Associative property

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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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