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.
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
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
Two sets and are equal if they contain exactly the same elements. Thus,
The symbol means if and only if.
For example, if
Union, Intersection, Difference of Sets
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
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
The Set of Real Numbers ()
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 
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
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.