2/5 AN INTRODUCTION TO CYBERNETICS 



Ex. 1 : If the operands are the positive integers 1, 2, 3, and 4, and the operator 

 is "add three to it", the transformation is: 



I 1 2 3 4 

 ^4567 

 Is it closed? 



Ex. 2: The operands are those EngUsh letters that have Greek equivalents (i.e. 



excluding J, q, etc.), and the operator is "turn each EngUsh letter to its 



Greek equivalent". Is the transformation closed ? 

 ^A . 3 : Are the following transformations closed or not : 



^: j " * ^ ^ B:\^^P'I 



^ a a a a g J q P 



^'•^ g f q ^'- ^ g f 



Ex. 4: Write down, in the form of Ex. 3, a transformation that has only one 

 operand and is closed. 



Ex. 5 : Mr. C, of the Eccentrics' Chess Club, has a system of play that rigidly 

 prescribes, for every possible position, both for White and Black (except 

 for those positions in which the player is already mated) what is the player's 

 best next move. The theory thus defines a transformation from position 

 to position. On being assured that the transformation was a closed one, 

 and that C always plays by this system, Mr. D. at once offered to play C 

 for a large stake. Was D wise? 



2/5. A transformation may have an infinite number of discrete 

 operands; such would be the transformation 



I 1 2 3 4 ... 

 ^ 4 5 6 7 ... 



where the dots simply mean that the list goes on similarly without 

 end. Infinite sets can lead to difficulties, but in this book we shall 

 consider only the simple and clear. Whether such a transformation 

 is closed or not is determined by whether one cannot, or can 

 (respectively) find some particular, namable, transform that does 

 not occur among the operands. In the example given above, each 

 particular transform, 142857 for instance, will obviously be found 

 among the operands. So that particular infinite transformation is 

 closed. 



Ex. I : In ^ the operands are the even numbers from 2 onwards, and the trans- 

 forms are their squares : 



A-i^ 4 6... 

 • ^ 4 16 36... 

 Is A closed? 

 Ex. 2: In transformation B the operands are all the positive integers 1, 2, 3, 

 . . . and each one's transform is its right-hand digit, so that, for instance, 

 127 ^ 7, and 6493 -> 3. Is B closed? 



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