# 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 ).