TRANSFORMATIONS OF A LOGICAL PROPOSITION. 137 



We now go on to the application of this notation to 

 transitive relations. 



The following theorems are taken from De Morgan's 



paper already mentioned, page 16. I use L as the symbol 



of transitive relation, L being his symbol for relation 



generally. It follows from the definition of transitiveness, 



that 



LL=L, and L- 1 L- l =L^ 1 . 



If, then, L means ancestor and its converse Lr 1 means 

 descendant, these equations assert that the ancestor of an 

 ancestor is an ancestor, and the descendant of a descendant 

 is a descendant. I quote De Morgan's theorems in his own 

 words, with the same in my own notation above them ; 

 non-ancestor and non-descendant are indicated by I and 

 l~ l respectively: — 



(i) L<LiL~\ (2) Z^- I <I- I Z,Zr , . 



<i- I /Z~ I . <lil-\ 



<i- I L- , L. <L~ 1 iL. 



(3) l<liL. (4) l-*<i-*L-*l-\ 



<i~ l Ll. <l- l iLr\ 



(5) 1>L~H. (6) 1-*>LI~\ 



>llr\ >l~ 1 L. 



(1) "An ancestor is always an ancestor of all descendants, 

 a non-ancestor of none but non-descendants, a non-descen- 

 dant of all non-ancestors, and a descendant of none but 

 ancestors." 



(2) " A descendant is always an ancestor of none but 

 descendants, a non-ancestor of all non-descendants, a non- 

 descendant of none but non-ancestors, and a descendant of 

 all ancestors." 



SER. III. VOL. VIII. L 



