MATH 450A, Review for Midterm Test 1

Test topics

0. Introduction
   1. Logic and proofs (Zorich 1.1; lecture notes)
   2. Sets, elements, operations on sets (Zorich 1.2; lecture notes)
   3. Functions; Images (direct and inverse); Compositions. Inverse functions (Zorich 1.3; lecture notes)

1. The Real Number System
   1. Axioms of a field (1.1)
   2. Inequalities (1.3)
   3. Absolute value (A.1)
   4. Natural numbers and mathematical induction (1.4)

2. Continuity and Limits
   1. Continuity and limits. Definitions and examples (2.1)
   2. Properties of limits (2.2)
   3. One-sided limits (2.3)
   4. Limits at infinity; Infinite limits (2.3)
   5. Limits of sequences. Axiom of Completeness. The Monotone Convergence Principle. The Archimedian Principle. (2.5)

Important definitions/facts

• Logic: statements, operations, truth tables. Universal and existential quantifiers. Proofs. Proofs by contradiction
• Sets, elements, operations: union, intersection, difference, complement, Cartesian product
• Functions, relations; direct and inverse images
• One-to-one, onto and invertible functions
• Axioms of real numbers: Field Axiom, Axiom of Order; the Completeness Axiom. The Monotone Convergence Principle
• Absolute value; triangle inequality and its versions (with proofs)
• Natural numbers in an arbitrary ordered field. Principle of mathematical induction. Well-ordering principle
• Neighborhoods and punctured neighborhoods; interior and isolated points
• ε-δ-defintions of continuity and limits
• Properties of limits: Theorems 2.1 - 2.10
• One-sided neighborhoods and neighborhoods of infinity
• The Archimedian Principle

Particular types of questions for the test:

• Prove an abstract statement about logical statements, sets, or functions.
• For a given ε find a δ in the definitions of limits and/or continuity.
• Prove a property of arithmetic of numbers or inequalities. Justify each step based on a particular axiom of real numbers.
• Prove a statement using mathematical induction, including ones with multiple parameters.
• Find a limit, justify all steps
• Prove that a function is continuous at all values a in its domain
• Prove a theorem about properties of limits (of the type in Theorems 2.1-2.10, but with neighborhoods corresponding to one-sided limits or infinite limits/limits at infinity.)

General rules

• All tests are closed books/notes; no electronic devices.
• The duration of the test is 75 minutes.
• Problems will come in the same format as on the quizzes: you will need to write a concise proof to answer each question.