Let the alphabet $\backslash$Sigma $=\{$a$,$ b$\}.$ Recall that a string x in $\backslash$Sigma $$ is called a palindrome if x $=$ xr $($x reads the same

backward as forward$).$

$($a$)$ Consider the following binary relation R on $\backslash$Sigma $$ as follows: For all x$,$ y in $\backslash$Sigma $,$ xRy if and only if $|$x$|=|$y$|$ and

xyr is a palindrome.

i$.$ Show that R is an equivalence relation on $\backslash$Sigma $.$

ii$.$ For an arbitrary x in $\backslash$Sigma $,$ describe the equivalence class $[$x$]$R $($the equivalence class of R containing x$).$

Justify your answer.

$($b$)$ Consider the following binary relation R$$ on $\backslash$Sigma $$ as follows: For all x$,$ y in $\backslash$Sigma $,$ xR$$y if and only if xyr is a

palindrome. Is R$$ an equivalence relation on $\backslash$Sigma $?$ Justify your answer.