[Solved]: Show a TM-recognizable language of TMs can be expressed by TM-description language of equivalent TMs

Problem Detail: 

I am studying "An Introduction to the Theory of Computation" by Sipser -- there is a problem *3.17 (p.161) which I can not solve. Any hints (not answers) from which side to attack it?

Let $B=\{M_1, M_2, ...\}$ be a Turing-recognizable language consisting of TM descriptions. Show that there is a decidable language C consisting of TM descriptions s.t. every machine in B has an equivalent machine in C and vice versa.

Asked By : Ayrat

Answered By : Shaull

Technical hint: assume that every string is a legal encoding of a TM. Can you solve this now?

Intuitive hint: the language $C$ can contain many other encodings as well as those that are equivalent to the machines in $B$. Perhaps so many that $C$ becomes decidable.

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Question Source : http://cs.stackexchange.com/questions/14680