Set Notation
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 \(7?\)