CHANGE 2/10 



transformation that is single-valued but not one-one will be referred 

 to as many-one. 



Ex. 1: The operands are the ten digits 0, 1, ... 9; the transform is the third 

 decimal digit of logio («+4). (For instance, if the operand is 3, we find 

 in succession, 7, logio?, 0-8451, and 5; so 3-^5.) Is the transformation 

 one-one or many-one? (Hint: find the transforms of 0, 1, and so on in 

 succession ; use four-figure tables.) 



2/9. The identity. An important transformation, apt to be 

 dismissed by the beginner as a nullity, is the identical transformation, 

 in which no change occurs, in which each transform is the same as 

 its operand. If the operands are all different it is necessarily one- 

 one. An example is/ in Ex. 2/6/2. In condensed notation n'=n. 



Ex. 1 : At the opening of a shop's cash register, the transformation to be made 

 on its contained money is, in some machines, shown by a flag. What flag 

 shows at the identical transformation ? 



Ex. 2 : In cricket, the runs made during an over transform the side's score from 

 one value to another. Each distinct number of runs defines a distinct 

 transformation : thus if eight runs are scored in the over, the transformation 

 is specified by ii' = n + S. What is the cricketer's name for the identical 

 transformation ? 



2/10. Representation by matrix. All these transformations can 

 be represented in a single schema, which shows clearly their mutual 

 relations. (The method will become particularly useful in Chapter 

 9 and subsequently.) 



Write the operands in a horizontal row, and the possible transforms 

 in a column below and to the left, so that they form two sides of a 

 rectangle. Given a particular transformation, put a "-)-" at the 

 intersection of a row and column if the operand at the head of the 

 column is transformed to the element at the left-hand side; otherwise 

 insert a zero. Thus the transformation 



would be shown as 



ABC 

 A C C 



The arrow at the top left corner serves to show the direction of the 

 transitions. Thus every transformation can be shown as a matrix. 



15 



