CARDINALITY OF BINARY OPERATIONS: A REMARK ON THE UBIQUITOUS SUM
Ikc Leung, WK Ching · Far East Journal of Mathematical Education · 2009
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 $P_N$ 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 $P_N$ is closely related to the sum of consecutive numbers, the Ubiquitous Sum (Bezuszka and Kenney [2]). In particular, $P_N$ 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 $N \in \mathbb{N}$, $T_n = 1^n+2^n+3^n+ \cdots+ N^n, n=1,2,3, \dots $. We also discuss how the formulae are applicable in mathematics pedagogy.