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Study guide for the midterm
(MATH 6101-090, Fall 2014)

  1. Definitions and axioms to remember:
    Some fundamental notions you should know from lectures and/or prerequisites: equivalence relation, partial order, total order, well-ordered set, ring, field.
    From the book: Peano postulates (Axiom 1.2.1), natural numbers (Definition 1.2.2), definition of order on natural numbers (Definition 1.2.8), definition of integers and the basic operations on them (Definitions 1.3.1 and 1.3.3), positive and negative integers (Definition 1.3.6), constructing rational numbers, relations and operations on them (Definitions 1.5.1, 1.5.3, and 1.5.7), Dedekind cuts (Definition 1.6.1), rational Dedekind cuts (Definition 1.6.4), real numbers (Definition 1.7.1), relations and operations on real numbers (Definitions 1.7.2 and 1.7.3), upper bound and lower bound properties (Definitions 1.7.7, 2.2.2 and 2.2.3), ordered field (Definition 2.2.1), axiom for the real numbers (Axiom 2.2.4), additional operations on real numbers (Definition 2.3.1), positive and negative real numbers (Definition 2.3.4), closed and open, bounded and unbounded intervals (Definition 2.3.6), absolute value (Definition 2.3.8), inductive sets (Definition 2.4.1), natural numbers as a subset of real numbers (Definition 2.4.2), rational numbers as a subset of real numbers (Definition 2.4.11).

  2. Statements you should be able to prove:
    Uniqueness of a successor (Lemma 1.2.3), existence and uniqueness of addition and multiplication on natural numbers (Theorems 1.2.5 and 1.2.6), well-ordering principle (Theorem 1.2.10), the relation defining integers is an equivalence relation (Lemma 1.3.2), the operations and relations on integers are well-defined (Lemma 1.3.4), the relation defining rational numbers is an equivalence relation (Lemma 1.5.2), relations and operations on rational numbers are well-defined (Lemma 1.5.4), rational numbers define Dedekind cuts (Lemma 1.6.2), properties of complements of Dedekind cuts (Lemma 1.6.5), trichotomy of inclusion on Dedekind cuts (Lemma 1.6.6), Dedekind cuts have the greatest lower bound property (Theorem 1.7.8), on any totally ordered set the greatest lower bound property is equivalent to the least upper bound property (Theorems 1.7.9 and 2.6.4, note the reversal of order highlighted in class!) Peano postulates for natural numbers as a subset of real numbers (Theorem 2.4.4), least upper bound and greatest lower bound are unique (Lemma 2.6.2), Archimedean property (Theorem 2.6.7), Heine-Borel theorem (Theorem 2.6.14).

  3. Statements you should be able to state (without proof):
    Definition by recursion (Theorem 1.2.4), properties of addition and multiplication on natural numbers (Theorem 1.2.7), properties of the order (Theorem 1.2.9), properties of operations and relations on integers (Theorem 1.3.5), natural numbers may be considered a subset of integers (Theorem 1.3.7), further properties of integers (Lemma 1.3.8), properties of rational numbers (Theorem 1.5.5), integers may be considered a subset of rational numbers (Theorem 1.5.6), further properties of rational numbers (Lemma 1.5.8), we may define addition, negative, multiplication and inverse on Dedekind cuts (Lemma 1.6.8), real numbers form an ordered field (Theorem 1.7.6), rational numbers may be considered a subset of real numbers (Theorem 1.7.10), properties of real numbers (Lemmas 2.3.2, 2.3.3, 2.3.5), principles of mathematical induction (Theorems 2.5.1, 2.5.4, basically be able to prove any statement by induction), definition by recursion (Theorem 2.5.5).

  4. What to expect
    The exam will be closed book. You will have 80 minutes to answer about 10 questions. Some questions may ask you to state and prove a theorem from the list above, others may be like the exercises from your homework assignments. Even if a statement is listed "without proof" above, you must remember the proof of those parts of it that were on a homework assignment! There may be questions where you have to decide about an example whether it has certain properties. (E.g. "Is this ordered set well-ordered?" "Is this subset of the rational numbers a cut?" etc.)