Exercise · ZW-FINITE-SET-009 · ★★★ Advanced
Why is \(\mathbb{N}\) not finite?
Choose one option.
Show answer
- D. Because it is bijective with a proper subset of itself
Explanation
A finite set can never be put in bijection with a proper subset of itself. The map \(n\mapsto n+1\) sends \(\mathbb{N}\) onto \(\mathbb{N}\setminus\{1\}\), which shows \(\mathbb{N}\) is infinite.
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-004 If Asim\1,dots,n\ and Asim\1,dots,m\, what follows?
- 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?