2.-Sets_02_01.ppt

CHAPTE R 2 Sets Copyright © Cengage Learning. All rights reserved. Section 2.1 Basic Proper...

CHAPTE R 2 Sets Copyright © Cengage Learning. All rights reserved. Section 2.1 Basic Properties of Sets Copyright © Cengage Learning. All rights reserved. Sets 3 Sets Any group or collection of objects is called a set. The objects that belong in a set are the elements, or members, of the set. For example, the set consisting of the four seasons has spring, summer, fall, and winter as its elements. The following two methods are often used to designate a set. Describe the set using words. List the elements of the set inside a pair of braces, { }. This method is called the roster method. Commas are used to separate the elements. 4 Sets For instance, let’s use S to represent the set consisting of the four seasons. Using the roster method, we would write S = {spring, summer, fall, winter} The order in which the elements of a set are listed is not important. Thus the set consisting of the four seasons can also be written as S = {winter, spring, fall, summer} 5 Sets The following table gives two examples of sets, where each set is designated by a word description and also by using the roster method. Define Sets by Using a Word Description and the Roster Method Table 2.1 6 Example 1 – Use the Roster Method to Represent a Set Use the roster method to represent the set of the days in a week. Solution: 7 Example 2 – Use a Word Description to Represent a Set Write a word description for the set Solution: Set A is the set of letters of the English alphabet. 8 Sets The following sets of numbers are used extensively in many areas of mathematics. 9 Sets The set of natural numbers is also called the set of counting numbers. The three dots... are called an ellipsis and indicate that the elements of the set continue in a manner suggested by the elements that are listed. The integers... , –4, –3, –2, –1 are negative integers. The integers 1, 2, 3, 4,... are positive integers. Note that the natural numbers and the positive integers are the same set of numbers. The integer zero is neither a positive nor a negative integer. 10 Sets If a number in decimal form terminates or repeats a block of digits without end, then the number is a rational number. Rational numbers can also be written in the form where p and q are integers and q  0. For example, are rational numbers. 11 Sets The bar over the 27 means that the block of digits 27 repeats without end; that is, A decimal that neither terminates nor repeats is an irrational number. For instance, 0.35335333533335... is a nonterminating, nonrepeating decimal and thus is an irrational number. Every real number is either a rational number or an irrational number. 12 Example 3 – Use the Roster Method to Represent a Set of Numbers Use the roster method to write each of the given sets. a. The set of natural numbers less than 5 b. The solution set of c. The set of negative integers greater than –4 Solution: a. The set of natural numbers is given by {1, 2, 3, 4, 5, 6, 7,...}. The natural numbers less than 5 are 1, 2, 3, and 4. Using the roster method, we write this set as {1, 2, 3, 4}. 13 Example 3 – Solution cont’d b. Adding –5 to each side of the equation produces x = –6. The solution set of c. The set of negative integers greater than –4 is {–3, –2, –1}. 14 Definitions Regarding Sets 15 Definitions Regarding Sets A set is well defined if it is possible to determine whether any given item is an element of the set. For instance, the set of letters of the English alphabet is well defined. The set of great songs is not a well-defined set. It is not possible to determine whether any given song is an element of the set or is not an element of the set because there is no standard method for making such a judgment. 16 Definitions Regarding Sets The statement “4 is an element of the set of natural numbers” can be written using mathematical notation as 4  N. The symbol  is read “is an element of.” To state that “–3 is not an element of the set of natural numbers,” we use the “is not an element of ” symbol, , and write –3  N. 17 Example 4 – Apply Definitions Regarding Sets Determine whether each statement is true or false. Solution: a. Since 4 is an element of the given set, the statement is true. b. There are no negative natural numbers, so the statement is false. 18 Example 4 – Solution cont’d c. Since is not an integer, the statement is true. d. The word nice is not precise, so the statement is false. 19 Definitions Regarding Sets The empty set, or null set, is the set that contains no elements. The symbol  or { } is used to represent the empty set. As an example of the empty set, consider the set of natural numbers that are negative integers. Another method of representing a set is set-builder notation. Set-builder notation is especially useful when describing infinite sets. 20 Definitions Regarding Sets For instance, in set-builder notation, the set of natural numbers greater than 7 is written as follows: The preceding set-builder notation is read as “the set of all elements x such that x is an element of the set of natural numbers and x is greater than 7.” It is impossible to list all the elements of the set, but set-builder notation defines the set by describing its elements. 21 Example 5 – Use Set-Builder Notation to Represent a Set Use set-builder notation to write the following sets. a. The set of integers greater than –3 b. The set of whole numbers less than 1000 Solution: 22 Definitions Regarding Sets A set is finite if the number of elements in the set is a whole number. The cardinal number of a finite set is the number of elements in the set. The cardinal number of a finite set A is denoted by the notation n (A). For instance, if A = {1, 4, 6, 9}, then n (A) = 4. In this case, A has a cardinal number of 4, which is sometimes stated as “A has a cardinality of 4.” 23 Example 6 – The Cardinality of a Finite Set Find the cardinality of each of the following sets. Solution: a. Set J contains exactly two elements, so J has a cardinality of 2. Using mathematical notation, we state this as n (J) = 2. b. Only a few elements are actually listed. The number of natural numbers from 1 to 31 is 31. If we omit the numbers 1 and 2, then the number of natural numbers from 3 to 31 must be 31 – 2 = 29. Thus n (S) = 29. 24 Example 6 – Solution cont’d c. Elements that are listed more than once are counted only once. Thus n (T) = 3. 25 Definitions Regarding Sets The following definitions play an important role in our work with sets. For instance {d, e, f } = {e, f, d }. 26 Example 7 – Equal Sets and Equivalent Sets State whether each of the following pairs of sets are equal, equivalent, both, or neither. Solution: a. The sets are not equal. However, each set has exactly five elements, so the sets are equivalent. b. The first set has three elements and the second set has four elements, so the sets are not equal and are not equivalent. 27