# Trying to show if two languages are recognizable or not

I have two languages that I am trying to prove are recognizable or not:

Let $$L_1 = \{(\langle M\rangle, w) \mid \text{M is a TM that accepts w and doesn't accept \varepsilon}\}$$ where TM is short for "Turing Machine", $$w$$ is a string, and $$ε$$ is the empty string.

Is $$L_1$$ recognizable? Prove your answer is correct.

Let $$L_2 = \{(\langle M\rangle, w) \mid \text{M is a TM that accepts w and rejects \varepsilon} \}$$ Is $$L_2$$ recognizable? Prove your answer is correct.

I've been having trouble with these proofs using mapping reducibility. I know that the language $$A_\mathrm{TM} = \{\langle M, w \rangle \mid \text{M is a Turing machine that accepts the string w}\}$$ is undecidable but recognizable. My approach is to say that $$L = \{\langle M, \varepsilon\rangle \mid \text{M is a Turing machine that rejects \varepsilon} \}$$ is recognizable and, knowing that recognizable languages are closed under intersection, this shows $$L$$ is also recognizable.

I'm not too sure about the first question, however. Is there a better approach to proving these two languages are recognizable or not?

• Note that you can use LaTeX here to typeset mathematics in a more readable way. See here for a short introduction. – Raphael Mar 26 '16 at 10:43
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• Deleting your question after receiving answers is bad style. Don't do that! – Raphael Mar 27 '16 at 18:26