
05773284
j
2010e.00502
Leung, Issic K.C.
Ching, WaiKi
Cardinality of binary operations: a remark on the ubiquitous sum.
Far East J. Math. Educ. 3, No. 2, 127143 (2009).
2009
Pushpa Publishing House, Allahabad, Uttar Pradesh, India
EN
H20
H40
cardinality
binary operations
ubiquitous sum
combination
ME 2007a.00386
http://pphmj.com/abstract/4254.htm
Summary: 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 [{\it S. J. Bezuszka} and {\it M. Kenney}, That ubiquitous sum: Math. Teacher 98, No. 5, 316321 (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 $T_n = 1^n + 2^n + 3^n + \cdots + N^n, n = 1,2,3,\dots$. We also discuss how the formulae are applicable in mathematics pedagogy.