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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. MM150 Unit 2 Seminar Agenda. Welcome and Review Sections 2.1 - 2.4. Set Examples. - PowerPoint PPT PresentationTRANSCRIPT
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
MM150 Unit 2 Seminar Agenda
• Welcome and Review
• Sections 2.1 - 2.4
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 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.
5
6
Elements or Members of a Set
• Let F = {1, 2, 3, 4, 5}
6
Set-Builder Notation
• D = { x | Condition(s) }
• Set D is the set of all elements x such that the conditions that must be met
8
Change from set-builder notation to roster notation.X = {x | x is a vowel}.
9
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.T = {1, 2, 3, 4, 5, 6, 7}.
11
Change from roster notation to set-builder notation.T = {1, 2, 3, 4, 5, 6, 7}.
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
• 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
U V
a e i o u
Complement of a Set
20
U = {x | x is a letter of the alphabet}V = {a,e,i,o,u}
Shaded part is V’, or the complement of V.
UV
EVERYONE: What is the complement of G?
U = {1, 2, 3, 4, 5, 6, …, 100}
G = {2, 4, 6, 8, 10, …, 100}
21
IntersectionThe 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}
22
U
A B
246810
1 3 5
7 9
1214…100
111315…99
23
UnionThe 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}
23
U
A B
246810
1 3 5
7 9
1214…100
111315…99
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.
24
Page 94 #100At 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 bandB = 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
25
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} 26
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}
27
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. 28
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’.29
Venn Diagram with 3 Sets
30
U A B
C
I II III
IV
V
VI
VII VIII
General Procedure for Constructing Venn Diagrams with Three Sets A, B, and C
1. Determine the elements to be placed in region V by finding the elements that are common to all three sets, A ∩ B ∩ C.
2. 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.
3. 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.
4. 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.
31
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}
32
U
A B
C
NYRI
MA
NJ PA NH DE CT GA MD
SC
VA NC
DeMorgan’s Laws
(A ∩B)’ = A’ U B’
(A U B)’ = A’ ∩ B’
33
Verifying (A U B)’ = A’ ∩ B’
34
U A B
C
I II III
IV
V
VI
VII VIII
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