Exercise · ZW-COUNTABLE-SET-007 · ★★☆ Intermediate
Is \(\mathbb{N}\times\mathbb{N}\) countable?
Choose one option.
Show answer
- D. Yes: the product of two countable sets is also countable.
Explanation
The product of two countable sets is countable, so \(\mathbb{N}\times\mathbb{N}\) can be listed. The other answers are false: it does not contain \(\mathbb{R}\), and being a product of infinite sets does not prevent countability.
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-008 Is ℚ countable?
- ZW-COUNTABLE-SET-009 Which statement about subsets of countable sets is correct?