Section 2-1 – Numeration Systems
Having a counting system requires that we recognize a one-to-one correspondence between two
equivalent sets: the set of objects we are counting and the set of numbers we are using to count
them.
Numbers, Numerals, and Numeration Systems:
Number – The amount being counted or measured
Numeral – The symbol used to designate the amount (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
Numeration System – A collection of properties and symbols agreed upon to represent numbers
systematically.
The table on pg. 52 has the symbols used for different numeration systems
(Babylonian, Egyptian, Mayan, Greek, Roman, Hindu, Arabic, Hindu-Arabic)
Hindu-Arabic System – Base 10 System:
•No tally marks
•All numerals are constructed from the 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
•Place value is based on powers of 10 (the base of the system)
•Concept of Zero
•Expanded form
For example, the number 546 = 500 + 40 + 6
= (5 * 100) + (4 * 10) + (6 * 1)
= (5*102) + (4 * 101) + (6 * 100) (expanded form)
We can use Base-10 Blocks to represent numbers in base-10.
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Examples: Represent each of the following numbers using the Base-10 Blocks and then write each
number in expanded form.
a) 245
b) 104
c) 1,213
Example: Rewrite the following as a base-10 numeral:
(7 × 103) + (4 × 102) + (0 × 101) + (6 × 100) =
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OTHER NUMERATION SYSTEMS:
Egyptian Numeration System:
The Egyptian system is an example of an additive system (tallies and pictures) where the value of
the number is the sum of the numerals. It dates back to 3400 BCE.
They used powers of 10:
Symbol
Name
Astonished
Person
Polliwog
Pointing
Finger
Lotus
Flower
Scroll
Heelbone
Staff
Value
1,000,000
100,000
10,000
1000
100
10
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Example: Find the value for each of the following Egyptian numerals:
a)
b)
c)
Example: Write 2,103,412
using the Egyptian system
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Roman Numeration System
Symbol
M
D
C
L
X
V
I
Value
1000
500
100
50
10
5
1
Examples: What is the value of the following numbers?
a) MDLVII
b) DXI
Examples: Express the following in Roman numerals:
a) 1,109
b) 418
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Babylonian Numeration System:
The Babylonians used powers of 60 for their place values: ……
603
602
601
600
Symbol
Value
1
10
This is an example of a positional system because the value of a numeral depends on its position in
the number. (Earliest occurrence of the concept of place values in recorded history)
Examples: What is the value of the following numbers?
a)
b)
c)
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Examples: How would the Babylonians represent the following amounts?
a) 310
b) 4,281
THE CARAMEL FACTORY
Suppose you own a caramel factory, but are having a lot of trouble finding good help. In fact, the people you
hired are only able to count up to four. So when your caramel machine spews out the caramels, your
employees are unable to count how many there are. After some deliberation, you decide that the thing to do
is to package your product by fours. Any time an employee sees four caramels, he is to put them in a pack.
Any time he sees four packs he is to put them in a flat, and any time he sees four flats, he is to put them in a
box. That is, your caramels are available only in the following quantities:
Singles
Packs (4 singles)
Flats (4 packs)
Boxes (4 flats)
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1. Using these packing instructions, your employees are now able to report how many caramels the
machine has produced without ever counting beyond four. The first step is to package the caramels
shown in the rectangle below by fours. DRAW THE CARAMELS PACKAGED AS PACKS.
2. Now, using the packs you created in #1, create as many flats as you can.
3. Do you have enough flats to make a box?
4. Draw the FINAL packaging you were able to create from the original pile of caramels.
5. Now an employee can report the number of caramels as:
________ box(es)
_______ flat(s)
_______ pack(s)
_______single(s)
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6. The caramel symbols below would be recorded as 2313c.
How many single caramels does this represent?
7. How would the final packaging look for a pile of 250 single caramels?
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OTHER NUMBER BASE SYSTEMS:
Base-10 numbers are always of the form:
… 104
103
102
101 100
Example: Write out the first 21 numbers using base-10. (notice the pattern)
Base-5 numbers will be of the form:
… ______ ______ ______ ______ ______
Example: Write out the first 21 numbers using base-5.
Examples: Before/After
a) What number comes before:
405
3205
b) What number comes after:
245
4445
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Example: Convert the following numbers into base-10:
(use a chart and place values)
a) 12345
b) 5156
Example: Convert the following base-10 numbers into the base specified:
(use repeated division & keep track of remainders)
a) 341 into base-5
b) 2,239 into base-6
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Base-12:
What are the symbols that would be used for a base-12 system?
0
1
2
3
4
5
6
7
8
9
_____
_____
Example: Convert ET312 into base-10
Example: Convert 1,463 into a base-12 number
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Section 2-2 – Describing Sets
Set – A collection of objects.
Elements or Members – The individual objects in a set.
Notation: The symbol  means “is a member of”.
The symbol  means “is not a member of”.
For example, the set of vowels in the alphabet can be written V = {a, e, i, o, u}
Note that, a  V but f  V.
For sets to be useful in mathematics, they must be well-defined.
Example:
Are the following sets well-defined?
A = The set of odd numbers
B = The set of large dogs
C = The set of whole numbers less than 2
The cardinal number of a set X, denoted n(X) indicates the number of elements in the set X.
Example: What is the cardinal number of each of the following sets?
A = {0, 1, 2, 3, 4, 5}
B = {0, 1, 2}
n(A) = ______
n(B) = ______
A set that is empty, or has no elements, has a cardinal number of 0 and is called the empty set or
the null set. Denoted Ø or { }.
TYPES OF SETS (Finite vs. Infinite):
Finite Sets – The cardinal number of the set is a whole number.
Infinite Sets – The set has an unlimited number of members.
Finite Sets
A = {the students in Math 1350}
B = {2, 4, 6, 8}
Infinite Sets
N = {1, 2, 3, …}
W = {0, 1, 2, 3, …}
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Ways to Describe a Set:
1. Making a list
2. Using Set-Builder Notation
Set-Builder Notation has the form:
Examples:
x
description of x 
Describe each of the following sets using a list and set-builder notation:
a) The letters in the word “cat”
b) The natural numbers from (and including) 6 to 10.
c) The positive multiples of 10.
d) The positive even numbers.
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One-to-One Correspondence:
If the elements of sets A and B can be paired such that for each element of A there is exactly one
element of B (and vice versa), then A and B are in a one-to-one correspondence.
Example: Let set A = {Bob, Jane, Mary} and set B = {1st, 2nd, 3rd).
How many one-to-one correspondences are there? List them.
The Fundamental Counting Principle:
If event A can occur in a ways and event B can occur in b ways, then event A followed
by event B can occur in ab ways.
Example: How many ice cream arrangements are possible on a two-scoop cone if 3 flavors are
offered? List them.
Example: How many mix and match outfits can be made from 2 pair of jeans and 3 shirts?
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Equal vs. Equivalent
Two sets are equal if and only if they contain exactly the same elements.
Two sets are equivalent if and only if the elements can be placed in a one-to-one correspondence
(ie, they have the same number of elements).
For example, if A = {1, 2, 3} and B = {3, 1, 2} then
For example, if A = {1, 2, 3, 4} and B = {a, b, c, d} then
Use the following sets for the examples that follow:
A = {1, 2, 3, 4}
D = {4, 2, 1, 3}
B = {m, n, o}
E = {10, 11, 12}
C = {a, b, c, d}
Example: Are the following sets equal or not equal? (Fill in the blank with = or ≠)
A ______ B
A ______ C
A ______ D
Example: Are the following sets equivalent or not equivalent? (Fill in the blank with
B ______ E
A ______ C
A ______ D
or
)
D ______ E
Subset – A set that is part of another set.
If B is a subset of A, then it is denoted B  A.
If B is not a subset of A, we say B  A.
Note that this means that set B can be equal to set A.
The null set is a subset of every set.
Example: Suppose that set A = {0, 1, 2, 3, 4, 5}
Are the following sets subsets of A? (Fill in the blank with  or  )
B = {0, 1, 2}
C = {1, 3, 5}
D = {1, 2, 3, 6}
E = {0, 1, 2, 3, 4, 5}
B ______ A
C ______ A
D ______ A
E ______ A
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Example: List all possible subsets of the following set: X = {5, 7, 9}
Example: How many subsets can be made from a set containing n elements?
Number of Elements in the Set
Number of Subsets
0
1
2
3
4
…
…
N
So, if there are n elements in a set, then ________ subsets can be formed.
A subset B is called a proper subset of A if and only if every element of B is an element of A ,
but B ≠ A. Denoted B  A.
The null set is a proper subset of every set.
This can be shown with the following Venn diagram:
A
B
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Example: Suppose that set A = {0, 1, 2, 3, 4, 5}
Are the following sets proper subsets of A? (Fill in the blank with  or  )
B = {0, 1, 2}
C = {1, 3, 5}
D = {1, 2, 3, 6}
E = {0, 1, 2, 3, 4, 5}
B ______ A
C ______ A
D ______ A
E ______ A
Example: Suppose that set A = {a, b, c, d}
Fill in the blank with the appropriate symbol.
(more than one symbol may apply)
 , ,  ,  ,  , 
a __________ A
Ø __________ A
e__________ A
{a, b, c, d, e} __________ A
e __________ Ø
{c, b} __________ A
{a, b} __________ A
{a, b, c, d} __________ A
The Universal Set – The set that contains all elements being considered in a given discussion.
Usually denoted as U.
The complement of set F, denoted F , is all of the members of U that are NOT members of F.
ie, F   x x  U , x  F 
Suppose that set U = all students taking Math 1350 and
set F = the female students taking Math 1350.
The complement of set F, denoted F , is the set of students that are NOT female.
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One way to describe these sets is by using a Venn Diagram.
U
U
F
F
Example: Suppose that U = {w, x, y, z} and A = {x, y} and B = {w} .
Find A and B .
Section 2-3 – Set Operations and Their Properties
Recall the complement of set F: F   x x  U , x  F 
This only acts on one set at a time. What if you want to work with two sets at a time??
Two sets that have no common elements:
A
B
Two sets that have common elements:
A
B
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Intersections and Unions:
The Intersection of sets A and B is the set of all elements common to both A and B.
Denoted:
A  B   x x  A and x  B
The Union of sets A and B is the set of all elements that are in set A or B or both.
Denoted:
A  B   x x  A or x  B
Example: For each of the following, find A  B and A  B (express each as a set):
a) A = {1, 2, 3}
B = {2, 3, 4}
A
B
A
B
A
B
A B =
A B =
b) A = {1, 2, 3, 4}
B = {1, 2, 3, 4, 5}
A B =
A B =
c) A = {1, 2, 3, 4, 5}
B = {6, 7, 8}
A B =
A B =
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The Complement of A relative to B is the set of all elements in B that are not in set A.
Denoted:
B  A   x x  B and x  A
Example: For the sets A = {a, b, c, d, e, f, g} and B = {d, e, f, h} find A – B and B – A.
(express each as a set)
A–B=
A
B
B–A=
Example: Shade a Venn Diagram for B  A and B  A .
A
B
Are they the same?
A
B
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Another way to form a set from two other sets is by forming a Cartesian Product.
For any sets A and B, the Cartesian Product A×B is the set of all ordered pairs such that the first
component of each pair is an element of A and the second component is an element of B.
A  B  ( x, y ) x  A, y  B
Example: If A = {0, 1, 2} and B = {5, 7}, then find:
(express each as a set)
a) A  B =
b) B  A =
The Commutative Property of Set Union:
A B  B  A
The Commutative Property of Set Intersection:
A B  B  A
The Associative Property of Set Union:
A   B  C    A  B  C
The Associative Property of Set Intersection:
A   B  C    A  B  C
The Distributive Property of Set Intersection over Union: A   B  C    A  B    A  C 
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Example: The fifth grade students at Kennedy Elementary were asked whether they bat right or left
handed. Some students responded that they bat equally well with either hand. A few
said that they have never batted. The following are the survey results:
23 bat left-handed
75 bat right-handed
8 bat both right and left-handed
5 have never batted
Draw a Venn Diagram to illustrate the results of the survey and then answer the questions below.
a) How many students were surveyed?
b) How many students only batted left-handed?
c) How many students only batted right-handed?
d) How many students never bat left-handed?
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Example: Let H be the set of all people who play a horn. Let T be the set of all people who play
the trumpet, and let S be the set of all people who play the saxophone.
H
T
S
Region 1
Region
Region 3
2
Region 4
a) Use WORDS to describe the people represented by each of the numbered regions in the
Venn Diagram above.
Region 1 -
Region 2 -
Region 3 -
Region 4 -
b) Represent each numbered region using set operations (symbols)
Region 1 Region 2 Region 3 Region 4 -
c) Which region(s) would describe the following people:
The set of all horn players who play the trumpet OR saxophone.
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Example:
A group of students at University U were polled to see which classes they enrolled in.
U = the students at university U
M = the set of students taking Mathematics
E = the set of students taking English
B = the set of students taking Biology
Describe each of the following sets using symbols and then shade each one on a Venn Diagram.
The students enrolled in
Biology and Math.
M
E
B
M
E
B
The students enrolled in
Biology or English or Math.
M
The students enrolled in
Biology or English.
E
B
The students enrolled in
Biology and English and Math.
M
E
B
The students enrolled in English but not in Math. (English and not Math)
M
E
B
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