Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.