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

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}

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

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}

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

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

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

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}

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

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)

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)

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.

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

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

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

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

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)

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?