模糊集理論因未能滿足傳統邏輯定律，且基本運算的選擇缺乏準則，其理論基礎常受到質疑。採用機率結構可使得模糊集合遵守邏輯定律，模糊集合的交集、聯集運算可依照模糊集合彼此關係而決定，不再只是採用某組特定運算。過去已有對於一維模糊集合的機率解釋之研究，但二維模糊集合之機率解釋仍付之闕如。本文提出將二維模糊集合視為擴大樣本空間中事件的方法，並說明模糊關係矩陣(即二維模糊集合之歸屬函數)如何對應到六種條件機率，並分別討論其特性。應用實例較為常見的一種模糊關係，為明確函數的延伸。我們討論其在模糊推論系統中的應用方式，並指出其在模糊關係矩陣合併及模糊關係方程式求解之特性。

中文摘要i
英文摘要ii
誌謝iii
目錄iv
圖目錄vi
表目錄viii
符號說明ix
第一章 緒論1
1.1 研究動機與目的1
1.2 研究方法與成果2
1.3 章節提要3
第二章 集合理論中的集合與關係5
2.1 集合的基本概念5
2.2 關係的基本概念8
第三章 傳統模糊集理論中的模糊集合與模糊關係14
3.1 模糊集合的基本概念14
3.1.1 宇集14
3.1.2 模糊集合14
3.1.3 歸屬函數15
3.2 模糊關係的基本概念19
3.2.1 宇集19
3.2.2 二元模糊關係19
3.2.3 歸屬函數20
第四章 機率理論的基本概念24
4. 1 樣本空間24
4.2 事件25
4. 3 機率27
第五章 機率理論觀點中的一維模糊集合與二維模糊集合
(二元模糊關係) 31
5. 1 一維模糊集合的機率解釋31
5.1.1 擴大宇集/擴大樣本空間32
5.1.2 一維模糊集合視為擴大樣本空間中的事件36
5.1.3 一維模糊集合的歸屬函數視為條件機率37
5. 2 二維模糊集合(二元模糊關係)的機率解釋39
5.2.1 擴大宇集/擴大樣本空間40
5.2.2 二維模糊集合視為擴大樣本空間中的事件45
5.2.3 二維模糊集合的歸屬函數視為條件機率47
5.2.4 模糊關係的應用53
5.2.5 結論58
第六章 結論60
參考文獻62
附錄A 集合與proper class64
附錄B 以模糊集合形成表述空間65
附錄C Measure Space and Probability Space67

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