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Example:Practical Problem on Union & Intersection of Sets

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Venn diagram



Venn diagram showing the intersections of the Greek, Latin and Russian alphabet (upper case graphemes)

Venn diagrams or set diagrams are diagrams that show all possible logical relations between a finite collection of sets (aggregation of things). Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics and computer science



A Venn diagram is constructed with a collection of simple closed curves drawn in a plane.

Venn diagrams normally comprise overlapping circles. The interior of the circle symbolically represents the elements of the set, while the exterior represents elements which are not members of the set.

(Our solved example in mathguru.com uses this concept)




Basic operations

There are several fundamental operations for constructing new sets from given sets.




The union of A and B, denoted A  B

Two sets can be "added" together. The union of A and B, denoted by A  B, is the set of all things which are members of either A or B.


1.  {1, 2} {red, white} = {1, 2, red, white}.

2.  {1, 2, green} {red, white, green} = {1, 2, red, white, green}.

3.  {1, 2} {1, 2} = {1, 2}.

(Our solved example in mathguru.com uses this concept)

Some basic properties of unions:

1.  A  B = B  A.

2.  A  (B  C) = (A  B)  C.

3.  A  (A  B).

4.  A  A = A.

5.  A  A.

6.  A  B if and only if A  B = B.



A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A  B, is the set of all things which are members of both A and B. If A  B = , then A and B are said to be disjoint.


The intersection of A and B, denoted A  B.


1.  {1, 2} {red, white} = .

2.  {1, 2, green} {red, white, green} = {green}.

3.  {1, 2} {1, 2} = {1, 2}.

(Our solved example in mathguru.com uses this concept)


Some basic properties of intersections:

  • A  B = B  A.
  • A  (B  C) = (A  B)  C.
  • A  B  A.
  • A  A = A.
  • A  = .

       A  B if and only if A  B = A.




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.