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Chapter 2: The Basic Concepts of Set Theory

Slides created by Avis Proctor, Broward Community College , and Linda Padilla, Joliet Junior College. Chapter 2: The Basic Concepts of Set Theory. 2.1 Symbols and Terminology 2.2 Venn Diagrams and Subsets 2.3 Set Operations and Cartesian Products 2.4 Cardinal Numbers and Surveys

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Chapter 2: The Basic Concepts of Set Theory

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  1. Slides created by Avis Proctor, Broward Community College, and Linda Padilla, Joliet Junior College

  2. Chapter 2: The Basic Concepts of Set Theory 2.1 Symbols and Terminology 2.2 Venn Diagrams and Subsets 2.3 Set Operations and Cartesian Products 2.4 Cardinal Numbers and Surveys 2.5 Infinite Sets and Their Cardinalities

  3. Set Theory How many groups of 0 does a 4-element set have? How many groups of 1 does a 4-element set have? How many groups of 2 does a 4-element set have? How many groups of 3 does a 4-element set have? How many groups of 4 does a 4-element set have? Section 2.1

  4. Definitions and Symbols • Set: a collection of objects A = {3,4,6,8,-10} B = {Joe, Susan, Paul} • Element: a member of the set • Cardinal number of set A, n(A) n(A) represents the number of elements in set A n(A) = 5 n(B) = 3 Section 2.1

  5. Set Equality vs. Set Equivalence • Set Equality (=) A = B if every element of A is in B AND every element of B is in A. Does {m,i,s,p} = {m,i,s,s,i,s,s,i,p,p,i}? • Set Equivalence (~) A ~ B if n(A) = n(B) {w,x,y} ~ {10,22,91} Section 2.1

  6. Sets of Numbers • Naturals {1,2,3,…} • Wholes {0,1,2,3,…} • Integers {…,-3,-2,-1,0,1,2,3,…} • Rationals Any number that can be written as a ratio of two integers with a nonzero denominator • Irrationals Any number that can NOT be written as a ratio of two integers Section 2.1

  7. U U A A A B B B C U Venn Diagrams • Pictorial representation of sets and their relationships • The universal set, U, is represented by the rectangle Section 2.2

  8. A U Complement of set A, A’ • Describe the shaded region. • Elements of the universe that are not in A Section 2.2

  9. U A B C Subsets () and Proper Subsets () • B  A if every element of B is in A • B  A if B  A A  B Section 2.2

  10. How many subsets does a set have? Section 2.2

  11. What’s behind the number of subsets? Pascal’s Triangle Section 2.2

  12. Use Pascal’s Triangle to answer the following. • How many groups (subsets) of 0 does a 4-element set have? 1 • How many groups of 1 does a 4-element set have? 4 • How many groups of 2 does a 4-element set have? 6 • How many groups of 3 does a 4-element set have? 4 • How many groups of 4 does a 4-element set have? 1  {a},{b},{c},{d} {a,b},{a,c},{a,d},{b,c},{b,d},{c,d} {a,b,c},{a,b,d},{a,c,d},{b,c,d} 2 2 {a,b,c,d} How many subsets does a 4-element set have? a 5-element set? Section 2.2

  13. Number of Subsets and Proper Subsets Section 2.2

  14. Set Operations and Cartesian Products • Complement of set A (A’) • Union () • Intersection() • Difference (–) • Cartesian Product (x) Section 2.3

  15. U A B U A B Union () vs. Intersection (∩) “put everything together” “find what’s in common” Section 2.3

  16. U U A B A B Difference (–) “things of A not in B” “things of B not in A” Section 2.3

  17. Cartesian Product (x) Given U = {a,b,c,x,m,e,n} A={a,e,m},B={a,c,x,m,n}, C={b,e,n}, find: {(b,a),(b,e),(b,m),(e,a),(e,e),(e,m),(n,a),(n,e),(n,m)} Section 2.3

  18. Cardinal Number of A B • A = {1,3,4,5}, B = {3,5,7,8,10} A  B = {1,3,4,5,7,8,10} n(A  B) = n(A) + n(B) – n(A B) = 4 + 5 – 2 = 7 • A = {3,15,6}, B = {7,8,9,11} n(A  B) = n(A) + n(B) if A and B are mutually exclusive (disjoint). A  B = {3,6,7,8,9,11,15} n(A  B) = n(A) + n(B) = 3 + 4 = 7 since A & B are disjoint. Section 2.4

  19. Cardinal Numbers of Regions • At BCC, 250 students were surveyed about their weekend activities. 135 like swimming. 150 like dancing. 65 like jogging. 80 like swimming and dancing. 40 like swimming and jogging. 25 like dancing and jogging. 15 like all three. • How many students only liked swimming? • How many liked exactly two activities? • How many students liked none of these activities? • How many liked dancing and jogging, but did not like swimming? • How many liked swimming or jogging, but not both? • How many liked at least two of these activities? Section 2.4

  20. Cardinality of Infinite Sets • A set is countable if it is finite. • Every whole number can be matched with every natural number. • Therefore, the sets are equivalent. {1, 2, 3, 4, 5, 6, 7, … , n , …} Naturals {0, 1, 2, 3, 4, 5, 6, … , n - 1, …} Wholes Section 2.5

  21. Cardinality of Infinite Sets • So far, the {naturals} ~ {whole numbers}. • Let’s consider the integers, {…,-3,-2,-1,0,1,2,3,…}. Integers Rewritten {0, 1, -1, 2, -2, 3, -3, …, n , -n , … } {1, 2, 3, 4, 5, 6, 7, … , 2n, 2n+1, …} Section 2.5

  22. Cardinality of Sets (Finite/Infinite) Countable sets Finite or matched with naturals Section 2.5

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