In algebra, a binary operation is a rule for combining two elements of a set. Usually, the result of the operation is also a member of that set. Addition, subtraction, multiplication, and division are familiar binary operations of this type. The associative property is a property that binary operations may possess.
A familiar example of a binary operation that is associative (obeys the associative principle) is the addition (+) of real numbers. For example, the sum of 10, 2, and 35 is determined equally as well as (10 + 2) + 35 = 12 + 35 = 47, or 10 + (2 + 35) = 10 + 37 = 47. In these equations, the parentheses on either side of the defining equation indicate which two elements are to be combined first. Thus, the associative property states that combining a with b first, and then combining the result with c, is equivalent to combining b with c first, and then combining a with that result. The technical definition of the associative property is as follows: A binary operation (*) defined on a set S obeys the associative property if (a * b) * c = a * (b * c), for any three elements a, b, and c in S. When the associative property holds for all the members of S, every combination of elements must result in another element of S.
Multiplication of real numbers is another associative operation, for example, (5 × 2) × 3 = 10 × 3 = 30, and 5 × (2 × 3) = 5 × 6 = 30. However, not all binary operations are associative. Subtraction of real numbers is not associative, since in general (a–b)–c does not equal a– (b–c); for example (35 – 2)–6 = 33 – 6 = 27, while 35 – (2 – 6) = 35 – (–4) = 39. Division of real numbers is not associative either.