*72. MISCELLANEOUS PROPOSITIONS CONCERNING ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *72.
In this number we shall prove various propositions involving , , or , but not embodying fundamental properties of these classes of relations.
The present number begins with various propositions (*72·1-·191) showing that various special relations are one-many or one-one. The most useful of these are
*72·182.
*72·184.
We have next a set of propositions concerning when and are one-many, or when is one-one, and kindred matters. The most useful of these is
*72·241.
We have next a set of propositions (*72·3-·341) concerning products and sums of classes of relations; of these the one most used is
*72·32.
which is an extension of *71·24.
We have next a set of propositions (*72·4—·481) giving various relations of and when , or of and when . The more useful propositions of this set are those that have the hypothesis ; these are occasionally useful in arithmetic. We have
*72·401.
*72·411.
For example, the relation of son to father is many-one. Let = Cabinet Ministers, = fools; then assuming , it will follow that the sons of Cabinet Ministers and the sons of (male) fools have no common member. If we make the relation of son to parent (which is not many-one), it no longer follows that the sons of Cabinet Ministers and the sons of fools have no common member.
We have
*72·451.
The effect of this proposition is that if and are both contained in , and , then (using .
We next have a set of propositions concerned with the relations of and (, or, what comes to the same thing, with the circumstances under which and under which . We have
*72·502.
Thus for example the fathers of the children of wise fathers are the class of wise fathers; but the fathers of the children of wise parents are not all wise, and the parents of the children of wise parents are not all wise—the first because "" fails, the second because "" fails.
We have also
*72·52.
We have next a set of propositions (*72·59—·66) in which the relative product occurs if , or if . The most useful propositions in this set are
*72·591.
*72·601.
*72·66.
This is the "principle of abstraction." It shows that every relation which has the formal properties of equality, i.e. which is transitive and symmetrical, is equal to the relative product of a many-one relation into its converse; i.e. whenever the relation holds between and , there is a term such that , where is a many-one relation; and *72·64 shows that this term may be taken to be , which is equal to . This principle embodies a great part of the reasons for our definitions of the various kinds of numbers; in seeking these definitions, we always have, to begin with, some transitive symmetrical relation which we regard as sameness of number; thus by *72·64, the desired properties of the numbers of the kind in question are secured by taking the number of an object to be the class of objects to which the said object has the transitive symmetrical relation in question. It is in this way that we are led to define cardinal numbers as classes of classes, and ordinal numbers as classes of relations.
The remaining propositions of this number are of less importance, with the exception of
*72·92.
This proposition shows that every relation contained in a one-many relation is obtainable by a limitation of the converse domain. Thus e.g. every relation contained in that of father to son can be specified by specifying the class of sons who are to be its converse domain; for then all the fathers of these sons must be included to provide referents. But if we take the relation of parent and child, which is not one-many or many-one, a contained relation is not determinate even when both its domain and its converse domain are given; for the relation may relate some of the children in any one family to the father and some to the mother, and so long as all the children and both parents are each related to some one by the relation, the domain and converse domain remain unchanged by permutations within the family.
*72·1.
Dem.
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Dem.
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*72·121.
Dem.
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*72·131.
*72·132.