PROPERTIES OF CLASSES OF FUNCTIONS 49 



The ten composite properties 



(+ + + •); (- ■ • •), 



are all possessed by classes of functions on •>!? dual and the 16 com- 

 posite properties are all possessed by classes of functions on 'i? de- 

 numerable. 



46. The complete existential theory of the properties L; C; D; A; A; 

 Ki; K2; if 12*- — Certain general relations among these properties are 

 expressed by the propositions § 23alli2, 14i2, 17, 18.* The effect of 

 these propositions is to cut down the number of existent composite 

 properties from 256 to 148. In fact by § 23alli2 all existent composite 

 properties are of the form 



( -) or ( + + +), 



and by § 23al4i2, 17 none of these can be of the form: 



(•■ + • + + + -) or (++•-• + + +). 



The following table classifies the existent composite properties as 

 to L; C; D; A giving the number of existent composite properties for 

 '^ unrestricted, ^1.^ finite, '^.^ dual, and 'ip singular. 



9 9 9 4 



99 9 4 



9 9 9 4 



9 9 9 4 



9 9 9 4 



7 7 5 



10 9 8 



10 9 8 



10 9 8 



10 9 8 



10 or 9t 



9 



8or7t 



9 



10 or 9t 



10 



Total 145^146-'147^I48 88 82 20 



* § 23alS follows from § 2.3al4i2, 1". 



t We have neither theorem nor examples to show that the composite properties 



(+ + - + + + + -),(+ + + + + -),(+ h + + + -), are existent 



or non-existent. 



