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Venn diagram: Different Types of Venn Diagrams and Important Results of Venn Diagrams

Venn diagram: Different Types of Venn Diagrams and Important Results of Venn Diagrams

Just like we perform operations (addition, subtraction etc.) on mathematical numbers, in set theory also operations are performed on two or more sets to obtain a combination of elements, as per the operation performed on them. Also, these operations can be visualized easily with the help of pictorial representations called Venn diagrams.

Table of Contents

  • What is a Venn diagram?
  • Different types of Venn Diagrams
  • Important results from Venn diagram
  • Practice Problems
  • FAQ’s

What is a Venn Diagram?

Diagrams drawn in an illustration that uses circles to represent sets and the relationship among the sets are called Venn-Euler diagrams or simply Venn-diagrams.Venn diagrams help to represent the similarities and differences between two concepts.

Notation: Universal set 𝑈 is represented by a rectangle.Its Subsets are represented using circles within the rectangle.

Different Types of Venn Diagrams

 

 

 

 

 

  • A is the subset of B i.e. 𝐴 ⊂ 𝐵

                      

                        

 


 

 

 

 

 

  • A intersection B 

i.e.  A∩B=shaded region

                                   

 


 

 

 

  • A union B

i.e. A∪B=shaded region

                          


  • 𝐴 and 𝐵 do not have any common elements.

 


  • Complement of A

 i.e.Ac=U-A =shaded region

 


  • Complement of A ∩ B

 i.e. (A âˆ© B)′=shaded region

 


 

 

 

 

 

  • Complement of A U B

 i.e. (A UB)′=shaded region

 


  • Intersection of A and B complement i.e. A âˆ© B'=A-B=shaded region

 


  • A delta B i.e.

𝐴 ∆ B= (𝐴−𝐵) ∪ (𝐵−𝐴)=shaded region

 


  • Intersection of A, B and C 

i.e. A âˆ© B âˆ© C=shaded region


Important Results from Venn Diagram


<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#xA0;</mo><mo>&#x222A;</mo><mo>&#xA0;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>)</mo><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#xA0;</mo><mo>&#x2212;</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x2229;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mspace linebreak="newline"/><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#xA0;</mo><mo>&#x222A;</mo><mo>&#xA0;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>)</mo><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#xA0;</mo><mo>&#x27FA;</mo><mo>&#xA0;</mo><mo>&#x1D434;</mo><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><mo>&#x1D435;</mo><mo>&#xA0;</mo><mi>a</mi><mi>r</mi><mi>e</mi><mo>&#xA0;</mo><mi>d</mi><mi>i</mi><mi>s</mi><mi>j</mi><mi>o</mi><mi>i</mi><mi>n</mi><mi>t</mi><mo>&#xA0;</mo><mi>n</mi><mi>o</mi><mi>n</mi><mo>-</mo><mi>v</mi><mi>o</mi><mi>i</mi><mi>d</mi><mo>&#xA0;</mo><mi>s</mi><mi>e</mi><mi>t</mi><mi>s</mi><mo>&#xA0;</mo><mspace linebreak="newline"/><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#xA0;</mo><mo>&#x2212;</mo><mo>&#xA0;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>)</mo><mo>&#xA0;</mo><mo>&#x2212;</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#xA0;</mo><mo>&#x2229;</mo><mo>&#xA0;</mo><mo>&#x1D435;</mo><mo>)</mo></math>


<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#xA0;</mo><mo>&#x2206;</mo><mo>&#xA0;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>)</mo><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#x2212;</mo><mn>2</mn><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#xA0;</mo><mo>&#x2229;</mo><mo>&#xA0;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#xA0;</mo><mspace linebreak="newline"/><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x222A;</mo><mo>&#x1D435;</mo><mo>&#x222A;</mo><mo>&#x1D436;</mo><mo>)</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>)</mo><mo>+</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D435;</mo><mo>)</mo><mo>+</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D436;</mo><mo>)</mo><mo>&#x2212;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x2229;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#x2212;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D435;</mo><mo>&#x2229;</mo><mo>&#x1D436;</mo><mo>)</mo><mo>&#x2212;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D436;</mo><mo>&#x2229;</mo><mo>&#x1D434;</mo><mo>)</mo><mo>+</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x2229;</mo><mo>&#x1D435;</mo><mo>&#x2229;</mo><mo>&#x1D436;</mo><mo>)</mo></math>


<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x2032;</mo><mo>&#x222A;</mo><mo>&#x1D435;</mo><mo>&#x2032;</mo><mo>)</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x2229;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#x2032;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D448;</mo><mo>)</mo><mo>&#x2212;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x2229;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mspace linebreak="newline"/><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x2032;</mo><mo>&#x2229;</mo><mo>&#x1D435;</mo><mo>&#x2032;</mo><mo>)</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x222A;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#x2032;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D448;</mo><mo>)</mo><mo>&#x2212;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x222A;</mo><mo>&#x1D435;</mo><mo>)</mo></math>


<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mi>u</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo>&#xA0;</mo><mi>o</mi><mi>f</mi><mo>&#xA0;</mo><mi>e</mi><mi>l</mi><mi>e</mi><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo>&#xA0;</mo><mi>i</mi><mi>n</mi><mo>&#xA0;</mo><mi>e</mi><mi>x</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>l</mi><mi>y</mi><mo>&#xA0;</mo><mi>o</mi><mi>n</mi><mi>e</mi><mo>&#xA0;</mo><mi>o</mi><mi>f</mi><mo>&#xA0;</mo><mi>t</mi><mi>h</mi><mi>e</mi><mo>&#xA0;</mo><mi>s</mi><mi>e</mi><mi>t</mi><mi>s</mi><mo>&#xA0;</mo><mo>&#x1D434;</mo><mo>,</mo><mo>&#xA0;</mo><mo>&#x1D435;</mo><mo>,</mo><mo>&#xA0;</mo><mo>&#x1D436;</mo><mo>&#xA0;</mo><mspace linebreak="newline"/><mo>=</mo><mo>&#xA0;</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>)</mo><mo>+</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D435;</mo><mo>)</mo><mo>+</mo><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D436;</mo><mo>)</mo><mo>&#x2212;</mo><mn>2</mn><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x2229;</mo><mo>&#x1D435;</mo><mo>)</mo><mo>&#x2212;</mo><mn>2</mn><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D435;</mo><mo>&#x2229;</mo><mo>&#x1D436;</mo><mo>)</mo><mo>&#x2212;</mo><mn>2</mn><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D436;</mo><mo>&#x2229;</mo><mo>&#x1D434;</mo><mo>)</mo><mo>+</mo><mn>3</mn><mo>&#x1D45B;</mo><mo>(</mo><mo>&#x1D434;</mo><mo>&#x2229;</mo><mo>&#x1D435;</mo><mo>&#x2229;</mo><mo>&#x1D436;</mo><mo>)</mo></math>

Concept Video

Applying Concepts: Formula based on Cardinality of Sets | Maths | JEE

Practice Problems of Venn Diagram 

Example 1 :A school has 175 students. The following information shows the number of students selecting one or more subjects: Physical Education 100, Fine Arts 70, Biology 40, Physical Education & Fine Arts 30, Physical Education & Biology 28, Fine Arts & Biology 23, Physical Education & Fine Arts & Biology 18. How many students have opted for Physical Education alone?

Answer:

 

 

Step 1 :



 

Step 2 :


<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>P</mi><mi>E</mi><mo>&#x2229;</mo><mi>F</mi><mo>&#x2229;</mo><mi>B</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>18</mn><mo>&#xA0;</mo><mspace linebreak="newline"/><mi>d</mi><mo>+</mo><mo>&#xA0;</mo><mi>e</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>F</mi><mo>&#x2229;</mo><mi>B</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>23</mn><mo>;</mo><mo>&#xA0;</mo><mi>d</mi><mo>&#xA0;</mo><mo>=</mo><mn>5</mn><mo>&#xA0;</mo><mspace linebreak="newline"/><mi>e</mi><mo>+</mo><mi>g</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>P</mi><mi>E</mi><mo>&#x2229;</mo><mi>B</mi><mo>=</mo><mo>&#xA0;</mo><mn>28</mn><mo>;</mo><mo>&#xA0;</mo><mi>g</mi><mo>&#xA0;</mo><mo>=</mo><mn>10</mn><mo>&#xA0;</mo><mspace linebreak="newline"/><mi>e</mi><mo>+</mo><mi>f</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>F</mi><mo>&#x2229;</mo><mi>P</mi><mi>E</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>30</mn><mo>;</mo><mo>&#xA0;</mo><mi>f</mi><mo>=</mo><mo>&#xA0;</mo><mn>12</mn><mo>&#xA0;</mo><mspace linebreak="newline"/><mi>e</mi><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mi>f</mi><mo>+</mo><mi>g</mi><mo>+</mo><mi>c</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>P</mi><mi>E</mi><mo>=</mo><mo>&#xA0;</mo><mn>100</mn><mo>&#xA0;</mo><mspace linebreak="newline"/><mo>&#x21D2;</mo><mi>c</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>100</mn><mo>-</mo><mo>(</mo><mn>18</mn><mo>+</mo><mn>10</mn><mo>+</mo><mn>12</mn><mo>)</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>40</mn><mo>&#xA0;</mo></math>


<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>100</mn><mo>-</mo><mfenced><mrow><mn>18</mn><mo>+</mo><mn>12</mn><mo>+</mo><mn>10</mn></mrow></mfenced><mo>=</mo><mn>60</mn></math>

Hence, 60 students opted for physical education alone.

 

Example 2 : In a city of 20,000 families it was found that 40% families buy toothpaste P, 20% families buy toothpaste Q , 10% families buy toothpaste R, 5% families buy P and Q, 3% buy Q and R and 4%  buy P and R. If 2%  families buy all the three toothpastes. Find

a. The number of families which buy toothpaste P only.
b. The number of families which buy none of P,Q and R.

Answer:

Given, there are 20,000 families in a town.

Let n(X) be the percentage number

n(P)=40%, nQ=20%, nR=10%

n(P ∩ Q)=5%,  n(Q ∩ R)=3%,  n(P ∩ R)=4%n(P ∩ Q ∩ R)=2%

(a) Percentage of families which buy toothpaste P only

=nP-nP ∩ Q-nP ∩ R+nP ∩ Q ∩ R

=(40-5-4+2)% = 33%

∴  Number of families which buy toothpaste P only = 20000×33100 = 6600 

(b) Percentage of families which buy none of P,Q, and R

=nU- nP ∪ Q ∪ R     =n(U) –[ nP+nQ+nR- nP ∩ Q- nQ ∩ R-n(P ∩ R)+ n(P ∩ Q ∩ R)]=100 – [40+20+10-5-3-4+2]  

=100 – 60= 40% 

∴  Number of families which buy none of  P,Q and R =20000 ×40100  =8000   

Example 3 : Venn diagram for sets A and B is given in the figure. Find 𝐴 ∆ B


Answer:

From the given Venn diagram, A={1,2,3,4,6,12} and B={1,2,3,6,9,18}

Now, 𝐴 ∆ B=(A-B)∪(B-A)=(A∪B)-(A∩B)

⇒(A∪B)={1,2,3,4,6,9,12,18} & (A∩B)={1,2,3,6}

⇒A ∆ B={4,9,12,18}

Example : In the following diagram (A-B)∪(A∩B) is

  1. μ                         (b) ϕ                              (c) A                        (d) B

Answer:

From the given Venn diagram,

A-B={7,8} and A∩B={1,2}

⇒(A-B)∪(A∩B)={7,8}∪{1,2}={1,2,7,8}=A

Hence, the correct option is (c).

Example 4 : From the given diagram, the shaded portion represents :


  1. B-A                (b) B∩A=B         (c) B∪A=A            (d) A∩B=A∪B

Answer:

From the given Venn diagram it's clear that A⊂B

The shaded region represents elements of B excluding the elements of A i.e., B-A

Hence, option (a) is correct.

FAQ’s of Venn Diagram

Question 1.What are the further applications of laws of algebra of sets?

Answer: Laws of algebra of sets are widely used in Permutations & Combinations & Probability.

Question 2. How to draw Venn diagrams?

Answer: Once you have a basic idea about Venn diagrams, just read the sentences one by one, note the relations between them and draw circles based on the relations you formed.

Question 3.Is it always necessary to draw a Venn diagram?

Answer: It is not a compulsion to draw a venn diagram. But using Venn diagrams problems can be solved more easily and quickly.

Question 4. What are Venn diagrams used for?

Answer: Venn diagrams are used to represent relations among different sets pictorially.

Related Topics to Venn Diagram in Maths

NCERT Class 11 Maths Chapters

Sets Relations and Functions Triginometric Functions
Mathematical Induction Numbers and Quadriatic Equations Linear Inequalities
Premutations and Combinations Binomial Theorem Sequence and Series
Straight Lines Conic Sections 3 D Geometry
Limits and Derivatives Mathematical Reasoning Statistics
Probability  

 

 

 

 

 

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