Cartesian Products of Relations and Relational Structures1

In this article we present several logical schemes. The scheme FraenkelA2 deals with a non empty set A , a binary functor F yielding a set, and two binary predicates P , Q , and states that: {F (s, t);s ranges over elements of A , t ranges over elements of A : P [s, t]} is a subset of A provided the following condition is satisfied: • For every element s of A and for every element t of A holds F (s, t) ∈ A . The scheme ExtensionalityR deals with binary relations A , B and a binary predicate P , and states that: A = B provided the parameters meet the following conditions: • For all sets a, b holds 〈a, b〉 ∈ A iff P [a,b], and • For all sets a, b holds 〈a, b〉 ∈ B iff P [a,b]. Let X be an empty set. Note that π1(X) is empty and π2(X) is empty. Let X , Y be non empty sets and let D be a non empty subset of [:X , Y :]. Observe that π1(D) is non empty and π2(D) is non empty. Let L be a relational structure and let X be an empty subset of L. One can verify that ↓X is empty and ↑X is empty. Next we state several propositions: