Venn Diagrams

1. Introduction

• Venn diagrams are visual tools used to show relationships between different sets.

• Developed by John Venn in the 1880s.

• Very useful in Economics (e.g., market segmentation, resource allocation) and Animal Science (e.g., species distribution, common traits among animals).

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2. Basic Concepts

• Set: A collection of distinct objects (e.g., all dairy animals, or all consumers of organic products).

• Elements: Items inside a set.

• Universal Set (U): All possible elements under consideration.

• Subset: A set entirely contained within another set.

• Intersection ( ∩ ): Elements common to two or more sets.

• Union ( ∪ ): All elements from two or more sets (without repetition).

• Complement ( A' ): Elements not in set A.

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3. Components of a Venn Diagram

• Circles represent sets.

• Overlapping areas represent common elements (intersections).

• Non-overlapping areas represent unique elements.

• Rectangle often represents the Universal Set.

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4. How to Draw a Venn Diagram (2 Sets Example)

Suppose:

• Set A = {Consumers who buy local products}

• Set B = {Consumers who buy organic products}

Steps:

1. Draw two overlapping circles.

2. Label one circle A and the other B.

3. Fill in:

   • Overlap (A ∩ B) = Consumers who buy both local and organic products.

   • A only = Consumers who buy only local products.

   • B only = Consumers who buy only organic products.

4. Outside the circles = Consumers who buy neither.

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5. Examples in Economics and Animal Science

Economics Example:

• Set A: People investing in stocks.

• Set B: People investing in bonds.

• Venn diagram shows:

  • People investing in both.

  • Only stocks.

  • Only bonds.

  • Neither (saving in cash, etc.)

Animal Science Example:

• Set A: Animals with four legs.

• Set B: Animals that are herbivores.

• Venn diagram shows:

  • Animals that are both four-legged and herbivores (e.g., cows, goats).

  • Four-legged non-herbivores (e.g., lions).

  • Herbivores without four legs (e.g., certain birds).

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6. Key Formulas

For two sets A and B:

• Union:

  $$n(A ∪ B) = n(A) + n(B) - n(A ∩ B)$$

• Intersection:

  $$n(A ∩ B)$$

  (elements common to both)

• Complement:

  $$n(A') = n(U) - n(A)$$

Where:

• $n(A)$ = number of elements in set A

• $n(U)$ = number of elements in universal set U

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7. Three-Set Venn Diagram

For sets A, B, and C:

• More complex, includes 7 regions.

• Useful for showing overlaps among three categories (e.g., farmers who raise cattle, sheep, and goats).

Formula for Union:

$$n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)$$

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8. Practical Tips for Students

• Always define your sets clearly.

• Start by filling intersection regions first.

• Double-check that total counts match the Universal Set.

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9. Simple Practice Questions

1. In a university, 70 students study Economics, 50 study Animal Science, and 20 study both. How many students study at least one of these subjects?

2. Among a group of farm animals:

• 15 cows eat grass.

• 10 goats eat grass.

• 5 eat both grass and shrubs.

• Draw a Venn Diagram to represent this situation.

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