1. **Problem statement:** Find the supremum, infimum, maximal, and minimum elements of the set \(S = \{r \in \mathbb{Q} : 0 \leq r \leq \sqrt{2} \}\). 2. **Recall definitions:** - The **supremum** (least upper bound) of a set is the smallest number that is greater than or equal to every element in the set. - The **infimum** (greatest lower bound) is the largest number less than or equal to every element in the set. - A **maximum** is the greatest element in the set. - A **minimum** is the smallest element in the set. 3. **Analyze the set:** - The set consists of all rational numbers \(r\) such that \(0 \leq r \leq \sqrt{2}\). - Note that \(\sqrt{2}\) is irrational, so it is not in \(\mathbb{Q}\). 4. **Find infimum and minimum:** - The smallest element in the set is \(0\), which is rational and included. - So, \(\inf S = 0\) and \(\min S = 0\). 5. **Find supremum and maximum:** - Since \(\sqrt{2}\) is irrational, it is not in the set. - The set contains rationals arbitrarily close to \(\sqrt{2}\) from below. - Therefore, \(\sup S = \sqrt{2}\), but there is no maximum element because no rational equals \(\sqrt{2}\). **Final answers:** - \(\inf S = 0\) - \(\min S = 0\) - \(\sup S = \sqrt{2}\) - No maximum element exists in \(S\).