Transcript Sets and Operations

Sets and Operations
TSWBAT apply Venn diagrams in problem
solving; use roster and set-builder
notation; find the complement of a set;
apply the set operations of intersection and
union; identify a disjoint set
Think about it – How would you go
about finding out this answer?

Of 250 people surveyed, 175 have
brown hair, 110 have brown eyes,
and 85 have both brown hair and
brown eyes. How many of these
people have neither brown hair nor
brown eyes?
Venn Diagrams


Venn diagrams show relationships
between sets
Set


A collection of objects, called elements
Elements
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

Numbers, letters, or physical objects
They are written between braces {}
Two sets are equal ONLY if they have
the exact same elements
Drawing Venn Diagrams

Universe



The set of all elements being
considered
Illustrated by a rectangle
Subsets


A set contained within a set
Denoted by the symbol
Illustrated as circles within the
rectangle
U

Use a Venn diagram
175-85
90
Subset A – Brown hair
B U
Contains 110 elements
U
U
A
U
Contains 175 elements
85
110-85
25
Subset B – Brown eyes
50
This blue rectangle is the universe
U = 250
Different Notations

Roster notation

List the elements of a set in brackets

Example



The set of whole numbers greater than 10
{11, 12, 13, 14…}
… or a ellipsis is used to show that the set
continues without end following the
established pattern
Different Notations

Set-builder notation

Describes the set by stating the
properties that its elements must
satisfy

Example



The set of whole numbers greater than 10
{x|x is a whole number and x > 10}
Read as: the set of all x such that x is a
whole number and x is greater than 10
Symbols used

Є



Read as “is an element of”, or “belongs
to”
50 Є {x|x is a whole number and
x>10}
Є


Read as “is not an element of” or “does
not belong to”
7 Є {x|x is a whole number and x>10}
Empty Sets


A set that has no elements is called
and empty set or a null set
It can be denoted two ways



{}
Ø
Do NOT write 0, zero is a number
and empty set does not contain
zero as an element
Examples

List the elements of each set

{q|q is a whole number and q<0}


{t|t is an integer and t<0}
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
{…, -4, -3, -2, -1}
{v|v Є the set of integers and v>0}


{}
{21, 22, 23…}
{n|n Є the set of natural numbers and
n<0}

Ø
Classwork


Complete numbers 1-4 on page 19
of your workbook
Turn it in when you are finished
Complements

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When a subset is defined, its
complement is also defined
The complement is denoted by the’
symbol



Set A
Complement is A’
Complement

The set of all elements in the universe
(U) but do not belong to the set
Examples

If U = {0, 1, 2, 3, 4 ] and
A = {0, 1, 3}, what is the
complement of A?

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U = {integers}, A = {0}

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A’ = {2, 4}
{… -3, -2, -1, 1, 2, 3…}
U = {real numbers}, A = Ø

A’ = {real numbers}
Example
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U = {1, 2, 3}, A = {1, 2, 3}

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A’ = {}
In your workbook complete page
19, 5-10
Operations with sets
Intersection of sets
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The elements BOTH sets contain
Denoted with the symbol
U
A
U

U

B
Example of Intersection
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A = {1, 2, 3, 4, 5}
B = {2, 5, 6, 7}

A
U

B = {2, 5}
Operations with Sets
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Union of Sets


All elements that are in either or both sets
Denoted by the symbol U
U
A
U
U
B
Example of Union
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A = {1, 2, 3, 4, 5}
B = {2, 5, 6, 7}

A U B = 1, 2, 3, 4, 5, 6, 7}
Examples
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7
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4
15
11
B
9
8
2
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0
3
C
U
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20
U
U
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16
U
A
AUB
A C
BUC
A B C
A (B U C)
A U (B
C)
U
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Classwork
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Complete WB page 20, 11-18