Exercise 821Logic, sets and proofs › Countability and cardinalityL1Added 3 Nov 2026

Comparing cardinalities

#comparison#injection#bijection

Statement

State what it means for $|A|\le|B|$ and $|A|<|B|$ .

Hint

$|A|\le|B|$ iff there is an injection $A\hookrightarrow B$ .

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Solution

Definition.
- $|A|\le|B|$ iff there is an injection $A\hookrightarrow B$ .
- $|A|=|B|$ iff there is a bijection.
- $|A|<|B|$ iff $|A|\le|B|$ and $|A|\neq|B|$ .

Remark. $\le$ is a preorder; $<$ is a strict partial order on cardinalities.

Conclusion. Cardinality comparison uses injections.

Comparing cardinalities

Statement

Solution

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