Here are the solutions to the problems: 1. (a) Express $6.093$ to one decimal place. Step 1: Identify the digit in the first decimal place, which is $0$. Step 2: Look at the digit immediately to its right, which is $9$. Step 3: Since $9 \ge 5$, round up the digit in the first decimal place. $$6.093 \approx \boxed{6.1}$$ 1. (b) Evaluate $\frac{3}{4} + \frac{1}{2} - \frac{1}{3}$. Step 1: Find a common denominator for $4, 2,$ and $3$. The least common multiple (LCM) is $12$. Step 2: Convert each fraction to have a denominator of $12$. $$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$$ $$\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}$$ $$\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$$ Step 3: Perform the addition and subtraction. $$\frac{9}{12} + \frac{6}{12} - \frac{4}{12} = \frac{9 + 6 - 4}{12} = \frac{15 - 4}{12} = \boxed{\frac{11}{12}}$$ 2. Given the sets $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $B = \{2, 3, 6, 7\}$. 2. (a) Find $A'$. Assuming that $A$ is the universal set for this context, the complement of $A$ ($A'$) would be the set of elements in the universal set that are not in $A$. $$A' = A \setminus A = \boxed{\emptyset}$$ 2. (b) Find $A \cap B$. $A \cap B$ represents the intersection of sets $A$ and $B$, which includes all elements common to both sets. $$A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$$ $$B = \{2, 3, 6, 7\}$$ The elements common to both sets are $2, 3, 6, 7$. $$A \cap B = \boxed{\{2, 3, 6, 7\}}$$ 2. (c) Find $n(B)$. $n(B)$ represents the number of elements in set $B$. $$B = \{2, 3, 6, 7\}$$ There are $4$ elements in set $B$. $$n(B) = \boxed{4}$$ 3. Complete the truth table below. Step 1: Fill the $\sim P$ column by negating the truth values of $P$. Step 2: Fill the $\sim P \lor Q$ column by applying the OR operator to the values in the $\sim P$ column and the $Q$ column. The OR operator is true if at least one of its operands is true. | P | Q | $\sim P$ | $\sim P \lor Q$ | |---|---|----------|-----------------| | T | T | F | T | | T | F | F | F | | F | T | T | T | | F | F | T | T | That's 2 down. 3 left today — send the next one.