Syntactic congruence and Myhill Nerode equivalence

Question

What is the difference between these two demonstrated by a concrete example? I mean I know the definitions of them, and I know that former implies latter, and intuitively I believe this, but I don't have a good example in mind at the moment that stresses out the difference. Do you know one? Thanks!

Answer 1

Let $L\subseteq\{0,1\}^*$ be defined by the regular expression $(0+1)^*11$, let $x=01$, and let $y=1$. Then $x$ and $y$ are Myhill-Nerode equivalent: for any $w\in\{0,1\}^*$, $xw$ ends in $11$ if and only if $yw$ ends in $11$. However, $x$ and $y$ are not syntactically equivalent: if $u=1$ and $v=\lambda$ (the empty word), then $uxv=101\notin L$, but $uyv=11\in L$.