Paper 4, Section II, $7 \mathrm{E}$

(a) Let $f: X \rightarrow Y$ be a function. Show that the following statements are equivalent.

(i) $f$ is injective.

(ii) For every subset $A \subset X$ we have $f^{-1}(f(A))=A$.

(iii) For every pair of subsets $A, B \subset X$ we have $f(A \cap B)=f(A) \cap f(B)$.

(b) Let $f: X \rightarrow X$ be an injection. Show that $X=A \cup B$ for some subsets $A, B \subset X$ such that

$\bigcap_{n=1}^{\infty} f^{n}(A)=\emptyset \quad \text { and } \quad f(B)=B$

[Here $f^{n}$ denotes the $n$-fold composite of $f$ with itself.]

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