Language regularity implications

Question

I have to decide whether this implications are true or false and prove it. Will you help me?

$L.\{a,b\}^{*}$ is regular $\implies$ $L$ is regular

$L.\{a,b\}^{*}$ is not regular $\implies$ $L$ is not regular

Thanks

Answer 1

For the first, consider any irregular $L\subset \{a,b\}^*$ with $\epsilon\in L$. What do you conclude?

For the second, what can you say about $L.\{a,b\}^*$ if $L$ is regular?