3/5 AN INTRODUCTION TO CYBERNETICS 



one is equal to the corresponding component of the other. Thus if 

 there is a vector (w,x,y,z), in which each component is some number, 

 and if two particular vectors are (4,3,8,2) and (4,3,8,1), then these 

 two particular vectors are unequal; for, in the fourth component, 

 2 is not equal to 1. (If they have different components, e.g. (4,3,8,2) 

 and {H,T), then they are simply not comparable.) 



When such a vector is transformed, the operation is in no way 

 different from any other transformation, provided we remember 

 that the operand is the vector as a whole, not the individual com- 

 ponents (though how they are to change is, of course, an essential 

 part of the transformation's definition). Suppose, for instance, the 

 "system" consists of two coins, each of which may show either 

 Head or Tail. The system has four states, which are 



(//,//) {H,T) {T,H) and (r,T). 



Suppose now my small niece does not like seeing two heads up, 

 but always alters that to {T,H), and has various other preferences. 

 It might be found that she always acted as the transformation 



^. , {H,H) {H,T) {T,H) {T,T) 



' ^ (T,H) iT,T) {T,H) (H,H) 



As a transformation on four elements, A'^ differs in no way from those 

 considered in the earlier sections. 



There is no reason why a transformation on a set of vectors 

 should not be wholly arbitrary, but often in natural science the 

 transformation has some simplicity. Often the components change 

 in some way that is describable by a more or less simple rule. Thus 

 if M were : 



^. , iH,H) (H,T) (T,H) {T,T) 



■ ^ {T,H) {T,T) iH,H) {H,T) 



it could be described by saying that the first component always 

 changes while the second always remains unchanged. 



Finally, nothing said so far excludes the possibility that some or 

 all of the components may themselves be vectors ! (E.g. S.6/3.) But 

 we shall avoid such complications if possible. 



Ex. I : Using /IBCas first operand, find the transformation generated by repeated 

 application of the operator "move the left-hand letter to the right" (e.g. 

 ABC-^BCA). 



Ex. 2: (Continued.) Express the transformation as a kinematic graph. 



Ex. 3: Using (1,-1) as first operand, find the other elements generated by 



repeated application of the operator "interchange the two numbers and then 



multiply the new left-hand number by minus one ". 



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