Properties of First and Second Order Cutting of Binary Relations

This paper introduces some notions concerning binary relations according to [9]. It is also an attempt to complement the knowledge contained in the Mizar Mathematical Library regarding binary relations. We define here an image and inverse image of element of set A under binary relation of two sets A, B as image and inverse image of singleton of the element under this relation, respectively. Next, we define “The First Order Cutting Relation of two sets A, B under a subset of the set A” as the union of images of elements of this subset under the relation. We also define “The Second Order Cutting Subset of the Cartesian Product of two sets A, B under a subset of the set A” as an intersection of images of elements of this subset under the subset of the Cartesian Product. The paper also defines first and second projection of binary relations. The main goal of the article is to prove properties and collocations of definitions introduced in this paper.