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Sets A set is a well defined group of objects such as those shown below. A set can be finite or infinite. The members of the set are called elements “e”. If an element e belongs to a set S then we write e  S (e is an element of S). B = { 3,7,9,14 } A = { London, Rome, Paris, Cairo,… } C = { A, B, C, D, E,……Z } D = { 1,4,9,16,25,36,….. } Rome  A7  B Q  C 100  D If an element e does not belong to a set S then we write e  S (e is not an element of S). Munich  A20  B β  Cβ  C 63  D European Capitals Letters of the Alphabet Square Numbers

Sets Describe in words the sets below and state whether they are finite or infinite. Where possible, use set notation to write down one additional element that is a member of the set and one that isn’t. B = { 1,3,6,10,… } A = { red, orange, yellow, green, blue, indigo, violet } C = { Rebecca, Razia, Rose, Rhea, Razaanah, … } D = { August, January, April, … } E = { Monday, Tuesday, Wednesday, … } F = { 64, 27, 8, 125, …} G = { 2,3,5,7,11,13, …} Colours of the rainbow Triangular numbers (infinite) Girls’ names beginning with R. Months of the year Days of the week Cube numbers (infinite) Prime numbers (infinite)

Sets If all the elements of a set A are also elements of another set B then A is said to be a subset of B (A  B). A = { August, January, April, … } B = { Monday, Tuesday, Wednesday, … } C = { 64, 27, 8, 125, … } D = { 2,4,6,8,10,12, … } Subsets G = { September,October,November } H = { Saturday,Sunday } I = { 1,343,1000 } E = { 2,3,5,7,11,13, … } F = { 1,2,3,4,5,6, … } Write down some sets that are subsets of others from the list below. G  A H  B I  C I  F C  F D  F E  F

Sets Consider the set A consisting of three people, Alice, Stephen and Jane. Subsets A = { Alice, Stephen, Jane } We can list all subsets of A as follows: B = { Alice, Stephen, Jane } C = { Alice, Stephen } D = { Alice, Jane } E = { Stephen, Jane } F = { Alice } G = { Stephen } H = { Jane } I =  B  A B  A C  A D  A E  A F  A G  A H  A By convention, sets B and I are considered as subsets of A. We distinguish these from proper subsets by use of  symbol. Notation:  (Subset),  (Proper Subset)  (The empty Set)  (Not a subset)  (Not a proper subset) We can write F  A or G  D

Sets Question 1. If Set A are the whole numbers less than 20, list: (a) The subset B {odd numbers}, (b) The subset C {prime numbers}, (c) The subset D {Multiples of 5). Subsets Notation:  (Subset),  (Proper Subset)  (The empty Set)  (Not a subset)  (Not a proper subset) (a) B = {1,3,5,7,9,11,13,15,17,19}, (b) C = {2,3,5,7,11,13,17,19}, (c) D = {5,10,15). Question 2. List all subsets of {STU} indicating which are proper/improper subsets. {STU} and  are improper subsets. {S,T},{S,U},{T,U),{S},{T},{U} are proper subsets. Question 3. State whether the following statements are true or false. (a){4,5,6,7}  {4,5,6,7} (b) {Red, Yellow, Blue}  {Yellow, Blue, Red} (c) {1,2,3}  {prime numbers} (d) {A,B,C,D,E}  {B,E,D,F,E,A} False True False

Sets List the members of the given sets below and state n(S) in each case. B = { Even numbers  14 } A = { Odd numbers < 12 } C = { Square numbers between 20 and 70 } D = { Vowels } E = { Months with 30 days } F = { prime numbers p; 11  p  39 } G = { Cube numbers C: 10  C  50 } 1,3,5,7,9,11 n(A) = 6 2,4,6,8,10,12,14 n(B) = 7 25,36,49,64 n(C) = 4 a,e,i,o,u n(D) = 5 September, April, June, November n(E) = 4 11,13,17,19,23,29,31,37 n(F) = 8 27 n(G) = 1 The number of elements of a set S can be written as n(S).

Sets A = { 3,7,9,14 }, B = { 9,14,28 }, C= { 5,7,12,14,24 } Consider the sets A and B below where 9 and 14 are elements common to both sets. We write A  B = { 9,14 } (A intersection B). Intersections and Unions The union of both sets is all elements that are contained in A or B or both. We write A  B = { 3,7,9,14,28 } ( A union B ). Write the following sets: P = A  C Q = B  C R = B  C S = A  C P = { 7,14 } Q = { 5, 7,9,12,14,24,28 } R = { 14 } S = { 3, 5, 7,9,12,14,24 }

Sets A = { 3,7,9,10 }, B = { 1,2,3 }, C= { 2,4,6,8,10 } The universal set is the set that contains all elements under consideration for a particular problem. This set can be denoted by the letter E. The Universal Set and Compliments Consider the universal set shown together with some subsets. E = { 1,2,3,4,5,6,7,8,9,10 } We define the compliment of a set A as the set of elements A’ that are in E but not in A. So we can write A’ = { 1,2,4,5,6,8 } B’ = { 4,5,6,7,8,9,10 } C’ = { 1,3,5,7,9 } Write out B’ and C’

Sets C = { Months with four-letter names } The Universal Set and Compliments (a) E = { Months of the year } A = { July, August September } B = { Months with less than 31days } Question: (a) Give a suitable universal set E for the subsets below. (b) Find A  B (c) A  C (d) B’ (b) { September } (c) { June, July, August, September } (d) { January, March, May, July, August, October, December }

Sets The Universal Set and Compliments For questions 1 to 4 below find: (a) A  B (b) A  B 1. A = { 1,2,3 }, B = { 2,3,5,7 } 2. A = { 1,3,5,7 }, B = { 2,3,5,8 } 3. A = { Multiples of 4 less than 20 }, B = { Multiples of 6 less than 20 } 4. A = { Square numbers below 100 }, B = { Cube numbers below 100 } 1 (a) { 2,3 }, (b) { 1, 2,3,5,7 } 2 (a) { 3,5 }, (b) { 1, 2,3,5,7,8 } 3 (a) { 12 }, (b) { 4, 6,8,12,16,18 } 4 (a) { 1, 64 }, (b) { 1, 4, 8, 9,16,25,27,36,49,64,81 }

Venn Diagrams Sets Venn Diagrams Venn diagrams are a way of showing sets pictorially in diagrammatic form. The universal set is represented by a rectangle and subsets are shown in circles within the rectangle. When drawing Venn diagrams we must ensure that the number of intersections fit the given data. A B A  B If E = { 1,2,3,4,5,6,7,8,9,10 } A = {1,3,5,7,9,10} B = {3,4,5,7,8} Example 1 E

Sets Venn Diagrams Venn diagrams are a way of showing sets pictorially in diagrammatic form. The universal set is represented by a rectangle and subsets are shown in circles within the rectangle. When drawing Venn diagrams we must ensure that the number of intersections fit the given data. E A B If E = { 1,2,3,4,5,6,7,8,9,10 } A = {2,4,6,8,10} B = {4,6,8} Example

Sets Venn Diagrams Venn diagrams are a way of showing sets pictorially in diagrammatic form. The universal set is represented by a rectangle and subsets are shown in circles within the rectangle. When drawing Venn diagrams we must ensure that the number of intersections fit the given data. E If E = { 1,2,3,4,5,6,7,8,9,10 } A = {2,4,6,8,10} B = {1,3,5} Example 3 A B

Sets Venn Diagrams Venn diagrams are a way of showing sets pictorially in diagrammatic form. The universal set is represented by a rectangle and subsets are shown in circles within the rectangle. When drawing Venn diagrams we must ensure that the number of intersections fit the given data. E If E = { 1,2,3,4,5,6,7,8,9,10 } A = {2,4,5,8,10} B = {4,6,9,10} C = {1,4,6,} Example 4 A B C A  B  C

Sets Venn Diagrams Show each group of sets in a Venn diagram. (b) E = { 1,2,3,4,5,6,7 } A = {1,2,6} B = {2,4,6} (a) E = { 1,2,3,4,5,6,7 } A = {2,3,4,5} B = {2,5} (c) E = { 1,2,3,4,5,6,7 } A = {1,2,4} B = {3,6} (d) E = { 1,2,3,4,5,6,7,8 } A = {2,4,5,8} B = {1,2,7,8,} C = {2,4,6,}

Sets Venn Diagrams (a) Describe in words the elements of: (i) Set A (ii) Set B (iii) Set C (b) Copy and complete the following statements: (i) A  B = { … } (ii) A  C = { … } (iii) B  C = { … } (iv) A  B  C = { … } (v) A  B = { … } (vi) C  B = { … } A B C (a) (i) Odd numbers from 3 to 15 (ii) Multiples of 3 from 3 to 18 (iii) Some odd primes  19 (i) A  B = { 3,9,15 } (ii) A  C = { 3,7,11,13 } (iii) B  C = { 3 } (iv) A  B  C = { 3 } (v) A  B = { 3,5,6,7,9,11,12,13,15,18 } (vi) C  B = { 3,6,7,9,11,12,13,15,17,18,19 } (b)

Sets A B E A B E A B E A B E C A B E A E A E A B E C A B E C In each of the Venn diagrams below, describe the shaded area in terms of the subsets. (A  B)(A  B) B’ (A  B  C) B B B A’ or (A  B)’ B’ (A  B’  C’) (A  B  C’) (a) (b) (c) (d) (e) (f) (g) (h) (i)

Sets A B E Example Question: In the Venn diagram below E is the number of pupils in a year 7 class that attend an after school sporting activity at a local gym A = { students who play squash }, B = { students who play volley ball } (a) How many students are in this year 7 class? (b) How many students play squash? (c) How many students play both squash and volley ball? (d) How many students play neither squash nor volley ball? (a) n( E ) = = 30 (b) n(A) = = 11 (c) n(A  B) = 8 (d) n(A  B)’ = 12

Sets A B E Question: In the Venn diagram below E is the number of people that attended a local council meeting A = { people that voted }, B = { people that asked for tea } (a) How many people asked for tea? (b) How many people asked for tea and voted? (c) How many people neither asked for tea nor voted? (d) How many people attended the meeting? (d) n( E ) = = 109 (a) n(B) = = 53 (b) n(A  B) = 30 (c) n(A  B)’ = 20

Sets P C E PROBLEM SOLVING x 18 - x x Example Question: In a class of 30 students, some studied physics and some chemistry. If 20 studied physics, 18 studied chemistry and 4 studied neither, calculate the number of students that studied both subjects. 20 – x + x + 18 – x + 4 = x = 30 x =

Sets F R E PROBLEM SOLVING x x x Question: In a sports survey, 324 teenagers responded to a questionnaire. They were asked if they liked football and rugby. 180 said they liked football, 159 said they liked rugby and 30 said they liked neither. Calculate how many teenagers: (a) liked both football and rugby. (b) Liked only rugby. 180 – x + x – x + 30 = x = 324 x =