Cardinality Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Cardinality.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
The cardinality of a set is the number of distinct elements it contains, written |A| or n(A).
Cardinality answers "how many?" — count each distinct element once and you have the cardinality.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Cardinality measures the size of a set; infinite sets can have different cardinalities (e.g., |\mathbb{N}| < |\mathbb{R}|).
Common stuck point: Some infinities are bigger than others: |\text{integers}| < |\text{reals}|.
Sense of Study hint: Write out the distinct elements, then count them. For union problems, use |A| + |B| - |A intersect B| to avoid double-counting.
Worked Examples
Example 1
easySolution
- 1 (a) Count the distinct elements: 2, 4, 6, 8, 10 — five elements, so |A| = 5.
- 2 (b) The only natural number \le 0 is 0 (assuming 0 \in \mathbb{N}). So B = \{0\} and |B| = 1.
- 3 (c) C has three elements: the set \{1,2\}, the number 3, and the set \{4\}. So |C| = 3.
Answer
Example 2
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyExample 2
mediumBackground Knowledge
These ideas may be useful before you work through the harder examples.