PlanetMath: associative

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

associative

(Definition)

Let $(S,\phi)$ be a set with binary operation $\phi$ . $\phi$ is said to be associative over if

$\displaystyle \phi(a,\phi(b,c)) = \phi(\phi(a,b),c)$

for all $a,b,c \in S$ .

Examples of associative operations are addition and multiplication over the integers (or reals), or addition or multiplication over $n \times n$ matrices.

We can construct an operation which is not associative. Let be the integers. and define $\nu(a,b)=a^2+b$ . Then $\nu(\nu(a,b),c)=\nu(a^2+b,c)=a^4+2ba^2+b^2+c$ . But $\nu(a,\nu(b,c))=\nu(a,b^2+c)=a+b^4+2cb^2+c^2$ , hence $\nu(\nu(a,b),c) \ne \nu(a,\nu(b,c))$ .

Note, however, that if we were to take $S=\{0\}$ , $\nu$ would be associative over !. This illustrates the fact that the set the operation is taken with respect to is very important.

Example.

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.

"associative" is owned by akrowne.

(view preamble)