Venn Diagram Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Venn Diagram.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A diagram using overlapping circles to visually represent sets and their relationships such as union, intersection, and complement.

Each circle represents a set; overlapping regions show shared elements; the rectangle border is the universal set.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Each region represents a different combination of membership.

Common stuck point: For 3+ sets, diagrams get complex; not all regions may exist.

Sense of Study hint: Draw the circles, then place each element into the correct region by checking: is it in A? in B? in both? in neither?

Worked Examples

Example 1

easy
In a class of 30 students, 18 study French, 12 study Spanish, and 5 study both. Use a Venn diagram to find how many study neither language.

Solution

  1. 1
    Let F = French students, S = Spanish students. Place |F \cap S| = 5 in the overlap region.
  2. 2
    French only: 18 - 5 = 13. Spanish only: 12 - 5 = 7. So |F \cup S| = 13 + 5 + 7 = 25.
  3. 3
    Students studying neither: 30 - 25 = 5.

Answer

5 \text{ students study neither language}
A Venn diagram partitions the universal set into non-overlapping regions. The inclusion-exclusion principle gives |F \cup S| = |F| + |S| - |F \cap S|.

Example 2

medium
Given |A| = 20, |B| = 15, |A \cup B| = 28, find |A \cap B| using the inclusion-exclusion principle.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
In a survey of 50 people, 30 like tea, 25 like coffee, and 10 like both. How many like only tea? Only coffee? Neither?

Example 2

medium
In a class of 30 students, 18 study French, 12 study Spanish, and 5 study both. How many study neither language?
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Related Concepts

SetUnionIntersection

Background Knowledge

These ideas may be useful before you work through the harder examples.

setunionintersection