Let L be the language over $\{$a$,$b$\}$ containing all even$-$length strings where the first and second halves are equal. For example, these are all in L: $($lambda$,$ AA$,$ BB$,$ AAAA, ABAB, BABA, BBBB$,$ AAAAAA, AABAAB. In a proof that L is not a regular language we assume L is regular with pumping length p and then pick a string that would allow an easy contradiction using the pumping lemma.

What string would you choose?The pumping lemma says that your string can be broken into xyz where y is a non$-$empty substring of the first p characters of your string and that both xz and xyyz are also in L$.$

But either xz or xyyz is not in L because why?