Paper 4, Section II, 7E7 \mathrm{E}

Numbers and Sets | Part IA, 2019

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

(i) ff is injective.

(ii) For every subset AXA \subset X we have f1(f(A))=Af^{-1}(f(A))=A.

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

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

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

[Here fnf^{n} denotes the nn-fold composite of ff with itself.]

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