Ullman (TOC) Edition 3 Exercise 9.3 Question 7 (Page No. 400 - 401)

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Show that the following problems are not recursively enumerable:

1. The set of pairs $(M,w)$ such that $TM \ M$, started with input $w$, does not halt.
2. The set of pairs $(M_{1},M_{2})$ such that $L(M_{1}\cap L_(M_{2})=\phi$.
3. The set of triples $(M_{1},M_{2},M_{3})$ such that $L(M_{1}) = L(M_{2})L(M_{3})$ ; i.e., the language of the first is the concatenation of the languages of the two $TM's$.

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Tell whether each of the following are recursive, RE-but-not-recursive, or non-RE. The set of all $TM$ codes for $TM's$ that halt on every input. The set of all $TM$ codes for $TM’s$ that halt on no input. The set of all $TM$ codes for $TM's$ that halt on at least one input. The set of all $TM$ codes for $TM's$ that fail to halt on at least one input.
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