Course Information

Lectures: MWF at 9am in our Zoom Meeting Room (Main Lecture).

Textbook: The Art of Proof by M. Beck and R. Geoghegan. The textbook should be accessible through the UC Davis Equitable Access Program. In addition, there is a version available in the authors' website.

Syllabus: The course syllabus contains the basic guidelines for the course.

Office Hours: Mondays 4-5pm and Wednesday 4-5pm (Casals) at the corresponding Zoom Meeting Rooms. Always feel free to ask after class.
The only three exceptions this Fall are Mondays Oct 12, Nov 2 and Nov 23 where Office Hours will be 3-4pm.

Sari Ogami's Office Hours are 2-3pm on Mondays and 4:00-5:00pm on Tuesdays. Shanon Rubin's Office Hours are 11-12pm on Tuesdays and Thursdays.

Recitations: Thursdays at 3:10pm, for Section 1, at 4:10pm, for Section 2, at 5:10pm, for Section 3, and at 6:10pm for Section 4.

Important Dates: First Day (Sep 30), Midterm Test (Oct 30), Final Project (Due week of Dec 14).

Additional dates: Add Date (Oct 15), Drop Date (Oct 27), Veterans Day (Nov 11), Thanksgiving Holiday (Nov 27), Last Day of Instruction (Dec 11).

Problem Sets: Weekly assignments are due on Fridays at the beginning of class, to be submitted through Gradescope.

Problem Sets

1. Problem Set 1: Due Friday Oct 9 and available Tue Sep 29.
   
   
2. Problem Set 2: Due Friday Oct 16 and available Sun Oct 4.
   
   
3. Problem Set 3: Due Friday Oct 23 and available Wed Oct 14.
   
   
4. Practice Problems I: practice only, no due date, available Thu Oct 22.
   
   
5. Practice Problems II: practice only, no due date, available Thu Oct 22.
   
   
6. Mock Midterm: practice only, no due date, available Wed Oct 28.
   
   
7. Midterm Exam: Due Friday Oct 30, and available Fri Oct 30.
   
   
8. Problem Set 4: Due Friday Nov 13 and available Wed Nov 4.
   
   
9. Problem Set 5: Due Friday Nov 20 and available Wed Nov 11.
   
   
10. Problem Set 6: Due Friday Dec 4 and available Sun Nov 22.
    
    
11. Final Project: Due Sunday Dec 13 and available Wed Nov 11.
    
    

Solutions

1. Solutions Problem Set 1: Available Friday Oct 9.
   
   
2. Solutions Problem Set 2: Available Friday Oct 16.
   
   
3. Solutions Problem Set 3: Available Saturday Oct 24.
   
   
4. Solutions Practice Problems I: Available Sun Oct 25.
   
   
5. Solutions Practice Problems II: Available Sat Oct 24.
   
   
6. Solutions Midterm Exam: Available Fri Oct 30.
   
   
7. Solutions Problem Set 4: Available Tuesday Nov 17.
   
   
8. Solutions Problem Set 5: Available.
   
   

Math Diaries

1. Wednesday Sep 30: Introduction to MAT-108.
   Introduction to proofs. Examples of Mathematical Statements.
   The REP Principle. Anatomy of a Statement and Axioms.
   
   The Blackboards for Lecture 1.
   
   
2. Friday Oct 2. Proofs by contradiction.
   
   Axiomatic for the Integers. Proof of Prop. 1.6.
   Axiomatic for the Naturals.
   
   Textbook Reading (Oct 2): Chapter 1 and Section 2.1.
   The Blackboards for Lecture 2.
   
   
3. Monday Oct 5: Principle of Mathematical Induction.
   
   Ordering of the Integers. The Axiom of Induction (Axiom 2.15).
   Principle of Induction (Theorem 2.17). Example of Divisibility.
   
   Textbook Reading (Oct 5): Section 2.2 and Section 2.3.
   The Blackboards for Lecture 3.
   Informal OH Notes of Oct 5.
   
   
4. Wednesday Oct 7: Principle of Mathematical Induction II.
   
   Proof of the Principle of Induction (Theorem 2.17).
   Problem on Divisibility: Proof by Induction No.1.
   Closed Formula for the Sum of Odd Numbers: Proof by Induction No.2.
   
   Textbook Reading (Oct 7): Section 2.3.
   The Blackboards for Lecture 4.
   Informal OH Notes of Oct 7.
   
   
5. Friday Oct 9: Problems and Applications.
   
   The sum of odd Integers. The sum of odd Integers.
   Inequalities with induction. Proof by contradiction: infinitely many primes.
   
   Textbook Reading (Oct 9): Section 2.3 (and Optional 2.4).
   The Blackboards for Lecture 5.
   
   This concludes Chapters 1 and 2 of the Textbook. The most important tool you have learnt this far are Proofs By Induction.
   There are at least three types of situations in which induction can be applied: inequalities, divisibility problems and closed formulas for sums.
   
   
   1. Formulas and Inequalities depending on a natural number: Examples are Prop. 2.20, Prop. 2.24, Prop. 2.26, Prop. 2.27 and Problems 4 and 5 in Problem Set 2.
      We did more examples in class, such as the sum of the first n odd natural numbers and the sum of the first n even natural numbers.
      Some Real Life Examples are Problems 6 and 7 in Problem Set 2
      
      
   On Monday Oct 12, we will start Chapter 4 of the Textbook, in the topic of Recursion. (Chapter 3 in the textbook is only recommended reading.)
   
   
6. Monday Oct 12: Introduction to Recursion.
   
   Sequences. Discussion on next term versus closed formula.
   Examples of sequences. Linear Recursion. Sums and Products.
   Geometric sequences. Geometric Series. Proof of Proposition 4.13.
   
   Textbook Reading (Oct 12): Sections 4.1 and 4.2.
   The Blackboards for Lecture 6.
   Informal OH Notes of Oct 12.
   
   
7. Wednesday Oct 14: Binomial Theorem and Factorials.
   
   Definition of Factorial and its Combinatorial Interpretation.
   Definition of Binomial Coefficients and its Combinatorics.
   Combinatorial Interpretation of a Binomial Coefficient.
   Corollary 4.20. The Binomial Theorem (Theorem 4.21).
   
   Textbook Reading (Oct 14): Section 4.4.
   The Blackboards for Lecture 7.
   Informal OH Notes of Oct 14.
   
   
8. Friday Oct 16: Closed Formula For Linear Recursions.
   
   Linear recursions. Examples and Non-Example.
   Characteristic polynomial. Recipe for degree 2 linear recursions.
   Proof of Proposition 4.29: Closed Formula for Fibonacci Numbers.
   
   Textbook Reading (Oct 16): Section 4.6.
   The Blackboards for Lecture 8.
   
   
   
9. Monday Oct 19: Notions of Set Theory.
   
   Examples of Sets. Inclusions, Intersections and Unions.
   Equalities of Sets as double inclusions. Examples. Complements.
   De Morgan's Laws for Complements. Proof of Theorem 5.15.
   
   Textbook Reading (Oct 19): Sections 5.1 and 5.2.
   The Blackboards for Lecture 9.
   Informal OH Notes of Oct 19.
   
   
10. Wednesday Oct 21: Equivalence Relations.
    
    Equivalence Relations: Definitions, Examples and Non-Examples.
    The Equivalence Relation "Modulo n". Relations mod 2, mod 3 and mod 5.
    Proof of Proposition 6.24: Modulo n is an equivalence relation.
    
    Textbook Reading (Oct 21): Sections 6.1, 6.2 and 6.3.
    The Blackboards for Lecture 10.
    Informal OH Notes of Oct 21.
    
    
11. Friday Oct 23: Fermat's Little Theorem.
    
    Problems Solved by Modular Arithmetic. Sums and Products.
    Proposition 6.25: Arithmetic Operations Modulo n. Examples.
    Powers modulo p: Freshman's Dream Lemma (Prop. 6.34) and proof.
    Fermat's Little Theorem and Application to divisibility.
    
    Textbook Reading (Oct 23): Sections 6.3 and 6.4.
    The Blackboards for Lecture 11.
    
    
12. Monday Oct 26: Problem-solving with Modular Arithmetic.
    
    Problems with modular arithmetic. Last digit: modulo 10 arithmetic.
    Non-Existence of integral solutions of Diophantine Equations.
    Divisibility Problems under the light of modular arithmetic.
    
    Textbook Reading (Oct 23): Sections 6.3 and 6.4.
    The Blackboards for Lecture 12.
    Informal OH Notes of Oct 26.
    
    
13. Wednesday Oct 28: Review and Mock Midterm.
    
    Proofs by Contradicton. Infinitely Many Primes.
    Induction and Recursion. Problems and Examples.
    
    The Blackboards for Lecture 13.
    Informal OH Notes of Oct 28.
    
    
14. Friday Oct 30: Midterm Exam.
    
    
    
15. Monday Nov 2: The Real Numbers.
    
    Axioms for the real numbers (8.1-8.5). Positive Real Numbers.
    Theorem 8.42: Non-existence of smallest positive real. Lower and Upper bounds.
    
    Textbook Reading (Nov 2): Chapter 8.
    The Blackboards for Lecture 14.
    
    
16. Wednesday Nov 4: Bounds, supremum and infimum.
    
    Infimums and Supremums: Examples and uniqueness (Prop. 8.45).
    The Completeness Axiom (8.52). Bounds for different types of intervals.
    Proposition showing that sup([a,b])=sup([a,b))=b. Theorem 10.1.
    Proposition 10.4: for any e positive, there is N such that 1/N < e.
    
    Textbook Reading (Nov 4): Section 8.4 and Section 10.1.
    The Blackboards for Lecture 15.
    
    
17. Friday Nov 6: Limits.
    
    Sequences of real numbers and Examples.
    Definition of limit (with epsilon) and Example.
    Proposition 10.13. Examples of limits.
    
    Textbook Reading (Nov 6): Section 10.2-10.4.
    The Blackboards for Lecture 16.
    
    
18. Monday Nov 9: Monotone Convergence Theorem.
    
    Proposition 10.14: Uniqueness of limits.
    Increasing and decreasing sequences. Bounded sequences.
    The Monotone Convergence Theorem. Proof and Examples.
    
    Textbook Reading (Nov 9): Section 10.4.
    The Blackboards for Lecture 17.
    Informal OH Notes of Nov 9.
    
    
19. Friday Nov 13: Recursive Sequences and Square Roots.
    
    Recursive sequences of real numbers. Scheme of study.
    Proving convergence with MCT. Finding the limit with uniqueness.
    Two examples. Definition of Square Root of a positive real number.
    
    Textbook Reading (Nov 13): Sections 10.4 and 10.5.
    The Blackboards for Lecture 18.
    
    
20. Monday Nov 16: The Rational Numbers.
    
    Definition of Rational and Irrational Numbers. Examples.
    Density of Rational Numbers in the Real Line (Theorem 11.8).
    Proof that the square root of 2 is irrational (Proposition 11.10).
    
    Textbook Reading (Nov 16): Sections 11.1 and 11.2.
    The Blackboards for Lecture 19.
    Informal OH Notes of Nov 16.
    
    
21. Wednesday Nov 18: Irrational Numbers.
    
    Theorem: Any real number is a limit of a rational sequence.
    Number Systems. More examples of (algebraic) irrationals.
    Theorem: e is an irrational number. Proof of Theorem.
    
    Textbook Reading (Nov 18): Chapter 11.
    The Blackboards for Lecture 20.
    Informal OH Notes of Nov 18.
    
    
22. Friday Nov 20: Introduction to Cardinality.
    
    Cardinality: Naturals, Integers, Rationals and Reals.
    Definition of a function. Three examples of Functions.
    Three Properties: Injections, Surjections and Bijections.
    First Cardinality Theorem: Card(N)=Card(Z).
    
    Textbook Reading (Nov 20): Chapter 13, 13.1, 13.2.
    The Blackboards for Lecture 21.
    
    
    
23. Monday Nov 23: Countable & Uncountable Sets.
    
    Definition of finite, countable and uncountable sets.
    Finite Sets: Theorem 13.4 and 13.5 (Pigeonhole Principle).
    Countably Infinite sets: the rationals are countable.
    Properties of countable sets. Examples and Hierarchy of Infinities.
    
    Textbook Reading (Nov 23): Chapter 13.1, 13.2.
    The Blackboards for Lecture 22.
    Informal OH Notes of Nov 23.
    
    
24. Wednesday Nov 23: Final Project Discussion.
    
    
25. Monday Nov 30: The real numbers are uncountable.
    
    Uncountability of Real Numbers and Irrationals.
    Proof of Uncountability of Irrationals from the Reals.
    Cantor's Diagonal Argument. Binary Sequences are uncountable.
    Proof that the real numbers are uncountable through binary sequences.
    
    Textbook Reading (Nov 30): Chapter 13.3, 13.4.
    The Blackboards for Lecture 23.
    
    
    
26. Wednesday Dec 2: Continuity of Real Functions.
    
    Examples of Functions: Intuition for Continuity.
    The epsilon-delta definition of local continuity.
    Proofs, Examples and Non-Examples of Continuous Functions.
    
    Textbook Reading (Dec 2): Appendix A.
    The Blackboards for Lecture 24.
    Informal OH Notes of Dec 2.
    Informal OH Notes of Dec 3.
    
    
27. Friday Dec 4: Undergraduate Research at UC Davis: Tips.
    
    
28. Monday Dec 7: Final Project Discussions & OH.
    
    

Further Resources

Books: The "Book of Proof" by R. Hammack. The book "Mathematical Olympiad Treasures" by T. Andreescu and B. Enescu can be a good introduction to problem solving. I particularly like R. Vakil's "A Mathematical Mosaic" as a colorful invitation to the many facets of mathematics.

Online resources: "The Proof Page" by D.A.Kouba, "Tips for Reading Mathematics" by M.Tomforde, and M.Hutchings' Introduction to mathematical arguments. See also "First Experiences with Writing Proofs" by M.Tomforde and the contents of "Resources for Math Majors". There are many additional resources online, let me know if you are interested in more references for particular topic.

Solving Problems: "How to Solve It" by G. Polya. "The Art of Problem Solving" can be interesting website, here are tips on how to write solutions. In addition to the books above, G. Polya and J. Kilpatrick's "The Stanford Mathematics Problem Book" contains interesting challenging problems.