# Finite elements/Sets

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.

## Sets[edit | edit source]

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

## Subsets[edit | edit source]

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

## Equality of sets[edit | edit source]

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

The symbol means * if and only if*.

For example, if

then .

## Union, Intersection, Difference of Sets[edit | edit source]

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 .

## Countable Sets[edit | edit source]

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 , , .

## Cartesian Product[edit | edit source]

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[edit | edit source]

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

- The
*half-open intervals*and are defined as

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).

## Open and Closed Sets in *R*^{n}[edit | edit source]

^{n}

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.