2.Basic derivation rules 基本推导规则
费马定理证明过程
费马定理证明过程全文共四篇示例，供读者参考第一篇示例：费马定理是数论中的一个重要定理，由著名数学家费马在17世纪时提出并据一直引起数学界的广泛关注和研究。
费马定理又称费马大定理，其表述为：对于大于2的正整数n，不存在三个正整数a、b、c，使得满足a^n + b^n = c^n。
费马定理证明的过程是一个漫长而又复杂的数学推理过程，而直到1995年，英国数学家安德鲁·怀尔斯才最终给出了费马定理的证明。
费马定理的证明历经了数百年间许多数学家的探索和努力，费马本人曾在他的笔记本上写下了：“我找到了这个证明，但是这个空间太小，无法容纳这个证明。
”这句话也在一定程度上激发了后世数学家对这个问题的研究和探索。
费马定理的证明过程可以大致被分为三个阶段，分别是费马猜想的提出、证明的辅助工具的建立、以及最终的证明。
费马猜想的提出发生在17世纪，费马在一个边注中提出了这个猜想，称其为“我无法证明的定理”，这也给后世数学家提供了一个极大的挑战。
费马猜想的提出激发了许多数学家的研究热情，这个定理的证明一度被认为是不可能的。
随后的数百年间，许多数学家纷纷投入到费马定理的研究之中，他们提出了许多有关费马定理的猜想和假设。
于是，证明费马定理的难度立即从退化为一个普通的数学难题而变得异常复杂。
在费马定理的证明中，数学家们创立了许多重要的数学概念和工具，例如椭圆曲线、调和模形式等，这一系列的辅助工具为费马定理的证明提供了坚实的数学基础。
这些独立的数学概念在费马定理的证明过程中发挥了至关重要的作用。
最终，英国数学家安德鲁·怀尔斯于1995年成功地证明了费马定理，这也为整个数学界带来了一场轰动。
怀尔斯的证明过程异常复杂，包含了许多高深的数学知识和技巧，这也是费马定理证明过程中最为汗牵动人心的部分。
通过费马定理的证明过程，我们可以看到数学家们在对一个数学难题进行探索和研究的过程中所需付出的辛勤努力和不懈追求。
费马定理的证明，实际上也反映了数学研究的艰辛和复杂性。
代数基本定理
n(n−1) 2
=
2n−1q(2kq − 1)
=
zk−1q′ ,
其中
q′
=
q(2kq − 1)
为奇数。
在环 P [x] 中组成用这些元素 βij 为根且只用它们做根的多项式 g(x)：
∏ g(x) = (x − βij).
i<j
g(x) 的系数为 βij 的初级对称多项式，由（1）式知，它们是 α1, α2, ..., αn 的实系数对称多项式。 由对称多项式基本定理，多项式 g(x) 的系数是所给 f (x) 的系数的多项式（f (x) 系数为实数），故仍
2) 假设小于等于 k-1 时，命题成立。 设 P 为实数域上多项式 f (x) 的分裂域，且设 α1, α2, ..., αn 为域 P 中 f (x) 的根。选取 ∀c ∈ R, 且取 出域 P 中形如下列的元素：
βij = αiαj + c(αi + αj), i < j
(1)
元素
βij
的个数为
θ∈[0,2π]
在 Ω 内为常数。即 |f (z)| 在 Ω 内无局部最大模，除非 f (z) 恒为常数。
Theorem 3.2. （代数基本定理）n 为正整数，P (z) = zn + an−1zn−1 + ... + a1z + a0, 其中 ai ∈ C, i = 0, 1, ..., n − 1. 则 P (z) 至少有一个根。
+
ζ) |
≤
|1
+
C eiθ ζ l |
+
D|ζ |l+1
=
|1
−
C λl |
+
《语言学导论》重点整理
♦Applied linguistics: the study of applications of linguistics.
5. Some distinctions in linguistics
6. Phonology : the sound patterns of language
Difference Phone, phoneme, allophone
Phonemic contrast, complementary distribution, minimal pair
7. Phones, phonemes, and allophones
Narrow transcription窄式标音: transcription with letter-symbols and the diacritics
4. Classification of English consonants
5. Classification of English vowels
1 .An Introduction to Linguistics and language
1. What is Linguistics?
Linguistics is the scientific study of language. It endeavors to answer the question--what is language and how is represented in the mind? Linguists focus on describing and explaining language and are not concerned with the prescriptive rules of the language.
第二节·有理函数求导规则
( uv )′( x ) = lim Δy Δu ⎞ Δv ⎞ . ⎛ ⎛ lim ( ) = lim ⎜ v ( x + Δx ) + u x ⎜ Δx → 0 Δ x Δx → 0 Δx ⎟ Δx ⎟ ⎝ ⎠ Δx → 0 ⎝ ⎠
(
)
y=
( x − 1) ( x 2 − 2 x )
x4
x 3 − 3 x 2 + 2 x . −1 −2 −3 = = x − 3 x + 2 x . 4 x
Then use the Sum and Power Rules:
dy 1 6 6 . 2+ 3− 4. = − x − 2 − 3 ( − 2 ) x − 3 + 2 ( − 3 ) x −4 = − dx x x x
Derivation rules for sum, difference, product and quotient of functions
1 2 1 ( x + ). . x x 1 x2 + 1 . Solution: By rules (2) with u = and v =. : x x
Since v ( x ) is derivable at x , it must be continuous at x and hence Therefore Finish.
3
Δx → 0
lim v ( x + Δx ) = v ( x ) .
( uv )′( x ) = u′( x )v ( x ) + u( x )v ′( x ) .
4
2.基本定律规则和公式化简法
Y
不能改变原来的运算顺序 不属于单个变量上的反号应保留不变
反演规则的应用：求逻辑函数的反函数 将 Y 式中“.”换成“+”,“+”换成“.” “0”换成“1”,“1”换成“0” 原变量换成反变量，反变量换成原变量 例如：已知 则
Y
Y1 A( B C ) CD
运算顺序： 括号 与 或 不属于单个变量上 的反号应保留不变
消去法 运用吸收律 A AB A B ，消去多余因子。
Y AB AC BC AB ( A B )C AB ABC AB C Y AB AB ABCD ABCD A B AB CD( AB A B ) A B CD A B A B CD AB AB CD 配项法 通过乘 A A 1或加入零项 A A 0 进行配项，然后再化简。 Y AB BC AC D AB BC ACD ( B B) AB BC ABC D ABC D AB(1 C D) BC (1 AD) AB BC Y ABC ABC AB ABC ABC AB AB AB AB( AB C ) ABC AB AB ABC ABC AB ABC A B C
左 AB AC ( A A) BC AB AC ABC ABC AB AC
推广公式： AB AC BCD AB AC
摩根定律 （又称反演律）
推广公式：
[例]利用摩根定律求下列函数的反函数：
(1) Y1 AB AB
解：(1) Y1 AB AB
Y1 ( A BC ) ( C D )
已知 Y2 A B C D C 则
逻辑代数的公式与基本定理
序号
公式
序号
公式
规律
1
A·0=0
10
A+1=1
0-1律
2
A·1=A
11
A+0=A
0-1律
3
0 1； 1 0
12
4
A·A= A
13
A A
A+A=A
还原律 重叠律
5
A A 0
14
A A1
互补律
6
A·B=B·A
15
A+B=B+A
交换律
7
A·(B·C) = (A·B)·C
16 A+(B+C)=(A+B)+C 结合律
AB CD ABD(E F) AB CD
被吸收
2.反变量的吸收：
证明： A AB A AB AB
长中含反， 去掉反。
A B(A A) A B
例如： A ABC DC A BC DC 被吸收
3.混合变量的吸收：
证明： AB AC BC
1
AB AC ( A A)BC
2
A AB A B
3
AB AB A
4
A(A+B)= A
AB AC BC AB AC
5
AB AC BCD AB AC
6
AAB AB ； AAB A
规律 吸收律 吸收律
吸收律
1.原变量的吸收： A+AB=A 证明： A+A=BA(1+B=)A•1=A
长中含短，留下短。
利用运算规则可以对逻辑式进行化简。 例如：
8
A·(B+C)=A·B + A·C 17 A+B·C =(A+B) ·(A+C) 分配律
初中英语学科知识大纲 人教版
初中英语学科知识大纲人教版Chinese Version:初中英语学科是义务教育阶段学生接触并学习英语的初始阶段，其课程内容以人教版教材为主线，旨在通过系统性、科学性的教学，培养学生的英语语言基础知识、基本技能以及跨文化交际意识。
本大纲将详细阐述初中英语学科的知识体系，涵盖词汇、语法、听说读写四项基本技能、主题内容以及跨文化交际等方面。
English Version:Junior high school English is the initial stage where students encounter and study English within compulsory education. Centered around textbooks from the People's Education Press, the curriculum aims to systematically and scientifically cultivate students' foundational knowledge of the English language, essential skills, and intercultural communication awareness. This outline will comprehensively detail the knowledge structure of junior high school English, encompassing vocabulary, grammar, the four basic skills of listening, speaking, reading, and writing, thematic content, and cross-cultural communication.二、Vocabulary (词汇)Chinese Version:词汇学习是英语学习的基础。
《新编语言学教程》练习参考答案
说明应许多教师和学生的要求，我们编写了本教科书每章后面的习题答案。
但是，有两点必须说明一下。
第一，对术语的定义或解释，多是以本书相关章节为依据，所以会有些局限性。
好在目前市场上有多种语言学词典，大家可参考其他定义和解释。
第二，对开放式问答题的回答，多为提示性的，而且也是基于本书内容做的基本回答，往往显得过于简略。
一方面，本书是入门性的教程，对诸多问题的讨论仍欠深入；另一方面，每章后面附的参考书，可以告诉读者到哪里去查找相关信息，加深对某个问题的了解。
总之，这里提供的答案是参考性的，绝不是唯一的，更不是穷尽一切的。
充其量，它只能给教师和同学提供一点点方便而已。
如发现不当之处，敬请指教。
Chapter 1Introduction1. Define the following terms briefly.(1) linguistics: the scientific or systematic study of language.(2) language: a system of arbitrary vocal symbols used for human communication.(3) arbitrariness: the absence of similarity between the form of a linguistic sign and what it relates to in reality, e.g. the word dog does not look like a dog.(4) duality: the way meaningless elements of language at one level (sounds and letters) combine to form meaningful units (words) at another level.(5) competence: knowledge of the grammar of a language as a formal abstraction and distinct from the behavior of actual language use, i.e. performance.(6) performance: Chomsky’s term for actual language behavior as distinct from the knowledge that underlies it, or competence.(7) stylistics: the study of how literary effects can be related to linguistic features.(8) phatic communion: Language is used to establish an atmosphere or maintain social contact between the speaker and the hearer.(9) functionalism: the study of the forms of language in reference to their social function in communication.(10) formalism: the study of the abstract forms of language and their internal relations.(11) synchronic linguistics: the study of language and speech as they are used ata given moment and not in terms of how they have evolved over time.(12) diachronic linguistics: the study of linguistic change over time in contrast to looking at language as it is used at a given moment.2. No, language is human-specific. Human language has seven design features, including arbitrariness, duality, productivity, interchangeability, displacement, specialization and cultural transmission. These features are found utterly lacking in dogs’ or pigs’ noises and thus set human language apart from animal cry systems.3. Arbitrariness refers to the fact that there is no logical or intrinsic connection between a particular sound and the meaning it is associated with. For example, for the same animal dog, in English we call it /d0g/, in Chinese as “gou”, but “yilu”in Japanese; it barks wow wow in English but wang wang in Chinese. Of course, onomatopoetic words such as “quack-quack” and “bang” are exceptions, but words like these are relatively few compared with the total number of words in a language.4. A human baby does not speak any language at birth. What language the baby is going to speak is determined by the culture he is born into. A Chinese baby born and brought up in London by an English family will speak English, while an English child brought up in Beijing by a Chinese aunt will speak Chinese. That is to say, language cannot be transmitted through heredity. It is culturally transmitted.5. Firstly, linguistics describes languages and does not lay down rules of correctness while traditional grammar emphasizes correctness. Secondly, linguistics regards the spoken language as primary, while traditional grammar emphasizes the priority of the written language. Thirdly, traditional grammar is based on Latin and it tries to impose the Latin categories and structures on other languages, while linguistics describes each language on its own merits.6. A descriptive approach attempts to tell what is in the language while the prescriptive approach tells people what should be in the language. Most modern linguistics is descriptive, whereas traditional grammars are prescriptive.7. Synchronic linguistics studies language at one particular time while diachronic linguistics studies language developments through time. Synchronic linguistics focuses on the state of language at any point in history while diachronic linguistics focuses on the differences in two or more than two states of language over decades or centuries.8. No, human language has the design feature of specialization. It refers to the fact that man does not have a total physical involvement in the act of communication. For example, a mother can tell a story to her child while slicing up a cake. However, wolves can only respond to a stimulus and is totally involved physically in the communication process. Thus, a wolf cannot have a language similar to man’s, even though it could express a thousand different emotions. Besides, the aspect of productivity also distinguishes human language from wolf ’s postures.Chapter 2The Sounds of Language1.Define each of the following terms briefly.(1) articulator: the tongue, lips, and velum, which change the shape of the vocal tract to produce different speech sounds.(2) assimilation: a phonological process whereby a sound becomes phonetically similar (or identical) to a neighboring sound, e.g. a vowel becomes [+nasal] when followed by a [+nasal] consonant.(3) consonant: a speech sound produced by partial or complete closure of partof the vocal tract, thus obstructing the airflow and creating audible friction. Consonants are described in terms of voicing, place of articulation, andmanner of articulation.(4) elision: the leaving out of a sound or sounds in speech.(5) intonation: the variation in pitch and stress which gives beat and rhythm to the tune the voice plays in ordinary speech.(6) phoneme: the abstract element of a sound, identified as being distinctive in a particular language.(7) phonetics: the study of linguistic speech sounds, how they are produced, how they are perceived, and their physical properties.(8) phonology: the study of the abstract systems underlying the sounds of language.(9) stress: the prominence given to certain sounds in speech.(10) voicing: the vibration in the larynx caused by air from the lungs passing through the vocal cords when they are partly closed; speech sounds are saidto be either voiced or voiceless.(11) voiceless: When the vocal cords are spread apart, the airstream from the lungs is not obstructed at the space between vocal cords and passes freely.The sounds produced in this way are described as voiceless sounds.(12) vowel: a speech sound produced without significant constriction of the air flowing through the oral cavity.2. Minimal pair test or substitution test.Minimal pair test or substitution test is to see whether substituting one sound for another results in a different word. If it does, the two sounds represent different phonemes. For example, as to the English word bear, if we substitute p for b, we get the word pear, the two are different words. Then /b/ and /p/ represent different phonemes. Other examples are chunk/junk, ban/bin, bet/beat, fine/vine, side/ site, etc.3. Take the word big for example, in the word big /big/, /b/ is the onset, /i/ is the nucleus and /g/ is the coda. The difference between open syllables and closed syllables is whether the words have codas. If there are codas, they are closed syllables,such as pig, hat and at; if not, they are open syllables, such as do, I, tea and key.4. (1) stop, consonant(2) back, rounded, vowel5. (1) voiceless / voiced(2) bilabial / labiodental(3) close / semi-open(4) stop / nasal(5) alveolar / palatal(6) alveolar / dental(1) kit/git, bucker/bugger, bag/back(2) mark/nark, smack/snack, sum/sun(3) best/vest, ober/over, lib/live(4) bore/more, abate/amate, mob/mom(5) pat/fat, apt/aft (AmE), harp (BrE)/half7. (1) The stresses are placed on the second syllable except for “promise”. We mayeasily conclude that the verbs usually are stressed on the second syllable. (2) Syllable representations of the words:collide [k2#laid] elect [i#lekt] consider [k2n#sid2]Chapter 3Morphology1. Define the following terms briefly.(1) morphology: the study of the structure of words.(2) morpheme: the smallest unit of language that carries meaning or serves a grammatical function.(3) free morpheme: a morpheme that can stand alone as a word.(4) bound morpheme: a morpheme that can not stand alone as a word, e.g.-ment (as in establishment), and -er (as in painter).(5) morph: the smallest meaningful phonetic segments of an utterance on thelevel of parole.(6) allomorph: a phonetic form in which a morpheme is realized, e.g. -s, -es, and -en are all allomorphs (in writing) of the plural morpheme.(7) derivation: the formation of new words by adding affixes to other words or morphemes in morphology and word formation.(8) clipping: the process by which parts of a word of more than one syllable have been cut off, and reduced to a shorter form.(9) acronym: words which are composed of the first letter of a series of words and are pronounced as single words. Examples: NATO, radar and yuppy.(10) initialism: Some new words are composed of the first letters of a series of words and pronounced by saying each letter in them. Such words are called initialism.(11) blending: A single new word can be formed by combining two separateforms. Typically, blending is finished by taking only the beginning of oneword and joining it to the end of another word. For example, brunch isformed by the shortened forms of breakfast and lunch.(12) root: the morpheme that remains when all affixes are stripped from a complex word, e.g. system from un- + system + atic + ally.(13) stem: the base towhich one or more affixes are attached to create a morecomplex form that may be another stem or a word. For example, book is thestem of bookish.(14) prefix: Affixes can be joined to the beginning of the root or stem, in which case they are called prefixes.(15) suffix: Affixes can be joined to the end of the root or stem, in which case they are called suffixes.2. (3), (5), (7)3. (1) simple: fly tree suite(2) bound morpheme rootfly / fly reuse re-use spiteful -ful spitepreplan pre-plan desks -s desk triumphed -ed triumphsuite / suite optionality -ality option untie un-tiedelight de-light fastest -est fast prettier -ier prettytree / tree justly -ly just deform de-formmistreat mis-treat dislike dis-like payment -ment paydisobey dis-obey premature pre-mature4. (1) Column I: ablaut (vowel modification)Column II: suppletionColumn III: stress modification(2) The process in the Column I is finished by changing the vowel of each word, while in Column II, the process is finished by changing vowel and consonantof each word.(3) Column I: awake/awoke bear/bore arise/aroseblow/blew bite/bit hide/hidlie/lay know/knew foot/feetgoose/geese tooth/teeth louse/liceColumn II: bad/worse are/were many/moreColumn III: #combine/com#bine #compress/com#press#conduct/con#duct #insert/in#sert#insult/in#sult #intern/in#tern5. (1) Omitted.(2) Other examples:#rerun (n.) – re#run (v.) #contrast (n.) – con#trast (v.)#convert (n.) – con#vert (v.) #desert (n.) – de#sert (v.)#export (n.) – ex#port (v.) #increase (n.) – in#crease (v.)#conduct (n.) – con#duct (v.) #object (n.) – ob#ject (v.)#content (n.) – con#tent (v.) #protest (n.) – pro#test (v.)#insult (n.) – in#sult (v.) #produce (n.) – pro#duce (v.)When a word belongs to different word classes, the stress of the word will be sometimes placed on different syllables. When all the words above are stressed on the first syllables, they are nouns, but if they have the second syllables stressed, the words become verbs.6. (1) It means “the inhabitant of ”.(2) It means “the person who does”.(3) The morphological rule working here is “n. + -er –– n.”, and the last phoneme of the noun, which the suffix -er is added to, should be a consonant.(4) The rule in (3) doesn’t work in the word discoverer because the last phoneme of discoverer is a vowel /2/.7. (1) inflection (2) derivation (3) inflection (4) inflection (5) derivation Chapter 4Syntax1. Define the following terms briefly.(1) syntax: the term used to refer to the structure of sentences and to the study of sentence structure.(2) word class: a group of words which are similar in function; words which aregrouped into word classes according to how they combine with other words,how they change their forms, etc.(3) prescriptive approach: This view regards grammar as a set of rules for the “proper” use of a language, that’s to say, it tries to lay down rules to tell peoplehow to use a language.(4) descriptive approach: the approach of linguistic studies, with which linguists collect samples of the language they are interested in and attempt to describe the regular structures of the language as they are used, not according to some view of how they should be used.(5) IC analysis: the approach to divide the sentence up into its immediate constituents by using binary cutting until obtaining its ultimate constituents. For example, the immediate constituents of “The man bought a car” are theman and bought a car. The immediate constituents of the man are the andman, and so on until no further cuts can be made. The ultimate constituentsof “The man bought a car” at the word level are the, man, bought, a, and car.(6) structural analysis: a type of descriptive approach to study the distribution of linguistic forms in a language through such methods as the use of “test frames”.(7) immediate constituent: Linguistic units can be divided into small constituents, which can be further analyzed into smaller constituents. This processcontinues until no further divisions are possible. The first division or units are known as immediate constituents.(8) ultimate constituent: Linguistic units can be divided into small constituents, which can be further analyzed into smaller constituents. This process continues until no further divisions are possible. The final division or units areknown as ultimate constituents.(9) constituent structural grammar: It refers to a grammar which analyzes sentences using only the idea of constituency, which reveals a hierarchy of structural levels.(10) transformational grammar: a type of grammar which attempts to define and describe by a set of rules or principles all the grammatical sentences (without ungrammatical ones) of a language.(11) ideational function: the use of language as a means of giving structure to our experience of the real or imaginary world.(12) interpersonal function: the use of language for maintaining social roles and interacting with others.(13) textual function: to create written or spoken texts which cohere within themselvesand which fit the particular situation in which they are used.2. Yes. As we know, morphology is the study of the internal structure, forms and classes of words, while syntax focuses on the structure and ordering of components within a sentence. The major distinction between morphology and syntaxis that the former is concerned with the internal composition of a word, while the latter is concerned with the combination of words.3. (2) Instead of using the form “suggest somebody to do something”, we usually use “suggest + that-clause” or “suggest doing”, here we’d better substitute “advise” for “suggest”(4) The word “request” is a transitive verb which should take an object directly, so the word “for” should be omitted.(6) The word “donate” cannot be followed by double objects as “donate somebody something”. Instead we always use “donate something to somebody”.(10) The subject of the verb “write” is usually a human; an “article” cannot write itself. In this case the passive construction is normally used: The article was very well writen.(11) Usually we don’t use “be bored of something/somebody”, but “be bored with something/somebody” which means losing interest in somebody/something.(13) Here “myself ” is a reflexive pronoun, which can’t be used as subject, and it should be replaced by “I”.(14) The word “surprise” is usually used as a transitive verb, so the expression “…surprise for you” is ungrammatical, and it can be replaced by “surprise somebody (with something)” or “I was surprised by your getting married.”(15) The word “sleep” is usually used as an intransitive verb, which can’t take an object. The cases of “sleep” being used as a transitive verb are semantically limited, as in “to sleep a good sleep” or “the room can sleep 3 people”.4. It’s ungrammatical, because “us” is the objective case which can’t serve as the subject,while “she” is the subjective case which can’t serve as the object. The sentence should be “We visit her on Sundays”. The personal pronouns “you” and “it” have the same form whether used as the subject or object.5. (1) NP: A Guns “N” Roses concert, an arena, some 2500 fans, a full-fledged riot,A Guns “N” Roses concert at an arena , A Guns “N” Roses concert at an arena near ST. Louis ,The trouble, venue security, a camera, the front, the stage, the front of the stage. PP: at an arena, at an arena near ST. Louis, near ST. Louis, in disaster, near the front, of the stage, near the front of the stage. VP: stageda full-fledged riot, asked venue security, confiscate a camera.(2) N: Guns, Roses, concert, arena, ST. Louis, disaster, fans, riot, trouble, Axl Rose, venue, security, camera, front, stage. Prep: at, near, in, of. V: end, stage, start, ask, confiscate, see.6. (1) You mustn’t end a sentence with a preposition.You mustn’t split infinitives.7. (i)SNP VP AdvArt N V NP Prep NPArt N Art NThe dog bit the man in the car.(ii)S ,Art N ,Art N ,Prep NP ,NP VP, Art N V NP P. The dog bit the man in the car.8. (1)a. Terry loves his wife and I love his wife, too.b. I love my wife as well as Terry loves his wife.(2) a. It’s yesterday that they said she would go.b. She would go yesterday as they said.(3) a. The governor is a street fighter who is dirty.b. The governor is a fighter in a dirty street.(4) a. The design has squares and circles, both of which are big.b. The design has big squares, and it also has some circles. (The sizes of the circles are not mentioned.)Chapter 5Semantics1. Define the following terms briefly.(1) semantics: the study of linguistic meaning.(2) truth-conditional semantics: an approach that knowing the meaning of the sentence is the same as knowing the conditions under which the sentence istrue or false, and knowing the meaning of a word or expression is knowingthe part that it plays in the truth or falsehood of the sentence containing it.(3) naming theory: the view that the meaning of an expression is what it refers to, or names.(4) behaviorist theory: the view that the meaning of a linguistic form is defined as observable behaviors which is an approach drawing on psychology.(5) use theory: the semantic theory according to which the meaning of an expression is determined by its use in communication and more generally, insocial interaction.(6) sense: the inherent part of an expression’s meaning, together with the context, determines its referent. For example, knowing the sense of a nounphrase such as the president of the United States in 2004 allows one to determine that George W. Bush is the referent.(7) reference: (in semantics) the relationship between words and the things, actions, events and qualities they stand for. An example in English is the relationshipbetween the word tree and the object “tree” (referent) in the realworld.(8) conceptual meaning: It means the meaning of words may be discussed interms of what they denote or refer to, also called denotative or cognitive meaning. It is the essential and inextricable part of what language is and is widely regarded as the central factor in verbal communication. For instance, the conceptual meaning of “he” in English is any male person or male animal.(9) connotative meaning: It is the communicative meaning that a word or a combination of words has by virtue of what it refers to, over its purely conceptual meaning. For example, the connotative meaning of “woman” is emotional,frail, inconstant, irrational, etc.(10) semantic field: the organization of related words and expressions into asystem which shows their relationship to one another. For example, kinship terms such as father, mother, brother, sister, uncle, aunt belong to a semantic field whose relevant features include generation, sex, membership of the father’s or mother’s side of family.(11) lexical gap: the absence of a word in a particular place in a semantic field of a language. For instance, in English we have brother versus sister, son versus daughter, but no separate lexemes for “male” and “female” cousin.(12) componential analysis: (in semantics) an approach to the study of meaning which analyzes a word into a set of meaning components or semantic features.For example, the meaning of the English word boy may be shown as[+human][+male][-adult].(13) semantic feature: the smallest units of meaning in a word. The meaningof word may be described as a combination of semantic components orfeatures. For example, the feature [+male] is part of the meaning of father,and so is the feature [+adult], but other features are needed to make up the whole meaning of father. Often, semantic features are established by contrast and can be stated in terms of [+] and [-], e.g. woman has the semantic features [+human], [-male] and [+adult].(14) synonym: the sense relations of equivalence of meaning between lexical items, e.g. small/little and dead/deceased.(15) antonym: the sense relation of various kinds of opposing meaning between lexical items, e.g. big/small, alive/dead and good/bad.(16) hyponymy: the sense relation between terms in a hierarchy, where a more particular term (the hyponym) is included in the more general one (the superordinate): X is a Y, e.g. a beech is a tree, a tree is a plant.(17) meronym: the sense relation between body and its parts which are not only sections of the body but defined in terms of specific functions. For example, the head is the part of the body which carries the most important sense organs,i.e. eyes, ears, nose and tongue.(18) semantic role: the way in which the referent of a noun phrase is involved in the situation described or represented by the clause, for example as agent, patient, or cause.(19) entailment: the relationship between two sentences where the truth of one (the second) is inferred from the truth of the other, e.g. Corday assassinated Marat and Marat is dead; if the first is true, the second must be true.(20) presupposition: implicit assumptions about the world required to make an utterance meaningful or appropriate, e.g. “some tea has already been taken”is a presupposition of “Take some more tea!”2. (1) He waited by the bank.a. He waited by the financial institution which people can keep their moneyin or borrow from.b. He waited by the bank of the river.(2) Is he really that kind?a. Is he really that type of person?b. Is he really that kind-hearted?(3) We bought her dog biscuits.a. We bought dog biscuits for her.b. We bought biscuits for her dog.(4) He saw that gasoline can explode.a. He saw that gasoline container explode.b. He saw that gasoline may explode.(5) Fifty soldiers shot three wild foxes.a. Fifty soldiers shot three wild foxes in total.b. Each of the fifty soldiers shot three wild foxes.(6) He saw her drawing pencils.a. He saw her pencils for drawing.b. He saw her drawing the picture of pencils.3. (2) (4) (5) (8) are antonyms; (1) (3) (6) (7) are synonyms.4. charity: kindness, sympathy, church, helpfuliron: strong, brave, hard, determinedmole: traitor, betrayal, spysnow: pure, virgin, cleanstreet: homeless, living hard, pitiable5. (1) a. hoard b. scribble c. barn, method d. olfactory(2) a. acquire b. tell c. way d. smell(3) a. buy, win, steal. b. talk, tell c. road, way, path d. smellThese words are less marked in their sets because they are more usual andtend to be used more frequently. They consist of only one morpheme andare easier to learn and remember than others. They are also often broader in meaning and cannot be described by using the name of another member ofthe same field.6. homophones: sea-see, break-brake; polysemies: sea, break, prayer, mature, trace, househomonyms: ear.7. In a semantic field, not all lexical items necessarily have the same status. The less marked members of the same semantic field (1) are usually easier to learn and remember than more marked members; (2) consist of only one morpheme incontrast to more marked members; (3) cannot be described by using the name of another member of the same field; (4) tend to be used more frequently than more marked terms; (5) broader in meaning than more marked members; (6) are notthe result of the metaphorical usage of the name of another object or concept, but more marked are.8. (1) a. bachelor, man, son, paperboy, pope, chiefb. bull, rooster, drake, ram.The (a) and (b) words are male.The (a) words are human.The (b) words are animals.(2) a. ask, tell, say, talk, converseb. shout, whisper, mutter, drawl, hollerThe (a) and (b) words are realized by sounds.The (a) words are normal voice quality.The (b) words are produced by modifying one’s normal voice quality.(3) a. walk, run, skip, jump, hop, swimb. fly, skate, ski, ride, cycle, canoe, hang-glideThe (a) and (b) words are sports (movement).The (a) words are sports without instruments.The (b) words are sports with instruments.Chapter 6Pragmatics1. Define the following terms briefly.(1) pragmatics: a branch of linguistics that studies language in use.(2) deixis: the marking of the orientation or position of entities and situations with respect to certain points of reference such as the place (here/there) and time (now/then) of utterance.(3) reference: (in semantics) the relationship between words and the things, actions, events, and qualities they stand for.(4) anaphora: a process where a word or phrase (anaphor) refers back to another word or phrase which was used earlier in a text or conversation.(5) presupposition: implicit assumptions about the world required to make an utterance meaningful or appropriate, e.g. “some tea has already been taken”is a presupposition of “Take some more tea!”(6) Speech Act Theory: The theory was proposed by J. L. Austin and has been developed by J. R. Searle. Basically, they believe that language is not only used to inform or to describe things, it is often used to “do things”, to perform acts. In saying “Sorry”, you are performing an act of apology.(7) indirect speech act: an utterance whose literal meaning (location) and intended meaning (illocution) are different. For example, Can you pass thesalt? is literally a yes/no question but is usually uttered as a request or polite directive for action.(8) the Cooperative Principle: a principle proposed by the philosopher Paul Grice whereby those involved in communication assume that both partieswill normally seek to cooperate with each other to establish agreed meaning.It is composed of 4 maxims: quality, quantity, relation and manner.(9) the Politeness Principle: politeness is regarded by most interlocutors as a means or strategy which is used by a speaker to achieve various purposes,such as saving face, establishing and maintaining harmonious social relationsin conversation. This principle requires speakers to “minimize the expression of impolite beliefs”. It is composed of 6 maxims: Maxims of Tact, Generosity, Approbation, Modesty, Agreement and Sympathy.(10) conversational implicature: the use of conversational maxims in the Cooperative Principle to produce extra meaning during conversation.。
2.2.2反证法
警察局里有５名嫌疑犯，他们分别做了如下口供： Ａ说：这里有１个人说谎． Ｂ说：这里有２个人说谎． Ｃ说：这里有３个人说谎． Ｄ说：这里有４个人说谎． Ｅ说：这里有５个人说谎．
聪明的同学们，假如你是警察，你觉得谁说了真话？
你会释放谁？
请与大家分享你的判断！
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第29页，共29页。
第2章 推理与证明 2.2.2 间接证明（反证法）
第1页，共29页。
复习
利用已知条件和某些数学定义、公理、定理等,经
过一系列的推理论证,最后推导出所要证明的结论成 立,这种证明方法叫做综合法
用P表示已知条件、已有的定义、公理、定 理等,Q表示所要证明的结论.
则综合法用框图表示为:
P Q1
Q1 Q2 Q2 Q3
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练习1.求证方程2x＝3有且仅有一个实根． 证明：∵2x＝3， ∴x＝log23，这说明方程有一个根．
下面用反证法证明根的唯一性．
假设方程2x＝3有两个根b1，b2(b1≠b2)，则2b1＝ 3，2b2＝3，两式相除得2b1－b2＝1， 如果b1－b2>0，则2b1－b2>1，这与2b1－b2＝1相
常用反证法．
(2)常用的“原结论词”与“反设词”归纳如下表：
至多有一 至少有
原结论词 至少有一个
个
n个
至多有n个
反设词
一个也没有 至少有两 至多有
至少有n＋1个
(不存在)
个 n－1个
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5．已知 x，y>0，且 x＋y>2.
求证：1＋y x，1＋x y中至少有一个小于 2. 证明：假设1＋y x，1＋x y都不小于 2. 即1＋y x≥2，1＋x y≥2. ∵x>0，y>0，∴1＋x≥2y,1＋y≥2x. ∴2＋x＋y≥2(x＋y)， 即 x＋y≤2，这与已知 x＋y>2 矛盾． ∴1＋y x，1＋x y中至少有一个小于 2.
简明语言学整理笔记
第一章1.linguistics is generally defined as the scientific study of language2．The scope of linguisticsPhonetics-语音学phonology-音系学morphology-形态学syntax-句法学semantics-语义学pragmatics-语用学从语言形式划分：Sociolinguistics社会语言学，psycholinguistics心理语言学，applied linguistics应用语言学3. Important distinctions in linguisticsDescriptive &> prescriptive 规定性&描写性Synchronic & >diachronic 共时性&历时性Speech&> writing 口语&书写Langue & <parole 语言&言语Competence &< performance 语言能力&语言运用(Saussure and Chomsky think rule>language fact )Traditional grammer & modern linguistics4.What is language?Language is a system of arbitrary vocal symbols used for human communication5.Design features of language 语言的识别特征Charles Hockett①Arbitrariness(任意性)refers to the forms of linguistic signs bear no natural relationship to their meaning. (sounds and meanings)②Productivity/creativity(能产性）:Language is productive in that it makes possible the construction and interpretation of new signals by its users③Duality（双重性）:The property of having two levels of structures, such that units of the primary level are composed of elements of the secondary level and each of the two levels has its own principles of organization..④Displacement(移位性):Human Languages enable their users to symbolize objects, events and concepts which are not present (in time and space) at moment of communication.⑤ Cultural transmission（文化传承性）人独有。
逻辑代数的运算公式和规则.ppt
逻辑代数的运算公式和规则
• 公理、定律与常用公式
AA0•••010==0A= 0
A+ 1=1 A+ 0=A
0
+
0=0
A0•• A1==01 • 0A=+0A=1 0 + 1 =1 + 0 =1 A• A=A A+ A=A
A1••B1==A1+B A+ B1=A+B1 = 1
AB AC ABC ABC
AB(1 C) AC(1 B) AB AC 等式右边
由此可以看出：与或表达式中，两个乘积项分别包 含同一因子的原变量和反变量，而两项的剩余因子 包含在第三个乘积项中，则第三项是多余的
公式可推广： AB AC BCDE AB AC
逻辑代数的运算公式和规则
重叠律 反演律 还原律 合并律 吸收律 消因律 包含律
证明方法
利用真值表
例：用真值表证明反演律
A B AB A+ B A• B A+B
00 1
1
1
1
01 1
1
0
0
10 1
1
0
0
11 0
0
0
0
A• B= A+B A+ B=AB
利用基本定律
例：证明包含律 AB AC BC AB AC成立
AB AC BC AB AC (A A)BC
得到新函数式为原函数式F的对偶式F′，也称对偶函数
例• 如对：果偶两规个F则函：数A式B相 等A，C 则 1它•们B对应的对偶式也相
其对等数偶目。增式即加若F一'倍。F(1A=F2B) •则( AF1′=CF2)′•。(0使公B式)的
达梦数据库物化视图的设计与实现
华中科技大学硕士学位论文达梦数据库物化视图的设计与实现姓名：***申请学位级别：硕士专业：计算机软件与理论指导教师：***20070411摘要物化视图是一种将代价大、重复次数多的查询预先计算出来并用于回答用户查询的优化手段，在分析型应用环境中能够非常有效地提升数据库系统的响应速度。
为了满足联机分析处理(Online Analytical Processing, OLAP)、数据挖掘等分析性应用环境的要求，达梦(Da Meng, DM)数据库管理系统需要支持物化视图功能。
要实现对物化视图的支持，必须解决好视图维护和利用物化视图对用户查询进行优化这两个关键问题，并提供一个适合达梦数据库管理系统的实现方案。
针对分析性应用环境下物化视图引用的关系的变化量小的特点，采用增量的方式对物化视图进行维护比重新执行一次视图内容的效率要高很多。
为了在达梦数据库管理系统中实现物化视图的增量维护，并保证增量维护机制的正确性，为SPJ查询中的选择、投影、连接操作引出了8条的等价增量推导规则。
为了实现对分组聚集操作的维护，引入了两个专用附加操作符来更新由插入和删除操作所带来的聚集结果的变化。
利用关系代数理论对所给的增量维护机制的正确性进行了证明。
为了利用物化视图优化用户查询，首先利用系统中定义的物化视图生成所有与原查询等价的重写查询，然后在所生成的等价查询中选择一条代价最小的查询作为最终查询。
查询重写的过程分为对关系乘积、选择条件、分组属性列表和输出部分的重写4个阶段构成，为每个阶段定义了一个等价重写子过程，并对等价重写的正确性进行了证明。
在实现的过程中，对达梦数据库管理系统的框架进行了扩充和修改。
引入了物化视图对象并添加了物化视图定义器用于执行与物化视图相关的DDL(Data Definition Language)语句，维护保存物化视图元数据的数据字典。
添加了查询重写索引管理模块，将系统中的物化视图的定义用一棵称为查询重写索引(Query Rewrite Index, QTI)的多路搜索树组织起来，引入了利用物化视图进行查询重写的模块，对查询重写索引进行遍历生成所有可能的等价查询，利用代价估算器对等价查询的执行代价进行估算。
霍顿规则公式
霍顿规则公式霍顿规则公式是一种数学公式，用于计算复合概率。
通过这个公式可以精准地预测事件的概率，从而帮助我们做出更合理的决策。
在本文中，我们将详细介绍霍顿规则公式的概念、应用和计算方法。
一、概念霍顿规则公式是基于概率的分段函数。
该函数分为两个部分：一个是小概率事件，另一个是大概率事件。
基本上，该规则可以帮助我们将风险或概率分成两块：一个是低风险或低概率的部分，另一个是高风险或高概率的部分。
二、应用霍顿规则公式通常用于金融领域中，帮助我们计算投资的概率。
在具体应用中，霍顿规则公式可以用于预测股票、市场、债券和其他金融产品的走势。
此外，该公式也可以被用于预测一项新业务或产品的成功概率。
三、计算方法为了使用霍顿规则公式，我们需要有以下一些基本的数据：1. 恰好发生的事件实际次数；2. 该事件的样本空间大小；3. 该事件的期望发生次数。
然后，我们可以使用以下公式计算该事件的概率：P = （X+1）/（n+2）其中，P表示该事件概率；X表示事件实际发生次数；n表示样本空间大小。
在以上公式中，我们将事件的实际发生次数加上1，因为霍顿规则公式要求我们先分割样本，再计算每个子集的概率。
同时，我们将样本空间大小加上2，以考虑到样本的基本值和实际值之间的差异。
需要注意的是，在计算概率时，我们必须先对每个样本进行分类。
这就意味着我们需要将样本化为更容易处理的子集。
霍顿规则公式可以帮助我们将样本分为易于计算的小块，从而更好地评估每个子集的概率。
四、实例下面我们来看一个简单的例子，以更好地理解霍顿规则公式。
假设我们在一家银行借了一笔钱，并签署了一个五年的还款计划。
每个月，我们需要偿还500美元的本息，共还了60个月。
根据这些信息，我们可以使用霍顿规则公式来计算我们在未来五年内没有违约的概率。
在这个例子中，我们可以将“违约”事件分为两部分：小概率和大概率。
小概率事件的定义是：未来五年（60个月）内我们没有违约。
大概率事件的定义是：在未来五年（60个月）内，我们至少会违约一次。
欧几里得算法和贝祖等式-概述说明以及解释
欧几里得算法和贝祖等式-概述说明以及解释1.引言1.1 概述概述部分应该介绍欧几里得算法和贝祖等式的背景和基本概念，以及它们在数学领域中的重要性和应用。
以下是一种可能的写作方式：在数学领域，欧几里得算法和贝祖等式是两个重要的概念，它们在解决数学问题和计算问题上具有广泛的应用。
欧几里得算法，也被称为辗转相除法，是一种用于求解最大公约数的经典算法。
贝祖等式则是描述最大公约数和最小公倍数之间关系的一组等式。
欧几里得算法最早出现在古希腊数学家欧几里得的《几何原本》中，作为一个基本的算法，它在数学的各个领域都有重要的应用。
其核心思想是通过反复迭代地用较小数去除较大数，直到余数为零，而最后一个非零余数即为最大公约数。
这个算法简单高效，不仅可以用于求解最大公约数，还可以应用于模运算、寻找两个正整数的最小公倍数等各种问题。
而贝祖等式则是描述最大公约数和最小公倍数之间关系的一组等式。
其中最著名的一条等式是贝祖等式的核心内容，即对于给定的两个整数a 和b，存在整数x和y，使得ax + by = gcd(a, b)，其中gcd(a, b)表示a 和b的最大公约数。
贝祖等式不仅在数论和代数等纯数学领域中有广泛的应用，还可以解决诸如线性同余方程、如何表示整数线性组合等实际问题。
综上所述，欧几里得算法和贝祖等式作为数学领域中的两个基础概念，对于数学研究和实际问题求解具有重要的意义。
它们的应用价值不仅体现在数学理论的发展上，还能为我们解决实际问题提供有力的工具和方法。
1.2 文章结构文章结构部分的内容可以包括以下内容：文章结构的目的是为了给读者提供一个清晰的大局观，让他们能够更好地理解整篇文章的组织和内容安排。
本文分为引言、正文和结论三个部分。
接下来将详细介绍每个部分的内容。
引言部分包括概述、文章结构和目的。
- 概述部分简要介绍了文章要讨论的主题，即欧几里得算法和贝祖等式。
并提出了这两个主题的重要性和应用价值。
- 文章结构部分则是对整篇文章的整体安排进行了概述。
高中数学 第一章 常用逻辑用语 1.3 简单的逻辑联结词 德摩根定律素材 新人教A版选修21
德摩根定律德·摩根定律：在命题逻辑中存在着下面关系：非（P 且 Q）=（非 P）或（非 Q)非（P 或 Q）=（非 P）且（非 Q)2012年的逻辑真题形式逻辑相当多，而不少同学都觉得形式逻辑很难。
其实形式逻辑就是那几个公式。
1）否定词代入的命题等价转化2）p->Q 等价于非Q-》非p ，3）如果p 则q，只要p就q 等价于 p->q 等价于非p 或 Q只有p,才q 等价于 q->p除非p，否则q 等价于非q-》p4）相容选言和不相容选言的区别5）一些隐藏的形式逻辑的标志。
A必须B 等价于只有B 才有A =》A->B B是A的必要条件A是B的基础，A是B的前提，等价于只有有了A 才有B B->A A是必要条件A当且仅当B，A是B的唯一条件等价于 A->B所有的A 是B 等价于 A->BMBA逻辑知识点与记忆口诀汇总大秘送注意：逻辑要考察我们对语言文字的体察和敏感度。
逻辑知识点分三大类：一是逻辑推理能力，二是综合归纳能力，三是评价论证能力。
一、逻辑推理能力。
（20分）答案一定不用多看，但是要死记住口诀，全答对没问题。
包括11性质命题、12充分条件、13必要条件假言命题，14联言、15选言、16模态命题，17复合命题 18三段论二、综合归纳能力（10分）21语义解释题2-4分，22争论焦点，23推出结论8-10分。
三、评价论证能力：（30分以上）31假设、32支持、33削弱、34评价论证分析，35指出论证缺陷、论证方法。
11、性质命题：方图记住。
Especially:下反对关系中，可能同真，不可同假，一个为真，另一个真假不能确定，一个为假，另一个一定为真。
原命题等价于逆否命题。
同理可得，否命题等价于逆命题。
负命题就是矛盾命题。
排中律、同一律和矛盾律。
同一律是形式逻辑的基本规律之一，就是在同一思维过程中，必须在同一意义上使用概念和判断，不能混淆不相同的概念和判断．公式是：”甲是甲”或”甲等于甲”包括三方面的内容：(1)思维对象的同一。
逻辑代数基本公式与定律
反(原)变量
2. 反变量的吸收： A + A B = A + B
添冗余项
证明：左 式 A AB AB
A B(A A)
=右式
||
1
长中含反,
口诀： 去掉反。
（7）
互为反变量
3.混合变量的吸收： A B + A C + BC=AB+AC
证明：左式 AB AC BC AB AC (A A)BC
A•B AB
1
AB A•B
2
（10）
A•B AB
1
AB A•B
2
以（B·C）代入（1）式中B，以（B+C）代 入（2）式中B，则得到：
Α•(Β•C)Α(Β•C)Α Β C A(BC)A•(BC)A•B•C
注：代入定理还可以扩展其他基本定律 的应用范围！
F2 A• B A•C A• D•E
与或式
（14）
常用公式
1.消去公式：A+AB A B
2.吸收公式： A AB A
3.并项公式：AB AB A
4.多余项公式：AB AC BC AB AC
（15）
（16）
知识回顾 Knowledge
Review
0
0
1
1
0
1
1 相等 1
1
0
1
1
1
1
0
0
（5）
2.3.2 若干常用公式--几种形式的吸收律
吸收：多余（冗余）项，多余（冗余）因子被取消、去
掉 被消化了。
短项
长项
1.原变量的吸收： A + AB = A
逻辑代数基本定律规则及常用公式
逻辑代数基本定律规则及常⽤公式在四则运算中，我们知道有交换律、结合律以及分配律等。
那么在逻辑运算中，也有它⾃⼰的基本定律，下⾯将介绍逻辑代数运算中的基本定理。
逻辑代数基本定理1.0、1定律0、1定律描述的是单个变量A和0、1之间的运算规则。
其中有以下四条定律：（1）A·0=0，即A和0相与始终为0；（2）A·1=A，即A与1相与结果为A；（3）A+0=A，即A和0相或结果为A；（4）A+1=1，即A和1相或始终为1。
2.重叠律重叠率描述逻辑变量A和其⾃⾝的运算。
（1）A·A=A，即A和⾃⼰相与等于它本⾝；（2）A+A=A，即A和⾃⼰相或亦等于它本⾝。
3.互补律互补律描述A和⾃⾝的反变量¬A之间的关系。
（1）A·¬A=0，即A和⾃⾝反变量相与始终为0；（2）A+¬A=1，即A和⾃⾝反变量相或始终为1。
证明：由于A和¬A之间⾄少有⼀个为0，即⼆者不可能全为1，所以相与得0；同时，A和¬A之间⾄少有⼀个为1，满⾜或运算的“有1出1”，所以相或得0。
4.还原律A的反变量再取反，等于本⾝，即¬(¬A)=A。
5.交换律在此定律及之后的定律中，都将会涉及到两个及以上的逻辑变量。
交换律即两个逻辑变量运算时交换位置，结果不变。
（1）A·B=B·A，即A 与B等于B与A；（2）A+B=B+A，即A或B等于B或A。
6.结合律结合律指三个及以上变量相与或相或时，可以使任意两个变量先进⾏运算，再去和别的变量进⾏运算。
（1）(A·B)·C=A·(B·C)，即A与B后再与C，等于B与C后再与A。
（2）(A+B)+C=A+(B+C)，即A或B后再或C，等于B或C后再或A。
7.分配律逻辑代数的分配律和四则运算的分配律很类似，但是有⼀些不同。
（1）A·(B+C)=A·B+A·C，即A和B或C相与，等于A和B、C分别相与，然后进⾏或运算；（2）(A+B)·(A+C)=A+B·C，这⼀条定律显得有⼀些特殊，它的结果并不像四则运算中展开后有四项的形式，实际上，我们可以这样的得到：(A+B)·(A+C)=A·A+A·C+A·B+B·C=A+AC+AB+BC=A(1+B+C)+BC=A·1+BC=A+BC。
classifier free guidance推导解释
classifier free guidance推导解释
推导解释是指通过逻辑推理、演绎或归纳等方法，对一个问题或概念进行详细的解释和说明。
在推导解释中，我们从已知事实或前提出发，运用逻辑规则和推理方法，得出结论或解释。
这种推导过程可以使我们更加深入地理解问题的本质，并提供具体的证据或理由支持我们的解释。
推导解释通常包括以下几个步骤：
1. 提出问题或给出需要解释的概念。
2. 确定已知事实或前提。
3. 运用逻辑规则和推理方法，根据已知事实得出中间推论。
4. 进一步运用逻辑规则和推理方法，根据中间推论得出最终结论或解释。
5. 提供具体的证据、例子或论证，支持我们的结论或解释。
推导解释的目的是为了提供逻辑严密和有说服力的解释，使读者或听众能够更好地理解问题，并接受我们的观点或结论。
推导解释在科学研究、数学推理、哲学思考等领域都有广泛应用，可以帮助我们深入思考和分析问题，促进知识的进一步发展和推广。
离散数学定义定理
离散数学定义定理1.3.1命题演算的合式公式规定为：（1）单个命题变元本身是一个合式公式。
（2）如果A是合式公式，那么┐A是合式公式。
（3）如果A和B是合式公式，那么（A∨B）、（A∧B）、（A→B）、（A B）、都是合式公式。
（4）当且仅当有限次地应用（1）（2）（3）所得到的包含命题变元，连接词和圆括号的符号串是合式公式。
1.3.2 设A i是公式A的一部分，且Ai是一个合式公式，称A i是A的子公式。
1.3.3 设P为一命题公式，P1,P2,……，P n为出现在P中的所有命题变元，对P1,P2,……，P n指定一组真值称为对P的一种指派。
若指定的一种指派，使P的值为真，则称这组指派为成真指派。
若指定的一种指派，使P的值为假，则称这种指派为成假指派。
含n个命题变元的命题公式，共有2n个指派。
1.3.4 给定两个命题公式A和B，设P1,P2,……，P n为所有出现于A和B中的原子变元，若给P1,P2,……，P n任一组真值指派，A和B的真值都相同，称A和B是等价的，记做A <=>B。
1.3.5 设A为一命题公式，若A在它的各种指派情况下，其取值均为真，则称A为重言式或永真式。
1.3.6 设A为一命题公式，若A在它的各种指派情况下，其取值均为假，则称A为矛盾式或永假式。
1.3.7设A为一命题公式，若A在它的各种指派情况下至少存在一组成真指派，则称A为可满足式。
1.4.1 设X式合式公式A的子公式，若有Y也是一个合式公式，且X<=>Y，如果将Ａ中的Ｘ用Ｙ置换，得到公式Ｂ，则A<=>B。
1.4.2 设A，B为两个命题公式，A<=>B，当且仅当A ←→B为一个重言式。
P=>Q称做P蕴含Q或蕴含式，又称永真条件式。
蕴含式有下列性质：（1）对任意公式A，又A=>A；（2）对任意公式A，B和C，若A=>B，B=>C，则A=>C；（3）对任意公式A，B和C，若A=>B，A=>C，则A=>(B∧C);（4）对任意公式A，B和C，若A=>C，B=>C，则A∨B=>C.1.5.1 一个命题公式称为合取范式，当且仅当它具有形式：A1∧A2∧…∧A n(n≥1)，其中A1，A2，…，A n都是有命题变元及其否定所组成的析取式。