1. **Stating the problem:** We are given sets $B_k = \left[ \frac{5}{k}, \frac{5k+2}{k} \right]$ for $k=1$ to $5$ with universal set $\mathbb{R}$. We want to verify and understand the following: - The union $\bigcup_{k=1}^5 B_k = [1,7]$ - The intersection $\bigcap_{k=1}^5 B_k = [5,5.4]$ - The difference $B_3 - B_1 = \left[ \frac{5}{3}, 5 \right)$ 2. **Formula and rules:** - Union of intervals combines all points in any of the intervals. - Intersection of intervals contains only points common to all intervals. - Set difference $A - B$ contains points in $A$ not in $B$. 3. **Calculate each $B_k$ explicitly:** - $B_1 = \left[ \frac{5}{1}, \frac{5\cdot1 + 2}{1} \right] = [5,7]$ - $B_2 = \left[ \frac{5}{2}, \frac{10 + 2}{2} \right] = [2.5,6]$ - $B_3 = \left[ \frac{5}{3}, \frac{15 + 2}{3} \right] = \left[ \frac{5}{3}, \frac{17}{3} \right] = [1.666...,5.666...]$ - $B_4 = \left[ \frac{5}{4}, \frac{20 + 2}{4} \right] = [1.25,5.5]$ - $B_5 = \left[ \frac{5}{5}, \frac{25 + 2}{5} \right] = [1,5.4]$ 4. **Find the union $\bigcup_{k=1}^5 B_k$:** - The smallest lower bound is $\min\{5,2.5,1.666...,1.25,1\} = 1$ - The largest upper bound is $\max\{7,6,5.666...,5.5,5.4\} = 7$ - Since these intervals overlap and cover continuously from 1 to 7, the union is $[1,7]$ 5. **Find the intersection $\bigcap_{k=1}^5 B_k$:** - The largest lower bound is $\max\{5,2.5,1.666...,1.25,1\} = 5$ - The smallest upper bound is $\min\{7,6,5.666...,5.5,5.4\} = 5.4$ - Intersection is $[5,5.4]$ 6. **Find the difference $B_3 - B_1$:** - $B_3 = \left[ \frac{5}{3}, 5.666... \right]$ - $B_1 = [5,7]$ - The difference is points in $B_3$ not in $B_1$, i.e. $\left[ \frac{5}{3}, 5 \right)$ **Final answers:** - $\bigcup_{k=1}^5 B_k = [1,7]$ - $\bigcap_{k=1}^5 B_k = [5,5.4]$ - $B_3 - B_1 = \left[ \frac{5}{3}, 5 \right)$