8.6.6. Fuzzy Relation

An important concept needed in fuzzy set theory is that of a fuzzy relation which generalizes the conventional set-theoretic notion of relation. Let W1 and W2 be two universes. A fuzzy relation – has the membership function mR : W1 x W2 Æ [0, 1]. The projection of – on W1 is the marginal fuzzy set

m = sup {m–(w1, w2)‘w2 Œ W2}

for all w1 Œ W1 . If  1 is a fuzzy set on W1 the m 1 can be extended to W1 x W2 by

m = m 1(w1)

for all (w1, w2) ΠW1 x W2 .

Based on the above introduction, it can be seen that a fuzzy relation in R, the real number space is a fuzzy set in the product space R x R. For example, the relation denoted by x >> y, x, y Œ R' may be regarded as a fuzzy set  in R2 with the membership function of  , f having the following values:

f = 0 ;

f = 0.7 ;

f = 1 ; etc.