Venn Diagram Examples

Set Theory · Axiom Academy

Using diagrams to solve set problems 25 students take Mathematics (M) 10 students take both subjects How many students take neither subject? |U| = 40, |M| = 25, |P| = 20, |M ∩ P| = 10 Students taking only Mathematics = |M| − |M ∩ P| = 25 − 10 = 15 Students taking only Physics = |P| − |M ∩ P| = 20 − 10 = 10 |M ∪ P| = |M| + |P| − |M ∩ P| = 25 + 20 − 10 = 35 Neither subject = |U| − |M ∪ P| = 40 − 35 = 5 students Inclusion-Exclusion: |A ∪ B| = |A| + |B| − |A ∩ B| Only A: Elements in A but not B = |A| − |A ∩ B| Neither: |U| − |A ∪ B| gives elements outside both sets Venn diagrams help visualize overlapping regions Always subtract the intersection to avoid double-counting!

This is the written version of the interactive lesson above. See the full Set Theory course.