I have some questions as follow...
1) How could I prove transitive closure $t(R)=R^+$, where $R^+=\bigcup_{k=1}^{\infty}R^k$, $R\subseteq A\times A$?
2) Prove or disprove: For any subset $A'\subseteq A,$ we always have $A' \subseteq f^{-1}(f(A'))$?