*92. POWERS OF ONE-MANY AND MANY-ONE RELATIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *92.
If , it follows that, starting from a given term , there is only one series of terms such that
Thus for example the relation of son to father is a ; and starting from a given man, the series of ancestors in the direct male line (which is the above series , , ,...) is unique and determinate. A result of this property of many-one relations is that if, starting from a term , we go backwards a certain number of steps to a term , and then forward a greater number of steps to a term , we must pass through in going from to ; while if the number of steps from to is less than that from to , must lie on the road from to . These facts are expressed by the proposition:
In the present number, we have to establish various propositions of this kind.
We prove in this number various propositions which are used in the discussion of "families" in *96 and *97, and some which are used in the theory of finite and infinite. But on the whole the propositions of this number are not much used. The most important of them are the following:
*92·11.
with a similar proposition (*92·111) for .
*92·132.
with a similar proposition (*92·133) for .
*92·14.
On this proposition, compare the remarks on *91·271 in the introduction to *91. If is a serial relation, is the condition that the series may have no last term.
*92·31.
*92·311.
*92·1.
Dem.
*92·101.
*92·102.
*92·11.
Dem.
*92·111.
*92·112.
*92·113.
*92·12.
*92·121.
*92·13.
Dem.
*92·131.
In this number, when proofs have been given for , we shall omit the proofs of corresponding propositions for , as these are always exactly analogous to the proofs for .
*92·132.
*92·133.
*92·14.
Dem.
*92·141.
*92·142.
Dem.
*92·143.
*92·144.
Dem.
*92·145.
*92·146.