PROPERTIES OF CLASSES OF FUNCTIONS 13 



(2) 2)J . 3 . 9J?/. 



(3) an^ . 3 . Wu = m . m^ = m,,. 



(4) 2«^^ D . sou = 9)J- 



(5) 2«^' . 3 . s)^''''""". 9!)f*^^'. 



(6) W^ . 3 . 9J^ . 9J?/-™ . 2>?** = 3}U. 



10. Composition of classes of elements* — From two classes 



of elements we derive the product class 



%^'r' - Kp', p")] - [p'p"], 



i. e., the class '^ = [p] whose elements p = {p' , p") are bipartite, the 

 first part p' ranging over %V and the second part p" ranging over *i)3". 

 In practice the notation {p', p") is replaced by p'p". 

 The classes *ip', ^" may be the same. We write 



^^^;> = [(pi, P2)] ^ biP2]. 



Thus ;;', p" are generic elements of the respective classes %', '^" 

 conceptually distinct, but not necessarily actually distinct, while pi, 

 Pt are independent (conceptually distinct) generic elements of the 

 class '^. 



10a. Remark.] — It should be noticed that the theory of functions 

 and classes of functions on "^ to 21 is applicab'e to functions and 

 classes of functions on ^^''ip", etc., to 21. The class 'i)3 in its generality 

 includes classes '^'%", etc. 



11. Reduction of classes of elements and of functions, t — A reduction 

 R oi a, class '^ of elements is the transformation of ^ into a subclass 

 "^R = [pn] of 'ip. Any subclass ^i of ^ defines a reduction jB of "ip, 

 the transformation being the transformation of '^ into '^i. A re- 

 duction R transforms a function /j on *il3 to 21 into a function hk on 



(6) If 50! has the property D then 91 has the property D and 3)}^, has the properties 

 LCD and the ♦-extension of 9Ji, is the same as 93}*. 



For a more extensive bodj- of propositions than the above see I. G. A., §§ 22o; 26; 44a. 

 Proposition (6) above is of fundamental importance in the general theory. 



*I. G. A., §51. 



fl. G. A., §51a. 



tl. G. A., §52. 



