for all .
We can construct an operation which is not associative. Let be the integers. and define . Then . But , hence .
Note, however, that if we were to take , would be associative over !. This illustrates the fact that the set the operation is taken with respect to is very important.
We show that the division operation over nonzero reals is non-associative. All we need is a counter-example: so let us compare and . The first expression is equal to , the second to , hence division over the nonzero reals is not associative.
Remark. The property of being associative of a binary operation can be generalized to an arbitrary -ary operation, where . An -ary operation on a set is said to be associative if for any elements , we have
In other words, for any , if we set , then is associative iff for all . Therefore, for instance, a ternary operation on is associative if .
|Date of creation||2013-03-22 12:22:48|
|Last modified on||2013-03-22 12:22:48|
|Last modified by||CWoo (3771)|