# You fuzzyin' with me ?

"So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality." -Albert Einstein, Geometry and Experience

[Abstract]

[Fuzzy what ??!! ] [Boolean vs. Fuzzy] [Fuzzy Subset Theory] [Fuzzy Operations]

The third statement hence, define Boolean logic as a subset of Fuzzy logic.

A subset U of a set S can be defined as a set of ordered pairs, each with a first element that is an element of the set S, and a second element that is an element of the set { 0, 1 }, with exactly one ordered pair present for each element of S. This defines a mapping between elements of S and elements of the set { 0, 1 }. The value zero is used to represent non-membership, and the value one is used to represent membership. The truth or falsity of the statement

x is in U

is determined by finding the ordered pair whose first element is x. The statement is true if the second element of the ordered pair is 1, and the statement is false if it is 0.

Similarly, a fuzzy subset F of a set S can be defined as a set of ordered pairs, each with a first element that is an element of the set S, and a second element that is a value in the interval [ 0, 1 ], with exactly one ordered pair present for each element of S. This defines a mapping between elements of the set S and values in the interval [ 0, 1 ]. The value zero is used to represent complete non-membership, the value one is used to represent complete membership, and values in between are used to represent intermediate degrees of membership. The set S is referred to as the universe of discourse for the fuzzy subset F. Frequently, the mapping is described as a function, the membership function of F. The degree to which the statement

x is in F

is true is determined by finding the ordered pair whose first element is x. The degree of truth of the statement is the second element of the ordered pair.

This can be illustrated with an example. Let's talk about people and "youthness". In this case the set S (the universe of discourse) is the set of people. A fuzzy subset YOUNG is also defined, which answers the question "to what degree is person x young?" To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset YOUNG. The easiest way to do this is with a membership function based on the person's age.

young(x) = { 1, if age(x) <= 20.,

(30-age(x))/10, if 20 < age(x) <= 30,

0, if age(x) > 30 }

A graph of this looks like:

Given this definition, here are some example values:

Person    Age    degree of youth
--------------------------------------
Johan     10        1.00
Edwin     21        0.90
Parthiban 25        0.50
Arosha    26        0.40
Chin Wei  28        0.20
Rajkumar  83        0.00
So given this definition, we'd say that the degree of truth of the statement "Parthiban is YOUNG" is 0.50.

Note: Membership functions almost never have as simple a shape as age(x). They will at least tend to be triangles pointing up, and they can be much more complex than that. Furthermore, membership functions so far is discussed as if they always are based on a single criterion, but this isn't always the case, although it is the most common case. One could, for example, want to have the membership function for YOUNG depend on both a person's age and their height (Arosha's short for his age). This is perfectly legitimate, and occasionally used in practice. It's referred to as a two-dimensional membership function. It's also possible to have even more criteria, or to have the membership function depend on elements from two completely different universes of discourse.

X is LOW and Y is HIGH or (not Z is MEDIUM)

The standard definitions in fuzzy logic as suggested by Lotfi are:

1) Negate(negation criterion) : truth (not x) = 1.0 - truth (x)
2) Intersection(minimum criterion): truth (x and y) = minimum (truth(x), truth(y))
3) Union(maximum criterion): truth (x or y) = maximum (truth(x), truth(y))

In order to clarify this, a few examples are given. Let A be a fuzzy interval between 5 and 8 and B be a fuzzy number about 4. The corresponding figures are shown below.

The figure below gives an example for a negation. The blue line is the NEGATION of the fuzzy set A. Note that the negation criterion is used.

The following figure shows the fuzzy set between 5 and 8 AND about 4 (blue line). This time the minimum criterion is used.

Finally, the Fuzzy set between 5 and 8 OR about 4 is shown in the next figure (blue line). This time the maximum criterion is used.

These basic operations, provide guidelines to construct more complex ones which in turn can be used to create fuzzy machines.

© Shahariz Abdul Aziz ISE2 1996

References
1. Eric Horstkotte on Fuzzy Logic

Author: Peter Baur, Stephen Nouak, Roamn Winkler

2. Fuzzy Systems-A Tutorial