set theory chapter 2. day 1 set – collection school of fish gaggle of geese pride of lions pod of...
TRANSCRIPT
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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
Ac
Ab
Aa
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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
}4,|{ xNxx
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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
C
BA
BA
A
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Set A
A
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Complement of A
}|{ AxxA
A
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A intersect B
}|{ BxandAxxBA
BA
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A Union B
BA
}|{ BxorAxxBA
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Disjoint Sets
BA
BA
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Subsets
A is a subset of B if every element of A is also an element of B.
BA
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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,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
C
BA
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
CA
CA
BA
BA
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
?)( CBA
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
CB
CBA )(
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
)(
},{
CBA
urCB
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
?BA
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
BA
BA
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
BA
rqBA },{
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
?
},,,,,,,,{
BA
zyxwvutspBA
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
B
A
BA ?
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
BA
zyxwvpB
zyxwvutsA
},,,,,{
},,,,,,,{
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U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}
BABA
zyxwvBA
zyxwvutspBA
},,,,{
},,,,,,,,{
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C
BA
)( CBA
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C
BA
)()( CABA
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C
BA
)( CBA
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C
BA
CBA )(
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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.
CA
CA
~
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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.)
z
y
x
c
b
a
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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 – yb – y b – z b – x b - zc – z c – y c – z c – x
a – zb – xc – 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 – yb – y b – z b – x b - zc – z c – y c – z c – x
a – z a - zb – x b - yc – y c - x
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A = {x|x is a moon of Mars}
B = {x|x is a former U.S. president whose last name is Adams}
C = {x|x is one of the Bronte sisters of nineteenth-century literary fame}
D = {x|x 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 = {x|x is a moon of Mars}
A = {Deimos, Phobos}
n(A) =
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A = {Deimos, Phobos}
n(A) = 2
B = {x|x 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 = {x|x 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 = {x|x 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
H
ES
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Attribute Lab
Three attributes considered are• Size• Color• Shape
HexagonYellow
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Yellow andHexagon
HexagonYellow
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Yellow or Hexagon
HexagonYellow
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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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RECTANGLEBLUE
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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.
Karen’s 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 Ken’s 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 TreeMelissa has 4 flags colored red, yellow, green and blue. How many ways can she display them on a flagpole?
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Blue
Green
Yellow
Red
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Blue-Green
Blue-Yellow
Blue-Red
Green-Blue
Green-Yellow
Green-Red
Yellow-Blue
Yellow-Green
Yellow-Red
Red-Blue
Red-Green
Red-Yellow
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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 ProductC = {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
4.3.
2.1.
A
B C CB
A
A
B CCB
A
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8.7.
6.5.
A
B C CB
A
A
B CCB
A
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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, they’re a breeze;
Learn a few, then work on speed.
Times facts, you’ll be surprised
By just how fast you can memorize.
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3 time 7 is 21Now, at last we’ve all begun.4 times 7 is 28Let’s sing what we appreciate.
(Chorus)
5 times 7 is 35.Yes, by gosh, we’re still alive.6 times 7 is 42.I forgot what we’re supposed to do.
(Chorus)
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Venn Diagram Lab