ANSWERS TO THE EXERCISES 3/6 



c 

 2/17. l:(i) I {ii)f:^gp<^q. 2: It contains no arrows, just 



b^- a^r- d 

 isolated points. 3: Each is composed solely of isolated points and/or 

 simple rings without branches. 

 4: 9^2 5^6 



t I 

 4->0->l<-7^-8 if 4-figure logs are used 



t 

 3 



5:7,1,2,2. 6: No. 7: Yes. 9: No. 



3/1. 1: Possible answers are: (a) soft-boiled egg -^ hard-boiled ; (b) log -> 

 ash; (c) cylinder full of vapour and air -> full of flame; (d) unicellular 

 ovum -> two-celled; (e) cumulus cloud -> thunder storm; (f) oestrus -> 

 pregnancy; (g) low price (with short supply) ^ high price; (h) cat 

 seeing mouse -^ cat chasing mouse; (i) nebulae close -> nebulae 

 dispersed. 



3/4. l:n' = 2n. 2:2,4,8,16,32,64x103. 3: Graph (ii): 1000 -> 2000 ^ 

 4000^... 4: «'= 0-8/7. 5: (i) 800, 640, 510, 410, 330 x 10^; 

 (ii) Zero. 6: It would run to state 3 at which it remains; 3 is the only 

 state it can stop at. 7 : It runs to a cycle of states 2 and 8, between which 

 it oscillates incessantly. 8: Four; two with a state of equilibrium and 

 two with a cycle. 9: «' = 0-9/; + 1,000,000. 10: 20, 19, 181, 17-3, 

 X 106. 11: 10,000,000. 12: If / is its length, its change of length over 

 one interval of time is /' — /; so /' — / = 1 -2, and the transformation is 

 /' = /+l-2. 13: The increase in number (not the next number) 

 is n' — n; so n' — n = 10"8//(108 — n), and the transformation is 

 n' = n + 10-8/2(108 - /;). 14: 19, 34, 57, 81, 97 x lO^. 

 . , {ABC) (BCA) {CAB) 

 *• ^ {BCA) {CAB) {ABC) 

 2: {ABC) 



/ \ 

 {CAB) <- {BCA) 



3: (1,-1), (1,1), (-1,1), (-1,-1). 4: A cycle of four elements. 

 5: (2,3,5), (3,5,8), (5,8,13). 

 3/6. 1: (i,2), (2,-i), (-i,-2), (-2,1), (i,2), etc. 2: (1,2,0,2,2,). 

 3:(2,1,0,2,2,) :^ (1,2,0,2,2,). 4: Further cycles of two elements each, 

 and not connected, would be added. 5: (8, — 3,1). 6: (8,4) transforms 

 to (6,6), at which the system remains. 7: If the operand is {a,b), 

 a' = ^a + ^b, b' = ia + ib. 8: (30,34) -> (28,36) -> (24,40) -^ (16,48) 

 -^ (0,64) -^ ? What happens next cannot be decided until the per- 

 missibility of borrowing is decided. 9:a' = ^{3a — b), b' = i(36 — a). 

 10: Whoever started with most money. 11: m' = m — n, n' = 2n. 

 12: The vector {m,n). 13: (150,10) -> (140,20) ->...-> (0,160), after 

 which the algebraic events no longer parallel the zoological . 14 : .v = 1 0, 

 0, -5, -5, -2i, 0, U, U, I; no. 15: It is heavily damped. 16: If 

 wages are represented by x, and the price index by ;', then the first 

 statement says that x' — x = y — 100, and the second says that 

 y' = x; so the transformation is x' = x + y — \00, y' = x. 

 17: (110,1 10) -> (120,1 10) -> (130,120) ^...->(1540,990). 18: No, the 

 system is caught in a "vicious spiral". 19: (110,110) ^ (110,100) 



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