Difference between revisions of "Associative property"

 
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A [[binary operation]] <math>G: S\times S \to S</math> is said 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.
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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 $G: S\times S \to S$ is said to have the associative property or to be associative if $G(a, G(b, c)) = G(G(a, b), c)$ for all $a, b, c \in S$. Associativity is one of the most basic properties an operation can have.

For instance, the operation "$+$" on the real numbers is associative because $a + (b + c) = (a + b) + c$ for all real numbers $a, b, c$.

If we have an operation $\circ$ which is written between its arguments (like "$+$" or "$\times$" conventionally are), associativity tells us that we may write $a \circ b \circ c$ unambiguously -- it does not matter which pair we combine first.


For a non-example, consider the operation $\circ: \mathbb {R \times R \to R}$ given by $a\circ b = a + 2b$. This operation is not associative because $a\circ(b\circ c) = a \circ(b + 2c) = a + 2b + 4c$ while $(a \circ b)\circ c = (a + 2b)\circ c = a + 2b + 2c$ and those expressions are not equal for all choices of $a, b, c$ (in particular, they differ whenever $c \neq 0$).

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