Exercise · ZW-FINITE-SET-004 · ★★☆ Intermediate
If \(A\sim\{1,\dots,n\}\) and \(A\sim\{1,\dots,m\}\), what follows?
Choose one option.
Show answer
- A. \(n=m\)
Explanation
By the pigeonhole principle, the two bijections force \(n=m\). This is why \(|A|\) is well defined.
Review the concept: Finite set
A set \(A\) is finite if there is a bijection between \(A\) and \(\{1,\dots,n\}\) for some \(n\in\mathbb{N}\), where \(\{1,\dots,0\}=\varnothing\).
\[A\sim\{1,\dots,n\}, \quad |A|=n\]
More practice problems: finite set
- ZW-FINITE-SET-001 How is it defined that a set A is finite?
- ZW-FINITE-SET-002 How many elements does the empty set ∅ have?
- ZW-FINITE-SET-003 Let A=\a,b,c\. How many subsets does A have?
- ZW-FINITE-SET-005 Is every subset of a finite set also finite?
- ZW-FINITE-SET-006 Which of these sets is finite?
- ZW-FINITE-SET-007 If A and B are finite and disjoint, how many elements does A∪ B have?
- ZW-FINITE-SET-008 If fcolon A→ B is injective and B is finite, what follows about A?
- ZW-FINITE-SET-009 Why is ℕ not finite?