From f479d796807cadfddfc5e4e8981a07882350c8b8 Mon Sep 17 00:00:00 2001 From: JamesZhengYuyu <31480030+JamesZhengYuyu@users.noreply.github.com> Date: Sat, 22 Jun 2019 12:13:10 +0800 Subject: [PATCH] =?UTF-8?q?=E7=A6=BB=E6=95=A31/=E7=94=B5=E5=AD=90=E5=AD=A6?= =?UTF-8?q?=20(#16)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit * 离散数学(A) * 电子学基础2018-2019期末考试题 --- ...18\345\271\264\346\234\272\347\273\217.md" | 98 ++++++++++++++++++ ...8\345\271\264\346\234\272\347\273\217.pdf" | Bin 0 -> 213718 bytes ...3\350\200\203\350\257\225\351\242\230.zip" | Bin 0 -> 1577035 bytes 3 files changed, 98 insertions(+) create mode 100644 "\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/exam/\347\246\273\346\225\243\346\225\260\345\255\246 (A) 2018\345\271\264\346\234\272\347\273\217.md" create mode 100644 "\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/exam/\347\246\273\346\225\243\346\225\260\345\255\246 (A) 2018\345\271\264\346\234\272\347\273\217.pdf" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/exam/\346\234\237\346\234\253/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\2002018-2019\346\234\237\346\234\253\350\200\203\350\257\225\351\242\230.zip" diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/exam/\347\246\273\346\225\243\346\225\260\345\255\246 (A) 2018\345\271\264\346\234\272\347\273\217.md" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/exam/\347\246\273\346\225\243\346\225\260\345\255\246 (A) 2018\345\271\264\346\234\272\347\273\217.md" new file mode 100644 index 000000000..8ed8fd072 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/exam/\347\246\273\346\225\243\346\225\260\345\255\246 (A) 2018\345\271\264\346\234\272\347\273\217.md" @@ -0,0 +1,98 @@ +# 离散数学 (A) 2018年机经 + +by 相逢有时 @学习部 + +考试时间:2019年01月11日 + +## 总体情况 + +题量和16年差不多,难度比16年和样卷都要大得多。不仅考察了细枝末节的各类知识点(比如良序集、求闭包的顺序等,考点几乎没有死角),而且要求灵活运用所学知识,有较高思维难度,题目的综合性和交叉性很大。 + +考完基本上是跪了,所调查范围内的同学没有一位表示做完了题目,最后两题没做是常态,普遍表示整体难度都很大。(笔者也是在心态爆炸中艰难地写完本片机经的orz) + +某大佬的做题感言: + +> 五教气候温暖宜人,试卷印刷清晰,考试氛围良好,明年还会继续体验。 + +## 选择题 + +- 有一个关于命题的定义的,有点坑。答案似乎是选“含命题变项”的一项 + +- 有一个关于集合对称差的,简单 +- 有一个关于存在、任意两次的使用的,简单 +- 关于关系的题目,较难:设xRy表示x是y的父亲,xSy表示x是y的母亲。问以下哪种关系表示x是y的妻子? + 1. R o S^-1 + 2. S o R^-1 + 3. R^-1 o S + 4. S^-1 o S +- 关于集合运算的题目,选择错误的一项,较难。 +- 关于连续统假设和集合基数的问题,中等。 +- …… + + + +## 判断题 + +- 集合 A∈B 等价于 P(A) ∈ P(B) + +- 良序集不一定是全序集(虽然良序集好像不在考纲里,但他还是考了orzorz) +- …… + + + +## 填空题 + +- 波兰表达式:!P V (Q ^ !R) V !!Q(大概是) +- 基数运算题,简单。(如 ∩{23, 43} ) +- 给一个10元素的集合,给一种划分{{1,2,3,4}, {5,6,7}, {8,9}, {10}},问等价关系中有序对的个数。(30) +- 关系的计数,较难。问,对于元素个数为n的集合A,A上的以下关系分别有多少种?(用n表示) + - 自反关系 + - 反自反关系 + - 对称关系 + - 反对称关系 + - 非自反且非反自反关系 + - 自反并且对称的关系 + + + +## 大题 + +题量大,笔者写满了整个答卷orz。难度阶梯明显 + +- 有一个关于容斥原理,简单:给了三个集合A、B、C中A和B的元素个数,给了全集元素个数,给了ABC任意两个的交的元素个数,给了A交B交C的元素个数,叫你求C的元素个数。 + +- 有一个主合取、主析取范式的题目,较简单:命题很长,直接化简十分麻烦(含双箭头和很多单箭头),但是整个式子只含P、Q两个命题,所以可以直接打表(枚举4次就完了,很简单)或者写成极小式再析取。 + +- 有一个命题形式化的题目,第一个难度较大: + - 过平面上的不共线的三点,有且仅有一个圆。(注意三点不共线、有且仅有、三点共圆的形式化) + - 数列A(下标从0到n-1)是升序列。 + +- 证明: (A-B)对称差(A-C) = ∅ 等价于 A∩B = A∩C + +- 证明:左边是一个前束范式,右边是一个范式。简单。 + +- 哈斯图,难度较大。 + + 定义集合 $ A = \{2, 2^2, ..., 2^{12}\} $ 偏序关系 $ R = \{ | x \in \N_+ \and y \in \N_+ \and (\exist p) (p \in \N_+ \and y = x^p)\} ​$ + + 1. 求 $ R_1 = (A \times A) \cap R​$ 的哈斯图(建议表示成2的指数形式,这样容易发现指数之间是简单的整除关系) + 2. 求 $R_1$ 最小元、最大元、极小元、极大元。 + 3. 在 $ \N_+ $ 上求 $$ 的上下界、上下确界。(用最小公倍数和最大公约数来求解) + 4. 求所有最长链、最长反链(这个有点多,要仔细数) + + + +- 函数构造题,综合性、难度较大。(4分) + + 设 $\N \to \Z$ 的函数 $f(t) = a t^2 + c$ ,其中a、c都是可以调节的自然数。试构造 $\N \to \Z$ 的**不依赖于a、c的取值**的函数$g(t)$,使得对于**任意**的 $a, c \in \N$ ,始终存在 $t_0 \in \N$,使得 $f(t_0) = g(t_0)$ + + 例如,当c = 0时,可以构造 $g(t) = t^3​$,这样对于任意的a,可以取$t_0 = a​$,$f(t_0) = g(t_0) = a^3​$ + + 1. 设 $c \in \{0, 1\}​$,构造 g(t) + 2. 设 c 可以是任何自然数,构造 g(t) + + (思路大概是要构造 $\N \to \Z^2​$ 的双射,by耀神) + + + +- 用罗素公理系统证明 $ p \leftrightarrow p $ 是重言式。卷尾给了3条定义、4条公理、7条定理。较难。(5分) \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/exam/\347\246\273\346\225\243\346\225\260\345\255\246 (A) 2018\345\271\264\346\234\272\347\273\217.pdf" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/exam/\347\246\273\346\225\243\346\225\260\345\255\246 (A) 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