Question

Consider these functions from languages over \{0, 1\} to languages over \{0, 1\}: $K(A, B) = \{w \in \{0, 1\}^* \mid (w \notin A \& w \in A \circ B) \text{ or } w \in B\}$, $N(A, B) = \{w \in \{0, 1\}^* \mid (w \in A \& w \in A \circ B) \text{ or } w \in \overline{A}\}$. [$\circ$ is the concatenation operator.] i. Prove that the set of Turing-decidable languages is closed under both $K$ and $N$. That is, if $A$ and $B$ are Turing-decidable, so are $K(A, B)$ and $N(A, B)$.

          Consider these functions from languages over \{0, 1\} to languages over \{0, 1\}:
$K(A, B) = \{w \in \{0, 1\}^* \mid (w \notin A \& w \in A \circ B) \text{ or } w \in B\}$,
$N(A, B) = \{w \in \{0, 1\}^* \mid (w \in A \& w \in A \circ B) \text{ or } w \in \overline{A}\}$.
[$\circ$ is the concatenation operator.]
i. Prove that the set of Turing-decidable languages is closed under both
$K$ and $N$. That is, if $A$ and $B$ are Turing-decidable, so are $K(A, B)$ and
$N(A, B)$.

$Consider these functions from languages over {0, 1} to languages over {0, 1}: K(A, B) = {w ∈{0, 1}^* | (w ∉ A & w ∈ A ∘ B) or w ∈ B}, N(A, B) = {w ∈{0, 1}^* | (w ∈ A & w ∈ A ∘ B) or w ∈A}. [∘ is the concatenation operator.] i. Prove that the set of Turing-decidable languages is closed under both K and N. That is, if A and B are Turing-decidable, so are K(A, B) and N(A, B).$

Added by Miguel -Ngel A.

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