Basic Notations
We assume that you are familiar with the concepts represented by the following
symbols.
{XQ, x±,... , xn} the set containing Xo, x\,... ,xn and no other elements;
0 the empty set;
G the membership relation;
C subset;
C proper subset;
3, V, -i, A, V, —•, - quantifiers and logical connectives. Although Chapter 5 con-
tains a thorough discussion of these symbols in the context of formal languages,
you should already be at ease with their use.
xUy = {z : z G x\J z y} the union of two sets;
xC\y = {z : z £ x Az e y} the intersection of two sets;
x\y = {z : z e x Az £ y} the difference of two sets;
xAy = (x\y) U (y\x) = (xUy)\(xf)y) the symmetric difference of two sets;
(J X = {z : 3Y G X [z G Y)} union of a family of sets;
P| X = {z : VY G X (z G Y)} intersection of a family of sets;
f : X —Y function from X into Y\
f[W] = {yeY: 3xeW f(x) = y} image of W under / ;
f~lZ = {x G X : f(x) G Z} inverse image of Z under / ;
dom(f) domain of a function /;
rn9(f) range of a function /;
f\W restriction of a function / to a subset W of its domain;
fog composition of two functions;
(an : n G N) = (an)ne^ sequence indexed by natural numbers;
N = {0,1,2,... } the set of natural numbers;
Z the set of integers;
Q the set of rationals;
R the set of reals;
P = R\Q the set of irrationals;
C the set of complex numbers.
A the set of algebraic numbers
Note that we impose no restrictions on the style of letters that represent sets.
Each of the symbols x,X,X,X may stand for a set.
XVll
Previous Page Next Page