Difference between revisions of "Associative property"
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− | A [[binary operation]] <math>G: S\times S \to S</math> is said to | + | A [[binary operation]] <math>G: S\times S \to S</math> is said to have the '''associative property''' or to ''be associative'' if <math>G(a, G(b, c)) = G(G(a, b), c)</math> for all <math>a, b, c \in S</math>. Associativity is one of the most basic properties an operation can have. |
For instance, the operation "<math>+</math>" on the [[real number]]s is associative because <math>a + (b + c) = (a + b) + c</math> for all real numbers <math>a, b, c</math>. | For instance, the operation "<math>+</math>" on the [[real number]]s is associative because <math>a + (b + c) = (a + b) + c</math> for all real numbers <math>a, b, c</math>. |
Latest revision as of 16:09, 15 August 2006
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A binary operation is said to have the associative property or to be associative if
for all
. Associativity is one of the most basic properties an operation can have.
For instance, the operation "" on the real numbers is associative because
for all real numbers
.
If we have an operation which is written between its arguments (like "
" or "
" conventionally are), associativity tells us that we may write
unambiguously -- it does not matter which pair we combine first.
For a non-example, consider the operation given by
. This operation is not associative because
while
and those expressions are not equal for all choices of
(in particular, they differ whenever
).