MATH 347 - Fundamental Mathematics - Spring 2012

Office hours: 2:00-3:00 PM Wednesday (or by appointment) in 222A Illini Hall

Syllabus (exam dates are subject to change)

Announcements

• Final exam scores are posted in the usual place and final grades have been posted to self-service!
• Link with info about to Noyce Scholarship available for math majors interested in teaching.
• You can check your homework and exam grades here (requires login with NetID).
• New office hours are Wednesday 2:00-3:00 p.m. I am also available in my office on Monday afternoons, but please schedule in advance!
• In the homeworks words like "determine", "obtain", "construct", or "show" request a proof.

Homeworks

• Homework 9: Chapter 5: 7, 29, 36, 37; Chapter 10: 1(b)(c), 2, 14 (due Fri, 4/20) (solutions)
• Homework 8: Chapter 5: 4, 6, 8, 25, 32, 39 (due Fri, 4/13) (solutions)
• Homework 7: Chapter 14: 3, 4, 9, 10, 17, 47 (due Fri, 4/6) [You may find Proposition 14.10 helpful for problem 17] (solutions)
• Homework 6: Chapter 13: 3, 4, 22(a)(b), 26, 30 (due Wed, 3/14) (solutions) [Note: the proof of 13.26 seems to be wrong!]
• Homework 5: Chapter 4: 5, 11, 21, 33, 34, 42; Chapter 13: 7 (due Mon, 3/5) [The notation [n] means the set of naturals from 1 to n.] (solutions)
• Homework 4: Chapter 1: 46, 48, 52; Chapter 4: 1, 3, 14, 15 (due Mon, 2/27) [Don't do number 1.52. We didn't cover the appropriate material.] (solutions)
• Homework 3: Chapter 3: 2, 4, 15, 17, 22, 23, 58(b) (due Fri, 2/10) [We proved 58(a) in class. A similar technique should work for 58(b)] [The triangle inequality may help solve 22] (solutions)
• Homework 2: Chapter 2: 2, 3, 4, 23, 34, 40(a), 46 (due Fri, 2/3) [You might find Remark 2.20(c) in MT helpful for solving 4(c)(e)(f)] (solutions)
• Homework 1: Chapter 1: 1, 7, 15, 16, 27, 30, 31 (due Wed, 2/1) (solutions)

Suggested practice problems (optional)

• Exams: 1 2 3
• Practice Exam 3 (hints/partial solutions)
• Chapter 10: 3, 4, 9, 17 [for 4: relatively prime means that the two numbers have no common factors] (solutions)
• Chapter 5: 5, 27, 28, 38, 40 (solutions)
• Chapter 14: 1, 2, 13, 15, 16, 22 (solutions)
• Chapter 13: 5, 6, 8, 24, 25, 27, 29, 37 (solutions)
• Practice Exam 2 (hints/partial solutions)
• Chapter 4: (solutions)
  • Base q representation: 4.2, 4.3, 4.13, 4.16
  • Bijections and cardinality: 4.22, 4.31, 4.43, 4.48, 4.49
  • Injections and surjections: 4.6, 4.9, 4.12, 4.24, 4.25
• Practice Exam 1 [The practice exam is longer than the real exam!] (solutions)
• Chapter 3: 14, 16, 21, 24, 38, 39, 49 (solutions)
• Chapter 2: 10(a)(b)(c), 18, 21, 22, 38, 47 (solutions)
• Chapter 1: 13, 18, 21, 40, 43 (solutions)

Class Schedule

• Wed, 5/2: Final review

• Mon, 4/30: Final review

• Fri, 4/47: Return and go over exam 3

• Wed, 4/25: Exam 3

• Mon, 4/23: Exam 3 review

• Fri, 4/20: Pigeonhole principle: Erdős-Szekeres, etc. (See pg. 190-191 in MT)

• Wed, 4/18: Pigeonhole principle: examples, Ramsey Theory (Exercise 10.14 in MT)

• Mon, 4/16: Pigeonhole principle: simple examples (See pg. 189-190 in MT)

• Fri, 4/13: Combinations with repetition (See pg. 107-108 in MT)

• Wed, 4/11: Pascal's identity proofs, chairperson identity, summation identity (See pg. 106-109 in MT)

• Mon, 4/9: Binomial coeff identities, lattice paths, Pascal's triangle and identity (See pg. 104-106 in MT)

• Fri, 4/6: Number of selections (double count of permutations), poker hands examples, binomial theorem and simple applications (See pg. 103-104 in MT)

• Wed, 4/4: Combinatorics: product and sum rule. Permutations and selections (combinations). (See pg. 100-102 in MT)

• Mon, 4/2: The Cauchy Criterion. Sum of infinite geometric series. (See pg. 277-280 in MT)

• Fri, 3/30: Proposition 14.11. Cauchy sequences (See pg. 275-276 in MT)

• Wed, 2/38: Return and discuss exam 2.

• Mon, 3/26: Squeeze theorem, ε/2-arguments. (See pg. 271-274 in MT)

• Fri, 3/16: Exam 2.

• Wed, 3/14: Discuss practice exam 2 solutions.

• Mon, 3/12: Limit of the square of a convergent sequence and other convergence examples. (See pg. 260-263 in MT)

• Fri, 3/9: Monotone convergence theorem, limit of the square of a convergent sequence. (See pg. 260-263 in MT)

• Wed, 3/7: Sequences and limits, a subset S of the reals has sup(S) iff there is a sequence in S that converges to sup(S). (See pg. 259-261 in MT)

• Mon, 3/5: Upper/lower bounds, supremum and infimum. √2 is real (See pg. 256-258 in MT)

• Fri, 3/2: Schroeder-Bernstein Theorem, power set of [n] and set of 01-strings of length n have the same size (See pg. 82-83 and 91 in MT)

• Wed, 2/29: Injections and surjections (See pg. 83-87 in MT)

• Mon, 2/27: Bijections and cardinality: rationals are countable, reals are uncountable (See pg. 80-83, 87-89 in MT)

• Fri, 2/24: Scales problem. Introduction to bijections (See pg. 78-83 in MT)

• Wed, 2/22: Return and go over exam 1

• Mon, 2/20: Representation of natural numbers in different bases. (See pg. 76-80 in MT)

• Fri, 2/17: Exam 1.

• Wed, 2/15: Exam 1 review topics.

• Mon, 2/13: Statements with bounded functions: problem 1.49 in MT. Polynomials have at most as many roots as degree. (See pg. 55, 59 in MT)

• Fri, 2/10: Finish induction: representation of a natural as the sum of Fibonacci numbers. Function basics: bounded, strictly and monotone increasing/decreasing, etc. (See pg. 10-14 in MT)

• Wed, 2/8: Strong induction examples: stamps, n is a power of 2 times an odd, Fibonacci closed form (See pg. 63-66 in MT)

• Mon, 2/6: Strong induction: Fundamental Theorem of Arithmetic, Nim (See pg. 63-66 in MT)

• Fri, 2/3: More induction: adjusted base case, geometric series, L-tilings: problem 3.58(a) (See pg. 55-62 in MT)

• Wed, 2/1: Induction: Gauss formula, common induction mistakes. (See pg. 50-57 in MT)

• Mon, 1/30: Contradiction: Infinite many primes, √2 is irrational, problem 2.40(b) from MT (chessboard problem), etc.

• Fri, 1/27: Truth tables, equivalence of elementary proof techniques, de Morgan's Laws, √2 is irrational. (See pg. 31-39)

• Wed, 1/25: Mathematical statements, negation, quantified statements, elementary proof techniques (direct, contrapositive, contradiction). (See pg. 27-31, 35-39)

• Mon, 1/23: Prove penny problem. Introduce set theory notation, de Morgan's Laws. (See pg. 6-10 in MT)

• Fri, 1/20: AGM inequality, set equality, introduce penny problem. (See pg. 6-10 in MT, review set notation)

• Wed, 1/18: Discuss class information. Proofs of simple inequalities e.g. triangle inequality. (See pg. 4-6 in MT)