Exercise · ZW-COUNTABLE-SET-006 · ★★☆ Intermediate
Which of these functions is a bijection from \(\mathbb{N}\) onto \(\mathbb{Z}\)?
Choose one option.
Show answer
- A. \(f(n)=(-1)^n\lceil n/2\rceil\)
Explanation
\(f(n)=(-1)^n\lceil n/2\rceil\) takes the values \(0,-1,1,-2,2,\dots\), covering \(\mathbb{Z}\) without repetition. \(n-1\) never reaches \(-2\), \(n\) never reaches a negative number, and \(n^2\) neither.
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-007 Is ℕ×ℕ countable?
- ZW-COUNTABLE-SET-008 Is ℚ countable?
- ZW-COUNTABLE-SET-009 Which statement about subsets of countable sets is correct?