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. U V 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
Complement of a Set • 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.
Subset • 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
SUBSETS • 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 ?
INTERSECTION OF SETS • 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}
Union of Sets • 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 B 1 3 5 2 6 7 8 3 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’
SETS, COUNTING, WHOLE NUMBERS • 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 • Whole numbers are the cardinal numbers of a finite set. • W = {0,1,2,3,4,5,6…}
Physical & Pictorial Representation of Whole Numbers • Tiles • Cubes • Number Strips & Rods • Number Line
Showing Order • Show 4 < 7 using • Tiles • Cubes • Number Strips & Rods • Number Line
Example 2.9 Pg 94
Addition & Subtraction • Set Model of Addition • Measurement Model of Addition • Rods
Properties Of Whole Number Addition • 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
Subtraction of Whole Numbers • a – b = c • a is the minuend • b is the subtrahend • c is the difference of a and b
Subtraction Models • Take Away (sets) • Missing Addend • Comparison (how many more) • Number line
Multiplication • Multiplication as repeated addition • Sets • Number Line • Rectangular Area
Properties Of Whole Number Multiplication • 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
Division of Whole Numbers • 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
Laws of Exponents • 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
