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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 |
저자 소개 2
『어떻게 증명할 것인가』저
ㆍ서울대학교 사범대학 수학교육과 (이학사)
ㆍ전북대학교 대학원 수학과 (이학박사)
ㆍ일본 나고야대학 수학과 Post Doc
ㆍ미국 미시간 주립대학, 미주리 주립대학, 오래곤 주립 대학 연구교수
ㆍ전남대학교 사범대학 수학교육과 명예교수
ㆍ저서로 『수학영재교육 길잡이』, 『현대대수학』
ㆍ역서로 『갈루아 방정식원론』
박종률의 다른 상품
품목정보
- 발행일
- 2023년 10월 31일
- 쪽수, 무게, 크기
- 610쪽 | 188*257*35mm
- ISBN13
- 9791190854122
출판사 리뷰
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“매력적으로 저술되고, 그리고 명료하고 대단히 자세하게 많은 증명 기술들을 다루고 있다. 이 책은 내가 학생들에게 추천한 자료들의 한 위대한 추가 자료가 될 것이다.”
Steven J. Miller, Dept. of Mathematics, Williams College “수학적 증명들을 어떻게 할 것인지 어려워하는 학생들을 위한 나의 대단히 믿음직한 자료이다.” Rafael Frongillo, Dept. of Computer Science, University of Colorado-Boulder “나는 이 책을 지난 8년 동안 진지하게 이용해왔다. 그것은 증명 쓰기의 한 강력한 기반을 이룩하였다 나는 이 책을 계속 이용할 것이고 모든사람들, 교수들과 학생들 모두에게 추천한다.” Mihai Bailesteanu, Dept. of Mathematical Sciences, Central Connecticut State University 증명들은 고급수학들과 컴퓨터 과학 이론에 핵심적인 역할을 한다, 그렇지만 많은 학생들은 처음에 증명들이 중요한 역할을 하는 과목을 이수하는데 힘들어한다. 베스트 셀러 교재인 3판은 그들에게 증명들을 읽고 쓰는데 필요한 기술들을 가르쳐서 학생들이 문제 푸는 것으로부터 정리들을 증명하는 것으로 변화시키도록 도와줄 것이다. 150개 넘는 새로운 연습문제들과 수론에 대한 한 새로운 장을 포함하는, 이 증보 판에서 학생들에게 증명들의 숙달을 통해서 고급수학의 세상을 소개한다. 이 책은 수학적인 용어에 익숙해지고 그것들이 어떻게 해석하는지에 관한 논리의 기본개념과 집합론으로 시작한다. 이런 개념들은 단계적으로 복잡한 증명들을 해나가는데 사용될 수 있는 분석 기술에 대한 기초로 사용되고, 수들, 집합들, 관계들, 그리고 함수들에 대한 증명들의 절차를 노출 시키는 “메모작업 scratch work"부분에서 상세하게 사용된다. 보통 고등학교 수학을 넘는 배경지식이 없어도, 이 책은 논리와 증명들에 관심이 있는 모든 사람들: 컴퓨터 과학자들, 철학자들, 언어학자들, 그리고 물론, 수학자들에게 유용하게 사용될 것이다. · 증명들을 소개하는 과목, 또는 이산수학 과목의 보조 책으로, 또는 컴퓨팅의 기초 과목으로 자기주도학습 하기에 완벽하다. · 복잡한 증명을 구성하는데 다양한 기술들이 어떻게 체계적으로 철저하게 결합되는지를 보여준다. · 논리, 집합론, 관계들, 함수들, 그리고 집합의 원소 개수 세기 등을 다룬다 |
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