Prerequisite : Fuzzy Logic | Introduction
In this post, we will discuss classical sets and fuzzy sets, their properties and operations that can be applied on them.
Set: A set is defined as a collection of objects, which share certain characteristics.
Classical set
- Classical set is a collection of distinct objects. For example, a set of students passing grades.
- Each individual entity in a set is called a member or an element of the set.
- The classical set is defined in such a way that the universe of discourse is spitted into two groups members and non-members. Hence, In case classical sets, no partial membership exists.
- Let A is a given set. The membership function can be use to define a set A is given by:
- Operations on classical sets: For two sets A and B and Universe X:
- Union:
This operation is also called logical OR. - Intersection:
This operation is also called logical AND. - Complement:
- Difference:
- Properties of classical sets: For two sets A and B and Universe X:
- Commutativity:
- Associativity:
- Distributivity:
- Idempotency:
- Identity:
- Transitivity:
Fuzzy set:
- Fuzzy set is a set having degrees of membership between 1 and 0. Fuzzy sets are represented with tilde character(~). For example, Number of cars following traffic signals at a particular time out of all cars present will have membership value between [0,1].
- Partial membership exists when member of one fuzzy set can also be a part of other fuzzy sets in the same universe.
- The degree of membership or truth is not same as probability, fuzzy truth represents membership in vaguely defined sets.
- A fuzzy set A~ in the universe of discourse, U, can be defined as a set of ordered pairs and it is given by
- When the universe of discourse, U, is discrete and finite, fuzzy set A~ is given by
- Fuzzy sets also satisfy every property of classical sets.
- Common Operations on fuzzy sets: Given two Fuzzy sets A~ and B~
- Union : Fuzzy set C~ is union of Fuzzy sets A~ and B~ :
- Intersection: Fuzzy set D~ is intersection of Fuzzy sets A~ and B~ :
- Complement: Fuzzy set E~ is complement of Fuzzy set A~ :
- Some other useful operations on Fuzzy set:
- Algebraic sum:
- Algebraic product:
- Bounded sum:
- Bounded difference:
where “n” is a finite value.
Sources:
(1) http://staff.cs.upt.ro/~todinca/cad/Lectures/cad_fuzzysets.pdf
(2) Principles of Soft Computing
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