Formal Definition

A binary operation on a set $S$ is $associative$ if $(a*b)*c=a*(b*c)$ for all $a,b,c\in S$

Informal Definition

It can be shown that if $*$ is associative, then longer expressions such as $a*b*c*d$ are not ambiguous. Parentheses may be inserted in any fashion for purposes of computation; the final results of two such computations will be the same.


\begin{align} (2+4)+3=2+(4+3)\\ 6+3=2+7\\ 9=9 \end{align}

So, addition is associative.

\begin{align} (2-4)-3\neq2-(4-3)\\ -2-3\neq2-1\\ -5\neq1 \end{align}

So, subtraction is not associative.
Additional Comments
Some of the most important binary operations we consider are defined using composition of functions. It is important to know that this composition is always associative whenever it is defined.

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