Transitive closure and invertible function

Question

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 $A' \subseteq f^{-1}(f(A'))$?

Answer 1

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: $$ \forall x \in A' \\\rightarrow f(x) \in f(A') \\\rightarrow x \in f^{-1}(f(A')) \\\rightarrow A' \subseteq f^{-1}(f(A')) $$