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Set Notation

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!

Subsets

If \(S=\{2,6,8\},\) which of the following is the power set of \(S?\)

Let \[X=\{6, -\alpha\}, Y=\{\alpha^2+5,\alpha-8,1\}.\] If \(X \subseteq Y,\) what is the real number \(\alpha?\)

Given the set \[A=\{x \mid x \text{ is a positive integer } < 13\},\] how many subsets of \(A\) have no even numbers?

Details and assumptions

The empty set \(\emptyset\) has no elements, and hence, has no even numbers.

If \(A=\{5, 6, 7\}\) and \(B=\{6, 7, 9, 11, 15\},\) how many sets \(X\) are there satisfying

\[(A-B) \cup X = X, (A \cup B) \cap X = X?\]

If \(A\) is the set

\[A=\{5, 7, 11, 14\},\] how many subsets of \(A\) do not contain the element \(7?\)

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