Proof the following - language

Question

Theorem $4$. A language $A$ is regular iff there exists a regular expression $\alpha$ such that $A = L(\alpha)$.

Check whether the following equations are correct.

1. $\left((a\cup b)^*\right)^*=(a\cup b)^*$;

2. $(a\cup b)^*(a\cup b)^*=(a\cup b)^*$.

Answer 1

Hint. Your question has nothing to do with regular languages. Look carefully at the definition of $L^*$ (where $L$ is any language, regular or not) and try to prove the following properties

1. $(L^*)^* = L^*$
2. $L^*L^* = L^*$