CHAP. XIX.] OF STATISTICAL CONDITIONS. 299 



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



-(l)-(l) + *(y)-(l) + n(0) 



= nx -f ny ft(l). 



The minor limit of nxy is therefore either or n (x) + n (y) - w(l), 

 according as n (y) is less or greater than n (1 - #), or, which is an 

 equivalent condition, according as n (x) is greater or less than 

 n(\-y). 



Now as is necessarily a minor limit of the numerical value 

 of any class, it is sufficient to take account of the second of the 

 above expressions for the minor limit of w (#?/). We have, there- 

 fore, 



Major limit of n (xy) = least of values n (x) and n (y). 

 Minor limit of n (xy} = n (x) + n (y) - n (1).* 



PROPOSITION I. 



^* 



5. To express the major and minor limits of a class represented 

 by any constituent of the symbols x, y, z, -c., having given the va- 

 lues ofn (x), n (y), n (*), *c., and n (1). 



Consider first the constituent xyz. 



It is evident that the major numerical limit will be the least 

 of the values n(x), n(y), n(z). 



The minor numerical limit may be deduced as in the previous 

 problem, but it may also be deduced from the solution of that 

 problem. Thus : 



Minor limit of n (xyz) = n (xy) + n(z) - n (1). (1) 



Now this means that n (xyz) is at least as great as the expres- 

 sion n(xy) + n(z) - ra(l). But n(xy) is at least as great as 

 n (x) + n (y) - n (1). Therefore n (xyz} is at least as great as 



n (x) + n (y) -n(\) + n(z)-n (1), 

 or n (x) + n (y) + n(z) ~ 2n (1). 



* The above expression for the minor limit of nxy is applied by Professor 

 De Morgan, by whom it appears to have been first given, to the syllogistic form : 

 Most men in a certain company have coats. 

 Most men in the same company have waistcoats. 

 Therefore some in the company have coats and waistcoats. 



