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!

If \(\mathbb{R}\) denotes the set of real numbers, and \(S\) is the set

\[ S = \{ 7, 1+\sqrt{2}, \pi, \sqrt{-11}, -11, 0, (-6)^3, -\sqrt[3]{2} \} , \]

how many elements does \(\mathbb{R} \cap S \) have?

**Details and assumptions**

You may choose to read the summary page Set Notation.

If \(\mathbb{Q}\) denotes the set of rational numbers, and \(S\) is the set

\[ S = \{ 9, 1+\sqrt{2}, \pi, \frac{5}{19}, \sqrt{-14}, -9, 0, (-6)^3, -\sqrt[3]{2} \} , \]

how many elements does \(\mathbb{Q} \cap S \) have?

**Details and assumptions**

You may choose to read the summary page Set Notation.

How many positive integers are there in the set

\[ \{ x \mid -14 \le x < 66\}? \]

How many negative integers are there in the set

\[ \{ x \mid -15 \le x < 37\}? \]

How many elements are in the set \(\{10, 12, 14, \ldots, 48, 50\}\)?

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