Determining whether the following language is decidable

Question

{⟨M,N⟩ | All strings in L(M)∩L(N) begin with 110.}

I think that this language is decidable. We can make a Turing Machine TM, which takes as input . For every string that is in L(M)∩L(N), if the string starts with 110, after the first 3 digits, we halt and accept. If the first three digits are not 110, we halt and reject. I am unsure what we do if the string is not in L(M)∩L(N).

Also overall I am unsure if my Turing Machine is actually working or not. Could I get some feedback on this?

Answer 1

If M and N are Turing machines, then this language is not decidable. If it were, we could make N a TM that accepts all strings, and then we'd have a decider for {M | all strings in M begin with 110}. We can recognize this is not decidable since the condition is true for some TMs, false for others, and it's semantic in that it deals with the strings in the language; so Rice's theorem applies.