Chapter 1. Set Theory

Transcription

1 Chapter 1 Set Theory 1

2 Section 1.1: Types of Sets and Set Notation Set: A collection or group of distinguishable objects. Ex. set of books, the letters of the alphabet, the set of whole numbers. You can represent a set of elements by: listing the elements; for example, A = {1, 2, 3, 4, 5} using words or a sentence; for example, A = {all integers greater than 0 and less than 6} using set notation; for example, A = {x/ 0 < x < 6, x I} Element: An object in a set. Ex. 2 is an element of the set A. We write this as 2 A. (2 is a member of (or belongs to) the set A) The number of elements in the set A is denoted by n(a). In the above example n(a) = 5. Universal Set: Ex. A set of all the elements under consideration for a particular context (also called the sample space). A = {all letters of the alphabet} The set of digits D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 2

3 Subset: A set whose elements all belong to another set. Ex. The set of vowels V is a subset of A, the letters of the alphabet. Notation: V A. The set of even digits E = {0, 2, 4, 6, 8} is a subset of D, the set of digits. Notation: E D. Complement: Ex. All the elements of a universal set that do not belong to a subset of it. V' the set of all elements in the universal set that are not in V. So V' = {set of consonants} is the complement of V. E' is the set of all digits in the universal D that are not in E. So E' = {1, 3, 5, 7, 9} is the complement of E. The sum of the number of elements in a set and its complement is equal to the number of elements in the universal set: n(a) + n(a') = n(u) 3

4 Empty Set: A set with no elements. Notation {} or Ex. The set of months with 32 days. Disjoint Sets: The set of squares with 5 sides. Two or more sets having no elements in common. Ex. The set of odd numbers and the set of even numbers. The number of females and the number of males in a room. When two sets A and B are disjoint, n(a or B) = n(a) + n(b) The events that describe disjoint sets are mutually exclusive.they are two or more events that cannot occur at the same time. Ex. Flipping a coin and getting a head and a tail at the same time. The sum of two numbers cannot be less than 10 and greater than 10 at the same time. 4

5 A finite set is a set with a countable number of elements. Ex: The set of even numbers less than 10, is finite. E = { 0, 2, 4, 6, 8} An infinite set is a set with an infinite number of elements. Ex: The set of natural numbers, is infinite. N = { 1, 2, 3,... } 5

6 Examples 1. The universal set is defined as A, the set of all natural numbers. B is the set of all natural numbers from 1 to 5. A. List the members of each set. How many elements are in each set? B. Is one set a subset of the other? Why? C. Which elements of the universal set does not belong to the subset? D. What is the complement of B? 2. Mary created the sets P = {1, 2, 3} and Q = { 2, 3, 4, 5, 6}. John stated that P Q since the elements 2 and 3 are in both sets. Do you agree or disagree? Explain 6

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8 3. Natasha drew the Venn diagram below. G = {plants in her garden} P = {perennials} A = {annuals} E = {edible plants} A. Is E P? Is E G? Explain. B. List the disjoint sets, if there are any. C. Is P' equal to A? Explain. D. Determine n(p) using n(g) and n(a). E. List the elements in E '. 8

9 4. Consider the following S = {4, 5, 6, 8, 9, 11, 15, 17, 20, 24, 30, 32} A) Complete the following Venn Diagram. B) Why do the circles overlap? C) What do the elements in the intersection represent? D) Why are some numbers not in either circle? E) Add another circle to represent the multiples of 4. Complete the following Venn Diagram. S = {4, 5, 6, 8, 9, 11, 15, 17, 20, 24, 30, 32} 9

10 5. A. Indicate the multiples of 2, 4, and 11, using set notation. Then draw a Venn diagram to represent these sets: U = {natural numbers from 1 to 20 inclusive} T = {multiples of 2) F = {multiples of 4} S = {multiples of 11} B. List the disjoint subsets, if there are any. 10

11 C. Is each statement true or false? Explain. i) F T ii) T F iii) T T iv) T' = {odd numbers from 1 to 20} v) In this example, the set of natural numbers from 21 to 50 is { }. 11

12 vi) Explain what the following statement means: F T but T F. vii) Suppose you choose one number from U. Are the events choosing a number that is a multiple of 2 and a multiple of 11 mutually exclusive? Explain. viii) Is the following statement correct? n(t or S) = n(t) + n(s) ix) Determine the value of n(t or S). 12

13 6. Consider the following information: U = {natural numbers from 1 to 100} X U n(x) = 19 Determine n(x '), if possible. If it is not possible, explain why. 7. Consider the following information: U = {natural numbers from 1 to 500} A B U U n(a) = 200 Determine n(b), if possible. If it is not possible, explain why. 8. Determine n(u), the universal set, given n(a) = 19 and n(a ') =