PROPERTIES OF CLASSES OF FUNCTIONS 



57 



(6) + + --;+-: r''; Ap; m - (1, 1), (- 1, 0). 



(7) + ; - - : r''; A¥'; ^ - (l, 1), (l, 0), (0, - 1). 



(8) - + --;--: r'^-, Ai'- 5W - (- 1, 0). 



(9) ; - - : r^ Ap; W = (1, - 1), (- 1, 0). 



48/. LCD-A, 7 examples (§ 23al4i2, 17), on ^;V (§41.2), 5 on 

 ^"^ (§42.7), 7 on ^Vi\— 



(1) 

 (2) 

 (3) 

 (4) 

 (5) 

 (6) 



(7) 



- + + - 

 + - + - 



- + - 



- + — 



+ + — 

 + 



^1* ^2. 





[a, — a] = [all ju3/xi = — jus]. 



[a, - a]. 



[a, ~ a\. 



[a, - a]. 



[a, - o]. 



[a, h, \(a + 6)] 



= [all ;U 9/X3 = Kmi + Ma)]- 

 [a, 6, a — fe] 



= [all M 5 M3 = (mi — ^2)]- 



48?. -LC-DA, 10 examples (§23all,2), ow i^^ (§41.1), 8 on 

 ^n« (§ 42.812), 9 on ^^13 (§ 44.5), 10 on ^''\ 



Ap; go; 



a^"- A,"- gjj 



(1) + + + + 



(2) + - + - 



(3) - - + - 



(4) + + - - 



(5) + 



(6) - + - - 



(7) - + + + 

 (8) 



(9) - + + - 



(10) + + + - 



^1* K^if 

 + + 



- + 



- + 

 + - 



- + 



sj;n2 

 s-pra 



AP=; 93J - (1, 0), (0, 1).* 



AV- 93J - (1, 0), (0, 1). 



A\'- m - (I, 0), (0, I). 



A^,'- m - (1, 0), (0, 1). 



Ap- arj ^ (1, 0), (0, 1). 



AP; 93? ^ (1, 0), (0, 1). 



Al'=; 9« - (1/2), 0), (0, I). 



Aj'- aw - (1, 0), (0, 1). 



AJ^; gji -.(1,0,0), (0, 1,0). 



m = [all M 5 (3/J„ 9 Mp„ = 1 ■ Atp = (p + pj)] 



+ [all M 9 (ni = 3) . (p°*" + 1 . 3 . Mp = 0)] 

 + [all M 9 (mi = 2) . (p"'^" . 3 . Mp = 0)].t 



Proof of 93^'. — The three component classes of 93? are closed as 



* A developmental system is S(W) ? (S"-' = (1, 0), &"•'' = (0, 1) (m)). 

 t A developmental system is 



S(3}f) ^ («;' = 1 (P = . < = (p + (ml)). 



