i've encountered a quite tricky question that i don't understand, and would appreciate your help with:
it is said that by the pumping lemma, for a certain L there exists(guarenteed) n=2. also known that $aa \in L$. prove that $a^6 \in L$
so from what i understand, because it is guranteed that for some L, because of the pumping lemma, n=2, it means that for a word $w \in L$, w can be represented as such: w=abc, so that $|ab| \leq n$, $|b| \geq 1$ and for every $i \in N$: $ab^ic \in L$. so if $aa \in L$, then the boundary(first condition) is n=2 as given, but does it mean that it can only one letter, so that $a^6 \in L$? basically, i think that i should be using the third condition(for every $i \in N$ $ab^ic \in L$, but how can i ommit a and c so i could show that i=6, i.e $a^6 \in L$
tried to explain the question and my way of thinking the best i could. how can i show that $a^6 \in L$?
Ok, let's fix some easier notation:
We say that a language $L$ satisfies the pumping lemma for $p>0$ if for any word $w\in L$ with $|w|\geq p$ we have that $w=xyz$ with the properties: $$1. p\geq |xy|, |y|>0 $$ $$2. \forall i\in \Bbb N: xy^iz\in L $$ So in our case, $y$ (which is being 'pumped') would either be $a$ or $aa$.