2.1 sets and operations on sets - modesto junior...
TRANSCRIPT
Sets and Whole Numbers
2.1 Sets and Operations on Sets2.2 Sets, Counting, and the Whole Numbers2.3 Addition and Subtraction of Whole Numbers2.4 Multiplication and Division of Whole Numbers
2.1 Sets and Operations on Sets
A set is a collection of objects.
Sets must be well defined, which means:• there is a universe of objects that are allowed into consideration.• any object in the universe is either an element of the set or is not an element of the set.
1. Word Description:The set of even numbers between 2 and 10 inclusive.
2. Listing in Braces:{2, 4, 6, 8, 10}
3. Set Builder Notation:{x | x = 2n, 1 ≤ n ≤ 5, n N}
THREE WAYS TO DEFINE A SETEach set that follows is taken from the universe N of the natural numbers and is described either in words, by listing the set in braces, or with setbuilder notation. Provide the two remaining types of description for each set.
a. The set of natural numbers greater than 12 and less than 17.
b. {3, 6, 9, 12,…}
c. The set of the first 10 odd natural numbers.
Set of natural numbers = = { 1, 2, 3, 4, 5, . . . }
VENN DIAGRAMS
AU
Sets can be represented pictorially by Venn diagrams, named for the English logician John Venn.The universal set U is represented by a rectangle. Any set A within the universe is represented by a closed loop lying within the rectangle.
A
B
U A
B
C
U
Example
Draw the Venn diagram for the following sets:U = {1, 2, 3, 4, 5, 6, 7, 8}V = { 1, 3, 5, 7}W = { 2, 3, 4, 7}
THE COMPLEMENT OF A SET A
AU
The complement of set , written , is the set of elements in the universal set that are not elements of . That is,
Example
Let be the set of natural numbers.
If , find the complement, .
SUBSETThe set A is a subset of B, written , if, and only if, every element of A is also an element of B.
Example:
EMPTY SETA set which has no elements in it is called the empty set and is written or { }.
Example:
INTERSECTION OF SETSThe intersection of two sets A and B, written , is the set of elements common to both A and B. That is,
DISJOINT SETSTwo sets C and D are disjoint if C and D have no elements in common. That is, C and D are disjoint means that
UNION OF SETSThe union of sets A and B , written , is the set of all elements that are in A or B. That is,
Example:Let U = { +, , =, !, ?, %, $, #, @ }, A = { +, , = }, B = { !, }, C = { @, $, # }
Find the following:a. A B b. B C
c. B A d. A B
e. B C f. C B
g. A B h. B C
i. A ( B C ) j. B A C
PROPERTIES OF SET OPERATIONS AND RELATIONS
Transitivity of inclusion
Commutativity of union and intersection
Associativity of union and intersection
Properties of the empty set
Distributive properties of union and intersection
DeMorgan's Laws
1. A ∩ B = A ∪ B
2. A ∪ B = A ∩ B
ExampleProve DeMorgan's 1st Law using a Venn diagram.
2.2 Sets, Counting, and Whole NumbersTYPES OF NUMBERSNominal numbersA number can be an identification such as a ticket number.
Ordinal numbersOrdinal numbers communicate location in an ordered sequence, such as first, second, third.
Cardinal numbersA cardinal number communicates the basic notion of “how many,” such as four may tell us how many tickets we have for an event.
ONETOONE CORRESPONDENCEA onetoone correspondence between sets A and B is an assignment, for each element of A, of exactly one element of B in such as way that all elements of B are used. It can also be thought of as a pairing of elements between A and B such that each element of A is matched with one and only one element of B, and every element of B has an element of A assigned to it.
EQUIVALENT SETSSets A and B are equivalent if there is a onetoone correspondence between A and B.
When A and B are equivalent, we write We also say that equivalent sets match.
If A and B are not equivalent, we write
Example:Let A = { x | x is a moon on 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 the nineteenthcentury literary fame }
D = { x | x is a satellite of the fourthclosest planet to the sun}
Which of these relationships, =, ≠, ~, and ~, holds between distinct pairs of the four sets?
Finite vs. Infinite Sets
A finite set is a set that is either the empty set or a set equivalent to {1, 2, 3, 4, . . . , n }, for some natural number n.
An infinite set is a set that is not finite.
Example:Are the following sets finite or infinite?1. { 2, 4, 6, 8, 10, . . . }
2. { x | 25 < x < 31 and x is a natural number }
3. { x | 25 < x < 31 }
4. The number of blades of grass on a football field.
Example:100 students were asked if they spoke English, Spanish, and/or Portuguese. The results were as follows:
46 spoke Spanish36 spoke English42 spoke Portuguese11 spoke Spanish and Portuguese13 spoke Portuguese and English7 spoke Portuguese and Spanish3 spoke all three languages
a. How many students did not speak any of these languages?b. How many students only spoke one of these languages?c. How many students spoke English, but not Spanish?
WHOLE NUMBERS
The whole numbers are the cardinal numbers of finite sets. If A ~ {1, 2, 3, …, m}, then n(A) = m and n(Ø) = 0, where n(A) denotes the cardinality of set A. The set of whole numbers is written W = {0, 1, 2, 3, …}.
Let a = n(A) and b = n(B) be whole numbers, where A and B are finite sets.
If A matches a proper subset of B, we say that a is less than b and write a < b.
Ordering Numbers
Example:Use (a) sets, (b) tiles, (c) rods, and (d) the number line to show that 4 < 7.
2.3 Addition and Subtraction of Whole Numbers
Two Conceptual Models for the addition of whole numbers1. Set Model
Let a and b be any two whole numbers. If A and Bare any two disjoint sets for which a = n(A) and b = n(B), then the sum of a and b, written a + b, is given by
2. NumberLine (Measurement) Model
PROPERTIES OF WHOLE NUMBER ADDITIONIf a, b, and c are any whole numbers,
CLOSURE PROPERTYa + b is a unique whole number.
COMMUTATIVE PROPERTYa + b = b + a.
ASSOCIATIVE PROPERTYa + ( b + c ) = ( a + b ) + c.
ADDITIVEIDENTITY PROPERTY OF ZEROa + 0 = 0 + a = a.
SUBTRACTION OF WHOLE NUMBERSLet a and b be whole numbers. The difference of a and b, written a – b, is the unique whole number c such that a = b + c.
That is, a – b = c if, there is a whole number c, such that a = b + c.
FOUR CONCEPTUAL MODELS FOR SUBTRACTION OF WHOLE NUMBERS
• TakeAway Model
• Missing Addend Model
• Comparison Model
• NumberLine (Measurement) Model
TAKEAWAY MODEL Joshuah has 10 cookies. He gives 3 of the cookies to his little sister Isabella. How many cookies does Joel have left?
COMPARISON MODEL Roger has read 9 books this week. His friend Jake has read 6 books this week. How many more books has Roger read than Jake?
MISSINGADDEND MODEL Tiffany has saved $15. She needs $22 to buy a blouse she noticed last week. How much more money does she need to buy the blouse?
NUMBERLINE MODEL Illustrate 8 – 5 on the number line.
2.4 Multiplication and Division of Whole Numbers
Let a and b be any two whole numbers. Then the product of a and b, written a + b, is defined by
and by
FIVE CONCEPTUAL MODELS FOR MULTIPLICATION OF WHOLE NUMBERS
• Array Model
• Rectangular Area Model
• Skip Count Model
• Multiplication Tree Model
• Cartesian Product Model
ARRAY MODEL Leah planted 3 rows of tomato plants with 4 plants in each row. How many tomato plants does she have planted?
RECTANGULAR AREA MODEL Mr. Hu bought an 8 ft by 6 ft rug for his house. How many square feet are covered by this rug?
SKIP COUNT MODEL “Skip” by the number 2 six times:
2, 4, 6, 8, 10, 12
“Skip” by the number 3 five times:
3, 6, 9, 12, 15
MULTIPLICATION TREE MODEL Timmy has 3 tops (blue, red and green) and 2 shorts (jean and khaki) that can be mixed and matched. How many outfits does he have?
CARTESIAN PRODUCT OF SETS The Cartesian product of sets A and B, written , is the set of all ordered pairs whose first component is an element of set A and whose second component is an element of set B. That is,
CARTESIAN PRODUCT MODEL Timmy has 3 tops (blue, red and green) and 2 shorts (jean and khaki) that can be mixed and matched. How many outfits does he have?
Properties of Whole Number MultiplicationIf a, b, and c are any whole numbers,
CLOSURE PROPERTYa . b is a unique whole number.
COMMUTATIVE PROPERTYa . b = b . a.
ASSOCIATIVE PROPERTYa . ( b . c ) = ( a . b ) . c.
MULTIPLICATION BY ZEROa . 0 = 0 . a = 0.
MULTIPLICATIVE IDENTITY OF ONE
a . 1 = 1 . a = a.
DISTRIBUTIVE PROPERTY OF MULTIPLICATION (OVER ADDITION)
a . ( b + c ) = a . b + a . c = ( b + c ) . a.
THREE CONCEPTUAL MODELS FOR DIVISION OF WHOLE NUMBERS
• RepeatedSubtraction Model
• Partition Model
• MissingFactor Model
REPEATEDSUBTRACTION MODELMs. Rislov has 28 students in her class whom she wishes to divide into cooperative learning groups of 4 students per group.
PARTITION MODEL
MISSING FACTOR MODEL
28 ÷ 4 = 7 ⇒ 4 x 7 = 28
Computing Quotients with ManipulativesSuppose you have 78 number tiles. Describe how to illustrate 78 ÷13 with the tiles, using each of the three basic conceptual models for division. Repeated subtraction
Partition
Missing factor
THE DIVISION ALGORITHM
Let a and b be whole numbers with b ≠ 0. Then there is a unique whole number q called the quotient and a unique whole number r called the remainder such that
DIVISION BY ZERO
0 ÷ 4 = 4 ÷ 0 =
EXPONENTS (THE POWER OPERATION)
Let a and m be whole numbers where m ≠ 0.Then a to the mth power, written am, is defined by
and
MULTIPLICATION RULES OF EXPONENTIALS
Let a, b, m, and n be whole numbers where m ≠ 0 and n ≠ 0.
00 =