Exercise · ZW-CARDINAL-NUMBER-010 · ★★★ Advanced
Which of the following statements is false?
Choose one option.
Show answer
- C. If \(A\) is infinite, then \(|A|=|\mathbb{N}|\).
Explanation
\(\mathbb{R}\) is infinite but not countable, so its cardinal is greater than \(|\mathbb{N}|\). The other three are theorems: inclusion, Cantor-Schröder-Bernstein, and Cantor's theorem.
Review the concept: Cardinal number
The cardinal number of a set \(A\) is the least ordinal equinumerous with \(A\). It measures the size of \(A\) and also works for infinite sets.
\[|A| = \min\{\alpha \in \mathrm{Ord} : \alpha \approx A\}\]
More practice problems: cardinal number
- ZW-CARDINAL-NUMBER-001 What is the cardinal of \2,4,6,8\?
- ZW-CARDINAL-NUMBER-002 What is the cardinal of ℕ?
- ZW-CARDINAL-NUMBER-003 What does it mean for two sets A and B to be equinumerous?
- ZW-CARDINAL-NUMBER-004 What is the cardinal of the empty set ∅?
- ZW-CARDINAL-NUMBER-005 What is the cardinal of \1,2,2,3\?
- ZW-CARDINAL-NUMBER-006 What is the cardinal of P(\a,b,c\)?
- ZW-CARDINAL-NUMBER-007 There is an injection from A into B, but no bijection between them. What is the relation between |A…
- ZW-CARDINAL-NUMBER-008 What is the cardinal of the set of integers x with -3≤ x≤ 4?