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When Cantor introduced his classification of multiple infinities, he was vehemently rejected by most mathematicians. Ye be warned: contemplating the continuum hypothesis can drive anyone a little mad!

How many **sets A** exist for which \(\{1,2\} \subseteq A \subseteq \{1,2,3,4,5\}\) ?

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Solve for \(X\).

As a true connoisseur of capsaicin, you can taste the presence of every single one your \(X\) chili species distinctly in any chili sauce. So there are a LOT of chili sauces in your kitchen. In fact, your kitchen is basically filled with the \(8192\) tiny, semi-explosive chili sauce bottles you own. Why so many? -- because you have every kind of chili sauce that uses *any combination* of *any number of your chilies* (including one INSANE chili sauce that has all \(X\) of the kinds of pepper you love in it!!!! -- whenever you use it, there's a 20% chance of spontaneous combustion of the food it's put on.) You also have exactly one chililess chili sauce among the \(8192\), just to complete the set... but anyone who uses it is a dodo bird.

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Calculate |\( \mathcal{P}(\mathcal{P}(A))\)| if A is the Empty Set and \(\mathcal{P}(X)\) means "the **power set** of set \(X\)."

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X is a subset of Y and Y is a subset of Z. Which of the following statements must be true?

(A)\(\ \ \) If 1 is in Y, then 1 is in X.

(B)\(\ \ \) If 2 is in Z, then 2 is in X.

(C)\(\ \ \) If 3 is in Z, then 3 is in Y.

(D)\(\ \ \) If 4 is in X, then 4 is in Z.

(E)\(\ \ \) None of the above.

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