Associative

**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.

**Example(s)**

(1)

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

So, addition is associative.

**Non-example(s)**

(2)

\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.