https://suchaharcan.github.io/ 月更 技术博客 CXX优先 Zola en Tue, 11 Aug 2026 00:00:00 +0000 Tue, 11 Aug 2026 00:00:00 +0000 Suchaharcan https://suchaharcan.github.io/00Read_blog_Math/math-discrete/ https://suchaharcan.github.io/00Read_blog_Math/math-discrete/ <h1 id="chi-san-shu-xue-he-xin-zhang-jie">离散数学 · 核心章节</h1> <blockquote> <p>计算机科学的基础数学：<strong>逻辑 → 集合 → 关系/函数 → 图论 → 代数系统 → 组合计数</strong>。</p> </blockquote> <hr /> <h2 id="1-shu-li-luo-ji">1. 数理逻辑</h2> <h3 id="ming-ti-luo-ji">命题逻辑</h3> <table><thead><tr><th>联结词</th><th>记号</th><th>读法</th></tr></thead><tbody> <tr><td>否定</td><td>¬p</td><td>非 p</td></tr> <tr><td>合取</td><td>p∧q</td><td>p 且 q</td></tr> <tr><td>析取</td><td>p∨q</td><td>p 或 q</td></tr> <tr><td>蕴含</td><td>p→q</td><td>若 p 则 q</td></tr> <tr><td>等价</td><td>p↔q</td><td>p 当且仅当 q</td></tr> </tbody></table> <p><strong>注意</strong>：<code>p→q</code> 只有 <code>p 真 q 假</code> 时为假；<code>p→q ≡ ¬p∨q</code>。</p> <h3 id="deng-zhi-shi-chang-yong">等值式（常用）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>双重否定：¬¬p ≡ p</span></span> <span class="giallo-l"><span>德摩根：¬(p∧q) ≡ ¬p∨¬q；¬(p∨q) ≡ ¬p∧¬q</span></span> <span class="giallo-l"><span>吸收律：p∨(p∧q) ≡ p</span></span> <span class="giallo-l"><span>蕴涵等值式：p→q ≡ ¬p∨q</span></span></code></pre><h3 id="tui-li-gui-ze">推理规则</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>假言推理（MP）：p→q，p ⇒ q</span></span> <span class="giallo-l"><span>拒取式（MT）：p→q，¬q ⇒ ¬p</span></span> <span class="giallo-l"><span>析取三段论：p∨q，¬p ⇒ q</span></span> <span class="giallo-l"><span>假言三段论：p→q，q→r ⇒ p→r</span></span></code></pre><h3 id="wei-ci-luo-ji">谓词逻辑</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∀x P(x)：全称量词；∃x P(x)：存在量词</span></span> <span class="giallo-l"><span>¬∀x P(x) ≡ ∃x ¬P(x)；¬∃x P(x) ≡ ∀x ¬P(x)</span></span></code></pre> <hr /> <h2 id="2-ji-he-lun">2. 集合论</h2> <h3 id="yun-suan">运算</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>并 A∪B；交 A∩B；差 A−B；补 ∁A</span></span> <span class="giallo-l"><span>对称差 A⊕B = (A−B)∪(B−A)</span></span></code></pre><h3 id="heng-deng-shi">恒等式</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>德摩根：∁(A∩B) = ∁A∪∁B；∁(A∪B) = ∁A∩∁B</span></span> <span class="giallo-l"><span>A∪(B∩C) = (A∪B)∩(A∪C)（分配律）</span></span></code></pre><h3 id="ji-shu">基数</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>|A∪B| = |A|+|B|−|A∩B|</span></span> <span class="giallo-l"><span>|A∪B∪C| = |A|+|B|+|C|−|A∩B|−|A∩C|−|B∩C|+|A∩B∩C|（容斥）</span></span></code></pre> <hr /> <h2 id="3-guan-xi">3. 关系</h2> <h3 id="er-yuan-guan-xi-xing-zhi">二元关系性质</h3> <table><thead><tr><th>性质</th><th>定义</th></tr></thead><tbody> <tr><td>自反</td><td>所有 x，(x,x)∈R</td></tr> <tr><td>反自反</td><td>所有 x，(x,x)∉R</td></tr> <tr><td>对称</td><td>(x,y)∈R ⇒ (y,x)∈R</td></tr> <tr><td>反对称</td><td>(x,y)∈R 且 (y,x)∈R ⇒ x=y</td></tr> <tr><td>传递</td><td>(x,y),(y,z)∈R ⇒ (x,z)∈R</td></tr> </tbody></table> <h3 id="deng-jia-guan-xi-yu-hua-fen">等价关系与划分</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>自反 + 对称 + 传递 = 等价关系</span></span> <span class="giallo-l"><span>等价类 [x] = {y | xRy}；等价类集合构成集合的划分</span></span></code></pre><h3 id="pian-xu-guan-xi">偏序关系</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>自反 + 反对称 + 传递 = 偏序（≤）</span></span> <span class="giallo-l"><span>哈斯图（Hasse 图）画法：去掉自反环与传递边</span></span> <span class="giallo-l"><span>极大/极小元、最大/最小元、上/下界、上/下确界</span></span></code></pre><h3 id="guan-xi-bi-bao">关系闭包</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>自反闭包 r(R) = R ∪ E</span></span> <span class="giallo-l"><span>对称闭包 s(R) = R ∪ R⁻¹</span></span> <span class="giallo-l"><span>传递闭包 t(R) = R ∪ R² ∪ R³ ∪ …（Warshall 算法）</span></span></code></pre> <hr /> <h2 id="4-han-shu">4. 函数</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>f: A → B</span></span> <span class="giallo-l"><span>单射（一一）：不同自变量 → 不同函数值</span></span> <span class="giallo-l"><span>满射（到上）：值域 = B</span></span> <span class="giallo-l"><span>双射：既单又满</span></span></code></pre> <p><strong>鸽巢原理</strong>：<code>n+1</code> 个物体放入 <code>n</code> 个盒子，至少一个盒子 ≥ 2 个。</p> <hr /> <h2 id="5-tu-lun">5. 图论</h2> <h3 id="ji-ben-gai-nian">基本概念</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>G = (V, E)；无向图 / 有向图</span></span> <span class="giallo-l"><span>握手定理：Σdeg(v) = 2|E|（无向）</span></span> <span class="giallo-l"><span>有向图：Σdeg⁺ = Σdeg⁻ = |E|</span></span></code></pre><h3 id="te-shu-tu">特殊图</h3> <table><thead><tr><th>图</th><th>判定</th></tr></thead><tbody> <tr><td>完全图 Kₙ</td><td>n(n−1)/2 条边</td></tr> <tr><td>二部图</td><td>顶点可二分，边只跨部（无奇圈）</td></tr> <tr><td>欧拉图</td><td>所有顶点度数为偶（有向：出入相等）</td></tr> <tr><td>哈密顿图</td><td>必要条件（割点/度条件），充分条件 Dirac：<code>deg ≥ n/2</code></td></tr> </tbody></table> <h3 id="shu">树</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>n 个顶点的树：n−1 条边，连通无圈</span></span> <span class="giallo-l"><span>生成树：连通图包含全部顶点的树</span></span> <span class="giallo-l"><span>最小生成树：Prim（加点）/ Kruskal（加边，并查集判环）</span></span></code></pre><h3 id="ping-mian-tu">平面图</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>欧拉公式：V − E + F = 2（连通平面图）</span></span> <span class="giallo-l"><span>K₅ 与 K₃,₃ 是非平面图</span></span></code></pre><h3 id="tu-zhao-se">图着色</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>四色定理：平面图 4 色足够</span></span> <span class="giallo-l"><span>图着色 = 相邻顶点不同色，最少色数 χ(G)</span></span></code></pre> <hr /> <h2 id="6-dai-shu-xi-tong">6. 代数系统</h2> <table><thead><tr><th>系统</th><th>公理</th><th>例子</th></tr></thead><tbody> <tr><td>半群</td><td>封闭 + 结合</td><td>(N,+)</td></tr> <tr><td>独异点</td><td>半群 + 单位元</td><td>(N,·)</td></tr> <tr><td>群</td><td>独异点 + 逆元</td><td>(Z,+)、(Zₙ,+)</td></tr> <tr><td>阿贝尔群</td><td>群 + 交换</td><td>(Z,+)</td></tr> <tr><td>环</td><td>(R,+) 阿贝尔群 + 乘法结合/分配</td><td>(Z,+,·)</td></tr> <tr><td>域</td><td>环 + 乘法可逆（除零外）</td><td>(R,+,·)、(Zₚ,+,·) p 素数</td></tr> </tbody></table> <p><strong>拉格朗日定理</strong>：有限群中子群的阶整除群的阶。</p> <hr /> <h2 id="7-zu-he-ji-shu">7. 组合计数</h2> <h3 id="pai-lie-yu-zu-he">排列与组合</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>P(n,r) = n!/(n−r)!</span></span> <span class="giallo-l"><span>C(n,r) = n!/[r!(n−r)!]</span></span></code></pre><h3 id="chang-yong-heng-deng-shi">常用恒等式</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>C(n,r) = C(n, n−r)</span></span> <span class="giallo-l"><span>C(n+1,r) = C(n,r) + C(n,r−1)（帕斯卡）</span></span> <span class="giallo-l"><span>Σ C(n,k) = 2ⁿ</span></span></code></pre><h3 id="ji-shu-ji-qiao">计数技巧</h3> <ul> <li>隔板法：<code>n</code> 个相同物分 <code>k</code> 组非空：<code>C(n−1, k−1)</code>；</li> <li>错排数：<code>Dₙ = n!·Σ(−1)^k/k!</code>，如 <code>D₃=2, D₄=9</code>；</li> <li>生成函数 / 母函数处理受限组合。</li> </ul> <hr /> <h2 id="8-kao-yan-ji-shi-chang-kao-ti-xing">8. 考研 / 机试常考题型</h2> <ol> <li>命题公式等值演算与主范式；</li> <li>关系性质判断、等价类与划分；</li> <li>哈斯图与偏序元；</li> <li>图论：握手定理、欧拉/哈密顿判定、着色；</li> <li>最小生成树（Prim/Kruskal）与最短路（Dijkstra/Floyd）；</li> <li>群判定与子群；</li> <li>排列组合与容斥、错排。</li> </ol> <h3 id="li-ti-su-lan">例题速览</h3> <p><strong>例 1</strong>：证明 <code>(p→q) ∧ p ⇒ q</code>（假言推理）：</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>p→q ≡ ¬p∨q；(¬p∨q)∧p ≡ q∧p ⇒ q</span></span></code></pre> <p><strong>例 2</strong>：<code>K₅</code> 的边数：</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>5·4/2 = 10</span></span></code></pre> <p><strong>例 3</strong>：4 封信全装错信封的方法数：</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>D₄ = 4!·(1 − 1 + 1/2 − 1/6 + 1/24) = 9</span></span></code></pre> <hr /> <h2 id="9-yi-cuo-dian-qing-dan">9. 易错点清单</h2> <ul> <li>蕴含联结词 <code>→</code> 与日常“因果“含义不同；</li> <li>自反/反自反、对称/反对称不互斥；</li> <li>握手定理两边都算；</li> <li>欧拉图关注“一笔画“，哈密顿关注“回路遍历所有顶点“；</li> <li>群必须封闭、结合、有单位元、有逆元四条件齐备；</li> <li>错排与隔板法别套错模型。</li> </ul> <blockquote> <p>上一篇：<a href="https://suchaharcan.github.io/00Read_blog_Math/math-probability/">概率论</a></p> </blockquote>