The document covers about fuzzy set in computational mathematics.

1. COMPUTATIONAL
MATHEMATICS

2. Table Of Contents
● Basic Fuzzy set operation
● Properties of fuzzy set
● Crisp relation

3. What is Fuzzy set?
This function can be generalized to such that the values assigned
to the elements of the universal set for within a specified a range
and indicate the membership grade of these element in the set in
question. Larger values denote higher degrees of set membership
such a function is called a membership function and set the
defined by it a fuzzy set.
The most commonly used to range of values of membership
functions is the unit interval [0,1]

4. A Fuzzy set is a set whose elements have degrees of membership.
Fuzzy sets are an extension of the classical notion of set (known
as a Crisp Set). More mathematically, a fuzzy set is a pair (A, µA)
where A is a set and µA : A → [0, 1]. For all x ∈ A, µA(x) is
called the grade of membership of x
Fuzzy set

5. ● Union
● Intersection
● Difference
● Complement
Let (A, µA) and (B, µB) be a fuzzy sets.
● Union: C = A ∪ B, where µC = max x∈X
{µA(x),
µB(x)}
● Intersection: C = A ∩ B, where µC = min x∈X
{µA(x),
µB(x)}
● Difference: C = A − B, where µC(x) = max x∈X
{0,
µA(x) − µB(x)}
● Complementation: (¬A, µ¬A), where µ¬A = 1 − µA
Basic Fuzzy Set
Operations

6. Union: C = A ∪ B, where µC = max x∈X
{µA(x), µB(x)}
A = {'a': 0.2, 'b': 0.3, 'c': 0.6, 'd': 0.6}
B = {'a': 0.9, 'b': 0.9, 'c': 0.4, 'd': 0.5}
● µC(a) = max{0.2, 0.9} = 0.9
● µC(b) = max{0.3, 0.9} = 0.9
● µC(c) = max{0.6, 0.4} = 0.6
● µC(d) = max{0.6, 0.4} = 0.6
A ∪ B = {'a': 0.9, 'b': 0.9, 'c': 0.6, 'd': 0.6}
UNION

7. Intersection: C = A ∩ B, where µC = min x∈X
{µA(x),
µB(x)}
A = {'a': 0.2, 'b': 0.3, 'c': 0.6, 'd': 0.6}
B = {'a': 0.9, 'b': 0.9, 'c': 0.4, 'd': 0.5}
● µC(a) = min{0.2, 0.9} = 0.2
● µC(b) = min{0.3, 0.9} = 0.3
● µC(c) = min{0.6, 0.4} = 0.4
● µC(d) = min{0.6, 0.5} = 0.5
A ∩ B = {'a': 0.2, 'b': 0.3, 'c': 0.4, 'd': 0.5}
INTERSECTION

8. Difference: C = A − B, where µC(x) = max x∈X
{0, µA(x)
−µB(x)}
A = {'a': 0.2, 'b': 0.3, 'c': 0.6, 'd': 0.6}
B = {'a': 0.9, 'b': 0.9, 'c': 0.4, 'd': 0.5}
● µC(a) = max{0, 0.2-0.9} = max{0, -0.7} = 0
● µC(b) = max{0,0.3-0.9} = max{0, -0.6} = 0
● µC(c) = max{0,0.6-0.4} = max{0, 0.2} = 0.2
● µC(d) = max{0,0.6-0.5} = max{0, 0.1} = 0.1
A - B = {'a': 0, 'b': 0, 'c': 0.2, 'd': 0.1}
DIFFERENCE

9. Complementation: (¬A, µ¬A), where µ¬A = 1 − µA
A = {'a': 0.2, 'b': 0.3, 'c': 0.6, 'd': 0.6}
● µ¬A(a) = 1-0.2 = 0.8
● µ¬A(b) = 1-0.3 = 0.7
● µ¬A(c) = 1-0.6 = 0.4
● µ¬A(d) = 1-0.6 = 0.4
¬A = {'a': 0.8, 'b': 0.7, 'c': 0.4, 'd': 0.4}
COMPLEMENT

10. ● Commutativity
○ A ∪ B = B ∪ A
○ A ∩ B = B ∩ A
● Associativity
○ A∪(B∪C) = (A∪B)∪C
○ A∩(B∩C) = (A∩B)∩C
● Distributivity
○ A∪(B∩C) = (A∪B)∩(A∪C)
○ A∩(B∪C) = (A∩B)∪(A∩C)
● Idempotency
○ A∪A = A
○ A∩A = A
Properties of fuzzy set

11. ● Identity
○ A∪Φ = A and A∪U = A
○ A∩Φ = Φ and A∩U = A
● Involution (Double Negation)
○ (A’)’ = A
● Transitivity
○ If A⊆B⊆C then, A⊆C
● De Morgan’s Law
○ (A ∪ B)' = A' ∩ B'
○ (A ∩ B)' = A' ∪ B'
Properties of fuzzy set

12. Crisp relation is defined over the cartesian product of two crisp
sets. Suppose, A and B are two crisp sets. Then Cartesian
product denoted as A×B is a collection of ordered pairs, such
that
A × B = { (a, b) | a ∈ A and b ∈ B }
Example:
Let’s consider two sets: 𝐴 = { 1 , 2 , 3 } A={1,2,3} 𝐵 = { 𝑎 , 𝑏 , 𝑐 }
B={a,b,c}
A crisp relation 𝑅 R could be a subset of 𝐴 × 𝐵 such as:
𝑅 = { ( 1 , 𝑎 ) , ( 2 , 𝑏 ) }
This means:
The relation holds between 1 and 𝑎. The relation holds between 2 and 𝑏.
There is no relation between 1 and 𝑏, or between 2 and 𝑎, and so on.
Crisp Relation

13. THANK YOU