Introduction to Fuzzy Sets

Fuzzy logic is a superset of Boolean (conventional) logic that handles the concept of partial truth, which is truth values between "completely true" and "completely false". This section of the fuzzy logic page describes the basics of fuzzy sets.

• Basic Definition of Fuzzy Set
• Similarities and Differences of Fuzzy Sets with Traditional Set Theory
• Examples Illustrating the Concepts of Fuzzy Sets
• Logical Operation on Fuzzy Sets
• Hedging

Fuzzy Sets and Traditional Sets

A fuzzy set is a set whose elements have degrees of membership. That is, a member of a set can be full member (100% membership status) or a partial member (eg. less than 100% membership and greater than 0% membership). To fully understand fuzzy sets, one must first understand traditional sets.

A traditional or crisp set can formally be defined as the following:

• A subset U of a set S is a mapping from the elements of S to the elements of the set {0,1}. This is represented by the notation:
  U: S-> {0,1}
• The mapping is represented by one ordered pair for each element S where the first element is from the set S and the second element is from the set {0,1}. The value zero represents non-membership, while the value one represents membership.

Essentially this says that an element of the set S is either a member or a non-member of the subset U. There are no partial members in traditional sets.

Here is an example of a traditional set:
Consider a set X that contains all the real numbers between 0 and 10 and a subset A of the set X that contains all the real numbers between 5 and 8. Subset A is represented in the figure below.

In the figure, the interval on the x-axis between 5 and 8 has y-value of one. This indicates that any number in this interval is a member of the subset A. Any number that has a y-value of zero is considered to be a non-member of the subset A.

Now that you have a clearer understanding of traditional sets, let's get to the fun stuff! Below is the explanation of fuzzy sets:

Again a fuzzy set is a set whose elements have degrees of membership. These can formally be defined as the following:

• A fuzzy subset F of a set S can be defined as a set of ordered pairs. The first element of the ordered pair is from the set S, and the second element from the ordered pair is from the interval [0,1].
• The value zero is used to represent non-membership; the value one is used to represent complete membership, and the values in between are used to represent degrees of membership.

Here is an example describing a set of young people using fuzzy sets. In general, young people range from the age of 0 to 20. But, if we use this strict interval to define young people, then a person on his 20th birthday is still young (still a member of the set). But on the day after his 20th birthday, this person is now old (not a member of the young set).

How can one remedy this?

By RELAXING the boundary between the strict separation of young and old. This separation can easily be relaxed by considering the boundary between young and old as "fuzzy". The figure below graphically illustrates a fuzzy set of young and old people.

Notice in the figure that people whose ages are >= zero and <= 20 are complete members of the young set (that is, they have a membership value of one). Also note that people whose ages are > 20 and < 30 are partial members of the young set. For example, a person who is 25 would be young to the degree of 0.5. Finally people whose ages are >= 30 are non-members of the young set.








EXAMPLE 2

Here is an application demonstrating fuzzy membership: Consider the colorwheel. Remember that there are 3 primary colors: Red, Yellow, and Blue. These colors, separately, represent crisp sets. For example, true red is a non-member of true blue and of true yellow; true blue is a non-member of true yellow and of true red; yellow is a non-member of true red and of true blue. There is a crisp boundary between these primary colors.

But, as we also learned in elementary school, it is possible to mix these colors with varying amounts of the true colors resulting in different shades of non-true colors. For example, mixing true red with true blue in equal portions of each will result in violet with a membership degree of 0.5 in true red and 0.5 in true blue. Different amounts of true red and true blue will result in varied membership values for the violet. The different violets represent the fuzzy boundaries between true red and true blue!
Click here to try it for yourself!

Membership Functions

A membership function is a mathematical function which defines the degree of an element's membership in a fuzzy set.

The best way to illustrate this concept is with an example. This example describes a fuzzy set for tallness. Below in the membership function for tallness.

	tall(x)= {         0, 	        if height(x) < 5ft,
	  	   (height(x)-5ft)/2, 	if 5ft <= height(x) <= 7ft,
                           1, 	        if height(x) > 7ft
		 }			

Essentially this function calculates the membership value of a certain height. For example, if a person is less 4'9", then this person has a membership value of 0.0 and thus is not a member of the set tall. If a person is 7'6", then this person has a membership value of 1.0 and thus is a member of the set tall. Finally, if a person is 5'5", then this person has a membership value of 0.21 and is a partial member of the set tall.

Below is a graphical representation of the fuzzy set for tallness.

Now that we understand what fuzzy sets and membership functions are, we can discuss three basic operation on sets: negation, intersection, and union of fuzzy sets. In L.A. Zadeh first paper, he formally defined these operations in the following mann er:

• Negation
  • membership_value(not x)= 1- membership_value(x)
    where x is the fuzzy set being negated
• Intersection
  • membership_value(x and y) = minimum( membership_value(x), membership_value(y) )
    where x and y are the fuzzy set being negated
• Union
  • membership_value(x or y) = maximum (membership_value(x), membership_value(y) )
    where x and y are the fuzzy set being negated

These concepts are illustrated below. The definitions will also be explained below.

Negation

In this figure, the red line is a fuzzy set. To negate this fuzzy set, subtract the membership value in the fuzzy set from one. For example, the membership value at 5 is one. In the negation, the membership value at 5 would be zero (1-1=0). For exa mple, if the membership value is 0.4. In the negation, the membership value would be 0.6 (1-0.4=0.6).

Put the mouse over the image to see the negation of the fuzzy set (blue curve).






Intersection

In this figure, the red and green lines are fuzzy sets. To find the intersection of these sets take the minimum of the two membership values at each point on the x-axis (see the formal definition above). For example, in the figure the red fuzzy set has a membership of ZERO when x = 4 and the green fuzzy set has a membership of ONE when x = 4. The intersection would have a membership value of ZERO when x = 4 because the minimum of zero and one is zero.

Put the mouseover the image to see the intersection of the fuzzy sets (blue curve).





Union

In this figure, the red and green lines are fuzzy sets. To find the union of these sets take the maximum of the two membership values at each point on the x-axis (see the formal definition above). For example, in the figure the red fuzzy set has a membership of ZERO when x = 4 and the green fuzzy set has a membership of ONE when x = 4. The union would have a membership value of ONE when x = 4 because the maximum of zero and one is one.

Put the mouse over the image to see the union of the fuzzy sets (blue curve).

Here is another activity demonstrating the operations on fuzzy set.

The above information was collected from numerous sources, but some of the definitions and images were found at Bibliography #2, #4, #6, & #11 on our References page.

Much has been made about the relationship of Fuzzy Logic to the human thought process and the ability to handle imprecise conditions that may arise. One of the terms frequently seen in the Fuzzy Logic literature is the concept of Hedging.

Hedging can be described as the modifiers to a certain set, much like the way adjectives and adverbs modify statements in the English language.

When referring to a fuzzy set, hedges are used to adjust the characteristics of that fuzzy set by either:

• Approximating
• Complementing
• Diluting
• Intensifying

Some specific words and their effect on the fuzzy set include:

Key Word	Effect on set characteristics
about near close to approximately	Approximate the set
not	Complement the set
somewhat rather quite	Dilute the set
very extremely	Intensify the set

In general, when a hedge is used to dilute a set, the set is expanded. When a set is intensified with a hedge, the set is compressed.

For more information on hedging, check out Reference #3 on our References page.