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!

Which flag would be in the same category as that of South Korea?

Given the sets \[ \begin{align} U &=\{1,2,3,\ldots,8\},\\ A &=\{X \mid 8 \in X, X \subseteq U\},\\ B &=\{Y \mid 1 \in Y, Y \subseteq U\}, \end{align}\] what is \(\lvert{A \cup B}\rvert?\)

**Details and assumptions**

\(\lvert{S}\rvert\) denotes the number of members of the set \(S.\)

In counting the numbers from 1 to 99 (inclusive), what is the total number of times that the digit 3 is used?

**Details and assumptions**

The number \(12\) uses the digit 1 once and the digit 2 once.

The number \(111\) uses the digit 1 three times.

Given the set \(U=\{x \mid x \text{ is a positive integer } \leq 12\},\) let \(A=\{2, 4\}\) and \(B=\{2, 6, 9, 11, 12\}\) be two subsets of \(U.\) How many sets \(X\) are there such that \[\begin{align} \text{I}&. && X \subseteq U, \\ \text{II}&. && |B \cap X|=3, \\ \text{III}&. && A \text{ and } X \text{ are mutually disjoint?} \end{align} \]

**Details and assumptions**

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