What would be the complement of $L =\{a^{n}b^{m}a^{n}b^{m} | n,m \geq 1\}$

Question

I understand the complement of L would be all the strings not in L, but I'm having a hard time writing down the structure of all the strings not in L.

Answer 1

As well as all sentences which are not in the form $a^ib^ja^kb^l$, there's the set $\{a^ib^ja^kb^l\mid i\ne k \lor j \ne l\}$. The two inequalities are not mutually exclusive, but it should be clear that only if both inequalities are false (i.e. both equalities would be true) would the sentences be in $L$.