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```10/9/18
Working with Sets
Algebra
Section 3.5
Do you remember these
vocabulary words?
If not, go back through your notes or
textbook to refresh your memory.
Real numbers
Rational numbers
Whole number
Integers
Prime
Composite
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Introduction
A set is a collection of objects.
Example: { -4, -3, -2, -1, 0, 1, 2, 3, 4}
A subset is a subgroup of a set.
– A subset of the above set could be all even #s
{ -4, -2, 0, 2, 4}
– A subset of the above set could be all positive
#s
{ 1, 2, 3, 4}
The objects (each number) in a set are called
elements of the set.
Writing a Set
When writing a set of numbers, a set must:
1. be written using braces { }.
2. be labelled with a capital letter.
3. be written with the values listed from least to
greatest from left to right (just like on a number
line).
4. not have duplicate elements.
For example the following is labelled as
“Set Z”: Z = { …-3, -2, -1, 0, 1, 2, 3, …}
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Two Ways to Write a Set
There are two ways to write a set of
numbers;
Roster form or Set-builder Notation
Roster Form
Roster form is the method of describing
a set by listing each element of the set.
Example:
Let M = The set of all 8th grade math teachers at
Martino. Roster form is:
M = {Bahret, Lavey, Proff, Sterritt}
(Since there are only 4 names it can clearly
be listed as a set.)
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Roster Form
Another example…
Let set A = The set of odd numbers greater than
zero, and less than 10. Roster form is:
A = {1, 3, 5, 7, 9}
(**Make sure all number appear in order from
least to greatest.)
Roster Form
Sometimes we can’t list all the elements of a
set. For instance, Z = The set of all integer
numbers. We can’t write out all the integers
because there are an infinite number of
integers, so we then use dots. The dots mean
continue on in this pattern forever and ever.
Z = { …-3, -2, -1, 0, 1, 2, 3, …}
This is the set of whole numbers.
W = {0, 1, 2, 3, …}
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Set – Builder Notation
When it is not convenient or not possible to list all
the elements of a set, we use a notation called set
– builder notation.
Set-builder notation is basically a written
description of the elements in a set.
If possible, use an inequality sign as part of the
description.
Set – Builder Notation
Example:
Let set A = The set of all real numbers greater than 5.
Set-builder notation is:
A = {x | x > 5}
This set is read as, “all elements x such that x is all real
numbers greater than 5”.
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Set – Builder Notation
Another Example:
Let set D = The set of all prime numbers greater than
2. Set-builder notation is:
D = {x | x is prime and x > 2}
This set is read as, “all elements x such that x is prime and x
is greater than 2”.
Roster form vs. Set-builder
Notation
J = The set of all whole numbers greater.
than 5.
Roster Form: J = {6, 7, 8, 9,…}
Set-builder Notation: J = {x | x is all whole
number where x > 5}.
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Roster form vs. Set-builder
Notation
N = The set of natural numbers.
Roster Form: N = {1, 2, 3, …}
Set-builder Notation: N = {x | x is all natural
numbers where, x > 0}. You can write this
set in set-builder notation, however, is a lot
longer to write---it is more convenient to
write the set in roster form.
Roster form vs. Set-builder
Notation
Q = The set of rational numbers
Roster Form: Cannot write in roster form, as
you cannot list all rational numbers. The list
would go on forever.
Set-builder Notation:
Q ={x | x are all rational numbers}
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Review Problem 1 & Got it 1 on
pg. 194.
(Answers to the Got its are in
the back of the textbook.)
Review Problem 2 & Got it 2 on
pg. 195.
(Answers to the Got its are in
the back of the textbook.)
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Empty Set
An empty set is a special set. It contains no
elements. (It can also be called a null set.)
It is usually written as { } or
Æ
Do not be confused by this question:
Is the set {0} empty?
Of course, not! It is not empty! It contains the element
zero.
Universal Set
The Universal Set (U ):
• is the set of all possible elements used in a
problem.
• can also be called the universe.
• is always the largest set you are working with.
For the set of the students in this class, U would be all
the students at Martino (or perhaps all the students in
the world).
For the set of the vowels of the alphabet, U would be
all the letters of the alphabet.
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Universal Set and Subsets
When every element of one set is also an element
of another set, we say the first set is a subset.
Example A= {1, 2, 3, 4, 5} and B ={2, 3}
We say that B is a subset of A. The symbol for
subset is Í. We can then say, B ÍA.
Universal Set and Subsets
Another example:
Let S= {1,2,3}, list all the subsets of S.
The subsets of S are Æ , {1}, {2}, {3}, {1,2},
{1,3}, {2,3}, {1,2,3}.
**The empty set is always considered a
subset of any set.
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Review Problem 3 & Got it 3 on
pg. 195.
(Answers to the Got its are in
the back of the textbook.)
Complement of a Set
The complement of a set is the set of all
elements in the universal set that are NOT in
the set.
The complement of Set A would be written
as A’.
The complement of set S would be written
as S’ .
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Venn Diagrams
Venn Diagrams represents sets graphically
– The box represents the universal set, U
– Circles represent the set(s), S
S’
U
S
– Inside the box, but outside
the circle represents the
complement of the set, S’
Venn Diagram Example
Consider set S, which is the set of all vowels in the
alphabet.
Remember:
– The circle represents the set (all vowels)
– The box represents the
b c d f S’
universal set (all letters in
g h j
S
the alphabet)
k l m
– Inside the box, but outside
n p q
e
a
the circle represents the
r s
t
o
complement of the set
v w x
(all consonants)
y z
U
i
u
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Venn Diagram Example
The Venn Diagram can look a little too busy, so
usually the elements within the complement of the
set are not written out.
So the final Venn Diagram
would look like this.
U
S’
S
e
a
o
i
u
Review Problem 4 & Got it 4 on
pg. 196.
(Answers to the Got its are in
the back of the textbook.)
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Now you should be ready for
the homework. But keep this
PPT at your finger tips, in case
you need to reference it.
14
```
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