Linked from

The 36 pages that link to Bijection, each with the reason it gives.

• FunctionBroader topic: A bijection pairs every input with exactly one output and every codomain element with one input.

• PermutationNarrower topic: A permutation is precisely a bijection whose domain and codomain are the same set.

• Georg CantorRelated: Cantor used one-to-one pairings to define when two sets have the same size.

• Inverse functionRelated: A bijection has an inverse function from its entire codomain to its domain.

• CardinalityRelated: A bijection is the criterion for two sets to have the same cardinality.

• Countable setRelated: A bijection with the natural numbers witnesses that an infinite set is countable.

• Finite setRelated: A bijection with a natural-number set witnesses finiteness.

• HomeomorphismRelated: A homeomorphism must pair the points of its two spaces one-to-one and onto.

• Injective functionRelated: Bijectivity combines injectivity with the additional requirement of covering the codomain.

• Cantor's theoremRelated: Sets have equal cardinality exactly when a bijection exists between them.

• CodomainRelated: Bijectivity requires the codomain to match the image and each output to have one input.

• IsomorphismRelated: Isomorphisms must pair every element of one structure with exactly one element of the other.

• Surjective functionBroader topic: Adding injectivity to surjectivity gives a one-to-one correspondence between domain and codomain.

• DiagonalizationRelated: An enumeration of a set can be expressed as a surjective sequence, whose exhaustiveness diagonalization challenges.

• Cantor's diagonal argumentRelated: A set is countable when a bijection with the natural numbers exists.

• Catalan numberRelated: Bijections explain why unlike structures can have identical Catalan counts.

• Change of variablesRelated: A global change of variables typically needs a one-to-one correspondence between domains.

• Cardinal numberRelated: Sets have the same cardinality precisely when a bijection exists between them.

• Infinite setRelated: Bijections define when two infinite sets have the same size.

• Uncountable setBroader topic: A bijection with the natural numbers would make this set countable.

• Combinatorial proofRelated: A bijection directly shows that the two sets counted by an identity have equal size.

• Permutation matrixRelated: Each row and column containing one 1 encodes a bijection of the index set.

• Graph automorphismRelated: An automorphism must rearrange all vertices without merging or omitting any.

• Left inverseRelated: A function has both a left and right inverse precisely when it is bijective.

• Graph isomorphism problemRelated: An isomorphism must pair all vertices without omissions or collisions.

• Adjoint functorsRelated: Each adjunction gives bijections between corresponding hom-sets.

• Aleph numberRelated: Bijections define when two sets have the same cardinality.

• Cayley's formulaRelated: A bijection transfers the count of Prüfer sequences to the count of trees.

• Schröder–Bernstein theoremBroader topic: The theorem guarantees this stronger kind of map from the two injections.

• CodeRelated: Bijections formalize when a coding loses no information.

• Cardinality of the continuumRelated: Bijections give the precise criterion for the continuum's equality with other cardinalities.

• Skolem's paradoxRelated: Countability is defined through bijections with the natural numbers, including within the model.

• Axiom of limitation of sizeRelated: Equinumerosity in the axiom is expressed by a bijection with the universe of sets.

• Cantor–Dedekind axiomRelated: The axiom requires every line point to match exactly one real number.

• Combinatorial principlesRelated: It transfers a count between two different descriptions of the same possibilities.

• Frege's theoremRelated: Bijections formalize the one-to-one correspondence underlying equinumerosity.