Given any two strings, call them $x$ and $y$, over any language $L$ and given property such that if $xc \equiv L$ and $yc \equiv L $ (where $c$ is some string), then $x \equiv y$.
I would like to prove that $\equiv$ is reflexive. I tried $x \equiv x \rightarrow x-x \equiv0$ and $y(0) \rightarrow x(0)\equiv0$ Therefore, $\equiv$ is reflexive. But somehow I don't feel this is right. Can someone point out what I am doing wrong here?
The relation $\equiv_L$ is defined by: $x\equiv_L y$ iff $xz\in L\iff yz\in L$ for all strings $z$.
Clearly, $x\equiv_L x$ since $xz\in L\iff xz\in L$ for all strings $z$.