SETS
Lecture
by
JK Gondwe
Aug-11
prob-1
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OBJECTIVES
1. Define terms
2. Describe a set using standard forms of
notation.
3. Learn to use set notation
4. Identify relationships between sets.
5. Perform operations on sets
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2
Basic Set Theory Definitions
• A set is a collection of elements
• An element is an object contained in a set
• Notation: ∈ means “is an element of”
∉ means “is not an element of”
• Upper case designates set name
• Lower case designates set elements
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STANDARD FORMS OF SET NOTATION
Sets can be defined in two ways:
Roster notation - listing all of the elements.
Set-builder notation- defining the rules for
membership
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4
Set Theory
Roster notation
{1, 2, 3, 4, 5, 6, 7, 8,9}
Roster notation lists all of the elements (inside curly brackets.
Aug-11
Course 2
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5
Set Theory
Set-builder
{x | x is a positive even integer < 10}
Set-builder notation
gives a rule.
Read the notation {x|x is a positive integer <10} as “the set
of all x such that x is a positive integer <10.”
Aug-11
Course 2
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6
Examples
A = {1, 2, 3, 4}
1∈A
2∈A
6∉A
z∉A
B = {x | x is an even number ≤ 10}
2∈B
4∈B
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9∉B
z∉B
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Definitions cont’d
• If every element of Set A is also
contained in Set B, then Set A is a subset
of Set B
• A is a proper subset of B if B has more
elements than A does
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Subsets notation
• Note: a subset exists when a set’s members
are also contained in another set
A
B
notation:
⊆ means “is a subset of”
⊂ means “is a proper subset of”
⊄ means “is not a subset of”
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Universal Set
Defn:
The set of all elements relevant to
a given discussion and is designated
by the symbol U
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Universal Set- Example
The universal set is all COM students,
each student is an element in the
universal set.
Some subsets are:
- MBBS, Pharmacy, physiotherapy
MLS
The universal set is a deck of ordinary
playing cards, each card is an element
in the universal set.
Some subsets are:
– face cards, numbered cards,
suits
Aug-11
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Subset Relationships
• A = {x | x is a positive integer ≤ 8}
set A contains: 1, 2, 3, 4, 5, 6, 7, 8
• B = {x | x is a positive even integer < 10}
set B contains: 2, 4, 6, 8
• C = {2, 4, 6, 8, 10}
set C contains: 2, 4, 6, 8, 10
• Subset Relationships (True or False)
A⊆A
A⊄B
A⊄C
B⊂A
B⊆B
B⊂C
C⊄A
C⊄B
C⊆C
Aug-11
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Set Equality
Two sets are equal if and only if they contain precisely the
same elements i.e. if A and B are two sets then A=B if every
member of A is a member of B and every member of B is a
member of A
• The order in which the elements are listed is unimportant.
• Repetition of elements is irrelevant.
Example:
If A = {1, 2, 3, 4}
A = B and B = A but
Aug-11
B = {1, 4, 2, 3} C = {1, 2, 3}
A ≠ C and B ≠ C
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Cardinality of Sets
• Cardinality refers to the number of
elements in a set
• A finite set has a countable number of
elements
• An infinite set has at least as many
elements as the set of natural numbers
• notation: |A| or n(A) represents the
cardinality of Set A
Aug-11
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Finite Set Cardinality
Set Definition
Cardinality
A = {x | x is a lower case letter}
|A| = 26
B = {2, 3, 4, 5, 6, 7}
|B| = 6
C = {x | x is an even number < 10}
|C|= 4
D = {x | x is an even number ≤ 10}
|D| = 5
Aug-11
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Infinite Set Cardinality
Set Definition
Cardinality
A = {1, 2, 3, …}
|A| = ∞
B = {x | x is a point on a line} |B| =
∞
C = {x| x is a point in a plane} |C| = ∞
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The Empty Set
• Any set that contains no elements is called the
empty set
• the empty set is a subset of every set
including itself
• notation: { } or φ
Examples ~ both A and B are empty
A = {x | x is a positive number < 0 }
B = {x | x is a MBBS MLS student}
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Set Theory Notation
Symbol
Meaning
Upper case
Lower case
{ }
designates set name
designates set elements
enclose elements in set
∈ or ∉
⊆
⊂
⊄
| or :
| | or n(A)
Aug-11
is (or is not) an element of
is a subset of (includes equal sets)
is a proper subset of
is not a subset of
such that (if a condition is true)
the cardinality of a set
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The Power Set ( P )
• The power set is the set of all subsets that can
be created from a given set
• The cardinality of the power set is 2 to the
power of the given set’s cardinality
• notation: P (set name)
Example:
A = {a, b, c}
where |A| = 3
P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, φ}
and |P (A)| = 8
In general, if |A| = n, then |P (A) | = 2n =23
Aug-11
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Special Sets
• Z represents the set of integers
– Z+ is the set of positive integers and
– Z- is the set of negative integers
• N represents the set of natural numbers
• ℝ represents the set of real numbers
• Q represents the set of rational
numbers
Aug-11
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Natural Numbers
• Counting numbers
– Begin with 1
– Each successive number is found by adding 1 to
the previous number
– {1,2,3,4,5,6,7,…}
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Whole Numbers
• The set of whole numbers is almost the same
as the set of natural numbers, the difference is
the addition of the number 0.
• Each successive number is found by adding 1
to the previous number.
• {0,1,2,3,4,5,…}
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Integers
• The integers consist of the set of whole
numbers and their opposites.
• Opposite=Additive inverse
– Example: The opposite of 3 is -3
• {…,-4,-3,-2,-1,0,1,2,3,4,…}
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Rational Numbers
• The set of fractions, repeating decimals, and
terminating decimals.
• Any number that can be represented as a
fraction with an integer numerator and
integer denominator.
• Cannot use roster notation
•
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Irrational Numbers
• The set of all non-terminating, non-repeating
decimals.
• Numbers cannot be expressed as fractions with
integer numerators and integer denominators.
• Examples:
–
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0.13482472454...,0.1212212221..., 2 , π , e
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Real Numbers
• The set of all natural, whole, rational, and
irrational numbers.
• Includes all of the sets previously described.
•
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Real Numbers
Real Numbers
Rational Numbers
Integers
Whole Numbers
Natural Numbers
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Irrational Numbers
Every set is contained in the set above, this
means that all of the natural numbers are
in the whole numbers, all of the whole
numbers are in the integers, all of the
integers are in the rational numbers and all
of the rational numbers are in the real
numbers. Additionally all of the irrational
numbers are in the real numbers.
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COMPLEMENT OF A SET
The rectangle represents the
universal set, U, while the portion
bounded by the circle represents set
A.
A
U
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Cont’d
The colored region inside U and outside the circle is
labeled A' (read “A prime”). This set, called the
complement of A, contains all elements that are
contained in U but not in A.
A′
A
U
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Complement of a Set
For any set A within the universal set U,
the complement of A, written A', is the set
of all elements of U that are not elements
of A. That is
A′ = {x | x ∈ U and x ∉ A}.
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INTERSECTION OF SETS
Notation: Intersection (
I ) ( and )
The elements which are common to both
sets.
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INTERSECTION OF SETS -example
A = { 1, 2, 3 ,4, 5}
B = { 2, 4, 6, 8, 10}
C = { 5, 6, 7, 8}
1) A I B ={ 2, 4}
4) (A I B) I C
{ 2, 4} I { 5, 6, 7, 8}
{ } or “empty set”
2) B I C ={ 6, 8}
3) A I C ={ 5 }
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or O
or “null set”
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UNION OF SETS
Notation: Union ( U ) ( or )
- the combined set of elements from
two sets with no duplication of elements.
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UNION OF SETS -example
Given A = { 1, 2, 3 ,4, 5} B = { 2, 4, 6, 8, 10}
C = { 5, 6, 7, 8}
1) A U B = { 1, 2, 3, 4, 5, 6, 8, 10}
2) B U C = { 2, 4, 5, 6, 7, 8, 10}
3) A U C = { 1, 2, 3, 4, 5, 6, 7, 8}
Aug-11
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Example 1
Set Operations with Three Sets
• Given
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4, 5}
B = {1, 2, 3, 6, 8}
C = {2, 3, 4, 6, 7}
• Find A ∩ (B U C’)
• Solution
Find C’
= {1, 5, 8, 9}
• Find (B U C’)
= {1, 2, 3, 6, 8} U {1, 5, 8, 9}
= {1, 2, 3, 5, 6, 8, 9}
• Find A ∩ (B U C’)
= {1,2,3,4, 5} ∩{1, 2, 3, 5, 6, 8, 9}
= {1, 2, 3, 5}
Aug-11
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