Introductiona property .that is vacuously satisfied. For example,0 = { x : x E ~' x^2 = -1}
3The empty set 0 is regarded as a subset of every set, so that 0 C A for
every set A.
Let X be a given set.' The set whose elements are all the subsets of
Xis called the set of parts of X, or the power set of X, denoted P(X).
In symbols, P(X) = {A: ACX} of course, P(X) contains 0 as well as
the set X itself.
For instance, if X = {p, q, r }, then
P(X) = {0, {p}, {q}, {r}, {p, q}, {p, r}, {q, r}, {p, q, r}}
The relations a E X and {a} E P(X) are clearly equivalent, as well as
A c X and A E P(X)..
The union, intersection, difference (or relative complement), and
Cartesian product of two sets are defined as follows:Union:
Intersection:
Difference:
Cartesian product:
A U B = { x : x E A or x E B}
A n B = { x : x E A and x E B}
A - B = {x : x E A and x rf B}
Ax B = {(x, y) : x EA and y EB}Thus the Cartesian product of the sets A and B (in that order) is defined
to be the set of all ordered pairs ( x, y) such that x E A and y E B.
0.2 Mappings
A set mapping (or simply a mapping, also called a correspondence or an
application) consists of three objects: two non-empty sets X and Y (which
may or may not be equal) and a rule f that assigns to each set A of a certain
class G of subsets of X a well-determined subset B of Y. For instance, we
may consider two different planes X and Y and a conical projection that
assigns to each circle in X a conic section in Y.
The set B that corresponds in Y to a given set A C G is called the
image of A under f, and we write B = f(A) (Fig. 0.1).
The rule f itself is sometimes called a mapping, and also a transformation
or all operator, depending on the particular theory under consideration. For
example, a differential operator D assigns to each function g (in the class
of differentiable functions) another function h = D g.
r;I:'he class G C P(X) is called the domain of the mapping, and the set
P(Y) is the codomain. The range of f, denoted f[G], is the set of all
images, that is, J[G] = {f(A): A E G}.