Let $L$ be a language over an alphabet $\Sigma$. The equivalence relation $\sim_{\mathrm{L}}$ on the set $\Sigma^{\ast }$ of finite strings over $\Sigma$ is defined by $u \sim_{L} v$ if and only if for all $w \in \Sigma^{\ast }$ it is the case that $u w \in L$ if and only if $v w \in L$.
Suppose that $\mathrm{L}=\mathrm{L}(\mathrm{M})$ is the language accepted by a deterministic finite automaton $\mathrm{M}$. For each $\mathrm{u} \in \Sigma^{\ast }$, let $\mathrm{s}(\mathrm{u})$ be the unique state of $\mathrm{M}$ reached from the initial state after inputting the string $\mathrm{u}$.
Now consider the following statements :
- $\mathrm{s}(\mathrm{u})=\mathrm{s}(\mathrm{v})$ implies $\mathrm{u} \sim_{\mathrm{L}} \mathrm{v}$.
- $\mathrm{u} \sim_{\mathrm{L}} \mathrm{v}$ implies $\mathrm{s}(\mathrm{u})=\mathrm{s}(\mathrm{v})$.
Which of the above statements is correct?
- Only $1$
- Only $2$
- Both
- None