Set Theory
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Set
- The term set is undefined. It is a term which accepted as a primitive
concept. We can gain an intuitive feeling for the meaning of the term set
by noting synonyms such as collection. The term object is also
undefined. It also goes by the synonym element. Thus we can speak of the
set A, the set of all the real numbers. Then any real number is an element
of the set A. The notation x
Ξ A
is read, x is an element of the set A. Similarly the
notation x Ο
A is read, x is not an element of A.
It is also possible for sets to be elements of another set. The term space
is sometimes used as a synonym for set.
Union The union of two sets A and B is
another set whose elements consist of the elements of both A and B.
The notation of the union of the sets A and B is
Eq.
(1) is read the union of A and B. As an example consider A
= {1, 3, 5, 7} and B = {2, 4, 6, 8}. Then A Θ
B = {1, 2, 3, 4, 5, 6, 7, 8}
Intersection
- The intersection of two sets A andd B is another set whose
elements consist of the elements that are common to A and B. That
is to say, an element is in the intersection of A and B if and
only if the element is in both A and B. The notation of the
intersection of the sets A and B is
and
is read the intersection of A and B. As an example of the
intersection of two sets define A = {1, 3, 5, 7} and B = {2, 5, 7,
8}. Then A Η
B = {5, 7}
It is sometimes convenient to generalize the operations of union and intersection. The generalization of the union of sets is
The generalization of the
intersection of sets is
Notation The notation for the union and intersection of
two or more sets is defined above. There are more notations that are important
to know. The notation
reads
A is a subset of B which means that all of the elements of A
are also in B. However it does not follow from this that B is a
subset of A. Consider the sets A = {a, b, g, h} and B = {a,
b, c, d, e, f, g, h, i, j). It is easily seen that A Μ B. The notation A Λ B is read A is not a subset
of B.
Rn
The space Rn
is the usual n-dimensional space of vector algebra: a point in Rn
is a sequence of n real numbers (x1,
x2,
, xn)
also called an n-tuple of real numbers. For example, R0
Ί R is the set of all real numbers. R2
is the set of pairs of real numbers such as (6, 12.4).
Neighborhoods
In order to state certain properties of sets, such as whether the set is
open or not, one must employ a distance function of which there are many.
A very important distance function defined for x
= (x1,
x2,
, xn)
and y = (y1,
y2,
, yn)
is
Let
n = 2. Then the equation d(x, y) < 1 defines a
neighborhood in R2
which is the interior of a disk of radius 1. The circle itself is not part of
the neighborhood. In such cases the set is said to be open.
The
distance function d(x, y) is used here to define
neighborhoods and thereby open sets. In such cases we say that d(x,
y) has induced a topology on R2.
This distance
function need not be as defined in Eq. (6) to establish the same topology since
other distance functions can be defined which would define the same topology.