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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?

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  • $\begingroup$ Show your work please, it will help you get more attention to your problem $\endgroup$ – Fareed AF Apr 24 at 19:55
  • $\begingroup$ I just want to know how I can show the equivalence of the statements? What is equivalent to what? $\endgroup$ – Analysis Apr 24 at 20:15
  • $\begingroup$ They are all equivalent. Have you already shown certain implications? Are you stuck on a specific part? $\endgroup$ – Mark Kamsma Apr 24 at 21:13
  • $\begingroup$ Hello Mark Kamsma, I have already proven (i) ⇔ (ii), (i) ⇔ (iii). I still need to proof (i) ⇔ (iv) and (i) ⇔ (v). Would I then have proven that they're all equivalent? $\endgroup$ – Analysis Apr 24 at 21:50
  • $\begingroup$ Yes sure since by doing this you'll prove that (i) is equivalent to all the others $\endgroup$ – Fareed AF Apr 25 at 8:20

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