MATH 350, Review for Midterm Test 1

Test topics

0. Introduction
   1. Logic and proofs (lecture notes)
   2. Sets and operations on them (Kirkwood 1-1; lecture notes)
   3. Relations, functions, direct and inverse images (Kirkwood, 1-1; lecture notes)
   4. Composition of functions; one-to-one, onto and inverse functions (Kirkwood, 1-1; lecture notes)

1. The real number system
   1. Axioms of a field (1.1)
   2. The Order Axiom (Axiom I); Inequalities (1.2)
   3. Absolute value (A.1)
   4. Natural numbers and mathematical induction (1.4)
   5. The Completeness axiom (lecture notes)

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 (2.4)

Important definitions/facts

• Sets and operations on sets: union, intersection, difference, complement, Cartesian product
• Relations, functions; direct and inverse images
• One-to-one, onto and invertible functions
• List of axioms for a field, axiom of order; the completeness axiom
• Absolute value; triangle inequality and its versions (with proof)
• Natural numbers; principle of mathematical induction
• Neighborhoods and punctured neighborhoods; interior points of a set
• ε-δ-defintions of continuity and limits
• Properties of limits: arithmetic operations, limit of composition, inequalities, "sandwiching principle" (with proofs)
• ε-δ-defintions for one-sided limits, infinite limits and other varieties

Types of questions that will be on the test:

• Prove facts about arithmetic operations and/or inequalities, using axioms of fields and the axiom of order
• For a given ε find a δ in the definitions of limits and/or continuity
• Prove a statement using mathematical induction
• Prove a theorem about properties of limits / one-sided limits / limits at infinity / infinite limits
• Find limits / determine continuity using properties of limits

General rules

• All tests are closed books/notes; graphing calculators, cell phones or other electronic devices are not permitted. A basic scientific calculator (example: TI 30X IIS or similar) may be helpful but it is not required.
• The duration of the test is 60 minutes.
• Problems will come in the same format as on the quizzes: you will need to write a concise proof to answer each question.