Which of the following four regular expressions are equivalent ?

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Which of the following four regular expressions are equivalent ?
(e = empty string)
(i) (00)*(e+0)
(ii) (00)*
(iii) 0*0*
(iv) 0(00)*

Ravi Garg edited question
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    Somak Sengupta (anonymous)

    (i) and (iii) are equivalent.

    Somak Sengupta answered
      0
      Debasish Ray Chawdhuri (anonymous)

      i and iv I guess.

      Debasish Ray Chawdhuri answered
        0
        Somak Sengupta (anonymous)

        No, its (i) and (iii)

        (00)*(e+0)= any number of zeros(even+0dd)+ empty string
        (00)*= empty string + even number of zeros
        0*0*=any number of zeros+empty string
        0(00)*=odd n of zeros

        Somak Sengupta answered
          0
          Debasish Ray Chawdhuri (anonymous)

          Hey Somak Sengupta,
          What about this proof ?
          (00)*(e+0) = (00)*0 [ e+P = P ]
          (00)*0 = 0(00)* [ (PQ)*P = P(QP)* ]
          This way (i) & (iv) are equivalent.

          Debasish Ray Chawdhuri answered
            0
            Debasish Ray Chawdhuri (anonymous)

            Oops,

            i think i used one step wrong:
            empty string + p = p
            erom formula nei……

            Debasish Ray Chawdhuri answered
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