Associative property

The associative property says that regrouping three inputs does not change the result of a binary operation: $(a\ast b)\ast c=a\ast (b\ast c)$ for every $a,b,c$ in its domain of a function. It changes parentheses while keeping the order of the inputs. Commutativity instead exchanges two inputs; the two properties are different.

For addition modulo three, $(2+1)+1\equiv 2+(1+1)\equiv 1(\mathrm{mod}3)$. For subtraction modulo three,so the second binary operation fails the associative property. One failed triple disproves associativity; checking a few successful triples does not establish it for all inputs.

The useful consequence is that a finite ordered sequence can be combined in chunks. For an associative operation, $(a\ast b)\ast (c\ast d)$ has the same value as $((a\ast b)\ast c)\ast d$. Intermediate results can therefore summarize parts of a computation without retaining their internal parentheses. The order must still be preserved when the binary operation is not commutative. Function composition and concatenation provide examples where grouping is harmless but exchanging entries changes the result.

The associative property is a fundamental property of certain mathematical operations that describes how the grouping of numbers affects the result of the operation. It states that when performing an operation on three or more numbers, the way the numbers are grouped does not change the result. The associative property applies to both addition and multiplication.