General Description (概要)
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Foundation of sets and algebraic structures basic to modern mathematics. Includes sets and mappings, equivalence relations and equivalence classes, cardinal, and algebraic structures. One period of lecture and two periods of recitation weekly.
現代数学の基本概念のなかから，集合および代数系の基礎を学ぶ。集合と写像，同値関係と同値類，濃度 および代数系。毎週講義1時限，演習2時限。
Students will learn sets and algebraic structures, which are among the basic concepts of
modern mathematics necessary to learn them as well as mathematical sciences.
Basic trainings will be given in order to properly use definitions and notation in mathematics
and to carry out proofs with clear logic.
This course is not only for students majoring in mathematics.
It offers hints for formulating questions properly in suitable terms.
数学および数理科学を学ぶために必要な現代数学の基礎概念の中から、
集合および代数系の基礎を学ぶ。特に、数学における定義と記号の明確な使用、
厳密な論理による証明の基礎的訓練を行なう。
数学を専門とする人の為のみならず、適切な問題設定の仕方を考える際のヒントがある。
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Learning Goals(学習目標)
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Students learn basics of sets, maps, equivalence classes, formal logic,
and also elementary properties of cardinality of infinite sets,
sets with algebraic structures. In the course, one becomes accustomed to
the formalism of definitions, propositions and proofs which is indispensable
to develop modern mathematics.
集合・写像・同値類・記号論理などの基本事項を学び、無限を扱う集合の濃度、
代数的構造の付いた集合(代数系)についての基本的な性質を学ぶ。
同時に、現代数学を展開するのに必要な定義・命題・証明の形式を学ぶ。
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Contents(内容)
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1. Sets and logic
2. Direct Proof and Proof by Contrapositive
3. Mathematical Induction
4. Relations, Equivalence Relations
5. Functions
6. Cardinalities of Sets
7. Division and Congruences of Integers
Please vist the homepage given below.
1. 集合と論理
2. 直接証明と、対偶による証明
3. 数学的帰納法
4. 関係・同値関係
5. 写像
6. 集合の濃度
7. 初等整数論
詳細は、下記ホームページを参照。
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Language of Instruction(教授言語の詳細)
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Lecture: English
Readings/Materials: Mostly English
Tests/Quizzes/Assignments: English
Discussions/Presentations/Other learning activities: Use of English is encouraged, but questions and presentations in Japanese are acceptable.
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Communication with the instructor:
We learn how to think logically and mathematically and write mathematical proofs.
Most of the lectures will be given in English. Textbook, blackboard writings, assignments, final will be given in English. At recitations, homework and final, students are encouraged to use English but also allowed to use Japanese.
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Grading Policy(成績評価基準)
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Grades are decided based on homework (20%), explanation of solutions at recitation (30%) and a final exam (50%).
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Expected study hour outside class(授業時間外学習)
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140 minutes per week.
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References(参考文献)
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Textbook: "Mathematical Proofs, A Transition to Advanced Mathematics, Third Edition" by Gary Chartrand, Albert D. Polimeni, Ping Zhang, Pearson International Edition (ISBN13: 978-0321782519. Similar to Second Edition)
Textbook will be available at the University Bookstore, 0. Communicating Mathematics, 1. Logic, 2. Sets, 3. Direct Proof and Proof by Contrapositive, 4. More on Direct Proof and Proof by Contrapositive, 5. Existence and Proof by Contradiction, 6. Mathematical Induction, 7. Prove or Disprove, 8. Equivalence Relation, 9. Functions, 10. Cardinalities of Sets, 11. Proofs in Number Theory, 12. Proofs in Calculus, 13. Proofs in Group Theory. We plan to cover Chap. 1-11. Please read Chapter 0 by the first class. We will discuss Chapter 2 on Logic, and Chapter 3 on Sets on the first day and solve problems in Chapter 3.
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Learning Support Resources for Students (学生のための学修支援リソース)
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If there are learning support resources that are especially recommended for this course, they will be listed below.
Here (ICU Internal page) is the list of learning support resources available at ICU.
このコースで特に利用を推奨する学修支援リソースがある場合、以下に記載されます。
ICUで利用可能なリソースの一覧は こちらです （学内ウェブサイト）。
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Notes(注意事項)
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Basically, in every week, the fifth period is for lecture and recitation during the 6th and the seventh periods.
Connection with other subjects:
The contents of Basic Concepts in Math series are foundation to all area
in modern mathematics, and the key course of a transition to advanced mathematics.
Basic Concepts in Math (BCM) I, dealing with "sets" and "algebraic structures",
is the first course to start. In the second and third terms, BCM II is for the foundation of analysis starting from the continuity of real numbers,
and BCM III is for "general topology" providing as the starting point of modern geometry, are offered.
This course is strongly recommended for those who are to take BCM II or III.
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Schedule(スケジュール)
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5/W,(6/W,7/W)
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URL
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https://icu-hsuzuki.github.io/science/class/bcmm1/index.html
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