Venn Diagram Math Example 2

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Example 2

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

Solution

  1. 1
    Apply inclusion-exclusion: ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|.
  2. 2
    Substitute: 28=20+15−∣A∩B∣28 = 20 + 15 - |A \cap B|.
  3. 3
    Solve: ∣A∩B∣=35−28=7|A \cap B| = 35 - 28 = 7.

Answer

∣A∩B∣=7|A \cap B| = 7
The inclusion-exclusion formula prevents double-counting elements in the overlap. It is a fundamental counting tool when working with Venn diagrams.

About Venn Diagram

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

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