Regular expressions, is it always true that (r U s)* = r* U s* U (rs)*? If r and s are any two regular expressions, then (r ∪ s)* = r* ∪ s* ∪
  (rs)*.


I think this is not true. And I believe this would always be true : 


  (r ∪ s)* = r* ∪ s*


I wanted to clarify this because of (rs)*. What do you think the right hand side would be if r* ∪ s* ∪ (rs)* ?
Consider the alphabet $\Sigma = \{a,b\}$, and the regular expressions $r = a$ and $s = b$. Then $(r+s)^* = \Sigma^*$ whereas $a^*+b^*$ and $a^*+b^*+(ab)^*$ are both smaller.