[Solved]: On the language of Turing machines not accepting their own encoding

Problem Detail: 

Let $\text{NOT-SELF} = \{\langle M \rangle : M \text{ is a TM that does not accept } \langle M \rangle \}$. How do you prove the following claim?

If $Q$ is a TM so that $L(Q) \subseteq \text{NOT-SELF}$, prove that $\langle Q \rangle \in \text{NOT-SELF}$.

Asked By : user2977105

Answered By : Yuval Filmus

Hint: If $\langle Q \rangle \notin \text{NOT-SELF}$ then $\langle Q \rangle \notin L(Q)$, and so $\langle Q \rangle \in \text{NOT-SELF}$.

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Question Source : http://cs.stackexchange.com/questions/17882