Question

Suppose that for every (including the empty set and the whole
set) subset X of a finite set S there is a subset X∗
of S and
suppose that if X is a subset of Y then X∗
is a subset of Y∗
.
Show that there is a subset A of S satisfying A∗ = A

Answer 1

Hold on, I'm getting confused here.

Answer 2

It says that "for every subset \(X \subseteq S\) there is a subset \(X* \subseteq S\). Thus, if \(A \subseteq S\) then \(A* \subseteq S\). We also know by the second hypothesis that \(A* \subseteq S*\). But this is basically just restating the hypotheses.

Answer 3

i m confused too....

Answer 4

Could we just let \(A = \emptyset \) ? Then \(A* = \emptyset\) and since the empty set is contained in every set, the claim is true.