2/7 AN INTRODUCTION TO CYBERNETICS 



We shall often require a symbol to represent the transform of such 

 a symbol as n. It can be obtained conveniently by adding a 

 prime to the operand, so that, whatever n may be, n -> n . Thus, . 

 if the operands of Ex. 1 are «, then the transformation can be written 

 as«' = « + 10 (« = 1,2,3). 



Ex. 3 : Write out in full the transformation in which the operands are the three 



numbers 5, 6 and 7, and in which //' = « — 3. Is it closed? 

 Ex. 4: Write out in full the transformations in which: 



Ci)n' = 5n {n = 5,6,7); 

 in)n' = 2«2 (« = - 1,0, 1). 



Ex. 5: If the operands are all the numbers (fractional included) between and 1, 

 and «' = ^ti, is the transformation closed? (Hint: try some representative 

 values for /?: |, |, i, 001, 0-99; try till you become sure of the answer.) 



Ex. 6: (Continued) With the same operands, is the transformation closed if 

 n' = l/(« + D? 



2/7. The transformations mentioned so far have all been character- 

 ised by being "single-valued". A transformation is single-valued 

 if it converts each operand to only one transform. (Other types 

 are also possible and important, as will be seen in S.9/2 and 12/8.) 

 Thus the transformation 



. A B C D 

 ^B A A D 



is single-valued; but the transformation 



, A B CD 



^ Box D A Box C D 



is not single-valued. 



J 

 2/8. Of the single-valued transformations, a type of some import- 

 ance in special cases is that which is one-one. In this case the trans- 

 forms are all different from one another. Thus not only does each 

 operand give a unique transform (from the single-valuedness) but 

 each transform indicates (inversely) a unique operand. Such a 

 transformation is 



.ABCDEFGH 

 ^ F H K L G J E M 



This example is one-one but not closed. 



On the other hand, the transformation of Ex. 2/6/2(e) is not one-one, 



for the transform "1" does not indicate a unique operand. A 



14 



