Probability
# 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.** $\{ \{2\}, \{6\}, \{8\}, \{2,6\}, \{2,8\},\{6,8\} \}$

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 $7?$