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:
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (18031882)
“Its a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was mine.”
—Jane Adams (20th century)
“Mothers often are too easily intimidated by their childrens negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)