ABSTRACT
This chapter presents concepts critically important to a multitude of mathematical areas: equivalence relations and order relations. Graphs naturally model relations between objects, with the objects being represented by the vertices of the graph. Graphs consisting of totally connected components induce equivalence relations on the set of their vertices. Relations between certain sets naturally give rise to bipartite graphs. The domain of the relation and the range of the relation form the two partitioning sets of the graph’s vertices. A major application of relations is in the definition of a function, which are relations satisfying particular requirements. The chapter investigates a particular relation stemming from the Quotient-Remainder Theorem. Anyone familiar with computer science has undoubtedly encountered concepts such as sorting, trees and task scheduling. These are just some of the applications of partial and linear orderings.
