Intersection of Injective Image with Relative Complement

Theorem
Let $f: S \to T$ be a mapping.

Then $f$ is an injection iff:
 * $\forall A \subseteq S: f \left({A}\right) \cap f \left({\complement_S \left({A}\right)}\right) = \varnothing$

Proof
From Intersection with Relative Complement, we have that $A \cap \complement_S \left({A}\right) = \varnothing$.

From Image of Intersection under Injection we have that:
 * $\forall A, B \subseteq S: f \left({A \cap B}\right) = f \left({A}\right) \cap f \left({B}\right)$

iff $f$ is an injection.

Hence the result.