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 3 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
 3 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

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KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.1Hold on, I'm getting confused here.

KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.1It 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.

vishal_kothari
 3 years ago
Best ResponseYou've already chosen the best response.1i m confused too....

KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.1Could we just let \(A = \emptyset \) ? Then \(A* = \emptyset\) and since the empty set is contained in every set, the claim is true.
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