2/11 



AN INTRODUCTION TO CYBERNETICS 



If the transformation is large, dots can be used in the matrix if 

 their meaning is unambiguous. Thus the matrix of the transforma- 

 tion in which n' = n -\- 2, and in which the operands are the positive 

 integers from 1 onwards, could be shown as 



(The symbols in the main diagonal, from the top 

 have been given in bold type to make clear the pos 



left-hand corner, 

 tional relations.) 



Ex. 1 : How are the +'s distributed in the matrix of an identical transformation? 

 Ex. 2: Of the three transformations, which is (a) one-one, (b) single-valued but 

 not one-one, (c) not single-valued ? 



(i) (ii) (iii) 



a matrix with (a) a row entirely of 



= 2« and the integers 



Ex. 3: Can a closed transformation have 

 zeros ? (b) a column of zeros ? 



Ex. 4: Form the matrix of the transformation that has //' 



as operands, making clear the distribution of the -|-'s. Do they lie on a 

 straight line? Draw the graph of >- = 2x; have the lines any resemblance? 



Ex. 5 : Take a pack of playing cards, shuffle them, and deal out sixteen cards 

 face upwards in a four-by-four square. Into a four-by-four matrix write 

 + if the card in the corresponding place is black and if it is red. Try 

 some examples and identify the type of each, as in Ex. 2. 



Ex. 6: When there are two operands and the transformation is closed, how 

 many different matrices are there? 



Ex. 7: (Continued). How many are single- valued? 



REPEATED CHANGE 



2/11. Power. The basic properties of the closed single-valued 

 transformation have now been examined in so far as its single 

 action is concerned; but such a transformation may be applied 

 more than once, generating a series of changes analogous to the 

 series of changes that a dynamic system goes through when active. 



16 



