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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 Sections 2.1 - 2.4. Definition of Set.

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Welcome to mm150
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
MM150 Unit 2 Seminar Agenda

  • Welcome

  • Sections 2.1 - 2.4


Definition of set
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)


Roster form notation
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


Order of elements in sets
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


Natural numbers in roster notation
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


Elements or members of a set
Elements or Members of a Set

  • Let F = {1, 2, 3, 4, 5}

7


Set builder notation
Set-Builder Notation

  • D = { x | Condition(s) }

  • Set D is the set of all elements x such that the conditions that must be met



Change from set builder notation to roster notation x x x is a vowel1
Change from set-builder notation to roster notation.X = {x | x is a vowel}.

  • X = {a, e, i, o, u}

10




Equality of sets
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


Cardinal number
Cardinal Number

  • For a set A, symbolized by n(A)

  • Let B = {Criminal Justice, Accounting, Education}

  • n(B) = 3

14


Equivalence of sets
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


Subsets
Subsets

  • M = {m, e}

  • N = {o, n, e}

  • P = {t, o, n, e}

  • Which set is a subset of another?

16


Proper subset
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


Distinct subsets of a finite set
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


Subsets of team from slide 17
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


Venn diagrams
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


Complement of a set
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


Everyone what is the complement of g
EVERYONE: What is the complement of G?

U = {1, 2, 3, 4, 5, 6, …, 100}

G = {2, 4, 6, 8, 10, …, 100}


Intersection
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


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


The relationship between n a u b n a n b and n a b
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.


Page 94 100
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


Difference
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}


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
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}


U 100 200 300 400 1000 a 100 200 300 400 500 b 500 1000
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.


U 20 21 22 23 24 25 26 27 28 29 30 r 22 23 26 28 29 s 21 22 24 28 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’.


Venn diagram with 3 sets
Venn Diagram with 3 Sets

U

A

B

I

II

III

V

IV

VI

VII

VIII

C


General procedure for constructing venn diagrams with three sets a b and c
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.


Page 100 10
Page 100 #10 Sets A, B, and C

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


Demorgan s laws
DeMorgan’s Laws Sets A, B, and C

(A ∩B)’ = A’ U B’

(A U B)’ = A’ ∩ B’


Verifying a u b a b
Verifying (A U B)’ = A’ ∩ B’ Sets A, B, and C

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