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**

- 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 bywhere “n” is a finite value.

- 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**:

Sources:

(1) http://staff.cs.upt.ro/~todinca/cad/Lectures/cad_fuzzysets.pdf

(2) Principles of Soft Computing