Transitive Closure of Fuzzy Relations

In this paper X, Y denote non empty sets. Let X be a non empty set. Observe that every membership function of X is real-yielding. Let f , g be real-yielding functions. The predicate f ⊑ g is defined by: (Def. 1) dom f ⊆ dom g and for every set x such that x ∈ dom f holds f(x) ¬ g(x). Let X be a non empty set and let f , g be membership functions of X. Let us observe that f ⊑ g if and only if: (Def. 2) For every element x of X holds f(x) ¬ g(x). We introduce f ⊆ g as a synonym of f ⊑ g. Let X, Y be non empty sets and let f , g be membership functions of X, Y . Let us observe that f ⊑ g if and only if: (Def. 3) For every element x of X and for every element y of Y holds f(〈x, y〉) ¬ g(〈x, y〉). One can prove the following propositions: