Cardinality of Sets

The cardinality of a set expresses how many elements it contains. Cardinality is one of the most important characteristics because it shows whether the set is finite or infinite, and we can even distinguish between infinite sets.

The |A| denotes the number of elements in set A. If A = {2,4,6,8}, then |A| = 4. This is a simple case since the set contains a finite number of elements.

Finite Cardinality

The cardinality of finite sets is always a specific natural number. We can simply count them. Examples:

• {a, b, c} has cardinality 3.
• The empty set ∅ has cardinality 0.

Infinite Cardinality

Infinite sets have infinitely many elements. We distinguish between countable infinite (can be listed in a sequence) and uncountable infinite (cannot be listed).

• Countable infinite: natural numbers ℕ, |ℕ| = ℵ₀ (aleph-null).
• Uncountable infinite: real numbers ℝ, |ℝ| = 2^ℵ₀ (continuum).

Countable vs. Uncountable Infinite

Countable infinite sets can be put into one-to-one correspondence with natural numbers. Uncountable ones cannot, like the reals (Cantor's diagonal argument).

• Countable: integers ℤ, rationals ℚ.
• Uncountable: reals ℝ, power set of naturals P(ℕ).

Everyday Examples

• Children in a family → finite cardinality.
• Days in a year → finite cardinality (365 or 366).
• Positive integers → countable infinite.
• Real numbers between 0 and 1 → uncountable infinite.

Summary

The cardinality of a set can be finite or infinite. Among infinite sets, we distinguish between countable and uncountable infinite. This concept is fundamental in higher mathematics, such as analysis and set theory.

Practice Exercise