Prove images of injection are disjoint.

Question

Let $A, B, C$ be sets such that $A \subseteq B \subseteq C$, and suppose that there is an injection $f : C \rightarrow A$. Define the sets $D_0, D_1, D_2, ...$ recursively by setting $D_0 := B \setminus A$, and then $D_{n+1} := f(D_n)$ for all natural numbers $n$. Prove that they're all disjoint from each other, i.e., $D_n \cap D_m = \emptyset$ whenever $m \neq n$.

Answer 1

Wlog $m>n$ and we write $m=n+k$ with $k>0$. The claim is clear for $n=0$ because $D_m=D_k=f(D_{k-1})\subseteq A$. The rest follows by induction because by injectivity $D_n\cap D_{n+k}=\emptyset$ implies $D_{n+1}\cap D_{n+1+k}=f(D_n)\cap f(D_{n+k})=\emptyset$.