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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.


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.