If you are looking everywhere to find Solved Questions on Venn Diagrams then you have come the right way. We use Venn Diagrams to Visualize Set Operations in Set Theory . Refer to Solved Questions of Venn Diagrams and learn how to find Union, Intersection, Complement, etc. using the Venn Diagrams. Use the Practice Problems provided and get a good grip on the concepts involving Sets easily. You can use the below existing questions as a quick reference to solve any kind of problem-related to Sets using Venn Diagrams.
1. From the following Venn diagram, find the following sets.
(i) A
(ii) B
(iii) ξ
(iv) A’
(v) B’
(vi) C’
(vii) C – A
(viii) B – C
(ix) A – B
(x) A ∪ B
(xi) B ∪ C
(xii) A ∩ C
(xiii) B ∩ C
(xiv) (B ∪ C)’
(xv) (A ∩ B)’
(xvi) (A ∪ B) ∩ C
(xvii) A ∩ (B ∩ C)
Solution:
Given Sets are A = {1, 2, 3, 4, 6, 9, 10}, B = {1, 3, 4, 9, 13, 14, 15}, C= {1, 2, 3, 6, 9, 11, 12, 14, 15}, ξ or U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
(i) A = {1, 2, 3, 4, 6, 9, 10}
(ii) B = {1, 3, 4, 9, 13, 14, 15}
(iii) ξ or U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
(iv) A’
A’ = U -A
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} – {1, 2, 3, 4, 6, 9, 10}
= { 5, 7, 8, 11, 12, 13, 14, 15}
(v) B’
B’ = U -B
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} – {1, 3, 4, 9, 13, 14, 15}
= { 2, 5, 6, 7, 8, 10, 11, 12}
(vi) C’
C’ = U – C
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} – {1, 2, 3, 6, 9, 11, 12, 14, 15}
= {4, 5, 7, 8, 10, 13}
(vii) C – A
C-A = {1, 2, 3, 6, 9, 11, 12, 14, 15} – {1, 2, 3, 4, 6, 9, 10}
= {11, 12, 14, 15}
C- A is the Elements that are in Set C but doesn’t belong to Set A.
(viii) B – C
B-C = {1, 3, 4, 9, 13, 14, 15} – {1, 2, 3, 6, 9, 11, 12, 14, 15}
= {4, 13}
(ix) A – B
A-B = {1, 2, 3, 4, 6, 9, 10} – {1, 3, 4, 9, 13, 14, 15}
= {2, 6, 10}
(x) A ∪ B
A ∪ B = {1, 2, 3, 4, 6, 9, 10} ∪ {1, 3, 4, 9, 13, 14, 15}
= {1, 2, 3, 4, 6, 9, 10, 13, 14, 15}
(xi) B ∪ C
B U C = {1, 3, 4, 9, 13, 14, 15} U {1, 2, 3, 6, 9, 11, 12, 14, 15}
= {1, 2, 3, 4, 6, 9, 11, 12, 13, 14, 15}
(xii) A ∩ C
A ∩ C = {1, 2, 3, 4, 6, 9, 10} U {1, 2, 3, 6, 9, 11, 12, 14, 15}
= { 1, 2, 3, 6, 9}
(xiii) B ∩ C
B ∩ C = {1, 3, 4, 9, 13, 14, 15} ∩ {1, 2, 3, 6, 9, 11, 12, 14, 15}
= { 1, 3, 9, 14, 15}
(xiv) (B ∪ C)’
(B ∪ C)’ = U – (B U C)
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} – {1, 2, 3, 4, 6, 9, 11, 12, 13, 14, 15}
= { 5, 7, 8, 10}
(xv) (A ∩ B)’
Firstly, find the (A ∩ B) i.e. {1, 2, 3, 4, 6, 9, 10} ∩ {1, 3, 4, 9, 13, 14, 15}
= {1, 4, 9}
(A ∩ B)’ = U – (A ∩ B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} – {1, 4, 9}
= {2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15}
(xvi) (A ∪ B) ∩ C
(A ∪ B) ∩ C = {1, 2, 3, 4, 6, 9, 10, 13, 14, 15} ∩ {1, 2, 3, 4, 6, 9, 11, 12, 13, 14, 15}
= { 1, 2, 3, 4, 6, 9, 13, 14, 15}
(xvii) A ∩ (B ∩ C)
A ∩ (B ∩ C) = {1, 2, 3, 4, 6, 9, 10} ∩ { 1, 3, 9, 14, 15}
= {1, 3, 9}
2. Find the following sets from the given Venn Diagram?
(i) F
(ii) H
(iii) B
(iv) F U H
(v) B ∩ F
(vi) F U H U B
Solution:
(i) F = {9, 12, 13, 15}
(ii) H = {12, 14, 15}
(iii) B = {13, 14, 15, 20}
(iv) F U H
F U H = {9, 12, 13, 15} U {12, 14, 15}
= {9, 12, 13, 14, 15}
(v) B ∩ F
B ∩ F = {13, 14, 15, 20} ∩ {9, 12, 13, 15}
= { 13, 15}
(vi) F U H U B
F U H U B = (F U H) U B
= {9, 12, 13, 14, 15} U {13, 14, 15, 20}
= { 9, 12, 13, 14, 15, 20}