Generalized Rough Sets

This chapter concerns construction of a new rough set structure for an ideal ordered topological spaces and ordered topological filters. The approximation space approached depend on general binary relation, partially order relation, ideal and filter concepts. Properties of lower and upper approximation are extended to an ideal order topological approximation spaces. The main aim of the rough set theory is reducing the bouwndary region by increasing the lower approximation and decreasing the upper approximation. So, in this chapter different methods are proposed to reduce the boundary region. Comparisons between the current approximations and the previous approximations (El-Shafei et al.,2013) are introduced. It's therefore shown that the current approximations are more generally and reduce the boundary region by increasing the lower approximation and decreasing the upper approximation. The lower and upper approximations satisfy some properties in analogue of Pawlak's spaces (Pawlak, 1982). Moreover, we give several examples for comparison between the current approach and (El-Shafei et al., 2013).

Theoretical Evaluation of Ultrasonic Velocities in Binary Liquid Mixtures of Anisaldehyde with Some Alcoxyethanols at Different Temperatures

Ultrasonic velocities and densities of the binary liquid mixtures of anisaldehyde with alcoxyethanols like methoxyethanol (MOE), ethoxyethanol (EOE), and butoxyethanol (BOE) have been measured at temperatures 303.15 K, 308.15 K, 313.15 K, and 318.15 K over the entire composition range of mole fractions. The theoretical values of ultrasonic velocity were evaluated using Nomoto's relation (), impedance relation (), ideal mixing relation (), Jungie's relation (), and Rao’s specific velocity relation (). The molecular interaction parameter () has been evaluated from the values of experimental and theoretical velocities. The variation of this interaction parameter with the composition mixture has been discussed in terms of molecular interactions.