self doubts

Question

{w1 x w2|w1,x,w2∈(a+b)*,w1=w2}

it is regular ?????

Comments

yes it is regular

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it is nothing but $(a+b)^{*}$

Answer 1

Suppose,

w1= aa

w2= aa

x=ba

string becomes aabaaa

since x is of infinite length as mentioned in question

we could rewrite the string as       

w1 = a

x= abaa

w2= a

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w1 =w2

=> Language is "Strings starting and ending with same alphabet."

=> it is regular.

However, if x is of finite length then the language would not be regular.

Answer 2

Yes it is regular,

Because we can construct dfa of this given language by keeping starting and ending same symbol.

Regular expression of this language can be described as below

a(a+b)* a  +  b(a+b)* b