The pumping lemma guarantees that w can be written in the form xyz. So x is a prefix, y is a middle part and z is a suffix of w. The part that is pumped is y. We need both x and z to describe how the word looks after pumping. The prefix x and the suffix y stay unchanged when pumping. Deleting z could lead to a word that is not in the language: think for instance of a language L = { a^n b | n >= 0 } where every word in L ends with b. Then deleting a suffix of these words leads immediately out of L.
When you pick i=0, doesn't that make |y|=0 and thus violate the assumption that |y|>=1? Or is it assumed that |y|>=1 iff when y is used (i.e. i>0)?
@@user-be1jx7ty7n that's obvious, but I was concerned about the assumption we have, namely: |y| > = 1!!
What does z was used for in the Lemma? Can we delete it?
The pumping lemma guarantees that w can be written in the form xyz. So x is a prefix, y is a middle part and z is a suffix of w. The part that is pumped is y. We need both x and z to describe how the word looks after pumping. The prefix x and the suffix y stay unchanged when pumping. Deleting z could lead to a word that is not in the language: think for instance of a language L = { a^n b | n >= 0 } where every word in L ends with b. Then deleting a suffix of these words leads immediately out of L.