MA 225-001

Foundations of Advanced Mathematics

Mathematics

Homework Assignments

Assigned Jan. 10

Sec. 1.1, problem 7. To discuss Wednesday; don't turn in.

Assigned Jan. 12

Sec. 1.1, problems 2cg. Don't turn in.
Sec. 1.1, problems 3gk (use truth tables), 4gh (do so I can follow your work), 5bi, 11a (use truth tables). Turn in Friday Jan 21.

Assigned Jan. 14

Sec. 1.2, problems 7f, 8bf, 9d.
Also: Use truth tables to show that the propositional forms "P or Q" and "(not P) implies Q" are equivalent.
Regarding 8bf: (i) Logical connectives connect propositions, which must be sentences. (ii) Use parentheses as needed for clarity.
Turn in Friday Jan 21.

Assigned Jan. 24

Sec. 1.3, problems 1 c - f, j and 2 c - f, j. Turn in Friday Jan. 28.

Assigned Jan. 26

Sec. 1.4, problems 4bc, 6a, 7, 8ab. Turn in Friday Feb. 4.

Assigned Jan. 28

In the following problem, "ge" means greater than or equal to, and "le" means "less than or equal to."
Prove the following: Let b ge 0. If -b le x le b, then |x| le b.
Turn in Friday Feb. 4.

Assigned Jan. 31

Sec. 1.5, problems 3cd. Turn in Friday Feb. 4.

Assigned Feb. 7

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. To give a proof by contradiction that P implies Q, you assume P and assume not Q, and derive a contradiction.
Sec. 1.5, problem 8. See example on pages 38 - 39. Here we are trying to prove that something does not exist by assuming it does exist and deriving a contradiction.
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 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. 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 Feb. 11.

Assigned Feb. 9

Sec. 1.6, problems 1d, 5def. Turn in Friday Feb. 11.

Assigned Feb. 18

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).) Turn in Friday Feb 25.

Sec. 2.1, problems 1abc, 2ac, 3acegi, 5ac. Don't turn in.

Assigned Feb. 21

Sec. 2.1, problem 11. Suggestion: Proof by contradiction. The statement you are to prove has the form: If P then Q. For a proof by contradiction, assume P and not Q, and try to derive a contradiction. Turn in Friday Feb 25.

Assigned Feb. 23

Sec. 2.2, problems 8n,o. Turn in Friday Feb. 25.
Sec. 2.2, problems 8q, 9gh, 10bd, 11d, 13ab. Turn in Friday Mar. 4.
Sec. 2.2, problems 14abf. Don't turn in.

Assigned Mar. 4

Sec. 2.4, problems 7ab, 8adfglo, 9a. Sec. 2.5, problems 2, 6b. Turn in Friday Mar. 18.

Assigned Mar. 14

Sec. 2.3, problems 1djn, 5b, 6b, 11c. Do not turn in. To discuss Monday, Mar. 21.

Assigned Mar. 28

Sec. 3.1, problem 3d. Turn in Friday Apr. 8.

Assigned Mar. 30

Sec. 3.2, problems 1gi, 4aej, 8. Turn in Friday Apr. 8.

Assigned Apr. 4

Sec. 4.1, problems 8af. Justify your answers. Turn in Friday Apr. 8.

Assigned Apr. 8

Sec. 4.2, problems 1j, 4.
Sec. 4.3, problems 1hl.
Turn in Friday Apr. 15.

Assigned Apr. 11

Sec. 4.3, problems 2ehl, 4, 9a. Turn in Friday Apr. 15.

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