# How to negate this one?

How can I negate the following sentence: For all words x from L with |x|>= n , exists decomposition x = uvw with |uv| <= n and |v| >= 1, so for all i >= 0 , is valid that u(v)^iw in L is.

• Did you try en.wikipedia.org/wiki/De_Morgan's_laws ? What did you get? – chi May 19 '17 at 19:10
• For at least one word x from L with |x|>= n , exists decomposition x = uvw with |uv| <= n and |v| >= 1,so for at least one i >= 0 , is valid that u(v)^iw not in L is. Is this the right way ? – unnamed May 19 '17 at 19:15
• The part "For at least one word x from L with |x|>= n , " is correct. But "exists decomposition such that ..." must be negated, so it becomes "for all decompositions, we do not have ..." (and then we push the negation further). – chi May 19 '17 at 19:27
• Is this actually the negation of the pumping lemma ? – unnamed May 19 '17 at 19:34
• Write it as formula. Slap $\lnot$ in front. Done. – Raphael May 20 '17 at 5:15

The negation is: There exists a word $x \in L$ satisfying $|x| \geq n$ such that for all decompositions $x = uvw$ with $|uv| \leq n$ and $|v| \geq 1$ there exists $i \geq 0$ such that $uv^iw \notin L$.