## vishal_kothari 4 years ago 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

1. KingGeorge

Hold on, I'm getting confused here.

2. KingGeorge

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.

3. vishal_kothari

i m confused too....

4. KingGeorge

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