# Venn Diagram Reasoning

The geometric shapes expressing the sets are called venn diagrams .

The questions asked in this chapter are mainly of two types-

(i) Classify groups of items given through diagrams.
(ii) Finding the number of items or numbers under a particular class through numbers or letters.

Classification of groups of items given through diagrams
Under this, the first questions are given in groups of some objects and some diagrams are given. you have to find that Venn Diagram through the Venn diagrams given in question, which classify the group of items given in the question in complete and correct manner or denote the relation between them.

The purpose of such questions is to examine the ability of the candidates to understand certain sections and illustrate them in a graphic interpretation. The candidate generally has to decide whether the given fact is related to the size of the set or the subset or it will be related to any other set. The facts can also be related to aggregates or subsets. Although the signs and language of the theory set is not used in the questions

Now take a closer look at the relations below.

(a) The given diagram indicates that one class is fully contained in another but is not mixed.

(b) The diagram illustrates that no class is fully contained in each other but there are some common members in both.

(c) The diagram provided indicates that no members are equal.

Find out the number of items or numbers coming under a particular class through numbers and letters: Few Venn Diagrams are given under this, each of which is contains different numbers, letters or sign at various places. in different Categories. Each diagram represents the different class. Candidates have to find the number of items or items in a particular class through the Venn or the numbers given in it. This method is used on the basis of differences and elements. These are governed by different executive rules. This separate set of elements is given only when it is related to specific operating rules or to set up a new set based on more than two separate sets, their union and Intersection of two types. If two sets are P and Q then P Union Q means that all elements of P and Q will be included in that set, whereas P Intersection Q will contain only those that are identical (Common) in P and Q. When a notification is presented as a set, then the process of Data Presentation starts. A set is a collection of elements, which is operated by the same operating rules. For example, the set of tennis players is different from the football players’ set.
For now, to clarify the draft of the questions and to clarify the above facts, please take a closer look at the key examples below.

Type 1

Example 1. Which of the following given Venn diagrams, represent the relation between the three students given?

Solution: (a) Some students may be cricket players and fans of tennis. Some cricket players and tennis fans students can be there. Some tennis fans, cricket players students can also be there. So the diagram given in the option (a) is appropriate.

→ Instructions (e.g. 2-5): In each of the questions given below, three types of items have been given. You have to find out that one of the four Venn diagrams below is a Venn diagram, which correctly denotes the relation of the group of three classes given in question.

2. Science, Physics, Chemistry
3. Doctors, Men, Artists
4. Creatures, humans, planets
5. Gold, Jewelry, Silver

Solution: 2. (a) Because both physics and chemistry are separate subjects but both come under science.

3. (c) Some male can be artists and some male can be doctors , some artists can be doctors and some doctors can be the artist.

4. (b) Man is an organism, while the planet is a member of the solar system different from these two.

5. (d) Jewel is made of gold or silver and some ornaments are made of other metals too.

Type 2

Example: If the ‘circle’ denotes tall persons, ‘triangle’ denotes soldiers and ‘square’ denotes strong persons, then what is the number given under the following diagram denotes only ‘Strong soldiers’?

Solution: (c) The circle represents the tallest persons, the ‘triangle’ represents soldiers and the ‘square’ represents strong persons, while we have to find the number that is the strongest soldier, i.e. the strongest and the soldiers Denotes, So in the given Venn diagram we will find the number which is common in the square and the triangle. Here we find that such a number is only 6 which is common in both the square and the triangle. Hence the number ‘6’ will represent strong soldiers.

Type 3

Example: In the Venn diagram below, the combined triangle, square and circle are represented While dissecting each other respectively. In the following Venn diagram, which of the areas given from A to G marked in the following diagram denotes such individuals, who are ‘urban and educated but not hard-working?’
Find that area from the given option.

Solution: (c) Here are the people who are urban and educated but not hard-working, that means we have to know such a person who is only urban and educated as in diagram urban is represented by ‘Triangle’ and city if defined by ‘circle’, so to find the urban and educated person, we will analyze carefully that which area is common in the triangle and the circle in the given diagram, we have found that the letter D is common to the triangle and the circle. Hence the area “D” denotes the urban and educated person.

Type 4

→ Instructions (e.g. 1-4) In the following diagram, the ‘Circle’ has been defined by the unemployed, rectangle to the hard-working class, the triangle to the rural and the intelligent persons from the rectangle. Carefully study the following diagram and answer the questions based on these.

1. People who are unemployed, hardworking and intelligent, but not rural, is denoted by which area in the diagram?
(a) 10 (b) 11 (c) 12 (d) 9

2. Such people who are not rural and neither unemployed nor intelligent but hardworking, by which field of diagrams they have been denoted?
(a) 11 (b) 10 (c) 12 (d) 8

3. Such rural, which are hardworking and unemployed but are not intelligent, by which field of diagrams they are denoted?
(a) 4 (b) 3 (c) 1 (d) 2

4. A rural hardworking person who is neither unemployed nor intelligent, by what field of diagrams have been defined?
(a) 3 (b) 2 (c) 14 (d) not in any region

Solution:
1. (d) Here we have to know such an area, which represents the unemployed, hardworking and intelligent person. So we will find the area which is only in the ‘circle’, ‘square’ and ‘rectangle’. Here we find that such area is just ‘9’ which comes under ‘circle’, ‘square’ and ‘rectangle’, hence the unemployed, hardworking and intelligent person has been defined by area “9”.

2. (c) Here we only have to find the area which denotes only the hard-working person. In the diagram, the hardworking person has been denoted by ‘square’. So we will see that there are such areas in the diagram that come under square only. Here we are seeing that such an area in the diagram is only ’12’ which comes under the square. Hence the hardworking person has been denoted by the area marked ’12’.

3. (d) Here we have to know the area which is rural, hardworking and unemployed but not intelligent, that is, we should know the area that represents rural, hardworking and unemployed, all three areas. Since the ‘rural’ defined in the diagram by the triangle, the ‘Hardworking’ is defined by the square and the ‘unemployed’ is a circle, so in the diagram we will find the number representing the area which is common to the triangle, square and circle. here we find that such a number is only ‘2’. Thus, in the diagram, the area ‘2’ represents a group of rural, hard-working and unemployed people.

4. (d) Here we have to find the area which denotes rural hardworking who are neither unemployed nor intelligent. Therefore, we will find the area which represents only ‘rural’ and ‘hard working’ individuals. Since the diagram has defined the ‘triangle’ to the villagers and the ‘square’ to the hard working. So we will find the area in the diagram, which is common to the triangle and square only. When we carefully analyse here, we find that there is no such symbolic area which is common to the triangle and square. Therefore, only the rural and hardworking person has not been described together in any field.

→ Note: The above questions can also be solved on the basis of the following chart, under which the common number is to be included and which is not common, will not include the number coming under it.
Circle-unemployed-1, 2, 3, 6, 7, 8, 9, 15
Rectangle-wise 1, 8, 9, 10, 11, 14
Class-diligent -1, 2, 3, 9, 11, 12, 14

Example 5. By carefully studying the diagram given below, let’s find out that the young man who works but is not educated, who is the following?

(a) 3 or 7 or 6 (b) 3 or 4 (c) 6 or 4 (d) 2 or 5 or 7

Solution: (a) With the question, we have to find a young man who does a job but is not educated. So in the given diagram we will see a number which is similar in only two diagrams. Here, we find that such numbers are only ‘3’, ‘7’ and ‘6’ which are similar in two diagrams, 3 or ‘7’ or ‘6 are young people who are employed but educated Are not there.

Type 5

→ Instructions (e.g. 1 – 2) In the given question, answer the following questions on the basis of the given chart in which large triangle represents politician, small triangle represents teacher, circle represents graduate, the quadrilateral represents the member of parliament.

1. Which of the following politicians is a graduate but not a member of parliament?
(a) IV, IX (b) I, VIII, IX (c) IX, II (d) II, III

2. Which of the following politicians, neither teachers nor graduates?
(a) III, IV (b) IX, VIII (c) IX, II (d) II , III

Solution:

1. (d) Because here we have to find the area which represents a politician who is a graduate but not a member of the Parliament. Therefore, we have to know the region which represents the politician and the undergraduate area except the member area of the Parliament. It is not said in the question whether a politician is a teacher or not; therefore, a politician graduate in the field of small triangle addressing the teacher will be valid, but he should not be in the area of the quadrilateral, i.e. Parliament, thus, We find that only area II and I indicates that politicians are graduates but not a member of the Parliament.

2. (b) Because here we have to know the area that displays such a politician who is neither a teacher nor a graduate. Therefore, we have to leave the teacher and graduate to find the area of the politician. In the above diagram, the teacher has been defined by ‘small triangle’, graduation with ‘circle’ and politician by ‘big triangle’. But we have to find the area in the big triangle except the ‘small triangle’ and ‘circle’. Thus we find that such areas are only IX and VIII. Therefore, IX and VIII are the areas which represents a politician who is neither a teacher nor a graduate.