Venn Diagrams – Set Operations

What it shows

A Venn diagram places every element of the universal set U = {1; 2; …; N} in exactly one region, according to which of the sets A, B and C contain it. Union, intersection, difference and complement then become shaded regions whose elements can be listed and counted. Counting reveals the inclusion–exclusion principle, n(A ∪ B) = n(A) + n(B) − n(A ∩ B): without the subtraction, elements in the overlap would be counted twice. It also shows that A \ B and B \ A are different sets, and that the complement of A contains n(U) − n(A) elements.

How to use

Type the elements of Set A and Set B (and Set C when using three sets), choose N, and check that each number lands in the right region. Click the operation buttons to shade A ∪ B, A ∩ B, A \ B, B \ A or U \ A, and read the listed elements and n(·). Then click regions directly to build other sets, such as (A ∩ B) \ C, and use Clear selection to start again.

Parameters you can change

• Number of sets 2 sets (A, B), 3 sets (A, B, C)
• Initial operation A ∪ B, A ∩ B, A \ B, B \ A, Complement of A in U, A ∪ B ∪ C, A ∩ B ∩ C
• U = {1; 2; …; N} 6–20
• Set A (numbers separated by commas)
• Set B
• Set C (when using 3 sets)

Questions to explore

1. Which regions of the diagram make up the set (A ∩ B) \ C?
2. When is n(A ∪ B) equal to n(A) + n(B)?
3. Is the complement of A ∪ B the same as the intersection of the complements of A and B?