Let $L\mid$ be a regular language and
$L_1| = \{x|\mid\text{there exist y}\mid \text{so that xy} \in L| \text{ and} \mid x \mid = 2 \mid y\mid \mid \}$
$L_2| = \{x|\mid\text{there exist y}\mid \text{so that yx} \in L| \text{ and} \mid x \mid = 2 \mid y\mid \mid \}$
Then which of the following is necessarily correct?
- $L_1|$ is regular but $L_2|$ is not.
- $L_2|$ is regular but $L_1|$ is not.
- Both $L_1|$ and $L_2|$ are regular.
- Both $L_1|$ and $L_2|$ are not regular.