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.
Like union, the ∩ symbol was introduced by Giuseppe Peano in his 1888 book Calcolo Geometrico. The underlying idea of “shared outcomes”, however, is much older and grew out of one of the most famous exchanges in the history of mathematics: the 1654 letters between Blaise Pascal and Pierre de Fermat, sparked by a gambler's question about dividing stakes fairly in an interrupted game of dice. Their correspondence is usually credited as the birth of probability theory, and reasoning about which outcomes two events have in common – exactly what intersection now formalises – was central to solving it.
Intersection is everywhere in modern computing: a database “JOIN” finds rows that satisfy both of two conditions, and typing two keywords into a search engine returns the intersection of pages matching each one.
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: Intersection Cardinality
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