# Soft Sets Extensions: Innovating Healthcare Claims Analysis
## 4 Soft Sets Extensions
### TreeSoft Set  
A TreeSoft Set introduces a structured framework for modeling uncertain or imprecise information, where each attribute is organized in a hierarchical tree-like structure, associating each node with a set of potential elements from the universe of discourse. This hierarchical approach enables the systematic representation and manipulation of uncertain data, facilitating various computational tasks such as decision-making, pattern recognition, and data analysis with a focus on hierarchical relationships and dependencies. Definition The TreeSoft Set is an innovative extension that introduces a hierarchical structure to soft sets, providing a comprehensive framework for modeling complex systems with multiple levels of attributes. Here is a refined explanation: We begin with a universe of discourse, denoted as \(U\), and a non-empty subset \(H\) of \(U\), along with its powerset, \(P(H)\), which encompasses all possible subsets of \(H\). Next, we define a set of attributes, denoted as \(A\), which consists of parameters, factors, and other relevant characteristics. This set is organized hierarchically into levels: first-level attributes \(A\) = {\(A_{1}\), \(A_{2}\), \(\ldots\), \(A_{n}\)}, for integer \(n\geq 1\), where \(A_{1}\), \(A_{2}\), \(\ldots\), \(A_{n}\) are considered attributes of first level (since they have one-digit indexes). Each attribute \(A_{i}\), \(1\leq i\leq n\), is formed by sub-attributes: \[A_{1} = \{A_{1,1},A_{1,2},\ldots\}\] \[A_{2} = \{A_{2,1},A_{2,2},\ldots\}\] \[................................\] \[A_{n} = \{A_{n,1},A_{n,2},\ldots\}\] where the above \(A_{i;j}\) are sub-attributes (or attributes of second level) (since they have two-digit indexes). Again, each sub-attribute \(A_{ij}\) is formed by sub-sub-attributes (or attributes of third level): \[A_{ijk}\]And so on, with as much refinement as needed going into each application, up to sub-sub-...-sub-attributes (or attributes of \(m\)-level (or having \(m\) digits into the indexes):  
\[A_{i1,i2},\ldots,_{im}\]  
This hierarchical structure forms a graph-tree, denoted as \(\mathit{Tree}(A)\), with \(A\) as the root node (level zero), followed by nodes at levels 1 to \(m\), where \(m\) represents the maximum level of refinement. The leaves of this graph-tree are terminal nodes that have no descendants. The TreeSoft Set, denoted as  
\[\mathit{F:P(Tree(A))\to P(H)},\]  
maps subsets of the graph-tree \(\mathrm{Tree}(A)\) to subsets of H. The powerset P(Tree(A)) encompasses all possible subsets of the graph-tree. All node sets of the TreeSoft Set of level m are  
\[\mathit{Tree}(A)=\{A_{i1}\,|_{i1}=1,2,\ldots\}\]  
The sets within the TreeSoft Set correspond to nodes at each level of the graph-tree: the first set consists of nodes at level 1, the second set consists of nodes at level 2, and so on, up to the last set comprising nodes at level \(m\). If the graph-tree has only two levels (\(m=2\)), then the TreeSoft Set simplifies to a MultiSoft Set [7]. In summary, the TreeSoft Set provides a structured approach for representing and analyzing complex systems with hierarchical attributes. By incorporating a hierarchical organization, it enhances the flexibility and expressiveness of soft set-based methodologies, enabling more nuanced modeling and analysis of multi-level systems across various domains. An illustrative example of a classical tree is shown in Figure 2.  
This tree contains three levels as followed:  
Level 0 (the root) is the node Attributes;  
Level 1 is formed by the nodes: Diagnosis, Treatment;  
Level 2 is formed by the nodes Diabetes, Cancer, Medication, and Surgery;  
Level 3 is formed by the nodes Pills, Injections. Let us consider \(\mathrm{p}=\{patient_{1},patient_{2},\ldots,patient_{10}\}\) to be a set of patients, and \(P(p)\) to be the power set of \(p\). Figure 2: Schematic representation of a TreeSoft Set of level 3 framework, illustrating the incorporation of hyperparameters to capture complex relationships within healthcare claims datasets.