P. HISTORY OF ' AATHEMATICAL - School of Mathematics

310 A HISTORY OF MATHEMATICAL NOTATIONS
i.e., " 'the x satisfying (Pi exists' is to mean 'there is an object c
such that c/Jx is true when x is c but not otherwise.' "
R'y= ('Ix) (xRy) Df.
The inverted comma may be real "of" i R'y is read "the R of y."
For the converse of R, the authors have the notation
Cnv'R=R=xy(yRx) Df.
-i>
R'y= x (xRy) Df.
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R'x = y(xRy) Df.
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"If R is the relation of parent to child, R'y=the parents of y,
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R'x=the children of z.'
D'R=x{(3y)·xRy} Df.
D'R is the domain of R, i.e., the class of all terms that have the
relation R to y. The. converse domain of R is symbolized and defined
thus:
U'R=y{(3x) .xRy} Df.
The sum of the domain and the converse domain is called the
"field" and is represented by C'R,
C'R=D'R\...IU'R Df.
The relative product of two relations Rand S arises when xRy
and ySz; it is written R IS.
R IS=x~{(3y) .xRy.ySz} Df.
"Paternal aunt" is the relative product of "sister" and "father."
R"a is the class of terms x which have the relation R to some member
of a class a. We have,
R"a=x{(3Y).YEa.xRy} Df.
Thus, if R is inhabitingand a is towns, then R"a=inhabitants of
towns.
,'x is the class whose only member is z, Peano and Frege showed that
the class whose only member is x is not identical with x.
a 1 R is a relation R with its domain limited to members of a (Vol. I
[1910], p. 278).
R ~ {3 is a relation R with its converse domain limited to members of fl.
a 1 R ~ {3 is a relation R with both of these limitations.
a l' {3 is the-relation between x and y in which x is a member of a and
y is a member of {3i i.e.,
a l' (3= xy(xEa • YE(3) Df.
The foregoing list represents most of the symbols used by Whitehead
and Russell in the first volume of their Principia mathematica.
This volume, after an Introduction of sixty-nine pages, is devoted to