Approximation Properties on a Set Based on Equivalence Relations and Dominance Relations

Dian Winda Setyawati (Department of Mathematics, Faculty of Science and Data Analytics, Sepuluh Nopember Institute of Technology, Surabaya)
Soleha Soleha (Department of Mathematics, Faculty of Science and Data Analytics, Sepuluh Nopember Institute of Technology, Surabaya)
Rinurwati Rinurwati (Department of Mathematics, Faculty of Science and Data Analytics, Sepuluh Nopember Institute of Technology, Surabaya)

CAUCHY: Jurnal Matematika Murni dan Aplikasi
Vol 11, No 1 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI

An equivalence relation on a set forms equivalence classes so that the concept of approximation is formed on that set (rough set). The concept of approximation on a set is developing very rapidly. Some papers replace the equivalence relation with other relations, one of which is the dominance relation. This paper provides the approximation properties that hold to the equivalence and dominance relations using the concept of three types of approximations defined on a set. In addition, it identifies the properties that hold in the equivalence relations but do not necessarily hold in the dominance relations. Furthermore, this paper proves that the three types of approximations on a set w.r.t an equivalence relation are identical, but this result does not necessarily hold for a dominance relation, especially for the third type.