The inclusion-exclusion principle is a counting rule for finding how many objects are in the union of overlapping sets. It matters because simple addition can double count objects that belong to more than one set. A Venn diagram makes this visible by showing shared regions between sets.
The principle gives a reliable way to add the right parts and subtract the overcounted parts.
Key Facts
- For two sets: |A ∪ B| = |A| + |B| - |A ∩ B|
- For three sets: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
- The union A ∪ B means all elements in A, in B, or in both.
- The intersection A ∩ B means elements that are in both A and B.
- If sets A and B do not overlap, then |A ∩ B| = 0 and |A ∪ B| = |A| + |B|.
- Number outside the union = total number in the universal set - |A ∪ B ∪ C|.
Vocabulary
- Set
- A set is a collection of distinct objects or elements.
- Union
- The union of sets is the collection of elements that are in at least one of the sets.
- Intersection
- The intersection of sets is the collection of elements that are shared by the sets.
- Cardinality
- Cardinality is the number of elements in a set, written with vertical bars such as |A|.
- Universal Set
- The universal set is the complete group of elements being considered in a problem.
Common Mistakes to Avoid
- Adding |A| + |B| without subtracting |A ∩ B|, which counts every shared element twice instead of once.
- Subtracting the triple intersection in the three-set formula, which is wrong because elements in all three sets are removed too many times and must be added back.
- Confusing union with intersection, which leads to counting all elements in at least one set when the problem asks only for shared elements, or the reverse.
- Using pairwise overlaps that include or exclude the triple overlap inconsistently, which makes the formula fail unless the meanings of |A ∩ B|, |A ∩ C|, and |B ∩ C| are clear.
Practice Questions
- 1 In a class, 18 students play soccer, 14 play basketball, and 6 play both. How many students play soccer or basketball?
- 2 A survey of 100 students finds that 45 like math, 38 like physics, 30 like chemistry, 12 like both math and physics, 10 like both math and chemistry, 8 like both physics and chemistry, and 5 like all three. How many students like at least one of the three subjects, and how many like none of them?
- 3 Explain why the three-set inclusion-exclusion formula adds |A ∩ B ∩ C| at the end instead of subtracting it.