# Power Sets: What's wrong with this reasoning?

1. Feb 8, 2006

### AKG

Let $\mathcal{P}(X)$ denote the power set of a set $X$.

$$1.\ \{x\} \in \mathcal{P}(\cup X_i) \Leftrightarrow x \in \cup X_i \Leftrightarrow (\exists i)(x \in X_i) \Leftrightarrow (\exists i)(\{x\} \in \mathcal{P}(X_i)) \Leftrightarrow \{x\} \in \cup\mathcal{P}(X_i)$$

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.

$$3.\ S \subset \cup X_i \Leftrightarrow S \in \mathcal{P}(\cup X_i) \Leftrightarrow S \in \cup\mathcal{P}(X_i) \Leftrightarrow (\exists i)(S \in \mathcal{P}(X_i)) \Leftrightarrow (\exists i)(S \subset X_i)$$

However, let i range over {1, 2}, let Xi = {i}. Let X denote the union of the Xi. Now X is the union of the Xi, and hence is contained in the union of the Xi, but X is not contained in any single Xi, 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.