In algebra, a binary operation is a rule for combining the elements of a set two at a time. In most important examples that combination is also another member of the same set. Addition, subtraction, multiplication, and division are familiar binary operations. A familiar example of a binary operation that is associative (obeys the associative principle) is 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. 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. 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. 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. When the associative property holds for all the members of a set, every combination of elements must result in another element of the same set.
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