If ℕ⊆ℚ and there is an injection from ℚ into ℕ, what is the conclusion?

Exercise · ZW-SCHRODER-BERNSTEIN-THEOREM-009 · ★★☆ Intermediate

If \(\mathbb{N}\subseteq\mathbb{Q}\) and there is an injection from \(\mathbb{Q}\) into \(\mathbb{N}\), what is the conclusion?

Choose one option.

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C. \(|\mathbb{N}|=|\mathbb{Q}|\)

Explanation

The inclusion gives an injection from \(\mathbb{N}\) into \(\mathbb{Q}\), and the given one goes from \(\mathbb{Q}\) into \(\mathbb{N}\). By Schröder-Bernstein, \(|\mathbb{N}|=|\mathbb{Q}|\).

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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 \(B\).

\[|A| \le |B| \land |B| \le |A| \Rightarrow |A| = |B|\]

If \(\mathbb{N}\subseteq\mathbb{Q}\) and there is an injection from \(\mathbb{Q}\) into \(\mathbb{N}\), what is the conclusion?

Explanation

Review the concept: Schröder–Bernstein theorem

More practice problems: schröder–bernstein theorem