Slide Copyright © 2009 Pearson Education, Inc. Welcome to MM150 – Unit 2 Seminar Unit 2 Seminar Instructor: Larry Musolino – Set Concepts 2.2 – Subsets 2.3 –Venn Diagrams and Set Operations 2.4 – More Venn Diagrams

Slide Copyright © 2009 Pearson Education, Inc. Reminder on Seminars Reminder on Seminars:  Flex Seminars Available (attend any one): Weds 8:00pm ET or Saturday 2:00pm ET  Seminar this week on Friday at 8pm is one-time exception due to Memorial Day Holiday 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.

Slide Copyright © 2009 Pearson Education, Inc. Weekly Responsibilities Readings and Video Lectures Attend Weekly Flex Seminar  If you cannot attend the live seminar then take Seminar Option 2 Quiz. 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).

Slide Copyright © 2009 Pearson Education, Inc. 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 !!!!

Slide Copyright © 2009 Pearson Education, Inc. 2.1 Set Concepts

Slide Copyright © 2009 Pearson Education, Inc. 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.”

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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: Well-Defined Set:  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.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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:

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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:

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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:

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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, …}

Slide Copyright © 2009 Pearson Education, Inc. 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

Slide Copyright © 2009 Pearson Education, Inc. 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} Equal Sets

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. Empty (or Null) Set A null set (or empty set ) contains absolutely NO elements. Symbol:

Slide Copyright © 2009 Pearson Education, Inc. Universal Set The universal set contains all of the possible elements which could be discussed in a particular problem. Symbol: U

Slide Copyright © 2009 Pearson Education, Inc. You Try It #3 Given that: Find n(C), n(D), n(E)

Slide Copyright © 2009 Pearson Education, Inc. 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

Slide Copyright © 2009 Pearson Education, Inc. 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

Slide Copyright © 2009 Pearson Education, Inc. 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

Slide Copyright © 2009 Pearson Education, Inc. 2.2 Subsets

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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: Determining Subsets A B .A C .

Slide Copyright © 2009 Pearson Education, Inc. 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

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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 }

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. Number of Distinct Subsets The number of distinct subsets of a finite set A is 2 n, 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 }.

Slide Copyright © 2009 Pearson Education, Inc. Solution: Since there are 4 elements in the given set, the number of distinct subsets is 2 4 = = 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}, { } Number of Distinct Subsets continued

Slide Copyright © 2009 Pearson Education, Inc. You Try It #6 Given that: (a) How many distinct subsets can be made from Sets C, D, E? (b) How many distinct proper subsets can be made from Sets C, D, E? Reminder: The number of distinct subsets of a set A is 2 n, where n is the number of elements in set A.

Slide Copyright © 2009 Pearson Education, Inc. You Try It #6 - Solutions Given that: (a) How many distinct subsets can be made from Sets C, D, E? Set C: Number Distinct Subsets = 2 6 = 64 Set D: Number Distinct Subsets = 2 1 = 2 Set E: Number Distinct Subsets = 2 0 = 1 (b) How many distinct proper subsets can be made from Sets C, D, E? Set C: Number Distinct Proper Subsets = 2 6 – 1 = 63 Set D: Number Distinct Proper Subsets = 2 1 – 1 = 1 Set E: Number Distinct Proper Subsets = 2 0 – 1 = 0 Based on this you can see that the Number of Distinct Proper Subsets for any set is 2 n – 1 where n is number elements.

Slide Copyright © 2009 Pearson Education, Inc. 2.3 Venn Diagrams and Set Operations

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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 JamesAlice MikeJames AliceHarry BobJen A B Set of students taking math only Set of students taking both math and history Set of students taking history only Mike Bob James Alice Harry Jen

Slide Copyright © 2009 Pearson Education, Inc. 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

Slide Copyright © 2009 Pearson Education, Inc. Intersection The intersection of two given sets contains only those elements common to both of those sets. Symbol:

Slide Copyright © 2009 Pearson Education, Inc. 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:

Slide Copyright © 2009 Pearson Education, Inc. Subsets When every element of B is also an element of A. Circle B is completely inside circle A.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. The Meaning of and and or and is generally interpreted to mean intersection or is generally interpreted to mean union

Slide Copyright © 2009 Pearson Education, Inc. 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)

Slide Copyright © 2009 Pearson Education, Inc. You Try It #7 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  A  C  A  B  A’  B  (A  B)’  n(A), n(B), n(A  B), n(A  B)

Slide Copyright © 2009 Pearson Education, Inc. 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 } (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

Slide Copyright © 2009 Pearson Education, Inc. 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) = = Solution Using Venn Diagram :

Slide Copyright © 2009 Pearson Education, Inc. 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. BA U IIIIII IV

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 2.4 Venn Diagrams with Three Sets And Verification of Equality of Sets

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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 A B C V I III VII VI IV VIII II

Slide Copyright © 2009 Pearson Education, Inc. 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 }

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. Example: Constructing a Venn diagram for Three Sets continued The Venn diagram is then completed.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. Example: Equality of Sets Determine whether (A  B) ’ = A ’  B ’ for all sets A and B.

Slide Copyright © 2009 Pearson Education, Inc. Solution Draw a Venn diagram with two sets A and B and label each region. BA U IIIIII IV Find (A  B) ’Find A ’  B ’ SetRegions AI, II BII, III A  BA  B II (A  B)’(A  B)’ I, III, IV SetRegions A’A’ III, IV B’B’ I, IV A’ B’A’ B’ I, III, IV

Slide Copyright © 2009 Pearson Education, Inc. Solution 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. SetRegions AI, II BII, III A  BA  B II (A  B)’(A  B)’ I, III, IV SetRegions A’A’ III, IV B’B’ I, IV A’ B’A’ B’ I, III, IV

Slide Copyright © 2009 Pearson Education, Inc. 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 ’