1. The problem asks: What are the 3 brackets for in the set $B = \{\{\{1, 4, 5, 3, 1\}\}\}$? 2. Each pair of brackets represents a level of set containment. 3. The innermost brackets $\{1, 4, 5, 3, 1\}$ define a set containing the elements 1, 4, 5, 3, and 1 (duplicates ignored, so effectively $\{1, 3, 4, 5\}$). 4. The next pair of brackets $\{\{1, 4, 5, 3, 1\}\}$ means a set whose only element is the inner set $\{1, 3, 4, 5\}$. 5. The outermost brackets $\{\{\{1, 4, 5, 3, 1\}\}\}$ mean a set whose only element is the set $\{\{1, 3, 4, 5\}\}$. 6. So, the three brackets indicate three levels: a set containing a set containing a set of numbers. 7. This is important in set theory to distinguish between elements and subsets. Hence, the 3 brackets show nested sets, each bracket pair representing one level of containment.