ANSWERS TO THE EXERCISES 5/6 



4/12. 1: (with boxes omitted for simplicity) : (i) >" ^- x; y dominates x; 



(ii) V a system with feedback; 



/\ 

 X -^ z 



(iii)M^A: v:^:^;'; the "whole" actually consists of two unconnected 

 parts; (iv) ii -^ x -^ y -^ z; a chain of action; 

 u 



/ 

 (v) V -> x\ \ dominates all the other three; 



' \ 



z 

 u 



\ 



(vi) X -^ z; z is dominated by the other three. 2: When y is zero. 



/ 



y 



4/13. 1 : z dominates x, y is independent of both. 



4/14. 1 : (iii) only. 



4/15. 1 : If the variables are S = Singing, L = Laughter, X = Organ- 

 playing, Y = Incense-burning, and each take the values or 1 for 

 inactive or active respectively, then the machine with input is soon 

 found to be 



One way to (0,0) is : Stop the incense burning for one minute ; next stop 

 the incense and play the organ; finally, start burning the incense again; 

 in future keep it burning and never play the organ. 2: Yes, for the L's 

 transitions are affected by 5"s values. 3: No. 4: X-^ S:^L<- Y. 



4/19. 1 : A possible method is roll a die and let its first number give the trans- 

 form of 5i, and so on. 2: A possible method is to number six cards 

 from 1 to 6, shuffle them, deal in a row, and then fill in the states in the 

 same order. 5: See 5.4/20. 



4/20. 1 : Yes, no, no. 



5/3. 2: No. 3: The only one is (0,0). 4: All the points on the ^'-axis are 

 equilibrial. 5: j = 0, k= —1. 6: Yes. 7: No. 8: Every arrow 

 returns to its state of origin, so the representative point is immobile. 

 9: Identity. 10: Yes. 11: Yes. 



_,. ^c^,.,abcdefg 3: No. 4: No. 5: No. 6: Every 

 5/4. 1 : Such is i , , ^ 



trajectory is a cycle. 7: No. 



5/5. 1: 6 + c -I- gonly. 2: Yes. 3: Yes. 4: Yes. 



5/6. 1: Yes; the sequence D{c), TD(c), T^D{c), T^D{c), . . . is d, a, c, c, . . . 

 2: No; the limit is not e. 3: The system, though displaced out of the 

 set, always comes back to it. 



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