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

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Start by writing down the definitions of transitive closure, and of $f^{-1}(f(A'))$, and try to work from there. (This sounds like pretty obvious advice, but it's surprising how often people don't even do that before deciding they don't know how to approach a problem.) –  Tara B Oct 20 '12 at 11:36
    
Thanks! I take your advice and prove it. –  lucasKoFromTW Oct 24 '12 at 23:35
    
What are you stuck on with the first one? Do you know the definition of transitive closure and of $R^k$? –  Tara B Oct 25 '12 at 21:09

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

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I think you have understood it, although your way of writing maths isn't completely intelligible to me. What do you mean by $\rightarrow$? My first guess would be 'implies', but then the beginning of the proof would be 'For all $x$ in $A'$ implies $\ldots$', which doesn't make sense. –  Tara B Oct 25 '12 at 21:12
    
So how should I prove it intelligibly? I don't have any idea with the structure or steps while proving this kind of question. –  lucasKoFromTW Oct 26 '12 at 14:06
    
I added my way of writing it underneath your answer. The edit hasn't come through yet though, so I don't know whether it will be accepted. In general I would suggest writing a few words in your proofs rather than using only symbols. Try reading aloud what you have written to check whether it makes sense. –  Tara B Oct 26 '12 at 17:41
    
Well, my edit was rejected, because people thought it should be an answer rather than a comment. I didn't really want to put it as a separate answer, but I guess I will when I get time. –  Tara B Oct 29 '12 at 14:08
    
Thank you! I'll try to write down some description after writing down the definition. –  lucasKoFromTW Oct 31 '12 at 0:12

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