Exercise · ZW-SCHRODER-BERNSTEIN-THEOREM-006 · ★★☆ Intermediate
If \(|A|\le|B|\) and \(|B|\le|A|\), what follows?
Choose one option.
Show answer
- B. \(|A|=|B|\)
Explanation
Together, the two inequalities give \(|A|=|B|\), that is, equinumerosity. They do not imply \(A=B\), since sets can be different and still be equinumerous.
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-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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