1. **Problem statement:** Given universal set $U=(-\infty,+\infty)$, $M=(-2,7]$, and $K=[0,13]$, find: 2. **Formulas and rules:** - Union: $M \cup K$ is all elements in $M$ or $K$ or both. - Intersection: $M \cap K$ is all elements in both $M$ and $K$. - Set difference: $M - K$ is elements in $M$ not in $K$. - Set difference: $K - M$ is elements in $K$ not in $M$. - Complement: $M'$ is elements in $U$ not in $M$. - Complement: $K'$ is elements in $U$ not in $K$. 3. **Step 1: Find $M \cup K$** - $M=(-2,7]$, $K=[0,13]$ - The union covers from $-2$ (exclusive) to $13$ (inclusive) because $M$ starts at $-2$ and $K$ extends to $13$. - So, $M \cup K = (-2,13]$ 4. **Step 2: Find $M \cap K$** - Intersection is overlap of $(-2,7]$ and $[0,13]$ - Overlap is from $0$ (inclusive) to $7$ (inclusive) - So, $M \cap K = [0,7]$ 5. **Step 3: Find $M - K$** - Elements in $M$ but not in $K$ - $M=(-2,7]$, $K=[0,13]$ - $M$ starts at $-2$ to $7$, $K$ starts at $0$ - So elements less than $0$ in $M$ are not in $K$ - Hence, $M - K = (-2,0)$ 6. **Step 4: Find $K - M$** - Elements in $K$ but not in $M$ - $K=[0,13]$, $M=(-2,7]$ - $K$ extends beyond $7$ to $13$ - So elements from $7$ (exclusive) to $13$ (inclusive) are in $K$ but not in $M$ - Hence, $K - M = (7,13]$ 7. **Step 5: Find $M'$ (complement of $M$)** - $M=(-2,7]$ - Complement is everything in $U$ not in $M$ - So $M' = (-\infty,-2] \cup (7,+\infty)$ 8. **Step 6: Find $K'$ (complement of $K$)** - $K=[0,13]$ - Complement is everything in $U$ not in $K$ - So $K' = (-\infty,0) \cup (13,+\infty)$ **Final answers:** - $M \cup K = (-2,13]$ - $M \cap K = [0,7]$ - $M - K = (-2,0)$ - $K - M = (7,13]$ - $M' = (-\infty,-2] \cup (7,+\infty)$ - $K' = (-\infty,0) \cup (13,+\infty)$