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Suppose there are sets $S\subseteq Q, T\subseteq Q$ such that $T=\epsilon (T),S=\epsilon (S)$.

Prove $\epsilon(S\cap T)\subseteq S \cap T$

Definition of $\epsilon$- closure for epsilon NFA is:

$\epsilon : 2^Q \rightarrow 2^Q $

a) $S \subseteq \epsilon (S)$ Base case

b) If $q \in \epsilon (S)$ then $\delta(q,\epsilon )\subseteq \epsilon (S)$ Recursive case

c) and nothing else is in $\epsilon (S)$

And also, S is a set of all states in epsilon-NFA.

My proof:

Is this a correct reasoning?

Epsilon NFA example

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  • $\begingroup$ I don’t understand your reasoning. Try instead to use the definitions. $\endgroup$ – Yuval Filmus Oct 18 '19 at 22:56
  • $\begingroup$ I included a picture with an example hopefully it clarifies. $\endgroup$ – Mandy Oct 18 '19 at 23:11
  • $\begingroup$ A picture is not a proof. $\endgroup$ – Yuval Filmus Oct 18 '19 at 23:18
  • $\begingroup$ The sets $S,T$ could definitely have outgoing $\epsilon$-transitions. $\endgroup$ – Yuval Filmus Oct 18 '19 at 23:19
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    $\begingroup$ As I explained in my answer to your previous question, you can't really prove anything using your "definition" of epsilon closure. $\endgroup$ – Yuval Filmus Oct 19 '19 at 8:04
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Since your definition of $\epsilon$-closure isn't really a definition, it is impossible to prove anything using it. Instead, let me use the following definition: the $\epsilon$-closure of a set $S \subseteq Q$ consists of all states $x \in Q$ which are reachable from a state in $S$ by a (possibly empty) $\epsilon$-path (which is a path consisting of $\epsilon$-transitions).

Suppose that $\epsilon(S) = S$ and $\epsilon(T) = T$, and let $q \in \epsilon(S \cap T)$. Thus $q$ is reachable by a state $r \in S \cap T$ via an $\epsilon$-path. Since $r \in S \cap T$, in particular $r \in S$, and so, by definition, $q \in \epsilon(S) = S$. Similarly, $q \in \epsilon(T) = T$. Therefore $q \in S \cap T$.

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