## Mathematics 201B: Introduction to Proofs

Instructor: Jonathan Smith, 496 Carver, 4-8172 (voice mail)

e-mail: jdhsmithATmathDOTiastateDOTedu (substitute punctuation)

Office Hours: Mon. 10am, 3:10 pm, 5:30pm; Wed. 10am, 3:10pm (subject to change)

Finals Week office hours: Mon. 12/10, 9:15-10:45am and 2:30-4pm.

Grading: 80% for in-class assignments (at random times), homework, and class participation; 20% for the final.

Textbook: T.A. Sundstrom, Mathematical Reasoning: Writing and Proof, 2nd ed., Pearson Prentice Hall, ISBN 0-13-187718-6

Study Plan: Regular attendance and participation in class activities are the prerequisites for success. Except in extreme circumstances, no accommodation can be given for failure to meet this responsibility.

Assignments will be given each week. They will not be graded every week, but it is absolutely essential to solve each homework problem in order to understand the material and develop the necessary skills.

Communication devices must remain switched off during the class periods and final.

Syllabus:

• Introduction to writing proofs.
• Logical reasoning: statements, operations, predicates, sets, quantifiers, etc.
• Constructing proofs: directly, by contradiction, induction, and so on.
• Proofs in set theory, algebra, number theory, analysis.

### Assignments

Click here for a copy of the Practice Final in Portable Document Format.

1. Prove that  n/(n2+1)  tends to zero as  n  tends to infinity.
2. Let  x  and  y  be real numbers. Prove that if  x + y  is irrational, then  x  is irrational or  y  is irrational.
3. Suppose that  s  is an irrational number. Prove that for any real number  x , either  s + x  is irrational or  s - x  is irrational.
1. Prove that  4n3  tends to infinity as  n  tends to infinity.
2. Prove that  ( - n )3  does not tend to infinity as  n  tends to infinity.

10/31 for 11/7 (graded): Sec. 5.1, Ex. 3(b); Sec. 5.2, Ex. 4, 12.
In Ex. 12, use "pi radians" rather than "180 degrees." Also, "Part (1)" and "Part (2)" mean "Part (a)" and "Part (b)." Recall that a polygon is convex if for any two points A and B inside the polygon, the straight line segment joining A and B stays entirely inside the polygon.

10/29 for 10/31: Sec. 5.1, Ex. 3(a).

10/19 for 10/24 (graded): Sec. 3.2, Ex. 4, 5(b)(c), 15 [use contrapositive for 15, not the "Hint" ! ].

10/8 for 10/15 (graded): Sec. 3.1, Ex. 8(a), 15(b)(c).

10/1: First in-class test.

Click here for a copy of Practice Test #1 in Portable Document Format.

9/17 for 9/21 (graded): Sec. 2.3, Ex. 7; Sec. 2.4, Ex. 1(b),(e),(f).

Click here for a copy of Quiz #1 in Portable Document Format.

9/5 for 9/10 (graded): Sec. 1.2, Ex. 7; Sec. 2.1, Ex. 6.

8/29 for 8/31: Read pp.16 - 23, do Ex. 4(a) for Section 1.2.

8/27 for 8/29: Progress Check 1.4 (p. 10).

8/22 for 8/27: Activity 1.5 (p. 10): 1, 3.