associative operation

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Any dyadic operation ∘ that satisfies the law

x ∘ (yz) = (xy) ∘ z

for all x, y, and z in the domain of ∘. The law is known as the associative law. An expression involving several adjacent instances of an associative operation can be interpreted unambiguously; the order in which the operations are performed is irrelevant since the effects of different evaluations are identical, though the work involved may differ. Consequently parentheses are unnecessary, even in more complex expressions.

The arithmetic operations of addition and multiplication are associative, though subtraction is not. On a computer the associative law of addition of real numbers fails to hold because of the inherent inaccuracy in the way real numbers are usually represented (see floating-point notation), and the addition of integers fails to hold because of the possibility of overflow.

Subjects: Computing.

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