Unions and Intersections of sets

empty set, intersection of sets, union of sets, common elements

A set is a collection of things or objects.  It can be anything, but clearly, when we are talking Math, often times we are talking about a set of numbers.  An example of a set is the following

A={4, 6, 8, *, %, #}

A set is denoted by the symbols {}.  And the members of a set are called elements of the set.  Thus, the set we used as an example above, Set A, has elements 4, 6, 8, *, %, and #.

If we have two sets, we have two operations, Union and Inetersection of sets.  We will define and examine them respectively.

Unions.  Symbol "U" is upper case U.  

: the Union of two sets A and B is the set C that contains all the elements of A and all the elements of B writting all the elements without repetition in the even that the two elements have common elements.

Example.

Set A= { 3, 5, 9, 0, *} and Set B={ 0, *, & ,^, % ,10}

Then the Union of Set A and B is:

A U B ={3, 5, 9, 0, *, &, ^,%,10}=Set C

Notice here that 0 and * are common elements to both sets A and B.  However, in the set Union of A and B, that is, Set C, we did not write 0 and * twice.

Intersection:  

Symbol ∩ is an upside down U, and denotes Interesection of two sets.

: the intersection of two sets A and B is the set C that contains the common elements of set A and Set B.  These are precisely the numbers that we want to avoid repeatedely writting when we are trying to find the Union of two sets.

Thus, if we use the same two sets as above.  

Set A= { 3, 5, 9, 0, *} and Set B={ 0, *, & ,^, % ,10}

then:

A ∩ B = {0, *} = Set C

Thus, the intersection of two sets comprises of the common elements.

** Note that if two sets are distinct from one another, that is, there is no element that the two sets have in common and, their intersection is the Empty Set.

The Empty Set is granted so much importance that it has its own symbol and for our purposes here and in most elementary Mathematical texts, it is denoted by:

Empty Set:  {}, set with no element in it.  

However, if two sets are not both the Empty set, then, observe that their Union can never be an empty set.

empty set, intersection of sets, union of sets, common elements