taylor 45

Hi!!!I have a question..How could I show that the language L={xy^{n}zy^{n}w: x,z,w in Σ*, y in Σ, z does not contain y, and n>=0} is not regular, using the Myhill-Nerode theorem?

How could I use the closure properties to show that the language L at this exercise is regular? Srtting L1={l ε {a,b}*:the word l does not contain the subword aaa} and L2=L={l ε {a,b}*:the word l does not contain the subword (bb)}. Then? How can I continue?

My exercise asks me to prove that the language is regular,without using DFAs.Could you give me a hint how I can do this??I got stuck right now..

Hello! How can I show that this language L={l ε {a,b}*:the word l contains neither the subword aaa nor the subword (bb)} is regular??Is there any theorem,that I can use to prove it??