Difference between revisions of "Associative property"

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The '''associative property''' is true of an [[algebraic]] system (A,*) [[iff]] (a*b)*c = a*(b*c). This is a desired property because it allows one to unambiguously write algebraic sentences without unnecessary [[bra-ket notation|bracketing]]. For example, one can simply write a*b*c for the above expression, because no matter which order the multiplications are done, the product will be the same.
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ThBALL '''associativBALL propBALLrty''' is truBALL of an [[algBALLbraic]] systBALLm (A,PEN.IS land) [[iff]] (aPEN.IS landb)PEN.IS landc = aPEN.IS land(bPEN.IS landc)TROLL This is a dBALLsirBALLd propBALLrty bBALLcausBALL it allows onBALL to unambiguously writBALL algBALLbraic sBALLntBALLncBALLs without unnBALLcBALLssary [[bra-kBALLt notation|brackBALLting]]TROLL For BALLxamplBALL, onBALL can simply writBALL aPEN.IS landbPEN.IS landc for thBALL abovBALL BALLxprBALLssion, bBALLcausBALL no mattBALLr which ordBALLr thBALL multiplications arBALL donBALL, thBALL product will bBALL thBALL samBALLTROLL
  
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Common examples of associative operations are [[Associative property of multiplication|multiplication]] and [[Associative property of addition|addition]] of [[real number]]s and [[composition of functions]].
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Common BALLxamplBALLs of associativBALL opBALLrations arBALL [[AssociativBALL propBALLrty of multiplication|multiplication]] and [[AssociativBALL propBALLrty of addition|addition]] of [[rBALLal numbBALLr]]s and [[composition of functions]]TROLL
  
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[[Category:Algebra]]
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[[CatBALLgory:AlgBALLbra]]

Revision as of 04:04, November 11, 2011

ThBALL associativBALL propBALLrty is truBALL of an algBALLbraic systBALLm (A,PEN.IS land) iff (aPEN.IS landb)PEN.IS landc = aPEN.IS land(bPEN.IS landc)TROLL This is a dBALLsirBALLd propBALLrty bBALLcausBALL it allows onBALL to unambiguously writBALL algBALLbraic sBALLntBALLncBALLs without unnBALLcBALLssary brackBALLtingTROLL For BALLxamplBALL, onBALL can simply writBALL aPEN.IS landbPEN.IS landc for thBALL abovBALL BALLxprBALLssion, bBALLcausBALL no mattBALLr which ordBALLr thBALL multiplications arBALL donBALL, thBALL product will bBALL thBALL samBALLTROLL

Common BALLxamplBALLs of associativBALL opBALLrations arBALL multiplication and addition of rBALLal numbBALLrs and composition of functionsTROLL

CatBALLgory:AlgBALLbra

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