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Some notation

In order to introduce some concepts in the following chapters, we now give some definitions.

Definition 7   The cardinality of an unidimensional variable $X$ is defined as follows:

$$Card(X) = \left\{ \begin{array}{ll} (max-min) & \mbox{ if } X=[min,max] \\ ... ... } X=<v_0,v_1,...,v_{m-1}> \\ k & \mbox{ if } X=[min:max):k \end{array}\right.$$ (2.1)

Definition 8   The order of a variable set of unidimensional variables $\Omega=\{X_1,X_2,...,X_n\}$ is denoted as $O(\Omega)$ and it is defined as $O(\Omega)= Card(X_1)*Card(X_2)...*Card(X_n)$.

Definition 9   Let $x$ be a value of $X$ then $index(x)$ denotes the position of $x$ on $X$. Particularly we have that

$$index(x) = \left\{ \begin{array}{ll} (x-min) & \mbox{ if } X=[min,max] \\ i... ...i\\ j & \mbox{ if } X=[min:max):k \mbox{ and } x\in Ai\in X \end{array}\right.$$ (2.2)