REGULAR OR NOT

Question

Please mention reason with answer:-

Comments

all three are regular..

1- it is set of all even length strings..

2- take x =(a+b)*, w1=epailon, w2=epsilon, it will become (a+b)*, and it absorb all other strings..

3-take x=(a+b)*, w=epsilon, it will also become (a+b)*, and will absorb all other strings..

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In (1) it is an even length string what it means as we have to remember number of counts  w1 to match with w2

example:-

         w1:- ababababababababbbbbaaaaba.........

         then w2 should have remembered the count of w1  to match the equal number of string...

but as we know in regular there is not too much memory to store that much information so,

how regular please explain in detail.

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Or do you mean even length string to below diagram

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you can construct a DFA which can check a string is regular or not.
Accept 0 length string i.e final state
Reject 1 lenght string i.e non final state
Accept 2 lenght string i.e final state
Reject 3 lenght string i.e non final state and so on

Answer 1

1 ) Yes its regular, it indicate set of all even length string.

2) Yes its regular, take $ w_{_1} = \epsilon$ and $w_{_2} = \epsilon$, so we have end up with x accepting everything, which is regular

3) Yes its regular, take $w = \epsilon$, so we end up with x accepting everything, which is regular.