MTH 535 Fall 06 -- Tips for Exam 1

Exam 1 is scheduled for Friday, October 20 , 3-5, Tyler 106.

Exam 1 covers classes 1-10 and homework assignments 1-5. You are expected to know and be able to state all definitions, propositions, and theorems given in class and in homework assignments as well as their proofs. You should know all of the important examples studied in class. You are expected to know solutions to all homework problems. Some problems on the exam will ask you to state and prove a theorem proved in class; some will ask you to state a definiton and give an example; some will be similar to homework problems.

While I was browsing my notes and homework assignments, the following questions stood out as good candidates for exam questions. (I am indicating below theorems, propositions, examples studied in class without stating precisely all the assumptions. Consult your notes for that. I am using numbers as we did in class, but of course, on the exam I will not expect you to remember numbers of defintions, propositions, theorem etc.) Items below are listed in no particular order. Some overlap with the homework problems listed below.

-- State Def.10.1. Prove Prop. 10.2.
-- State Def. 9.2. Prove Prop. 9.1 or Prop.9.2.
-- State Th 10. 1. Prove it. Prove Col. 10.1 and 10.2.
-- State Th 9.3. Prove Step 1 and Step 2 (assuming Lemma 9.1 is known).
-- State the Heine-Borel Theorem. Use it to prove Prop 16 page 45.
-- Define the Borel sigma-algebra B in R. Prove (b) or (c) or (d) of Th.8.1. (or any other part of it).
-- Show that every countable subset of R is Borel.
-- State the definiton of an open subset of R (Def.5.2). Prove Prop. 5.1, Prop. 5.2. Give an example that Prop. 5.2 does not hold for an infinite family.
-- Give an example of a subset of R which is (a) both closed and open (b) neither closed nor open.
-- State Def. 7.1. Prove Prop. 7.1.
-- Prove Prop. 6.1 and 6.2.
-- State the definiton of an algebra od sets. Prove that finite-cofinite collection is an algebra.
-- State the definiton of a sigma-algebra od sets. Prove that countable-cocountable collection is a sigma-algebra.
-- State the Cantor theorem (Th. 3.3). Describe how the theorem can be used to show that B is not equal P(R).

From homework problems, the following seem particularly good:

H1: #1,#2,#6
H2: #1, #2, #3, #4
H3: #2,#4,#5
H4: #1, #3, #5, #6
H5: # 3, #4.

Consider it a practice exam (Of course, the actual exam will be much shorter). Some of the problems listed above will appear, some will not. A problem or two not listed above may appear as well. Who knows? I hope you all get As!