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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