--- aliases: [cardinality] --- #sets-functions ## Definition > [!tldr] Definition > The **cardinality** of a [[set]] is the set's size. If the set is finite, its cardinality is simply the number of elements it contains. If the set is infinite, we just say that the set "has infinite cardinality". If $A$ is a set, its cardinality is denoted $|A|$. Notes: - For infinite sets, the concept of "the number of elements in the set" does not make semantic sense because there is no number that describes the cardinality. Again, we just say that infinite sets have infinite cardinalities. - It is possible to distinguish between different types of infinite cardinalities; for example the cardinality of the set of real numbers has a "larger" cardinality than the set of integers. But this is beyond the scope of this course. ## Examples - The cardinality of the set $\{10, x, 4/5\}$ is $3$. - The cardinality of the set $\{\emptyset, \{1,2,3\}, 4, \{5,6\}\}$ is $4$. - The cardinality of $\{n \in \mathbb{N} \, : \, 24 \, \% \, n = 0 \}$ is $8$. That's because this set that is given in [[Set-builder notation|set-builder notation]] would, in [[Roster notation|roster notation]], be $\{1, 2, 3, 4, 6, 8, 12, 24\}$. - The cardinality of $\{x^2 \, : \, x \in \mathbb{N}\}$ is infinite. ## Resources <div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/606573917?badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Screencast 3.5: Power set and cardinality"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script> - Tutorial: [Cardinality](https://www.cuemath.com/algebra/cardinality/) - Video: [What is the cardinality of a set?](https://www.youtube.com/watch?v=SY0ixs-_IGc)