How to solve questions based on Venn Diagram

Venn diagram problems are used to calculate the differences and similarities between two or three sets of numbers using a visual diagram. Generally, in venn-diagram, all the sets are represented by circles with the area inside the circle being a part of the set and area outside the circle does not belonging to the set.

To start with, let us understand the meaning of set first. A set is a well-defined group of objects and all these objects are called elements. If A is a set and ‘a’ is an element of this set then it is said that ‘a’ belongs to A. And above all, all the elements of a set need to satisfy some quality as per the definition. A set can be a finite or infinite depending upon the definition. For example a set of all the natural numbers less than 100 will be a finite set where as a set of all the natural numbers will be an infinite set.

Some basic definition:

• Universal set is defined as a set of all the elements under consideration.
• An empty set is defined as a set which contains no element. It is represented by {}. We need to understand here that .
• Intersection of two sets is defined as all the common elements of two sets and is represented by symbol “ ”whereas the union of two sets is a set which contains all the elements of both the sets in question and is represented by the symbol “ ”. For example if A={1,3,5,7,9} and B={2,3,5,7}, then  .
• A-B is defined as all the terms of A which are not present in B. For example if A={1,3,5,7,9} and B={2,3,5,7}, then  .
• A is defined as the subset of B if all the elements of A are also present in B. In such a situation, B will also be called superset of A. An empty set is a sub set of all the sets whereas a universal set is super set of all the sets. At the same time, all sets are subset of it-self and super set of it-self. For any set with n elements, there will be exactly  subsets out of which one will be empty set and one will be the set it-self.

Pictorial Representation
 

Now let us look at questions which will help us understand these concepts a little better. Suppose, there is a school, wherein 60 drink tea, 70 drink coffee, 80 drink cold drink. Among the same group of students, there are 20 preferring tea and coffee, 25 prefer tea and cold drink and 30 prefer coffee and cold drink. In the same school, there are 5 students, who prefer tea, coffee and cold drink. Now, let us try to put in on a venn diagram.

First thing which we need to understand in that is 25 prefer tea and cold drink then it does not mean that these persons will not prefer coffee and more importantly, these 25 is a sub set of 60 who prefer tea and 80 who prefer cold drink. All these questions should start with the intersection of all the three.  Now, there are 5 who prefer all the three drinks. Thus 15 prefer only tea and coffee, 20 prefer only tea and cold drink and 25 prefer only coffee and cold drink. Now, coming to persons preferring only tea, out of 60 students, 5 prefer all the three drinks, 15 prefer tea and coffee and 20 prefer tea and cold drink thus, only 20 prefer only tea and so on an so forth.  The following data can be put on a venn diagram in the following manner.

We can also solve the given question with the help of formulae as


Thus total number of students, who prefer at least one of the three drinks, is 140.

Now, try to solve the following questions on the basis of the knowledge that you have gathered from this article.

Directions for questions 1 to 3:

There are 200 students in a school. 140 opt for Maths, 100 opt for Biology and all of them have atleast opted for one of the subjects.

1. How many of them opted for both the subjects?
a. 60
b. 80
c. 40
d. Cannot be determined
2. How many of them opted for Maths only?
a. 100
b. 60
c. 40
d. Cannot be determined
3. How many of them opted for Biology only?
a. 100
b. 60
c. 40
d. Cannot be determined

Solutions

1. Maths        Biology


As per the Venn-diagram,
There are 40 students who opted for both the subjects.
Hence (c)
2. 100 of them opted for Maths only.
Hence (a)
3. 60 of them opted for Biology only.
Hence (b)

Directions for questions 4 and 5:

A survey is conducted among 200 students in a college. 100 like Hiking, 120 like Rafting and 80 like Cycling. 60 students like Hiking and Rafting, 40 like Rafting and Cycling, 10 students liked all the three and 20 students did not like any of the three.

4. How many students liked at least two of the three?
a. 90
b. 100
c. 120
d. 80
5. How many students liked only two of the three?
a. 90
b. 100
c. 80
d. 120

Solutions

4. H = 100;   R = 120;  C = 80;    H R = 60; R C = 40; H R  C=10

Hiking      Rafting

             Cycling

a + b + d + e = 100
b + c + e + f = 120
d + e + f + g = 80
a + b + c + d + e + f + g = 180
b + e = 60
e + f = 40
e = 10
From these equations we can find;
a = 20; b = 50; c = 30; d = 20; e = 10; f = 30; g = 20
At least any two = b + d + e + g = 100
Hence (b)

5. Only two = b + d + f = 90
Hence (a)