Injective function

A function is injective when distinct inputs always have distinct outputs. Equivalently, if two outputs are equal, their inputs must be equal.

Function: A relation that assigns each element of a domain exactly one element of a codomain. Injectivity is a property of functions, defined by how their inputs correspond to outputs.
Horizontal line test: A graph test in which a function is injective if every horizontal line meets its graph at most once. It turns the distinct-input definition into a visual test for real-valued functions.
Surjective function: A function whose image equals its codomain. Surjectivity requires reaching every declared output, unlike injectivity's restriction on repeated outputs.
Cardinality: A measure of the size of a set, including comparisons between infinite sets. An injection from one set to another establishes a cardinality comparison.
Domain of a function: The set of inputs for which a function is defined. Injectivity depends on which inputs the function is allowed to compare.
Left inverse: A function that, when composed after another function, returns each input unchanged. Having a left inverse guarantees that the original function is injective.
Bijective function: A function that is both injective and surjective. Bijectivity combines injectivity with the additional requirement of covering the codomain.
Pigeonhole principle: A principle stating that a map from a larger finite set to a smaller one cannot be injective. It proves collisions must occur when there are more inputs than available outputs.
Codomain: The declared set in which a function's outputs lie. A function can be injective even when some elements of its codomain are never attained.
Inverse function: A function that reverses the input-output assignments of another function. An injective function has an inverse defined on its image.

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