Fuzzy Logic | Set 2 (Classical and Fuzzy Sets)

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

  1. Classical set is a collection of distinct objects. For example, a set of students passing grades.
  2. Each individual entity in a set is called a member or an element of the set.
  3. 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.
  4. Let A is a given set. The membership function can be use to define a set A is given by:
  5. 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:
  6. Properties of classical sets: For two sets A and B and Universe X:
    • Commutativity:
    • Associativity:
    • Distributivity:
    • Idempotency:
    • Identity:
    • Transitivity:

Fuzzy set:

  1. 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].
  2. Partial membership exists when member of one fuzzy set can also be a part of other fuzzy sets in the same universe.
  3. The degree of membership or truth is not same as probability, fuzzy truth represents membership in vaguely defined sets.
  4. A fuzzy set A~ in the universe of discourse, U, can be defined as a set of ordered pairs and it is given by
  5. When the universe of discourse, U, is discrete and finite, fuzzy set A~ is given by
  6. where “n” is a finite value.

  7. Fuzzy sets also satisfy every property of classical sets.
  8. 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~ :

  9. Some other useful operations on Fuzzy set:
    • Algebraic sum:
    • Algebraic product:
    • Bounded sum:
    • Bounded difference:

Sources:
(1) http://staff.cs.upt.ro/~todinca/cad/Lectures/cad_fuzzysets.pdf
(2) Principles of Soft Computing



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