Classical and Fuzzy Set Theory

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To begin this article, First let me tell you the difference between the Classical Set theory and Fuzzy Set Theory.

Classical Set Theory

Fuzzy Set Theory

1. This theory is a class of those sets having sharp boundaries.

1. This theory is a class of those sets having un-sharp boundaries.

2. This set theory is defined by exact boundaries only 0 and 1

2. This set theory is defined by ambiguous boundaries.

3. In this theory, there is no uncertainty about the boundary's location of a set.

3. In this theory, there always exists uncertainty about the boundary's location of a set.

4. This theory is widely used in the design of digital systems.

4. It is mainly used for fuzzy controllers.

Set

 A set is a term, which is a collection of unordered or ordered elements.
Following are the various examples of a set:
  1.  A set of all-natural numbers
  2. A set of students in a class.
  3. A set of all cities in a state.
  4. A set of upper-case letters of the alphabet.

Types of Set:

There are following various categories of set:
  1. Finite
  2. Empty
  3. Infinite
  4. Proper
  5. Universal
  6. Subset
  7. Singleton
  8. Equivalent Set
  9. Disjoint Set

Classical Set

It is a type of set which collects the distinct objects in a group. The sets with the
crisp boundaries are classical sets. In any set, each single entity is called an
element or member of that set.

Mathematical Representation of Sets

Any set can be easily denoted in the following two different ways:

1) Roaster Form: This is also called as a tabular form. In this form, the set is represented in the following way:

Set_name = { element1, element2, element3, ......, element N}
The elements in the set are enclosed within the brackets and separated by the
commas. Following are the two examples which describes the set in Roaster or Tabular
form:

Example 1: Set of Natural Numbers: N={1, 2, 3, 4, 5, 6, 7, ......,n).
Example 2: Set of Prime Numbers less than 50: X={2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}.
 
2) Set Builder Form: Set Builder form defines a set with the common properties of an element in a set. In this form, the set is represented in the following way:

A = {x:p(x)}

The following example describes the set in the builder form:

The set {2, 4, 6, 8, 10, 12, 14, 16, 18} is written as: B = {x:2 ≤ x < 20 and (x%2) = 0}


Operations on Classical Set

Following are the various operations which are performed on the classical sets:

  1. Union Operation
  2. Intersection Operation
  3. Difference Operation
  4. Complement Operation

1. Union:

This operation is denoted by (A U B). A U B is the set of those elements which
exist in two different sets A and B. This operation combines all the elements from both the sets and make a new set. It is also called a Logical OR operation.
It can be described as:

A ∪ B = { x | x ∈ A OR x ∈ B }.

Example: Set A = {10, 11, 12, 13}, Set B = {11, 12, 13, 14, 15}, then 
A ∪ B = {10, 11, 12, 13, 14, 15}
 

2. Intersection:

This operation is denoted by (A ∩ B). A ∩ B is the set of those elements which are common in both set A and B. It is also called a Logical OR operation.
It can be described as:

A ∩ B = { x | x ∈ A AND x ∈ B }.

Example: Set A = {10, 11, 12, 13}, Set B = {11, 12, 14} then
 A ∩ B = {11, 12}

3. Difference Operation

This operation is denoted by (A - B). A-B is the set of only those elements which
exist only in set A but not in set B.
It can be described as:

A - B = { x | x ∈ A AND x ∉ B }.

4. Complement Operation: 

This operation is denoted by (A`). It is applied on a single set. A` is the set of elements which do not exist in set A.
It can be described as:

A′ = {x|x ∉ A}.

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