Venn diagram questions with solutions are given here for students to practice various questions based on Venn diagrams. These questions are beneficial for both school examinations and competitive exams. Practising these questions will develop a skill to solve any problem on Venn diagrams quickly.
Venn diagrams were first introduced by John Venn to represent various propositions in a diagrammatic way. Venn diagrams are used for representing relationships between given sets. For example, natural numbers and whole numbers are subsets of integers represented by the Venn diagram:
Using Venn diagrams, we can easily understand whether given sets are subsets of each other or disjoint sets or have something in common.
Also Read:
Following are some set operations and their meaning useful while solving problems on the Venn diagram:
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Set Operations |
Meaning |
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A ⊂ B |
Set A is a proper subset of B, or A is contained in B. |
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A ⋃ B |
Set of all those elements which either belong to A or belong to B |
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A ∩ B |
Set of all those elements which belong to both A and B |
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AC or A’ |
Set of all those elements which are not in A |
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A – B |
Set of all those elements which only belong to A |
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A ⊝ B |
Symmetric difference: Set of all those elements which either belong to A or belong to B, but not in both. |
Some important formulae:
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Venn Diagram Questions with Solution
Let us practice some questions based on Venn diagrams.
Question 1: If A and B are two sets such that number of elements in A is 24, number of elements in B is 22 and number of elements in both A and B is 8, find:
(i) n(A ∪ B)
(ii) n(A – B)
(ii) n(B – A)
Solution:
Given, n(A) = 24, n(B) = 22 and n(A ∩ B) = 8
The Venn diagram for the given information is:
(i) n(A ∪ B) = n(A) + n(B) – n(A ∩ B) = 24 + 22 – 8 = 38.
(ii) n(A – B) = n(A) – n(A ∩ B) = 24 – 8 = 16.
(iii) n(B – A) = n(B) – n(A ∩ B) = 22 – 8 = 14.
Question 2: According to the survey made among 200 students, 140 students like cold drinks, 120 students like milkshakes and 80 like both. How many students like atleast one of the drinks?
Solution:
Number of students like cold drinks = n(A) = 140
Number of students like milkshake = n(B) = 120
Number of students like both = n(A ∩ B) = 80
Number of students like atleast one of the drinks = n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 140 + 120 – 80
= 180.
Question 3: In a group of 500 people, 350 people can speak English, and 400 people can speak Hindi. Find how many people can speak both languages?
Solution:
Let H be the set of people who can speak Hindi and E be the set of people who can speak English. Then,
n(H) = 400
n(E) = 350
n(H ∪ E) = 500
We have to find n(H ∩ E).
Now, n(H ∪ E) = n(H) + n(E) – n(H ∩ E)
⇒ 500 = 400 + 350 – n(H ∩ E)
⇒ n(H ∩ E) = 750 – 500 = 250.
∴ 250 people can speak both languages.
Questions 4: The following Venn diagram shows games played by the number of students in a class:
How many students like only cricket and only football?
Solution:
As per the given Venn diagram,
Number of students only like cricket = 7
Number of students only like football = 14
∴ Number of students like only cricket and only football = 7 + 14 = 21.
Question 5: In a class of 40 students, 20 have chosen Mathematics, 15 have chosen mathematics but not biology. If every student has chosen either mathematics or biology or both, find the number of students who chose both mathematics and biology and the number of students chose biology but not mathematics.
Solution:
Let, M ≡ Set of students who chose mathematics
B ≡ Set of students who chose biology
n(M ∪ B) = 40
n(M) = 20
n(B) = n(M ∪ B) – n(M)
⇒ n(B) = 40 – 20 = 20
n(M – B) = 15
n(M) = n(M – B) + n(M ∩ B)
⇒ 20 = 15 + n(M ∩ B)
⇒ n(M ∩ B) = 20 – 15 = 5
n(B – M) = n(B) – n(M ∩ B)
⇒ n(B – M) = 20 – 5 = 15
Question 6: Represent The following as Venn diagram:
(i) A’ ∩ (B ∪ C)
(ii) A’ ∩ (C – B)
Solution:
(i)
(ii)
Question 7: In a survey among 140 students, 60 likes to play videogames, 70 likes to play indoor games, 75 likes to play outdoor games, 30 play indoor and outdoor games, 18 like to play video games and outdoor games, 42 play video games and indoor games and 8 likes to play all types of games. Use the Venn diagram to find
(i) students who play only outdoor games
(ii) students who play video games and indoor games, but not outdoor games.
Solution:
Let V ≡ Play video games
I ≡ Play indoor games
O ≡ Play outdoor games
n(V) = 60, n(I) = 70, n(O) = 75
n(I ∩ O) = 30, n(V ∩ O) = 18, n(V ∩ I) = 42
n(V ∩ I ∩ O) = 8
Hence, by Venn diagram
Number of students only like to play only outdoor games = 35
Number of students like to play video games and indoor games but not outdoor games = 34
Note: Always begin to fill the Venn diagram from the innermost part.
Question 8: Using the Venn diagrams, verify (P ∩ Q) ∪ R = (P ∪ R) ∩ (Q ∪ R).
Solution:
The shaded portion represents (P ∩ Q) ∪ R in the Venn diagram.
Comparing both the shaded portion in both the Venn diagram, we get (P ∩ Q) ∪ R = (P ∪ R) ∩ (Q ∪ R).
Question 9: Prove using the Venn diagram: (B – A) ∪ (A ∩ B) = B.
Solution:
From the Venn diagram, it is clear that (B – A) ∪ (A ∩ B) = B
Question 10: In a survey, it is found that 21 people read English newspaper, 26 people read Hindi newspaper, and 29 people read regional language newspaper. If 14 people read both English and Hindi newspapers; 15 people read both Hindi and regional language newspapers; 12 people read both English and regional language newspaper and 8 read all types of newspapers, find:
(i) How many people were surveyed?
(ii) How many people read only regional language newspapers?
Solution:
Let A ≡ People who read English newspapers.
B ≡ People who read Hindi newspapers.
C ≡ People who read Hindi newspapers.
n(A) = 21, n(B) = 26, n(C) = 29
n(A ∩ B) = 14, n(B ∩ C) = 15, n(A ∩ C) = 12
n(A ∩ B ∩ C) = 8
(i) Number of people surveyed = n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) = 21 + 26 + 29 – 14 – 15 – 12 + 8 = 43
(ii) By the Venn diagram, number of people who only read regional language newspapers = 10.
Video Lesson on Introduction to Sets
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Practice Questions on Venn Diagrams
1. Verify using the Venn diagram:
(i) A – B = A ∩ BC
(ii) (A ∩ B)C = AC ∪ BC
2. For given two sets P and Q, n(P – Q) = 24, n(Q – P) = 19 and n(P ∩ Q) = 11, find:
(i) n(P)
(ii) n(Q)
(iii) n (P ∪ Q)
3. In a group of 65 people, 40 like tea and 10 like both tea and coffee. Find
(i) how many like coffee only and not tea?
(ii) how many like coffee?
4. In a sports tournament, 38 medals were awarded for 500 m sprint, 15 medals were awarded for Javelin throw, and 20 medals were awarded for a long jump. If these medals were awarded to 58 participants and among them only three medals in all three sports, how many received exactly two medals?
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