Intersection of Sets - Only What Both Sets Share
The intersection of two sets is a new set containing only the elements that belong to both sets simultaneously. Where union gives you everything in either group, intersection gives you only what the two groups share.
Notation and Definition
A ∩ B = {x : x ∈ A and x ∈ B}
The symbol ∩ is called the cap symbol. It stands for intersection. An element belongs to A ∩ B only if it is in A and also in B.
Disjoint Sets
Two sets are disjoint (or mutually exclusive) if they share no elements at all – their intersection is the empty set.
A ∩ B = ∅ ⇒ A and B are disjoint.
Key Properties of Intersection
| Property | Meaning | Example |
|---|---|---|
| Commutative | A ∩ B = B ∩ A | Order of sets does not matter |
| Associative | (A ∩ B) ∩ C = A ∩ (B ∩ C) | Grouping does not matter |
| Identity | A ∩ ξ = A | Intersecting with the universal set changes nothing |
| Null element | A ∩ ∅ = ∅ | Intersecting with the empty set gives the empty set |
| Idempotent | A ∩ A = A | Intersecting a set with itself gives the same set |
Worked Examples
Elements in both A and B: 3, 4, 5.
A ∩ B = {3, 4, 5}. |A ∩ B| = 3.
C = {4, 8, 12, 16, 20, 24}. D = {6, 12, 18, 24}.
C ∩ D = {12, 24}. These are the multiples of 12 up to 24.
E = {2, 3, 5, 7, 11, 13}. F = {2, 4, 6, 8, 10, 12, 14}.
E ∩ F = {2}. 2 is the only even prime.
|F ∩ G| = 10 (study both). Only French = |F| − |F ∩ G| = 30 − 10 = 20 students.
Intersection and the Inclusion-Exclusion Principle
The intersection ties directly into the inclusion-exclusion formula from the Union topic:
|A ∩ B| = |A| + |B| − |A ∪ B|
If you know the sizes of A, B, and A ∪ B, you can find the intersection without listing elements.
Key Takeaways
- A ∩ B contains only elements that are in both A and B simultaneously.
- If A ∩ B = ∅, the sets are disjoint (no shared elements).
- Intersection is commutative (A ∩ B = B ∩ A) and associative.
- |A ∩ B| = |A| + |B| − |A ∪ B| (rearrangement of the inclusion-exclusion principle).
Practice Questions
- A = {3, 6, 9, 12, 15} and B = {6, 12, 18, 24}. Find A ∩ B.
- C = {factors of 36} and D = {factors of 48}. Find C ∩ D.
- Are {odd numbers} and {even numbers} disjoint? Explain.
- |A| = 20, |B| = 15, |A ∪ B| = 28. Find |A ∩ B|.
- In a class of 40, 22 study art and 18 study music. 7 study both. Find the number who study only music and only art.