PROPERTIES OF CLASSES OF FUNCTIONS 65 



Thus we have the contradiction that either 



L a. + or L b, + 0. 



It now follows from § 351; § 20a2; § 27.10, that gjj'-^'^^"^"-. 



48g. Miscellaneous examples. — In this section we give examples 

 useful in connection with those already given for the characterizations 

 of the sequel. 



(1) -LCDAAK.KJC,.^; 



(2) -LCDAAK,K,K,,^; 



(3) LCDAAK,K,-K,,^; 



(4) -LC-DAAK,K^,,^; 



49. Characterizations of classes ^ of elements, of developments A, 

 and of classes 9}? of functions. — By comparison of previous theorems and 

 examples there arise many characterizations of classes ^^^ of elements, 

 of developments A, and of classes 9}? of functions. Of these character- 

 izations we indicate the principle ones which are evident. 

 49a. Characterizations* of classes "^ of elements. — 

 Theorem. In the following four sets (I, II12, II123, II) of properties 

 of a class ^ of elements, any two properties of the same set are equivalent. 



* Cf. I. G. A., § 49, for similar characterizations. 



t A° ^ A 5 every class $"'' is the class 5p. I. G. A., § 75. 



t The references are to the theorem and example which in view of the fact that a com- 

 posite property existent for a class *^i is existent for any superclass of ^i (cf. §48), establish 

 the characterization. 



