Venn Diagrams with Three Sets

Explanation of three-set Venn diagrams, regions, intersections, unions, complements, and clear examples.

1. Understanding Three-Set Venn Diagrams

A three-set Venn diagram uses three overlapping circles, usually labeled A, B, and C. The diagram helps show how the sets overlap individually, in pairs, and all together.

This diagram is very useful when working with problems that involve several conditions or groups.

2. Structure of a Three-Set Venn Diagram

When three circles overlap, they create multiple distinct regions. Each region represents a unique combination of membership among the three sets.

2.1. Regions Formed

  • Only A
  • Only B
  • Only C
  • A ∩ B (but not C)
  • B ∩ C (but not A)
  • A ∩ C (but not B)
  • A ∩ B ∩ C (common to all three)
  • Outside all three (part of the universal set)

3. Pairwise Intersections

Each pair of sets may have elements in common. These are shown in the overlapping sections of two circles.

3.1. Examples

  • \( A \cap B \)

    : elements common to A and B

  • \( B \cap C \)

    : elements common to B and C

  • \( A \cap C \)

    : elements common to A and C

4. Intersection of All Three Sets

The center region where all three circles overlap shows elements that belong to A, B, and C at the same time.

4.1. Notation

\( A \cap B \cap C \)

4.2. Example

If A, B, and C represent people who like three different activities, then the central region shows those who enjoy all three.

5. Union of Three Sets

The union combines all elements that belong to A, B, or C, including all overlaps.

5.1. Notation

\( A \cup B \cup C \)

5.2. Example

If A = {1,2}, B = {2,3}, C = {3,4}, then:

\( A \cup B \cup C = \{1,2,3,4\} \)

6. Differences Involving Three Sets

Differences help show what belongs to one set but not the others.

6.1. Examples

  • A − (B ∪ C): elements only in A
  • B − (A ∪ C): elements only in B
  • C − (A ∪ B): elements only in C

7. Complement of a Set

The complement of any set (like A′) includes everything in the universal set except the elements of that set. The region outside a circle represents its complement.

8. Example Problem Using Three-Set Venn Diagram

Suppose:

A = \{1,2,3\}, \quad B = \{3,4\}, \quad C = \{2,5\}

8.1. Visual Interpretation

  • A ∩ B = {3}
  • A ∩ C = {2}
  • B ∩ C = {}
  • A ∩ B ∩ C = {}

9. Important Points

Some useful ideas to remember about three-set Venn diagrams:

9.1. Key Ideas

  • Shows overlapping relationships clearly.
  • Useful for questions involving three categories.
  • Helps identify intersections, unions, and exclusive parts.
  • All sets lie inside the universal set U.