Let languages $A, B$ be defined as$$\begin{align}A &= \{\langle M\rangle\mid M(\langle M\rangle)=reject\}\\ B &= \{\langle M\rangle\mid M(\langle M\rangle)=accept\}\end{align}$$In other words, $A$ is the set of descriptions of TMs which reject their own descriptions and $B$ is defined similarly. The languages $A$ and $B$ are disjoint, and are both Turing-recognizable. Prove that there does not exist a decidable language $C \subseteq \Sigma^*$ such that $A \subseteq C$ and $B \subseteq \overline{C}$, the complement.
Two Disjoint Turing-recognizable languages do not have a decidable language
computability;turing machines;undecidability