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 $A' \subseteq f^{-1}(f(A'))$?
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 $A' \subseteq f^{-1}(f(A'))$?
Although I still have no idea of how to prove the first question,
I try to prove the second one with the advice from @Tara B
Here's how I prove it..
(1) Write down the definition:
$$ \because \forall x \in A' \\\rightarrow f(x) = y \in B \\\rightarrow A'\subseteq B \\\rightarrow f^{-1}(f(A)) = \left \{ a \in A | f(a) \in f(A') \right \} $$
(2) Then prove it: $$ \forall x \in A' \\\rightarrow f(x) \in f(A') \\\rightarrow x \in f^{-1}(f(A')) \\\rightarrow A' \subseteq f^{-1}(f(A')) $$