# Complement of $a^n b^n c^n$

I am trying to find the complement of the language $L = \{ a^n b^n c^n \mid n \ge 0\}$.

I know that one of the things I gotta do is take out $n \ge 0$ so $\{a^n b^n c^n \mid n > 0\}$ but I feel there is not enough.Is making it so $a^i b^j c^k$ so that $i \neq j \neq k$ enough and make it so that $i,j,k > 0$?

Or am I completely off track?

• 1) What about $bca$? 2) If you write "$i \neq j \neq k$", do you mean that also $i \neq k$? (Chains of $\neq$'s don't make sense, it's not a transitive relation!) 3) What does it mean to "find" $\overline{L}$"? A description in which form? Mathematically, $\overline{L} = \{ w \in \{a,b,c\}^* \mid w \not\in L \}$ is a perfectly correct description. (Note that you have to fix the alphabet in for the complement operator to make sense!) – Raphael Mar 16 '16 at 10:20

The complement is the set of all strings over alphabet $\{a,b,c\}$ that are not of the form $a^i b^i c^i$ for some $i\geq 0$. That includes all strings $a^ib^jc^k$ where $i\neq j$, $i\neq k$ and/or $j\neq k$, as you say, but also some other strings. Mouse-over the box below for a hint.
For example, $bacbabc$ is in the complement.