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離散數(shù)學(xué)及應(yīng)用 讀者對象:本書可作為普通本科院校和應(yīng)用型本科院校計算機(jī)科學(xué)與技術(shù)、軟件工程、智能科學(xué)與技術(shù)、物聯(lián)網(wǎng)工程、網(wǎng)絡(luò)工程等專業(yè)離散數(shù)學(xué)課程的教材。
本書內(nèi)容分為數(shù)理邏輯、集合論、代數(shù)結(jié)構(gòu)、圖論等四個主要部分,包含命題邏輯、一階謂詞邏輯、集合、二元關(guān)系、函數(shù)、代數(shù)結(jié)構(gòu)、特殊代數(shù)系統(tǒng)、圖、特殊圖及圖的應(yīng)用、樹等10個章節(jié)。本書以應(yīng)用型人才培養(yǎng)為目標(biāo),突出離散數(shù)學(xué)作為計算機(jī)及相關(guān)本科的專業(yè)基礎(chǔ)課這一定位,本書可作為計算機(jī)科學(xué)與技術(shù)、軟件工程、智能科學(xué)與技術(shù)、物聯(lián)網(wǎng)工程、網(wǎng)絡(luò)工程等專業(yè)離散數(shù)學(xué)課程的教材。
黃瑋,男,1978年生人,副教授,在讀博士,現(xiàn)任教于福州理工學(xué)院,具有知名企業(yè)的從業(yè)經(jīng)歷,曾獲"微軟金牌講師”等頭銜,長期從事軟件工程、虛擬現(xiàn)實研發(fā)與教學(xué)工作,主持和完成過各類省級、市級科研項目5項,獲軟件著作權(quán)14項,發(fā)表科研論文9篇。
目 錄
第 1 章 命題邏輯 ····································································································.1 1.1 命題與聯(lián)結(jié)詞 ·····························································································.1 1.2 命題公式及其賦值 ······················································································.5 1.3 等值式 ·······································································································.8 1.4 析取范式與合取范式 ·················································································.10 1.4.1 范式的基本概念 ··············································································.10 1.4.2 主范式及其表示 ··············································································.12 1.4.3 主范式的求解 ··················································································.13 1.4.4 主范式的數(shù)量 ··················································································.15 1.5 聯(lián)結(jié)詞的完備集 ························································································.16 1.5.1 真值函數(shù) ·························································································.16 1.5.2 聯(lián)結(jié)詞完備集 ··················································································.17 1.6 命題邏輯的推理理論 ·················································································.18 1.6.1 推理的基本概念 ··············································································.18 1.6.2 推理定律和推理規(guī)則 ·······································································.19 1.6.3 命題邏輯的推理方法 ·······································································.19 1.7 命題邏輯的應(yīng)用 ························································································.23 1.7.1 命題公式在計算機(jī)中的表示 ·····························································.23 1.7.2 程序優(yōu)化 ·························································································.24 1.7.3 資源分配 ·························································································.24 1.7.4 計算機(jī)硬件電路設(shè)計 ·······································································.25 1.8 命題邏輯實驗 ····························································································.25 習(xí)題 1 ···············································································································.27 第 2 章 一階邏輯 ···································································································.30 2.1 一階邏輯的基本概念 ·················································································.30 2.2 一階邏輯公式及其解釋 ··············································································.33 2.2.1 一階邏輯公式的符號 ·······································································.33 2.2.2 一階邏輯公式的結(jié)構(gòu) ·······································································.34 2.2.3 量詞的轄域 ·····················································································.34 2.2.4 公式的解釋 ·····················································································.35 2.3 一階邏輯等值式及演算 ··············································································.37 2.4 一階邏輯前束范式 ·····················································································.38 2.5 一階邏輯的推理理論 ·················································································.39 2.6 一階邏輯實驗 ····························································································.43 習(xí)題 2 ···············································································································.43 第 3 章 集合論 ·······································································································.47 3.1 集合的基本概念 ························································································.47 3.2 集合的基本運(yùn)算 ························································································.50 3.3 有限集的計數(shù) ····························································································.54 3.4 集合論實驗 ·······························································································.55 習(xí)題 3 ···············································································································.57 第 4 章 二元關(guān)系 ···································································································.60 4.1 有序?qū)εc笛卡兒積 ·····················································································.60 4.2 二元關(guān)系的定義 ························································································.62 4.3 關(guān)系的運(yùn)算 ·······························································································.63 4.4 關(guān)系的性質(zhì) ·······························································································.67 4.4.1 性質(zhì)的定義 ·····················································································.67 4.4.2 性質(zhì)的判定 ·····················································································.70 4.5 關(guān)系的閉包 ·······························································································.72 4.5.1 閉包的基本概念 ··············································································.72 4.5.2 閉包的性質(zhì) ·····················································································.74 4.5.3 用 WarShall 算法求傳遞閉包 ····························································.74 4.6 等價關(guān)系 ···································································································.76 4.6.1 等價關(guān)系的概念 ··············································································.76 4.6.2 等價類 ····························································································.76 4.6.3 商集與劃分 ·····················································································.77 4.7 偏序關(guān)系 ···································································································.79 4.7.1 偏序關(guān)系的概念 ··············································································.79 4.7.2 哈斯圖 ····························································································.79 4.7.3 偏序集中的特殊元素 ·······································································.80 4.7.4 偏序關(guān)系的應(yīng)用 ··············································································.81 4.8 二元關(guān)系實驗 ····························································································.83 習(xí)題 4 ···············································································································.84 第 5 章 函數(shù) ··········································································································.86 5.1 函數(shù)的定義與性質(zhì) ·····················································································.86 5.1.1 函數(shù)的定義 ·····················································································.86 5.1.2 函數(shù)的性質(zhì) ·····················································································.88 5.2 函數(shù)的復(fù)合與反函數(shù) ·················································································.90 5.2.1 函數(shù)的復(fù)合 ·····················································································.90 5.2.2 反函數(shù) ····························································································.91 5.3 函數(shù)實驗 ···································································································.92 習(xí)題 5 ···············································································································.92 第 6 章 代數(shù)結(jié)構(gòu) ···································································································.94 6.1 運(yùn)算及其性質(zhì) ····························································································.94 6.1.1 代數(shù)的構(gòu)成 ·····················································································.94 6.1.2 二元運(yùn)算的性質(zhì) ··············································································.97 6.2 代數(shù)系統(tǒng) ···································································································.99 6.3 半群與群 ·································································································.102 6.4 子群 ········································································································.104 6.5 代數(shù)結(jié)構(gòu)實驗 ··························································································.107 習(xí)題 6 ·············································································································.108 第 7 章 特殊代數(shù)系統(tǒng) ··························································································.111 7.1 環(huán)與域 ····································································································.111 7.2 格的定義與性質(zhì) ······················································································.113 7.2.1 格的定義 ·······················································································.113 7.2.2 格的性質(zhì) ·······················································································.114 7.2.3 特殊的格 ·······················································································.115 7.3 布爾代數(shù) ·································································································.116 7.4 布爾代數(shù)的應(yīng)用 ······················································································.118 7.4.1 邏輯門電路 ···················································································.118 7.4.2 電路的極小化 ················································································.118 7.5 特殊代數(shù)系統(tǒng)實驗 ···················································································.119 習(xí)題 7 ·············································································································.120 第 8 章 圖 ············································································································.123 8.1 圖的基本概念 ··························································································.123 8.1.1 圖的概念 ·······················································································.123 8.1.2 握手定理 ·······················································································.125 8.1.3 圖的同構(gòu) ·······················································································.127 8.1.4 圖的運(yùn)算 ·······················································································.128 8.1.5 子圖與補(bǔ)圖 ···················································································.128 8.2 通路與回路 ·····························································································.129 8.3 圖的連通性 ·····························································································.130 8.3.1 無向圖的連通性 ············································································.130 8.3.2 有向圖的連通性 ············································································.132 8.4 圖的矩陣表示 ··························································································.133 8.5 圖的實驗 ·································································································.135 習(xí)題 8 ·············································································································.137 第 9 章 特殊圖及其應(yīng)用 ·······················································································.139 9.1 歐拉圖 ····································································································.139 9.1.1 歐拉圖的概念 ················································································.139 9.1.2 歐拉圖的判別法 ············································································.140 9.2 漢密爾頓圖 ·····························································································.141 9.3 平面圖 ····································································································.143 9.3.1 平面圖的基本定義 ·········································································.143 9.3.2 歐拉公式 ·······················································································.145 9.3.3 歐拉公式的推廣 ············································································.146 9.3.4 平面圖的判斷 ················································································.147 9.4 圖的應(yīng)用 ·································································································.148 9.4.1 最短路徑問題 ················································································.148 9.4.2 中國郵遞員問題 ············································································.150 9.5 特殊圖實驗 ·····························································································.152 習(xí)題 9 ·············································································································.153 第 10 章 樹 ··········································································································.155 10.1 無向樹 ··································································································.155 10.2 生成樹及應(yīng)用 ························································································.157 10.2.1 生成樹的基本概念 ·······································································.157 10.2.2 最小生成樹 ··················································································.158 10.3 根樹及應(yīng)用 ···························································································.161 10.3.1 根樹的相關(guān)概念 ················································································.161 10.3.2 最優(yōu)二叉樹 ······················································································.162 10.3.3 樹的遍歷 ·························································································.165 10.4 樹的實驗 ·······························································································.166 習(xí)題 10 ···········································································································.167 模擬卷 A ················································································································.169 模擬卷 B ················································································································.172 參考文獻(xiàn) ···············································································································.176
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