1. Unit 2 Seminar Welcome to MM150! To resize your pods:
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2. MM150 Unit 2 Seminar Agenda
Welcome and Review
Sections

3. Set Examples H = {h, e, a, t, r} T = {t, o, d} O = {1, 3, 5, 7, ...}
S = {Elm, Oak, Palm, Fig} 3

4. 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. 4

5. Natural Numbers in Roster Notation
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. 5

6. Elements or Members of a Set
Let F = {1, 2, 3, 4, 5}
3 E F or we can write 3 E {1, 2, 3, 4, 5}
5 E F or we can write 5 E {1, 2, 3, 4, 5}
But 10 ∉ F
or we can write 10 ∉ {1, 2, 3, 4, 5} 6

7. Set-Builder Notation D = { x | Condition(s) }
Set D is the set of all elements x such that the conditions that must be met
L = {x | x ∈ N and 9 < x < 20}
This is read Set L is the set of all elements x such that x is greater than 9 and less than 20.
L = {10, 11, ..., 19} 7

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

9. Change from set-builder notation to roster notation
Change from set-builder notation to roster notation. X = {x | x is a vowel}.
X = {a, e, i, o, u} 9

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

11. T = { x | x ∈ N and 0 < x < 8}
Change from roster notation to set-builder notation. T = {1, 2, 3, 4, 5, 6, 7}.
T = { x | x ∈ N and 1 ≤ x ≤ 7} OR
T = { x | x ∈ N and 0 < x < 8} 11

12. 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. 12

13. Cardinal Number For a set A, symbolized by n(A)
Let B = {Criminal Justice, Accounting, Education}
n(B) = 3 13

14. 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? 14

15. Subsets
Set A is a subset of set B, A ⊆ B, if and only if all the elements of set A are also elements of set B.
M = {m, e}
N = {o, n, e}
P = {t, o, n, e}
Which set is a subset of another? 15

16. 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! 16

17. 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 17

18. 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! 18

19. Venn Diagrams U = {x | x is a letter of the alphabet} V = {a,e,i,o,u}19

20. 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

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

22. 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

23. 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} ,9,10,12,14,
B = {2, 4, 6, 8, 10, …, 100} , …, 100} U 11 13 15 … 99 12 14 …
100 2 4 6 8 10
3 5
7 9 A B 23

24. The Relationship between n(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.

25. 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) = n(A) = n(A) 20 students sang in chorus

26. 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}

27. 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}

28. 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.

29. The difference wants elements in R but not in S’.
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’.

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

31. 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.

32. 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

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

34. 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 34