Equivalence relations

Question

Having trouble proving this is an equivalence relation.

Is it suffice to say that let $x y z$ be any string in $\Sigma^*$,

$(xz \in L \iff yz \in L) \rightarrow (yz \in L \iff xz \in L)$ shows that $xRy \rightarrow yRx$?

Answer 1

For any $x\in \Sigma^*$, define $L(x)$ to be the language $\{z\in \Sigma^*\mid xz\in L\}$. Then the definition of $R$ can be rephrased as $$xRy \iff L(x)=L(y).$$ This is clearly reflexive, symmetric, and transitive, but assuming this is an elementary course you probably want to at least write out transitivity, making clear you understand why $xRy$ and $yRz$ imply $xRz$.

Answer 2

The specific situation of formal languages is unnecessary.

Suppose there is a relation called "?". Given $x$, define $??(x)$ be the set of $u$ such that $x?u$. Let $???$ be the relation "has the same value of ?? as", so that $a???b$ when $??(a)=??(b)$. Then ??? is clearly an equivalence relation. That's all that is going on here, for a particular choice of $?$.