Exercise · ZW-INFINITE-SET-010 · ★★★ Advanced
Let \(A\) be infinite and \(a\in A\). Is \(A\setminus\{a\}\) infinite?
Choose one option.
Show answer
- A. Yes: if it were finite, adding \(a\) back would give \(A\) as a finite set
Explanation
If \(A\setminus\{a\}\) were finite, adding the point \(a\) back would keep the set finite, contradicting that \(A\) is infinite.
Review the concept: Infinite set
A set \(A\) is infinite if it is not finite, that is, if there is no bijection between \(A\) and \(\{1,\dots,n\}\) for any \(n\in\mathbb{N}\).
\[A\not\sim\{1,\dots,n\}\ \ \forall n\in\mathbb{N}\]
More practice problems: infinite set
- ZW-INFINITE-SET-001 How is an infinite set defined?
- ZW-INFINITE-SET-002 Which of these sets is infinite?
- ZW-INFINITE-SET-003 If A⊆ B and A is infinite, what follows about B?
- ZW-INFINITE-SET-004 Can an infinite set be a subset of a finite set?
- ZW-INFINITE-SET-005 Is [0,1] a finite set?
- ZW-INFINITE-SET-006 If A is infinite, is there a bijection between A and \1,dots,n\ for some n∈ℕ?
- ZW-INFINITE-SET-007 Let A be finite and B infinite. What can be said about A∪ B?
- ZW-INFINITE-SET-008 Is the set of even natural numbers infinite?