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
Famous quotes containing the word definition:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)