2/15 AN INTRODUCTION TO CYBERNETICS 



Ex. 10: What is the result of applying the transformation n' = 1/n twice, 

 when the operands are all the positive rational numbers (i.e. all the frac- 

 tions) ? 



Ex. 1 1 : Here is a geometrical transformation. Draw a straight line on paper 

 and mark its ends A and B. This line, in its length and position, is the 

 operand. Obtain its transform, with ends A' and B', by the transformation- 

 rule R: A' is midway between A and B; B' is found by rotating the line 

 A'B about A' through a right angle anticlockwise. Draw such a line, 

 apply R repeatedly, and satisfy yourself about how the system behaves. 



*Ex. 12: (Continued). If familiar with analytical geometry, let A start at 

 (0,0) and B at (0,1), and find the limiting position. (Hint: Build up A's 

 final AT-co-ordinate as a series, and sum; similarly for A's j-co-ordinate.) 



2/15. Notation. The notation that indicates the transform by the 

 addition of a prime (') is convenient if only one transformation is 

 under consideration; but if several transformations might act on 

 «, the symbol n' does not show which one has acted. For this 

 reason, another symbol is sometimes used: if « is the operand, and 

 transformation T is applied, the transform is represented by T{n). 

 The four pieces of type, two letters and two parentheses, represent 

 one quantity, a fact that is apt to be confusing until one is used to it. 

 T{n), really n' in disguise, can be transformed again, and would be 

 written T{T(n)) if the notation were consistent; actually the outer 

 brackets are usually ehminated and the T's combined, so that n" 

 is written as T-(n). The exercises are intended to make this notation 

 familiar, for the change is only one of notation. 



1 2 3 

 Ex. 1 : If/: i T 1 2 



whatis/(3)?/(l)?/2(3)? 



Ex. 2: Write out in full the transformation g on the operands, 6, 7, 8, if ^(6) = 8, 



^(7) = 7, ^(8) = 8. 

 Ex. 3 : Write out in full the transformation h on the operands a, p, y, S, if h(a) 



= y, fi2(a) = j3, /j3(a) = §, /i4(a) = „. 



Ex. 4: If A{n) is n + 2, what is /i(I5)? 



Ex. 5: If/(/0 is -//2 + 4, what is/(2)? 



Ex. 6: If Tin) is 3n, what is T'^Ui)'] (Hint: if uncertain, write out T in extenso.) 



Ex. 7: If / is an identity transformation, and / one of its operands, what is /(O? 



2/16. Product. We have just seen that after a transformation T 

 has been applied to an operand n, the transform T(n) can be treated 

 as an operand by T again, getting T(T(n)), which is written T^in). 

 In exactly the same way Tin) may perhaps become operand to a 



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