The Ordered Pair Proposition

The main noticeable difference between modern Mathematics and that of Mathematics before the twentieth century is the role of set theory in unifying the entire discipline. The axiomatic Zermel-Fraenkel- Skolem Set Theory with its ten axioms, including the Axiom of Choice, is able to provide a unified basis for Mathematics by objectifying almost all mathematical patterns, notions, and operations, into sets and membership of sets Thus, the Ordered Pair Proposition allows us to reformulate the key concept in all of Mathematics - relation - as a set of ordered pairs of sets. Three most important types of mathematical relations, namely, function, equivalence, and order, are shown to be able to reproduce the most diverse concepts and objects across Mathematics. Significantly, the ZFC Set Theory enable us to deal rigorously with the concept of the Infinite. A simple proof is provided of Cantor- Bernstein Theorem.