Associative law

the associative law (lat. associare -, connects, links, interlaces), on German linkage law or also connecting law , is a rule from mathematics combines. (Two digit) a linkage is associative, if the sequence of the execution does not play a role. Differently said:the clasping of several associative linkages is arbitrary.

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definition

associative law = clammy law

in a humming or a product term may one the addends or factors at will with clipsconnect. This applies also to more than three addends or factors.

A binary linkage <math> *: A \ times A \ tons of M< /math> on a quantity <math> A< /math> into a quantity <math> M< /math> means associative, if for all <math> A, b, C \ in A< /math> applies

< math> A * \ left (b * C \ right) = \ left (A * b \ right) * C </math> (Assoziativität)

a simplified notation without brackets can consequences with validity of the associative law be introduced. Because of

< math> \ (A + b \ right) + C left= A + \ (b + C \ right) /math <left>

is the expression

< math> a+b+c </math>

clearly, since from any clasping always the same result follows.

classification

the associative law belonged to the group axioms, becomes however alreadyfor the weaker structure of a half-group demanded.

examples

as linkages on the real numbers are addition and multiplication associatively, but not subtraction and division, because it is z. B.

<math> 2 - (3 - 1) =0 \ quad \ neq \ quad -2 = (2 - 3) - 1 </math>.

Also the power is not associatively, there z. B.

<math> 2^ {(2^3)} = 2^8 = 256 \ quad \ neq \ quad 64 = 4^3 = (2^2) ^3< /math>

applies.

see also

 

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