14 PITCftER: INTERRELATIONS OF 



^s to 21. Thus 



We speak of the reduced class f<n and the corresponding reduced 

 function hr. 



In case a property P of functions is possessed by hb the function 

 IX is said to possess the property P on the class 'iPjf, in notation: 

 f^J'<VE\ Similarly in case a property P of classes of functions is 

 possessed by a class We = [mJ, the class 9)} = [lu] is said to possess 

 the property ^T? on ^]?jj, in notation: W^^'J'K 



11a. Propositions* — The following propositions are relative to a 

 class '^, a reduction R applicable to *p, and classes SOJ and 2 of func- 

 tions on 'ip to 31. 



(1) mL)R^mE)L. (4) 5m^.3.a«/. 



(2) t ®*»™. 3 . @/»^'«. (5) ®Z^- . 3 . 9(rJ/'. 



(3) S[R^ .'z) . a«^-\ (6)t 9JJ^.3.5W/. 



12. Composition of classes of functions.^ — From any two classes: 



W - [m']; »r' - W'], 



of functions : 



m' = (m,''!7j'); m^' = Cm;.'"!?^"), 



on the classes ^'; ^" respectively to 21, arises by multiplication of 

 constituent functions the product class: 



Wm" ^ Wix"], 

 of functions: 



mV" = ii^',,' iJ-','" I p'p"), 



on the product class ^''X'" to 31. 



It is to be noticed that while the notion of composition of classes 

 of functions is very different from that of composition of classes of 

 elements, the same notations are used in each case. This need cause 

 no confusion. 



We desire to call attention to the remark of §10a. The theory 

 previously given relative to classes of functions 90? on '"13 to 31 applies 



*I. G. A., §52a. 



t The property belonging to is an example of a bipartite property of classes of functions 

 invariant under a reduction of classes. I. G. A., § 526. 



J The properties A, L, D, Di are examples of unipartite properties invariant under 

 reduction. I. G. A., § 526. 



§ I. G. A., § 53. 



