Study guide for the midterm
(MATH 6101-001, Fall 22)
This document will be continuously updated until we review for the
midterm on October 5
Last update: Wednesday, October 5, 2022
- 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), intervals (Definition 3.2.1), subset (Definition
3.2.2), proper subset and equality of sets (Definition 3.2.5), power
set (Definition 3.2.8), union and intersection (Definition 3.3.1),
disjoint sets (Definition 3.3.4), set difference (Definition 3.3.5),
Cartesian product (Definition 3.3.9).
- 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. Write a set in set notation, using a defining property,
or by listing its elements.
- 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). Properties
of the subset relation (Lemma 3.2.4), laws on union and intersection
(Theorem 3.3.3), laws involving set difference (Theorem 3.3.8),
properties of the Cartesian product (Theorem 3.3.12).
- 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), 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). Regarding statements involving
matrices, study the statement types and the proof techniques
only. Similar but simpler questions (involving radical or linear
equations) may be asked. The focus will be on the logic, not the
algebra.
- 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)
- 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!