Thinking Mathematically Seventh Edition Chapter 2 Set Theory Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 1 Section 2.1 Basic Set Concepts Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved

Slide - 2 Objectives 1. Use three methods to represent sets. 2. Define and recognize the empty set. 3. Use the symbols and . 4. Apply set notation to sets of natural numbers. 5. Determine a sets cardinal number. 6. Recognize equivalent sets.

7. Distinguish between finite and infinite sets. 8. Recognize equal sets. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 3 Sets A set is collection of objects whose contents can be clearly determined. Elements or members are the objects in a set. A set must be well-defined, meaning that its contents can be clearly determined. The order in which the elements of the set are listed

is not important. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 4 Methods for Representing Sets Capital letters are generally used to name sets. A word description can designate or name a set. Use W to represent the set of the days of the week. Use the roster method to list the members of a set. W ={Monday, Tuesday, Wednesday, Thursday, Friday,

Saturday, Sunday}. Commas are used to separate the elements of the set. Braces, , are used to designate that the enclosed elements form a set. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 5 Example 1: Representing a Set Using a Description Write a word description of the set: P = {Washington, Adams, Jefferson, Madison,

Monroe}. Solution Set P is the set of the first five presidents of the United States. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 6 Example 2: Representing a Set Using the Roster Method Write using the roster method: Set C is the set of U.S. coins with a value of

less than a dollar. Express this set using the roster method. Solution C = penny, nickel, dime, quarter, half-dollar Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 7 Set-Builder Notation We read this notation as Set W is the set of all elements x such that x is a day of the week.

Before the vertical line is the variable x, which represents an element in general. After the vertical line is the condition x must meet in order to be an element of the set. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 8 Example 3: Converting from SetBuilder to Roster Notation Express set A x x is a month that begins with the letter M using the roster method. Solution

There are two months, namely March and May. Thus, A {March, May}. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 9 The Empty Set The empty set, also called the null set, is the set that contains no elements. The empty set is represented by { } or . These are examples of empty sets:

Set of all numbers less than 4 and greater than 10 x x is a fawn that speaks Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 10 Example 4: Recognizing the Empty Set (1 of 2) Which of the following is the empty set? a. 0 No. This is a set containing one element. b. 0

No. This is a number, not a set. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 11 Example 4: Recognizing the Empty Set (2 of 2) Which of the following is the empty set? c. x x is a number less than 4 or greater than 10 No. This set contains all numbers that are either less than 4, such as 3, or greater than 10, such as 11.

d. x x is a square with three sides Yes. There are no squares with three sides. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 12 Notations for Set Membership The Notations and The symbol is used to indicate that an object is an

element of a set. The symbol is used to replace the words is an element of. The symbol is used to indicate that an object is not an element of a set. The symbol is used to replace the words is not an element of. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 13 Example 5: Using the Symbols Epsilon and Epsilon Crossed Out Determine whether each statement is true

or false: a. r {a, b, c, , z} True b. 7 {1, 2,3, 4,5} True c. {a} {a, b} False. a is a set and the set a is not an element of the set a, b . Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved

Slide - 14 Sets of Natural Numbers The Set of Natural Numbers N {1, 2, 3, 4, 5,} The three dots, or ellipsis, after the 5 indicate that there is no final element and that the list goes on forever. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved

Slide - 15 Example 6: Representing Sets of Natural Numbers Express each of the following sets using the roster method: a. Set A is the set of natural numbers less than 5. A {1, 2, 3, 4} b. Set B is the set of natural numbers greater than or equal to 25. B 25, 26, 27, 28, c. E { x | x N and x is even}.

E {2, 4, 6, 8, } Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 16 Inequality Notation and Sets (1 of 2) Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 17 Inequality Notation and Sets (2 of 2)

Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 18 Example 7: Representing Sets of Natural Numbers Express each of the following sets using the roster method: a. x | x N and x 100 Solution: {1, 2, 3, 4, , 100} b. x | x N and 70 x < 100 Solution: {70, 71, 72, 73, , 99} Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved

Slide - 19 Cardinality and Equivalent Sets Definition of a Sets Cardinal Number The cardinal number of a set A, represented by n( A) is the number of distinct elements in symbol n( A) is read n of A. set A. The Repeating elements in a set neither adds new elements to the set nor changes its cardinality. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved

Slide - 20 Example 8: Cardinality of Sets Find the cardinal number of each of the following sets: a. A {7, 9,11,13} n( A) 4 b. B 0 n( B ) 1 c. C {13,14,15, , 22, 23} n(C ) 11 Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved

Slide - 21 Equivalent Sets (1 of 3) Definition of Equivalent Sets Set A is equivalent to set B means that set A and set B contain the same number of elements. For equivalent sets, n( A) n( B ). Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 22

Equivalent Sets (2 of 3) These are equivalent sets: The line with arrowheads, indicate that each element of set A can be paired with exactly one element of set B and each element of set B can be paired with exactly one element of set A. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 23 Equivalent Sets (3 of 3)

One-to-One Correspondences and Equivalent Sets 1. If set A and set B can be placed in one-to-one correspondence, then A is equivalent to B: n( A) n( B ). 2. If set A and set B cannot be placed in one-to-one correspondence, then A is not equivalent to B: n( A) n( B ). Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 24

Finite and Infinite Sets Finite Sets and Infinite Sets Set A is a finite set if n( A) 0 (that is, A is n( A) is a natural number. the empty set) or A set whose cardinality is not 0 or a natural number is called as infinite set. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 25

Equal Sets Definition of Equality of Sets Set A is equal to set B means that set A and set B contain exactly the same elements, regardless of order or possible repetition of elements. We symbolize the equality of sets A and B using the statement A = B. Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 26 Example 10: Determining Whether

Sets Are Equal Determine whether each statement is true or false: a. 4, 8, 9 {8, 9, 4} True b. 1, 3, 5 {0,1, 3, 5} False Copyright 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 27