어떻게 증명할 것인가

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