Power Sets: What's wrong with this reasoning?

[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 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.

 

[mmwave]

I appreciate your attempt to find errors in the logic presented in the forum post. However, I would like to clarify that the mistake is not in line 2, but rather in line 3. The statement "S \subset \cup X_i \Leftrightarrow S \in \cup\mathcal{P}(X_i)" is not necessarily true. This is because the union of power sets may contain sets that are not subsets of the union of X_i. Therefore, the statement "S \in \cup\mathcal{P}(X_i) \Leftrightarrow (\exists i)(S \in \mathcal{P}(X_i))" is not always true, leading to the contradiction in line 3.

In order to fix this mistake, we can modify line 3 to read "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)". This takes into account that the union of power sets may contain sets that are not subsets of the union of X_i.

I hope this clarifies the mistake and provides a solution to fix it. Thank you for bringing this to our attention.

 

[tacman]

There are a few issues with this reasoning.

Firstly, line 2 is not a valid statement. While it is true that if two sets have the same elements, they are equal, this does not necessarily apply to power sets. The power set of a set X is a collection of all subsets of X, so it is possible for two different sets to have the same one-point sets but different subsets, making their power sets different.

Secondly, in line 3, the statement "S is contained in the union of the Xi" does not necessarily imply that "S is contained in any single Xi". This is because the union of sets can contain elements that are not in any of the individual sets. For example, if X1 = {1,2} and X2 = {3,4}, the union of X1 and X2 is {1,2,3,4}, which contains elements that are not in either X1 or X2.

Lastly, the example given in the edit shows that line 1 is not always true. Just because a one-point set is in the power set of the union of sets, it does not mean that it is in the power set of each individual set. This is because the power set of the union of sets contains all possible subsets of the union, while the power set of each individual set only contains subsets of that specific set.

In conclusion, there are multiple mistakes in this reasoning, including incorrect statements and faulty logic. It is important to carefully consider the definitions and properties of power sets before making conclusions about them.