MA 225-003

Foundations of Advanced Mathematics

Mathematics

Homework Assignments

Assigned Aug. 18

Sec. 1.1, problems 1abdefg (just look at), 2 ji, 3 f (make a truth table for the first, then compare to your answer for 2 j). Turn in Friday Aug. 27.

Assigned Aug. 20

Sec. 1.1, problems 4fj, 5gi, 11 ac (use truth tables). Turn in Friday Aug. 27.

Assigned Aug. 23

Sec. 1.2, problems 7af, 8bc, 9cd. Regarding 8bc: (i) Logical connectives connect propositions, which must be sentences. (ii) Use parentheses as needed for clarity. Turn in Friday Aug. 27.

Assigned Aug. 25

Sec. 1.3, problems 1 b - f, h, j. Turn in Friday Sept. 3.

Assigned Aug. 27

Sec. 1.3, problems 2 b - f, h, j. Turn in Friday Sept. 3.

Assigned Aug. 30

Sec. 1.4, problems 4ce, 7. Turn in Friday Sept. 3.

Assigned Sept. 1

Sec. 1.4, problems 5e (assume b >= 0) (DUE DATE ON THIS PROBLEM CHANGED TO FRIDAY SEPT. 17), 6jk, 8bc. (8c: Assume the quadratic formula. You may want to consider separately the cases b>0 and b<0.) Turn in Friday Sept. 10.

Assigned Sept. 3

Sec. 1.5, problems 3dg and 8. (For problem 8, see example on pages 38 - 39.) Turn in Friday Sept. 10.

Assigned Sept. 10

1. Sec. 1.4, problem 5e.
2. Prove that if a and b are positive real numbers and a < b, then 1/b < 1/a.
3. Prove that if a < b and c < d, then a+c < b+d. (real numbers)
4. Prove that if a < b and c < d, then a-d < b-c. (real numbers)
5. Prove that if a is a rational number other than 0, and b is an irrational number, then a/b is irrational. (real numbers) (I suggest a proof by contradiction.)
On all these problems, you may, as always, use anything we have done in class or homework.
Turn in Friday Sept. 17.

Assigned Sept. 15

Sec. 1.6, problems 1b, 2a, 5d.
Prove that if we are given five points in a square of side length 1, two of them are within a distance of sqrt(2)/2 of each other.
Without using calculus, prove: There is a unique real number x such that x > 1/sqrt(3) and x^3-x-6=0.
Turn in Wednesday Sept. 22.

Assigned Sept. 17

Sec. 1.6, problems 2d.
Prove: For every b > 0 there exists a natural number M such that if x > M then sqrt(x) > b. (Real numbers. You may assume: If 0 < x < y then 0 < sqrt(x) < sqrt(y).
Turn in Wednesday Sept. 22.

Assigned Sept. 29

Sec. 2.1, problems 1be (first look at the text's answers for 1ac), 2be, 3bdh, 11. Turn in Wednesday Oct. 6.

Assigned Oct. 1

Sec. 2.2, problems 8noq, 9dgh. Turn in Wednesday Oct. 6.

Assigned Oct. 4

Sec. 2.2, problems 10bd, 14bdf. Turn in Friday, Oct. 15.

Assigned Oct. 6

Sec. 2.3, problems 1djn, 11c. Turn in Friday, Oct. 15.

Assigned Oct. 11

Sec. 2.3, problems 5b, 6b. Turn in Friday, Oct. 15.

Assigned Oct. 15

Sec. 2.4, problems 7ab, 8adfglo, 9ac. Sec. 2.5, problems 2, 6b. Turn in Wednesday, Oct. 20.

Assigned Oct. 27

Sec. 3.1, problems 3d. Sec. 3.2, problems 1 gil, 4aegj. Turn in Friday, Nov. 5.

Assigned Oct. 29

Sec. 3.2, problems 8a, 15. Turn in Friday, Nov. 5.

Assigned Nov. 5

Sec. 4.1, problems 2e, 5c, 8af (justify answers to 8af). Sec. 4.2, problems 1j, 4. Turn in Friday Nov. 12.

Assigned Nov. 9

Sec. 4.3, problems 1hl. Turn in Friday Nov. 12.

Assigned Nov. 12

Sec. 4.3, problems 2ehl, 4, 9a. Turn in Wednesday Nov. 17.

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