Exercise · ZW-COUNTABLE-SET-008 · ★★☆ Intermediate
Is \(\mathbb{Q}\) countable?
Choose one option.
Show answer
- C. Yes: \(\mathbb{Q}\) is countable even though it is dense in \(\mathbb{R}\).
Explanation
The rationals can be listed, for example by running through fractions \(p/q\) along diagonals and skipping repeats. Density does not imply uncountability.
Review the concept: Countable set
A set is countable if it is finite or equipotent to \(\mathbb{N}\), that is, if its elements can be listed in a sequence.
\[A \text{ numerable} \iff A \text{ finito} \lor A\sim\mathbb{N}\]
More practice problems: countable set
- ZW-COUNTABLE-SET-001 What does it mean for a set A to be countable?
- ZW-COUNTABLE-SET-002 Which of these sets is countable?
- ZW-COUNTABLE-SET-003 What is a set equipotent to ℕ called?
- ZW-COUNTABLE-SET-004 What is an enumeration of a countably infinite set A?
- ZW-COUNTABLE-SET-005 Is a finite set countable?
- ZW-COUNTABLE-SET-006 Which of these functions is a bijection from ℕ onto ℤ?
- ZW-COUNTABLE-SET-007 Is ℕ×ℕ countable?
- ZW-COUNTABLE-SET-009 Which statement about subsets of countable sets is correct?