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\begin{document}
\title{\textbf{Pumping Lemma - an Example}}
\author{Wang-Shiuan Pahngerei}
\date{April 27, 2012}
\maketitle

Show that $L:=\{x\in\{a,b\}\,\,:\,\,x\ne x^R \}$ is non-regular.

\begin{proof}
Assuming regularity, but whenever $m\in\mathbb{N}$, one might choose 
\begin{align}
w=a^m ba^N \in L,\notag
\end{align}
where $N= m + m!$. Then if $w$ is decomposed as $w=xyz$, with $|xy|\leq m$ and $|y|\geq 1$, the string
\begin{align}
w_j := xy^j z,\notag
\end{align}
where $j=1+\frac{m!}{|y|}$, is in the form $a^K ba^N$. Note that $x,y$ looks like this form $a^r$, and $z$ in the form $a^s ba^N$. So the number $K$ of $a$'s at the left hand side of the middle $b$ of $w_j$ is 
\begin{align}
K = m-|y| + |y|\cdot j = m + m! .\notag
\end{align}
Therefore it is symmectric, and then $w_j\notin L$. This contradicts the Pumping Lemma.
\end{proof}

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