One-one relations give a correlation of two classes, term for term, so that each term in either class has its correlate in the other. Such correlations are simplest to grasp when the two classes have no members in common, like the class of husbands and the class of wives; for in that case we know at once whether a term is to be considered as one from which the correlating relation goes, or as one to which it goes. It is convenient to use the word referent for the term from which the relation goes, and the term relatum for the term to which it goes. Thus if and are husband and wife, then, with respect to the relation [Pg 48] "husband," is referent and relatum, but with respect to the relation "wife," is referent and relatum. We say that a relation and its converse have opposite "senses"; thus the "sense" of a relation that goes from to is the opposite of that of the corresponding relation from to . The fact that a relation has a "sense" is fundamental, and is part of the reason why order can be generated by suitable relations. It will be observed that the class of all possible referents to a given relation is its domain, and the class of all possible relata is its converse domain. But it very often happens that the domain and converse domain of a one-one relation overlap. Take, for example, the first ten integers (excluding 0), and add 1 to each; thus instead of the first ten integers we now have the integers These are the same as those we had before, except that 1 has been cut off at the beginning and 11 has been joined on at the end. There are still ten integers: they are correlated with the previous ten by the relation of to , which is a one-one relation. Or, again, instead of adding 1 to each of our original ten integers, we could have doubled each of them, thus obtaining the integers Here we still have five of our previous set of integers, namely, 2, 4, 6, 8, 10. The correlating relation in this case is the relation of a number to its double, which is again a one-one relation. Or we might have replaced each number by its square, thus obtaining the set On this occasion only three of our original set are left, namely, 1, 4, 9. Such processes of correlation may be varied endlessly. The most interesting case of the above kind is the case where our one-one relation has a converse domain which is part, but [Pg 49] not the whole, of the domain. If, instead of confining the domain to the first ten integers, we had considered the whole of the inductive numbers, the above instances would have illustrated this case. We may place the numbers concerned in two rows, putting the correlate directly under the number whose correlate it is. Thus when the correlator is the relation of to , we have the two rows: When the correlator is the relation of a number to its double, we have the two rows: When the correlator is the relation of a number to its square, the rows are: