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}