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AKG

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Let [itex]\mathcal{P}(X)[/itex] denote the power set of a set [itex]X[/itex].

[tex]1.\ \{x\} \in \mathcal{P}(\cup X_i) \Leftrightarrow x \in \cup X_i \Leftrightarrow (\exists i)(x \in X_i) \Leftrightarrow (\exists i)(\{x\} \in \mathcal{P}(X_i)) \Leftrightarrow \{x\} \in \cup\mathcal{P}(X_i)[/tex]

2. If two power sets share the same one-point sets, then they are the same. In particular, the power set of the union is the union of the power sets.

[tex]3.\ S \subset \cup X_i \Leftrightarrow S \in \mathcal{P}(\cup X_i) \Leftrightarrow S \in \cup\mathcal{P}(X_i) \Leftrightarrow (\exists i)(S \in \mathcal{P}(X_i)) \Leftrightarrow (\exists i)(S \subset X_i)[/tex]

However, let i range over {1, 2}, let X

EDIT: Oh, I think I see the problem. Line 2 doesn't apply to line 1. That is, the union of the power sets does contain the same one point sets as the power set of the unions, but the union of the power sets is not generally a power set, so there's no reason for it to equal the power set of the union.

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[tex]1.\ \{x\} \in \mathcal{P}(\cup X_i) \Leftrightarrow x \in \cup X_i \Leftrightarrow (\exists i)(x \in X_i) \Leftrightarrow (\exists i)(\{x\} \in \mathcal{P}(X_i)) \Leftrightarrow \{x\} \in \cup\mathcal{P}(X_i)[/tex]

2. If two power sets share the same one-point sets, then they are the same. In particular, the power set of the union is the union of the power sets.

[tex]3.\ S \subset \cup X_i \Leftrightarrow S \in \mathcal{P}(\cup X_i) \Leftrightarrow S \in \cup\mathcal{P}(X_i) \Leftrightarrow (\exists i)(S \in \mathcal{P}(X_i)) \Leftrightarrow (\exists i)(S \subset X_i)[/tex]

However, let i range over {1, 2}, let X

_{i}= {i}. Let X denote the union of the X_{i}. Now X is the union of the X_{i}, and hence is contained in the union of the X_{i}, but X is not contained in any single X_{i}, contradicting line 3. So somewhere in either line 1, 2, or 3, there is a mistake. Where is it?EDIT: Oh, I think I see the problem. Line 2 doesn't apply to line 1. That is, the union of the power sets does contain the same one point sets as the power set of the unions, but the union of the power sets is not generally a power set, so there's no reason for it to equal the power set of the union.

This thread can be deleted.

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