Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

166 MATHEMATICAL PHILOSOPHY
or the y-terms first; that is, for example, no essential
difference between (i) 2x+3y-I=0, or x=2(I -3y)
and (I') 3y+2-I =0 or y==(I-2X); (i) and (i'), for
example, represent the same straight line; but in relationtheory it is essential to take account of the order in which
the variables occur; the relations determined respectively
by (I) and (i') are not the same; for example, the former
relation contains the couple (0, 1) but the latter does not;
on the other hand, the latter contains the couple (1,0)
but the former does not; if we denote the former relation
by R and the latter one by R', we then write OR3 to say
that " 0 has the relation R to " and write 1 R'0 to say" 3
has the relation R' to O "; but we have neither OR'3 nor
RO, for both of these propositions are false; in general,
as you see, if we have xiRy1, then we have ylR'xi, and
conversely, but not yiRxi nor xiR'yl (except in very
special cases); here we encounter the important notion
of the converse of a relation-two relations, R and R',
are each the other's converse if they are such that, whenever one of them holds between the terms of a couple
(t1, t2), the other holds between the terms of the inverse
couple (t2, t1); thus the relations determined by the propositional functions (i) and (i') are each the other's converse.
It is sufficiently obvious that every relation has a converse.
(In the case of some important relations,-such as equality, for example, or similarity or diversity or identity,the relation and its converse are the same.) The converse of the relation determined by (2) is that determined
by the propositional function (2'), y is a child of x, so that,
if the couple (John Jones, Mary Jones) be a constituent of
the former relation, the couple (Mary J, John J) is a constituent of the latter.
You are now in a fairly good position to see that the