**Definition**

Formally, a binary operation on a set *S* is called **associative** if it satisfies the **associative law**:

*Using * to denote a binary operation performed on a set*

*An example of multiplicative associativity*

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing *any* number of operations. Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

However, it is important to remember that changing the order of operations does not involve or permit moving the operands around within the expression; the sequence of operands is always unchanged.

The associative law can also be expressed in functional notation thus : .

Associativity can be generalized to n-ary operations. Ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcde with any three adjacent elements bracketed. N-ary associativity is a string of length n+(n-1) with any n adjacent elements bracketed.

Read more about this topic: Associative Property

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