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Show that the regular languages are closed under the following operations$:$

  1. $ min(L)=\big\{\text{w|w is in L, but no proper prefix of w is in L}\big\}.$
  2. $ min(L)=\big\{\text{w|w is in L, and  for no x other than $\in$  is wx in L}\big\}.$
  3. $ init(L)=\big\{\text{w| for some x, wx is in L}\big\}.$
in Theory of Computation by Veteran (59.5k points)
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