errata p. 34, line 15, for "x" read "R."
34 INTRODUCTION [CHAP. diversity, agreement or disagreement in any respect, are sYInmetrical relations. A relation is called asymmetrical when it is incompatible with its converse, v . i.e. when R n R = A, or, what is equivalent, xRy. ::>x,y . t’.) (yRx). Before and after, greater and less, ancestor and descendant, are asym- metrical, as are all other relations of the sort that lead to series. But there are many asynunetrical relations which do not lead to series, for instance, that of wife’s brother.. A relation may be neither symmetrical nor asymmetrical; for example, this holds of the relation of inclusion between classes: a C f3 and f3 C ex will both be true if ex = (3, but otherwise only one of them, at most, will be true. The relation brother is neither symmetrical nor asymmetrical, for if x is the brother of y, y may be either the brother or the sister of x. In the propositional function xRy, we call x the referent and y the relatlm. The class tc (xRy), consisting of all the x’s which have the relation R to y, is called the class of referents of y wl’th respect to x he c lass y (xRy), consisting of all the y’s to which x has the relation R, is ca lTed t h e class of relata of x wth respect to R. These two classes are denoted
respectively by R’y and R’x. Thus -+ R’y = tc(xRy) Df,
’:'J. V
R’x=y(yRx) Df. The arrow runs towards y in the first case, to show that we are concerned with things having the relation R to y; it runs away from x in the second
case to show .that the relation R goes from x to the members of R’x. It runs in fact ft.om a referent and towards a relatum. -+ The notations R’y, R’x are very important, and are used constantly. If -+ R is the relation of parent to child, R’y = the parents of y, R’x = the children of x. We have and -+ I- : a; € R’y . = . xRy
I- : y € R’x . = . xRy. These equivalences are often embodied in common language. For example, we say indiscriminately" x is an inhabitant of London" or" x inhabits London." Ifwe put "R" for "inhabits," ’(x inhabits London" is "xR London," while "x -+ is an inhabitant of London" is C( X € R’ London."
- This relation is not strictly asymmetrical, but is so except when the wife’s brother is also
the sister’s husband. In the Greek Church the relation is strictly asymmetrical. I ,&
