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 15:09, 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 
).
