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Set Notation
Set- is a collection or aggregate of definite,
distinct objects.
 A well-defined set means that it is possible to
determine whether an object belongs to a
given set.
 Elements of a Set- the objects or members
of a set.

• Symbol:
ϵ (epsilon)- use to denote the element of a
set. Ex: r ϵ A
ϵ – use to denote that an element is not an
element of the given set. Ex: b ϵ A
Set Notation
Ex. of well-defined sets (defined)
1. Set of ace cards
2. Colors of the rainbow
3. Days of the week
Ex. of not well-defined sets (undefined)
1. Set of cards
2. Set of books
3. Set of beautiful women in Asia
Set Notation

Different symbols are used when dealing
with sets:
1. A pair of braces { } – is used to
represent the idea of a set.
2. Capital letters of the English alphabet
– are used to name sets.
Example: A = { a, b, c, d, e }
B = { 2, 4, 6, 8, 10 }

Letters with subscripts can also be used,
e.g. A1, A2, A3,…etc.
Methods of Listing the Elements of a Set
1. Roster or Tabular Method – listing all
the elements, enclosed it in braces and
separated by comma.
Ex:
C = { a, b, c }
B = { 1, 2, 3, 4, 5 }
2. Rule Method – a conditional way of
listing method by writing and
description using a particular variable.
3. Set Builder Notation – a modification
of the rule method.
Methods of Listing the Elements of a Set

Examples:
Rule
Set-builder
1. A = { counting numbers less than 5}
A = { x/x is a counting number less than
5}
2. B = { days of the week that begin with
letter S }
B = { d/d is a day of the week beginning
with letter S }

The symbol ( / ) means “wherein” or “such
that.”
Cardinality
Refers to the number of elements
contained in a set.
Symbol: n(A) or |A|
 Examples:
1. A = {a, b, c, d}
Ans: |A| = 4
2. Z = {x/x is a set of integers}

Ans: n(Z) = ∞ or |Z| = ∞
Kinds of Sets
Subset
Subset – it is a part of a given set. Let A and B
be sets. B is a subset of A if each element of B
is an element of A.
In symbols: B ⊂ A or B ⊆ A read as “B is a
subset of A.”
 Two kinds of subset:
1. Proper Subset – a part of a set, symbol ( ⊂ )
2. Improper (Strict) Subset – the given set is
equal to that set, symbol ( ⊆ )

Subset
Example:
Let P = {5, 6, 7, 8, 9, 10}
Q = {5, 7, 9}
R = {5, 9, 10}
S = {1, 2}
True or False
1. Q ⊆ P
2. R ⊂ P
3. P ⊂ Q
4. S ⊂ P
5. R ⊂ Q
6. Q ⊆ Q
Super set
If B ⊆ A, then A is a super set of B.
In symbols: A ⊇ B read as “A is a
superset of B.”
Examples:
A
1. A ⊇ B

B
2. R ⊇ Q ⊇ Z
Kinds of Sets


Empty or Null Set – sets having no
elements. Symbol: { } or ∅
Note: An empty set is a subset of any
set.
Universal Set (U) – also called the
general set, is the sum of all sets or the
totality of elements under consideration
or a particular discussion.
Example:
U = {a, b, c, d, x, y, z}
A = {a, b, c, d} B = {x, y, z}
Kinds of Sets



Unit Set – set having only one element.
Ex: D = {y/y is a day of the week that
begins with letter M}
Finite Set – sets having a limited or
countable number of elements.
Ex: set of counting numbers less than 5
set of colors of the rainbow
Infinite Set – sets having an unlimited or
uncountable number of elements.
Ex: set of counting numbers
Kinds of Sets


Equal Sets – sets having the same
elements. Symbol: (=)
Note: Two sets A and B are said to be
equal if and only if A ⊆ B and B ⊆ A.
Ex: A = {2, 4, 6} and B = {4, 6, 2}
A = B and B = A
Equivalent Sets – sets having the same
number of elements or cardinality, that is
|A| = |B| . Symbol: (~)
Ex: X = {1, 2, 3} Y = {a, b, c}
|X| = |Y|, so X ~ Y and Y ~ X
Kinds of Sets


Joint Sets – sets that have elements in
common.
Ex: M = {5, 6, 7, 8} and N = {4, 6, 7, 9}
are joint sets.
Disjoint Sets – sets that have no
elements in common. They are mutually
exclusive.
Ex: C = {c, a, t} and D = {d, o, g} are
disjoint sets or mutually exclusive.
Power Set
If A is a set, the power set of A is the set of
all subsets of A denoted by:
P(A) = {X/X ⊆ A}
 Number of Subsets of a Given Set:
- If a set contains n number of elements then
the number of subsets is 2ⁿ

Examples:
1. What is the power set of O = {a, b, c}?
2. If B = {1, 2}, describe P(B).
3. What is P(ø)?
Venn Diagram
It is a graphical representation, usually
circular in nature. It is one way of showing
the relationships of two or more sets by the
use of pictures.
 This method was developed by John Venn
(1834-1923) thus, the name Venn
Diagram.
 It consists of a rectangle representing the
universal set and circles that represent the
sets. Sometimes, circles can also represent
the universal set.

Venn Diagram
A
B
a, b, c
d, e, f
Set Operations
1.
Union – it shows the unity of two or more
sets. It is the joining of sets. ( ∪ )
In symbols: A ∪ B = {x/x ϵ A v x ϵ B}
Examples: Find the union of sets and
draw Venn diagrams
a) A = {1, 2, 3}
B = {4, 5, 6}
b) S = {a, b, c, d}
T = {c, d, m, n, o}
Set Operations
2.
Intersection – it shows the intersection
of the common elements of sets. ( ∩ )
In symbols: A ∩ B = {x/x ϵ A ʌ x ϵ B}
Example: Find the intersection of sets
and draw the Venn diagram.
a) B = {m, o, p, q} C = {m, p, r, s}
b) F = {1, 3, 5, 7}
G = {2, 4, 6, 8}
Set Operations
3.
Complement of a Set – it is the set
whose elements are in the universal set
but not in a set or a given set. ( ʼ )
In symbols: A’ = {x ϵ U / x ϵ A}
Ex: Let U be the universal set.
U = {2, 4, 6, 8,10}
A = {6, 8, 10} B = {2, 4, 6}
Find: a) A’
b) B’
Set Operations
4.
5.
Relative Difference – set of elements found in
a set but not belong or found in other set. ( – )
In symbols: A – B = {x/x ϵ A ʌ x ϵ B}
Ex: A = {1, 2, 3, 4, 5} B = {3, 4, 5, 6, 7}
Find: a) A – B b) B – A
Symmetric Difference – the symmetric
difference of A and B is denoted by:
A  B = (A – B) U (B – A)
Ex: A = {a, e, i, o, u} B = {e, o, n, s}
Find: A  B
Set Operations
6.
Cartesian Product – For any sets A and
B, the Cartesian product A x B is the set
of all ordered pairs (a, b) where a ϵ A and
b ϵ B. A x B is defined by:
A x B = {(a, b) / a ϵ A ʌ b ϵ B}
Examples:
Let A = {0, 1} B = {x, y, z}
Find: a) A x B
b) B x A
c) A2
Exercises
Given: Let U be the universal set
U = {a, b, c, d, e, f, g, h}
A = (a, b, c}
C = {a, g, h}
B = {d, e, f}
D = { b, c, d, e, f}
Find the ff. and draw Venn diagrams.
1) A’ ∩ B’
6) P(A)
2) (A U B)’
7) B x C
3) (C’ ∩ D) U A
8) A – C
4) (A ∩ B)’
9) D – B
5) A’ U B’
10) A  D
Applications of Set Theory
Examples
1. In a class, 15 are taking English, 20 are taking
Filipino and 10 are taking both English and Filipino.
How many students are there in all?
2. Of 1000 applicants for a mountain-climbing trip in
the Himalayas, 450 get altitude sickness, 622 are
not in good enough shape, and 30 have allergies.
An applicant qualifies if and only if this applicant
does not get altitude sickness, is in good shape,
and does not have allergies. If there are 111
applicants who get altitude sickness and are not in
good enough shape, 14 who get altitude sickness
and have allergies, 18 who are not in good enough
shape and have allergies, and 9 who get altitude
sickness, are not in good enough shape, and have
allergies, how many applicants qualify?
Do Worksheet 6