Exercise · ZW-SCHRODER-BERNSTEIN-THEOREM-010 · ★★★ Advanced
Which of the following statements is false?
Choose one option.
Show answer
- B. If \(f\colon \mathbb{N}\to\mathbb{N}\) is injective, then \(f\) is bijective.
Explanation
The function \(n\mapsto n+1\) from \(\mathbb{N}\) to itself is injective but not surjective, since \(0\) has no preimage. The other statements are true: the Schröder-Bernstein theorem, and the facts that a set injectable into a finite set is finite and that a subset of a finite set is finite.
Review the concept: Schröder–Bernstein theorem
The Schröder-Bernstein theorem states that if there are injections from \(A\) into \(B\) and from \(B\) into \(A\), then there is a bijection between \(A\) and \(B\).
\[|A| \le |B| \land |B| \le |A| \Rightarrow |A| = |B|\]
More practice problems: schröder–bernstein theorem
- ZW-SCHRODER-BERNSTEIN-THEOREM-001 What hypothesis does the Schröder-Bernstein theorem require?
- ZW-SCHRODER-BERNSTEIN-THEOREM-002 What does the Schröder-Bernstein theorem conclude?
- ZW-SCHRODER-BERNSTEIN-THEOREM-003 If there is an injection from A into B, what cardinal relation follows?
- ZW-SCHRODER-BERNSTEIN-THEOREM-004 What does it mean for fcolon A→ B to be injective?
- ZW-SCHRODER-BERNSTEIN-THEOREM-005 Which of these functions is a bijection from ℕ onto the even numbers?
- ZW-SCHRODER-BERNSTEIN-THEOREM-006 If |A|≤|B| and |B|≤|A|, what follows?
- ZW-SCHRODER-BERNSTEIN-THEOREM-007 Let gcolon ℤ→ℕ be given by g(z)=2z if z≥ 0 and g(z)=-2z-1 if z<0. What is g(-2)?
- ZW-SCHRODER-BERNSTEIN-THEOREM-008 If |A|=5 and |B|=4, can there be an injection from A into B?