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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} - PowerPoint PPT Presentation
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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 DiagramsPictorial representation of sets.
Rectangle is used to represent the universal set.
Circles represent a set within the universe.
B C D R S T
F G H A E V W
I O U
J K L
M N P Q X Y Z
U is the set of letters. V is the set of vowels.
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.
U
B
A
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 A ?C A ?D A ?B D ?
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 B
A B = {1,3,5,7}
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}
1 3 5 2 6 7 83 9
A BU
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,}
UA
B
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’
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
Whole numbers are the cardinal numbers of a finite set.
W = {0,1,2,3,4,5,6…}
Tiles Cubes Number Strips & Rods Number Line
Show 4 < 7 using Tiles Cubes Number Strips & Rods Number Line
Example 2.9 Pg 94
Set Model of Addition
Measurement Model of Addition
Rods
Closure: if a and b are two whole numbers then a + b is a whole number
Commutative Property: a + b = b +a
Associative Property: a + (b + c) = (a + b) + c
Additive Identity: a + 0 = 0 + a = a
Associative Property with Rods
Commutative Property with Rods
Associative Property with Number Line
Commutative Property with Number Line
a – b = c
a is the minuend b is the subtrahend c is the difference of a and b
Take Away (sets)
Missing Addend
Comparison (how many more)
Number line
Multiplication as repeated addition
Sets
Number Line
Rectangular Area
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
Repeated Subtraction
Partition
Missing Factor
Division by Zero is Undefined There is no unique number such that a ÷ 0 = c because this means a = 0 X c
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