Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/57

 Instead of $$\scriptstyle{\overset{\rightarrow}{R}}$$ and $$\scriptstyle{\overset{\leftarrow}{R}}$$ we sometimes use $\scriptstyle{\text{sg}'R}$, $\scriptstyle{\text{gs}'R}$, where "sg" stands for "sagitta," and "gs" is "sg" backwards. Thus we put These notations are sometimes more convenient than an arrow when the relation concerned is represented by a combination of letters, instead of a single letter such as R. Thus e.g. we should write $\scriptstyle{\text{sg}'(R\dot\cap S)}$, rather than put an arrow over the whole length of $\scriptstyle{(R\dot\cap S)}$.

The c1ass of all terms that have the relation R to something or other is called the domain of R. Thus if R is the relation of parent and child, the domain of R will be the class of parents. We represent the domain of R by "D'R." Thus we put Similarly the class of all terms to which something or other has the relation R is called the converse domain of R; it is the same as the domain of the converse of R. The converse domain of R is represented by "ꓷ'R"; thus The sum of the domain and the converse domain is called the field, and is represented by C'R: thus

The field is chiefly important in connection with series. If R is the ordering relation of a series, C'R will be the class of terms of the series, $$\scriptstyle{\text{D}'R}$$ will be all the terms except the last (if any), and ꓷ$$\scriptstyle{'R}$$ will be all the terms except the first (if any). The first term, if it exists, is the only member of $\scriptstyle{\text{D}'R\cap\lnot}$ꓷ$\scriptstyle{'R}$, since it is the only term which is a predecessor but not a follower. Similarly the last term (if any) is the only member of ꓷ$\scriptstyle{'R\cap\lnot\text{D}'R}$. The condition that a series should have no end is ꓷ$\scriptstyle{'R\subset\text{D}'R}$, i.e. "every follower is a predecessor"; the condition for no beginning is $\scriptstyle{\text{D}'R\subset}$ꓷ$\scriptstyle{'R}$. These conditions are equivalent respectively to $$\scriptstyle{D'R=C'R}$$ and ꓷ$\scriptstyle{'R=C'R}$.

The relative product of two relations R and S is the relation which holds between x and z when there is an intermediate term y such that x has the relation R to y and y has the relation S to z. The relative product of R and S is represented by $R; thus we put Thus "paternal aunt" is the relative product of sister and father; "paternal grandmother" is the relative product of mother and father; "maternal