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Final Exam

Interesting stuff relevant to our class

- Aug. 26:
- Read syllabus (quiz next class)
- Study chapter 1.1 and 1.4 of our text (possible open-notes reading quiz next class)
- Do chapter 1.1 #2ac, 3b, 5a and chapter 1.4 #3ab, 5abc, 8, 12

- Aug. 28:
- Read chapter 1.2, 1.5
- Do 1.4 #1a (hint: find some way to use the distributive law), 13,14,15
- Do 1.2 #3,4,11
- Suppose that f is a function from A to D, and B and C are subsets of A. Prove that if B C then f(B)f(C).
- Suppose that f is a function from A to D, and E and G are subsets of D. Prove that if E G then f
^{ -1}(E)f^{ -1}(G). - Prove that if a,b with a<b then a < (a+b)/2 < b.
- Note: If A={1,2,3,4} then sup A = 4. If B = [2,4] then sup B = 4. But if U = (0,), then no sup for U!

- Sept. 2:

- Catch up
- Show, using only the existence of the set P of positives from p. 20-21, and its properties (O1) and (O2), that if a<b and b<c, then a<c, the transitivity property of inequalities. [I'm surprised this isn't mentioned in Stoll!]
- Expand on the proof of theorem 1.2.5(b) in the text (the text version is very concise).
- Do 1.2 #8,9

- Sept. 4:

- Show that if f:AB and EB, then f
^{ -1}(E^{C}) = [f^{ -1}(E)]^{C}. - Finish up the in-class exercise: Write up a proof that f:AB is a one-to-one function if and only if

for all A_{1}, A_{2}A, f (A_{1}A_{2}) = f (A_{1})f (A_{2}) . This write up IS done in groups. - Read 1.5 & 1.6, and start on chapter 1.5 #1,3,4,5,6a
- Turn in next class meeting: chapter 1.1 #5a, 1.2 #9,11, 1.4 #5abc,12, and the write-up for the in-class exercise above.

- Sept. 9:

- Do 1.5 #7; you may use the lemma that 2
^{n}> n n without proving the lemma. - Give decimal values for each of: 0.1010
_{(2)}, 0.101010..._{(2)}, which repeats the block "10" forever [I don't have a way to "overline" in html], and

0.1010_{(3)}, 0.101010..._{(3)}, which also repeats the block "10" forever.

- Find the ternary (base 3) expansion of 1/4 (recall that 4
_{(10)}= 11_{(3) }). You can do this by long-division in ternary

or by "building up" 1/4 from negative powers of 3, e.g. "How many 1/3's fit in 1/4, then how many more 1/9's, then how many more 1/27's" and so on. Show work somehow.

Hint: the expansion will repeat.

- Check your answer for 1/4 above by working out a geometric series calculation.
- Now find a ternary expansion of 1/7 (recall that 7
_{(10)}= 21_{(3) }). - Read chapter 1.6

- Sept. 11:

- Read 1.7, 2.1
- Do 1.7 #1,3 (hint for 3 - every natural number has a unique factorization into primes. That should get you started.)
- Do 1.7 #17 (hint: Find a first point in (-,) - then note why the point you found can't be the largest such point. Then find another, and continue.)
- Give a direct proof that the union of two disjoint, countable sets is countable. It may be convenient to use enumerations here.
- Use the prior problem to do 1.7 #13
- do 1.7#14 by using 1.7.7

- Sept. 16:

- Show that if A~B and C~D, then A×C ~ B×D (recall A~B means there is a bijection between these two sets).

- Show that ×(×) is countable, finding a way to use the prior problem and another result established in class (don't reinvent the wheel here!)
- The set of zero-degree polynomials with integer coefficients is countable because: (give your answer, then:)

- Every 1st degree polynomial with integer coefficients can be formed by adding mx (for some m in the integers) to a zero-degree polynomial. Explain why, then use this to prove that the 1st degree polynomials are countable.
- Show that for any n in , the nth degree polynomials with integer coefficients are countable
- Show that the set of all polynomials with integer coefficients is countable (note that polynomials necessarily have finitely many terms).
- Make sure you know how to do 1.4 #15 (that inf/sup problem I mentioned in class).

- Sept. 18:

- Quiz 1
- Prove
that whenever a real number in [0,1] has two ternary expansions, at
least one of them contains the digit "1", so if a number is in the
Cantor set, it has only one expansion consisting of only 0's and
2's. (This should be short). You may take as given the fact
that if a number has two ternary expansions, one of them ends in a
string of repeating 2's.

- Bonus: Find a number in the Cantor set that is not an endpoint of one of the removed subintervals.

- Read chapter 2.1 in Intro. to Real Analysis (our regular textbook).
- Do 2.1 #2,3,6a,b, start to think about #8a.

- Read p.1-2 of the Notes on Metric Spaces handout from class (also available on the class WISE site).
- Do problems 1-3 on p.2 of the handout.

- Sept. 23:
- Sept. 25:

- Sept. 30:
- Oct. 2:

- Oct. 7:
- Oct. 9:

- Oct. 14:

- Oct. 16:
- Oct. 17: Mid-semester day - no classes

- Oct. 21:
- Oct. 23:

- Oct. 28:

- Oct. 30:

- Nov. 4:

- Nov. 6:

- Nov. 11:

- Nov. 13:

- Nov. 18:
- Nov. 20:

- Nov. 25:

<>Nov. 27:

- Dec. 2:

- Dec. 4:

- Thursday, Dec. 11, 2014 from 2-5 p.m.
**Final Exam**

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