Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I came across this relation betwen tww sets of languages formed from the alphabet V.


The relation is

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

I am confused how this is derived. Any pointer?

share|cite|improve this question

It is not true in general, for example take $A=\{a\},B=\{b\}$.

$A^*\cup B^*=\{a\}^*\cup\{b\}^*$ - the language of all words that have only $a$'s in them or only $b$'s in them (including only zero $a$'s i.e $\epsilon$ is also in this language)

$A\cup B={a,b}\implies (A\cup B)^*=\{a,b\}^*=(\{a,b\}^*)^*$ - and this is the language of all words over $\{a,b\}$ and it has, for example, the word $ab$ that is not in $A^*\cup B^*$

share|cite|improve this answer

It’s not generally true that $A^*\cup B^*=((A\cup B)^*)^*$. (I really don’t understand the function of the second star on the righthand side, since $(S^*)^*=S^*$ always.)

It’s clear that $A^*\cup B^*\subseteq(A\cup B)^*=((A\cup B)^*)^*$, but the reverse inclusion is not in general true. Suppose that $a\in A\setminus B$ and $b\in B\setminus A$. Then $ab\in(A\cup B)^*$, but $ab\notin A^*\cup B^*$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.