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

p. 7 #1 – 3, 7, 9 – 13, 24 – 27

Homework Review1a) No1b) No1c) Yes2a) Yes2b) No2c) Yes3) {a, b, c}

7a) Yes7b) No7c) Yes7d) No7e) No7f) Yes9a) {u, v, x, 1, 3}9b) {2, 3, u, v, x, y, z}9c) {u, z}

Homework Review10a) {y, z, 1, 2}10b) ø10c) {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) D13a) x owns a GM car and works for GM13b) x works for GM and does not own a GM car

Homework Review13c) x owns a GM car or works for GM and owns stock in GM13d) x is the president of GM and owns a GM car24) {(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 DiagramsO Representing the universal set, U,

and its subsets by using geometric shapes

DeMorgan’s LawsO (A U B)’ = A’ ∩ B’O (A ∩ B)’ = A’ U B’

Set ExampleO 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 RelationshipsO 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

Numbering of SetsO 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)

Partition PrincipleO If a set X is partitioned into subsets

X1, X2, …, Xk, thenn(X) = n(X1) + n(X2) + … + n(Xk)

O If they each have the same number of elements, then the equation can be simplified to

n(X) = kn(X1)

Partition ExampleOSuppose 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 ProductsO If A and B are sets, then

n(A x B) = n(A) ∙ n(B)

Homework AssignmentO Work on p. 13 #1 – 4, 6 – 8, 11 – 14,

18, 19, 27, 28 O Quiz TomorrowO Test on Friday

Sizes of SetsCLE 3182.4.6 Analyze arguments

with quantifiers through the use of Venn diagrams

ACT QuestionAfter 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. 9B. 1 E. 17C. 8

Size of SetsO A set U with nondisjoint subsets A

and B has the following:n(U) = 10, n(A) = 7, n(B) = 6, n(A ∩ B) = 4Find n(A U B)O We need to use a Venn DiagramO 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 QuestionO The operator of a radio station surveyed 100

listeners by asking them two questionsO 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?