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Given $A, B$ languages over $\Sigma,$ Prove/Disprove: If $A _{\leq M} B$ and $B _{\leq M} A$ then $A=B$.

I would like to disprove this claim, with the languages $H_{TM}$ and $H_\epsilon = \{\langle M \rangle\ |\ M \ halts\ on\ empty\ tape\}$

I can prove $H_{TM} {_{\leq M}} H_\epsilon$ by taking $\langle M,w \rangle$, and build new machine $M’_w$ that writes $w$ on tape and then simulates $M$. Then $\langle M,w \rangle \in H_{TM} \iff \langle M'_w \rangle ∈ H_\epsilon$

I got stuck proving the other direction.

Any help is appreciated.

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Let $\Sigma = \{0,1\}$, $A=\{0\}$, $B=\{1\}$.

Define the following reductions:

$f(0)=1$ and $\forall x\ne0 . f(x)=0$,

$g(1)=0$ and $\forall x\ne1 . g(x)=1$.

Then you can easily show that $A \le_{M}B$ (using $f$) and $B \le_{M}A$ (using $g$) but obviously $A\ne B$.

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