Operations on sets
A set is a collection of well-defined objects. The union A ∪ B contains elements in A or B (or both); the intersection A ∩ B contains elements in both; the complement A′ contains elements of the universal set 𝓤 not in A; the difference A \ B contains elements in A but not in B.
| Law | Statement |
|---|---|
| Commutative | A ∪ B = B ∪ A; A ∩ B = B ∩ A |
| Associative | (A ∪ B) ∪ C = A ∪ (B ∪ C) |
| Distributive | A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |
| De Morgan | (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′ |
| Complement | A ∪ A′ = 𝓤 and A ∩ A′ = ∅ |
Cardinality
The cardinality n(A) is the number of elements in A. For two sets and three sets:
Worked exampleA class survey
In a class of 40, 25 take Physics, 20 take Chemistry and 8 take both. How many take neither?
- Union.
- Neither.
Answer: 3 students.
Try itTwo sets
Choose the numbers in a group of 100.
Key terms
- Union
- Elements in either set.
- Intersection
- Elements in both sets.
- Complement
- Elements not in the set.
- Cardinality
- The number of elements in a set.
Summary
- Union, intersection, complement and difference are the basic operations.
- De Morgan’s laws link complements with unions and intersections.
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- Venn diagrams help solve counting problems.