ANSWERS TO THE EXERCISES 7/13 



6/11. 1: Think of .y as the price of butter and y as the price of sugar; their 

 difference now is .v — v; tomorrow's difference is (x — y)'; and this is 

 the same as tomorrow's price of butter less tomorrow's price of sugar, 

 x' - y'. 



6/12. 1: It is if the one-one transformation is regarded as simply an extreme 

 case of the many-one. 



6/13. 1: Even -|- Even = Even, E+0 = 0,0 + E=0, + 0=E. 

 2: (Let "x + >■" mean "merge x and .v"). The systems are: (i) a + b, 

 (ii) c + d, (ill) a + b and c + d, (iv) b + c -\- d, {v) a -\- b -\- c -\- d, and 

 (vi) (ex officio) the original system with none merged. 3: The states 

 (a-,.v') and (— x,.v) can be merged, for the change of x's sign does not 

 alter the next state {x',y'); thus, to be told only that the present state is 

 ( ±4,-2), without specification of x's sign, is still sufficient to show that 

 the next state must be the single one of (+2,-1-14). 



6/16. 1: System and model would be indistinguishable. 2: It persists, so 

 does the brain; they are isomorphic at the lowest level. 3: (i) a,b+c+d 

 is isomorphic with p,q+r; (ii) a+b+c+d is isomorphic withp + q + r. 



7/6. 1: 26 X 26 X 26, which is 17,576. 2: 16. 3: 11. 4: 2 x 2 x 2 . . . 

 ten times, i.e. 1024. 5: S'^ must be not less than 2 x 10^ so, taking 

 logs to any convenient base (10 will do): 



xlogS > log 2 + 9 log 10 



.-. X > (log 2 + 9 log 10)/Iog 5 

 > 13-3; 



so at least 14 such tests would be necessary. 6: (i) 27, (ii) 21. 7: 27. 

 8: 33 = 27 and 3^ = 81 ; so, to select one from 52, four indications would 

 be necessary. 9: None, the father's group can only be O. 



7/7. 1: One bit. 2: (i) 2-32 bits, (ii) 30-9 bits. 3: 4-7 bits. 4:5x4-7 = 

 23-5 bits. 5: (i) 1 bit, (ii) 20 bits. 6: 220, i.e. 1,048576. 7: The re- 

 placement of each question mark has variety of log26 bits, so the whole 

 has variety of 6 log2 6 bits, i.e. 15-5 bits. 8: n \0g2n bits. 9: 12000 

 bits. 10: A page of 5000 words would carry about 50,000 bits — more 

 than the record. 11: Other things being equal the varieties must be 

 equal. 12: That of "all possible pamphlets that are printed in Enghsh 

 and that take ten minutes in the reading". The variety belongs not to 

 the pamphlet but to this set. 13: Certainly; it has only to be distinct 

 from the other possibilities. 



7/12. 1: No, for all combinations of past marital state and present marital 

 state are included. 2: Yes; four possibilities are missing. 



7/13. 1: Three, so far as the quantities mentioned are concerned. 2: Yes, 

 if the hands are accurately set; thus the hour-hand being midway 

 between two numbers implies that the minute-hand is at the "half-past". 

 3: One; for the information given by the minute hand is implied by that 

 given by the hour-hand. 4: The chameleon's have four; Man has a 

 little more than two, for his eyes can move with slight independence. 

 5: Two. 6: One, for its variety cannot exceed that of a; it would still 

 be 1 however many components the vector had. 7: Before the graph 

 is given, y might, for given x, have any value over >''s range; but after 

 the graph is drawn /s value, for given x, is limited to some one value. 

 8: Six. 



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