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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 

Authors  
Keywords  Cardinality Binary operations Ubiquitous sum Combination 
Issue Date  2009 
Publisher  Pushpa Publishing House. The Journal's web site is located at http://pphmj.com/journals/fjme.htm 
Citation  Far East Journal of Mathematical Education, 2009, v. 3 n. 2, p. 127143 How to Cite? 
Abstract  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. 
Persistent Identifier  http://hdl.handle.net/10722/75180 
ISSN 
DC Field  Value  Language 

dc.contributor.author  Leung, IKC  en_HK 
dc.contributor.author  Ching, WK  en_HK 
dc.date.accessioned  20100906T07:08:42Z   
dc.date.available  20100906T07:08:42Z   
dc.date.issued  2009  en_HK 
dc.identifier.citation  Far East Journal of Mathematical Education, 2009, v. 3 n. 2, p. 127143  en_HK 
dc.identifier.issn  09735631  en_HK 
dc.identifier.uri  http://hdl.handle.net/10722/75180   
dc.description.abstract  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.   
dc.language  eng  en_HK 
dc.publisher  Pushpa Publishing House. The Journal's web site is located at http://pphmj.com/journals/fjme.htm  en_HK 
dc.relation.ispartof  Far East Journal of Mathematical Education  en_HK 
dc.subject  Cardinality   
dc.subject  Binary operations   
dc.subject  Ubiquitous sum   
dc.subject  Combination   
dc.title  Cardinality of Binary Operations: A Remark on the Ubiquitous Sum  en_HK 
dc.type  Article  en_HK 
dc.identifier.openurl  http://library.hku.hk:4550/resserv?sid=HKU:IR&issn=02570203&volume=3&spage=127&epage=143&date=2009&atitle=Cardinality+of+Binary+Operations:+A+Remark+on+the+Ubiquitous+Sum  en_HK 
dc.identifier.email  Ching, WK: wching@hkucc.hku.hk  en_HK 
dc.identifier.authority  Ching, WK=rp00679  en_HK 
dc.identifier.hkuros  164011  en_HK 
dc.identifier.volume  3   
dc.identifier.issue  2   
dc.identifier.spage  127   
dc.identifier.epage  143   
dc.publisher.place  India   