Nonlinear 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.

SetsEdit

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

 

SubsetsEdit

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 setsEdit

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 SetsEdit

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 SetsEdit

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 ProductEdit

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

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 Edit

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.