Download Sets

Set: a well defined collection of objects.
Universe: only those objects that will be
considered.
Three ways of describing a set:
Words: The set of first 3 presidents of
the U.S.
Listing in Braces: { G Washington, T.
Jefferson, J Adams}
Set Builder: { x| x is one of the first 3
presidents of the U.S.}
The set of natural numbers greater than 12
and less than 17.
{13,14,15,16}
{x | x=2n and n = 1,2,3,4,5}
{2,4,6,8,10}
{3,6,9,12….}
The set of multiples of 3.
Venn Diagrams
Pictorial representation of sets.
Rectangle is used to represent the
universal set.
Circles represent a set within the
universe.
U is the set of letters. V is the set of vowels.
B C
S T
D
R
F G H
VW
A
E
I
O
U
J K L
M N
P
Q
X Y Z
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Complement of Set A, written A’ or A, is the
set of elements in the universal set U that are
not elements of set A.
A’ = { x | x Є U and x Є A}
If set A is the Green section then the yellow
section is the complement of A.
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The set A is a subset of B written A B, if and
only if every element of A is also an element of
B.
U
B
A
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A = {1,2,3,4,5,6,7,8,9,10}
B = {2,4,6,8,}
C={1,3,5,7,9}
D = {2,4,6,8,10,12}
B
C
D
B
A?
A?
A?
D?
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The intersection of two sets A and B written
A  B
It is the set of elements common to both A and
B. (The set of elements that are in both A and B
at the same time).
A  B = { x | x Є A and x Є B}
A
B
The yellow section is the
intersection of sets A and B.
A = {1,2,3,4,5,6,7,8}
B = { 1,3,5,7,9,11,13,15}
A
A  B = {1,3,5,7}
B
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The UNION of sets A and B, written A  B, is
the set of all elements that are in set A or in set
B.( All the elements that are in either set but
don’t repeat them.)
A  B = { x| x Є A or x Є B }
A = { 1,2,3,4,5,6}
B = {5,6,7,8,9}
A B = { 1,2,3,4,5,6,7,8,9}
U
A
1 3
2
3
B
5
6
78
9
U = { p,q,r,s,t,u,v,w,x,y}
A = {p,q,r}
B = { q,r,s,t,u}
C = { r,u,w,}
U
B
A
C
U = {1,2,3,4,5,6,7,8,9}
A = {1,2,3}
B = {2,3,4,5,6}
C = { 3,6,9}
AC
AC
A B
AB
B’
C’
A  B’
A  C’
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The student with ticket 507689 has just won
second prize - four tickets to the Bills game.
Three types of numbers
Identification –Nominal numbers – sequence of
numbers used as a name or label (telephone #)
Ordinal Number – relative position in an
ordered sequence – first second, etc
Cardinal Number – number of objects in a set
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Whole numbers are the cardinal numbers of a
finite set.
W = {0,1,2,3,4,5,6…}
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Tiles
Cubes
Number Strips & Rods
Number Line
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Show 4 < 7 using
Tiles
Cubes
Number Strips & Rods
Number Line
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Example 2.9 Pg 94
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Set Model of Addition
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Measurement Model of Addition
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Rods
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Closure: if a and b are two whole numbers then
a + b is a whole number
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Commutative Property: a + b = b +a
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Associative Property: a + (b + c) = (a + b) + c
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Additive Identity: a + 0 = 0 + a = a
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Associative Property with Rods
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Commutative Property with Rods
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Associative Property with Number Line
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Commutative Property with Number Line
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a–b=c
a is the minuend
b is the subtrahend
c is the difference of a and b
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Take Away (sets)
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Missing Addend
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Comparison (how many more)
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Number line
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Multiplication as repeated addition
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Sets
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Number Line
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Rectangular Area
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Closure: if a and b are two whole numbers then
a X b is a whole number
Commutative Property: a X b = b X a
Associative Property: a X (b X c) = (a X b) X c
Multiplication by Zero: a X 0 = 0 X a = 0
Multiplicative Identity: a X 1 = 1 X a = a
Distributive Property: a X (b +c ) = a X b + a X c
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Repeated Subtraction
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Partition
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Missing Factor
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Division by Zero is Undefined
There is no unique number such that
a ÷ 0 = c because this means
a=0Xc
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a1 = a
a0 = 1
am = a X a X a X a … M factors of a
am X an = a m+n
am / an = a m-n
(am)n = a mXn