4.1. Theory of Training Let U represent the universe, a finite set of objects, and A denotes a set of condition attributes. For x, y ∈ U, Wortmannin clinical trial we say that x and y are indiscernible

by the set of condition attributes A if ρ(x, q) = ρ(y, q) for every q ∈ A where ρ(x, q) denotes the information function. A set that has objects within it that are indiscernible by the set of condition attributes A is called elementary set. The family of all elementary sets is denoted by A*. It represents the smallest partitions of objects by the specified condition attributes so that objects belonging to different elementary sets are discernible and those belonging to the same elementary sets are indiscernible. The lower approximation of X (X⊆U), denoted by A_X, and the upper approximation of X, denoted by A¯X, are defined as A_X=∪P P∈A∗,P⊆X,A¯X=∪P P∈A∗,P∩X≠∅. (1) The lower approximation contains all objects that certainly belong to that category. The upper approximation consists of all objects that possibly belong to that category. A rough set is thus any subset defined through its lower and upper approximation. Figure 1 is a graphical representation of this concept.

Each indiscernible set is displayed by a pixel. The subset of objects we want to approximate is drawn as a dashed line that crosses pixel boundaries and cannot be defined in a crisp manner. The lower and upper approximations are drawn as thick gridlines. Figure 1 Approximation of sets. For example, five mode choice cases, described with four attributes, age,

car ownership, purpose, and mode choice, are given in Table 2. Table 2 Examples of mode choice cases with describing features. Mode choice case 1, for instance, is characterized by the following statement: IF (age = young) AND (car ownership = yes) AND (purpose = work) THEN (mode choice = bus). The above statement is called a rule in rough sets theory. The attributes in “THEN” part are called decision attribute which is the concept of concern, and attributes in “IF” part are called condition attributes which are the information we observe. The three condition attributes, age, car ownership, and purpose, form four elementary sets: 1,3, 2, 4, 5. It represents that cases 1 and 3 are indiscernible while other cases are characterized uniquely with condition attributes. Since cases 1 and 3 are indiscernible Cilengitide and lead to different mode choices, they are called boundary-line cases representing those that cannot be properly classified with the available information. Therefore, the bus mode choice is described with the lower approximation set, 2, and the upper approximation set, 1,2, 3. Similarly, the concept of car mode choice is characterized with its lower approximation set, 4,5, and upper approximation set, 1,3, 4,5.