1. **Problem:** Write the set $\{0,1,2,3,\ldots,10\}$ in set builder notation. 2. **Formula and rules:** Set builder notation describes a set by a property that its members satisfy. For example, $\{x \mid P(x)\}$ means the set of all $x$ such that property $P(x)$ holds. 3. **Step-by-step solution:** 1. The set $\{0,1,2,3,\ldots,10\}$ contains all integers from 0 to 10 inclusive. 2. We can express this as $\{x \mid x \in \mathbb{Z} \text{ and } 0 \leq x \leq 10\}$. 3. Here, $\mathbb{Z}$ denotes the set of all integers. 4. **Explanation:** This notation reads as "the set of all $x$ such that $x$ is an integer and $x$ is between 0 and 10 inclusive." This precisely captures the original set. **Final answer:** $$\{x \mid x \in \mathbb{Z} \text{ and } 0 \leq x \leq 10\}$$