How do I prove languages Regular / Not regular?

[dionysus]

Let L be a regular set. Is the following regular: why or why not?

CYCLE (L) = { x_1 x_2 | x_2 x_1 is in L for strings x_1 and x_2 }
MAX (L) = { x in L | for no y other than epsilon is xy in L }
MIN (L) = { x in L | no proper prefix of x is in L }

[Sanju]

the way to detect if a language is regular is by testing it with pumping lemma

CYCLE (L) = { x_1 x_2 | x_2 x_1 is in L for strings x_1 and x_2 }
this language fails pumping lemma. Hence it is not regular.
I am still working on the other two. Will post them when I get done

[dionysus]

Err. I'm not sure you understand.

It is given that L is regular already. You can't apply the pumping lemma here.

[Sanju]

Oops, I misinterpreted the question ..

[Sanju]

Have a look at this link. The second question of HW1 is kinda similar to the one that you posted. I however have not been able to work my way through the ones that you posted , See if this helps.
http://www.cs.cornell.edu/courses/cs...ml#assignments

[blah321]

solution is here (All are regular)
cs.cornell.edu/courses/cs381/2005fa/solutions/hw5/381-hw5-solns.pdf