Exercise · ZW-CARDINAL-NUMBER-007 · ★★☆ Intermediate

There is an injection from \(A\) into \(B\), but no bijection between them. What is the relation between \(|A|\) and \(|B|\)?

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Explanation

The injection gives \(|A|\le|B|\). Since there is no bijection, the cardinals are not equal, so \(|A|<|B|\).

The cardinal number of a set \(A\) is the least ordinal equinumerous with \(A\). It measures the size of \(A\) and also works for infinite sets.

\[|A| = \min\{\alpha \in \mathrm{Ord} : \alpha \approx A\}\]