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STAT 111 Chapter Zero A Review of Set Notation

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Definition: A set is a well-defined collection (possible empty) of objects, such as the set of letters in the alphabet, or the set of students in a classroom. We denote a set by a capital letter such as A,B,C and an element by a lower case letter such a,b,c. Notes: If an element a belongs to a set C we write a C. If an element a does not belong to C we write a C.

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Presentation Sets may be described by listing the elements or by describing some property held by all members. For example the set P of positive integers might be described by: 1. P={1,2,3,…} 2. P={x : x is a positive integer}

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Example Let A be the set of all real numbers whose squares are equal to 25. Show how to describe A by a) Listing its elements, A={-5,+5} since x 2 =25 x=-5 or x=+5 b) By describing some property held by all members. A={X : X 2 =25}

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Example Determine which of the following statement are true and which are false. 1) {2}=2 False (since 2 is areal number while {2} is a set which consists of the real number 2) 2) True (since every elements is assumed to be equal to itself) 3) True (since no number satisfy which is a subset of every set

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Notations and Venn Diagram The universal set S is the set of all elements under consideration. Using Venn diagram it consists of a rectangle. The null or empty set, denoted by Φ, is the set consisting of no elements. Thus Φ is a subset of every set.. For any two sets A and B we will say that A is a subset of B, or A is contained in B (denoted A B), if every element in A is also in B. Remarks: Every set is a subset of itself If A B and B A then A=B A B

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Notations and Venn Diagram S A B S A B The union of A and B, denoted by A U B, is the set of all elements which belong to either A or B or both. That is, the union of A and B contain all elements that are in at least one of the sets. The key word for expressing the union of two sets is or (meaning A or B or both). A U B The intersection of A and B, denoted by A ∩ B, is the set of all elements which belong to both A and B. The key word for expressing the intersections is and (meaning A and B simultaneously). A ∩ B

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Notations and Venn Diagram If A is a subset of S, then the complement of A, denoted by A c (or A ), is the set of elements that are in S but not in A. the shaded area in S but not in A is A c. Note that A U A c =S. Note also, (A c ) c =A S A AcAc

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Notations and Venn Diagram S A B Two sets, A and B, are said to be disjoint or mutually exclusive if A B= . That is, mutually exclusive sets have no elements in common. Note that, it is easy to see that A and A c are mutually exclusive sets for any set A. S A B The difference of A and B, denoted by A-B, is the set of all elements of A which do not belong to B. Note that A-B=A ∩ B c Note also A=(A∩B)U(A∩B c ), and, B=(B∩A)U(B∩A c )

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Example If S= {1/2,0,л,5, - √ 2, - 4 } and subsets of S are given by A={- √2, л,0}, B={ 5, ½,-√ 2,- 4} and C={ 1/2,-4 } find A ∩ B, A U B,(A U B ) ∩ C, A- B, ( B ∩ C ) c, B c U C c and (A ∩ C ) U ( B ∩ C ) Solution: A ∩ B={- √ 2 } A U B= {1/2,0,л,5, - √ 2, - 4 } (A U B ) ∩ C ={ 1/2,-4 } A-B={л,0} B c ={л,0} C c ={0,л,5, - √ 2} B c U C c ={0,л,5, - √ 2} = ( B ∩ C ) c (A ∩ C ) U ( B ∩ C )= U { 1/2,-4 } = { 1/2,-4 }

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Example From a survey of sixty students attending a university, it was found that nine were living off campus, thirty-six were undergraduates and three were undergraduates and living off campus. a) Find the number of these students who were undergraduates, or living off campus, or both. n (LU U )=n(L)+n( U )-n(L∩ U )=9+36-3=42

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b) Find the number of these students who were undergraduates and living on campus. LLcLc Sum U336 UcUc Sum960 33 51 246 18 n(U U L c )=33 c) Find the number of these students who were graduate students living on campus. n(U c U L c )=18 S L U 6 33 18 3

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Example يوجد في معهد الامل للصم 300طالب يدرسون لغة الاشارة بالانجليزية أو بالعربية, 150يدرسون لغة الاشارة بالعربية فقط, 80يدرسون لغة الاشارة بالانجليزية فقط. I. ما هو عدد الذين يدرسون لغة الاشارة بالانجليزية والعربية؟ II. ما هو عدد الذين يدرسون لغة الاشارة بالعربية؟ III. ما هو عدد الذين يدرسون لغة الاشارة بالانجليزية؟ IV. ما هو عدد الذين لا يدرسون لغة الاشارة بالانجليزية ولا بالعربية؟ S E A 80 150 0 70

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Some Theorems Involving Sets 1)A U B = B U A (commutative law for unions) 2) A ∩ B = B ∩ A (commutative law for inter sections) 3) A U ( B U C ) = (A U B ) U C = (associative law for unions) 4)A ∩ (B ∩ C)= (A∩B)∩C = A ∩ B ∩ C (associative law for inter section 5) A∩( B U C) = (A ∩ B)U (A∩C) (first distributive law) 6) A U (B ∩ C) = (A U B)∩ (A U C) (second distibutive law) 7) A -B = A ∩ B c 8) if A B then B c A c 9) A U Ø = A, A ∩ Ø = Ø 10 ) A U S = S, A ∩ S = A 11) De Morgran's law a) ( A U B ) c = A c ∩ B c b) (A ∩ B ) c = A c U B c (De Morgan's laws can be generalized to any relation between For Example A ∩ B c = ( A c U B ) c, and (A c U B) c = A ∩ B c 12 ) A = ( A ∩ B ) U ( A ∩ B c )

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Generalizations of Definitions The union of the sets A 1,A 2,A 3,…, denoted A 1 U A 2 U A 3,… is the set of all elements which are in at least one member of the collection. The intersection of the sets A 1,A 2,A 3,…, denoted, A 1 ∩A 2 ∩A 3,.. is the set of all elements which are in every member of the collection. The collection A 1,A 2,A 3,…, is a pair wise disjoint if A i ∩ A j whenever i ≠ j

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Generalizations of Some Theory Involving Sets 1-B ∩ (A 1 U A 2 U A 3 …) = (B ∩ A 1 ) U ( B ∩ A 2 ) U (B ∩ A 3 ) … 2-B U (A 1 ∩A 2 ∩A 3 …)= ( B U A 1 ) ∩ ( B U A 2 ) ∩ (B U A 3 ) ∩ … 3-De Morgan's laws

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Cartesian Products The cartesian product of the sets A 1, A 2,…,A n, denoted by A 1 A 2 … A n, is the set of all n-tuples with the ith coordinate drawn from the ith set, i=1,2,…,n, i.e., A 1 A 2 … A n ={(a 1, a 2,…,a n ): a i A i, i=1,2,…,n} Example let A 1 = {1,2,3},, A 2 = {a, b }, then A 1 × A 2 = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)} وعدد العناصر لحاصل الضرب هو n (A 1 × A 2 ) = n (A 1 ) × n(A 2 ) The Cartesian product may be plotted in 2-dimensional space using points on the horizontal axis for the elements of A 1,, and points of the vertical axis for the elements of A 2, the resulting ordered pairs represent

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Power Sets The power set of A is the set of all subsets which may be formed from the elements of A. Aجميع المجموعات الجزئية المكونة من Example Let A= { a, b, c}. Find the power set of A Power set of A= { Ø, A, {a }, { b }, { c }, {a,b}, {a,c }, { b,c}}

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STAT 111 Chapter One Combinatorics

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In many experiment the number of outcomes in S is so large that a complete listing of these outcomes is too difficult. In such experiment, it is convenient to have a mathematical method of determining the total number of outcomes in the space S and in various events in S without compiling a list of all these outcomes. In the following, some of these methods will be presented. طرق العد هي مجموعة من الطرق الرياضية التي تساعد علي معرفة عدد مرات حدوث تجربة ما دون الحاجة لكتابة عناصر هذه المجموعة

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The Basic Principle of Couting (Multiplication Rule ) If an experiments consists of r steps, of which the first can be made in n 1 ways, for each of these the second step can be made in n 2 ways, and so forth, then the whole experiment can be made in n 1 × n 2 × …× n r

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Example 1: A drug for the relief of asthma can be purchased from 5 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. In how many different ways can a doctor prescribe the drug for a patient suffering from asthma 5x3x2 = 30 Example 2: In how many ways can 8 teaching assistants be assigned to 6 sections of a statistic course if no teacher is assigned to more than one section? 8x7x6x5x4x3 = 20160 The Basic Principle of Couting (Multiplication Rule )

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Example 3: a) How many three-digit numbers can be formed from the digits 0,1, 2,3,4,5, and 6 if each digit can be used only once(التكرار غير مسموح) 6x 6 x 5 = 180 b)How many of these are odd numbers? 5x5x3 = 75 c) How many are greater than 330? 1x3x5 + 3x6x5 = 15 + 90 = 105 The Basic Principle of Couting (Multiplication Rule ) لايتم اختيار الصفر في هذه الخانة الأعداد الأكبر من 3 وهي 6 ، 5 ، 4

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Example 4: If a multiple-choice test consists of 5 questions each with 4 possible answers of which only one is correct, How many different ways can a student check off one answer to each question ? 4 × 4 × 4 × 4 × 4 = ( 4 ) 5 = 1024 How many ways can a student check off one answer to each question and get all the question wrong ? 3× 3 × 3 × 3 × 3 = ( 3 ) 5 = 243 The Basic Principle of Couting (Multiplication Rule )

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Permutations Frequently, we are interested in situations where the outcomes are the different orders or arrangements that are possible for a group of objects. Different arrangements like these are called permutations. Note: n!=n(n-1)(n-2)…3x2x1 0!=1 Definition A permutation is an ordered arrangement of a set of distinct objects. The number of permutations of n distinct objects taken r at a time is denoted by where =n! / (n-r)!= n(n-1) (n-2)….(n-r+1) التباديل هي عدد طرق الاختيار عند توفر شرطين I. الترتيب مهم ( جوائز مختلفة مراكز مختلفة ) II. التكرار غير مسموح

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Examples Example A president, treasurer (أمين صندوق), and a secretary, all different, are to be chosen from a club consisting of 10 people. How many different s of officers are possible? Example Five separate awards are to be presented to selected Students from a class of 30. How many different outcomes are possible if A student can receive any number of awards ; Each student can receive at most 1 award?

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notes In if r= n then which represents the different ways to arrange n objects in a line. The number of different ways to arrange n objects in a circle is ( n – 1 ) !

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Examples 1. In how many different ways can the five letters a, b, c, d and e be arranged? 5! = 120 2. In how many ways can 3 Arabic books, 2 Mathematics books, and 1 Chemistry book be arranged on a bookshelf if; The books can be arranged in any order; 6! = 720 the Arabic books must be together but the other books can be arranged in any order ; 3! 4! = 144 the Mathematics books must be together and the Arabic books must be together ? 3 ! 3 ! 2 ! = 72

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Example 3. a) In how many ways can 5 persons line up to get on a bus. 5! = 120 b) In how many ways can they line up, if 3 persons insist on following each other? 3! 3! =36 c) In how many ways can they line up, if 3 persons refuse to follow each other n(A) = n(S)-n(A c ) - 120 - 3! 3! = 84 4. a) In how many ways may four women and four children be seated in a row of chairs if the women and children to occupy alternate seats'? 2 (4 x 4 x 3 x 3 x 2 x 2 x l x l) = 2 (4! 4!) b) Repeat a) when they are seated at a round table. (4-1)! 4! 5. In how many ways can 7 people be seated at a round table if a) They can sit anywhere; (7-1)! = 6! = 720 b)Two particular people must not sit next to each other? 6! - 2! 5! = 480

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Permutations When Some Elements Are Alike Given n objects of which n 1 are alike, n 2 are alike,..., n r are alike, then the total number of permutations is Example 1 How many different ways can we arrange the letters in the word a) Tool 4!/2! a) Statistics 10!/(3!3!2!) Example 2 How many different arrangement are there of the following set {a, α, α, β, β, γ }. 6!/(3!2!)

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Combinations In many problems we are interested in the number of selecting r objects from n without regard to order (the order is unimportant), these selections are called combinations. التوافيق هي عدد طرق الاختيار عندما يكون الترتيب غير مهم التكرار غير مسموح Definition The number of combinations of r objects is The number is also denoted by the symbol. Note that represents the number of ways we can draw a sample of size r from a set of n distinct elements without replacement and without regard to the order in which these r elements were chosen. It is defined that = 0 whenever n is a positive integer and r is a positive integer greater than n (clearly, there is no way in which we can select a subset which contains more elements than the set itself if r n ) الفرق بين التباديل والتوافيق هو الترتيب ( إذا تم اختيار شخصين من A, B, C الترتيب مهم AB, BA, AC, CA, BC, CB الترتيب غير مهم AB, AC, BC

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Examples Example 1 In how many different ways can a committee of 4 be selected from among the 64 staff members of a hospital? Example 2 A committee of 7, consisting of 2 doctors, 2 teachers, and 3 engineers, is to be chosen from a group of 5 doctors, 6 teachers, and 4 engineers. How many committees are possible?

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Examples Example 3 From a group of 4 teachers and 5 doctors, how many committees of size 3 are possible; a) with no restrictions; b) with 1 teacher and 2 doctors; c) With 2 teachers and 1 doctor if a certain teacher must be on the committees; d) with at least 1 teacher: e) with at most 2 doctors?

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Examples From a group of 8 statisticians and 6 economists a committee consisting of 3 statisticians and 3 economists is to be formed. How many different committees are possible if 1- Two of the statisticians refuse to work together. 2- Two of the economists will only work together. 3- One statistician and one economist refuse to work together.

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Theorem for any positive integers n and r = 0, 1... n. Result:

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The Multinomial Theorem The number of possible divisions of n distinct objects into r distinct groups elements in the first group, n2 elements in the second, and so forth, is جميع الطرق الممكنة لتقسيمn من الأشياء أو الأشخاص إلي rمن المجموعات بحيث تحتوي علي من العناصر n 1, n2, …,nr

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Multinomial Example 1: If 12 People are to be divided into 3 committees of respective sizes 3,4 and, 5 how many divisions are possible ? In how many ways can a man divide 7 gifts among his 3 children if the eldest to receive 3 gifts and the other 2 each ?

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The Binomial Theorem The number is often referred to as the binomial coefficient because it arises in the binomial theorem which may be started as follow : For any number x and y and any positive integer n,

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Examples Example 1 Expand Example 2 Expand Example 3 Find the constant term in the expansion of Since constant term correspond to the one with 3 k – 12 =0, therefore, K = 4 and the constant term

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Example 4 Find the constant term in the expansion of Since constant term correspond to the one with 3 k – 12 =0, therefore, K = 4 and the constant term

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