SAMPLE EXAM I
- Explain all arguments clearly. You may use the back of each page if you need more room.
Problem 1)
Make
a truth table for
p; q; p or q; (not p)
and use it to verify "{(p or q) and (not p)} implies q"
Problem 2)
Find Union and Intersection of the sets B_n = (1 , 2 + 1/n) where n ranges over all natural numbers.
Problem 3)
Let f be a function from A to B. Let C be a subset of A. Prove if f is injective, then f^{-1}(f(C))=C; that is, the pre-image by f of f(C) is C.
Problem 4) Let S and T be two subsets. Prove if S\T and T\S are equinumerous then S and T are equinumerous.
Problem 5) State the converse and contrapositive of the following statement: "The square of an odd integer has remainder 1 under division by 8".
SAMPLE EXAM II
-
Problem 1) Prove by induction, explaining each step carefully, that the sum of the first 2n odd positive integers is equal to 4n^2.
- Problem 2) Prove that any finite union of compact sets is compact.
- Problem 3) a) Given a subset S of the reals, carefully define the boundary set of S. (This will be graded on accuracy and clarity of exposition as well).
b) If S contains its boundary, prove the complement of S in the reals is open
Rough gradelines through two exams:
- 77-100..............A
- 65-77...............B
- 47-65...............C
- Below 47............D or F
FINAL
- My understanding is that our final is will be given on Thursday, December 18, 1:30-3:30. The room is the classroom Fraser Hall 101.