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Thinking Mathematically Chapter 2 Set Theory Basic Set Concepts A set is a collection of objects. Each object is called an element, or member of the set. Often the objects in a set are listed and are enclosed in “braces.” For example the set of integers that fall between 1 and 5 can be written {2 , 3 , 4}. Representing Sets • Word Description: Describe the set in your own words, but be specific so the elements are clearly defined All the whole numbers from 1 to 20 • Roster Method: List each element, separated by commas, in braces {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, .12, 13, 14, 15, 16, 17, 18, 19, 20} {1, 2, 3, . ., 20} • Set-Builder Notation: {x | x is … word description} {x | x is a whole number and 1 ≤ x ≤ 20} The Empty Set The empty set, also called the null set, is the set that contains no elements. The empty set is represented by {} or by We will use { } most of the time, because it's easier to understand. Elements of a set • The symbol is used to indicate that an object is an element of a set. The symbol is used to replace the words: is an element of, or belongs to • The symbol is used to indicate that an object is not an element of a set. The symbol is used to replace the words: is not an element of, or does not belong to Elements of a set • 8{2,4,6,8} • you{x | x is a student in this class} • Dr. Landis {x | x is a student in this class} • f { a, b, c, d, e, g, h, i, k , l, m} Counting and the Natural Numbers • Sets are collections of elements. Sometimes we want to count how many elements a set has. • The natural numbers are the "counting numbers": N = {0, 1, 2, 3, 4, 5,…} Definition of a Set’s Cardinal Number The cardinal number of set A, represented by n(A), is the number of elements in set A. The symbol n(A) is read “n of A”. Cardinal Number If A ={a, b, c}, what is n(A)? n(A) = 3 0 1 2 3 If A = {}, what is n(A)? n(A) = 0 0 If A = {a, b, c, d, a, e}, what is n(A)? n(A) = 5 0 1 2 3 4 5 no 4 5 element can be counted twice, even if it's accidentally listed twice!!!!! Definition of a Finite Set Set A is a finite set if n(A) is a natural number. A set that is not finite is called an infinite set. The set of natural numbers, for example, is itself an infinite set. Definition of Equality of Sets Set A is equal to set B means that set A and set B contain exactly the same elements, regardless of order. We symbolize the equality of sets A and B using the statement A = B. Examples: Equal Sets • Working left to right, cross out every occurrence of the given element. If anything is left over, the sets are not equal. • {a, b, c} {b, a, c} • {a, b, c, d} {a, b, c, d, e} • {a, b, c, d}{a, c, d, d, b, a, b} Definition of Equivalent Sets Set A is equivalent to set B means that set A and set B contain the same number of elements. For equivalent sets, n(A) = n(B). The sets {G. Washington, J. Adams, T. Jefferson, J. Madison} and {1789, 1797, 1801, 1809} are equivalent because they both have a cardinality of 4. Definition of Equality of Sets • Remember: – Equal means the sets are exactly the same. – Equivalent means that the sets have the same number of elements. – Every semester, lots of students lose points on an exam because they forget this!! Section Quiz • Problem 1: Represent the set {x | 1 < x ≤ 9} using the roster method. {2, 3, 4, 5, 6, 7, 8, 9} • Problem 2: Are the sets {a, b, c} and {c, a, b, a} equivalent? yes. they both have the cardinality 3 • Problem 3: Are the sets {a, b, c} and {c, a, b, a} equal? yes. they contain exactly the same elements Thinking Mathematically Section 2 Subsets Definition of a Universal Set • When we talk about a set, we ask what's in the set and what's not in the set. • Well, pretty much anything you can think of might not be in the set. • We limit ourselves to what makes sense. • The universal set is one that contains all of the elements that are included in the discussion. Definition of a Universal Set • Examples: – When talking about the set {a, b, c, e} our universe most likely would be the set of English language lowercase letters. – When talking about students in this classroom, my universe might be all FDU students taking Comprehensive Math. It might also be all students at FDU. It might be all people in the U.S. Definition of the Complement of a Set The complement of a set is the collection of all the objects in the universal set that are not in the given set. The complement of a set A is written A´. A´= {x | x U and x A }. I have made those apostrophes blink to emphasize that you should always watch out for them, especially on exams. Examples • Suppose I said "Consider the set of students who are actually listening to me now." • The universal set, or universe under discussion, would be the set of all students in this classroom. I'm not interested in chairs, tables, books, or even students in other classrooms. • The complement of the set would be anyone not listening to me. Definition of a Subset of a Set Set A is a subset of set B, expressed as AB if every element in set A is also in set B. Note that the set A could be equal to the set B. That's why there's a line at the bottom of the symbol. Think about how ≤ means less than or equal. Definition of a Proper Subset of a Set Set A is a proper subset of set B, expressed as A B, if set A is a subset of set B and sets A and B are not equal ( A B ). Note that the set A can not be equal to the set B. That's why there isn't a line at the bottom of the symbol. The Empty Set as a Subset 1. For any set B, { } B. 2. For any set B other than the empty set, { } B. 3. Of course, { } might also be written as . Number of subsets • How many subsets does {a, b, c} have? • Let's count: choose a choose b choose c 1. {} no no no 2. {a } yes no no 3. {b } no yes no 4. {c} no no yes 5. {a, b} yes yes no 6. {a, c} yes no yes 7. {b, c} no yes yes 8. {a, b, c} yes yes yes Number of subsets • For each element of the set, we could either choose or not choose that element. • Every set of different choices forms a different subset. • Since there are two choices for a, two choices for b, and two choices for c, there are 2 2 2 choices. • 2 2 2 = 23 = 8 Number of Subsets and Proper Subsets • The number of subsets of any set is given by: 2n 2n means 2 x 2 x 2 x . . . x 2, n times. The number of proper subsets of any set is given by: 2n - 1 Section Quiz • Problem 1: If the universe is {1, 2, 3, 4, 5, 6, 7, 8} and A = {1, 3, 5, 7}, what is A' ? • Problem 2: True or false? {2, 4, 6, 8} – {} {a, b, c} true – {} {a, b, c} false – {b} {a, b, c} true Thinking Mathematically Section 3 Venn Diagrams and Set Operations Venn Diagrams U Disjoint sets have no A B elements in common. U A The set B is a proper B subset of A. U The sets A and B have A B some common elements. Venn Diagrams • A general Venn Diagram looks like the one below, with the understanding that the purple center region might be empty or that one set might be inside the other. A B Venn Diagrams • Consider the case the universe of {1, 2, ..., 8}, with A = {1,2, 3, 4} and B = {2, 4, 6, 8}. Let's see how these values get placed in the Venn Diagram below: but 2 is also B but 4 is also in in B! 1 2 3 4 5 A B 6 7 8 Venn Diagrams • The area representing those elements of A that don't belong to B is the region: A B Venn Diagrams • The area representing those elements that are both in A and in B is: A B Venn Diagrams • The area representing those elements of B that don't belong to A is the region: A B Venn Diagrams • The area representing those elements that don't belong to either A or B is: A B Definition of Intersection of Sets The intersection of sets A and B, written AB is the set of elements common to both set A and set B. This definition can be expressed in set builder notation as follows: A B = { x | x A AND x B} Definition of Intersection of Sets The intersection of sets A and B A B = { x | x A AND x B} A B Definition of Union of Sets The union of sets A and B, written AB is the set of elements that are members of set A or of set B or of both sets. This definition can be expressed in set builder notation as follows: A B = { x | x A OR x B} Definition of Union of Sets The union of sets A and B, written A B = { x | x A OR x B} A B DeMorgan's Laws • Remember what happened when we considered those don't belong to A or to B? That's the complement of A or B, namely (A B)' DeMorgan's Laws The the region is sae ••The purple region therepresents B' Therefore A'redB' =below B)' represents A' Now,blue region (A below as what If we combine these Venn Diagrams, was left blue = purple, slide: since red + in our previousthe purple region represents A' B' A' B' DeMorgan's Laws • DeMorgan's Laws state that (A B)' = A' B' and (A B)' = A' B' Section Quiz • Problem 1: Given the Venn Diagram 8 1 4 A 2 5 B 6 3 7 – Describe set A in roster notation. {2, 3, 4, 5} – Describe set A' in roster notation. {1, 6, 7, 8} – Describe A B' in roster notation {3, 4} – Describe A' B in roster notation. {1, 2, 5, 6, 7, 8} Thinking Mathematically Section 4 Set Operations and Venn Diagrams with Three Sets Venn Diagrams - Two Sets IV: In not B, A, Region III:A A of B, (A B)' A', B The RegionregionsBandtheAB, A BDiagram four I:II:NotbutA ornotVennB', A - B - A Region InIn in but in in B A B I II III IV Venn Diagrams - Three Sets Region VII: In Aand CC VennCC Region III: V: Inand not B B or Diagram regionsC but not in A The eightRegionIn ABof thebutandorCBA Region IV: VIII: butand A, notorin C RegionVI: In B Not Bbut B in B RegionII: I: Region A in in not A A B I II III VIII V IV VI VII C Venn Diagrams - Three Sets Region I: In A but not in B or C U Region II: In A and B but not in C A B Region III: In B but not in A or C I II III Region IV: In A and C but not in B V IV VI Region V: In A and B and C VII VIII Region VI: In B and C but not in A C Region VII: In C but not in A or B Region VIII: Not in A, B or C Example: Blood Typing Blood is characterized by examining components called antigens. Two of these antigens are called type A and type B. We name a person's blood on whether or not they have these antigens: A: has only antigen type A B: has only antigen type B AB: has both O: has neither. Example: Blood Typing Look at the diagram below: has only A has only B A B AB has both A and B O has neither A nor B Example: Blood Typing But there's a third antigen as well: the Rh antigen. Blood with Rh is said to be positive: + Blood without Rh is said to be negative: - Example: Blood Typing Look at the diagram below: - A B A AB- B AB B- A- AB+ B+ O A+ + O+ O- Rh Example: Blood Typing When you receive blood, your blood must have all the A B AB- antigens found in A- B- the donor's blood. AB+ A+ B+ Who can receive O+ ANY type of blood? O- Rh Universal recipient: AB+ Example: Blood Typing When you donate blood, the B acceptor's blood A AB- must have all the A- B- antigens found in AB+ B+ your blood. A+ O+ Who can donate to O- Rh everyone? Universal donor: O- Section Quiz • Problem: Suppose the Venn diagram below represented A: people who like candy, B: people who like soda, and C: people who like liver. – what region(s) represent people who like all three? AA B V I II III VIII IV V VI VII C Section Quiz • Problem: Suppose the Venn diagram below represented A: people who like candy, B: people who like soda, and C: people who like liver. – what region(s) represent people who like only one of the three? AA B I, III I II III VIII IV V VI and VII VII C Section Quiz • Problem: Suppose the Venn diagram below represented A: people who like candy, B: people who like soda, and C: people who like liver. – what region(s) represent people who like candy, but not soda? AA B I and I II III VIII IV V VI IV VII C Thinking Mathematically Section 5: Surveys and Cardinal Numbers Cardinal Number of the Union of Two Sets • Suppose a class has 16 students with brown hair and that it has 12 students who wear glasses. • How many students in the class either have brown hair or wear glasses? • Since a student in this group can either have brown hair (16 students) or wear glasses (12 students) a good guess is that there are 16 + 12 = 28 students in this group. • Let's count! Blink for wearing wear glassesleave have I haven't mentioned, for Everyoneme if youglasses, blink ANDthe Everyone with brown hair, blink for me. room. me. hair. brown 11 1 2 2 3 2 4 3 3 5 6 4 7 5 8 6 74 2 9 5 10 11 36 8 9 12 7 13 10 14 15 8 16 9 11 17 18 10 + - 19 12 11 4 13 20 21 14 22 15 23 5 16 28 12 23 Cardinal Number of the Union of Two Sets • What happened there? • When counting heads, we counted the intersection twice, first as having brown hair and then again as wearing glasses. • We have to take that into account. • We subtract that number in the intersection from the total. Cardinal Number of the Union of Two Sets • The number of students who have brown hair or who wear glasses is the UNION of two sets. (Remember, or means union.) • The number of students who have brown hair and who wear glasses is the INTERSECTION of two sets. (Remember, and means intersection.) Cardinal Number of the Union of 16 = 5 + 11 Two Sets 12 = 5 + 7 brown hair glasses 5 16 12 11 7 n(brown hair) = 16 n(glasses) = 12 n(brown hair glasses) = 5 n(brown hair glasses) = 11 + 5 + 7 = 23 Formula for the Cardinal Number of the Union of Two Sets n(A B) = n(A) + n(B) - n(A B) To find the cardinal number in the union of sets A and B, add the number of elements in sets A and B and then subtract the number of elements common to both sets. Formula for the Cardinal Number of the Union of Three Sets n(A B C) = n(A) + n(B) + n(C) - n(A B) - n(A C) - n(B C) + n(A B C) This one is tougher, and you don't have to memorize it if you don't want to. You'll see we can solve problems without it. Solving Survey Problems 1. Use the survey’s description to define sets and draw a Venn diagram. 2. Use the survey’s results to determine the cardinality for each region in the Venn diagram. Start with the intersection of the sets, the innermost region, and work outward. 3. Use the completed Venn diagram to answer the problem’s questions. Solving Survey Problems 1. It's easier if we do an example. 2. Sixty people were contacted and responded to a movie survey. The following results were obtained. 1. 6 people liked comedies, dramas AND sci-fi. 2. 13 people liked comedies and dramas. 3. 10 people liked comedies and sci-fi. 4. 11 people liked dramas and sci-fi. 5. 26 people liked comedies. 6. 21 people liked dramas. 7. 25 people liked sci-fi. workin the center: start finally..... out. continue further and outwards: Step 1: Draw a Venn diagram. Survey Sixtypeople liked comediesand sci-fi. 21 11 people were contacted and responded 25people liked comedies, dramas AND sci-fi.to a movie survey. 26 people liked comedies and dramas. comedies. dramas. 13people liked sci-fi. 6 10 people liked dramas and sci-fi. these two numbers must add up to 13 9 comedy drama 7 9 7 7 43 9 7 3 3 6 64 +5 +4 6 66 +5 4 4 55 ---- 5 ---- 18 17 +10 15 10 16 ---- 10 21 26 44 sci fi 25 -18 -17 U -15 ---- 60 ---- -449 3 10 ---- Solving Survey Problems 1. 6 people liked comedies, dramas AND sci-fi. 2. 13 people liked comedies and dramas. 3. 10 people liked comedies and sci-fi. 4. 11 people liked dramas and sci-fi. 5. 26 people liked comedies. 6. 21 people liked dramas. 7. 25 people liked sci-fi. How many How many people liked people don't like one type only movies atof U 16 C C D movie? 9 7 3 all? 65 9 + 3 + 10 = 4 16 22 10 SF Solving Survey Problems A class has 28 students. Of the 15 female students, 8 wear glasses. Half the class (14 students) wear glasses. How many students are either male or don’t wear glasses? Venn Diagrams How many Females with glasses students are glasses Females, no either male or don’t wear glasses? Female15 14 Glasses 8 7 6 glasses 7 Males with glasses Males, no 28 students 7+7+6=20 15 female students 14 wear glasses 8 female students wear glasses Or Use DeMorgan's Law • Male = not Female = F ' • don’t wear glasses = G ' • F ' G' = (F G)' • n(F G) = 8 • n((F G)') = 28 – 8 = 20 Section Quiz 50 students were contacted and responded to a school survey. The following results were obtained. 1. 3 students liked math, history and literature. 2. 5 people liked math and literature. 3. 15 people liked history and literature. 4. 6 people liked history and math. 5. 30 people liked history. 6. 8 people liked math. 7. 31 people liked literature. A. How many students like only math? B. How many students like either history or literature, but not math? C. How many students don't like any of these subjects? Section Quiz 50 students were contacted and responded to a school survey. The following results were obtained. 1. 3 students liked math, history and literature. 2. 5 people liked math and literature. 3. 15 people liked history and literature. 4. 6 people liked history and math. 5. 30 people liked history. 6. 8 people liked math. 7. 31 people liked literature.. U 4 M 3 H 0 0 12 3 12 2 38 14 4 L Section Quiz • During a survey of 100 people who were asked to name their three favorite flavors of ice cream, the following was noted. – 15 people liked vanilla, chocolate and strawberry. – 35 people liked vanilla and chocolate – 27 people liked vanilla and strawberry – 25 people liked chocolate and strawberry – 60 people liked vanilla – 70 people liked chocolate – 40 people liked strawberry • How many people liked only vanilla? Section Quiz • During a survey of 100 people who were asked to name their three favorite flavors of ice cream, the following was noted. – 15 people liked vanilla, chocolate and strawberry. – 35 people liked vanilla and chocolate – 27 people liked vanilla and strawberry – 25 people liked chocolate and strawberry – 60 people liked vanilla – 70 people liked chocolate – 40 people liked strawberry strawberry vanilla • How many people liked only vanilla? 12 13 15 10 20 13 chocolate