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Arxiv / №8 · 2026 / Maqola
01.00.00 – FIZIKA-MATEMATIKA FANLARI

MATEMATIK MANTIQ AMALLARINING FUZZY MANTIQDAGI MATEMATIK ASOSLARI VA AMALIY QOʻLLANILISHI

Annotatsiya
O'ZBEKCHA

Ushbu maqolada matematik mantiq amallarining fuzzy mantiqdagi matematik asoslari hamda ularning amaliy qo‘llanilishi tahlil qilingan. Klassik va fuzzy mantiqdagi AND, OR, NOT hamda implication amallarining matematik ifodalanishi qiyosiy o‘rganilgan. Tadqiqotda matematik modellashti-rish va qiyosiy tahlil usullaridan foydalanilgan. Natijada fuzzy mantiq amallarini T-norm va S-norm ope-ratorlari asosida ifodalashning o‘ziga xos xususiyatlari yoritilib, ularning noaniq ma’lumotlarni qayta ish-lashdagi afzalliklari asoslandi. Shuningdek, fuzzy mantiq amallarining sun’iy intellekt, ekspert tizimlari va avtomatik boshqaruv tizimlaridagi amaliy qo‘llanilish imkoniyatlari ko‘rsatildi.

Kalit so'zlar: matematik mantiq, fuzzy mantiq, fuzzy to‘plamlar, a’zolik funksiyasi, T-norm, S-norm, mantiqiy amallar, matematik interpretatsiya, matematik modellashtirish, amaliy qo‘llanilish.
РУССКИЙ

В данной статье рассмотрены математические основы логических операций в нечеткой логике и особенности их практического применения. Проведен сравнительный анализ ма-тематического представления логических операций AND, OR, NOT и импликации в классической и нечеткой логике. В исследовании использованы методы математического моделирования и сравни-тельного анализа. В результате раскрыты особенности представления логических операций в не-четкой логике на основе операторов T-норм и S-норм, а также обоснованы их преимущества при обработке неопределенной информации. Кроме того, показаны возможности практического при-менения логических операций нечеткой логики в системах искусственного интеллекта, экспертных системах и системах автоматического управления.

Ключевые слова: математическая логика, нечеткая логика, нечеткие множества, функция принадлежности, T-норма, S-норма, логические операции, математическая интерпретация, мате-матическое моделирование, практическое применение.
ENGLISH

This paper examines the mathematical foundations of logical operations in fuzzy logic and their practical applications. A comparative analysis of the mathematical representation of the AND, OR, NOT, and implication operations in classical and fuzzy logic is presented. The study employs mathematical modeling and comparative analysis methods. The results reveal the distinctive features of representing fuzzy logical operations based on T-norm and S-norm operators and demonstrate their advantages in processing uncertain information. Furthermore, the practical applications of fuzzy logical operations in artificial intelligence, expert systems, and automatic control systems are discussed.

Keywords: mathematical logic, fuzzy logic, fuzzy sets, membership function, T-norm, S-norm, logical operations, mathematical interpretation, mathematical modeling, practical application.
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