Take time to review yesterdays Homework with your desk group p. 7 #1 3, 7, 9 13, 24 27.
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Transcript of Take time to review yesterdays Homework with your desk group p. 7 #1 3, 7, 9 13, 24 27.
![Page 1: Take time to review yesterdays Homework with your desk group p. 7 #1 3, 7, 9 13, 24 27.](https://reader035.fdocuments.net/reader035/viewer/2022081512/5a4d1b677f8b9ab0599b15a0/html5/thumbnails/1.jpg)
Take time to review yesterday’s Homework with your desk group
p. 7 #1 – 3, 7, 9 – 13, 24 – 27
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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}
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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
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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}
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Venn Diagrams and Partitions
CLE 3182.4.6 Analyze arguments with quantifiers through the use of
Venn diagrams
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Venn DiagramsO Representing the universal set, U,
and its subsets by using geometric shapes
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DeMorgan’s LawsO (A U B)’ = A’ ∩ B’O (A ∩ B)’ = A’ U B’
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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}
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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
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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)
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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)
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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.
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Cartesian ProductsO If A and B are sets, then
n(A x B) = n(A) ∙ n(B)
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Homework AssignmentO Work on p. 13 #1 – 4, 6 – 8, 11 – 14,
18, 19, 27, 28 O Quiz TomorrowO Test on Friday
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Sizes of SetsCLE 3182.4.6 Analyze arguments
with quantifiers through the use of Venn diagrams
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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
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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)
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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?