Sets and Probability - Grade 9 Maths Past Paper PDF

UNIT 8 SETS AND PROBABILITY A. DEFINING A SET A collection of well - defined objects is called a set. For instance, the following sets are well-defined: The set of days of the week. The set of natural numbers. The set of prime numbers less...

UNIT 8 SETS AND PROBABILITY A. DEFINING A SET A collection of well - defined objects is called a set. For instance, the following sets are well-defined: The set of days of the week. The set of natural numbers. The set of prime numbers less than 50. However, the following sets are not well-defined: The good students in your class. Some of the letters of the alphabet. The set of humanists. Each object in a set is called an element or a member of the set. For example, in the set A = {a, b, c, d}, The letter b is an element of set A and we write b Î A. We say, b is an element of set A, or b is in A. The letter f is not an element of set A and we write f Ï A. We say, f is not an element of set A, or f is not in A. B. REPRESENTATION OF SETS 1. The List Method If we represent a particular set using the list method, we write all the elements of the set inside a pair of set brackets { } separated by commas. For example: A = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}; B = {a, b, c, d, e}; C is the set of letters of MATHEMATICS; C = {M, A, T, H, E, I, C, S} Each element can be written only once. D = {1, 2, 3,... ,1000}; E = {2, 3, 5, 7, 11, 13, 17, 19}; F = {5, 10, 15, 20, 25,... ,200} 2. The Defining-Property Method If there is a common property among the elements of a set, then we can use the defining-property or set builder method. To represent a set by this method, we write the common property inside braces. For example: A = {The days of the weekend}; B = {x | x is a day of the weekend}; C = {x | x is a season of the year}; D = {y | y is a natural number less than 25 and divisible by 3}; E = {x | x is a learner in Grade 9 A}; F = {y | y is a two digit prime number}; G = {x | x is a multiple of 5}; H = {z | z is a blood type}. Grade 9 > Mathematics Paper 1 > 115 Unit 8 Sets and Probability 3. The Venn Diagram Method                 Venn diagrams are used to represent the elements of the set in a closed shape. Each element of the set is represented by a point inside the closed shape. NUMBER OF ELEMENTS OF A SET The number of elements in a set A is called the cardinal number of the set A and is denoted by n(A). ii EXAMPLE: NOTES ii Find the number of elements in each set: ii A = {1, 2, 3, 4, 5, 6, 7, 8, 9}, ii n(A) = 9. ii B = {a, 5, c, d, 12}, ii n(B) = 5. 116 > Unit 8 > Sets and Probability  Assessment TASK-40 1. Determine whether each of the following B = {Tuesday, Thursday} sets is well-defined or not. A = {x|x is a counting number less than 7} C = {East, West, North, South} B = {x|x is a day of the week} C = {x|x is a child} 5. Express the sets A, B and C using the list method.     2. The set A = {1, a, 3, b, c, 6} is given.  Complete the statements below using    the symbol Î or Ï.   2.1 a........A 2.2 3........A   2.3 10........A 2.4 c........A  ©... Publications 2.5 6........A 3. Express the following sets using the list method. M = {The letters in the word MARMARA} 6. Find the number of the elements of the T = {The odd numbers between 12 and 24} following sets. A= {x|x is a natural number less than 9} B = {The digits of the number 214561302} 4. Express the following sets using the defining property method. C = {1, 2, a, 0, x, 3} A = {12, 14, 16, 18, 20, 22, 24, 26} Grade 9 > Mathematics Paper 1 > 117 Unit 8 Sets and Probability C. OPERATION ON SETS In arithmetic, we know that there are four basic operations: addition, subtraction, multiplication and division. We can perform similar operations on sets. These operations are union, intersection, and complement. Sample Space or Universal Set is the set of all elements under consideration in a given discussion. In a Venn diagram, the universal set is represented by a rectangle which includes other sets (events). All the other sets are to be drawn inside this rectangle.     A = {a, b, c}   B = {c, d, e}   C = {f, g, h}        So, U = {a, b, c, d, e, f, g, h, i, j}. A set which has no elements is called an Empty Set (Æ) An empty set is like nothing which is opposite of sample set as everything is in the sample (universal) set. Sometimes sets can contain other sets. Set A is a subset of set B if every element of A is also an element of B.    A = {1, 3, 5} and    B = {1, 2, 3, 4, 5, 6} then   U = {1, 2, 3, 4, 5, 6, 7, 8}     1. UNION OF SETS The union of sets A and B is the set of all elements which belong to A or to B (or to both). We write A or B.      If A = {a, b, c, d, e} and    B = {a, c, f, g, h} then    A or B = {a, b, c, d, e, f, g, h}  118 > Unit 8 > Sets and Probability  Unit 8 Sets and Probability ii EXAMPLE: ii Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, ii A = {1, 3, 5, 7, 9}, ii B = {1, 2, 3, 4, 5} and ii C = {2, 4, 6, 8}. ii Find: Solution: You should draw Venn Diagram to see these better ii a. A or B a. {1, 2, 3, 4, 5, 7, 9} ii b. A or C b. {1, 2, 3, 4, 5, 6, 7, 8, 9} ii c. B or C c. {1, 2, 3, 4, 5, 6, 8} ii d. A or B or C d. {1, 2, 3, 4, 5, 6, 7, 8, 9} ii e. A or A e. {1, 3, 5, 7, 9} equals to A ii f. S or U f. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} equals to S 2. INTERSECTION OF SETS Let A and B be two sets: NOTES A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8, 10}.         5   As we can see, 2 and 4 are elements of both sets. The set formed by these common elements is called the intersection of sets A and B. The intersection of two sets A and B is the set of all elements that are in both A and B. We write A and B. Grade 9 > Mathematics Paper 1 > 119 Unit 8 Sets and Probability ii EXAMPLE: ii Let A = {1, 2, 3, 4}, B = {2, 3, 5, 6}, and C = {3, 4, 5, 7} be three sets. Find the following sets: ii a. A and B b. A and C   ii c. C and B d. A and B and C    ii e. A and A  4  ii SOLUTION  ii Let’s draw Venn diagram: ii a. A and B = {2, 3} b. A and C = {3, 4}  ii c. C and B = {3, 5} d. A and B and C = {3} ii e. A and A = {1, 2, 3, 4} equals to A ii Now you may draw Venn diagram by starting with A and B and C, from the most inner part. OO Two sets are called disjoint if they don’t have any intersection. ii EXAMPLE: NOTES ii Let ii A = {a, b, c, d}, ii B = {b, d, e, f}, and ii C = {a, b, e, g}. ii Find: ii A or (B or C), n(A), n(B and C) ii Examples may be done either by drawing Venn diagram or just analytically ii Anything that in A or B or C , A or (B or C) = {a, b, c, d, e, f, g} ii n(A) means number of elements in A = 4 ii B and C = {b, e}, n(B and C) = 2 120 > Unit 8 > Sets and Probability  Unit 8 Sets and Probability 3. COMPLEMENT OF A SET NOTES The set of elements of Sample Space which are not the elements of set A is called the complement of A. We write õA or A‘ or not A, and say the complement of set A. The shaded region in Venn diagram shows complement of A.    ii EXAMPLE: ii For the diagram below: ii a. A‘ a. A‘= {3,4, 5, 6, 7, a, b}  ii b. B‘ b. B‘= {1, 2, a, b}   ii c. (A and B)‘ c. (A and B)‘ = {1, 2, 5, 6, 7, a, b}    ii d. A‘ and B d. A‘ and B = {5, 6, 7 }    ii e. A and B ‘ e. A and B‘ = {1, 2}  ii f. (A or B)‘ f. (A or B)‘ = {a, b}  ii g. A‘ or B‘ g. A‘ or B ‘= {1, 2, 5, 6, 7, a, b}  Grade 9 > Mathematics Paper 1 > 121 Unit 8 Sets and Probability D. SURVEY PROBLEMS NOTES When we collect data and organize it into sets, we then usually want to manipulate the information to answer questions about those sets. These types of problems are often called survey problems. Let us learn how to use set operations in survey problems. Look at the following example: i EXAMPLE: i In a class, every student plays at least one game: chess or table tennis. 15 students play chess, 25 play table tennis, and 5 play both. i a. How many students play only table tennis? i b. How many students play only chess? i c. How many students are there in this class? i SOLUTION: i Let us express the students who play chess by C and table tennis by T. i Then, n(C) = 15, n(T) = 25, n(C and T) = 5 ii   a. Only T =25-5 = 20 students ii b. Only C=15-5 = 10 students ii    c. 35 students NOTE: Every student plays at least one game means there is no one plays nothing. So in the Venn diagram, the number to be written inside the  rectangle but outside the circles must be zero, 0.   122 > Unit 8 > Sets and Probability  Unit 8 Sets and Probability The following strategies are useful in solving survey problems; 1. Draw a Venn diagram in a universal set. 2. Label the sets. 3. Write the given information in the regions: a. Start with the intersection of all sets. b. Write the sets only A, only B and only C. 4. Use the following property: n(A or B) = n(A) + n(B) – n(A and B) if necessary.  only       only  only Only  ii EXAMPLE: ii In a class, the students speak at least one language: Russian, English or Turkish. If 10 students speak Russian, 8 students speak English, and 15 students speak Turkish, 6 students speak Russian and Turkish, 5 students speak Russian and English, 4 students speak Turkish and English, and 3 students speak Russian, English and Turkish, then how many students are there in this class? ii SOLUTION: ii n(R and E and T) = 3   ii n(R and T) = 6 2   ii n(R and E) = 5  ii n(T   and E) = 4 ii n(R) = 10  ii n(E) = 8  ii n(T) = 15  ii n(R or E or T) = 2+2+2+3+3+1+8 = 21 ii There are 21 students in this class. Grade 9 > Mathematics Paper 1 > 123  Assessment TASK-41 QUESTION 1 QUESTION 3 Use the given Venn diagram to find; Two hundred teenagers had to answer about their favorite music. The results were:   160 like hip-hop (H), 140 like pop (P),    108 like both (H and P)   3.1 Represent results on Venn Diagram.         1.1 The Sample Space 3.2 How many like hip-hop only? 1.2 A 1.3 A and B 3.3 How many like pop music only? 1.4 B or C 1.5 A and B and C 3.4 How many like Hip-hop or Pop? 1.6 (A or B)' QUESTION 4 QUESTION 2 S={natural number less then 18}  A={first seven prime numbers}  B={factors of 16} C={first five multiples of 3}  2.1 List the three sets A, B and C.  2.2 Represent them on a Venn Diagram. 4.1 I f n(A) = 6 and n(A or C) = 15 then determine n(C). 4.2 If n(A) = 7, n(B) = 5, n(A or B) = 10 then 2.3 Which two sets are disjoint? determine n(A and B). 2.4 List A and B 4.3 If n(A) = 5, n(A and B) = 2, n(A or B) = 8 then determine n(B). 2.5 List (A or B or C)' 4.4 2. n(A and B) = n(A or B) and 2.6 Determine n(A or B) n(A or B) = 8 then determine n(A and B). 124 > Unit 8 > Sets and Probability  Assessment TASK-42 QUESTION 1 QUESTION 3 In group of 85 girls, 48 play hockey, 43 play In a class of 40 learners, 32 take Maths and tennis and 12 do not play hockey or tennis. 23 History and 4 do not take any of them. 1.1 D raw a Venn diagram to illustrate this 3.1 Represent information on Venn Diagram information and to determine how many girls play hockey and tennis. (Hint: let x to be number of girls playing hockey and tennis) 3.2 How many take both?  1.2 How many of them play hockey only? 3.3 How many don’t take math? 1.3 How many of them doesn’t play tennis only? 3.4 How many take History but not Maths? 1.4 How many of them plays hockey or tennis? QUESTION 4 QUESTION 2 S = (x|x £ 10, x Î N) A={x|x £ 6 ; x Î N0(natural numb + zero)} A = (1, 2, 3, 4, 5) B={x|x = p2 ; p Î {0,1,2,3,4}} B = (2, 4, 6, 8, 10) S={x|x £ 16; x Î N0(natural numb + zero)} 4.1 Represent sets on a Venn Diagram 2.1 List the two sets A and B. 2.2 Represent them on Venn Diagram. 4.2 Determine n(A and B) 2.3 Find n(A). 4.3 Find (A or B)' 2.4 List A and B. 2.5 Find n(A and B). 2.6 Find n(õ B). Grade 9 > Mathematics Paper 1 > 125