Cardinality

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′ = ∅

The cardinality n(A) is the number of elements in A. For two sets and three sets:

Inclusion–exclusion for two sets.

For 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.

Only Physics 17, both 8, only Chemistry 12, neither 3 (total 40).

Try itTwo sets

Choose the numbers in a group of 100.

Check yourself (A ∪ B)′ equals

Check yourself If n(A) = 12, n(B) = 9 and n(A ∩ B) = 4, then n(A ∪ B) =

Check yourself A ∩ A′ =

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.