Math

Understanding Sets and Set Notations

 

A set is a collection of distinct objects. One can use {} to represent a set. Some sample sets are listed below:

• Set of natural numbers less than 12: 

• Set of even numbers less than or equal to 8: 

• Set of multiples of 5 between 11 and 21: 

 

Union of Sets

The union of two sets is defined as a new set that includes all of the elements that are in at least one of the sets. The symbol U is used to represent the union of sets. 

 

Example: A is the set of odd numbers less than or equal to 19 and B is the set of multiples of 3 less than 15. Write down the elements of A and B and find their union.

The union of these sets is a set than contains all the elements of A and B. If an element belongs to both A and B, we do NOT write it twice in the union set.

 

Intersection of Sets

The intersection of two sets A and B is a set that contains elements that are in both A and B.  The symbol  is used to represent the intersection of sets. 

 

Example: Find the intersection of the following sets:

The intersection of the above two sets is a set of the elements that are in both A and B:

 

The Universal Set

A Universal Set is a set of all elements involve in a given problem. The universal set is usually denoted by U.  For example is set  and , the universal set associated with this problem is 

 

Complement of a Set

Complement of a Set represented byis a set of elements in the universal set that are NOT in A. 

 

Example: Find the complement of A:

               .

 

In order to find the complement of A, we have to subtract the elements of A from the universal set. 

 

Venn Diagrams

A Venn diagram is a diagram that represents the universal set and all the sets that are defined in a specific problem. The universal set is a rectangle that contains the sets defined in a problem. For each set a circle is drawn inside the universal set. 

 

Example: Consider the universal set , and  and  Represent the sets in a Venn diagram.

                                                            

Note that for sets that do NOT intersect, we use disjoint circles.

 

Example: Show the following sets on a Venn diagram.

, , 

                                                       

 

Note that for two sets that intersect we use overlapping circles in a Venn diagram.

 

Empty Set

An empty set is a set that includes no elements. We use Ø to represent an empty set.

 

Subset

Set A is a subset of the set B if B includes all elements of A and more.

 

Example: Is A a subset of B?

          

A is NOT a subset of B because B does not include 8.