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
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
If is not a subset of , we write
Two sets and are equal if they contain exactly the same elements.
The symbol means if and only if.
For example, if
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 .
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 ()[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
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 | edit source]
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.