Counting Overlapping Sets

Combinatorics · Axiom Academy

INTRO Counting Overlapping Sets When sets overlap, simple addition leads us astray - discover why we need a smarter approach. Let's start with something easy. Imagine you have two completely separate groups of students. ✅ When groups are separate: 5 + 7 = 12 students Now let's add some students who are in BOTH clubs. Click on the overlapping region to reveal them. 👉 Click the overlapping region to discover who's in both clubs 🧪 Experimenting with Overlaps Use the slider to change how many students are in both clubs. Watch what happens to our count! Now you've discovered the key insight! Let's see the formula that fixes our counting problem. When sets overlap, members belong to multiple groups. Simple addition counts these members multiple times, leading to an inflated total. The amount we overcount equals exactly the size of the overlap. Each person in the intersection gets counted once for each set they belong to. Subtract the overlap to correct for double-counting. This principle is called the Inclusion-Exclusion Principle , and it extends to three or more overlapping sets!

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