Regular language properties

Question

For regular languages $R, S$ and $T$, which of the following are true?

1. $R \cup S = S \cup R$
2. $(R \cup S) \cdot T = RT \cup ST $
3. $R^* \cdot S^* = (R \cup S)^*$

Answer 1

1. True
2. True
3. False

To find the proofs, proceed as follows:

1. Show that any string $x \in R \cup S$ belongs to $S \cup R$, and vice versa.
2. Ditto 1.
3. Find a string $x$ that is in $(R \cup S)^*$ but not the LHS. Hint: what about $rsr$ where $r \in R, s \in S$?

Answer 2

1. $x\in R\cup S \iff x\in R \vee x \in S \iff x\in S \vee x \in R \iff x\in S\cup R$

2. $x\in(R\cup S)T \iff \exists u, v \mid (x=uv)\wedge (u \in R\cup S)\wedge (v\in T) \iff \exists u, v \mid (x=uv)\wedge (u \in R \vee u \in S)\wedge (v\in T) \iff \exists u, v \mid (x=uv)\wedge (u\in R \wedge v\in T) \vee (u\in S \wedge v\in T) \iff \exists u, v \mid (x=uv)\wedge (x\in RT)\vee (x\in ST)\iff x\in RT\cup ST$.

3. Let $R=\{0\}$ and $S=\{1\}$.