Definition:Injection

A mapping $$f$$ is an injection, or injective, or one-one, or one-to-one iff:

$$\forall x_1, x_2 \in \mathrm{Dom} \left({f}\right): f \left({x_1}\right) = f \left({x_2}\right) \Longrightarrow x_1 = x_2$$

That is, it is a mapping such that the output uniquely determines its input.

Alternatively, this can be put:

$$\forall x_1, x_2 \in \mathrm{Dom} \left({f}\right): x_1 \ne x_2 \Longrightarrow f \left({x_1}\right) \ne f \left({x_2}\right)$$