Question

Problem 33. Prove that the language A = \{x \in \{0,1\}^* | #(01,x) = #(10,x)\} is regular. (Here, for a, b \in \{0,1\}, #(ab, x) is the number of a's in x that are immediately followed by b.) Recall that the canonical equivalence relation of a language A \subset \Sigma^* is the binary relation $=_A$ on \Sigma^* defined by x $=_A$ y if and only if, for all z \in \Sigma^*, [xz \in A \iff yz \in A].

          Problem 33. Prove that the language
A = \{x \in \{0,1\}^* | #(01,x) = #(10,x)\}
is regular. (Here, for a, b \in \{0,1\}, #(ab, x) is the number of a's in x that are immediately followed
by b.)
Recall that the canonical equivalence relation of a language A \subset \Sigma^* is the binary relation $=_A$ on
\Sigma^* defined by
x $=_A$ y if and only if, for all z \in \Sigma^*, [xz \in A \iff yz \in A].

$Problem 33. Prove that the language A = {x ∈{0,1}^* | #(01,x) = #(10,x)} is regular. (Here, for a, b ∈{0,1}, #(ab, x) is the number of a's in x that are immediately followed by b.) Recall that the canonical equivalence relation of a language A ⊂Σ^* is the binary relation =A on Σ^* defined by x =A y if and only if, for all z ∈Σ^*, [xz ∈A yz ∈A].$

Added by Thomas W.

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