Logical Venn Diagrams

Logical Venn Diagrams
Venn Diagrams :-

                    A Venn diagram  or logic diagram is a diagram or Euler Venn diagram  
                    that shows all possible

                  logical relations between a finite collection of different sets.Logical

                   A Venn diagram is the area of each shape is proportional to the number of

                   elements it contains is called an area-proportional or scaled Venn diagram.

In Other words

                    It is a process of shows relationship between  various geometric strictures.

                    Intersection between two geometric structures indicate that they have

                    something in common .
Representation of Venn diagram
                             (1)

                           (2)

Basic Formula of Venn diagram
                     (1). n(A⋃B) = n(A) +n(B) -n(A⋂B)
                     (2).n(A⋃B⋃C) = n(A) +n(B)+n(C) -n(A⋂B)- n(B⋂C)- n(B⋂C)+n(A⋂B⋂C)

Examples 1.The triangle circle and rectangle shown below representation smart ,

                      hardworking and educated persons  respectively  which one the areas marked

                      1 to 6 is represented by the hardworking educated persons who are not smart. 

                    

                It is clear that the required region is the one which is common in circle and rectangle

                but not in triangle i.e. region 5 which shows in co lour  red .

Examples 2.The rectangle triangle circle rectangle and square shown below representation

                     males ,educated,urban and civil services.

                    Answers the following questions

                   (1). How many peoples are educated male not an urban resident.

                   (2).How many are educated male an urban resident. 

Ans. (1). 11            (2).4

Examples 3.

                   The  triangle circle  and square shown below representation

                     educated people,employed people backward people .

                    Answers the following questions

                   (1). How many backward people are educated.

                   (2).How many backward people are not educated.. 

            

Ans. (1). 14            (2).22.

Examples 4.

                  The circle represents players

                  The triangle represents outdoor games

                  The rectangle represents indoor games

                  The square represents national level  players

Answer the following

(1) Which letter represent the players who play indoor games at national level.

(2) Which letter represent the outdoor game as well as indoor game players who

       do not  play at national level.

Examples 5.In a class of 35 students, 24 like to play cricket and 16 like to play football. Also,

                    each student likes to play at least one of the two games. How many students like to

                    play both cricket and football

Explanation :-Let C denote student play cricket,F denote student play football,

                       Then n(C)=24,n(F)=16,n(C⋃F)= 35  ,n(C⋂F) = ?

                       By the formula

                      n(A⋃B) = n(A) +n(B) -n(A⋂B)

                       n(C⋃F) = n(C) + n(F) - n(C⋂F)

                        35 = 24 + 16 -n(C⋂F)

                      n(C⋂F) = 40 - 35 = 5

                      Hence  5  students like to play both cricket and football.

Examples 6 A sport club  awarded 38 medals in handball, 15 in basketball and 20 in football. 

                   If these medals went to a total of 58 men and only three men got medals in

                   all the three sports, how many received medals in exactly two of the three sports ?

Explanation :-Let H denote  men play handball ,B denote men  play  basketball ,

                         F denote men  play  football,

                       Then n(H)=38,n(B)=15, n(F)=20,

                      n(H⋃B⋃F) = 58 ,n(H⋂B⋂F) = 3

                         n (H∪ B ∪ F ) = n ( H ) + n ( B ) + n ( F ) – n (H ∩ B ) – n (B ∩ F ) – n (F ∩ H ) + n (H∩ B ∩ F ), 

                          n (H ∩ B ) + n (B ∩ F ) +n (F ∩ H ) = n ( H ) + n ( B ) + n ( F ) + n (H∩ B ∩ F )-n (H∪ B ∪ F )

.                                                                      = 38 +15 + 20 +3 -58

                                                                       =  18 (Red Shaded region)