1. Unit 2: Sets Prof. Carolyn Dupee July 3, 2012

2. HOW DO YOU WRITE SETS? P. 69 Ex. 2 Set A is the set of all natural numbers (counting numbers) less than 5. -A= {1,2,3,4} Set B is the set of natural numbers less than or equal to 75. -B= {1,2,3,4,......75}... Means the numbers keep going 2

3. EXPRESS THE FOLLOWING IN ROSTER FORM. The set of natural numbers between 4 and 9. -A= {4,5,6,7,8,9} The set of natural numbers between 4 and 9, inclusive. -B= {4,5,6,7,8,9 } 3

4. SET BUILDER NOTATION D = {x | x  N and x> 10} Set D is x such that x is a natural number and x is greater than 10. 4

5. WRITE SET THE SET: Set c= {North America, South America, Asia, Australia, Africa, Antartica} in set builder notation -A= {x | x is a continent} Write in words how you would read set C in set builder notation. -Set C is the set of all elements x such that x is a continent. 5

6. COMPARING SETS Set A = {1,2,3}Set B = {3,2,1} -Set A equals set B because the two sets have the same elements. -Set A is equivalent to set B if the have the same cardinality (# of things in the set). -{ } would be an empty set with no numbers in it Poll Question 1: -Set S= {North Carolina, Georgia, South Carolina, Florida} --Set C= {Columbia, Raleigh, Tallahasse, Atlanta} 6

7. SECTION 2.2 SUBSETS Determine whether set A is a subset of set B. Problem 1: A= {marigold, pansy, geranium} B= {marigold, pansy, begonia, geranium} -All of the elements of set A are contained in set B, so A ⊆ Problem 2: A= {2,3,4,5}B= {2,3} -The elements 4 and 5 are in set A, but not in set B, so A ⊆ B (A is not a subset of B). In this example, however, all the elements of set B are contained in set A; therefore, B ⊆ A. 7

8. SECTION 2.2: SUBSETS CONT. Problem 3: A: {x| x is a yellow fruit}B= {x | x is a red fruit} -There are fruits, such as bananas, that are in set A that are not in set B, so A ⊆ B. (Poll question 2) Problem 4: A= {vanilla, chocolate, rocky road} B= {chocolate, vanilla, rocky road} 8

9. PROPER SUBSET A ⊂ B, iff (if and only if) all the elements of set A are elements of set B and set A doesn’t equal set B (Set B must contain at least one element not in set A). Ex. 2, p. 78 -Determine whether set A is a proper subset of set B. -A= {jazz, pop, hip hop} -B= {classical, jazz, pop, rap, hip hop} -All elements of set A are contained in set B, and sets A and B are not equal so A ⊂ B. 9

10. PROPER SUBSETS- POLL 3 A= {a, b, c, d}B= {a, c, b, d} Is set A a proper subset of set B? 10

11. NUMBER OF SUBSETS (CARDINALITY) P. 79 Formula for number of subsets= 2 n, where n is the number of elements in the set A. -{ } 2 0 = 1 -{ a } 2 1 = 2 -{a, b} 2 2 = 4 -{a, b, c}= 2 3 = 8 11

12. NUMBER OF SUBSETS (CARDINALITY EX. 4) Determine the number of distinct subsets for the set {S,L,E,D} -2 n = 2 4 =16 List all the distinct subsets for the set {S,L,E,D} -{S,L,E,D}{S,L,E}; {S,L,D}; {S,E,D}; {L,E,D} -{S,L}; {S,E}; {S,D}; {L,E}, {L, D}; {E,D}{S}; {L}, {E}, {D} { } How many of the distinct subsets are proper subsets? -There are 15 proper subsets. {S,L,E,D} can’t be a proper subset of itself. 12

13. UNDERSTANDING VENN DIAGRAMS Example 1, p. 84: Given U (universal set) = {1,2,3,4,5,6,7,8} and A= {1,3,4} find A’ (complement or NOT A) and illustrate the relationship among sets U,A, and A’ in a Venn Diagram. 1.What’s in set A? 1, 3, 4 2.What’s not in set A or in A’? 2, 5, 6, 7, 8 3.What doe the sets U and A have in common? 1,3,4 4.Draw the Venn Diagram (See next slide) 13

14. UNDERSTANDING VENN DIAGRAMS CONT. 14 U 134134 A A’ 2,5, 6,7,8

15. INTERSECTION OF VENN DIAGRAMS A ∩ B where elements are shared by sets A and B. Example 2, p. 85: U= 50 states in the United States, B= set of states with a population of more than 10 million people as of 2005, B= set of states that have at least one city with a population more than 1 million people as of 2005. Draw a Venn Diagram illustrating the relationship between sets A and set B. 15

16. INTERSECTION OF VENN DIAGRAMS ∩ STEP 1 16 Population > 1 millionOne city > 1 million California CA Texas TX New York NY Florida FL Illinois IL Pennsylvania PA Ohio OHArizona AZ Michigan MI

17. INTERSECTION OF VENN DIAGRAMS ∩ STEP 2 U 17 A B I II CA TX NY IL PA III AZ OH MI FL All other U.S. states

18. INTERSECTION OF SETS NOTATION PRACTICE U = {1,2,3,4,5,6,7,8,9,10} A={1,2,3,8}B= {1,3,6,7,9} C= { } Question A) A ∩ B= What’s in common to both A and B? A AND B -Answer: A={1,2,3,8}B= {1,3,6,7,9} The elements 1 and 3. -Notation {1,3) Question B) A ∩ C= What’s in common to both A and C? A AND C -Answer: A={1,2,3,8} C= { } -This is a trick question, the elements are inside the brackets and they don’t have any elements in common in this case. 18

19. CONT. INTERSECTION OF SETS PRACTICE U = {1,2,3,4,5,6,7,8,9,10} A={1,2,3,8}B= {1,3,6,7,9} C= { } Question C) A’ ∩ b= What’s common to NOT A and B? Not A and B -Answer: not A= 4,5,6, 7,9B= 1,3,6,7,9 -Notation: {6,7,9} Question D) (A ∩ B)’ -A ∩ B = 1, 3 -NOT A ∩ B = NOT A AND NOT in B -{2,4,5,6,7,8,9,10} 19

20. UNION (OR) Where do the elements unite? A ∪ B P. 86 Ex. 4: Use the Venn Diagram to determine the sets. (See next slide). 20

21. UNION PRACTICE (P. 86 EX. 4) UAB I IIIII 9 3 ? 7 IV 8 # 21

22. UNION PRACTICE P. 86, EX. 4 A) What’s the Universal Set? -U= {9, triangle, 8, #, square, circle, 3, ?, 7} -You can use the symbols when you’re doing MML graded practice they will show up in your box to the left. B) What’s in set A? -A= {9, triangle, square, circle} C) What’s the complement of set B (B’)? -{9, triangle, 8, #} 22

23. UNION PRACTICE P. 86, EX. 4 D) A ∩ B -{square, circle} E) A ∪ B -{9, triangle, square, circle, 3, ?, 7} F) (A ∪ B)’ = {#, 8} G) n (A ∪ B) This is asking for the number of elements. -2 + 2+ 3 = 7 23

24. USING VENN DIAGRAMS WITH 3 CIRCLES P. 96, Ex. 1: Construct a Venn Diagram illustrating the following sets: -U= {1,2,3,4,5,6,7,8,9,10,11,12,13,14} -A= {1,5,8,9,10,11} -B= {2,4,5,9,10,13} -C= {1,3,5,8,9,11} What’s in common? -A + C: 1, 5, 8-A + B + C: 5, 9-A + B: 5, 9, 10B + C: 5, 9 -Not in any only U: 6, 7, 14 24

25. DRAWING VENN DIAGRAMS WITH 3 CIRCLES UA B Be ready for some poll questions!! I II III 13 12 10 2 4 IV 5,9 1 8 V VI VII 3 11VIII 6,7,14 C 25