1. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 2.1 Set Concepts 2.1-1

2. Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Equality of sets Application of sets Infinite sets 2.1-2

3. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Set A set is a collection of objects, which are called elements or members of the set. Three methods of indicating a set: Description Roster form Set-builder notation 2.1-3

4. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Well-defined Set A set is well defined if its contents can be clearly defined. Example: The set of U.S. presidents is a well defined set. Its contents, the presidents, can be named. 2.1-4

5. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Description of Sets Write a description of the set containing the elements Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. 2.1-5

6. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Description of Sets Solution The set is the days of the week. 2.1-6

7. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Roster Form Listing the elements of a set inside a pair of braces, { }, is called roster form. Example {1, 2, 3,} is the notation for the set whose elements are 1, 2, and 3. (1, 2, 3,) and [1, 2, 3] are not sets. 2.1-7

8. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Naming of Sets Sets are generally named with capital letters. Definition: Natural Numbers The set of natural numbers or counting numbers is N. N = {1, 2, 3, 4, 5, …} 2.1-8

9. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Roster Form of Sets Express the following in roster form. a)Set A is the set of natural numbers less than 6. Solution: a) A = {1, 2, 3, 4, 5} 2.1-9

10. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Roster Form of Sets Express the following in roster form. b) Set B is the set of natural numbers less than or equal to 80. Solution: b) B = {1, 2, 3, 4, …, 80} 2.1-10

11. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Roster Form of Sets Express the following in roster form. c)Set P is the set of planets in Earth’s solar system. Solution: c) P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} 2.1-11

12. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Set Symbols The symbol ∈, read “is an element of,” is used to indicate membership in a set. The symbol ∉ means “is not an element of.” 2.1-12

13. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Set-Builder Notation (or Set-Generator Notation) A formal statement that describes the members of a set is written between the braces. A variable may represent any one of the members of the set. 2.1-13

14. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: Using Set-Builder Notation a)Write set B = {1, 2, 3, 4, 5} in set-builder notation. b) Write in words, how you would read set B in set-builder notation. 2.1-14

15. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: Using Set-Builder Notation Solution a) or b)The set of all x such that x is a natural number and x is less than 6. 2.1-15

16. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 6: Set-Builder Notation to Roster Form Write set in roster form. Solution A = {2, 3, 4, 5, 6, 7} 2.1-16

17. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Finite Set A set that contains no elements or the number of elements in the set is a natural number. Example: Set B = {2, 4, 6, 8, 10} is a finite set because the number of elements in the set is 5, and 5 is a natural number. 2.1-17

18. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Infinite Set A set that is not finite is said to be infinite. The set of counting numbers is an example of an infinite set. 2.1-18

19. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Set A is equal to set B, symbolized by A = B, if and only if set A and set B contain exactly the same members. Example: { 1, 2, 3 } = { 3, 1, 2 } Equal Sets 2.1-19

20. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Cardinal Number The cardinal number of set A, symbolized n(A), is the number of elements in set A. Example: A = { 1, 2, 3 } and B = {England, Brazil, Japan} have cardinal number 3, n(A) = 3 and n(B) = 3 2.1-20

21. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Equivalent Sets Set A is equivalent to set B if and only if n(A) = n(B). Example: D={ a, b, c }; E={apple, orange, pear} n(D) = n(E) = 3 So set A is equivalent to set B. 2.1-21

22. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Equivalent Sets - Equal Sets Any sets that are equal must also be equivalent. Not all sets that are equivalent are equal. Example: D ={ a, b, c }; E ={apple, orange, pear} n(D) = n(E) = 3; so set A is equivalent to set B, but the sets are NOT equal 2.1-22

23. Copyright 2013, 2010, 2007, Pearson, Education, Inc. One-to-one Correspondence Set A and set B can be placed in one- to-one correspondence if every element of set A can be matched with exactly one element of set B and every element of set B can be matched with exactly one element of set A. 2.1-23

24. Copyright 2013, 2010, 2007, Pearson, Education, Inc. One-to-one Correspondence Consider set S states, and set C, state capitals. S = {North Carolina, Georgia, South Carolina, Florida} C = {Columbia, Raleigh, Tallahassee, Atlanta} Two different one-to-one correspondences for sets S and C are: 2.1-24

25. Copyright 2013, 2010, 2007, Pearson, Education, Inc. One-to-one Correspondence S = {No Carolina, Georgia, So Carolina, Florida} C = {Columbia, Raleigh, Tallahassee, Atlanta} S = {No Carolina, Georgia, So Carolina, Florida} C = {Columbia, Raleigh, Tallahassee, Atlanta} 2.1-25

26. Copyright 2013, 2010, 2007, Pearson, Education, Inc. One-to-one Correspondence Other one-to-one correspondences between sets S and C are possible. Do you know which capital goes with which state? 2.1-26

27. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Null or Empty Set The set that contains no elements is called the empty set or null set and is symbolized by 2.1-27

28. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Null or Empty Set Note that { ∅ } is not the empty set. This set contains the element ∅ and has a cardinality of 1. The set {0} is also not the empty set because it contains the element 0. It has a cardinality of 1. 2.1-28

29. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Universal Set The universal set, symbolized by U, contains all of the elements for any specific discussion. When the universal set is given, only the elements in the universal set may be considered when working with the problem. 2.1-29

30. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Universal Set Example If the universal set is defined as U = {1, 2, 3, 4,,…,10}, then only the natural numbers 1 through 10 may be used in that problem. 2.1-30