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Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004 Topic 3 – Logic, Sets and Probability 3.5 – Set Theory and Venn Diagrams Basic Set Theory • A set is a collection of numbers or objects. - If A = {1, 2, 3, 4, 5} then A is a set that contains those numbers. • An element is a member of a set. - 1, 2, 3, 4 and 5 are all elements of A. - means ‘is an element of’ hence 4 A. - means ‘is not an element of’ hence 7 A. - means ‘the empty set’ or a set that contains no elements. Basic Set Theory • If P and Q are sets then: – P ⊆ Q means ‘P is a subset of Q’. – Therefore every element in P is also an element in Q. For Example: {1, 2, 3} ⊆ {1, 2, 3, 4, 5} or {a, c, e} ⊆ {a, b, c, d, e} Basic Set Theory • P Q is the union of sets P and Q meaning all elements which are in P or Q. • P ∩ Q is the intersection of P and Q meaning all elements that are in both P and Q. A = {2, 3, 4, 5} AB= A∩B= and B = {2, 4, 6} Basic Set Theory M = {2, 3, 5, 7, 8, 9} and N = {3, 4, 6, 9, 10} • True or False? I. 4 M II. 6 M • List: I. M ∩ N II. M N • Is: I. M ⊂ N ? II. {9, 6, 3} ⊂ N? Basic Set Theory Reals Rationals R Q (fractions; decimals that repeat or terminate) Integers Z (…, -2, -1, 0, 1, 2, …) Irrationals (no fractions; decimals that don’t repeat or terminate) , 2, etc. Natural N (0, 1, 2, …) Basic Set Theory • N = {0, 1, 2, 3, 4, …} is the set of all natural numbers. • Z = {0, + 1, + 2, + 3, …} is the set of all integers. • Z+ = {1, 2, 3, 4, …} is the set of all positive integers. • Z- = {-1, -2, -3, -4, …} is the set of all negative integers. • Q = { p / q where p and q are integers and q ≠ 0} is the set of all rational numbers (fractions). • R = {real numbers} is the set of all real numbers. All numbers that can be placed on a number line. Basic Set Theory Set Builder Notation describes the properties of the elements of a set, using mathematical notation. • The ’∣’ symbol means “such that.” Example: • D = { x ∣ 0 ≤ x ≤ 5 } – This is read “D is the set of all x such that x is greater than or equal to 0 or less than or equal to 5.” • It is important to note that this example lacks precision as it does not specify what sort of numbers the set elements should be (integers, rationals, etc.)? Basic Set Theory The number of elements in set A is denotes as n(A) Example: • G = { x ∣ x is a square number less than 50 } • Assuming x is an integer then set G has 7 elements. Thus, n(G) = 7 When a set has no elements, such as F = { p ∣ p is a prime number and a multiple of 10} then n(F) = 0. This is called an empty set. A set such as B = { 4, 5, 6, 7} contains a finite number of elements, n(B) = 4, and is termed a finite set. However, n(Z+) = ∞, so Z+ is termed an infinite set. Basic Set Theory - Summary Sets ⊆ N Z Q R Basic Set Theory - Summary Sets - is an element of - is not an element of ⊆ - is a subset of - union (everything) - intersection (only what they share) N – natural numbers Z - integers Q – rational numbers R – real numbers Venn Diagrams The universal set (denoted U) must be stated to make a set well defined. U The universal set U is shown in diagrammatic as a rectangle. A This type of set is called a Venn Diagram. Any set under consideration is shown as a circle inside the universal set U. Venn Diagrams Example U A Tuesday Thursday A universal set that contains all the days of the week that start with T or S. Saturday Sunday Set A is defined as all the days that start with T. Venn Diagrams Subsets If every element in a given set, M, is also an element of another set, N, then M is a subset of N, denoted M ⊆ N A proper subset of a given set is one that is not identical to the original set. If M is a proper subset of N (denoted M⊂ N) then: 1. Every element of M also lies in N and 2. There are some elements in N that do not lie in M Venn Diagrams U N M • • • • If M is a proper subset of N then we write M ⊂ N If M could be equal to N then we write M ⊆ N Clearly, M and N are both subsets of the universal set U In this Venn Diagram M ⊂ N ⊂ U Venn Diagrams Subsets Venn Diagrams Practice Let U = {months of the year that end (in English) with ‘…ber’} Let A = {months of the year that begin with a consonant} Let B = {months of the year that have exactly 30 days} Draw a Venn diagram to show: a. Sets U and A b. Sets U and B c. Sets U, A and B Venn Diagrams Practice U A October September November December Venn Diagrams Practice U B October September November December Venn Diagrams Practice U A September B November December October Venn Diagrams Intersection The intersection of set A and B (denoted A ⋂ B) is the set of all elements that are in both A and B. A ⋂ B is the shaded Venn Diagrams Practice Given the sets: A = {1, 2, 3, 4, 5} B = {x ∣0 < x ≤ 5, x ∈ Z} C = {p ∣ p is a prime number and a multiple of 10} D = {4, 5, 6, 7} E = {x ∣ x is a square number less than 50} Write down the sets: a. b. c. d. A⋂D A⋂B D⋂E C⋂D Venn Diagrams Union The union of set A and B (denoted A ∪ B) is the set of all elements that are in either A or B or both. A ∪ B is the shaded region Venn Diagrams Practice Given the sets: A = {1, 2, 3, 4, 5} B = {1, 2, 3, 4, 5} C={∅} D = {4, 5, 6, 7} E = {1, 4, 9, 16, 25, 36, 49} Write down the sets: a. A ∪ D b. A ∪ B c. C ∪ D Venn Diagrams Complement The complement of set M (denotes M’) is the set of all the elements in the universal set that do not lie in M M N M’ is the shaded region U Venn Diagrams Example: M ∩ N’ M N U Venn Diagrams Example: M ∪ N’ M N U Venn Diagrams Complement Venn Diagrams Practice U = {4, 5, 6, 7, 8, 9, 10}, F = {4, 5, 6, 7}, G = {6, 7, 8, 9} a. Draw a Venn diagram for F , G and U b. Write these sets: i) F’ ii) F ∩ G’ iii) (F ∩ G)’ Venn Diagrams U Practice In this Venn diagram each dot represents an element. Write down: a. n(G) b. n(F) c. n(G ∩ F) d. n(H’) e. n(F ∩ H) f. n(G ∩ H) Are these statements true or false? g. n(F ∪H) = n(F) + n(H) h. n(G ∪ H) = n(G) + n(H) F H G Venn Diagrams U 3 Sets B A Shade the region on the Venn diagram that shows the sets: a. (A ∪ B) ∩ C b. A ∪ (B ∩ C) C Venn Diagrams U 3 Sets B A a. (A ∪ B) ∩ C C Venn Diagrams U 3 Sets B A b. A ∪ (B ∩ C) C Venn Diagrams Problem Solving A teacher asks her class how many of them study chemistry. She finds there are 15. She then asks how many study biology and finds there are 13. She also finds out that 5 students do not take any science. There are 26 total students in the class. 15 + 13 + 5 = 33. Has she miscounted? Venn Diagrams Problem Solving U Represent this in a Venn diagram with: Chemistry = Set C = n(C) = 15 Biology = Set B = n(B) = 3 Total Class = Set U = n(U) = 26 No Science = n(B’ ∩ C’) = 5 B C 13 6– x x 7 158– x 5 Thus: (13 – x) + x + (15 – x) + 5 = 26 33 – x = 26 x=7 Venn Diagrams Practice In a class of 29 students, 19 study German, 14 study Hindi and 5 study both languages. Work out the number of students that study neither language. Venn Diagrams n(G’∩H’) = 29 – (14+5+9) = 1 Practice n(U) = 29, n(G) = 19, n(H) = 14 and n(G∩H) = 5 U G H 19 - 5 14 14 - 5 5 9 Venn Diagrams Practice 145 people answered a survey to find out which flavor of fruit juice, orange, apple or grape the preferred. The replies showed: • 15 liked none of the three • 55 liked grape • 80 liked apple • 75 liked orange • 35 liked orange and apple • 20 liked orange and grape • 30 liked apple and grape Find the number of people who liked all three types of juice Basic Set Theory - Review Notation • set: A = {1, 3, 5} • subset: P Q – ‘P is a subset of Q’ {2, 3, 5} {1, 2, 3, 4, 5, 6} Basic Set Theory - Review Notation • ‘is an element of’ – 3 {2, 3, 5} • ‘is not an element of’ – 7 {2, 3, 5} • n(A) number of elements in set A. Basic Set Theory - Review Notation • Infinite Sets – sets with an infinite number of members. o P = {0, 1, 2, 3, …} o n(P) = ∞ • Finite Sets – sets that contain a finite number of members. o A = {1, 2, 3} o n(A) = 3 • Note: n( ) = 0 o = empty set Basic Set Theory - Review Set Builder Notation A x x Z , 2 x 4 “the set of all x such that x is an integer between -2 and 4, inclusive.” Basic Set Theory - Review Complementary Sets If the universal set is: U = {1, 2, 3, 4, 5, 6, 7, 8} and A = {1, 3, 5, 7, 8} then the complement of A, denoted A’ is A’ = {2, 4, 6} Basic Set Theory - Review Notation A A' 0 A A' U n(A’) = n(U) – n(A) Practice Practice Venn Diagrams - Review Diagrams used to represents sets of objects, numbers or things. U = {2,3,5,7,8} A = {2,7,8} A’ = {3,5 Venn Diagrams - Review Disjoint Sets Practice Practice Venn Diagrams - Review A is shaded A∩B is shaded B’ is shaded A∩B’ is shaded Practice