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Chapter 3 - Sets Combinatorics and Probability Probability...

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Unformatted text preview: Sets, Combinatorics and Probability Probability Sets Counting Permutations and Combinations Probability Basic Set Concepts Basic A set is a collection of objects. Each object set is called an element of the set. element 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 Representing Word Description: Describe the set in Description your own words, but be specific so the elements are clearly defined elements All the whole numbers from 1 to 20 Roster Method: List each element, Method separated by commas, in braces separated {1, 2, 3, 4, 5, 6, 7, 8,{1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} 9, 2, 3, . . ., 20} Set-Builder Notation: Notation {x | x is … word description} {x | x is a whole number and 1 ≤ x ≤ 20} The Empty Set The The empty set, also called the null set, empty null set is the set that contains no elements. is elements. The empty set is represented by The {} or by or Φ We will use { } most of the time, because it's easier to understand it's Elements of a set Elements The symbol ∈ is used to indicate that an object is an element of a set. The symbol is used to replace the words: is is an element of is The symbol ∉ is used to indicate that an object is not an element of a set. The symbol is used to replace the words: symbol is not an element of not Standard Number Sets Standard e ˆ e ˆ e ˆ e ˆ e ˆ : integers integers : natural numbers (non-negative) natural : rational numbers (fractional) rational : real numbers real : complex numbers complex Definition of a Set’s Cardinal Number Definition The cardinal number of set A, The cardinal represented by |A|, is the number of |A| elements in set A. Cardinal Number Cardinal If A ={a, b, c}, what is |A|? If }, |A 0 1 23 |A| = 3 If A = {}, what is |A|? If |A 0 |A| = 0 If A = { a, b, c, d, a , e}, what is |A|? |A| = 5 If d, 01 2 3 45 4 5 no element can be counted no twice, even if it's accidentally listed twice!!!!! twice!!!!! Definition of a Finite Set Definition Set A is a finite set if |A| is a Set finite |A| natural number. A set that is not finite is called an infinite set. infinite The set of natural numbers, for example, is itself an infinite set. example, Definition of Equality of Sets Definition Set A is equal to set B means that set Set equal 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. Definition of Equivalent Sets Definition Set A is equivalent to set B means that set A Set equivalent and set B contain the same number of elements. For equivalent sets, |A| = |B|. |A| The sets {George Washington, John Adams, Thomas Jefferson, James Madison} and {1789, 1797, 1801, 1809} are equivalent because they both have a cardinality of 4. Definition of a Universal Set Definition When we talk about a set, we ask what's When in the set and what's not in the set. in Well, pretty much anything you can think Well, of might not be in the set. of We limit ourselves to what makes sense. The universal set is one that contains all universal of the elements that are included in the discussion. discussion. Definition of a Universal Set Definition Examples: When talking about the set {a, b, c, e} our When universe most likely would be the set of English language lowercase letters. English When talking about students in this When classroom, my universe might be all FDU students enrolled for this course. It might also be all students at FDU. It might be all people in the U.S. all Definition of the Complement of a Set Definition The complement of a set is the collection complement 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}. Definition of a Subset Definition Set A is a subset of set B, expressed as subset A⊆ B 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 Definition Set A is a proper subset of set B, Set proper expressed as A ⊂ B, if set A is a if 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 The 1. 2. For any set B, { } ⊆ B. For For any set B other than the empty For set, { } ⊂ B. Of course, { } might also be written as Of Φ 3. Sets of Sets Sets Number of subsets Number How many subsets does {a, b, c} have? Let's count: 1. 2. 3. 4. 5. 6. 7. 8. choose a no {} yes {a} no {b} no {c} yes {a, b} yes {a, c} no {b, c} yes {a, b, c} choose b no no yes no yes no yes yes choose c no no no yes no yes yes yes Number of Subsets and Proper Subsets Number The number of subsets of any set is The Why is this correct? For each element of the set it is either contained or not contained in a subset. Two choices. n elements. Venn Diagrams Venn Disjoint sets have no elements in common. The set B is a proper subset of A. The sets A and B have some common elements. U A B U A U A B B Venn Diagrams Venn A general Venn Diagram looks like the general one below, with the understanding that the purple region might be empty or that one set might be inside the other. one A B Venn Diagrams Venn Consider the case the universe of {1, 2, ..., 8}, Consider with A = {1,2, 3, 4} and B = {2, 4, 6, 8}. Let's see how these values get placed in the Venn but 2 is but Diagram below:also in B! Diagram 4 is also in B 1 2 3 4 5 6 7 A 8 B Venn Diagrams Venn The area representing those elements of A The that don't belong to B is the region: is A B Venn Diagrams Venn The area representing those elements of B The that don't belong to A is the region: is A B Venn Diagrams Venn The area representing those elements that The are both in A and in B is: A B Venn Diagrams Venn The area representing those elements that The don't belong to either A or B is: A B Definition of Intersection of Sets Definition The intersection of sets A and B, The intersection written A∩ B is the set of elements common to both set A and set B. This definition can be This expressed in set builder notation as follows: A ∩ B = { x | x ∈ A AND x ∈ B} Definition of Intersection of Sets Definition The intersection of sets A and B The intersection A ∩ B = { x | x ∈ AAND x ∈ B} what blinked both times? A B Definition of Union of Sets Definition The union of sets A and B, written The union written A∪ B 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 Definition The union of sets A and B The union A ∪ B = { x | x ∈ A OR x ∈ B} A B DeMorgan's Laws DeMorgan's Remember what happened when we Remember considered those don't belong to A or to B? That's the complement of A or B, namely (A ∪ B)' DeMorgan's Laws DeMorgan's I Now, we blue red region below represents fThe combine these Venn Diagrams,' A' • Now, the region below represents B Therefore (region is = A' ∩ B' in )' The red + blue∪ B)' the same as purple A = purple, the purple since our previous slide: region represents A' ∩ B' DeMorgan's Laws DeMorgan's DeMorgan's Laws state that and (A ∪ B)' = A' ∩ B' )' (A ∩ B)' = A' ∪ B' )' Definition of the Cartesian Product of Sets Sets The Cartesian product of sets A and B, The Cartesian written Ae B ˆ is the set of all pairs of elements taken from A and B. This definition can be expressed in This set builder notation as follows: A e B = { (x, y) | e x ∈ A and y ∈ B} ˆ ˆ Definition of the Cartesian Product of Sets Sets Countable and Uncountable Sets Countable A denumerable set is called countable. denumerable countable The set of all integers is also denumerable. The (What is the mapping?) The set of rational numbers is also denumerable. (What is the mapping?) denumerable. Countable and Uncountable Sets Countable Speaking informally, there are two types of Speaking "infinity." An infinite set is one whose infinite cardinality is boundless, (for example, the integers, the real numbers, the set of all strings over the alphabet {a,b}). A set is denumerable if we can place it into denumerable a one-to-one correspondence with the nonone-to-one negative integers. Clearly the non-negative negative integers is a denumerable set. integers Countable and Uncountable Sets Countable The set of rational numbers is also denumerable. The (What is the mapping?) (What 1/1 1/2 1/3 1/4 1/5 … 1/1 2/1 2/2 2/3 2/4 2/5 … 2/1 3/1 3/2 3/3 3/4 3/5 … 3/1 4/1 4/2 4/3 4/4 4/5 … 4/1 . .. .. . .. .. . .. .. Venn Diagrams - Two Sets Venn RegionIV:In Inautb B B in Region III: A nd o I: Not The RegionregionsinofnotnotVenn Diagram four II: In A bB Autr in B A the A I II B III IV Venn Diagrams - Three Sets Venn The eight regionsCofutCnotBinCorC C Region III: InNotbnd Andutrnot or Diagram RegionIV: In AInin a , b in and in B Region I: V:B a ut the o BA CA Region In b not Venn VIII: VI: A A B B A VII: II: A I II B IV V III VI VII C VIII Example: Blood Typing Example: Blood is characterized by examining Blood components called antigens. antigens Two of these antigens are called type A Two type and type B. type We name a person's blood on whether We or not they have these antigens: or A: has antigen type A has B: has antigen type B has AB: has both O: has neither. has Example: Blood Typing Example: Look at the diagram below: has type A has type B has both A and B has neither A nor B A AB O B Example: Blood Typing Example: But there's a third antigen as well: the But Rh antigen. Rh antigen. Blood with Rh is said to be positive: + Blood Rh Blood without Rh is said to be negative: Blood Rh - Example: Blood Typing Example: Look at the diagram below: A A ABB BAB AAB+ B+ A+ O O+ ORh B + Example: Blood Typing Example: When you receive blood, your blood must have all the antigens found in the donor's blood. Who can receive ANY type of blood? A AA+ O- ABAB+ B BB+ O+ Rh Universal acceptor: AB+ Example: Blood Typing Example: When you donate blood, the acceptor's blood must have all the antigens found in your blood. Who can donate to everyone? A AA+ O- ABAB+ B BB+ O+ Rh Universal donor: O- Cardinal Number of the Union of Two Sets Two Suppose a class has 16 students with brown Suppose hair and that it has 12 students who wear glasses. glasses. How many students in the class either have How brown hair or wear glasses? brown Since a student in this group can either have Since brown hair (16 students) or wear glasses (12 students) a good guess is that there are 16 + 12 = 28 students in this group. 12 Let's count! Everyone with brown hair, blink for the Everyone haven't glasses, blink for me. Everyone I wearingmentioned, AND have Blink for me if you wear glasses leaveme. room. brown hair. 11 1 4 3 6 4 94 2 7 13 7 19 12 5 10 10 14 11 4 13 20 11 36 8 15 8 21 14 16 9 22 15 2 2 3 5 5 7 12 9 17 11 18 10 + 23 5 16 28 12 23 8 6 2 3 Cardinal Number of the Union of Two Sets Two What happened there? When counting heads, we counted the When intersection twice, first as having brown hair and then again as wearing glasses. hair We have to take that into account. We We subtract that number in the We intersection from the total. intersection Cardinal Number of the Union of Two Sets Two The number of students who have brown The hair or who wear glasses is the UNION of or two sets. (Remember, or means union.) or union The number of students who have brown The hair and who wear glasses is the and INTERSECTION of two sets. INTERSECTION (Remember, and means intersection.) (Remember, and intersection Cardinal Number of the Union of Two Sets Two brown hair 5 glasses 11 16 12 7 n(brown hair) = 16, n(glasses) = 12, Region 1: 11, Region II: 5 and n(brown hair e glasses) = 5 Region III: 7. 11 + 5 + 7 = 23 students 16 + 12 – 5 = 23 students. Solving Survey Problems Solving A class has 28 students. Of the 15 class female students, 8 wear glasses. Half the class (14 students) wear glasses. How many students are either male or don’t wear glasses? don’t Venn Diagrams Venn Female15 14 Glasses 8 6 7 7 28 students 15 female students 14 wear glasses 8 female students wear glasses Venn Diagrams Venn Female 8 7 7 Glasses 6 students who are male or don't wear glasses include all the male students (7+6, or 28 - 15) and all the female students who don't wear glasses (7) = Or Use DeMorgan's Law Or Male = not Female = F ' don’t wear glasses = G ' don’t F ' ∪ G' = (F ∩ G)' |F ∩ G) = 8 |(F ∩ G)') = 28 – 8 = 20 Venn Diagrams - Three Sets Venn Region III: In AInnd anotutinA, rCor C RegionIV: In BC ut not in andBr C Region I: V:A a butnd B BA in A Region In b A C b A o VI: VII: II: The set of elements Bnotinnot o CB A I VIII II B IV V III VI VII C Venn Diagrams - Three Sets Venn Region I: In A but not in B or C Region II: In A and B but not in C Region III: In B but not in A or C Region IV: In A and C but not in B Region V: In A and B and C Region VI: In B and C but not in A Region VII: In C but not in A or B Region VIII: Not in A, B or C U A I IV II V VII VI B III VIII C Solving Survey Problems Solving 1. Sixty people were contacted and responded Sixty to a movie survey. The following results were obtained. were 1. 2. 3. 4. 5. 6. 7. 6 people liked comedies, dramas AND sci-fi. 13 people liked comedies and dramas. 10 people liked comedies and sci-fi. 11 people liked dramas and sci-fi. 26 people liked comedies. 21 people liked dramas. 25 people liked sci-fi. Solving Survey Problems Solving 60 people took the survey 1. 6 people liked comedies, dramas AND scifi. 2. 13 people liked comedies and dramas. 3. 10 people liked comedies and sci-fi. 4. 11 people liked dramas and sci-fi. 5. 26C people likedD comedies. start in the work towards U 16 60 = 9 ctenter. he outside 9 7 liked dramas. + 7 6. 21 people 3 +3+4+6 6 liked sci-fi. +15= 61+ + 21 = 7.+ 25 people 5 26 =7 +5 7 4 1 + 05 25 = 4 + 5 +6+30 4+6+9 +6+1 10 SF 10 = 6 + 4 13 = 6 + 7 16 Solving Survey Problems Solving 1. 6 people liked comedies, dramas AND scifi. 2. 13 people liked comedies and dramas. 3. 10 people liked comedies and sci-fi. How many How many 4. 11 people liked dramas and sci-fi. people liked people don't 5. 26 people liked comedies. only movies atof like one type U 16 21C C 6. people likedD dramas. mll? 7 a ovie? 9 3 7. 25 people liked sci-fi. 6 9 + 3 + 10 = 4 5 10 SF 16 22 Formula for the Cardinal Number of the Union of Two Sets of Principle of Inclusion and Exclusion Principle (Two Sets) |A e B|| = |A| + |B| - |A ˆ B|| ˆB |B |A B To find the cardinal number in the union of To sets A and B, add the number of elements add 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 of Principle of Inclusion and Exclusion Principle (Three Sets) ||A e B e C|| = |A| + |B| + |C| Aˆ ˆ C |A| |B |C - |A e B| - |A e C| - |B e C| ˆ ˆ ˆ + |A e B e C| ˆ ˆ Counting Counting Addition Principle If two events are mutually exclusive, that If mutually that is the events do not overlap, then there are n + m ways of performing one or the other event. other Counting Counting Multiplication Principle Whenever two independent events are to be Whenever performed in sequence, if there are n ways of performing the first and m ways of performing the second, there are ne m ways ˆ of performing the sequence. of Example Example Car manufacturers are now experimenting Car with lightweight three-wheeled cars, designed for a driver and one passenger, and considered ideal for city driving. Suppose you could order such a car with a choice of 9 possible colors, with or without air-conditioning, with or without a removable roof, and with or without an onboard computer. In how many ways can this car be ordered in terms of options? options? Solution Solution This situation involves making choices This with four groups of items. with color - air-conditioning - removable roof color computer computer 9 × 2 × 2 × 2 = 72 Thus the car can be ordered in 72 different Thus ways. ways. Example A Multiple Choice Test Example You are taking a multiple-choice test that You has ten questions. Each of the questions has four choices, with one correct choice per question. If you select one of these options per question and leave nothing blank, in how many ways can you answer the questions? the Solution Solution We use the Fundamental Counting We Principle to determine the number of ways you can answer the test. Multiply the number of choices, 4, for each of the ten questions ten 4× 4× 4× 4× 4× 4× 4× 4× 4× 4 =1,048,576 =1,048,576 Permutations Permutations A permutation is an arrangement of permutation arrangement objects. objects. No item is used more than once. The order of arrangement makes a The difference. difference. Example Counting Permutations Example You need to arrange seven of your You favorite books along a small shelf. How many different ways can you arrange the books, assuming that the order of the books makes a difference to you? books Solution Solution You may choose any of the seven books for You the first position on the shelf. This leaves six choices for second position. After the first two positions are filled, there are five books to choose from for the third position, four choices left for the fourth position, three choices left for the fifth position, then two choices for the sixth position, and only one choice left for the last position. last 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040 There are 5040 different possible permutations. Factorial Notation Factorial If n is a positive integer, the notation n! is If is the product of all positive integers from n down through 1. down n! = n(n-1)(n-2)…(3)(2)(1) note that 0!, by definition, is 1. 0!=1 Permutations of n Things Taken r at a Time The number of permutations possible if The r items are taken from n items: n! (n – r)! n! = n(n – 1) (n – 2) (n – 3) . . . (n – r + 1) (n - r) (n - r - 1) . . . (2)(1) (n – r)! = (n - r) (n - r - 1) . . . (2)(1) Problem Problem A math club has eight members, and it must math choose 5 officers --- president, vice-president, secretary, treasurer and student government representative. Assuming that each office is to be held by one person and no person can hold more than one office, in how many ways can those five positions be filled? positions We are arranging 5 out of 8 people into the five distinct offices. Any of the eight can be president. Once selected, any of the remaining seven can be vice-president. Clearly this is an arrangement, or permutation, problem. 85 P = 8!/(8-5)! = 8!/3! = 8 · 7 · 6 · 5 · 4 = 6720 Permutations with duplicates. Permutations In how many ways can you arrange the In letters of the word minty? minty That's 5 letters that have to be arranged, That's so the answer is 5P5 = 5! = 120 But how many ways can you arrange the But letters of the word messes? messes You would think 6!, but you'd be wrong! Permutations with duplicates. Permutations How many ways can you arrange the How letters of the word messes? messes The problem is that there are three s''s and s 2 e''s. It doesn't matter in which order the s. s''s are placed, because they all look the s same! same! This is called permutations with This duplicates. duplicates. Permutations with duplicates. Permutations Well, there are 3! = 6 ways to arrange the Well, s''s. So there will be 6 permutations that s. should count as one. Same with the e''s. s. There are 2! = 2 permutations of them that should count as 1. that So we divide 6! by 3! and also by 2! There are 6!/3!2! = 720/12 = 60 ways to There arrange the word messes. messes Permutations with duplicates. Permutations In general if we want to arrange n items, of In n! m1! m2 ! m3 ! Problem Problem A signal can be formed by running signal different colored flags up a pole, one above the other. Find the...
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