1. **State the problem:** Determine whether each set C is a subset of set D for the given pairs. 2. **Recall the definition of subset:** A set $C$ is a subset of $D$, written $C \subseteq D$, if every element of $C$ is also an element of $D$. 3. **Analyze each part:** **a.** $C = \{x \mid 0 \leq x \leq 10\}$ and $D = \{x \mid -10 < x < 20\}$ - Since $C$ includes all $x$ from 0 to 10 inclusive, and $D$ includes all $x$ strictly between -10 and 20, all elements of $C$ lie within $D$. - Therefore, $C \subseteq D$ is **true**. **b.** $C = \{x \in \mathbb{Z} \mid 2 \leq x \leq 5\}$ and $D = \{x \mid 2 \leq x < 5\}$ - $C$ includes integers 2, 3, 4, 5. - $D$ includes all real numbers $x$ with $2 \leq x < 5$, so it includes 2, 3, 4 but not 5. - Since 5 is in $C$ but not in $D$, $C \subseteq D$ is **false**. **c.** $C = \{x \in \mathbb{Z}^- \mid x > -7\}$ and $D = \{x \in \mathbb{Z} \mid -6 \leq x \leq 0\}$ - $\mathbb{Z}^-$ means negative integers. - $C$ includes negative integers greater than -7, i.e., $-6, -5, -4, -3, -2, -1$. - $D$ includes integers from -6 to 0 inclusive. - All elements of $C$ are in $D$, so $C \subseteq D$ is **true**. **d.** $C = \{x \in \mathbb{Q} \mid 0 < x < 1\}$ and $D = \{x \in \mathbb{Q} \mid 0 \leq x \leq 1\}$ - $C$ includes all rational numbers strictly between 0 and 1. - $D$ includes all rational numbers between 0 and 1 inclusive. - Since $D$ includes all elements of $C$ plus the endpoints, $C \subseteq D$ is **true**. **Final answers:** - a) True - b) False - c) True - d) True