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| [[File:lighterstill.jpg]][[File:Associativity_colors.jpg|right|frame]] | | [[File:lighterstill.jpg]][[File:Associativity_colors.jpg|right|frame]] |
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− | In [[mathematics]], '''associativity''' is a property of some [http://en.wikipedia.org/wiki/Binary_operation binary operations]. It means that, within an [[expression]] containing two or more occurrences in a row of the same associative operator, the order in which the operations are [[performed]] does not matter as long as the [[sequence]] of the [http://en.wikipedia.org/wiki/Operand operands] is not changed. That is, rearranging the parentheses in such an expression will not [[change]] its [[value]]. Consider for instance the [[equation]] | + | In [[mathematics]], '''associativity''' is a property of some [https://en.wikipedia.org/wiki/Binary_operation binary operations]. It means that, within an [[expression]] containing two or more occurrences in a row of the same associative operator, the order in which the operations are [[performed]] does not matter as long as the [[sequence]] of the [https://en.wikipedia.org/wiki/Operand operands] is not changed. That is, rearranging the parentheses in such an expression will not [[change]] its [[value]]. Consider for instance the [[equation]] |
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| [[File:Associativity1.jpg]] | | [[File:Associativity1.jpg]] |
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− | Even though the parentheses were rearranged (the left side requires adding 5 and 2 first, then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5), the [[value]] of the [[expression]] was not [[Change|altered]]. Since this holds true when [[performing]] addition on any [http://en.wikipedia.org/wiki/Real_number real numbers], we say that "addition of real numbers is an associative operation." | + | Even though the parentheses were rearranged (the left side requires adding 5 and 2 first, then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5), the [[value]] of the [[expression]] was not [[Change|altered]]. Since this holds true when [[performing]] addition on any [https://en.wikipedia.org/wiki/Real_number real numbers], we say that "addition of real numbers is an associative operation." |
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− | Associativity is not to be confused with [http://en.wikipedia.org/wiki/Commutativity commutativity]. Commutativity justifies changing the order or [[sequence]] of the operands within an expression while associativity does not. For example, | + | Associativity is not to be confused with [https://en.wikipedia.org/wiki/Commutativity commutativity]. Commutativity justifies changing the order or [[sequence]] of the operands within an expression while associativity does not. For example, |
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| [[File:Associativity2.jpg]] | | [[File:Associativity2.jpg]] |
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| is not an example of associativity because the operand sequence changed when the 2 and 5 switched places. | | is not an example of associativity because the operand sequence changed when the 2 and 5 switched places. |
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− | Associative operations are [[abundant]] in [[mathematics]]; in [[fact]], many [http://en.wikipedia.org/wiki/Algebraic_structure algebraic structures] (such as [http://en.wikipedia.org/wiki/Semigroup_(mathematics) semigroups] and categories) explicitly require their binary operations to be associative. | + | Associative operations are [[abundant]] in [[mathematics]]; in [[fact]], many [https://en.wikipedia.org/wiki/Algebraic_structure algebraic structures] (such as [https://en.wikipedia.org/wiki/Semigroup_(mathematics) semigroups] and categories) explicitly require their binary operations to be associative. |
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− | However, many important and interesting operations are non-associative; one common example would be the [http://en.wikipedia.org/wiki/Vector_cross_product vector cross product].[http://en.wikipedia.org/wiki/Associativity] | + | However, many important and interesting operations are non-associative; one common example would be the [https://en.wikipedia.org/wiki/Vector_cross_product vector cross product].[https://en.wikipedia.org/wiki/Associativity] |
| ==See also== | | ==See also== |
| *[[Association]] | | *[[Association]] |
| ==References== | | ==References== |
− | # Dudek, W.A. (2001), "On some old problems in n-ary groups", Quasigroups and Related Systems 8: 15–36, http://www.quasigroups.eu/contents/contents8.php?m=trzeci . | + | # Dudek, W.A. (2001), "On some old problems in n-ary groups", Quasigroups and Related Systems 8: 15–36, https://www.quasigroups.eu/contents/contents8.php?m=trzeci . |
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| [[Category: Mathematics]] | | [[Category: Mathematics]] |