L$\_(2).$

Prove that L$\_($eq$)=$L$\_($eq$)$@L$\_($eq$).$ Is it true that L$\_($eq$)=$L$\_($eq$)^(*)?$

Recall that for any two regular languages L$\_(1),$L$\_(2)$ their union, intersection and

concatenation are regular as well.

Give an example of two non$-$regular languages L$\_(1),$L$\_(2)$ such that their

concatenation is non$-$regular; Give an example of a non$-$regular lan$-$

guages L$\_(1)$ and a regular language L$\_(2)$ such that L$\_(1)\backslash$cup L$\_(2)$ is non$-$regular.

Bonus problem: Give an example of two non$-$regular languages

L$\_(1),$L$\_(2)$ such that their concatenation is regular.

Hint: the choice of an alphabet is up to you, choose the simplest.