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Study guide for the midterm
(MATH 6100-001, Fall 2015)
This document will be continuously updated until we review for the midterm on September 29
Last update: Tuesday, September 29, 2015

  1. Definitions and notions to remember:
    conjunction, disjunction, negation, conditional, biconditional, tautology, contradiction (Section 1.1); P implies Q, P is logically equivalent to Q (Section 1.2), bound variable, existential quantifier, universal quantifier, universal instantiation, existential instantiation, universal generalization, existential generalization, even and odd integers (Definition 2.1.2), divisibility (Definition 2.2.1), partial order (from your notes), proofs by contrapositive and by contradiction (Section 2.3), rational and irrational numbers (Definition 2.3.3), prime and composite numbers (Definition 2.3.6).

  2. Exercises you should be able to perform:
    Translate a statement into a formula, build a truth table, formally derive a conclusion from premises (using the rules of inference listed in Fact 1.3.1). If the formula does not involve quantifiers and the conclusion does not follow, then you should be able to show this using truth tables.

  3. Statements you should be able to prove:
    Any of the logical implications listed in Fact 1.3.1 and any logical equivalence listed in Fact 1.3.2 (I will provide the lists, you have to work out the truth table), facts about even and odd integers (Theorem 2.1.3), implications and equivalences listed in Fig. 1.5.1, divisibility on nonnegative integers is a partial order (Theorem 2.2.2 and your notes), any integer divides zero (Theorem 2.2.3), an integer has the same parity as its square (Theorem 2.3.1 and Exercise 2.2.4), only Pythagorean triple with consecutive integers is (3,4,5) (Theorem 2.3.2), the square root of 2 is irrational (Theorem 2.3.5), there are infinitely many prime numbers (Theorem 2.3.7), n2+n is even (Theorem 2.4.1).

  4. Statements whose proof techniques you should be able to reproduce to prove similar statements:
    If xy is irrational then x or y is irrational (Theorem 2.4.2), a divides b and b divides a if and only if a=b or a=-b (Theorem 2.4.3), mn is odd iff both m and n are odd (Theorem 2.4.4), equivalent conditions for a 2x2 upper triangular matrix with integer entries to have unit determinant (Theorem 2.4.5), there is a 2x2 matrix with prescribed trace and determinant (Proposition 2.5.1), solving radical equations (Section 2.5), 2x2 matrix with nonzero determinant has unique inverse, for all a there is a b such that a2-b2+4=0 (Proposition 2.5.3), there is an x such that (3-x)(y2+1)>0 for all y (Proposition 2.5.4).

  5. Statements you should be able to state (without proof):
    Completeness theorem (from your notes), analogues of De Morgan's laws for quantifiers (Fact 1.5.1)

  6. What to expect
    The exam will be closed book. I will provide a list of logical implications and equivalences listed in Fact 1.3.1 and 1.3.2. 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!