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I need help understanding the language L above.

These are my understanding:
- w = uu is a concatenation of u
- La(1*01*) can be expressed in regular language i.e. r1 = 1*01* so La(r1)
- also we have regular expression property of concatenation: L(r1r2) = L(r1)L(r2)

So base on my understanding the language L can be simplified to

I'm not quite sure about this.

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No, your simplification is wrong.

Consider, as a simpler case, $A=\{a,b\}$ and $L'=\{ w \mid w=uu , u\in A\}$. Then, $L'=\{aa,bb\}$.

Instead, if we apply your simplification and write $L''=AA$ then we have $L''=\{aa,ab,ba,bb\}$ which is a larger language, since we forgot that both $u$'s in $w=uu$ must be the same word.

Keeping that into account, your original language $L$ seems to be $\{1^n01^m1^n01^m \mid n,m\geq 0 \}$

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  • $\begingroup$ So we ignored ab and ba in the set because uu is an element of A and must be the same character. $\endgroup$
    – tramyardg
    Dec 1, 2018 at 21:39
  • $\begingroup$ @tramyardg It must be the same word, repeated. In this case $A$ only contains single-letter words, but in general it can contain longer words, as in your example. $\endgroup$
    – chi
    Dec 1, 2018 at 22:08

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