Volume 1 / Part 1 — *33. DOMAINS, CONVERSE DOMAINS, AND FIELDS OF RELATIONS.
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*33. DOMAINS, CONVERSE DOMAINS, AND FIELDS OF RELATIONS.
Summary of *33.
If is any relation, the domain of , which we denote by , is the class of terms which have the relation to something or other; the converse domain, , is the class of terms to which something or other has the relation ; and the field, , is the sum of the domain and the converse domain. (Note that the field is only significant when is a homogeneous relation.)
The above notations , , are derivative from the notations , , for the relations, to a relation, of its domain, converse domain, and field respectively. We are to have hence we define , , as follows:
*33·01.
*33·02.
*33·03.
The letter is chosen as the initial of the word "campus." We require one other definition, namely of the relation of to when is a member of the field of . This relation, which we will call , is defined as follows:
*33·04.
We shall find that . will be the relation of a relation to its domain, will be the class of relations having for their domain. Similar remarks apply to and . The field of a relation is specially important in connection with series.
The propositions of this number are constantly used throughout the remainder of the work. The ideas of the domain, converse domain, and field are very general, and have somewhat different uses for relations of different kinds. Consider first the sort of relation that gives rise to a descriptive function . For this we require that should exist whenever there is anything having the relation to , i.e. that there should never be more than one term having the relation to a given term . In this case, the values of for which exists will constitute the "converse domain" of , i.e. , and the values which assumes for various values of will constitute the "domain" of , i.e. . Thus the converse domain is the class of possible arguments for the descriptive function , and the domain is the class of all values of the function. Thus, for example, if is the relation of the square of an integer to , then =the square of , provided is an integer. In this case, is the class of integers, and is the class of perfect squares. Or again, suppose is the relation of wife to husband; then =the wife of , =married men, =married women. In such cases, the field usually has little importance; and if the values of the function are not of the same type as its arguments, i.e. if the relation is not homogeneous, the field is meaningless. Thus, for example, if is a homogeneous relation, and are not homogeneous, and therefore "" and "" are meaningless.
Let us next suppose that is the sort of relation that generates a series, say the relation of less to greater among integers. Then = all integers that are less than some other integer = all integers, = all integers that are greater than some other integer = all integers except 0. In this case, = all integers that are either greater or less than some other integer = all integers. Generally, if generates a series, = all members of the series except the last (if any), = all members of the series except the first (if any), and = all members of the series. In this case, "" expresses the fact that is a member of the series. Thus when generates a series, becomes important, and the relation is likely to be useful.
We shall have occasion to deal with many relations having some of the properties of series, and with many propositions which, though only important in connection with serial relations, hold much more generally. In such cases, the field of a relation is likely to be important. Thus in the section on Induction (Part II, Section E), where we are preparing the way for the construction of serial relations by means of a certain kind of non-serial relation, and throughout relation-arithmetic (Part IV), the fields of relations will occur constantly. But in the earlier parts of the work, it is chiefly domains and converse domains that occur.
Among the more important properties of domains, converse domains and fields, which are proved in the present number, are the following.
We have always , , (*33·12·121·122). (The last of these, however, is only significant when is homogeneous.)
*33·13.
*33·131.
*33·132.
*33·14.
*33·16.
*33·2·21·22. The converse domain of a relation is the domain of its converse, the domain of a relation is the converse domain of its converse, and the field of a relation is the field of its converse.
*33·24.
*33·4.
with corresponding propositions (*33·41·42) for and .
*33·43.
*33·431.
*33·5.
*33·51.
The proofs of propositions concerning and are usually similar to those for , and are therefore often omitted.
*33·01.
*33·02.
*33·03.
*33·04.
*33·1.
*33·101.
*33·102.
*33·103.
*33·11.
*33·111.
*33·112.
*33·12.
*33·121.
*33·122.
*33·123.
*33·124.
*33·125.
*33·13.
*33·131.
*33·132.
*33·14.
Dem.
*33·15.
Dem.
*33.151.
*33·152.
*33·16.
Dem.
*33·161.
*33·17.
*33·18.
Dem.
*33·181.
Dem.
*33·182.
If is the sort of relation which generates a series, so that "" may be read " precedes ," then is the condition that the series may have no last term, since it states that every term which follows some term precedes some other term, and is therefore not the last of the series.
*33·2.
Dem.
*33·21.
*33·22.
Dem.
*33·24.
Dem.
*33·241.
*33·25.
Dem.
*33·251.
*33·252.
*33·26.
Dem.
*33·261.
*33·262.
*33·263.
Dem.
*33·264.
*33·265.
*33·27.
Dem.
*33·271.
*33·272.
*33·28.
Dem.
*33·29.
*33·3.
Dem.
*33·31.
The three following propositions are used in the theory of selections (*80, *83 and *85). The second of them is also used in the theory of greater and less (*117) and in the theory of transitive relations (*201).
*33·32.
The converse of this proposition is not true.
Dem.
*33·33.
*33·34.
Dem.
*33·35.
Dem.
*33·351.
*33·352.
Dem.
The two following propositions (*33·4·41) are very frequently used.
*33·4.
Dem.
*33·41.
*33·42.
Dem.
*33·43.
Dem.
*33·431.
Dem.
*33·432.
Dem.
*33·44.
Dem.
*33·45.
Note that by our conventions as to denoting expressions, the scope of both and in the above is "," and is to have the larger scope.
Dem.
*33·46.
*33·47.
Dem.
*33·48.
*33·5.
Dem.
*33·51.
is useful in ordinal arithmetic, where we are concerned with a series generated by a relation , and "" expresses the fact that is a member of this series. The above two propositions (*33·5·51) will be much used in Part IV, where we deal with the foundations of ordinal arithmetic, but will not often be referred to elsewhere.
*33·6.
Dem.
*33·61.
*33·62.