1. Take time to review yesterday’s Homework with your desk group

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 Analyze arguments with quantifiers through the use of Venn diagrams

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

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

8. Set Example
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 What is the set represented by A U E U S?
What are the intersections between each pair of sets?
Sets in a collection are said to be pairwise disjoint if every pair of sets in the collection is disjoint
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
Let A be a set with a finite number of elements. The number of elements in A is denoted by n(A)
In our previous example, what is n(X)?
How about n(A)? How about n(E)? How about n(S)?
Notice n(X) = n(A) + n(E) + n(S)

11. Partition Principle
If a set X is partitioned into subsets X1, X2, …, Xk, then
n(X) = n(X1) + n(X2) + … + n(Xk)
If they each have the same number of elements, then the equation can be simplified to
n(X) = kn(X1)

12. Partition Example
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
If A and B are sets, then
n(A x B) = n(A) ∙ n(B)

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

15. Sizes of Sets
CLE 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. 0 D. 9
B. 1 E. 17
C. 8

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