Principles of Analysis (Both sections)

Exam #1 (Fall 2002)

Friday, Oct. 11

The first exam covers Sections 1.1, 1.2, 1.3, 1.4, 1.5, 2.1 and the material on cardinality and infinite sets. You should go over homework problems and class notes. The exam will be designed to take the full class period (45-50 minutes). No calculators required or allowed.

The following concepts, definitions, theorems and axioms are crucial material for the exam. You WILL BE ASKED to give a precise definition or carefully state a particular theorem. (eg. Give the definition of a least upper bound of a set.) I will expect your answers to be concise and specific, with little margin for error. You may also be asked to compute something, like a least upper bound, or the intersection of two sets, or a solution to an equation, etc.

Definitions and terminology:

1. Basic Set Theory: null set, union, intersection, complement, A - B, universal set, interval notation
2. Definition of subset, definition of when two sets are equal (A = B)
3. Function, domain, range, image, pre-image, one-to-one (injection), onto (surjection), inverse, absolute value function
4. Maximum, minimum, upper and lower bounds, least upper bound (lub), greatest lower bound (glb)
5. n choose k, Pascal's Triangle
6. Natural numbers N, integers Z, rationals Q, irrationals Q^c, reals R
7. Cardinality, bijection, finite and infinite sets, countable sets, uncountable sets
8. Convergent sequence, divergent sequence
Axioms and Theorems:
1. Axioms of the real number system (commutative, associative, etc.) You do not have to know the particular numbers of each axiom, although you should know which facts are axioms (assumed) and which facts are theorems (deduced from the axioms).
2. Well Ordering Principle, Principle of Induction
3. Binomial Theorem
4. Least Upper Bound Axiom
5. Every interval contains an infinite number of both rationals and irrationals (each set is dense in R)
6. Theorems on Countable sets, for example, The rationals are countable.
7. The Big Limit Theorem

Proofs:

You will have to do a couple of proofs on the exam. These will be similar to homework problems. In addition, I will ask you to prove one or more of the following:

1. The square root of 2 is irrational.
2. The set of real numbers is an uncountable set. (Cantor's Diagonalization argument)
3. Use the epsilon-n_0 definition to prove a given sequence converges. (#2 a-h in Section 2.1)

Practice Problems:
Some sample practice problems similar to homework problems are listed below. Solutions will be located in a binder in the Math/Science Library. Note that the solutions from the solution manual are very terse and leave out much of the motivating details I would expect you to include in your proofs.

Chapter 1

• Section 1.1 : 4, 5
• Section 1.2 : 3, 13a
• Section 1.3 : 1c, 8, 9
• Section 1.4 : 2b, 3, 7, 13a
• Section 1.5 : 1, 6, 9d, 9e
Cardinality and Infinite Sets
Problems 2 and 4 from the class handout.

Chapter 2

• Section 2.1 : 2e, 2f, 2i, 12