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. $init(L)=\big\{\text{w| for some x, wx is in L}\big\}.$