Problem 4, 2018

For a set \(X\) let \(\mathcal{P}(X) = \{Y : Y \subseteq X\}\) denote its power set. For example \(\mathcal{P}(\{1\}) = \{\varnothing, \{1\}\}\), since the subsets of \(\{1\}\) are \(\varnothing\) and \(\{1\}\), and \(\mathcal{P}(\varnothing) = \{\varnothing\}\), since \(\varnothing\) has exactly one subset, namely \(\varnothing\) itself. Write \(\mathcal{P}^{n}(X)\) for \(\mathcal{P}(\mathcal{P}(\ldots \mathcal{P}(X) \ldots))\), where \(\mathcal{P}\) is applied \(n\) times.

Find all two-element subsets \(A\) of the set

for which \(A \subseteq \mathcal{P}(A)\).