Free Online Preparation for CAT with Minglebox eCAT Prep. Cover basic concepts of Venn Diagrams under Data Interpretation for MBA Entrance Exam Preparation with Study material, solved examples and tests prepared by CAT coaching experts.
Venn diagrams are a geometrical representation of sets. The geometrical representation makes the solution easier to understand. Given below are the geometrical shapes used to represent sets
Universal set is represented by a rectangle
Any other set is represented by a circle, lying inside the rectangle,as any other set is a subset of Universal set
Representation of Union, Intersection
Representation of Union, Intersection, Difference;Complement, etc of given number of sets in Venn diagrams. Let us consider the following sets:
Set U, which the set of Natural numbers less than 10.
Set A, set of prime numbers less than 10.
Set B, set of even numbers less than 10.
We first list the elements of each set mentioned above,
U = {1,2,3,............9}
A = {2,3,5,7}
B = {2,4,6,8}
now = {2} ;the information is represented in Venn diagram as follows.
As seen above the boundary within which 3,5,7 lie belong to set A only whereas the number '2' lies in a region which belongs to both A and B And 4,6,8 lie in region which belongs to set B only.
The region which belongs to set A only is the set AB. The region which belongs to set A and B is the set . The region which belongs to set A only or set A and set B or set B only represents the set .
Let us consider the Venn diagram drawn below and understand which set is represented by what region.
Region 1 Represents count of those elements which belong to set A only.
Region 2 Represents count of those elements which belong to set only I..e common to set A and set B but not set C.
Region 3 Represents count of those elements which belong to set B only.
Region 4 Represents count of those elements which belong to set only.
Region 5 Represents count of those elements which belong to all the 3 sets A,B and C
i..e ( )
Region 6 Represents count of those elements which belong to set only.
Region 7 Represents count of those elements which belong to set C only.
Region 8 Represents count of those elements which belong to neither of A,B or C.
Note – Each of these regions are nonoverlapping regions they do not have anything in common.
Representation of few sets in Venn diagrams
The Shaded region represents the set in Venn – Diagrams
A' A  B
Question
In a tourist but consisting of 40 passengers, 35 speak German language and 20 speak French. If all the tourists speak at least one of the 2 languages, Find
(i)How many speak both the languages ?
(ii)How many speak exactly one language ?
Solution
Let G denote the set of tourists speaking German language and F denote the set of tourists speaking French language then as each tourist speaks one of the 2 languages hence Universal set is same as the Union of the 2 sets. Therefore we need not draw the rectangle for Universal set. The Venn diagram will appear as follows.
Let x number of tourists speak both the languages then 35x speak German only and 20x speak French only. Sum total of all the 3 categories Should be equal to total number of tourists
35x + x+20x = 40 => x = 15
Hence (i) 15 speak both the languages.

Count of exactly one language speakers is = 35x + 20x = 25
Question
In a Housing Society, out of 100 children, 32 play chess, 64 play ludo and 40 play Chinese checkers, 14 play chess and ludo, 15 play ludo and Chinese checkers, 13 play chess and Chinese checkers. Only 6 children play all the 3 games. If all the children play at least one of the 3 games then :
(a) How many Children play only one games ?
(b) How many Children play at least two game ?
How many Children play chess or ludo but not Chinese Checkers.
Solution
We draw the Venn diagrams for the problem
If 6 Children play all 3 games and 14 play chess and ludo it means 8 play only chess and ludo similarly we can find the number of children who play ludo and Chinese checkers only, I.e 9 likewise we can reason out all the numbers in the Venn – diagram. Now we can answer all the questions
(a) 11 + 41 + 18 = 70
(b) (8 + 7 + 9) + 6 = 30
(c) (11 + 4) + 8 = 60