1. The problem asks to explain why the three brackets indicate nested sets and their importance in set theory. 2. In set theory, a single pair of brackets $\{ \}$ denotes a set containing elements. 3. When you have multiple pairs of brackets like $\{\{\{1, 4, 5, 3, 1\}\}\}$, each pair represents a new level of containment. 4. The innermost set $\{1, 4, 5, 3, 1\}$ is a set of numbers (duplicates ignored, so effectively $\{1, 3, 4, 5\}$). 5. The next level $\{\{1, 4, 5, 3, 1\}\}$ is a set whose only element is the inner set. 6. The outermost level $\{\{\{1, 4, 5, 3, 1\}\}\}$ is a set whose only element is the set containing the inner set. 7. This nesting is crucial because it distinguishes between an element and a subset. For example, $1$ is an element, but $\{1\}$ is a set containing that element. 8. Without these brackets, it would be unclear whether we are referring to elements directly or sets containing those elements. 9. Therefore, the three brackets show three levels of sets, helping us understand the structure and hierarchy in set theory clearly.