Exercise · ZW-SCHRODER-BERNSTEIN-THEOREM-005 · ★★☆ Intermediate
Which of these functions is a bijection from \(\mathbb{N}\) onto the even numbers?
Choose one option.
Show answer
- C. \(n\mapsto 2n\)
Explanation
The function \(n\mapsto 2n\) is injective and its image is exactly the even numbers. The others do not have exactly the even numbers as their image, or are not injective.
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-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?
- ZW-SCHRODER-BERNSTEIN-THEOREM-009 If ℕ⊆ℚ and there is an injection from ℚ into ℕ, what is the conclusion?
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