SET THEORY

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SET THEORY

• Chapter 2

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DAY 1
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• Set collection
• School of fish
• Gaggle of geese
• Pride of lions
• Pod of whales
• Herd of elephants

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• Set usually named with a capital letter.
• Well defined
• A is the set of the first three lower case
  letters of the English alphabet.

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• Elements of the set
• A is the set of the first three lower case
  letters of the English alphabet.
• a, b, and c are elements of set A

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• Natural Numbers (Counting Numbers)
• N 1, 2, 3, . . .

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• Three ways of defining a set
• List
• A 1,2,3
• Description
• A is the set of the first three counting
  numbers.
• Set Builder Notation

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• Universe
• Empty set

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Example

• The set of natural numbers greater than 12 and
  less than 17.

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Example

• x x 2n and n 1, 2, 3, 4, 5

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Example

• 3, 6, 9, 12, . . .

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Example

• The set of the first 10 odd natural numbers.

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Venn Diagrams
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Set A
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Complement of A
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A intersect B
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A Union B
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Disjoint Sets
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Subsets

• A is a subset of B if every element of A is also
  an element of B.

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List all the subsets of a,b,c

• a
• b
• c
• a,b
• a,c
• b,c
• a,b,c

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List all the subsets of a,b,c

• Proper Subsets
• , a , b , c , a,b , a,c , b,c
• THE Improper Subset
• a,b,c

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Subset Notation

• Let A a,b,c
• a A
• The set of a is a subset of A.
• (think The set of a is a proper subset OR IS
  EQUAL TO A.)
• a A
• The set of a is a proper subset of A.

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True or False? A b,c,f,g

• b,f A
• b,f A

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True or False? A b,c,f,g

• b,f A True
• b,f A True

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True or False? A b,c,f,g

• b,d A

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True or False? A b,c,f,g

• b,d A False
• Because d A

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True or False? A b,c,f,g

• b,c,f,g A

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True or False? A b,c,f,g

• b,c,f,g A True
• b,c,f,g A

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True or False? A b,c,f,g

• b,c,f,g A True
• b,c,f,g A False
• Because b,c,f,g A

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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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U p,q,r,s,t,u,v,w,x,y,zA p,q,r, B
q,r,s,t,u, C r,u,w,y
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DAY 2
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Homework QuestionsPage 83
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Three types of numbers.

• Nominal
• Ordinal
• Cardinal
• The student with ticket 50768-973 has just won
  second prize four tickets to the big game this
  Saturday.

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Three types of numbers.

• Nominal name or label for identification
• Ordinal tells what order it comes in relation
  to the rest.
• Cardinal Answers the question how many?
• The student with ticket 50768-973 has just won
  second prize four tickets to the big game this
  Saturday.

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Cardinality of the Set

• If a cardinal number answers the question how
  many? then the cardinality of a set will tell us
  how many elements are in the set.
• The notation for the cardinality of set A (or
  the number of elements in A) is
• n(A)

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Equal Sets

• Two sets are equal if the have the exact same
  elements.
• Example
• A a,b,c and B c,a,b
• then A B

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• Consider A a,b,c and C x,y,z
• They are not equal because they do not have the
  same exact elements.
• What characteristic do they share?

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Equivalent Sets

• A and C have the same number of elements. Their
  cardinality is the same.
• n(A) 3 and n(C) 3
• n(A) n(C)
• A and C are equivalent sets.

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• If two sets are equivalent, you can set up a
  one-to-one correspondence between them. (That
  is, you can match them up in pairs.)

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• There are actually 6 different one-to-one
  correspondences you can set up between these two
  sets. (6 ways that you can make pairs.)
• A a,b,c and C x,y,z
• (make an orderly list)

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• 6 different one-to-one correspondences
• A a,b,c and C x,y,z
• a x
• b y
• c z

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• 6 different one-to-one correspondences
• A a,b,c and C x,y,z
• a x a - x
• b y b - z
• c z c - y

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• 6 different one-to-one correspondences
• A a,b,c and C x,y,z
• a x a x a - y
• b y b z b - x
• c z c y c - z

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• 6 different one-to-one correspondences
• A a,b,c and C x,y,z
• a x a x a y a - y
• b y b z b x b - z
• c z c y c z c - x

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• 6 different one-to-one correspondences
• A a,b,c and C x,y,z
• a x a x a y a y
• b y b z b x b - z
• c z c y c z c x
• a z
• b x
• c y

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• 6 different one-to-one correspondences
• A a,b,c and C x,y,z
• a x a x a y a y
• b y b z b x b - z
• c z c y c z c x
• a z a - z
• b x b - y
• c y c - x

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• A xx is a moon of Mars
• B xx is a former U.S. president whose last
  name is Adams
• C xx is one of the Bronte sisters of
  nineteenth-century literary fame
• D xx is a satellite of the fourth-closest
  planet to the sun
• Which of these sets are equal and which are
  equivalent?
• What do we need to know about each set to answer
  this question?

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• A xx is a moon of Mars
• A Deimos, Phobos
• n(A)

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• A Deimos, Phobos
• n(A) 2
• B xx is a former U.S. president whose last
  name is Adams
• B John Adams, John Quincy Adams
• n(B)

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• A Deimos, Phobos
• n(A) 2
• B John Adams, John Quincy Adams
• n(B) 2
• C xx is one of the Bronte sisters of
  nineteenth-century literary fame
• C Anne, Charlotte, Emily
• n(C)

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• A Deimos, Phobos
• n(A) 2
• B John Adams, John Quincy Adams
• n(B) 2
• C Anne, Charlotte, Emily
• n(C) 3
• D xx is a satellite of the fourth-closest
  planet to the sun
• D Deimos, Phobos
• n(D)

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• A Deimos, Phobos
• n(A) 2
• B John Adams, John Quincy Adams
• n(B) 2
• C Anne, Charlotte, Emily
• n(C) 3
• D Deimos, Phobos
• n(D) 2

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Finite/Infinite

• Whole numbers?
• Real numbers between 0 and 1?
• Factors of 20?
• Multiples of 20?
• Number of grains of sand on the earth?

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Example 2.9Page 94
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• n(U) 60
• n(S) 24
• n(E) 22
• n(H) 17
• 5 both S and E
• 4 both S and H
• 3 both E and H
• 2 all three

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Attribute Lab

• Three attributes considered are
• Size
• Color
• Shape

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• A and B You must get through the first door AND
  the second door. (more restrictive)
• A or B You may go in the first door OR the
  second door. (more generous)

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Day 3
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Homework Questions Page 97
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Binary Operations

• Addition
• Subtraction
• Multiplication
• Division

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• __________ __________ __________

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• Addend Addend Sum
• __________ - __________ __________

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• Addend Addend Sum
• Minuend Subtrahend Difference
• __________ X __________ __________

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• Addend Addend Sum
• Minuend Subtrahend Difference
• Factor X Factor Product
• __________ __________ __________

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• Addend Addend Sum
• Minuend Subtrahend Difference
• Factor X Factor Product
• Dividend Divisor Quotient

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PropertiesPages 104 and 120

• Closure

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• Counting Numbers 1, 2, 3, . . .
• Whole Numbers 0, 1, 2, 3, . . .

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Closure Examples

• Is the set of Whole Numbers closed with respect
  to
• Addition?
• Subtraction?
• Multiplication?
• Division?

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Closure Examples

• Is the set of Even Counting Numbers closed with
  respect to
• Addition?
• Subtraction?
• Multiplication?
• Division?

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Closure Examples

• Is 0, 1 closed with respect to
• Addition?
• Subtraction?
• Multiplication?
• Division?

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PropertiesPages 104 and 120

• Closure
• Commutative
• Associative
• Identity Element for Addition
• Identity Element for Multiplication
• Multiplication-by-Zero Property
• Distributive Property of Multiplication over
  Addition

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Examples

• 2 (3 4) 5 4

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Examples

• 2 (3 4) 5 4 Associative
• 2 (3 4) 7 2

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Examples

• 2 (3 4) 5 4 Associative
• 2 (3 4) 7 2 Commutative
• 2(3 4) 6 8

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Examples

• 2 (3 4) 5 4 Associative
• 2 (3 4) 7 2 Commutative
• 2(3 4) 6 8 Distributive
• 2(3 4) (7)2

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Examples

• 2 (3 4) 5 4 Associative
• 2 (3 4) 7 2 Commutative
• 2(3 4) 6 8 Distributive
• 2(3 4) (7)2 Commutative

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Conceptual Models

• Addition
• Set Model

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Conceptual Models

• Addition
• Subtraction (page 108)
• Take-away
• Missing Addend
• Comparison
• Number-line

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Take-away - Missing AddendComparison -
Number-line

• Identify which model would illustrate the
  problem best.
• Mary got 43 pieces of candy. Karen got 36
  pieces. How many more pieces does Mary have than
  Karen?

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Take-away - Missing AddendComparison -
Number-line

• Identify which model would illustrate the
  problem best.
• Mary gave 20 pieces of her 43 pieces of candy to
  her brother. How many pieces does she have left?

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Take-away - Missing AddendComparison -
Number-line

• Identify which model would illustrate the
  problem best.
• Karens older brother collected 53 pieces. How
  many more pieces would Karen need to have as many
  as her brother?

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Take-away - Missing AddendComparison -
Number-line

• Identify which model would illustrate the
  problem best.
• Ken left home and walked 10 blocks east. The
  last 4 blocks were after crossing Main Street.
  How far is Main Street from Kens house?

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Conceptual Models

• Addition
• Subtraction
• Multiplication (page 115)
• Repeated Addition
• Number-line
• Rectangular Array
• Multiplication Tree

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Multiplication Tree

• Melissa has 4 flags colored red, yellow, green
  and blue. How many ways can she display them on
  a flagpole?

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Conceptual Models

• Addition
• Subtraction
• Multiplication
• Repeated Addition
• Number-line
• Rectangular Array
• Multiplication Tree
• Cartesian Product

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Cartesian Product

• The Cartesian Product of A and B is a set of
  ordered pairs written A X B, and read A cross
  B.
• A X B (a,b) a A and b B

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Cartesian Product

• A X B (a,b) a A and b B
• Example
• A 5, 6, 7 B 6, 8
• A X B (

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Cartesian Product

• A X B (a,b) a A and b B
• Example
• A 5, 6, 7 B 6, 8
• A X B (5,6), (5,8), (6,6), (6,8), (7,6), (7,8)

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Cartesian Product

• Example
• A 5, 6, 7 B 6, 8
• A X B (5,6), (5,8), (6,6), (6,8), (7,6),
  (7,8)
• NOTE
• n(A) 3 , n(B) 2 and n(AXB) 6

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• How many different things can you order at the
  yogurt shop if you must choose from a waffle cone
  or a sugar cone and either vanilla, chocolate,
  mint, or raspberry yogurt?
• C w, s, Y v, c, m, r

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Cartesian Product

• C w, s, Y v, c, m, r
• C X Y (w, v), (w, c), (w, m), (w, r), (s, v),
  (s, c), (s, m), (s, r)
• n(C X Y) 8

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Conceptual Models

• Addition
• Subtraction
• Multiplication
• Division (Page 121)
• Repeated Subtraction
• Sharing
• Missing Factor

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Division Example

• Describe how you would divide 78 by 13 using
  counters and each of the following models.
• Repeated Subtraction
• Sharing
• Missing Factor

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Family of Facts

• 20 4 5 5 X 4 20
• and
• 20 5 4 4 X 5 20

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Family of Facts

• 0 4 0 and 0 X 4 0
• 4 0 ??
• and ?? X 0 4

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• Division by Zero is Undefined.

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Extra Practice Worksheet
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DAY 4
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HomeworkPages 111 and 130
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Worksheet Answers
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Math and MusicThe Magical Connection!

• Scholastic Parent and Child Magazine
• Spelling
• Phone Numbers
• School House Rock

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Skip to My Lou

• Chorus Times facts, theyre a breeze
• Learn a few, then work on speed.
• Times facts, youll be surprised
• By just how fast you can memorize.

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• 3 time 7 is 21
• Now, at last weve all begun.
• 4 times 7 is 28
• Lets sing what we appreciate.
• (Chorus)
• 5 times 7 is 35.
• Yes, by gosh, were still alive.
• 6 times 7 is 42.
• I forgot what were supposed to do.
• (Chorus)

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Print Review for Test
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Venn Diagram Lab