We dealt in the preceding chapter with a supremely important class, namely, serial relations. Each of the three properties which we combined in defining series—namely, asymmetry, transitiveness, and connexity—has its own importance. We will begin by saying something on each of these three. Asymmetry, i.e. the property of being incompatible with the converse, is a characteristic of the very greatest interest and importance. In order to develop its functions, we will consider various examples. The relation husband is asymmetrical, and so is the relation wife; i.e. if is husband of , cannot be husband of , and similarly in the case of wife. On the other hand, the relation "spouse" is symmetrical: if is spouse of , then is spouse of . Suppose now we are given the relation spouse, and we wish to derive the relation husband. Husband is the same as male spouse or spouse of a female; thus the relation husband can [Pg 42] be derived from spouse either by limiting the domain to males or by limiting the converse to females. We see from this instance that, when a symmetrical relation is given, it is sometimes possible, without the help of any further relation, to separate it into two asymmetrical relations. But the cases where this is possible are rare and exceptional: they are cases where there are two mutually exclusive classes, say and , such that whenever the relation holds between two terms, one of the terms is a member of and the other is a member of —as, in the case of spouse, one term of the relation belongs to the class of males and one to the class of females. In such a case, the relation with its domain confined to will be asymmetrical, and so will the relation with its domain confined to . But such cases are not of the sort that occur when we are dealing with series of more than two terms; for in a series, all terms, except the first and last (if these exist), belong both to the domain and to the converse domain of the generating relation, so that a relation like husband, where the domain and converse domain do not overlap, is excluded.