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$$L=\{M\mid \text{there exist }x,y\in\Sigma^* \text{ s.t. }x \in L(M)\text{ and } y \notin L(M)\}\,.$$

I think it's not recursively enumerable because this language reduces to complement of the membership problem. $\langle M,w\rangle$ would be input where $y$ would become $w$ and $x$ belongs to $L(M)$.

Please tell me whether I'm correct or not, and help to solve if I'm wrong.

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    $\begingroup$ "either A and B" doesn't make sense. Do you mean "both A and B" or "either A or B"? $\endgroup$ – David Richerby Feb 8 '17 at 11:09
  • $\begingroup$ @DavidRicherby since the answer was accepted, may the question be edited accordingly? $\endgroup$ – dave Feb 8 '17 at 15:48
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    $\begingroup$ @dave I guess so. I went ahead and did it, since I have enough rep to edit unilaterally, whereas you'd have to propose the edit and get it accepted. $\endgroup$ – David Richerby Feb 8 '17 at 15:53
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In other words, $L=\{\langle M \rangle:L(M)\notin \{\phi,\Sigma^*\}\}$. you can show explicit reduction from a language you already know is not in $RE$. for example $\overline{HP}$ (it is a very standard reduction). or you may use the rice theorem to show that.

I did not understand the reduction you suggested. The reduction from the language you suggested could be as follows: $$f(\langle M \rangle,x)=\langle M_x \rangle$$

$M_x$ on input w:

  1. run $M$ on x for |w| steps.
  2. if $M$ accepted then reject, otherwise accept.

Note that $\epsilon \in L(M_x)$ in anycase, so $L(M_x)\neq \phi$. and $L(M_x)\neq\Sigma^{*}\iff x\in L(M)$. the correctness follows.

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