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1. Welcome to MM150! Unit 2 Seminar To resize your pods: Place your mouse here. Left mouse click and hold. Drag to the right to enlarge the pod. To maximize chat, minimize roster by clicking here

2. MM150 Unit 2 Seminar Agenda • Welcome • Sections 2.1 - 2.4

3. Definition of Set • Set: Collection of objects which are called elements or members of the set. For example: United States is a collection of 50 states and District of Columbia. (50 states and District of Columbia are the elements/members)

4. Roster Form Notation • When elements are listed inside a pair of curly braces { } separated by a comma. Sets are generally named with capital letters. For example: • H = {h, e, a, t, r} • T = {t, o, d} • O = {1, 3, 5, 7, ...} • S = {Elm, Oak, Palm, Fig} 4

5. Order of Elements in Sets • D = {Lab, Golden Retriever, Boxer} Can the elements of D be rewritten as • D = {Boxer, Golden Retriever, Lab}? Yes! Order of elements in a set is not important. 5

6. Natural Numbers inRoster Notation • N = {1, 2, 3, 4, 5, ...} If we do not put the elements in increasing order, how would we handle it to make sense? • N = {5, 2, 4, 1, 3, ...} In this case the ellipses are meaningless as there is no pattern to follow. 6

7. Elements or Members of a Set • Let F = {1, 2, 3, 4, 5} 7

8. Set-Builder Notation • D = { x | Condition(s) } • Set D is the set of all elements x such that the conditions that must be met

9. Change from set-builder notation to roster notation.X = {x | x is a vowel}.

10. Change from set-builder notation to roster notation.X = {x | x is a vowel}. • X = {a, e, i, o, u} 10

11. Change from roster notation to set-builder notation.T = {1, 2, 3, 4, 5, 6, 7}.

12. Change from roster notation to set-builder notation.T = {1, 2, 3, 4, 5, 6, 7}. 12

13. Equality of Sets • N = {n, u, m, b, e, r} • M = {r, e, b, m, u, n} • Does N = M? • Yes, they have exactly the same elements. Remember, order does not matter. 13

14. Cardinal Number • For a set A, symbolized by n(A) • Let B = {Criminal Justice, Accounting, Education} • n(B) = 3 14

15. Equivalence of Sets • Set A is equivalent to set B if and only if n(A) = n(B). • A = {Oscar, Ernie, Bert, Big Bird} • B = {a, b, c} • C = {1, 2, 3, 4} • EVERYONE: Which two sets are equivalent? 15

16. Subsets • M = {m, e} • N = {o, n, e} • P = {t, o, n, e} • Which set is a subset of another? 16

17. Proper Subset • N ⊂ P • Every element of N is an element of P and N ≠ P. • REMEMBER: the empty set is a subset of every set, including itself! 17

18. Distinct Subsets of a Finite Set • 2n, where n is the number of elements in the set. • To complete a project for work, you can choose to work alone or pick a team of your coworkers: Jon, Kristen, Susan, Andy and Holly. How many different ways can you choose a team to complete the project? • There are 5 coworkers so n = 5. • 25 = 32 18

19. Subsets of Team from slide 17 • { } • Subsets with 1 element {J}, {K}, {S}, {A}, {H} • Subsets with 2 elements {J, K}, {J, S}, {J, A}, {J, H}, {K, S}, {K, A}, {K, H}, {S, A}, {S, H}, {A, H} • Subsets with 3 elements {J, K, S}, {J, K, A}, {J, K, H}, {J, S, A}, {J, S, H}, {J, A, H}, {K, S, A}, {K, S, H}, {K, A, H}, {S, A, H} • Subsets with 4 elements {J, K, S, A}, {J, K, S, H}, {J, K, A, H}, {J, S, A, H}, {K, S, A, H} • Subsets with 5 elements {J, K, S, A, H} • only 1 set is not proper, the set itself! 19

20. Venn Diagrams U = {x | x is a letter of the alphabet} V = {a,e,i,o,u} V U a e i o u 20

21. Complement of a Set U = {x | x is a letter of the alphabet} V = {a,e,i,o,u} Shaded part is V’, or the complement of V. U V

22. EVERYONE: What is the complement of G? U = {1, 2, 3, 4, 5, 6, …, 100} G = {2, 4, 6, 8, 10, …, 100}

23. Intersection The intersection of sets A and B, symbolized by A ∩ B, is the set containing all the elements that are common to both set A and set B. U = {1, 2, 3, 4, 5, …, 100} A ∩ B = {2, 4, 6, 8, 10} A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B = {2, 4, 6, 8, 10, …, 100} U 11 13 15 … 99 12 14 … 100 2 4 6 8 10 • 3 5 • 7 9 A B

24. Union The union of set A and B, symbolized by A U B, is the set containing all the elements that are members of set A or of set B or of both. U = {1, 2, 3, 4, 5, …, 100} A U B = {1,2,3,4,5,6,7, A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 8,9,10,12,14, B = {2, 4, 6, 8, 10, …, 100} 16, …, 100} U 11 13 15 … 99 12 14 … 100 2 4 6 8 10 • 3 5 • 7 9 A B 24

25. The Relationship betweenn(A U B), n(A), n(B)and n(A ∩ B) • For any finite sets A and B, n(A U B) = n(A) + n(B) – n(A ∩ B) By subtracting the number of elements in the intersection, you get rid of any duplicates that are in both sets A and B.

26. Page 94 #100 At Henniger High School, 46 students sang in the chorus or played in the stage band, 30 students played in the stage band, and 4 students sang in the chorus and played in the stage band. How many students sang in the chorus? A = sang in chorus A U B = chorus or band B = played in stage band A ∩ B = chorus and band n(A U B) = n(A) + n(B) – n(A ∩ B) 46 = n(A) + 30 - 4 46 = n(A) + 26 20 = n(A) 20 students sang in chorus

27. Difference The difference of two sets A and B, symbolized A – B, is the set of elements that belong to set A but not to set B. A – B = {x | x E A and x ∉ B} U = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 2, 3, 4} B = {1, 3, 5, 7} A – B = {2, 4}

28. U = {x | x is a letter of the alphabet}A = {a, b, c, d, e, f, g, h}B = {r, s, t, u, v, w, x, y, z} A’ ∩ B = A’ = {I,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z} B = {r,s,t,u,v,w,x,y,z} Union wants what is common to both A’ ∩ B = {r,s,t,u,v,w,x,y,z}

29. U = {100, 200, 300, 400, …, 1000}A = {100, 200, 300, 400, 500}B = {500, 1000} (A U B)’ = A U B = {100, 200, 300, 400, 500, 1000} The union wants the elements in one set or both. (A U B)’ = {600, 700, 800, 900} The complement wants what is in the Universal set, but not in the union.

30. U = {20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}R = {22, 23, 26, 28, 29}S = {21, 22, 24, 28, 30} R – S’ R = {22, 23, 26, 28, 29} S’ = {20, 23, 25, 26, 27, 29} R – S’ = {22, 28} The difference wants elements in R but not in S’.

31. Venn Diagram with 3 Sets U A B I II III V IV VI VII VIII C

32. General Procedure for Constructing Venn Diagrams with Three Sets A, B, and C • Determine the elements to be placed in region V by finding the elements that are common to all three sets, A ∩ B ∩ C. • Determine the elements to be place in region II. Find the elements in A ∩ B. The elements in this set belong in regions II and V. Place the elements in the set A ∩ B that are not listed in region V in region II. The elements in regions IV and VI are found in a similar manner. • Determine the elements to be placed in region I by determining the elements in set A that are not in regions II, IV, and V. The elements in regions III and VII are found in a similar manner. • Determine the elements to be placed in region VIII by finding the elements in the universal set that are not in regions I through VII.

33. Page 100 #10 Construct a Venn diagram illustrating the following sets. U = {DE, PA, NJ, GA, CT, MA, MD, SC, NH, VA, NY, NC, RI} A = {NY, NJ, PA, MA, NH} B = {DE, CT, GA, MD, NY, RI} C = {NY, SC, RI, MA} U DE CT GA MD NJ PA NH A B NY RI VA NC MA SC C

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

35. Verifying (A U B)’ = A’ ∩ B’ A U B -> regions II, V A’ -> regions III, VI, VII, VIII (A U B)’ -> regions I, III, IV, VI, VII, VIII B’ -> regions I, IV, VII, VIII A’ U B’ - >regions I, III, IV, VI, VII, VIII U A B I II III V IV VI VII VIII C