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4. VENN DIAGRAM’S

**4.1 INTRODUCTION **

This chapter deals with questions which aim at analysing a candidate’s ability to relate a certain given group of items and illustrate it diagrammatically.

Here are a few different types of Venn diagramms with their implications made clear.

Suppose you are given a group of three items. Then,

(1) If the items evidently belong to three different groups, the Venn diagram representing it would be as shown alongside.

Ex. Doctors, Engineers, Lawyers

These three items bear no relationship to each other. So, they are represented by 3 disjoint figures as shown in Fig. 1

(2) If one item belongs to the class of the second and the second belongs to the class of third,

then the representation is in the form of three concentric circles, as shown in Fig. 2

Ex. Seconds ,Minutes, Hours

Clearly, seconds are a part of minutes and minutes are a part of hours. So, the Venn diagram would be as shown in

the adjoining figure with circle A representing Seconds, Circle B representing Minutes and Circle C representing Hours.

(3) If two separate items belong to the class of the third, they are represented by two disjoint circles inside a bigger circle as shown in Fig. 3

Ex. Table, Chair, Furniture

Clearly, table and chair are separate items but both are items of furniture.

So they would be represented as in the adjoining figure

with circle A representing Table, circle B representing Chair and

circle C representing Furniture.

(4) All the numbers: If two items belong to the class of the third such that some items of each of these two groups are

common in relationship, then they are represented by two intersecting circles enclosed within a bigger circle.

Ex. Males, Fathers, Brothers

Clearly, some fathers may be broghers. So, fathers and broghers would

be represented by two intersecting circles. Also both fathers and brothers

are males. So, the diagrammatic representation would be as shown in

fig, 4, with circle A representing Fathers, circle B representing Brothers

and circle C representing Males.

(5) If two items are partly related to the third, and are themselves independent of each

other they are represented by three intersecting circles in a line.

Ex. Dogs, Pets, Cats

Clearly, some dogs and some cats are pets. But, all the pets are not dogs

or cats.. Also, dogs and cats are not related to each other. So, the given

items would be represented as shown in fig. 5 with circle A representing

Dogs, circle B representing Pets and circle C representing Cats.

(6) If the three items are partly related to each other, they are represented as shown in the adjoining figure.

Ex. Clerks, Government Employees, Educated Persons

Clearly, some clerks may be government employees and some

may be educated. Similarly, some government employees may be

clerks and some may be educated. Also, some educated persons

may be clerks and some may be government employees. So, the

given items may be represented as shown in Fig. 6 with three

different circles denoting the three classes

(7) If one item belongs to the class of second while third item is entirely different from the two,

then they may be represented by the adjoining diagram.

Ex. Engineers, Human Beings, Rats

Clearly, all engineers are human beings. This would be represented

by two concentric circles. But the class of rats is entirely different

from these two. Thus, these items would be represented as shown

in Fig. 7 with circle A representing Engineers, circle B representing

Human Beings and circle C representing Rats.

(8) If one item belongs to the class of second and the third item is partly related to these two, they are represented as shown alongside.

Ex. Females, Mothers , Doctors

Clearly, all mothers are females. This would be represented by two concentric circles. But, some females and some mothers can be doctors.

So, the circle representing doctors would intersect the two concentric circles. Thus, the diagram becomes as shown in

Fig 8 with circle A representing Mothers, circle B representing. Females and circle C representing Doctors.

**Solved Examples**

In this type of questions, generally a Venn Diagram is given. Each geometrical figure in the diagram represents a certain class.

The candidate is required to study and analyse the figure carefully and then answer certain questions regarding the given data.

**Example 1**

In the following diagram, three classes of population are represented

by three figures. The triangle represents the scfhool teachers,

the square represents the married persons and the circle represents

the persons living in joint families.

(i) Married persons living in the joint families but not working as school teachers are represented by

(A) C ( B) F (C) D (D) A

(ii) Persons who live in joint families, are unmarried and who do not work as school teachers are represented by

(A) C (B) B (C) E (D) D

(iii) Married teachers living in joint families are represented by

(A) C (B) B (C) D (D) A

(iv) School teachers who are married bu do not live in joint families are represented by

(A) C (B) F (C) A (D) D

(v) School teachers who are neither married nor do live in joint families are represented by

(A) F (B) C (C) B (D) A

**Solution**

(i) Married persons living in joint families are represented by the regin common to the square and the circle i.e. D and B.

But, according to the given conditions, the persons ahould not be school teachers. So, B is tto be thexcluded. Hence,

the required condition is denoted by region D. So, the answer is (C).

(ii) Persons living in joint families are represented by the circle. According to the given conditions, the person should be

unmarried and not working as school teachers. So, the region should not be a part of either square or the triangle. thus,

the given condition are satisfied by the region E. So, the answer is (C).

(iii) Married teachers are represented by the region common to the square and the triangle. i.e. B and C.

But according to the given conditions, the persons chould be living in joint families. So, the required region should be

a part of the circle. Since B lies inside the circle, so the given conditions are satisfied by the persons denoted by the region B. Hence, the answer is (B).

(iv) As in the above question, married teachers are represented by B and C. But, here, the given conditions lay down that the

persons should not be living in joint families. So, the required region should lie outside the circle. Since C lies outside the circle,

so the given conditions are satisfied by the persons denoted by the region C. Hence the answer is (A).

(v) School teachers are represented by the triangle.But according to the given conditions, persons are neither married nor do they

live in joint families. So, the region should not be a part of either the square or the circle. Such a region is F. Hence, the answer is (A).

Example 2

In the following diagrame, the square represents girls,

the circle tall persons, the triangle is for tennis players and

the rectangle stands for the swimmers.

On the basis of above diagram, answer the following questions.

(i) Which letter represents tall girls who are swimmers but don’t play tennis ?

(A) C (B) D (C) G (D) H

(ii) Which letter represent girls who are swimmers, play tennis but are not tall ?

(A) B (B) E (C) F (D) None of these

(iii) Which letter represetns tall girls who do not play tennis and are not swimmers ?

(A) C (B) D (C) E (D) G

(iv) Which letter represents tall persons who are gents and swimmers but do not play tennis ?

(A) I (B) J (C) K (D) L

**Solution**

(i) Tall girls, who are swimmers are represented by the region common to the square, circle and the rectangle, i.e.

G and H. But, according to the given conditions, the girls should’t be tennis players, So the required region should not

be the part of the triangle i.e. H should be excluded. thus, the region representing the persons satifying the given conditions is G. Hence, the answer is (C).

(ii) Girls who are swimmers and play tennis are represetned by the region common to the suqare, triangle and rectangle, i.e.

H. But, it is given that the girls Since H is a part of the circle, so the answer is (D).

(iii) Tall girls are represented by the region common to the square and the circle i.e. D, C, G and H. But, according to the conditions,

the girls are neither tennis players nor swimmers. So, the required region should be neither a part of the rectangle nor the triangle.

G lies inside the rectangle, C inside the triangle and H is common to both. So. the answer is (B).

(iv) The tall persons are represented by regions inside the circle i.e. C, D, G, H, I, J and K. Since the persons are not girls and do not

play tennis. so the region should not be a pert of either the square or the triangle. Thus, C, D, G, H should be excluded. Also,

according to the given conditions, the persons should be swimmers

. So, the required region should be a pert of the rectangle and such a region is K. Hence, the answer is (C).

**Example 3**

The following questions are based on the diagram given below.

(A) The rectangle represents government employees

(B) The triangle represents urban people.

(C) The circle represents graduates.

(D) The square represents clerks.

(i) Which of the following statements is true ?

(a) All government employees are clerks.

(b) Some government employees are graduates as well as clerks.

(c) All Government employees are graduates.

(d) All clerks are government employees but not graduates.

(ii) Which of the folloiwng statements is true ?

(a) All urban people are graduates.

(b) Some clerks are government employees but not urban.

(c) All goverment employees are clerks.

(d) Some urban people are not graduates.

(iii) Choose the correct statement :

(a) Some clerks are government employees

(b) No clerk is urban.

(c) All graduates are urban

(d) All graduates are government employees.

**Solution**

(i) The above cases may be considered as under:

For statement (a) to be true, the rectangle should lie inside the square. This is not true. Hence, (a) is false.

For statement (b) to be true, there should be a region common to the rectangle, circle and the square. Such a region is 6. Hence (b) is true.

Further for statement (c) to be true, the rectangle should lie inside the circle. So, (c) is false.

Forse statement (d) to be true, square should lie wholly inside the rectangle, with no region common to the circle. This is not true. So, (d) is false.

(ii) For the validity of condition (a), the triangle should lie inside the circle. This is not true. So, (a) is false.

For the validity of statement (b), there should be a region which is common to the square and the rectangle bu is

not a part of the triangle. Sinche no such region exists, (b) is false

For the validity of statement (c), the rectangle should lie inside the square. this is not true. So, (c) is false.

For the validity of statement (d), some region of the triangle should lie outside the circle. Since this is true, so, (d) is true.

(iii) For the validity of statemnt (a), there should be a region common to the square and rectangle.

Such regions are 6 and 7, So (a) is true.

Further, for statement (b) to be true, there should be no region common to the square and the triangle.

But since square lies wholly inside the triangle, (b) is false.

For statement (c) to be true, circle should lie inside the triangle. Clearly, (c) is false. For the validity of statement

(d), the circle should lie inside the rectangle. Clearly, (d) is false.

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