Venn Diagram

 A Venn Diagram is a visual tool used to show logical relationships between different sets or groups using overlapping circles.

• Each circle represents a group/category.

• Overlapping regions represent common elements.

• Non-overlapping regions represent distinct elements.

🔑 Importance in Reasoning

Venn Diagrams are used to:

• Classify elements into sets

• Solve problems involving group-based relationships

• Understand subset, intersection, disjoint sets

• Solve syllogism, set theory, and categorization questions

📐 Important Methods (How to Solve)

🧩 Step-by-Step Approach:

Step 1: Understand the categories/groups

• Identify all sets given (e.g. "Boys", "Students", "Cricketers")

Step 2: Determine the relationship

• Are the groups completely overlapping (subset)?

• Partially overlapping (intersection)?

• No overlap (disjoint)?

Step 3: Draw circles accordingly

• One circle for each group

• Overlap or keep separate based on their relation

Step 4: Use symbols/numbers if required

• Fill in numbers in appropriate regions

• Avoid double-counting

📊 Types of Venn Diagram Relationships

Type	Description	Diagram Example
Subset	One group is completely within another	Boys ⊆ Students
Intersection	Groups partially overlap	Doctors ∩ Players
Disjoint (No Relation)	No common elements	Mango ∩ Train = Ø
All Same (Identical)	All circles represent the same group	Triangle = Shape = Polygon

🖼️ Diagrams & Examples

🔹 Example 1: Subset

Q: Which diagram represents: "All apples are fruits, but all fruits are not apples"?

✅ Answer:

    _______
   |       |
   |Fruit  |
   |   ____|___
   |  |Apple  |
   |  |_______|
   |__________|

🔹 Example 2: Intersection

Q: Which diagram represents: "Some teachers are writers"?

✅ Answer:

   _______       _______
  /       \     /       \
 / Teacher \ ∩ / Writer  \
 \_________/   \_________/

🔹 Example 3: Disjoint

Q: Which diagram shows: "Books, Fans, Flowers"?

✅ Answer: (No relation)

   (Book)     (Fan)     (Flower)
     O          O          O

🧮 Venn Diagram with Numbers (Set Theory)

🔹 Example 4: Numerical Problem

Q: In a group of 100 students:

• 60 like English

• 40 like Maths

• 25 like both

How many like only English, only Maths, and neither?

✅ Diagram:

       English        Maths
         _______     _______
        /       \   /       \
       |   35   |25|   15    |
        \_______/   \_______/

🧠 Calculation:

• Only English = 60 - 25 = 35

• Only Maths = 40 - 25 = 15

• Both = 25

• Total = 35 + 15 + 25 = 75

• Neither = 100 - 75 = 25

✅ Answer:

• Only English: 35

• Only Maths: 15

• Both: 25

• Neither: 25

✍️ Practice Question

Q: Which diagram represents: "All Tigers are Animals, but some Animals are Birds"?

Answer:

   _______
  |       |_______
  |Animal |       |
  |   ____|____   |
  |  |Tiger   |  |
  |  |_______ |  |
  |        \__|__|
  |         Bird |
  |______________|

📚 Applications of Venn Diagrams in Exams

• Syllogism

• Set Theory

• Data Interpretation

• Logical Categorization

📝 Summary Chart

Concept	Symbol	Meaning
∪ (Union)	A ∪ B	All elements in A or B
∩ (Intersection)	A ∩ B	Common elements in A and B
⊆ (Subset)	A ⊆ B	All elements of A are in B
∅ (Null)	A ∩ B = ∅	No common element

🎯Practice Question - Venn Diagram

Q1. Which diagram represents:

"All Dogs are Animals, and all Animals are Living beings"?

Answer: A ⊂ B ⊂ C (Subset within subset)

✅ Diagram:

    ___________
   | Living     |
   |  ________  |
   | | Animal | |
   | |  ____  | |
   | | |Dog | | |
   | | |____| | |
   | |________| |
   |___________ |

Q2. Which diagram shows the relationship among "Men", "Women", and "Humans"?

Answer: All are humans; Men and Women are distinct categories.

✅ Diagram:

       _________
      |         |
      | Humans  |
      |  ___ ___|
      | |   |   |
      |Men Women|
      |_________|

Q3. Which diagram represents: "Some Engineers are Doctors, some Doctors are Teachers"?

Answer: Three circles partially overlapping.

✅ Diagram:

    ______
   /      \   
  |Engineer|
   \__  __/
      \/        _______
      /\       /       \
   __/  \_____/ Doctor  \
  /         \   \_______/
 | Teacher  |
  \_________/

Q4. Which diagram shows: "Fruits, Apples, Mangoes"?

Answer: Apples and Mangoes ⊂ Fruits

✅ Diagram:

     ____________
    |   Fruits   |
    |  ____   ___|
    | |App| |Man|
    | |___| |___|
    |___________|

Q5. Which diagram shows: "Vehicles, Scooters, Bicycles"?

Answer: Scooter and Bicycle are part of Vehicles, but separate types.

✅ Diagram:

     ____________
    |  Vehicles  |
    |  ____  ____|
    | |Sco| |Bi |
    | |___| |___|
    |___________|

Q6. "Pen", "Pencil", "Stationery"?

Answer: Pen and Pencil are part of Stationery.

✅ Diagram:

     ______________
    |  Stationery  |
    |  ____   ____ |
    | |Pen| |Penc| |
    | |___| |____| |
    |______________|

Q7. "Circle, Polygon, Square"?

Answer: Square ⊂ Polygon; Circle ≠ Polygon

✅ Diagram:

     _________         _________
    | Polygon |       | Circle  |
    |  ____   |       |_________|
    | |Sqre|  |
    | |____|  |
    |_________|

Q8. "Teacher", "Player", "Women"?

Answer: Some Teachers/Players may be Women

✅ Diagram:

    _______      _______
   /       \    /       \
  | Teacher |  | Player |
   \___  ___/    \___  _/
        \/            \
        /\             \
     __/  \_____        \
    | Women     |        |
    \___________/        |
                         |

Q9. "Doctors", "Engineers", "Lawyers"?

Answer: No overlap; all are different professions.

✅ Diagram:

     (Doctors)   (Engineers)   (Lawyers)
         O            O             O

Q10. In a class of 100 students:

• 70 play Cricket

• 60 play Football

• 40 play both

Find how many play only Cricket, only Football, and neither.

✅ Solution using Venn Diagram:

• Total = 100

• Cricket only = 70 - 40 = 30

• Football only = 60 - 40 = 20

• Both = 40

• Total playing = 30 + 40 + 20 = 90

• Neither = 100 - 90 = 10

✅ Diagram:

         _________
        /         \
       |   30      |  
       |  ∩ (40)   | ← Both
       |      20   |
        \_________/

       Cricket   Football

✅ Answer:

• Only Cricket: 30

• Only Football: 20

• Both: 40

• Neither: 10