1. Take time to review yesterday’s Homework with your desk group p. 7 #1 – 3, 7, 9 – 13, 24 – 27

2. Homework Review 1a)No 1b)No 1c)Yes 2a)Yes 2b) No 2c)Yes 3) {a, b, c} 7a)Yes 7b)No 7c)Yes 7d)No 7e)No 7f)Yes 9a){u, v, x, 1, 3} 9b){2, 3, u, v, x, y, z} 9c){u, z}

3. Homework Review 10a){y, z, 1, 2} 10b)ø 10c){x, 1, 3} 10d){y, 1, 2} 10e){1} 11a){a, b, c, 2, 3} 11b){2, 3} 11c){a, 2, 3} 11d){1, 2, 3} 11e){b, c} 11f){a, b, c} 12)D 13a)x owns a GM car and works for GM 13b)x works for GM and does not own a GM car

4. Homework Review 13c)x owns a GM car or works for GM and owns stock in GM 13d)x is the president of GM and owns a GM car 24){(s,r), (s,s), (s,t), (t,r), (t,s), (t,t), (u,r), (u,s), (u,t)} 25a){(a,a), (a,b), (a,d), (b,a), (b,b), (b,d), (c,a), (c,b), (c,d)} 25b){(a,a), (a,b), (b,a), (b,b)} 26)X: {1, 4} Y: {2, 4} Z: {4, 6} 27)A: {a, b} B: {1,2,3}

5. Venn Diagrams and Partitions CLE 3182.4.6 Analyze arguments with quantifiers through the use of Venn diagrams

6. Venn Diagrams O Representing the universal set, U, and its subsets by using geometric shapes

7. DeMorgan’s Laws O (A U B)’ = A’ ∩ B’ O (A ∩ B)’ = A’ U B’

8. Set Example O Let X = {Austria, Brazil, Chile, Denmark, Egypt, France}. Find subsets A, E, and S of X such that A = {x: x is a country in Africa} E = {x: x is a country in Europe} S = {x: x is a country in South America}

9. Set Relationships O What is the set represented by A U E U S? O What are the intersections between each pair of sets? O Sets in a collection are said to be pairwise disjoint if every pair of sets in the collection is disjoint O A partition of a set X is a collection of nonempty subsets which are pair-wise disjoint and whose union is the entire set, X

10. Numbering of Sets O Let A be a set with a finite number of elements. The number of elements in A is denoted by n(A) O In our previous example, what is n(X)? O How about n(A)? How about n(E)? How about n(S)? O Notice n(X) = n(A) + n(E) + n(S)

11. Partition Principle O If a set X is partitioned into subsets X 1, X 2, …, X k, then n(X) = n(X 1 ) + n(X 2 ) + … + n(X k ) O If they each have the same number of elements, then the equation can be simplified to n(X) = kn(X 1 )

12. Partition Example O Suppose that a set X is divided into 3 partitions: A, B, and C. B has two times as many elements as A. C has four times as many elements as A. If n(X) = 63, then find how many elements are in each partition.

13. Cartesian Products O If A and B are sets, then n(A x B) = n(A) ∙ n(B)

14. Homework Assignment O Work on p. 13 #1 – 4, 6 – 8, 11 – 14, 18, 19, 27, 28 O Quiz Tomorrow O Test on Friday

15. Sizes of Sets CLE 3182.4.6 Analyze arguments with quantifiers through the use of Venn diagrams

16. ACT Question After polling a class of 20 music students by a show of hands, you find that 8 students play the guitar and 9 students play the piano. Given that information, what is the minimum number of students in this class who play both? A.0D.9 B.1E.17 C.8

17. Size of Sets O A set U with nondisjoint subsets A and B has the following: n(U) = 10, n(A) = 7, n(B) = 6, n(A ∩ B) = 4 Find n(A U B) O We need to use a Venn Diagram O If the sets are not partitioned, then we will use the following equation: n(A U B) = n(A) + n(B) – n(A ∩ B)

18. Example Question O The operator of a radio station surveyed 100 listeners by asking them two questions O Do you prefer to have a weather update once an hour or more frequently? O Do you prefer to have commercial breaks three times an hour (with more commercials) or spread evenly throughout the hour? O 60 prefer the weather update once an hour, 45 prefer three commercials per hour, and 25 prefer both O How many wanted both more frequent weather updates and spread commercials?