离散1知识点总结
约 4725 字大约 16 分钟
2026-05-12
本word文档是个人根据大连理工大学离散数学的PPT和课本进行的知识点总结,仅限参考,如有造成知识点空缺,本人概不负责。
该word文档的所有解释权归本人所有。
该word文档拒绝以盈利性质分享。
该word文档不会更新,所以有错误请自行更正并及时告知诸位。
命题逻辑
命题和联结词
1.1.1命题的概念
所谓命题,是指具有非真必假性质的陈述句
命题的判断结果称作命题的真值,真值只有两个取值------真(1)或假(0)
命题常用大写字母表示
一个命题不能再分解为更简单的命题,这个命 题称为原子命题。
由多个原子命题和联结词组成的命题成为复合命题
1.1.2联结词
{width="6.367547025371828in" height="2.244915791776028in"}
设P是一个命题,则P的否定是一个新的命题, 记作" ¬P",读作"非P"
{width="1.5000765529308837in" height="1.6250831146106737in"}
设P 和Q是命题,则用 PΛQ 表示 " P并且 Q"
{width="1.4861876640419946in" height="2.1320538057742784in"}
设P和Q是命题,则用 P∨Q表示命题"P或者Q"
{width="1.4861876640419946in" height="2.1181649168853895in"}
设P和Q是命题,则用P∇Q表示命题"P异或Q"
{width="1.4792432195975502in" height="2.15288823272091in"}
设P和Q是命题,则用P→Q表示命题"如果P那么Q"
{width="1.4931321084864393in" height="2.138999343832021in"}
在命题逻辑中,一个条件式的前提并不要求与结论有任何关系,这种条件式称为实质条件命题。
设P和Q是命题,则用P↔Q表示命题"P等值于Q"
{width="1.4931321084864393in" height="2.1320538057742784in"}
{width="5.768055555555556in" height="2.6555555555555554in"}
1.2合式公式与真值表
1.2.1合式公式
{width="5.768055555555556in" height="3.902083333333333in"}
1.2.2真值表
{width="4.562734033245844in" height="3.4515660542432194in"}
1.3永真式和等价式
1.3.1永真式
永真式/重言式:不依赖命题变元真值指派而总是取值为真(1)的命题公式
永假式/矛盾式:不依赖命题变元真值指派而总是取值为假(0)的命题公式
可满足式:至少存在一组命题变元的真值指派使命题公式为真
永真式的否定是矛盾式,矛盾式的否定是重言式。
永真式的析取、合取、单条件和双条件都是重言式。
重言式可以产生许多有用的恒等式。
1.3.2等价式
设A、B是两个命题公式,如果A、B在任意解释下(任意命题变元真值指派组合下),其真值都是相同的(A↔B为重言式),则称A和B等价。
常用等价逻辑式:
{width="4.80580271216098in" height="3.1112707786526683in"}
{width="3.3960083114610673in" height="3.6668547681539807in"}
{width="5.5558409886264215in" height="3.1529396325459316in"}
个人认为常用的有但不限于:1、2、3(交换律) 4、5、6(结合律) 7、8、9(分配律) 11、12、13(摩根律) 14(逆反律) 27(如何把蕴含拆掉) 29、30(吸收律)
1.3.3代入规则和替换规则
{width="4.722465004374453in" height="1.076444663167104in"}
{width="5.041925853018372in" height="0.7361493875765529in"}
1.4对偶式与蕴含式
1.4.1对偶式
{width="4.514120734908136in" height="0.8542104111986002in"}
{width="4.569679571303587in" height="1.3264566929133859in"}
对偶原理↓
{width="4.562734033245844in" height="0.6319772528433946in"}
{width="4.437728565179353in" height="0.6250317147856518in"}
1.4.2蕴含式
{width="5.701681977252844in" height="3.4237871828521436in"}
常用永真蕴含公式:
{width="4.116132983377078in" height="5.0022451881014875in"}
个人认为常用的有:9(析取三段论) 12(假言三段论) 10(假言推论) 1、2、3、4(化简附加式)
{width="5.768055555555556in" height="2.316666666666667in"}
{width="5.528062117235345in" height="3.0626574803149604in"}
1.5范式和判定问题
1.5.1析取范式与合取范式
{width="4.958588145231846in" height="1.2570089676290463in"}
{width="5.048870297462817in" height="1.7362007874015748in"}
P∧¬P永假 P∨¬P永真
1.5.2主析取范式与主合取范式
{width="4.333556430446194in" height="1.215340113735783in"}
{width="4.0974332895888015in" height="3.2779461942257218in"}
{width="4.694686132983377in" height="2.2223359580052495in"}
{width="4.778023840769904in" height="2.930706474190726in"}
{width="4.514120734908136in" height="3.3335050306211724in"}
{width="4.757189413823272in" height="3.458511592300962in"}
对于任意的命题公式A,其真值表中,真值取值为真(1)的真值指派组合都是主析取范式中的一个极小项
{width="4.423838582677165in" height="1.1181135170603675in"}
{width="4.07659886264217in" height="3.159884076990376in"}
{width="4.604402887139107in" height="2.1737226596675416in"}
{width="4.73635498687664in" height="2.930706474190726in"}
{width="4.729409448818898in" height="3.4237871828521436in"}
对于任意的命题公式A,其真值表中,真值取值为假(0)的真值指派组合都是主合取范式中的一个极大项
{width="3.7710269028871393in" height="1.2292300962379703in"}
{width="4.500230752405949in" height="0.6597561242344707in"}
{width="4.743299431321085in" height="2.8543132108486438in"}
1.6命题演算的推理理论
{width="4.180769903762029in" height="0.9792169728783902in"}
{width="4.312722003499562in" height="3.159884076990376in"}
你可以不用看懂下边这个定义(出于礼貌,我写进来了),你只要会写题就行↓
{width="5.146097987751531in" height="2.2292814960629923in"}
这就是前边写到的公式,可以略过↓
{width="4.937753718285214in" height="3.1876640419947506in"}
{width="5.173876859142607in" height="3.2501673228346455in"}
{width="5.243324584426946in" height="3.409897200349956in"}
{width="5.222490157480315in" height="3.069601924759405in"}
{width="4.36133530183727in" height="1.5764698162729658in"}
{width="4.653017279090114in" height="0.6250317147856518in"}
极速版------当你要用引入一个新的条件时(P规则)
当你要用前边的等价公式推出新的条件时(T规则)
当你要证的结论是蕴含式/你需要引入结论作为前提时(CP规则,记得写附加前提)
当你要用反证法,取结论的反为前提时(F规则,记得写假设前提)
以下用三道例题分别演示,普通方法、CP法、反证法
普通方法:
{width="5.114726596675416in" height="4.0653258967629045in"}
CP法:
{width="5.213273184601925in" height="4.268840769903762in"}
反证法:
{width="5.066856955380578in" height="4.2972080052493435in"}
(反证法是引入相反的结论作为前提,推出和现有前提的矛盾,例如该题的第7步)
自己写题时可以省略部分文字,但必须要包含的是每条结论前边的序号、结论、使用的规则、规则涉及到的结论的序号
谓词逻辑
2.1基本概念和表示
2.1.1个体、谓词和谓词形式
{width="5.118319116360455in" height="2.076496062992126in"}
{width="4.5418996062992125in" height="3.5349037620297463in"}
{width="4.611348425196851in" height="2.2709503499562556in"}
{width="5.625288713910761in" height="3.8751990376202974in"}
{width="5.041925853018372in" height="1.7848140857392827in"}
2.1.2量词
{width="4.764133858267717in" height="1.6598075240594925in"}
{width="4.625237314085739in" height="1.701476377952756in"}
存在唯一量词∃!,表示"恰有一个"、"唯一存在"。辖域等概念同上
存在唯一量词应该是不列入考纲的(我上了一学期翻书头一次看见它)
{width="4.611348425196851in" height="3.458511592300962in"}
谓词逻辑翻译时非常重要的注意事项↓
{width="6.98576990376203in" height="1.5780041557305338in"}
{width="5.906235783027122in" height="0.8491393263342082in"}
{width="4.458562992125985in" height="1.944543963254593in"}
一般来讲,量词的先后顺序不可随意交换
当量词辖域中的个体变元互不干涉时才可随意交换。
2.1.3合适谓词公式
{width="5.790522747156605in" height="3.7034241032370954in"}
2.1.4自由变元和约束变元
{width="5.637601706036746in" height="3.903611111111111in"}
{width="4.597458442694663in" height="3.1321052055993in"}
{width="4.514120734908136in" height="2.8751476377952754in"}
2.2谓词逻辑的翻译与解释
2.2.1谓词逻辑的翻译
把一个文字叙述的命题,用谓词公式表示出来,成为谓词逻辑的翻译或符号化
{width="6.803221784776903in" height="1.6667683727034122in"}
该部分十分重要,以下用两道例题做演示
{width="4.887288932633421in" height="2.305903324584427in"}
{width="5.746010498687664in" height="3.014300087489064in"}
2.2.2谓词公式的解释
{width="5.854673009623797in" height="2.6196587926509185in"}
对于一个谓词公式A,其个体域为D,若在D中怎么解释A,A的真值总为真(1),则称A在D中永真。(永假和可满足 类似定义)
{width="5.764074803149606in" height="3.473118985126859in"}
2.3谓词逻辑的等价式与蕴含式
非常重要的三个公式↓
{width="6.95538167104112in" height="1.8116765091863518in"}
{width="6.612941819772528in" height="3.1037970253718283in"}
{width="6.972617016622922in" height="1.8375863954505687in"}
以下是一些等价式和永真蕴含式:
{width="5.473236001749782in" height="2.543030402449694in"}
{width="6.077483595800525in" height="3.0905653980752406in"}
{width="5.892485783027122in" height="1.6729461942257218in"}
个人认为常用的有:E39、E40、I17、I18、I19、I20(抛开上边提到的三个公式)
以下是量词转换
{width="4.0835433070866145in" height="2.8543132108486438in"}
{width="4.041874453193351in" height="3.368228346456693in"}
2.4谓词逻辑中的推理理论
2.4.1推理规则
P、T、CP、F规则在谓词逻辑推理中仍然使用(对,就是使用,离不开的)
{width="5.803463473315835in" height="3.6648797025371826in"}
{width="5.895334645669291in" height="2.688124453193351in"}
{width="6.063930446194226in" height="2.2279615048118986in"}
{width="6.094599737532809in" height="2.7182491251093612in"}
下边这个US、ES、UG、EG是最重要的,做题时经常会用到的(注意相应规则的使用条件)
{width="5.474481627296588in" height="4.010182633420823in"}
{width="5.676160323709536in" height="4.010600393700788in"}
2.4.2推理实例
谓词逻辑的推理相较于命题逻辑复杂的多,所以给出四道例题做演示,希望能总结一些经验。(不是我不想总结,是我真的不知道做题的思路怎么写)
{width="6.016476377952756in" height="4.10254593175853in"}
{width="6.179438976377953in" height="1.3024803149606299in"}
{width="6.173464566929134in" height="2.472970253718285in"}
{width="5.961624015748032in" height="3.2135476815398074in"}
{width="5.726669947506561in" height="3.8119772528433944in"}
{width="5.82588145231846in" height="2.6409350393700786in"}
{width="6.062364391951006in" height="3.535622265966754in"}
{width="6.150605861767279in" height="2.337787620297463in"}
{width="5.335250437445319in" height="3.559488188976378in"}
{width="5.1944017935258096in" height="5.038764216972878in"}
2.5谓词逻辑中公式范式
2.5.1前束范式
{width="6.507597331583552in" height="2.7670100612423445in"}
{width="6.315646325459317in" height="4.221466535433071in"}
{width="6.365205599300087in" height="4.376621828521435in"}
2.5.2斯柯林范式
{width="6.736552930883639in" height="2.164073709536308in"}
{width="6.379068241469816in" height="4.972793088363955in"}
{width="6.115433070866142in" height="4.827156605424322in"}
个人感觉:谓词逻辑的范式考察的可能性不大,稍微记一下看看例题就行
集合论
集合论概念较多,所以该部分总结时以概念为主,习题方面大量减少
3.1集合的概念及其表示
{width="5.667139107611549in" height="1.31751968503937in"}
{width="7.775002187226597in" height="2.231582458442695in"}
集合中的元素也可以是集合!!!!!!!!(套娃)
{width="6.263953412073491in" height="4.117491251093614in"}
{width="5.801211723534558in" height="3.7833989501312337in"}
(外延和内涵不考,但是可以看看,不过也没什么b用)
{width="5.970681321084864in" height="3.3801312335958005in"}
(基数在离散1基本上用不到,但是是很重要的概念)
{width="5.640515091863517in" height="2.320977690288714in"}
(请务必玩懂空集,在这里会非常绕弯)
{width="5.874300087489064in" height="1.9542902449693789in"}
{width="6.40044728783902in" height="1.828698600174978in"}
{width="6.591123140857393in" height="0.851007217847769in"}
{width="6.490875984251969in" height="2.8985126859142607in"}
{width="6.452781058617672in" height="2.2067530621172353in"}
(真包含只是在包含的基础上去掉了相等的情况)
{width="6.161608705161854in" height="1.542992125984252in"}
(用定义证明相等可行)
(用A包含B同时B包含A也可以证明相等)
(也可以用第五章的特征函数进行相等的证明)
{width="6.226924759405074in" height="2.083532370953631in"}
{width="6.112554680664917in" height="2.2428740157480314in"}
{width="6.486984908136483in" height="1.456913823272091in"}
{width="6.538260061242345in" height="1.7370625546806648in"}
幂集的元素都是集合
{width="6.934195100612423in" height="0.6810378390201225in"}
{width="5.765649606299212in" height="3.1111297025371827in"}
3.2集合的运算与恒等式
交集并集在此不赘述,大家应该高中都讲过
{width="6.244656605424322in" height="4.828991688538933in"}
{width="6.143255686789152in" height="3.141087051618548in"}
{width="6.311938976377952in" height="1.3981627296587926in"}
{width="6.449724409448819in" height="1.3730030621172353in"}
{width="6.472979002624672in" height="1.0802591863517061in"}
{width="6.764317585301837in" height="1.2519838145231845in"}
{width="5.577841207349081in" height="4.602083333333334in"}
{width="5.768502843394575in" height="2.7501006124234473in"}
以下是常用的集合定律:
{width="5.1747550306211725in" height="5.784069335083115in"}
{width="5.511435914260717in" height="6.023884514435696in"}
{width="5.551083770778653in" height="5.449039807524059in"}
{width="5.871937882764654in" height="2.704711286089239in"}
个人认为常用的有:9、10(分配律) 19、20(吸收律) 21-26(摩根律) 31 32 33
3.3有穷集的计数和包含排斥原理
{width="5.312773403324584in" height="2.3890113735783025in"}
{width="6.09494750656168in" height="1.0558180227471565in"}
包含排斥原理↓
{width="6.329468503937008in" height="2.5049201662292213in"}
以下用两道例题做演示:
{width="4.992841207349081in" height="3.274773622047244in"}
{width="5.668597987751531in" height="3.430716316710411in"}
{width="6.19628937007874in" height="2.1039643482064743in"}
{width="5.993040244969379in" height="3.83038167104112in"}
二元关系
4.1多重序元与笛卡尔乘积
{width="5.602261592300962in" height="1.1083070866141733in"}
{width="4.882194881889764in" height="2.1876126421697286in"}
{width="5.382221128608924in" height="3.604351487314086in"}
(我们只学二重序偶,所以多重序偶带过)
{width="6.190793963254593in" height="2.393011811023622in"}
|A|=m,|B|=n则|A*B|=m*n
A*∅=∅ 般来说,笛卡尔乘积不满足交换律
切记,笛卡尔乘积对应的是集合运算
笛卡尔乘积满足分配律
{width="5.639666447944007in" height="2.430100612423447in"}
{width="6.099318678915136in" height="4.082096456692914in"}
A⊆C∧B⊆D=>A*B⊆C*D
4.2关系的基本概念
{width="5.464402887139108in" height="3.5050557742782154in"}
我们只研究二元关系
{width="5.437779965004374in" height="2.5487423447069117in"}
很重要的恒等关系↓
{width="7.003498468941382in" height="1.5969838145231845in"}
4.3关系的运算
{width="5.322176290463692in" height="2.133558617672791in"}
{width="6.404314304461942in" height="1.9943963254593176in"}
{width="6.496886482939632in" height="1.261007217847769in"}
{width="6.510038276465441in" height="1.4882841207349082in"}
{width="5.909783464566929in" height="3.7201312335958003in"}
{width="5.382221128608924in" height="4.2224387576552935in"}
{width="5.7820833333333335in" height="1.7546445756780402in"}
{width="5.768055555555556in" height="2.6352285651793528in"}
{width="5.768055555555556in" height="1.7888692038495189in"}
限制用的很少↓
{width="6.024291338582677in" height="2.1025109361329832in"}
{width="6.51136811023622in" height="2.0899715660542433in"}
以下定理很少用到↓
{width="6.778437226596675in" height="0.8610454943132109in"}
{width="6.900772090988626in" height="1.3734547244094488in"}
4.4关系的性质
{width="5.215545713035871in" height="2.215391513560805in"}
极速版
自反:对于x∈A,则<x,x>∈R
反自反:对于x∈A,则<x,x>∉R
{width="5.444724409448819in" height="1.9375995188101487in"}
极速版
对称:对于x、y∈A, <x,y>∈R,则<y,x>∈R
反对称:对于x、y∈A, <x,y>∈R,则<y,x>∉R,或当且仅当,x=y <y,x>∈R
注意的是, x和y可以相等
{width="5.5558409886264215in" height="2.4098458005249346in"}
极速版
可传递:对于x、y、z∈A, <x,y>、<y,z>∈R,则<x,z>∈R
不可传递:对于x、y、z∈A, <x,y>、<y,z>∈R,则<x,z>∉R
注意的是,x和y和z可以任意两个相等,甚至三个相等
{width="5.768055555555556in" height="3.4895833333333335in"}
{width="5.668397856517935in" height="2.674274934383202in"}
对于有限集合A,|A|=n
{width="4.875250437445319in" height="2.791809930008749in"}
4.5关系的表示
{width="5.81005249343832in" height="1.9629615048118985in"}
{width="5.892590769903762in" height="3.696459973753281in"}
自反关系:每个节点都有指向自己的有向边
对称关系:每两个有连线的节点必然是相互指向的
{width="5.598429571303587in" height="4.294019028871391in"}
{width="5.856628390201225in" height="4.105860673665791in"}
{width="5.5558409886264215in" height="3.833530183727034in"}
{width="5.271104549431321in" height="3.0904363517060367in"}
{width="5.768055555555556in" height="0.7916666666666666in"}
{width="5.319717847769029in" height="3.7849168853893262in"}
4.6关系的闭包运算
{width="5.943712817147857in" height="3.6735979877515312in"}
{width="6.010637576552931in" height="1.8632075678040245in"}
人话版
R自反------r(R)=R
R对称------s(R)=R
R可传递------t(R)=R
闭包的计算↓
{width="6.793850612423447in" height="1.9748906386701661in"}
{width="4.528010717410324in" height="1.3959055118110235in"}
{width="6.3231353893263345in" height="1.7663823272090988in"}
{width="5.222490157480315in" height="1.6181386701662293in"}
比较重要的性质↓
{width="6.707309711286089in" height="1.5822375328083988in"}
4.7特殊关系
4.7.1集合的划分和覆盖
{width="5.2502701224846895in" height="2.3334536307961504in"}
{width="5.146097987751531in" height="3.7224136045494314in"}
{width="5.768055555555556in" height="2.352777777777778in"}
{width="5.768055555555556in" height="3.6979166666666665in"}
{width="5.5419510061242345in" height="0.7500382764654419in"}
4.7.2等价关系
{width="5.166932414698163in" height="2.4167913385826774in"}
非常重要的模m同余关系↓
{width="5.069704724409449in" height="1.3542366579177603in"}
模m同余是等价关系
{width="5.361386701662292in" height="3.0001541994750656in"}
{width="5.575825678040245in" height="2.6342465004374453in"}
人话版
对于a∈A,与a有关系的所有元素构成的一个集合,称为等价类
{width="6.372419072615923in" height="1.7868875765529308in"}
{width="3.5626826334208226in" height="0.6597561242344707in"}
我也不知道这个3是什么,所以不能给出回答。
{width="3.124831583552056in" height="0.7952077865266841in"}
{width="6.707487970253719in" height="3.1745352143482064in"}
{width="6.026314523184602in" height="3.230433070866142in"}
全域关系将一个集合作为一个划分,即秩为1
恒等关系将集合中每一个元素作为一个等价类,即秩为|X|
{width="6.467756999125109in" height="1.8067825896762906in"}
{width="6.6874300087489065in" height="1.0797397200349956in"}
"划分"的概念和"等价关系"的概念本质 上是相同的。
4.7.3相容关系
{width="5.768055555555556in" height="2.240972222222222in"}
相容关系相较于等价,缺少了可传递的性质
{width="5.768055555555556in" height="0.4270833333333333in"}
{width="5.768055555555556in" height="3.8319444444444444in"}
{width="5.768055555555556in" height="3.8875in"}
{width="5.502679352580928in" height="1.0419772528433946in"}
{width="6.3573239282589675in" height="1.9830194663167104in"}
{width="2.493183508311461in" height="1.1597823709536308in"}
{width="5.354442257217848in" height="3.9863156167979in"}
{width="4.757189413823272in" height="4.041874453193351in"}
{width="5.166932414698163in" height="2.555686789151356in"}
{width="5.187766841644795in" height="4.152991032370954in"}
{width="5.5419510061242345in" height="3.4793460192475942in"}
{width="5.333607830271216in" height="2.9932097550306214in"}
4.7.4次序关系
{width="5.146097987751531in" height="2.257060367454068in"}
偏序关系的性质------自反 反对称 可传递
{width="5.166932414698163in" height="2.6945833333333336in"}
人话版
<P, ≤>是一个偏序集合,其中≤所表示的关系不定,根据题意可以随意翻译
注意:取反不影响关系(即取反前是偏序,取反后仍是偏序)
"整除"和"整倍数"互为逆关系, 它们都是I+ 中的偏序关系。
{width="5.885094050743657in" height="3.5045188101487312in"}
拟序条件的性质------反自反 反对称 可传递
实数集合中的大于小于、集合中的真包含都是拟序关系
{width="6.116729002624672in" height="1.6500174978127733in"}
{width="5.486393263342082in" height="2.9029265091863516in"}
全序集合的性质------自反 反对称 可传递
{width="5.0974846894138235in" height="1.9306550743657043in"}
全序集合是在偏序集合的基础上得来的,相较于偏序集合,全序集合中任意两个元素总有关系成立
{width="5.583619860017498in" height="1.9375995188101487in"}
人话版:
对于一个字符串,字母序就是从第一位开始比较大小(判断关系成立与否),若当前比较位置相同,则比较下一个位置,直到某一个字符串先遍历完,先遍历完的小于为遍历完的。
{width="6.62805227471566in" height="3.19036854768154in"}
4.7.5偏序集合与哈斯图
{width="5.625288713910761in" height="1.2847878390201224in"}
{width="5.361386701662292in" height="1.9862128171478566in"}
{width="5.4169455380577425in" height="3.3612839020122482in"}
{width="5.247227690288714in" height="1.2080993000874891in"}
{width="5.444724409448819in" height="1.944543963254593in"}
{width="5.354442257217848in" height="1.3195122484689414in"}
{width="5.4030555555555555in" height="1.2986778215223098in"}
{width="5.514172134733158in" height="1.9792683727034122in"}
{width="5.729460848643919in" height="2.3195636482939634in"}
{width="5.764185258092739in" height="2.0973304899387575in"}
{width="5.604454286964129in" height="0.6736461067366579in"}
{width="2.7501410761154856in" height="1.2222845581802275in"}
最小(大)成员是唯一的
{width="4.778023840769904in" height="0.7361493875765529in"}
{width="3.291836176727909in" height="0.33335083114610675in"}
{width="5.768055555555556in" height="2.736111111111111in"}
函数
5.1函数的基本概念与性质
{width="5.768055555555556in" height="1.2881944444444444in"}
{width="5.768055555555556in" height="2.5194444444444444in"}
{width="5.768055555555556in" height="1.39375in"}
{width="5.768055555555556in" height="3.1284722222222223in"}
{width="6.632700131233595in" height="1.5260159667541557in"}
{width="5.964227909011374in" height="4.041216097987752in"}
{width="5.768055555555556in" height="3.1256944444444446in"}
{width="5.722516404199475in" height="4.0835433070866145in"}
{width="5.768055555555556in" height="3.415277777777778in"}
很重要的一个定义(小心看不懂题)
{width="6.517666229221347in" height="1.5800404636920384in"}
{width="4.36613845144357in" height="1.1133650481189852in"}
5.2函数的合成和合成函数的性质
{width="5.768055555555556in" height="2.6347222222222224in"}
{width="5.576675415573053in" height="2.4098458005249346in"}
第二条性质非常重要
{width="6.611089238845144in" height="1.2642399387576553in"}
{width="6.79540135608049in" height="0.7559536307961505in"}
5.3特殊函数
{width="5.768055555555556in" height="3.966666666666667in"}
{width="1.9862128171478566in" height="0.4375229658792651in"}
{width="5.729460848643919in" height="1.826482939632546in"}
{width="3.9168678915135606in" height="1.5209109798775153in"}
{width="3.5487937445319333in" height="0.7778182414698163in"}
{width="5.768055555555556in" height="2.245138888888889in"}
{width="5.768055555555556in" height="1.58125in"}
{width="5.7086264216972875in" height="2.0417716535433073in"}
{width="5.768055555555556in" height="1.132638888888889in"}
5.4反函数
{width="6.217444225721785in" height="3.073540026246719in"}
一个函数的左逆和右逆不一定是唯一的
{width="6.493662510936133in" height="0.7583497375328084in"}
{width="6.491476377952756in" height="1.6058759842519685in"}
{width="6.889711286089239in" height="0.5326213910761155in"}
{width="6.861054243219598in" height="1.2043602362204724in"}
5.5特征函数
{width="6.3966896325459315in" height="3.3585312773403326in"}
{width="6.516552930883639in" height="2.63533573928259in"}
都挺重要的,尽力记一下↓
如果记不全,可以挑几个重要的,比如7、8、9、10
{width="6.007402668416448in" height="6.13200021872266in"}
特征函数可以用来证明函数的相等亦或包含关系,也可以进行函数双射等性质的推导
事例如下:
{width="5.768055555555556in" height="4.427777777777778in"}
{width="6.331088145231846in" height="3.6008595800524934in"}
5.6基数
{width="6.860474628171478in" height="1.1125765529308835in"}
{width="5.552903543307087in" height="1.9823261154855643in"}
等势满足交换律和传递性
{width="6.432088801399825in" height="4.238240376202975in"}
{width="6.734394138232721in" height="1.2056397637795275in"}
↑如何证明一个集合可数
查看是否有函数能实现A到N的双射(可以考虑用特征函数证明),否则可以考虑将A中的元素一一对应N中的元素
{width="6.503666885389326in" height="0.5183519247594051in"}
{width="6.7370603674540686in" height="1.164750656167979in"}
{width="6.60548665791776in" height="0.6744061679790027in"}
每取一次幂集,阿列夫数字+1,例如自然数集合N取幂集则从阿列夫0变成阿列夫1
{width="6.245932852143482in" height="1.4638899825021872in"}
{width="6.778319116360455in" height="1.3488353018372703in"}