# Showing equivalences (functions, injective)

I wonder which of these statements are equivalent to each other.

X,Y are Quantities. $$f: X\rightarrow Y$$ is a function. Show the equivalence of the following statements:

(i) f injective

(ii) $$f^{-1}(f(A))=A$$ for every $$A⊂X$$

(iii) $$f(A \cap B)=f(A) \cap f(B)$$ for all $$A,B⊂X$$

(iv) $$f(A) \cap f(B)=\emptyset$$ for all $$A,B⊂X$$ with $$A \cap B=\emptyset$$

(v) $$f(A\backslash B)=f(A)\backslash f(B)$$ for alle $$A,B⊂X$$ with $$B⊂A$$

I know that (i) ist equivalent to (ii). I guess that (iii) and (iv) are equivalent and (iv) and (v).

Is that true?