*31. CONVERSES OF RELATIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *31.
If is a relation, the relation which has to when is called the converse of . Thus greater is the converse of less, before of after, husband of wife. The converse of identity is identity, and the converse of diversity is diversity. The converse of is written (read "-converse"). When , is called a symmetrical relation, otherwise it is called not-symmetrical. When is incompatible with , is called asymmetrical. Thus "cousin" is symmetrical, "brother" is not-symmetrical (because when is the brother of , may be either the brother or the sister of , and "husband" is asymmetrical.
The relation of to is called "." It will be shown that every relation has one, and only one, converse; hence, applying the notation of *30, that one is . Thus . We have thus two notations for the converse of ; the second is more convenient for the converse of a relation not denoted by a single letter.
The more important propositions of the present number are the following:
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I.e. any relation has a converse. Hence the relation "" verifies the hypothesis , i.e. we have .
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I.e. two relations are identical when, and only when, their converses are identical.
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I.e. any relation is the converse of its converse.
Very many of the subsequent uses of the notion of the converse of a relation require only the propositions which embody the definitions of and , namely
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and
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Dem.
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Dem.
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Dem.
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Dem.
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Dem.
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Dem.
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