PROPERTIES OF CLASSES OF FUNCTIONS 



55 



one needs to show that for no representative system does there exist a 

 developmental sj^stem !j)(9Jf)> belonging to the class 9)?. 



If a particular class 9)f' has the property A';^ (i = 1, 2, 12) the 

 proof of the same may be made either by recognizing that other 

 properties possessed by 9K' imply K',^ or, if this is not true, then by 

 actual computation and identification of the classes C^l"^")^ and 

 gi = [all ^x.'"''iK")-B<i'""(!''>] where W is an arbitrary class possessing 

 the properties L; C; D. The class gi2 is usually best computed as 

 the greatest common subclass of the classes %i and go- If for a 

 particular W"'^^^ the class {WW)^ is not the same as the class 

 [all ^J^i'»'(-^'").Bo«'"(p'i] then m' does not have the property A';.*. 

 The absence of A;^ in the examples given is best shown by taking 

 93?" as 31 and showing that W^ and [all ^' ' "' ] are not the same. 



48a. LC DA, 9 examples (§ 23alli2, 17), 4 on ^;^' (§41.6, 7, 12), 

 9 on 'iP"^— 



'^^■, M; 'm ^ 0.* 



'^'] M; m ^ 0. 



i^^; A}; m ^ 0. 



^;V; A.^; 9)? ^ [a] ^ W = 21. 



<j3n=; AY^; ^l ^ [0, a] ^ [all M J Ml = 0]. 



i^i- AF- m = [0, a]. 



^]?"- Ap; m = [a, b] ^ W. 



%^"'-, Ar%- 9JJ = [a, 6]t. 



^Vi- AJ'- 9)J ^ [0, a]. 



486. "L CDil, 9 examples (§23alli.2, 17), 4 on "ip' (§41.7, 14), 

 9 on '^^'K— 



A A', A'j A',2, ; A„ A'j* 



* 9)! consists of the zero function. 



t For this case a developmental system is: 



3)(»!) 9 (5'"i = (1, 0) (»(.'■"■■) . «"> = (1, 1) im"^^)). 

 i SR consists of a single function whose functional value is 1. 

 § 9)J consists of a single function m such that iii = 1, m2 =0. 



