fuzzy and crisp relations

A crisp relation represents the presence or absence of association, interaction, or interconnectedness between the elements of two or more sets. This concept can be generalized to allow for various degrees or strengths of relation or interaction between elements. Degrees of association can be represented by membership grades in a fuzzy relation in the same way as degrees of set membership are represented in the fuzzy set. In fact, just as the crisp set can be viewed as a restricted case of the more general fuzzy set concept, the crisp relation can be considered to be a restricted case of the fuzzy relations.

Crisp Relation

A crisp relation is used to represents the presence or absence of interaction, association, or interconnectedness between the elements of more than a set. This crisp relational concept can be generalized to allow for various degrees or strengths of relation or interaction between elements.

Operations on Crisp Relations

Let A and B be two relations defined on X x Y and are represented by relational matrices. The following operations can be performed on these relations A and B
Union
A ∪ B (x,y) = max [ A (x,y) , B (x,y) ]

Intersection
A ∩ B (x,y) = min [ A(x,y) , B (x,y) ]

Read more: http://tech-wonders.blogspot.com/2010/08/more-crips-relation-terms.html#ixzz2Sc2ZWnPl
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Fuzzy relation

Degrees of association can be represented by grades of the membership in a fuzzy relation in the same way as degrees of set membership are represented in the fuzzy set. In fact, just as the crisp set can be viewed as a restricted case of the more general fuzzy set concept, the crisp relation can be considered to be a restricted case of the fuzzy relations.