214 



MR. JOSEPH JOHM MURPHY ON ADDITION 



We have now these four relations : — 



A=LB, or A is included in B ; 

 A=L~^B, or A includes B ; 

 A = NB, or Nothing is both A and B ; 

 A=MB, or Every thing is either A or B. 



These are expressed in Boole^s and Jevons^s systems by the 

 following : — 



A=AB, or Ab = o 

 AB = B^ or dB = o; 



A=A6^ or AB = o; 



d = aB, or db = o. 



If we multiply these four relatives into each other^ ex- 

 changing the places of iV" and M in the column of multi- 

 pliersj we obtain the symmetrical result shown in the 

 following Table. The products / and Z~' ought properly 

 to be omitted from this table_, and the squares containing 

 them left blank, because these products are not of the same 

 form with any of the factors; but they are inserted in 

 order to facilitate comparison with the larger table further 

 on. As we shall see, / is the denial of L, and l~^ oi L~^ ; 

 that is to say, 



and 



A=ZB means some A is not B, 



A=/~^B means some B is not A. 





L 



i-' 



N 



M 



L 



L 



i° 



N 





i-' 



(i-0° 



i-i 



1 



M 



M 



M 





i- 



M° 



N 



h' 



N 



i\-° 



L 



