Some New Results on Binary Relations

TL;DR

This paper extends classical function properties to binary relations on finite sets, deriving formulas for counting various relation types and analyzing their probabilities.

Contribution

It introduces new formulas for counting specific types of binary relations and explores their probabilistic characteristics.

Findings

Formulas for counting right total, right unique, left total, and left unique relations.

Counts of relations that are both right and left unique or total.

Probability estimates for randomly selected relations being right unique or right total.

Abstract

It is well known that if a function from set A to set B has a right inverse then the function is a surjection and the right inverse is an injection. For finite sets, the number of functions, injections, and surjections can also be counted. Relations generalize functions: do similar results exist for relations? This paper proves several new results concerning binary relations. For finite sets, we derive formulas for the number of right total, right unique, left total, and left unique relations. We also provide formulas that count the number of relations that are both right unique and left unique; right unique and right total; and left unique and left total. We conclude by discussing the probability that a relation selected at random is right unique or right total.