Math 242 Principles of Analysis

Exam #1

Thursday, Oct. 8, In Class

The first exam covers Chapter 1 and Sections 2.1 through 2.4 of the course text. You should go over homework problems (as well as the solutions posted), quizzes and your class notes. The exam will be designed to take the full class period (45-50 minutes). No calculators are required or allowed.

Exam Review Session: We will review for the exam during Tuesday's class on Oct. 6. Please come prepared with specific questions. Some practice problems are available here.

The following concepts, definitions, theorems and axioms are crucial material for the exam. As with the quizzes, you will be asked to give a precise definition or carefully state a particular theorem. I will expect your answers to be concise and specific, with little margin for error. You may also be asked to compute a particular quantity such as a supremum or infimum, or an infinite intersection of sets, etc. Note: Infinite series will not be covered on this exam.

Definitions and terminology:

  1. Basic Set Theory: null set, union, intersection, complement, A - B, subset, def. of when two sets are equal, De Morgan's Laws,
  2. Natural numbers N, integers Z, rationals Q, irrationals I, reals R
  3. Logic: negation, truth tables, "or" versus "and", implications, converse, contrapositive, if and only if
  4. Function, domain, range, image, pre-image, f(A), f^{-1}(B), one-to-one, onto, bijection, inverse
  5. Bounds: maximum, minimum, upper and lower bounds, the supremum and infimum of a set
  6. Cardinality: infinite sets, A ~ B, countable sets, uncountable sets
  7. Sequences: convergent sequence, divergent sequence, increasing, decreasing and monotone sequences, recursive sequences, Fibonacci sequence, triangle inequality, bounded sequence
Axioms and Theorems:
  1. Fundamental Theorem of Arithmetic, Square root of a prime is irrational
  2. Principle of Induction
  3. Axiom of Completeness, The Sup (and Inf) Lemma
  4. Nested Interval Property
  5. Archimedean Property, The rationals and irrationals are each dense in R.
  6. Theorems on Countable sets (eg. the rationals are countable, the reals are uncountable)
  7. Theorems on Sequences: The Big Limit Theorem, Every convergent sequence is bounded, Order Limit Theorem, Monotone Convergence Theorem, the Squeeze 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 at least one of the following:

  1. The square root of 2 is irrational (be able to generalize to other natural numbers).
  2. The set of real numbers is an uncountable set (Cantor's Diagonalization argument).
  3. Use the epsilon-N definition to prove a given sequence converges.