Languages and their operations: determining equivalence I'm currently taking a subject on theoretical computer science. This is not a homework question.

I'm having trouble figuring out to show the equivalence or disprove the equivalence. Consider any two languages $A$ and $B$. Is the following statement true or false?

$A^* \cup B^* = (A \cup B)^*$

Honestly, I use the above example mainly to ensure my question is less subjective. However I'm more concerned about the generalised approach to determining whether languages are equivalent (or subsets of each other and such) when operated on by various operators (kleene star, union, intersection, difference, concatenation). If I know that they are not equivalent I can basically just start bruteforcing for counter-examples, however I can't really think of any structured and consistent approach to answering questions such as the above.

Thanks.
Nope. Let $A = \{a\}$, $B= \{b\}$. Then $A^* = \{\epsilon,a,aa,
\ldots\}$ and similarly for $B$. So, $A^*\cup B^*$ contains strings of repeated "a" or repeated "b". But $(A\cup B)^*$ contains strings like $ab$, which you can't get from $A^*$ or $B^*$.

I'd suggest looking at a book like Sipser for a good coverage of languages like these and how to prove properties like equivalence. In particular, he has a great number of exercises to work through. Good luck!