Injectivity

In set theory, injectivity is a property of a function. Given a function $f : A \rightarrow B$, if when supplied with every input in $A$ $f$ produces every possible output in $B$ then $f$ is injective.

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• Range
  • In Maths[^1], the range of a function is the set of all possible outputs within the Codomain. This might be smaller than the codomain itself if the function is not injective.
• A pair of types with the same cardinality will always be isomorphic
  • There will always be a way of exhaustively mapping each value of type $a$ to type $b$, injectively and surjectively when viewing the domain and the codomain as a set.
• Bijection
  • In the category of sets, Set, a bijection is a morphism which is both injective and surjective. In it's generalised form, this is isomorphism.