Schröder–Bernstein theorem

If there is an injection from each of two sets into the other, then there is a bijection between them. The theorem requires no choice of a preferred injection.

Injection: A function that maps distinct inputs to distinct outputs. The theorem's two hypotheses are injections, one in each direction.
Surjection: A function whose image equals its codomain. Surjectivity is the other half of bijectivity, which the theorem concludes.
Felix Bernstein: A German mathematician whose work included an early proof of the theorem now bearing his name. The theorem's name records Bernstein's contribution to its proof.
Cantor's theorem on cardinality: The result that a set's power set has strictly greater cardinality than the set itself. It combines with the theorem to distinguish strict cardinal inequalities from mutual injections.
Bijection: A function that is both injective and surjective, pairing every element of its domain with exactly one codomain element. The theorem guarantees this stronger kind of map from the two injections.
Equinumerosity: The relation between sets that can be paired by a bijection. The theorem says mutual injectability already guarantees this relation.
Ernst Schröder: A German mathematician known for work in logic and algebra. The theorem is also named for Schröder, whose earlier proof was incomplete.
Cantor's diagonal argument: A method of proving that no proposed list captures every member of certain infinite collections. Diagonal arguments often establish one-way cardinal bounds that the theorem can complete.
Cantor–Bernstein proof: A proof of the Schröder–Bernstein theorem that partitions a set into chains generated by two injections. It constructs the promised bijection by alternating between the given maps.
Cardinality: A measure of the size of a set defined by its bijection class. The theorem converts mutual size comparisons into equality of cardinalities.

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