volume_mute
Match Each Set to Its Cardinality Type
publish date: 2026/05/23 21:45:36.339440 UTC
volume_muteMatch each set to its correct cardinality classification.
To complete the line match
- Click on an item in the first group
- Click on the match in the second group
To delete a match, double click on a line
Set
\(\{a, b, c, d, e\}\)
\(\mathbb{N} = \{1, 2, 3, 4, ...\}\)
All points on the real line \(\mathbb{R}\)
All subsets of \(\mathbb{R}\)
Cardinality
Uncountable (cardinality \(2^c\))
Finite (cardinal number 5)
Uncountable (cardinality \(c\))
Countably infinite (cardinality ℵ₀)
Correct Answer
(1) \(\{a, b, c, d, e\}\),Finite (cardinal number 5)
(2) \(\mathbb{N} = \{1, 2, 3, 4, ...\}\),Countably infinite (cardinality ℵ₀)
(3) All points on the real line \(\mathbb{R}\),Uncountable (cardinality \(c\))
(4) All subsets of \(\mathbb{R}\),Uncountable (cardinality \(2^c\))
Explanation
The hierarchy: finite sets have finite cardinal numbers. \(\mathbb{N}\) is the prototype countably infinite set with \(\aleph_0\). \(\mathbb{R}\) has cardinality \(c = 2^{\aleph_0}\). The power set \(P(\mathbb{R})\) has cardinality \(2^c\), larger than \(c\). By the axiom of power sets, the cardinal of any set is strictly less than the cardinal of its power set.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012