어떻게 증명할 것인가
$51.75
SKU
9791190854122
[Free shipping over $100]
Standard Shipping estimated by Fri 12/6 - Thu 12/12 (주문일로부 10-14 영업일)
Express Shipping estimated by Tue 12/3 - Thu 12/5 (주문일로부 7-9 영업일)
* 안내되는 배송 완료 예상일은 유통사/배송사의 상황에 따라 예고 없이 변동될 수 있습니다.
| Publication Date | 2023/10/31 |
| Pages/Weight/Size | 188*257*35mm |
| ISBN | 9791190854122 |
| Categories | 대학교재 > 공학계열 |
Description
이 책은 Daniel J. Velleman 교수의 저서인 어떻게 증명할 것인가 -구조적 접근 How to prove It-A structured Approach의 제3판(2019)을 번역한 것이다. 벨만 교수가 1994년에 초판을 발간했으며, 2006년에 제2판을 발간하면서 200여개의 연습문제들을 추가했고 증명 설계자 Proof Designer소프트웨어를 사용하여 문제풀이를 하도록 했다. 그리고 제3판에서는 수론에 대한 한 새로운 장과 역시 150개 이상의 많은 연습문제들을 추가했다.
이 책은 수학적 명제들이 가질 수 있는 다양한 형식들에 학생들이 친근하게 접근하게 하는 기초적인 논리로 시작한다. 제1장은 논리적 연결사들을 논의하고, 제2장은 한정기호들을 소개했다. 이 장들에서 집합론의 기초개념들을 제시한다. 그것들이 이 책의 나머지 부분에서 사용되는 중요한 주제이기 때문이다. 제3장은 한 조직적인 방법으로 수학적 명제들이 가질 수 있는 다양한 양식들을 통하여 적절한 증명구조들을 논의하고 그 증명들은 수학적으로 더 복잡해지고 또한 더 흥미로워질 것이다. 제4 그리고 5장에서는, 관계들과 함수들에 대한 것으로, 학생들이 3장으로부터 증명을 쓰는 기술들을 실행하는 주제를 제공하고, 수학 전체 영역에서 사용되는 약간의 기본적인 개념들을 소개한다. 제6장은 수학과 컴퓨터 과학에서 아주 중요한 증명방법인 수학적 귀납법을 다룬다. 마지막으로, 제7장과 제8장에서는 정수론의 기초개념과 무한집합의 위수 정리를 소개하고 있다. 이것들은 학생들에게 가장 흥미로운 정리들을 함께 가져다 줄 것이다.
저자는 수학 문제해결과 정리들의 증명에서 학생들이 증명의 구성과 관련된 밑바닥에 있는 원리들을 밝혀냄으로써 수학하는 방법에 대한 해답들을 주고자 의도하고 있다. 증명들은 고급수학들과 컴퓨터 과학 이론에 핵심적인 역할을 한다, 많은 학생들은 처음에 증명들이 중요한 역할을 하는 과목을 이수하는데 힘들어 한다. 그들에게 증명들을 읽고 쓰는데 필요한 기술들을 가르쳐서 학생들이 문제 푸는 것으로부터 정리들을 증명하는 것으로 변화시키도록 도와주고 증명들의 숙달을 통해서 고급수학의 세상을 소개한다. 이 책은 수학적인 용어에 익숙해지고 그것들이 어떻게 해석하는지에 관한 논리의 기본개념을 단계적으로 소개하고, 복잡한 증명들을 해나가는데 사용될 수 있는 분석 기술과 증명들의 절차를 “메모작업 scratch work"부분에서 상세하게 소개한다.
또한, 컴퓨터 과학에서 일찍이 프로그램잉 언어들의 지침에 번호가 붙은 목록으로 쉽게 접근하도록 가르친다. 이 책에서도 증명들에 대한 논의와 증명쓰기 에서도 유사하게 “구조적인 증명”을 받아드리도록 소개한다. 과학 문명의 발전과 더불어 우리의 삶 속에서 바르게 사고하고 거짓된 논증과 모호한 주장에 동조하지 않는 지적 정직성을 갖추도록 논리적인 증명방법을 알야야한다.
보통 고등학교 수학을 넘는 배경지식이 없어도, 이 책은 논리와 증명들에 관심이 있는 모든 사람들: 컴퓨터 과학자들, 철학자들, 언어학자들, 그리고 물론, 수학자들에게 유용하게 사용되기를 기대한다.
이 책은 수학적 명제들이 가질 수 있는 다양한 형식들에 학생들이 친근하게 접근하게 하는 기초적인 논리로 시작한다. 제1장은 논리적 연결사들을 논의하고, 제2장은 한정기호들을 소개했다. 이 장들에서 집합론의 기초개념들을 제시한다. 그것들이 이 책의 나머지 부분에서 사용되는 중요한 주제이기 때문이다. 제3장은 한 조직적인 방법으로 수학적 명제들이 가질 수 있는 다양한 양식들을 통하여 적절한 증명구조들을 논의하고 그 증명들은 수학적으로 더 복잡해지고 또한 더 흥미로워질 것이다. 제4 그리고 5장에서는, 관계들과 함수들에 대한 것으로, 학생들이 3장으로부터 증명을 쓰는 기술들을 실행하는 주제를 제공하고, 수학 전체 영역에서 사용되는 약간의 기본적인 개념들을 소개한다. 제6장은 수학과 컴퓨터 과학에서 아주 중요한 증명방법인 수학적 귀납법을 다룬다. 마지막으로, 제7장과 제8장에서는 정수론의 기초개념과 무한집합의 위수 정리를 소개하고 있다. 이것들은 학생들에게 가장 흥미로운 정리들을 함께 가져다 줄 것이다.
저자는 수학 문제해결과 정리들의 증명에서 학생들이 증명의 구성과 관련된 밑바닥에 있는 원리들을 밝혀냄으로써 수학하는 방법에 대한 해답들을 주고자 의도하고 있다. 증명들은 고급수학들과 컴퓨터 과학 이론에 핵심적인 역할을 한다, 많은 학생들은 처음에 증명들이 중요한 역할을 하는 과목을 이수하는데 힘들어 한다. 그들에게 증명들을 읽고 쓰는데 필요한 기술들을 가르쳐서 학생들이 문제 푸는 것으로부터 정리들을 증명하는 것으로 변화시키도록 도와주고 증명들의 숙달을 통해서 고급수학의 세상을 소개한다. 이 책은 수학적인 용어에 익숙해지고 그것들이 어떻게 해석하는지에 관한 논리의 기본개념을 단계적으로 소개하고, 복잡한 증명들을 해나가는데 사용될 수 있는 분석 기술과 증명들의 절차를 “메모작업 scratch work"부분에서 상세하게 소개한다.
또한, 컴퓨터 과학에서 일찍이 프로그램잉 언어들의 지침에 번호가 붙은 목록으로 쉽게 접근하도록 가르친다. 이 책에서도 증명들에 대한 논의와 증명쓰기 에서도 유사하게 “구조적인 증명”을 받아드리도록 소개한다. 과학 문명의 발전과 더불어 우리의 삶 속에서 바르게 사고하고 거짓된 논증과 모호한 주장에 동조하지 않는 지적 정직성을 갖추도록 논리적인 증명방법을 알야야한다.
보통 고등학교 수학을 넘는 배경지식이 없어도, 이 책은 논리와 증명들에 관심이 있는 모든 사람들: 컴퓨터 과학자들, 철학자들, 언어학자들, 그리고 물론, 수학자들에게 유용하게 사용되기를 기대한다.
Contents
도입 Introduction
1장. 문장제 논리 Sentential Logic ············································································· 17
1.1. 연역적 추론 Deductive Reasoning과
논리적 연결사 Logical Connectives ····································································· 18
1.2. 진리표 Truth Tables ······························································································ 25
1.3. 변수들과 집합 Variables and Sets ······································································· 40
1.4. 집합의 연산 Operations on Sets ·········································································· 51
1.5. 조건부 그리고 쌍조건부 연결사들
The Conditional and Biconditional Connectives ················································· 62
2장. 한정작용소 논리 Quantificational Logic ······························································ 77
2.1. 한정작용소들 Quantifiers ······················································································· 78
2.2. 복잡한 한정작용소들의 동치들 Equivalences Involving Quantifiers ··················· 89
2.3. 집합들에서 더 많은 연산들 More Operations on Sets ····································· 101
3장. 증명 Proof ··········································································································· 115
3.1. 증명의 전략들 Proof Strategies ·········································································· 116
3.2. 부정명제들과 조건문들이 포함된 증명들
Proofs involving Negations and Conditionals ················································· 129
3.3. 한정작용소가 관련된 증명들 Proofs Involving Quantifiers ······························· 146
3.4. 논리곱(합접)과 쌍조건문이 관련된 증명들
Proofs Involving Conjunctions and Biconditionals ··········································· 169
3.5. 이접(논리합)이 관련된 증명들 Proofs Involving Disjunctions ·························· 185
3.6. 존재성과 유일성의 증명들 Existence and Uniqueness Proofs ·························· 200
3.7. 증명들에 대한 추가 예들 More Examples of Proofs ······································· 212
4장. 관계 Relations ····································································································· 225
4.1 순서쌍과 곱집합 Ordered Pairs and Cartesian Products ··································· 226
4.2. 관계 Relations ····································································································· 236
4.3 더 많은 관계들 More about Relations ······························································ 248
4.4 순서 관계들 Ordering Relations ·········································································· 260
4.5 동치관계 Equivalence Relations ·········································································· 279
5장. 함수 Functions ·································································································· 297
5.1. 함수 Functions ··································································································· 298
5.2. 일대일 함수와 위에로의 함수 One-to-one and Onto ········································ 311
5.3. 역함수 Inverses of Functions ··········································································· 324
5.4. 폐포들 Closures ··································································································· 337
5.5 상과 역상 Images and Inverse Images:
한 연구 계획 A Research Project ······································································ 349
6장. 수학적 귀납법 Mathematical Induction ····························································· 355
6.1. 수학적 귀납법에 의한 증명 Proof by Mathematical Induction ························ 356
6.2. 더 많은 예들 More Examples ············································································ 365
6.3. 점화식 Recursion ································································································· 382
6.4. 강한 귀납법 Strong Induction ············································································ 396
6.5. 폐포를 다시보자. Closures again ········································································ 412
7장. 정수론 Number Theory ···················································································· 421
7.1. 최대공약수 Greatest Common Divisors ····························································· 422
7.2. 소인수분해 Prime Factorization ········································································ 432
7.3. 모듈러 산술 Modular Arithmetic ······································································· 445
7.4. 오일러의 정리 Euler's Theorem ·········································································· 458
7.5. 공개-키 암호론 Public-key Cryptography ·························································· 470
8장. 무한 집합 Infinte Sets ························································································ 487
8.1. 대등한 집합 Equinumerous Sets ········································································· 488
8.2. 가산집합과 비가산집합 Countable and Uncountable Sets ································ 501
8.3. 칸토어-쉴뢰더-베른쉬타인 정리
The Cantor-Schroder-Bernstein Theorem ··························································· 512
부록1: 선택된 연습문제 풀이 모음 Solutions to Selected Exercises ······················· 523
부록2: 증명 설계자 Proof Designer ··········································································· 599
부록3: 참고문헌 Suggestions for Further Reading ··················································· 601
부록4: 증명 기술들의 정리 Summary of Proof Techniques ···································· 602
1장. 문장제 논리 Sentential Logic ············································································· 17
1.1. 연역적 추론 Deductive Reasoning과
논리적 연결사 Logical Connectives ····································································· 18
1.2. 진리표 Truth Tables ······························································································ 25
1.3. 변수들과 집합 Variables and Sets ······································································· 40
1.4. 집합의 연산 Operations on Sets ·········································································· 51
1.5. 조건부 그리고 쌍조건부 연결사들
The Conditional and Biconditional Connectives ················································· 62
2장. 한정작용소 논리 Quantificational Logic ······························································ 77
2.1. 한정작용소들 Quantifiers ······················································································· 78
2.2. 복잡한 한정작용소들의 동치들 Equivalences Involving Quantifiers ··················· 89
2.3. 집합들에서 더 많은 연산들 More Operations on Sets ····································· 101
3장. 증명 Proof ··········································································································· 115
3.1. 증명의 전략들 Proof Strategies ·········································································· 116
3.2. 부정명제들과 조건문들이 포함된 증명들
Proofs involving Negations and Conditionals ················································· 129
3.3. 한정작용소가 관련된 증명들 Proofs Involving Quantifiers ······························· 146
3.4. 논리곱(합접)과 쌍조건문이 관련된 증명들
Proofs Involving Conjunctions and Biconditionals ··········································· 169
3.5. 이접(논리합)이 관련된 증명들 Proofs Involving Disjunctions ·························· 185
3.6. 존재성과 유일성의 증명들 Existence and Uniqueness Proofs ·························· 200
3.7. 증명들에 대한 추가 예들 More Examples of Proofs ······································· 212
4장. 관계 Relations ····································································································· 225
4.1 순서쌍과 곱집합 Ordered Pairs and Cartesian Products ··································· 226
4.2. 관계 Relations ····································································································· 236
4.3 더 많은 관계들 More about Relations ······························································ 248
4.4 순서 관계들 Ordering Relations ·········································································· 260
4.5 동치관계 Equivalence Relations ·········································································· 279
5장. 함수 Functions ·································································································· 297
5.1. 함수 Functions ··································································································· 298
5.2. 일대일 함수와 위에로의 함수 One-to-one and Onto ········································ 311
5.3. 역함수 Inverses of Functions ··········································································· 324
5.4. 폐포들 Closures ··································································································· 337
5.5 상과 역상 Images and Inverse Images:
한 연구 계획 A Research Project ······································································ 349
6장. 수학적 귀납법 Mathematical Induction ····························································· 355
6.1. 수학적 귀납법에 의한 증명 Proof by Mathematical Induction ························ 356
6.2. 더 많은 예들 More Examples ············································································ 365
6.3. 점화식 Recursion ································································································· 382
6.4. 강한 귀납법 Strong Induction ············································································ 396
6.5. 폐포를 다시보자. Closures again ········································································ 412
7장. 정수론 Number Theory ···················································································· 421
7.1. 최대공약수 Greatest Common Divisors ····························································· 422
7.2. 소인수분해 Prime Factorization ········································································ 432
7.3. 모듈러 산술 Modular Arithmetic ······································································· 445
7.4. 오일러의 정리 Euler's Theorem ·········································································· 458
7.5. 공개-키 암호론 Public-key Cryptography ·························································· 470
8장. 무한 집합 Infinte Sets ························································································ 487
8.1. 대등한 집합 Equinumerous Sets ········································································· 488
8.2. 가산집합과 비가산집합 Countable and Uncountable Sets ································ 501
8.3. 칸토어-쉴뢰더-베른쉬타인 정리
The Cantor-Schroder-Bernstein Theorem ··························································· 512
부록1: 선택된 연습문제 풀이 모음 Solutions to Selected Exercises ······················· 523
부록2: 증명 설계자 Proof Designer ··········································································· 599
부록3: 참고문헌 Suggestions for Further Reading ··················································· 601
부록4: 증명 기술들의 정리 Summary of Proof Techniques ···································· 602