Analysis based Venn diagrams

In these types of questions, generally a Venn diagram comprising of different geometrical figures is given. Each geometrical figure in the diagram represents a certain class. The candidate is required to study and analyze the figure carefully and then answer the given questions based on it.

To solve such questions, a student should have solid grasp on these six conceptual terms:

1. AND – It’s intersection (A Ո B)
2. OR – It’s union (A U B)
3. Only
4. Not
5. At least
6. At most

Let’s understand them using a couple of examples.

Let us consider a scenario where we have people speaking one or more of the given three languages - English, Sanskrit and German.
Let us analyse the given Venn diagram.

Number of people who speak:

• only English = 35

• English = 35 + 13 + 9 + 12

• German AND English = 12 + 9

• German AND English only = 12

• German OR English = 35 + 24 + 12 + 13 + 19 + 9

• German OR English only = 35 + 12 + 24

• German only OR English only = 24 + 35

• not German OR English = (German OR English)’ = 27

• at least 2 languages = 12 + 13 + 19 + 9 (i.e. those who speak 2 or 3 languages)

• at most 2 languages = 35 + 24 + 27 + 12 + 19 + 13 or Total – 9 (i.e. those who speak 2 or 1 languages)

Q. The following figure shows the viewership of two leading news channels, as found in a sample survey. Find out the number of people who do not watch any news channel.
(a) 175   (b) 75   (c) 195   (d) 135

Explanations :

Explanation 1:

People who watch Republic OR ZEE News (i.e. atleast one channel) = 85 + 200 - 60 = 225
(we subtracted 60 as otherwise it would have been counted twice)

So, people who do not watch any news channel = (Republic OR ZEE News)' = Total number of people - 225 = 420 - 225 = 195

Explanation 2:

People who watch only Republic = 85 - 60 = 25
People who watch only ZEE News = 200 - 60 = 140

So, people who watch Republic OR ZEE News (i.e. atleast one channel) = 25 + 140 + 60 = 225

And people who do not watch any news channel = (Republic OR ZEE News)'= Total number of people - 225 = 420 - 225 = 195