MadeEasy Test Series : Regular Expression

Question

Consider the following regular expressions:
I. 0(0+1)*
II. 0* 10*1(0 +1)*
III (0+10)*(1+€)
IV.[(0*10* 10*)* +0*]10*
A language L whose regular expression is r is said to be reverse isomorphic if L(r)= L(r^R). How many of the above regular expressions are reverse isomorphic?

Comments

What does it mean by $r^R$ ?

How can we reverse a regular expression?

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They mean to say $[L(r)]^R$.

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I think only  II. 0* 10*1(0 +1)* is reverse isomorphic

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Well, what we 'think' may not always be correct. If you say that II is reverse isomorphic and the others are not, you must provide a justification as to why II is, and the others aren't.

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given regular expressions are along with there meaning

I. 0(0+1)*   all strings starting wiith 0

II. 0* 10*1(0 +1)* all strings containing atleast two 1's

III (0+10)*(1+€)  all strings containing no consecutive 1's

IV.[(0*10* 10*)* +0*]10*  now i don't know exactly what this is representing ,but it does not necessarily start with 1

Reversed expressions can be writtten as below 

1.(0+1)*0    all strings ending  wiith 0

2.(0 +1)*10*10*   all strings containing atleast two 1's

3.(1+€) (0+01)*  all strings containing no consecutive 1's

4.  0*1[(0*10* 10*)* +0*]      

@Lakshay Kakkar  thanks for rectifying ,pls verfify now  that 2nd and 3rd are reverse isomorphic and what about 4th?

 

 

 

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You have not reversed the 3rd regular expression correctly. And the exact same mistake has been repeated in 4th aswell.

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@Lakshay Kakkar    2nd is reverse isomorphic I have written above

and according to you what should be correct reversed expressions for 3rd and 4th pls tell 

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Let's take III.

Correct R.E. : (1+€) (0+01)*

Your R.E.: (1+€) (0+10)*

Can you spot the mistake now?

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Yes. It is rev isomorphic as well. Odd number of 1s.

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@Lakshay Kakkar thanks bro!!

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What was the answer? I feel following:

L1^R = (0+1)*0
L2^R = (0+1)*10*10*
L3^R = (1+e)(0+01)*
L4^R = 0*1[0*+(0*10*10*)*]

L1 can have strings ending with 1, but L1^R cannot, so not reverse isomorphic.

Cant find such things for rest. So I guess answer is 3. What was the solution?