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Trying to answer the many questions you have here to the best of my ability:
what is R(M)? They say it is representation of turing machine but what is it exactly? Is it tuples of turing machine?
It is some representation of some Turing Machine $M$. The exact format of the representation is not important for this proof, just that it unambiguously specifies ...
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This statement is incorrect.
If $\overline{A_{TM}}$ is co-recognizable, then we have a TM that recognizes the set of words not accepted by some TM $M$.
The fact that $\overline{A_{TM}}$ is co-recognizable means that there is a TM $T$ that recognizes all words not in $\overline{A_{TM}}$.
Equivalently, $T$ recognizes all words in $\overline{\overline{A_{TM}}}...
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