1. **State the problem:** We are given sets $B_k$ for $k=1$ to $5$ and need to find three things: the union $\bigcup_{k=1}^5 B_k$, the intersection $\bigcap_{k=1}^5 B_k$, and the set difference $B_3 - B_1$ for two different definitions of $B_k$. 2. **Recall definitions:** - The union $\bigcup_{k=1}^5 B_k$ contains all elements that are in at least one of the sets $B_1, B_2, ..., B_5$. - The intersection $\bigcap_{k=1}^5 B_k$ contains only elements common to all sets $B_1, B_2, ..., B_5$. - The set difference $B_3 - B_1$ contains elements in $B_3$ that are not in $B_1$. 3. **Part (a):** $B_k = \{0, 1, 2, ..., 2k\}$ - Calculate each $B_k$: - $B_1 = \{0,1,2\}$ - $B_2 = \{0,1,2,3,4\}$ - $B_3 = \{0,1,2,3,4,5,6\}$ - $B_4 = \{0,1,2,3,4,5,6,7,8\}$ - $B_5 = \{0,1,2,3,4,5,6,7,8,9,10\}$ - **Union:** The union is all elements appearing in any $B_k$, so it is $\{0,1,2,3,4,5,6,7,8,9,10\}$. - **Intersection:** The intersection is elements common to all $B_k$. Since $B_1$ is the smallest set, intersection is $B_1 = \{0,1,2\}$. - **Set difference:** $B_3 - B_1$ means elements in $B_3$ not in $B_1$. - $B_3 = \{0,1,2,3,4,5,6\}$ - $B_1 = \{0,1,2\}$ - So difference is $\{3,4,5,6\}$. 4. **Part (b):** $B_k = \{k-1, k, k+1\}$ - Calculate each $B_k$: - $B_1 = \{0,1,2\}$ - $B_2 = \{1,2,3\}$ - $B_3 = \{2,3,4\}$ - $B_4 = \{3,4,5\}$ - $B_5 = \{4,5,6\}$ - **Union:** Combine all elements: - $\{0,1,2,3,4,5,6\}$ - **Intersection:** Find elements common to all sets: - No element appears in all five sets, so intersection is empty $\emptyset$. - **Set difference:** $B_3 - B_1$: - $B_3 = \{2,3,4\}$ - $B_1 = \{0,1,2\}$ - Difference is $\{3,4\}$. This step-by-step approach uses the definitions of union, intersection, and set difference applied to the explicit sets $B_k$ to find the answers.