, oranges} Answer: Neither
Page 76 Problem 83
Set A is the set of letters in word tap Answer: Equivalent,
Set B is the set of letters in word ant Not Equal

27. 2.2
Subsets

28. Subsets
A set is a subset of a given set if and only if all elements of the subset are also elements of the given set.
Symbol: A B, means A is a subset of B.
To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B. The symbol for “not a subset of” is
Notice that for any Set A, A A
That is, every set is also a subset of itself.

29. Determining Subsets A B Í . A C Í . Example:
Determine whether set A is a subset of set B.
Also determine whether set A is subset of set C.
A = { 3, 5, 6, 8 }
B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C = {1, 2, 3, 4, 5, 6, 7}
Solution:
All of the elements of set A are contained in set B, so:
All of the elements of set A are not contained in set C, so:
A B Í .
A C Í .

30. Proper Subset
All subsets are proper subsets except the subset containing all of the given elements, that is, the given set must contain one element not in the subset (the two sets cannot be equal).
Written Another Way:
Set A is proper subset of Set B if and only if all elements of set A are elements of set B and set A  B
Symbol for Proper Subset: A B
Symbol for Not Proper Subset: A B

31. Determining Proper Subsets
Example:
Determine whether set A is a proper subset of set B.
A = { dog, cat }
B = { dog, cat, bird, fish }
Solution:
All the elements of set A are contained in set B, and sets A and B are not equal, therefore A B.

32. Determining Proper Subsets continued
Example:
Determine whether set A is a proper subset of set B.
A = { dog, bird, fish, cat }
B = { dog, cat, bird, fish }
Solution:
All the elements of set A are contained in set B, but sets A and B are equal, therefore A B.

33. You Try It #5 Determine whether set A is a proper subset of set B.
A = { x | x  N and x < 6 }
B = { x | x  N and 1  x  5 } 33

34. You Try It #5 - Solution
Determine whether set A is a proper subset of set B.
A = { x | x  N and x < 6 }
B = { x | x  N and 1  x  5 }
Solution:
All the elements of set A are contained in set B, but sets A and B are equal, therefore A B,
that is, set A is not a proper subset of set B. 34

35. Number of Distinct Subsets
The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set A.
Example:
Determine the number of distinct subsets for the given set { t , a , p , e }.
List all the distinct subsets for the given set: { t , a , p , e }.

36. Number of Distinct Subsets continued
Solution:
Since there are 4 elements in the given set, the number of distinct subsets is
24 = 2 • 2 • 2 • 2 = 16 subsets.
{t,a,p,e},
{t,a,p}, {t,a,e}, {t,p,e}, {a,p,e},
{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e},
{t}, {a}, {p}, {e}, { }

37. You Try It #6
Given that:
How many distinct subsets can be made from Sets C, D, E?
How many distinct proper subsets can be made from Sets C, D, E?
Reminder: The number of distinct subsets of a set A is 2n, where n is the number of elements in set A.

38. You Try It #6 - Solutions Set C: Number Distinct Subsets = 26 = 64
Given that:
How many distinct subsets can be made from Sets C, D, E?
Set C: Number Distinct Subsets = 26 = 64
Set D: Number Distinct Subsets = 21 = 2
Set E: Number Distinct Subsets = 20 = 1
How many distinct proper subsets can be made from Sets C, D, E?
Set C: Number Distinct Proper Subsets = 26 – 1 = 63
Set D: Number Distinct Proper Subsets = 21 – 1 = 1
Set E: Number Distinct Proper Subsets = 20 – 1 = 0
Based on this you can see that the Number of Distinct Proper Subsets for any set is 2n – 1 where n is number elements.

39. Venn Diagrams and Set Operations
2.3
Venn Diagrams and Set Operations

40. Venn Diagrams
A Venn diagram is a technique used for picturing set relationships.
A rectangle usually represents the universal set, U.
The items inside the rectangle may be divided into subsets of U and are represented by circles.

41. Disjoint Sets
Two sets which have no elements in common are said to be disjoint.
The intersection of disjoint sets is the empty set.
Disjoint sets A and B are
drawn in this figure. There
are no elements in common
since there is no overlap-
ping area of the two circles.

42. Overlapping Sets
For sets A and B drawn in this figure, notice the overlapping area shared by the two circles.
This section represents the elements that are in the intersection of set A and set B.
Elements in the intersection are elements that are in common for Sets A and B.

43. Setting up Venn Diagram with Two Overlapping Sets (See page 85)
Notice Region I consists of elements which appear only in SET A.
Notice Region III consists of elements which appear only in SET B.
Notice Region II (Middle Region) consists of elements which appear in both Set A and Set B.

44. Example of Venn Diagram for Two Overlapping Sets
To fill in a Venn Diagram, Start with the Middle Region and then work outward.
Students Taking Math Class
Students Taking History Class
James
Alice
Mike
Harry
Bob
Jen A B
Set of students taking math only
Set of students taking both math and history
Set of students taking history only
James
Alice
Harry
Jen
Mike
Bob

45. Complement of a Set
The set known as the complement contains all the elements of the universal set, which are not listed in the given subset.
Symbol: A’
Stated another way:
The complement of Set A is all of the elements in U which are not in A

46. Intersection
The intersection of two given sets contains only those elements common to both of those sets.
Symbol:

47. Union
The union of two given sets contains all of the elements for those sets.
The union “unites” that is, it brings together everything into one set.
Symbol:

48. Subsets When every element of B is also an element of A.
Circle B is completely inside circle A.

49. Equal Sets
When set A is equal to set B, all the elements of A are elements of B, and all the elements of B are elements of A.
Both sets are drawn as one circle.

50. The Meaning of and and or
and is generally interpreted to mean intersection
or is generally interpreted to mean union

51. The Relationship Between n(A  B), n(A), n(B), n(A  B)
To find the number of elements in the union of two sets A and B, we add the number of elements in set A and B and then subtract the number of elements common to both sets.
n(A B) = n(A) + n(B) – n(A  B)
n(A or B) = n(A) + n(B) – n(A and B)

52. You Try It #7 Given following sets: Find the following:
A = { 1, 2, 4, 6 }
B = { 1, 3, 6, 7, 9 }
C = { }
Find the following:
A  B
A  C
A  B
A’  B
(A B)’
n(A), n(B), n(A  B), n(A  B)

53. You Try It #7 - Solutions Given following sets: Find the following:
A = { 1, 2, 4, 6 }
B = { 1, 3, 6, 7, 9 }
C = { }
Find the following:
A  B = { 1, 2, 3, 4, 6, 7, 9 }
A  C = { 1, 2, 4, 6 } (Note that A  C = A)
A  B = { 1, 6 }
A’  B = { 3, 5, 7, 8, 9, 10 }  { 1, 3, 6, 7, 9 }
= { 1, 3, 5, 6, 7, 8, 9, 10 }
(A  B)’ = { 5, 8, 10 }
n(A) = 4, n(B) = 5, n(A  B) = 2, n(A  B) = 7
Notice that n(A B) = n(A) + n(B) – n(A  B)
= = 7

54. Solution Using Venn Diagram:
You Try It #7 - Solutions
Given following sets:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 4, 6 }
B = { 1, 3, 6, 7, 9 }
C = { }
Find the following:
A  B = { 1, 2, 3, 4, 6, 7, 9 }
A  C = { 1, 2, 4, 6 }
A  B = { 1, 6 }
A’  B = { 3, 5, 7, 8, 9, 10 }  { 1, 3, 6, 7, 9 }
= { 1, 3, 5, 6, 7, 8, 9, 10 }
(A  B)’ = { 5, 8, 10 }
n(A) = 4, n(B) = 5, n(A  B) = 2, n(A  B) = 7
Notice that n(A B) = n(A) + n(B) – n(A  B)
= = 7
Solution Using Venn Diagram: 3 2 1 7 4 6 9

55. Difference of Two Sets
The difference of two sets A and B symbolized A – B, is the set of elements that belong to set A but not to set B. Region 1 represents the difference of the two sets. U A B I II
III IV

56. Cartesian Product
The Cartesian product of set A and set B, symbolized A  B, and read “A cross B,” is the set of all possible ordered pairs of the form (a, b), where a  A and b  B.
Select the first element of set A and form an ordered pair with each element of set B. Then select the second element of set A and form an ordered pair with each element of set B. Continue in this manner until you have used each element in set A.

57. 2.4
Venn Diagrams with Three Sets
And
Verification of Equality of Sets

58. General Procedure for Constructing Venn Diagrams with Three Sets
Determine the elements that are common to all three sets and place in region V, A  B  C.

59. General Procedure for Constructing Venn Diagrams with Three Sets continued
Determine the elements for region II.
Find the elements in A  B. The elements in this set belong in regions II and V.
Place the elements in the set A  B that are not listed in region V in region II.
The elements in regions IV and VI are found in a similar manner.

60. General Procedure for Constructing Venn Diagrams with Three Sets (continued)
Determine the elements to be placed in region I by determining the elements in set A that are not in regions II, IV, and V.
The elements in regions III and VII are found in a similar manner.

61. General Procedure for Constructing Venn Diagrams with Three Sets (continued)
Determine the elements to be placed in region VIII by finding the elements in the universal set that are not in regions I through VII. U I II
III A V B IV VI
VII C
VIII

62. Example: Constructing a Venn diagram for Three Sets
Construct a Venn diagram illustrating the following sets.
U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 5, 8}
B = { 2, 4, 5}
C = {1, 3, 5, 8}
Solution:
Find the intersection of all three sets and place in region V, A  B  C = { 5 }

63. Example: Constructing a Venn diagram for Three Sets continued
Determine the intersection of sets A and B and place in region II.
A  B = {2, 5}
Element 5 has already been placed in region V, so 2 must be placed in region II.
Now determine the numbers that go into region IV.
A  C = {1, 5, 8}
Since 5 has been placed in region V, place 1 and 8 in region IV.

64. Example: Constructing a Venn diagram for Three Sets continued
Now determine the numbers that go in region VI. B  C = {5}
There are no new numbers to be placed in region VI.
Since all numbers in set A have been placed, there are no numbers in region I. The same procedures using set B completes region III, in which we must write 4. Using set C completes region VII, in which we must write 3.

65. Example: Constructing a Venn diagram for Three Sets continued
The Venn diagram is then completed. U A B C V I
III
VII VI IV
VIII II 2 4 5
1,8 3
6 7

66. Verification of Equality of Sets
To verify set statements are equal for any two sets selected, we use deductive reasoning with Venn Diagrams.
If both statements represent the same regions of the Venn Diagram, then the statements are equal for all sets A and B.

67. Example: Equality of Sets
Determine whether (A  B)’= A’ B’ for all sets A and B.

68. Solution Find A’  B’ B A U I II III IV
Draw a Venn diagram with two sets A and B and label each region.
Find (A  B)’
Find A’  B’
Set
Regions A
I, II B
II, III
A  B II
(A  B)’
I, III, IV
Set
Regions A’
III, IV B’
I, IV
A’ B’
I, III, IV

69. Solution
Set
Regions A
I, II B
II, III
A  B II
(A  B)’
I, III, IV
Set
Regions A’
III, IV B’
I, IV
A’ B’
I, III, IV
Both statements are represented by the same regions, I, III, and IV, of the Venn diagram. Thus, (A  B)’ = A’  B’ for all sets A and B.

70. De Morgan’s Laws
A pair of related theorems known as De Morgan’s laws make it possible to change statements and formulas into more convenient forms.
(A  B)’ = A’ B’
(A  B)’ = A’  B’