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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!

# Sets - Subsets

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

A. $$\{ \{2\}, \{6\}, \{8\}, \{2,6\}, \{2,8\},\{6,8\}, \{2,6,8\} \}$$

B. $$\{ \emptyset, \{2\}, \{6\}, \{8\}, \{2,6\}, \{2,8\},\{6,8\}, \{2,6,8\} \}$$

C. $$\{ \emptyset, \{2\}, \{6\}, \{8\}, \{2,6\}, \{2,8\},\{6,8\} \}$$

D. $$\{ \{2\}, \{6\}, \{8\}, \{2,6\}, \{2,8\},\{6,8\} \}$$

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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