Let the rotational closure of language A be RC(A) = {yx

Step 1 of 2

Given,  rotational closure of language \(A\) is \(R C(A)=\{y x \mid x y \in A\}\).

We have to show that for any language \(A\), we have \(R C(A)=R C(R C(A))\).

First, for any language \(A\), let's consider a string \(w \in A\). Then set of all possible strings over the alphabet is given by \(\Sigma^{*}\). Then \(w \in \Sigma^{*}\) and will contain the empty string \(\varepsilon\) and thus \(\varepsilon \in A\).

Let \(A=w \varepsilon\)

We know that \(w \varepsilon=\varepsilon w=w\).

Now, consider LHS, \(R C(A)\)

So, \(R C(A)=R C(w \varepsilon)=\varepsilon w=w\).

RHS is given by, \(R C(R C(A))\)

\(R C(R C(A))=R C(R C(w \varepsilon))=R C(\varepsilon w)=w \varepsilon=w\).

Thus LHS = RHS. Hence \(R C(A)=R C(R C(A))\).