Cardinality of Binary Operations: A Remark on the Ubiquitous Sum

Abstract

We establish the sufficient conditions to determine how many binary operations can possibly take place between any two arbitrary elements from a given set, provided that the operation is well defined. If we mark and collect each of such operations in another set S, we call the number �the cardinality of the set S of binary operations between any two elements for a given set of N elements. We find that such number �is closely related to the sum of consecutive numbers, the Ubiquitous Sum (Bezuszka and Kenney [2]). In particular, �is simply the combination of selecting from N distinct objects, two at a time. This idea can be generated to look for the cardinality of a set of ternary operations. We have verified that this cardinality is the same as the combination of selecting from N distinct objects, three at a time. The results can be generalized to derive the formulae of factorization for, when N ε N, Tn = 1n + 2n +3n + ... + Nn, n = 1, 2, 3, ... We also discuss how the formulae are applicable in mathematics pedagogy.