**Introduction:**

A Venn diagram is a diagram that shows all possible logical relations between a finite collections of different sets.

These diagrams depict elements as points in the plane, and sets as regions inside closed curves.

#### Example 1:

Individual sets are independent of each other but, contain in a universal set.

The dogs, cows, and horses are all animals but, have no relation with respect to each other.

#### Example 2:

Elements in the first set are related to second set and the elements in the second set are related to the third.

The hundreds set includes the tens and in turn the tens set include the set of units (ones).

#### Example 3:

Women is a universal set. Some mothers are sisters and vice versa. However, all sisters and mothers are women.

#### Example 4:

Some elements in two (or more) sets are related to another but, not completely.

The set of students contains boys and girls. But, all boys and girls are not students.

#### Example 5:

Within the universal set, all the three sets are related to one another but not completely.

#### Example 6:

Two sets are related to each other completely and the third set is entirely different from first two.

Lions are sub-set of carnivorous animals but, cows are herbivorous.

#### Example 7:

First set is partially related to the second but third is entirely different from the first two sets.

Some boys are students. No animal can be a boy or student.

#### Note:

The combinations are infinite and the students should analyze carefully.

**Problem 1:**

Which of the following diagrams indicates the best relation between Men, Fathers and Doctors?

#### Solution:

Answer: Option “A”

All Fathers are Men and some Fathers and some Men may be Doctors.

**Problem 2:**

Which of the following diagrams indicates the best relation between Passengers, Train and Bus?

#### Solution:

Answer: Option “C”

Bus and Train are different from each other but some passengers travel by bus and some travel by train.

**Problem 3:**

Which number indicates the boys who are athletic and are disciplined?

#### Solution:

Answer: Portion “2”

It is at the intersection of rectangle (boys), circle (athletes) and square (disciplined). Also, it should be independent of triangle (girls).

**Problem 4:**

In the above diagram, number 3 indicates:

#### Solution:

Number 3 indicates the boys and girls who are un-disciplined athletes.

**Problem 5:**

In the above diagram, rectangle represents Women, Triangle represents Educated, Circle represents Rural and square represents Government Employees.

Which one of the following represents the Educated Women who are from non-Rural areas?

#### Solution:

Answer: Portion “11”

We should find a number in the intersection of Triangle (Educated), Rectangle (Women) but, independent of Circle (Rural).

**Problem 6:**

In the above diagram, number 6 indicates:

#### Solution:

The number 6 is in the intersection of Rectangle (Women), Square (Government Employees) and Circle (Rural) and independent of the Triangle (Educated).

Hence, the number 6 indicates the Un-educated Women Government Employees who are from rural areas.

**Problem 7:**

In a room there are 63 students. 33 students know Panjabi, 25 Guajarati and 26 Bengali. 10 students are fluent in both Panjabi and Guajarati, 9 in Guajarati and Bengali and 8 are comfortable with Panjabi and Bengali. Same numbers of students know all the three languages as those who know none of the three languages.

How many students know (i) all the three languages (ii) only one of the three languages?

#### Solution:

Let the set P1 represents the students who know only Punjabi and, P – who know Punjabi

C1 represents the students who know only Gujarati and, C – who know Gujarati

B1 represents the students who know only Bengali and, B– who know Bengali.

We have to start from the maximum intersection region to minimum intersection region.

Maximum intersection region: students who know all three languages then, two languages and finally only one language.

Let the number of students who know all the three languages = the number of students who don’t know any of the languages = ‘x’.

The equations are:

P1+x+(10-x)+(8-x) = P = 33 (or) P1-x = 15 …(i)

C1+x+(10-x)+(9-x) = C = 25 (or) C1-x = 6… (ii)

B1+x+(8-x)+(9-x) = B = 26 (or) B1-x = 9 … (iii) and,

[P1+x+(10-x)+(8-x)+C1+(9-x)+B1]+x = 63 (or)

[P1+C1+B1]-x = 36 … (iv)

Where, (10-x) – Number of students who know both Punjabi and Gujarati

(8-x) – Number of students who know both Punjabi and Bengali

(9-x) – Number of students who know both Bengali and Gujarati

Adding equations (i), (ii) and (iii); P1+C1+B1-3x = 30 … (v)

Subtracting (v) from (iv); 2x =6 (or) x = 3.

The Venn diagram can now be completed.

Using (i); P1-3 = 15 and, P1 = 18.

Using (ii); C1-3 = 6 and, C1 = 9

Using (iii); B1-3 = 9 and, B1 = 12.

Number of students who know only one of the three languages = P1+C1+B1 = 18+9+12 = 39.

Number of students who know all the three languages = x =3.

**About Author**

#### Dr. BASKAR .A,

a former Research Scientist Government of India, presently working as a Professor in the Dept of Mechanical Engineering, Authored books on Kinematics and Dynamics of Machinery, Also Published Numerous Research papers moreover a Geek in Mathematics