Venn Diagram/Overlapping Sets

Venn diagrams is a convenient way of representing data

We will discuss below representing data using the method of Venn diagrams for 2 groups and 3 groups:


From the above figure, consider the following data:

The box denotes a class having N students. The N students are divided as below:

Number of students in one group is A

Number of students in another group is B,

Number of students in both the groups is x,  

Number of students who are in neither A nor B is p,

Now, the following formula arise from the above data i.e. using venn diagrams for 2 groups:

Number of students in:

1. At least one group

(i.e., number of students who belong to any one group or either A or B) is:

A + B – x

2. Only one group

(i.e. in only A or in only B) is:

A + B – 2x

3. Total strength of the class:

(including those who belong to any one group and also none of the two groups) is

A + B – x + p, which N is, i.e.,

N = A + B – x + p

Let us solve a question to illustrate the above concept of Venn diagrams used to represent data comprising 2 groups:


In a class of 100 students, 50 like singing, 35 painting and 25 like neither singing nor painting. How many like both the activities?


In the data given, take:

N as 100, A as 50, B as 35 and p as 25.

What needs to be found is x, i.e., those who like both singing and painting.

Apply N = A + B – x + p

100 = 50 + 35 – x +25,

On solving the above equation, p = 10

Therefore, in the class, 10 like either singing or painting,


In the above group, the respective terms (letters) denote the following data:

Number of students in:

One group is A,

a 2nd group is B,

a third group is C,

all three groups is x,

A or B but not C is a,

A or C but not A is b,

B or C but not A is c,

a', b’ and c’ denote those who belong to only A, only B and only C but not the other two groups in each respectively.

From the data as constructed in the above Venn diagram, the following formulae arise:

Number of students who belong to

1. At least one group is

A + B + C – (a + b + c +2x)

2. Only two groups but not the third group is

a + b + c

3. The entire class of N students who belong to any one group or none of the three groups is

N = A + B + C – (a + b + c + 2x) +p

(p denotes number of students who do  not belong to any group)

Let us solve a question applying the above formula:


In a class of 90 students, 50 can speak English; 40 german; and 35 french. 10 students can speak all of the three languages. How many of the students can speak only any of the two languages?


Apply the third formula above for 3-group Venn diagrams:

N = A + B + C – (a + b + c + 2x) + p

What is required is a + b + c

90 = 50 + 40 + 35 – ((a + b + c) + 2 * 10)

a + b + c = 15

Therefore, 15 students can speak any of only two languages.