## Math 4433## (Intro. to Analysis I)## Summer 2009 |
Yellow Flax |

Review sheet for the third exam

Solutions to problems on the second exam.

Review sheet for the second exam

Solutions to problems on the first exam.

Review sheet for the first exam.

Solutions to problems 15 and 16 in section 3.1.

Step-by-step write-up of proofs for problems 2.3.5 and 2.4.6.

The syllabus for this course is here.

Sometimes I post an assignment in advance but change it in class the day before it's due. If you miss a class you should check this web page after class for the final version of the next day's assignment.

## Assignment |
## Due Date |
## Problems |

1 | Wednesday, June 10 | 2.3.6, 2.3.10 |

2 | Thursday, June 11 | 2.3.5, 2.4.6 |

3 | Friday, June 12 | 2.3.9, 2.4.1 |

4 | Monday, June 15 | 3.1.4 |

5 | Wednesday, June 17 | 3.1.5(a,d) |

6 | Thursday, June 17 | 3.1.15, 3.1.16 |

7 | Friday, June 18 | 3.2.6(a,b,c,d) |

8 | Wednesday, June 24 | 3.2.3 |

9 | Thursday, June 25 | 3.2.5, 3.2.7 |

10 | Friday, June 26 | 3.2.13, and for extra credit: 3.2.14 |

11 | Wednesday, July 1 | 3.3.1, 3.3.2 |

12 | Thursday, July 2 | 3.3.7 |

13 | Monday, July 6 | 3.3.11, 3.3.13 |

14 | Tuesday, July 7 | 3.4.1 |

15 | Wednesday, July 8 | 3.4.14 |

16 | Monday, July 13 | 4.1.4, 4.1.10(a), 4.1.11(a) |

17 | Tuesday, July 14 | 4.1.7, 4.1.12 |

18 | Wednesday, July 15 | 4.2.1(b), 4.2.2(c,d) |

19 | Tuesday, July 21 | 4.2.11(a,b,d) |

20 | Wednesday, July 22 | 4.1.14, 5.1.5 |

21 | Thursday, July 23 | 5.1.10 |

22 | Friday, July 24 | 5.2.10 |

23 | Monday, July 27 | 5.2.8, 5.3.1 |

24 | Tuesday, July 28 | 6.1.1(b), 6.1.2 |

25 | Wednesday, July 29 | 6.1.3, 6.1.4 |

26 | Friday, July 31 | 6.2.7, 6.2.13 |

- M.I.T. Open Courseware: ocw.mit.edu. Click on "Mathematics" and then "Analysis I, Fall 2006".
- Wikipedia entry for "set theory": en.wikipedia.org/wiki/Set_theory.
- There are many interesting articles on set theory and foundations of mathematics at the online Stanford Encyclopedia of Philosophy.
- "A Primer for Logic and Proof", by H. P. Hirst and J. L. Hirst: www.mathsci.appstate.edu/~jlh/primer/hirst.pdf.
- The website "Intro to Logic" by Ian Barland et al. contains a course on logic, and in particular a discussion of the proper use of quantifiers such as "for every" and "there exists". In our analysis course, we don't delve too much into the details of the rules of logic, and rely on our common sense to tell us whether a proof or argument is logically correct. But when you have to explain to someone else what you think is wrong with their proof, it's sometimes difficult: everybody has common sense, but not everybody can explain common sense to others. Courses in logic such as this one aim at making "common sense" rules explicit, so you can communicate with others about them.
- Here is a site where fundamental theorems in many fields of mathematics are proved in complete detail, with proofs that are verifiable (and have been verified) by computer: us.metamath.org.
- "Understanding Analysis", a textbook by Stephen Abbott.
- "Analysis, Vol. 1", a textbook by Terence Tao.

The founder of the subject of analysis, as we learn it in this class, was the French mathematician Augustin-Louis Cauchy (1789-1857). Here is a nice article about what Cauchy did, and why.

The distinction between rational and irrational numbers has a long history, going back to the Greeks in the time of Plato and before, but the Greek mathematicians of those times also thought about numbers very differently than we do today. In particular, they did not have our modern concepts of multiplication and division: it would not have made sense to them to think of 2/3 as a number that could be added to 7/8. In "The Mathematics of Plato's Academy", by D. H. Fowler, it's suggested that the Greeks of Plato's day would have based their idea of the ratio of two quantities on the process of "anthyphairesis", or successively removing squares from rectangles in the way we demonstrated in class. (The meaning of the Greek word "anthyphairesis" seems to be something like "two things taking turns removing parts from each other".) A Google search on "anthyphairesis" turns up several interesting commentaries on this suggestion, including a couple of favorable ones ( Plato's Theory of Number, by I. Bulmer-Thomas, and F. Gouvea's review of Fowler's book) and a not so favorable one (a review of Fowler's book by S. Unguru).