298 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



will be a superior limit of the number of individuals contained in 

 the given class. 



2nd. A minor limit of the number of individuals in any class y 

 will be found by subtracting a major numerical limit of the con- 

 trary class, I-?/, from the number of individuals contained in the 

 universe. 



To exemplify these principles, let us apply them to the fol- 

 lowing problem : 



PROBLEM. Given, ft(l), n(x)^ and n(y), required the su- 

 perior and inferior limits of nxy. 



Here our data are the number of individuals contained in the 

 universe of discourse, the number contained in the class #, and 

 the number in the class ?/, and it is required to determine the 

 limits of the number contained in the class composed of the indi- 

 viduals that are found at once in the class x and in the class y. 



By Principle i. this number cannot exceed the number con- 

 tained in the class #, nor can it exceed the number contained in 

 the class y. Its major limit will then be the least of the two va- 

 lues n(x) and (y). 



By Principle n. a minor limit of the class xy will be given by 

 the expression 



n (1)- major limit of {x(\ -y) + y(\ -#) + (!-#) (l-y)J,(l) 



since x (1 - y) + y (1 - x) + (1 - #) (1 - y) is the complement of 

 the class xy, i. e. what it wants to make up the universe. 



Now x (1 - y) + (1 - x) (1 - y) = 1 - y. We have there- 

 fore for (1), 



n (1) - major limit of { 1 -y + y (1 - x)} 

 = n (1) - n (1 - y) - major limit of y (1 - #). (2) 



The major limit of ?/(! - x) is the least of the two values n(y) 

 and n(\ - x). Let n (y) be the least, then (2) becomes 



n(\)-n(\-y)-n(y) 

 = (1) -*(!) + fi (y)-n(y) = 0. 

 Secondly, let n (1 - x) be less than n (#), then 

 major limit of ny (1 - x) - n (1 - x) ; 

 therefore (2) becomes 



