64 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
361. Two classes U and Z are considered which are such that whatever class A 
may be, we have 
U + A— U, and therefore also UA=A 
Z-fi A=A, and therefore also ZA =Z 
so that U contains every class, and Z is contained in every class. U is the class 
called the “universe,” Z is that called the “non-existent’ class. The class U need 
not be taken as the actual universe of thought, but as that class which contains all 
classes considered in an investigation. So Z may be taken to be that class which in 
any investigation is contained in all others considered, and is ignored, or regarded as 
having the quality of “ non-existence.” 
362. Every class A is accompanied by a class A' which may be called its obverse , 
and is such that 
A-f A'=U 
AA = Z 
so that A is the obverse of A'. A' is “not A.” U and Z are obverses of each other. 
We have 
(A-j- B+C-f D-f . . .)'=A'B / C'D / . . . 
(ABCD . . .) / =A'+B / +C / +D'+ . . . 
363. Now, consider a collection of n classes a, b, c, d, . . . such that the product of 
every two is a class z; i.e., such that no two classes of the collection have any part in 
common except that which is common to all the classes, and may therefore, when our 
attention is confined to the classes of the collection and their aggregates, be regarded as 
immaterial, and therefore be treated as non-existent. Such a collection may be said 
to consist of separated classes. Taking these, their sums two, three, four, _ &c. 
together, and the class z, we have a collection of 2 n classes, which may be called a 
full set, and maybe said to be derived from the collection of n separated classes.* All 
the classes of the set are contained in a class u of the set which is the sum of the 
separated classes from which the full set is derived, and may be called the universe of 
the set. The class z may be called the zero of the set. The whole system of classes 
involved in any inquiry is a system of 2 m classes where in is the number of separated 
classes, the product of each two of these being the non-existent class Z. In discussing 
a system of classes we have under consideration a number of full sets with their 
accompanying universes and zero classes. 
364. If a be any class of a full set of which u is the universe and z the zero class, 
we have 
a-fu—u and an —a 
ct-j-z —a and az —z 
* A more general meaning is given to the expression “fall set” in sec. 381, and it must not therefore 
he assumed that conversely every full set can be derived in the manner just indicated. 
