Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

*97. ANALYSIS OF THE FIELD OF A RELATION INTO FAMILIES.

Summary of *97.
In this number, we consider not only the posterity of a term, but the
— ) 4 -ancestry and posterity together, i.e. R*'x u R*'x. We put
4-) -- 4 -R'x = n-x (l'x n C'R) v R'x Df.
Thus the whole family of a term, i.e. its ancestry and posterity together,
is R*'x. The most important case here is when R e 1 -; in this case
families are mutually exclusive, i.e. we have
4-): R e 1 -- 1. D. R*"C'R e Cls ex2 excl.
In case R e I- 1 and y belongs to a family which has a beginning, i.e.
— ) 4
in case a! BR'y n B'R, the whole family of y consists of the posterity of the
beginning, i.e. we have
-: Re 1 -- 1. xBR. xR*y. D. R'y R'x,
whence
*97-21. F: R e 1 - 1. ). R*"s'gen'R = R*"B'R
When R e l-1, the relation of gen'R to R*"B'R may be pictured as
the relation of rows to columns. E.g. let the field of R consist of the dots
R* * * B~ * *
*. \0 y. * 0
in the accompanying rectangle, and let each dot have the relation R to the
dot below it. Then the top row is B'R, the second row is ('R-R (R2, the