1. The problem asks to describe each given roster (list of elements) using set-builder notation. 2. Set-builder notation describes a set by stating the properties that its members must satisfy. 3. For each roster, identify the pattern or rule that defines the elements. 4. Write the set as $\{x \mid \text{condition on } x\}$, meaning "the set of all $x$ such that the condition holds." **a)** $\{...,-3,-2,-1\}$ - These are all integers less than or equal to $-1$. - Set-builder: $\{x \mid x \in \mathbb{Z} \text{ and } x \leq -1\}$ **b)** $\{1.0, 1.1, 1.2, 1.3\}$ - These are decimal numbers starting at $1.0$ increasing by $0.1$ up to $1.3$. - Set-builder: $\{x \mid x = 1 + 0.1n, n \in \{0,1,2,3\}\}$ **c)** $\{3,4\}$ - Two integers 3 and 4. - Set-builder: $\{x \mid x \in \{3,4\}\}$ **d)** $\{-5,-4,-3,-2\}$ - Integers from $-5$ to $-2$ inclusive. - Set-builder: $\{x \mid x \in \mathbb{Z} \text{ and } -5 \leq x \leq -2\}$ **e)** $\{20,21,22,23,...\}$ - Integers starting at 20 and increasing without bound. - Set-builder: $\{x \mid x \in \mathbb{Z} \text{ and } x \geq 20\}$ **f)** $\{...,-1,0,1,2,...\}$ - All integers. - Set-builder: $\{x \mid x \in \mathbb{Z}\}$ **g)** $\{3,4,5,6\}$ - Integers from 3 to 6 inclusive. - Set-builder: $\{x \mid x \in \mathbb{Z} \text{ and } 3 \leq x \leq 6\}$ **h)** $\{-7,-6,-5,-4\}$ - Integers from $-7$ to $-4$ inclusive. - Set-builder: $\{x \mid x \in \mathbb{Z} \text{ and } -7 \leq x \leq -4\}$ **i)** $\{2.0, 2.1, 2.2, 2.3, ...\}$ - Decimal numbers starting at 2.0 increasing by 0.1 without bound. - Set-builder: $\{x \mid x = 2 + 0.1n, n \in \mathbb{N}_0\}$ where $\mathbb{N}_0$ is the set of whole numbers including zero. This completes the set-builder notation descriptions for all rosters.