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 [S. J. Bezuszka and M. Kenney, That ubiquitous sum: Math. Teacher 98, No. 5, 316–321 (2005; ME 2007a.00386)]. 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, when T n =1 n +2 n +3 n +⋯+N n ,n=1,2,3,⋯. We also discuss how the formulae are applicable in mathematics pedagogy. Copyright © 2009 Pushpa Publishing House.
|Journal||Far East Journal of Mathematical Education|
|Publication status||Published - Jun 2009|
CitationLeung, I. K. C., & Ching, W.-K. (2009). Cardinality of binary operations: A remark on the ubiquitous sum. Far East Journal of Mathematical Education, 3(2), 127-143.
- Binary operations
- Ubiquitous sum