ABSTRAK

Himpunan fuzzy merupakan pengembangan dari himpunan klasik, yakni fungsi karakteristiknya memiliki nilai diinterval 0 sampai 1. Perkembangan konsep himpunan tersebut menghasilkan dasar penelitian model aljabar fuzzy lainnya, seperti himpunan multi-fuzzy. Konsep subhimpunan multi-fuzzy merupakan perumuman dari teori himpunan fuzzy, yakni fungsi keanggotaannya merupakan himpunan fuzzy berdimensi k dan memiliki nilai diinterval 0 sampai 1. Kemudian perpaduan subhimpunan multi-fuzzy dengan struktur grup, menghasilkan konsep subgrup multi-fuzzy dengan levelnya. Tujuan penelitian ini adalah membuktikan sifat elementer dari subgrup multi-fuzzy, membuktikan syarat perlu dan syarat cukup suatu subhimpunan multi-fuzzy merupakan subgrup multi-fuzzy, membuktikan sifat-sifat subgrup multi-fuzzy terkait operasi irisan dan operasi gabungan yang berlaku pada subgrup multi-fuzzy, dan membuktikan hubungan antara multi-level dan subgrup multi-level darisubgrup multi-fuzzy dengan grupnya. Prosedur dari penelitian ini yaitu mengkaji definisi dan teorema dari himpunan, grup, subgrup, himpunan fuzzy dengan levelnya,subgrup fuzzy, subhimpunan multi-fuzzy dengan levelnya, serta membuktikan teorema terkait subgrup multi-fuzzy dengan levelnya. Hasil penelitian ini adalah terbuktinya sifat elementer dari subgrup multi-fuzzy, terbuktinya syarat perlu dan syarat cukup suatu subhimpunan multi-fuzzy merupakan subgrup multi-fuzzy, irisan dua subgrup multi-fuzzy merupakan subgrup multi-fuzzy, gabungan dari dua subgrup multi-fuzzy belum tentu merupakan subgrup multi-fuzzy, dan terbukti hubungan antara multi-level dan subgrup multi-level darisubgrup multi-fuzzy dengan grupnya.  

Kata kunci: Grup, Himpunan Fuzzy,Subhimpunan Multi-Fuzzy, Subgrup Multi-Fuzzy, Multi-Level.

ABSTRACT

Fuzzy subhimpunans is an extension of classical sets, where its characteristic function has values in the interval from 0 to 1. The development of this subhimpunan concept leads to the foundation of further research models in fuzzy algebra, such as multi-fuzzy subhimpunans. The concept of multi-fuzzy subhimpunans is a generalization of fuzzy subhimpunan theory, where its membership function is a fuzzy subhimpunans with dimension k and has values in the interval from 0 to 1. The combination of multi-fuzzy subhimpunans with group structures results in the concept of multi-fuzzy subgroups with their levels. The aim of this research is to prove the elementary properties of multi-fuzzy subgroups, to establish the necessary and sufficient conditions for a multi-fuzzy subhimpunan to be a multi-fuzzy subgroup, to prove the properties of multi-fuzzy subgroups related to intersection and union operations that apply to multi-fuzzy subgroups, and to prove the relationship between multi-level and multi-level subgroup of multi-fuzzy subgroup and its group. The procedure of this research involves studying the definitions and theorems of sets, groups, subgroups, fuzzy subhimpunans with their levels, fuzzy subgroups, multi-fuzzy subhimpunans with their levels, and proving the theorems related to multi-fuzzy subgroups with their levels. The results of this research include the proof of the elementary properties of multi-fuzzy subgroups, proven necessary and sufficient conditions for a multi-fuzzy subhimpunan to be a multi-fuzzy subgroup, the intersection of two multi-fuzzy subgroups is a multi-fuzzy subgroup, the union of two multi-fuzzy subgroups is not necessarily a multi-fuzzy subgroup, and the relationship between multi-level and multi-level subgroups of multi-fuzzy subgroups and their groups is proven.

Keywords: Group, Fuzzy Subhimpunans,Multi-Fuzzy Subhimpunans, Multi-Fuzzy Subgroup, Multi-Level.