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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!

Rakesh is classifying flags of the world according to the presence of red or blue in them.

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

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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 \(S.\)

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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.

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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**

You may choose to read the summary page Set Notation.

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