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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!

If set \(S\) has \(|S|=3\) elements and satisfies \[\begin{align} \text{(i)} &\; 0 \in S \text{ and} \\ \text{(ii)} &\; \text{if } p \in S, q \in S \text{ and } p \neq q, \text{ then } (p+q) \in S \end{align}\] and \(10\) is an element of \(S,\) what is the other element of \(S\) excluding \(0\) and \(10?\)

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If \(A\) is the set

\[ A= \{13, 17, 9, 0, -13\},\]

what is \(\lvert A \rvert? \)

**Details and assumptions**

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

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