2.1 Sets and Operations on Sets
Sets and Whole Numbers
A set is a collection of objects.
2.1 Sets and Operations on Sets
2.2 Sets, Counting, and the Whole Numbers
2.3 Addition and Subtraction of Whole Numbers
2.4 Multiplication and Division of Whole Numbers
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.
Set of natural numbers = = { 1, 2, 3, 4, 5, . . . }
THREE WAYS TO DEFINE A SET
1. Word Description:
The set of even numbers between 2 and 10 inclusive.
2. Listing in Braces:
{2, 4, 6, 8, 10}
Each 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.
3. Set Builder Notation:
{x | x = 2n, 1 ≤ n ≤ 5, n N}
b. {3, 6, 9, 12,…}
c. The set of the first 10 odd natural numbers.
VENN DIAGRAMS
Example
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.
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}
U
A
U
A
A
U
B
B
C
THE COMPLEMENT OF A SET A The complement of set , written , is the set of elements in the universal set that are not elements of . That is,
U
A
Example
Let be the set of natural numbers. If
, find the complement, .
SUBSET
The set A is a subset of B, written , if, and only if, every element of A is also an element of B.
EMPTY SET
A set which has no elements in it is called the empty set and is written or { }.
Example: Example:
INTERSECTION OF SETS
The intersection of two sets A and B, written , is the set of elements common to both A and B. That is, DISJOINT SETS
Two sets C and D are disjoint if C and D have no elements in common. That is, C and D are disjoint means that 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
UNION OF SETS
The union of sets A and B , written , is the set of all elements that are in A or B. That is, PROPERTIES OF SET OPERATIONS AND RELATIONS
Transitivity of inclusion
DeMorgan's Laws
Commutativity of union and intersection
1.
A ∩ B = A ∪ B
2.
A ∪ B = A ∩ B
Example
Prove DeMorgan's 1st Law using a Venn diagram.
Associativity of union and intersection
Properties of the empty set
Distributive properties of union and intersection
2.2 Sets, Counting, and Whole Numbers
TYPES OF NUMBERS
Nominal numbers
A number can be an identification such as a ticket number.
Ordinal numbers
Ordinal numbers communicate location in an ordered sequence, such as first, second, third.
ONE­TO­ONE CORRESPONDENCE
A 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.
Cardinal numbers
A cardinal number communicates the basic notion of “how many,” such as four may tell us how many tickets we have for an event.
EQUIVALENT SETS
Sets 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 }
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 }
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?
3. { x | 25 < x < 31 }
4. The number of blades of grass on a football field.
WHOLE NUMBERS
Example:
100 students were asked if they spoke English, Spanish, and/or Portuguese. The results were as follows:
46 spoke Spanish
36 spoke English
42 spoke Portuguese
11 spoke Spanish and Portuguese
13 spoke Portuguese and English
7 spoke Portuguese and Spanish
3 spoke all three languages
a.
b.
c.
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, …}.
How many students did not speak any of these languages?
How many students only spoke one of these languages?
How many students spoke English, but not Spanish?
2.3 Addition and Subtraction of Whole Numbers
Two Conceptual Models for the addition of whole numbers
1.
Set Model
Let a and b be any two whole numbers. If A and B
are 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 Ordering Numbers
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.
Example:
Use (a) sets, (b) tiles, (c) rods, and (d) the number line to show that 4 < 7.
2.
Number­Line (Measurement) Model
PROPERTIES OF WHOLE NUMBER ADDITION
If a, b, and c are any whole numbers,
CLOSURE PROPERTY
a + b is a unique whole number. COMMUTATIVE PROPERTY
a + b = b + a. ASSOCIATIVE PROPERTY
a + ( b + c ) = ( a + b ) + c. ADDITIVE­IDENTITY PROPERTY OF ZERO
a + 0 = 0 + a = a. SUBTRACTION OF WHOLE NUMBERS
Let 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?
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
MISSING­ADDEND MODEL and by
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.
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 FIVE CONCEPTUAL MODELS FOR MULTIPLICATION OF WHOLE NUMBERS • Array 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 • Rectangular Area Model
“Skip” by the number 2 six times:
• Skip Count Model
“Skip” by the number 3 five times:
2, 4, 6, 8, 10, 12
3, 6, 9, 12, 15
• Multiplication Tree Model
• Cartesian Product Model
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 Multiplication
If a, b, and c are any whole numbers,
CLOSURE PROPERTY
a . b is a unique whole number. COMMUTATIVE PROPERTY
a . b = b . a. ASSOCIATIVE PROPERTY
a . ( b . c ) = ( a . b ) . c. MULTIPLICATION BY ZERO
a . 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. REPEATED­SUBTRACTION MODEL
THREE CONCEPTUAL MODELS FOR DIVISION OF WHOLE NUMBERS • Repeated­Subtraction Model
Ms. Rislov has 28 students in her class whom she wishes to divide into cooperative learning groups of 4 students per group.
• Partition Model
• Missing­Factor Model
PARTITION MODEL
MISSING FACTOR MODEL
28 ÷ 4 = 7 ⇒ 4 x 7 = 28
Computing Quotients with Manipulatives
Suppose 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
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
Partition
Missing factor
EXPONENTS (THE POWER OPERATION) DIVISION BY ZERO
Let a and m be whole numbers where m ≠ 0.
Then a to the mth power, written am, is defined by
0 ÷ 4 =
4 ÷ 0 =
and MULTIPLICATION RULES OF EXPONENTIALS
Let a, b, m, and n be whole numbers where m ≠ 0 and n ≠ 0. 00 =