id: 05773284
dt: j
an: 2010e.00502
au: Leung, Issic K.C.; Ching, Wai-Ki
ti: Cardinality of binary operations: a remark on the ubiquitous sum.
so: Far East J. Math. Educ. 3, No. 2, 127-143 (2009).
py: 2009
pu: Pushpa Publishing House, Allahabad, Uttar Pradesh, India
la: EN
cc: H20 H40
ut: cardinality; binary operations; ubiquitous sum; combination
ci: ME 2007a.00386
li: http://pphmj.com/abstract/4254.htm
ab: 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, 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 $T_n =
1^n + 2^n + 3^n + \cdots + N^n, n = 1,2,3,\dots$. We also discuss how
the formulae are applicable in mathematics pedagogy.
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