Cardinality of Sets
Comparing sizes of infinite sets.
Key idea
Comparing sizes of infinite sets.
In depth
Cardinality is the size of a set. Two sets have the same cardinality if there is a bijection between them. This extends counting to infinite sets, giving the surprising result that some infinities are larger than others.
Worked example
The positive integers and the even integers have the same cardinality via n ↦ 2n.
Real-world uses
- Explaining why there are as many even numbers as whole numbers in set theory
Common mistake
Believing a proper subset must be smaller for infinite sets.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Cardinality of Sets?
Comparing sizes of infinite sets. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Cardinality of Sets used in real life?
Real-world examples: Explaining why there are as many even numbers as whole numbers in set theory.
What should I learn before Cardinality of Sets?
Make sure you are comfortable with Injective, Surjective, Bijective, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Cardinality of Sets?
Believing a proper subset must be smaller for infinite sets.
How can I practise Cardinality of Sets?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Cardinality of Sets?
Move on to Countable and Uncountable Sets next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.
- Combinations with Repetition – Stars and bars.