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