Nowadays, picture fuzzy set theory is a flourishing field in mathematics with
uncertainty by incorporating the concept of positive, negative and neutral
membership degrees of an object. A traditional crisp relation represents the
satisfaction or the dissatisfaction of relationship, connection or
correspondence between the objects of two or more sets. However, there are some
problems that can’t be solved through classical relationships, such as the
relationship between two objects being vague. In those situations, picture fuzzy relation over picture fuzzy sets is an important
and powerful concept which is suitable for describing correspondences
between two vague objects. It represents the strength of association of the
elements of picture fuzzy sets. It plays an important role in picture fuzzy
modeling, inference and control system and also has important applications in
relational databases, approximate reasoning, preference modeling, medical
diagnosis, etc. In this article, we define picture fuzzy relations over picture
fuzzy sets, including some other fundamental definitions with illustrations.
The max-min and min-max compositions of picture fuzzy relations are defined in
the light of picture fuzzy sets and discussed some properties related to them.
The reflexivity, symmetry and transitivity of a picture fuzzy relation are
described over a picture fuzzy set. Finally, various properties are explored
related to the picture fuzzy relations over a picture fuzzy set.
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