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# State/Injective functions have left inverses

Proposition: (Injective functions have left inverses) Suppose that $X$ and $Y$ are sets, and $f$ is a function from $X$ to $Y$. Then, $f$ is injective if and only if there exists a function $g \colon Y \rightarrow X$ such that $g \circ f = Id_X$, i.e., a left inverse of $f$.

## Logical Connections

This statement logically relies on the following definitions and statements: Def/Function, Def/Injective, Def/Inverse function, Def/Image, Def/Extension of functions, Def/Complement of a set

The following statements and definitions rely on the material of this page: State/Bijections are injective and surjective functions, and State/Composing injective surjective or bijective functions yields the same

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