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# Def/Injective

Definition of Injective: Suppose that $X$ and $Y$ are sets, and $f$ is a function from $X$ to $Y$. We say that $f$ is injective, or one-to-one if: $$(x_1, x_2 \in X) \wedge (f(x_1) = f(x_2)) \Rightarrow x_1 = x_2.$$

In other words, a injective function never maps different elements to the same element.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Function

The following statements and definitions logically rely on the material of this page: Def/Finite (set), State/Bijections are injective and surjective functions, State/Composing injective surjective or bijective functions yields the same, State/Injective functions have left inverses, and State/Kernel criterion for injectivity of group homomorphisms

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