MA 225-002

Foundations of Advanced Mathematics

Mathematics

Homework Assignments

Assigned Aug. 17

Sec. 1.1, problems 1a, b, c, g, h; 2e, g; 3a, c. To discuss Friday; don't turn in.

Assigned Aug. 19

Sec. 1.1, problems 3e, g; 7c, d; 9a, c; 11f, i, j, k. To discuss Monday; don't turn in.

Assigned Aug 22

Sec. 1.2, problems 8, 16f, 16g.
Also: Use truth tables to show that the propositional forms "P or Q" and "(not P) implies Q" are equivalent.
To discuss Wednesday; don't turn in.

Assigned Aug. 24

Sec. 1.2, problems 1 (all parts), 10d, g, j. To discuss Friday; don't turn in.

Assigned Aug. 26

Sec. 1.3, problems 1a, b, e, h, l, m, and 2 a, b, g, h. To discuss Monday; don't turn in.
Sec. 1.3, problems 1f, i, n, and 2i. Turn in Friday Sept. 2.

Assigned Aug. 29

Sec. 1.3, problems 1k, 2c, d, e, k, m, 9a, b, c, e, f. To discuss Wednesday; don't turn in.
Sec. 1.3, problems 2f, n. Turn in Friday Sept. 2.

Assigned Aug. 31

Sec. 1.4, problems 5c, f, i. Turn in Friday Sept. 9.

Assigned Sept. 2

Sec. 1.4, problems 11b, d, e. To discuss Wednesday; don't turn in.
Sec. 1.4, problems 6b, f, 7d, m. You should be able to do 6b with just two cases. Turn in Friday Sept. 9.

Assigned Sept. 9

Sec. 1.5, problems 12b, c, d. To discuss Monday; don't turn in.
Sec. 1.5, problems 3d, h, 4a, 5a, 6d. Turn in Friday Sept. 16.

Assigned Sept. 12

Prove: 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. Be sure your proof uses the assumption that a is not 0!
Sec. 1.6, problems 1b, 1g. In problem 1g you are trying to prove that there exists j such that m+2=4j-1.
Prove: 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. Suggestion: Suppose no two of the points are within a distance of sqrt(2)/2 of each other. Divide the square into four smaller squares, all of equal size. How many points can be in each of the smaller squares?
Turn in Friday Sept. 16.

Assigned Sept. 15

Sec. 1.6, problems 1d, 2b (ignore complicated answer in text), d, 4d, e, 7d (you may assume that sqrt(3) is irrational). To discuss Monday; don't turn in.

Assigned Sept. 16

Sec. 1.6, problems 1h, 2e, 6j. To discuss Monday; don't turn in.

Assigned Sept. 26

Prove: For every b > 0 there exists an M > 0 such that if x > M then sqrt(x) > b. (Real numbers. You may assume: if 0 < x < y then 0 < sqrt(x) < sqrt(y).)
Sec. 2.1, problem 7. Suggestion: proof by contradiction.
Turn in Friday Sept. 30.

Sec. 2.1, problems 1abd, 4acij, 6a, 14a, 17ghijkl. To discuss Wednesday; don't turn in.

Assigned Sept. 28

Sec. 2.2, problems 7n, o, 8g, h, 9b, d, 10d, 12b. Turn in Wednesday, Oct. 5.
Sec. 2.2, problems 11b, f. To discuss Friday; don't turn in.

Assigned Oct. 7

Sec. 2.3, problems 1dkh, 5b, 6b.
Sec. 2.4, problems 6bd, 7ae, 8b. 6b: left hand side is sum from 1 to n of (8i-5).
Sec. 2.5, problems 2, 6b.
Turn in Friday Oct. 14.

Assigned Oct. 14

Sec. 2.2, problems 13a, 15c.
Sec. 3.2, problems 1gj, 5aeh.
Additional problem: Let R be the relation on Zx(Z-{0}) (Z=integers) given by: (a,b)R(c,d) if and only if ad=bc. Show that R is an equivalence relation. Find some members of the equivalence class of (2,3). What do all members of the equivalence class of (2,3) have in common?
Turn in Friday Oct. 28.

Assigned Oct. 21

Sec. 4.1, problems 2, 3bg (domain and range only), 7, 11af. Turn in Friday Oct. 28.

Assigned Oct. 26

Sec. 4.2, problem 1j.
Sec. 4.3, problems 1hl.
Turn in Friday Nov. 4.

Assigned Oct. 28, corrected Nov. 4

Sec. 4.3, problems 2ehl, 5, 9a, 11d (first show that it is a function). Turn in Friday Nov. 4.

Assigned Oct. 31

Sec. 4.3, problem 9a continued: show that the range is (0,infinity). Think of f as a function from R to (0,infinity), and find the inverse function.
Sec. 4.4, problem 1a (also find the inverse function).
Turn in Friday Nov. 4.

Assigned Nov. 11

Sec. 5.1, problems 1, 6b, 7, 12, 17.
Turn in by Tuesday Nov. 22.

Assigned Nov. 27

Sec. 5.2, problems 1, 3b d f.
Sec. 5.3, problems 3 (ignore the hint), 6 (just do n=3), 13a, 14b.
Don't turn in. To be discussed Friday Dec. 2.

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