A familiarity with the notation of sets is essential for the student
who wants to read modern literature on finite elements. This handout
gives you a brief review of set notation. More details can be found
in books on advanced calculus.
A set is a well-defined collection of objects. As far as we are
concerned, these objects are mainly numbers, vectors, or functions.
If an object
is a member of a set
, we write

If
is not a member of
, we write

An example of a finite set (of functions) is

Another example is the set of integers greater than 5 and less than 12

If we denote the set of all integers by
, then we can
alternatively write

The set
of positive integers is an infinite set
and is written as

An empty (or null) set is a set with no elements. It is denoted
by
. An example is

If
and
are two sets, then we say that
is a subset of
if each element of
is an element of
.
For example, if the two sets are

we write

On the other hand, if
is a subset of
which may be the set
itself
we write

If
is not a subset of
, we write

Two sets
and
are equal if they contain exactly the same elements.
Thus,

The symbol
means if and only if.
For example, if

then
.
The union of two sets
and
is the set of all elements
that are in
or
.

The intersection of two sets
and
is the set of all elements
that are both in
and in
.

The difference of two sets
and
is the set of all elements that
are in
but not in
.

The complement of a set
(denoted by
) is the set of all
elements that are not in
but belong to a larger universal set
.

Suppose we have a set
. Such a set is called countable
if each of its members can be labeled with an integer subscript of the form

Obviously, each finite set is countable.
Some infinite sets are also countable. For instance, the set of integers
is countable because you can label each integer with an subscript that
is also an integer. However, you cannot do that with the real numbers
which are uncountable.
The set of functions

is countable.
The set of points on the real line

is not countable because the points cannot be labeled
,
,
.
The Cartesian product of two sets
and
is the set of all ordered
pairs
, such that

In general,
.
For example, if

then

and

The set of real numbers (
) can be visualized as an infinitely
long line with each real number being represented as a point on this line.
We usually deal with subsets of
, called intervals.
Let
and
be two points on
such that
. Then,
- The open interval
is defined as

- The closed interval
is defined as
![{\displaystyle [a,b]=\{x~|~x\in \mathbb {R} ,a\leq x\leq b\}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16a033bbd7658a18e66a871d536cee1de06c8ecf)
- The half-open intervals
and
are defined as
![{\displaystyle (a,b]=\{x~|~x\in \mathbb {R} ,a<x\leq b\}~{\text{and}}~[a,b)=\{x~|~x\in \mathbb {R} ,a\leq x<b\}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d70bc4e1ac974da23b3d6d760d333dd513e36f37)
Let
and
. Then the neighborhood of
is defined as the open interval

Let
. Then
is an interior point of
if we can find a nbd(
) all of whose points belong to
.
If every point of
is an interior point, then
is called an
open set. For example, the interval
is an open set. So is
the real line
.
A set
is called closed if its complement
is open.
The closure
of a set
is the union of
the set and its boundary points (a rigorous definition of closed sets
can be made using the concept of points of accumulation).
The concept of the real line can be extended to higher dimensions. In
two dimensions, we have
which is defined as

can be thought of as a two-dimensional plane and each
member of the set
represents a point on the plane.
In three dimensions, we have

In
dimensions, the concept is extended to mean

In the case of sets in
the concept of distance in
is
extended so that

where

The definition of interior point also follows from the definition
in
. Thus if
, then
is
an interior point if we can always find a nbd
, all of whose
points belong to
. If every point on
is an interior point,
then
is an open set.
As in the real number line, a closed set is the complement of an open set. One way of creating a closed set is by taking an open set
and its boundary
. This particular closed set is called
the closure
of
. A rigorous definition can once
again be obtained using the concept of points of accumulation.