CHANGE 2/4 



Another example of a transformation is given by the simple coding 

 that turns each letter of a message to the one that follows it in the 

 alphabet, Z being turned to ^; so CAT would become DBU. The 

 transformation is defined by the table: 



Y->Z 

 Z^A 



Notice that the transformation is defined, not by any reference to 

 what it "really" is, nor by reference to any physical cause of the 

 change, but by the giving of a set of operands and a statement of 

 what each is changed to. The transformation is concerned with 

 what happens, not with why it happens. Similarly, though we may 

 sometimes know something of the operator as a thing in itself (as 

 we know something of sunlight), this knowledge is often not essen- 

 tial; what we must know is how it acts on the operands; that is, we 

 must know the transformation that it effects. 



For convenience of printing, such a transformation can also be 

 expressed thus: 



, A B ... Y Z 

 ^ B C ... Z A 



We shall use this form as standard. 



2/4. Closure. When an operator acts on a set of operands it may 

 happen that the set of transforms obtained contains no element that 

 is not already present in the set of operands, i.e. the transformation 

 creates no new element. Thus, in the transformation 



I A B ... Y Z 

 ^ B C ... Z A 



every element in the lower line occurs also in the upper. When this 

 occurs, the set of operands is closed under the transformation. The 

 property of "closure" is a relation between a transformation and a 

 particular set of operands; if either is ahered the closure may alter. 

 It will be noticed that the test for closure is made, not by reference 

 to whatever may be the cause of the transformation but by reference 

 to the details of the transformation itself. It can therefore be applied 

 even when we know nothing of the cause responsible for the changes. 



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