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QUESTION 26 : Consider below the regular expressions over alphabet {0, 1, 2}

(i) (0 + 01 + 012))*
(ii) 0*(0 + 1)*(0+1+2)*
(iii) 0*(ε + 01)*(Φ + 01)*
(iv) 0* + (01)* + (012)*

Which of the above regular expressions represent same language as 0*(01)*(012)*

(A) i, ii and iii
(B) ii and iv
(C) iii
(D) iv

2 Answers

Best answer
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None of these for the given options 

$0^*(01)^*(012)^*$

i) ${\color{Purple} 012}\;{\color{DarkGreen} 01}$ can be derived from option i) but cannot derived from given regular expression 

ii) $2$ can be derived from option ii) but cannot derived from given regular expression.

iii) $012$ cannot be derived from option iii) but can derived from given regular expression 

iv)${\color{Purple} 01}\;{\color{DarkGreen} 012}$ cannot be derived from option iv) but can derived from given regular expression.

if we do some correction in option iii) as $0^*(\epsilon+01)^*(\phi+012)^*$, then it will be correct 

*Note $(\epsilon+R_1)^*= R_1^*\quad\text{and}\quad(\phi+R_2)^*= R_2^*$

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D ,because from expr 1 it produces 01201 but from given it is not .

similarly from expr 2 it produces 011 but from given not produces .

similarly in expr 3 their is not at all 2in input alphabets so this is not at all. remaining is option 4.

so ans is D.

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