Set, Union and Intersection

 

Motivation:

                          A set is consisted of a collection of elements.  If  there are many sets, we would like to figure out the relationship between  elements and those sets.  For example, an element might belong to set A and set B, but it does not belong to set C. Or the number of distinct elements of set A is 10, and the number of distinct elements of set B is 8,  but the total number of distinct elements of set A and set B is 12.  Is it possible to know that number of elements that belong to set A and set B at the same time ?

 

                      There exist more extreme cases that the number of elements in a set is not countable: for example, if we consider a set is composed by a line, then the number of points in this line is not countable.   Set theory is with rich material. Here, we only introduce some basic concepts.

 

 

Set and Its Members ( or Elements )

               Usually when we speak of a set,  we need to specify its member(s) or element(s) in this set and it can be done by enumerating those elements one by one or describing the characteristics of those members.   For example,

 

                     A= { 1, 2, 3, 4, 5 }

           

              It stands for set A is consisting of  5 elements: 1, 2, 3, 4, 5 .

             Or we can just use languages like:

 

                    “Set A  stands for all the people of age 20”

                    “Set B stands for all the males of age 20”

                    “Set C stands for all the females of age 20”

                     …

 

              There exists another way to describe a set:

 

                              A={ x | x is a person of age 20 }

                   Or

                              A={ x: x is a person of age 20 }

 

              That means : as long as we find a person of age 20,  we classify this person into set A.   We use the notation above to specify the elements in a set when we can not list all the elements of a set one by one.

 

              For an element e in set A ,  we use the following notation to indicate this relationship:

                                   e  A 

 

             This notation is used to identify the relationship between an element and a set.

 

Subset

          The concept of subset is to indicate the relationship that all the elements in a set A also belong to another set B.  And it is denoted as  A  B .   Let’s use the following example:

 

                           A={1,2,3,4,5}

                           B={1,2,3,4,5,8,9,10}

 

             For every element in A ,  it also belongs to B .  We just use the notation  A  B . Please notice that this notation indicates the relationship between two sets.

 

              If   C={ 1 } ,   then we can denote it as  C  A  because the element 1  A .

 

 

 

Union

             The union of two sets A and B  is the super set that contains all the elements of A and B .  We denote it as  A  B .     For example,

 

                                 A={1,2,3}

                                 B={ 3,4,5 }

            Then

                                 A  B = { 1,2,3,4,5 }

 

 

 

 

Intersection

              The intersection of two sets A and B is the set that contains the elements that belong to A and also belong to B .  It is denoted as A  B .   For example,

 

                                A={1,2,3}

                                B={3,4,5}

             Then

                                A  B = {3}