*73. SIMILARITY OF CLASSES.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *73.
Two classes and are said to be similar when there is a one-one relation whose domain is and whose converse domain is . We express " is similar to " by the notation "." When two classes are similar, they have the same cardinal number of terms: it is this fact which gives importance to the relation of similarity.
We have
The relation of similarity is that of the domain of a to the converse domain, i.e. it is the relative product of and , or, what comes to the same thing, it is the relative product of and .
Most of the properties of similarity result immediately from those of one-one relations and offer no difficulty of any kind.
When there are relations which correlate 's with 's so as to make similar to , we denote the class of such relations by "." Thus we have
When, as in this case, we have a descriptive double function closely connected with a relation, we shall make it a practice to distinguish the descriptive double function by a bar.
It is to be observed that "," like and and and , is ambiguous as to type, and only acquires a definite meaning when the types of its domain and converse domain are specified. The domain and the converse domain may or may not be of the same type, i.e. "" may or may not be a homogeneous relation. This enables us to speak of two classes of different types as having the same number of terms. We shall return to this point in connection with cardinal numbers (cf. especially *102—*106).
The propositions of the present number are important, and are very frequently referred to throughout cardinal arithmetic. In order to prove that two classes and have the same cardinal number of terms, it is generally necessary, in the fundamental arithmetical propositions with which we are concerned, actually to construct a relation such that . Such relation will be called a correlator of and . It will usually be obtained by taking some relation for which we have (, and limiting the converse domain to , so that is the required correlator. Very frequently we shall have , not , but will be such that .
Among the more important propositions of the present number are the following:
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I.e. is a correlator of and if (1) is one-one, (2) is contained in the converse domain of , (3) is the class of those terms which have the relation to members of .
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This results immediately from the definition.
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The above three propositions show that similarity is reflexive, symmetrical and transitive.
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Thus every class is similar to a class of higher type, and consisting wholly of unit classes.
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Thus 1 is the class of all classes similar to any unit class.
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Thus 0 is the class of all classes similar to the null-class.
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This proposition is very often useful. For arithmetical purposes, we often wish to obtain mutually exclusive classes. Now whether or not and be mutually exclusive, and are mutually exclusive provided . Thus by means of the above proposition we can always construct mutually exclusive classes each similar to a given class, i.e. each having some assigned number of members.
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This proposition is fundamental in the theory of addition.
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I.e. "if is similar to a part of , and is similar to a part of , then is similar to ." This is the Schröder-Bernstein theorem. The proof given below is due to Zermelo.
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Dem.
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