Examples on Venn Diagrams with Answers | Solved Problems on Venn Diagrams

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}