Does the Kleene star distribute over each element? (0+1)* = 0* + 1*?

Does the Kleene star distribute over each element? Is this true: $$(0+1)^* = (0^* + 1^*)$$?

• Interesting question. Have you tried a few examples? Have you tried proving the quality? Is 01 in $0^*+1^*$? – Apass.Jack Mar 22 at 22:02

You can verify that $$010$$ is in $$(0+1)^*$$ but not in $$(0^* + 1^*)$$. Therefore, $$(0 + 1)^* \neq (0^* + 1^*)$$.
$$(0+1)^* = (0^*1^*)^*$$
$$0^* + 1^*$$ would limit the amount of strings covered.