2/12 AN INTRODUCTION TO CYBERNETICS 



2/12. The trial in the previous exercise will make clear the import- 

 ance of closure. An unclosed transformation such as W cannot be 

 apphed twice; for although it changes h to k, its effect on k is 

 undefined, so it can go no further. The unclosed transformation is 

 thus like a machine that takes one step and then jams. 



2/13. Elimination. When a transformation is given in abbreviated 

 form, such as «' = // + 1, the result of its double application must 

 be found, if only the methods described so far are used, by re-writing 

 the transformation to show every operand, performing the double 

 application, and then re-abbreviating. There is, however, a quicker 

 method. To demonstrate and explain it, let us write out in full 

 the transformation T: n' — n + 1, on the positive integers, showing 

 the results of its double application and, underneath, the general 

 symbol for what lies above : 



r: j 1 2 3 ...... . 



2 3 4 ... n' ... 



^'- h 4 5 ... n" . . . 



n" is used as a natural symbol for the transform of n', just as n' is 

 the transform of n. 



Now we are given that n' = n + 1. As we apply the same 

 transformation again it follows that n" must be 1 more than /;'. 

 Thus«" = n + 1. 



To specify the single transformation T^ we want an equation that 

 will show directly what the transform n" is in terms of the operand 

 n. Finding the equation is simply a matter of algebraic elimination: 

 from the two equations n" = n' + 1 and n' = n -\- \, eliminate n'. 

 Substituting for n' in the first equation we get (with brackets to show 

 the derivation) n" = (n + 1) + 1, i.e. n" = n -\- 2. 



This equation gives correctly the relation between operand {n) 

 and transform {n") when T- is applied, and in that way T^ is specified. 

 For uniformity of notation the equation should now be re-written 

 as m' = m + 2. This is the transformation, in standard notation, 

 whose single application (hence the single prime on m) causes the 

 same change as the double application of T. (The change from 

 n to w is a mere change of name, made to avoid confusion.) 



The rule is quite general. Thus, if the transformation is 

 n' = 2n — 3, then a second application will give second transforms 

 n" that are related to the first by n" = 2n' — 3. Substitute for n\ 

 using brackets freely: 



//" = 2(2/2 - 3) - 3 

 = 4« - 9. 



18 



