Week 06 Set Theory Discrete Structures.pptx

(MA-216) - Discrete Structures
(Discrete Mathematics)
Spring 2025
Set Theory
Week- 06

Outline
Set
Application of Sets
Sets Notation
Set Membership
Cardinality of a Set
Types of Sets

• Databases
• Data-type or type in computer programming
• Constructing discrete structures
• Finite state machine
• Modeling computing machine
• Representing computational complexity of algorithms
Application of Sets

A set is an unordered collection of different elements. A set is usually denoted by a capital
letter, and its elements are listed within curly braces { }
Some Example of Sets
A set of all positive integers
A set of all the planets in the solar system
A set of all the states in USA
A set of all the lowercase letters of the alphabet
Sets

Set notation is a mathematical way to define and represent sets.
Roster Notation:
 The elements of a set are explicitly listed inside curly braces { }.
 Elements are separated by commas.
Example:
 Set of odd positive integer less than 10
O = {1,3,5,7,9}
Sets Notation

Set Builder Notation:
Instead of listing elements, it describes a property that all elements must satisfy.
Uses the format: O= {x ∈ 𝒁+ | x is odd and x<10}
where | (or :) means "such that."
Sets Notation (Cont.)

If an element is in a set, written as:
Example 1 −
V = {a, e, i, o, u} A = {x | x is a vowel in English alphabet}
• a is an element of the set V, denoted by a ∈ V.
• a is not an element of the set V, denoted by a ∉ V.
Set Membership

Example:
 I: {0, 1, 2, …,99}
50 ∈ I
100 ∉ I
 S: {a, 2, class}
2 ∈ S
room ∉ S
Set Membership (Cont.)

• Set of natural numbers
• 𝐍 = {1,2,3,…}
• Set of integers
• 𝐙 = {…,-2,-1,0,1,2,…}
• Set of positive integers
• 𝐙+ = {1,2,3,…}
• Set of rational numbers
• 𝐐 = {p/q | p ∈ 𝐙, q ∈ 𝐙, and q ≠ 0}
• Set of real numbers
• 𝐑
• Set of Whole Numbers
• W={0,1,2,3,4,5,…}
Important Sets

Cardinality of a set S, is the number of elements of the set and denoted by |S|. The
number is also referred as the cardinal number.
If a set has an infinite number of elements, its cardinality is ∞.
Example:
S = {1,4,3,5}
W = {1,2,3,4,5,…}
Cardinality:  | S | = 4,| W | = ∞
Cardinality of a Set

• Find cardinality of following sets.
• A = {x | x ∈ 𝐙+ , x is odd and x<10} A = {1,3,5,7,9}
|A| = 5
• B = Ø
|B| = 0
• C = {Ø}
|C| = 1
• R
R is infinite.
Cardinality of a Set (Cont.)

 Two sets A and B are equal if and only if they contain exactly the same elements,
regardless of the order.
 This is written as: A=B
 A and B are equal if and only if ∀x (x ∈ A ↔ x ∈ B).
"for every element x, if x is in A, then x must also be in B, and vice
versa."
Equality of Sets

Example 1:
Let:
A = {1,2,3}, B = {3,1,2}
Since both sets contain exactly the same
elements.
we say:
A=B
Equality of Sets (Cont.)
Example 2: Unequal Sets
A = {1,2,3}, B = {1,2,4}
Here: A contains 3, but B contains 4
instead.
Since they have different elements
we say:
A≠B

Sets can be classified into many types. Some of which are finite, infinite, subset,
universal, proper, singleton set, etc.
Finite Set
A set which contains a definite number of elements is called a finite set.
Example: S = { X | X ∈ N and 70 > X > 50}
Types of Sets

Infinite Set
A set which contains infinite number of elements is called an infinite set.
Example − S={x|x∈N and x >10}
Types of Sets (Cont.)

It is a collection of all elements in a particular context or application.
All the sets in that context or application are essentially subsets of this universal set.
Universal sets are represented as U.
Universal Set

Example − We may define U as:
The set of all animals on earth.
In this case, set of all mammals is a subset of U, set of all fishes is a subset of U, set of
all insects is a subset of U, and so on.
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}  A = {2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 9}
For the set of the vowels of the alphabet, U would be all
the letters of the alphabet
Universal Set (Cont.)

An empty set contains no elements. It is denoted by .
∅
As the number of elements in an empty set is finite, empty set is a finite set.
The cardinality of empty set or null set is zero.
Example:
S = ∅
S = {x|x∈N and 7<x<8}=∅
Empty Set or Null Set

Example:
• S = {x | x ∈ 𝑍+ and x < 0 } S = { } = Ø
• A set that has no elements called
empty set, or null set.
• Ø and {Ø}
• Ø ≠ {Ø}
Empty Set or Null Set (Cont.)

Singleton set or unit set contains only one element.
A singleton set is denoted by {s}.
Example:
S = {8}
S={x | x ∈ N, 7 < X < 9}
Singleton Set or Unit Set

• Set can contain other sets
• S = { {1}, {2}, {3} }
• T = { {1}, {{2}}, {{{3}}} }
• V = { { {1}, {{2}} }, { {{3}} }, { {1}, {{2}}, {{{3}}} } }
V has only 3 elements!
• Note that 1 ≠ {1} ≠ {{1}} ≠ {{{1}}}
• They are all different
Set Of Sets

• Let A and B be sets.
• A is a subset of B if and only if every element of A is also an element
of B.
• Denoted by A ⊆ B.
• A ⊆ B if and only if ∀x (x ∈ A → x ∈ B).
• A is a subset of B if and only if, for all x, if x is an element of A, then x is an
element of B
• ∀ set S, Ø ⊆ S, S ⊆ S
Subset

• A ⊆ B, ∀ x (x ∈ A → x ∈ B) and
• B ⊆ A, ∀ x (x ∈ B → x ∈ A)
then
• A = B, ∀ x (x ∈ A ↔ x ∈ B)
Subset and Equality

Example
• Q and R Q ⊆ R
• N and Z N ⊆ Z
• A = {x | x ∈ 𝐙+ and x<10}
B = {x | x ∈ 𝐙+ , x is even and x<10} B ⊆ A
Subset (Cont.)

Let A and B be sets.
• A is a proper subset of B if and only if A ⊆ B but A ≠B, denoted A ⊂ B.
• A ⊂ B if and only if ∀x (x ∈ A  x ∈ B) ˄ x (x ∈ B ˄ x ∉ A).
Proper Subset

• If S is a subset of T, and S is not equal to T, then S is a proper subset of T
• Let:
• T = {0, 1, 2, 3, 4, 5} and S = {1, 2, 3}
• S is not equal to T, and S is a subset of T
• Let Q = {4, 5, 6}. Q is neither a subset of T nor a proper subset of T
• The difference between “subset” and “proper subset” is like the difference
between “less than or equal to” and “less than” for numbers
Proper Subset (Cont.)

• Let S be a set.
• The power set of S is the set of all subsets of S, denoted by P(S).
• Example:
• S = {a, b}
• P(s) 
• P({ a ,b }) = {Ø ,{a} ,{b} ,{ a ,b }}
• Using:
The Power Set

• What is P({1,2,3})?
• Solution:
P({1,2,3}) = { Ø, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} }
• P(Ø) = ?
• P({Ø}) = ?
The Power Set (Cont.)

• Assume A is finite.
• |P(A)| = ?
Solution:
• A = {a}
• A = {a,b}
• A = {a,b,c}
P(A) = {Ø, {a}}
P(A) = {Ø, {a}, {b}, {a,b}}
|P(A)| = 2
|P(A)| = 4
P(A)={Ø,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}} |P(A)| = 8
• |P(A)| = 2|A|
The Cardinality of the Power Set

Two sets that have at least one common element are called overlapping sets. In case of
overlapping sets.
Example − Let, A={1,2,6} and B={6,12,42}.
There is a common element ‘6’, hence these sets are overlapping sets.
Overlapping Set

Two sets A and B are called disjoint sets if they do not have even one
element in common.
Example − Let, A={1,2,6} and B={7,9,14}
There is not a single common element, hence these sets are disjoint sets.
Disjoint Set

Venn diagram, invented in 1880 by John Venn, is a schematic diagram
that shows all possible logical relations between different
mathematical sets.
Venn Diagrams

Chapter Reading
Chapter # 2, Kenneth H. Rosen, Discrete Mathematics and Its Applications, Section 2.1
Exercise Questions
Question # 1,3,5,6,7,9,12,19,20,23,32,43,44

Week 06 Set Theory Discrete Structures.pptx

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