As we know, a mathematical relation between a two sets is a subset of their cartesian product. In such a definition members of the two sets are either related or not in a binary fashion.
Is there a concept that captures a relaxation of the membership of the relationship? In the sense that $R(a_1, b_1) \in [0,1]$?
I am looking at a relaxation of a bijection function, in a machine learning context, if that is relevant.
On
There is such a thing as a fuzzy set that is used in the context of machine learning. A fuzzy set $(X, \chi)$ has a characteristic function $\chi: X \to [0,1]$, and thus does not admit strictly binary outcomes for elements of $X$. Its members can be assigned any value in the interval $[0, 1]$. With this in mind, a fuzzy characteristic function may be defined for a relation $R \subset A\times B$ as $\chi_R : A\times B \to [0,1]$. See here: https://en.wikipedia.org/wiki/Fuzzy_set
I think the concept you're looking for is fuzzy relations which is an extension of fuzzy sets (sets whose elements have degrees of membership in the range $[0,1]$).