Answer
{ }, {a}, {e}, {i}, {o}, {a,e}, {a,i}, {a,o}, {e,i}, {e,o}, {i,o}, {a,e,i}, {a,e,o}, {a,i,o}, {e,i,o}, {a,e,i,o}
Work Step by Step
The empty set is a subset of all sets { }
List all the subsets with 1 member {a}, {e}, {i}, {o}
List all the subsets with 2 members {a,e}, {a,i}, {a,o}, {e,i}, {e,o}, {i,o}
List all the subsets with 3 members {a,e,i}, {a,e,o}, {a,i,o}, {e,i,o}
List all the subsets with 4 members {a,e,i,o}