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#### Article: Cardinality of Binary Operations: A Remark on the Ubiquitous Sum

Title Cardinality of Binary Operations: A Remark on the Ubiquitous Sum Leung, IKCChing, WK CardinalityBinary operationsUbiquitous sumCombination 2009 Pushpa Publishing House. The Journal's web site is located at http://pphmj.com/journals/fjme.htm Far East Journal of Mathematical Education, 2009, v. 3 n. 2, p. 127-143 How to Cite? 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. http://hdl.handle.net/10722/75180 0973-5631

DC FieldValueLanguage
dc.contributor.authorLeung, IKCen_HK
dc.contributor.authorChing, WKen_HK
dc.date.accessioned2010-09-06T07:08:42Z-
dc.date.available2010-09-06T07:08:42Z-
dc.date.issued2009en_HK
dc.identifier.citationFar East Journal of Mathematical Education, 2009, v. 3 n. 2, p. 127-143en_HK
dc.identifier.issn0973-5631en_HK
dc.identifier.urihttp://hdl.handle.net/10722/75180-
dc.description.abstractWe 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.-
dc.languageengen_HK
dc.publisherPushpa Publishing House. The Journal's web site is located at http://pphmj.com/journals/fjme.htmen_HK
dc.relation.ispartofFar East Journal of Mathematical Educationen_HK
dc.subjectCardinality-
dc.subjectBinary operations-
dc.subjectUbiquitous sum-
dc.subjectCombination-
dc.titleCardinality of Binary Operations: A Remark on the Ubiquitous Sumen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=0257-0203&volume=3&spage=127&epage=143&date=2009&atitle=Cardinality+of+Binary+Operations:+A+Remark+on+the+Ubiquitous+Sumen_HK
dc.identifier.emailChing, WK: wching@hkucc.hku.hken_HK
dc.identifier.authorityChing, WK=rp00679en_HK
dc.identifier.hkuros164011en_HK
dc.identifier.volume3-
dc.identifier.issue2-
dc.identifier.spage127-
dc.identifier.epage143-
dc.publisher.placeIndia-