C++ Neural Networks and Fuzzy Logic by Valluru B. Rao M&T Books, IDG Books Worldwide, Inc. ISBN: 1558515526   Pub Date: 06/01/95

Properties of Fuzzy Relations

A relation on a set, that is a subset of a Cartesian product of some set with itself, may have some interesting properties. It may be reflexive. For this you need to have 1 for the degree of membership of each main diagonal entry. Our example here is evidently not reflexive.

A relation may be symmetric. For this you need the degrees of membership of each pair of entries symmetrically situated to the main diagonal to be the same value. For example (Jeff, Mike) and (Mike, Jeff) should have the same degree of membership. Here they do not, so our example of a relation is not symmetric.

A relation may be antisymmetric. This requires that if a is different from b and the degree of membership of the ordered pair (a, b) is not 0, then its mirror image, the ordered pair (b, a), should have 0 for degree of membership. In our example, both (Steve, Mike) and (Mike, Steve) have positive values for degree of membership; therefore, the relation much_more_educated over the set {Jeff, Steve, Mike} is not antisymmetric also.

A relation may be transitive. For transitivity of a relation, you need the following condition, illustrated with our set {Jeff, Steve, Mike}. For brevity, let us use r in place of much_more_educated, the name of the relation:

     min (mr(Jeff, Steve) , mr(Steve, Mike) )[le]mr(Jeff, Mike)
     min (mr(Jeff, Mike) , mr(Mike, Steve) )[le]mr(Jeff, Steve)
     min (mr(Steve, Jeff) , mr(Jeff, Mike) )[le]mr(Steve, Mike)
     min (mr(Steve, Mike) , mr(Mike, Jeff) )[le]mr(Steve, Jeff)
     min (mr(Mike, Jeff) , mr(Jeff, Steve) )[le]mr(Mike, Steve)
     min (mr(Mike, Steve) , mr(Steve, Jeff) )[le]mr(Mike, Jeff)

In the above listings, the ordered pairs on the left-hand side of an occurrence of [le] are such that the second member of the first ordered pair matches the first member of the second ordered pair, and also the right-hand side ordered pair is made up of the two nonmatching elements, in the same order.

In our example,

     min (mr(Jeff, Steve) , mr(Steve, Mike) ) = min (0.2, 0.3) = 0.2
     mr(Jeff, Mike) = 0.7 > 0.2

For this instance, the required condition is met. But in the following:

     min (mr(Jeff, Mike), mr(Mike, Steve) ) = min (0.7, 0.6) = 0.6
     mr(Jeff, Steve) = 0.2 < 0.6

The required condition is violated, so the relation much_more_educated is not transitive.

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NOTE:  If a condition defining a property of a relation is not met even in one instance, the relation does not possess that property. Therefore, the relation in our example is not reflexive, not symmetric, not even antisymmetric, and not transitive.

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If you think about it, it should be clear that when a relation on a set of more than one element is symmetric, it cannot be antisymmetric also, and vice versa. But a relation can be both not symmetric and not antisymmetric at the same time, as in our example.

An example of reflexive, symmetric, and transitive relation is given by the following matrix:

          1     0.4   0.8
          0.4   1     0.4
          0.8   0.4   1

Similarity Relations

A reflexive, symmetric, and transitive fuzzy relation is said to be a fuzzy equivalence relation. Such a relation is also called a similarity relation. When you have a similarity relation s, you can define the similarity class of an element x of the domain as the fuzzy set in which the degree of membership of y in the domain is m_s(x, y). The similarity class of x with the relation s can be denoted by [x]_s.

Resemblance Relations

Do you think similarity and resemblance are one and the same? If x is similar to y, does it mean that x resembles y? Or does the answer depend on what sense is used to talk of similarity or of resemblance? In everyday jargon, Bill may be similar to George in the sense of holding high office, but does Bill resemble George in financial terms? Does this prompt us to look at a ‘resemblance relation’ and distinguish it from the ‘similarity relation’? Of course.

Recall that a fuzzy relation that is reflexive, symmetric, and also transitive is called similarity relation. It helps you to create similarity classes. If the relation lacks any one of the three properties, it is not a similarity relation. But if it is only not transitive, meaning it is both reflexive and symmetric, it is still not a similarity relation, but it is a resemblance relation. An example of a resemblance relation, call it t, is given by the following matrix.

Let the domain have elements a, b, and c:

               1     0.4   0.8
          t =  0.4   1     0.5
               0.8   0.5   1

This fuzzy relation is clearly reflexive, and symmetric, but it is not transitive. For example:

      min (mt(a, c) , mt(c, b) ) = min (0.8, 0.5) = 0.5 ,

but the following:

      mt(a, b) = 0.4 < 0.5 ,

is a violation of the condition for transitivity. Therefore, t is not a similarity relation, but it certainly is a resemblance relation.

Fuzzy Partial Order

One last definition is that of a fuzzy partial order. A fuzzy relation that is reflexive, antisymmetric, and transitive is a fuzzy partial order. It differs from a similarity relation by requiring antisymmetry instead of symmetry. In the context of crisp sets, an equivalence relation that helps to generate equivalence classes is also a reflexive, symmetric, and transitive relation. But those equivalence classes are disjoint, unlike similarity classes with fuzzy relations. With crisp sets, you can define a partial order, and it serves as a basis for making comparison of elements in the domain with one another.