Math Department (Thesis/2012-13)

# Sets, Relations, and Matrices

## Under the direction of: Susan Niefield

The study of sets and functions was one of the main topics in Math 199. Recall that a function from $X$ to $Y$ is formally defined to be a set of ordered pairs $(x,y)$ satisfying certain properties. A relation from $X$ to $Y$ is a subset $R \subseteq X \times Y$. Thus, a function is a special kind of relation. You may have studied other special relations, e.g., equivalence relations or partial orders, but what can be said about general relations?

Many of the topics considered in Math 199 extend to arbitrary relations. One can define the composition of a relation from $X$ to $Y$ and one from $Y$ to $Z$, and obtain all of the properties of composition of functions, and more. Every relation has an inverse or opposite relation, not just the bijection ones, as in the case of functions. In addition, since relations are merely sets, it makes sense to talk about unions and intersections of relations from $X$ to $Y$, and study the algebra of relations on sets. Moreover, this algebra can be directly related to the already familiar algebra of matrices. In particular, every relation between finite sets corresponds to a matrix consisting entirely of 0's and 1's, and there are operations on these matrices whic correspond to composition, inverse, union, and interjection.

One Term: Winter only, Two Term: Fall/Winter

 Math Department web pages Created: 15 Apr 2002 Last modified: 02 May 2012 09:19:56 Comments to: math@union.edu