set theory chapter 2. day 1 set – collection school of fish gaggle of geese pride of lions pod of...
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SET THEORY
Chapter 2
DAY 1
Set – collection
• School of fish
• Gaggle of geese
• Pride of lions
• Pod of whales
• Herd of elephants
• Set – usually named with a capital letter.
• Well defined
A is the set of the first three lower case letters of the English alphabet.
• 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
• Natural Numbers (Counting Numbers)
N = {1, 2, 3, . . . }
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
• Universe
• Empty set
Example
The set of natural numbers greater than 12 and less than 17.
Example
{x | x = 2n and n = 1, 2, 3, 4, 5}
Example
{3, 6, 9, 12, . . . }
Example
The set of the first 10 odd natural numbers.
Venn Diagrams
C
BA
BA
A
Set A
A
Complement of A
}|{ AxxA
A
A intersect B
}|{ BxandAxxBA
BA
A Union B
BA
}|{ BxorAxxBA
Disjoint Sets
BA
BA
Subsets
A is a subset of B if every element of A is also an element of B.
BA
List all the subsets of {a,b,c}
• { }• {a}• {b}• {c}• {a,b}• {a,c}• {b,c}• {a,b,c}
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}
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.”
True or False? A = {b,c,f,g}
{b,f} A
{b,f} A
True or False? A = {b,c,f,g}
{b,f} A True
{b,f} A True
True or False? A = {b,c,f,g}
{b,d} A
True or False? A = {b,c,f,g}
{b,d} A False
Because d A
True or False? A = {b,c,f,g}
{b,c,f,g} A
True or False? A = {b,c,f,g}
{b,c,f,g} A True
{b,c,f,g} A
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
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
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
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
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 )(
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
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
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
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 },{
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
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 ?
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
},,,,,{
},,,,,,,{
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
},,,,{
},,,,,,,,{
C
BA
)( CBA
C
BA
)()( CABA
C
BA
)( CBA
C
BA
CBA )(
DAY 2
Homework QuestionsPage 83
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.”
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.”
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)
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
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?
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
~
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
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)
6 different one-to-one correspondences:
A = {a,b,c} and C = {x,y,z}
a – x
b – y
c – z
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
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
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
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
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
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?
A = {x|x is a moon of Mars}
A = {Deimos, Phobos}
n(A) =
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) =
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) =
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) =
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
Finite/Infinite
• Whole numbers?
• Real numbers between 0 and 1?
• Factors of 20?
• Multiples of 20?
• Number of grains of sand on the earth?
Example 2.9Page 94
• 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
Attribute Lab
Three attributes considered are• Size• Color• Shape
HexagonYellow
Yellow andHexagon
HexagonYellow
Yellow or Hexagon
HexagonYellow
• 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)
RECTANGLEBLUE
Day 3
Homework Questions Page 97
Binary Operations
• Addition
• Subtraction
• Multiplication
• Division
__________ + __________ = __________
Addend + Addend = Sum
__________ - __________ = __________
Addend + Addend = Sum
Minuend – Subtrahend = Difference
__________ X __________ = __________
Addend + Addend = Sum
Minuend – Subtrahend = Difference
Factor X Factor = Product
__________ __________ = __________
Addend + Addend = Sum
Minuend – Subtrahend = Difference
Factor X Factor = Product
Dividend Divisor = Quotient
PropertiesPages 104 and 120
• Closure
Counting Numbers = {1, 2, 3, . . . }
Whole Numbers = {0, 1, 2, 3, . . . }
Closure Examples
Is the set of Whole Numbers closed with respect to
• Addition?
• Subtraction?
• Multiplication?
• Division?
Closure Examples
Is the set of Even Counting Numbers closed with respect to
• Addition?
• Subtraction?
• Multiplication?
• Division?
Closure Examples
Is {0, 1} closed with respect to
• Addition?
• Subtraction?
• Multiplication?
• Division?
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
Examples
2 + (3 + 4) = 5 + 4
Examples
2 + (3 + 4) = 5 + 4 Associative
2 + (3 + 4) = 7 + 2
Examples
2 + (3 + 4) = 5 + 4 Associative
2 + (3 + 4) = 7 + 2 Commutative
2(3 + 4) = 6 + 8
Examples
2 + (3 + 4) = 5 + 4 Associative
2 + (3 + 4) = 7 + 2 Commutative
2(3 + 4) = 6 + 8 Distributive
2(3 + 4) = (7)2
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
Conceptual Models
• Addition– Set Model
Conceptual Models
• Addition
• Subtraction (page 108)– Take-away– Missing Addend– Comparison– Number-line
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?
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?
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?
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?
Conceptual Models
• Addition
• Subtraction
• Multiplication (page 115)– Repeated Addition– Number-line– Rectangular Array– Multiplication Tree
Multiplication TreeMelissa has 4 flags colored red, yellow, green and blue. How many ways can she display them on a flagpole?
Blue
Green
Yellow
Red
Blue-Green
Blue-Yellow
Blue-Red
Green-Blue
Green-Yellow
Green-Red
Yellow-Blue
Yellow-Green
Yellow-Red
Red-Blue
Red-Green
Red-Yellow
Conceptual Models
• Addition
• Subtraction
• Multiplication– Repeated Addition– Number-line– Rectangular Array– Multiplication Tree– Cartesian Product
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}
Cartesian Product
• A X B = {(a,b) | a A and b B}
Example:
A = {5, 6, 7} B = {6, 8}
A X B = {(
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)}
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
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}
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
Conceptual Models
• Addition
• Subtraction
• Multiplication
• Division (Page 121)– Repeated Subtraction– Sharing– Missing Factor
Division Example
• Describe how you would divide 78 by 13 using counters and each of the following models.
– Repeated Subtraction– Sharing– Missing Factor
Family of Facts
20 4 = 5 5 X 4 = 20
and
20 5 = 4 4 X 5 = 20
Family of Facts
0 ÷ 4 = 0 and 0 X 4 = 0
4 ÷ 0 = ??
and ?? X 0 = 4
Division by Zero is Undefined.
Extra Practice Worksheet
DAY 4
HomeworkPages 111 and 130
Worksheet Answers
4.3.
2.1.
A
B C CB
A
A
B CCB
A
8.7.
6.5.
A
B C CB
A
A
B CCB
A
Math and MusicThe Magical Connection!
• Scholastic Parent and Child Magazine
• Spelling
• Phone Numbers
• School House Rock
“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.
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)
Print Review for Test
Venn Diagram Lab