Study guide for the final
(MATH 6101-001, Fall 2022)
This document will be continuously updated until we review for the
final on December 7
Last update: Wednesday, December 7, 2022
- Definitions and notions to remember:
Families of sets (Definition 3.4.2), infinite
unions and intersections (Definition 3.4.3), functions (Definition
4.1.1), special maps, restriction and extension (Definition 4.1.3),
image and inverse image (Definition 4.2.1), composing functions
(Definition 4.3.1), coordinate function (Definition 4.3.3), left and
right inverses (Definition 4.3.6), injections, surjections and
bijections (Definition 4.4.2), Relations (Definition 5.1.1), relation
class (Definition 5.1.3, I will provide this one, if needs to be
used), reflexive, symmetric and transitive
relations (Definition 5.1.5), congruence modulo n and
operations on congruence classes (Definitions 5.2.1,
5.2.8 and 5.2.10), equivalence relation (Definition 5.3.1),
equivalence classes (Definition 5.3.3), quotient of an equivalence
relation (Definition 5.3.6), canonical map (Definition 5.3.8),
partition (Definition 5.3.10), Peano postulates (Axiom 6.2.1),
cardinality (Definition 6.5.1), finite, countable and infinite sets
(Definition 6.5.4), smaller or equal sets (Definition 6.5.9),
cardinality of a finite set as a natural number (Definition 6.6.1).
- Exercises you should be able to perform:
Write a set in set notation, using a defining property, or by listing
its elements, provide left inverses and right inverses to functions,
prove statements by induction.
- Statements you should be able to prove:
Properties of image and inverse image (Theorem 4.2.4), associative and
identity laws for composing functions (Lemma 4.3.5), every left
inverse equals every right inverse (Lemma 4.3.7), composing injections
and surjections (Lemma 4.4.4), one sided inverses, injectivity and
surjectivity (Theorem 4.4.5), congruence modulo n is an
equivalence relation (Lemma 5.2.3), there are n congruence
classes modulo n (Theorem 5.2.4), congruence modulo n is
compatible with addition and multiplication (Lemma 5.2.11), equivalence classes form a partition of
the underlying set (Theorem 5.3.4), summation formula for the
arithmetic and geometric sequences (from your notes on November 9 see
also
Proposition 6.3.3), having the same cardinality is an equivalence
(Lemma 6.5.2), the set of natural numbers is infinite (Lemma 6.5.5), no
set has the same cardinality as its power set (Theorem 6.5.7), a subset of a
countable set is countable (Theorem 6.6.7, only basic ideas as we did
in class), the union of a countable family of countable sets is
countable (Theorem 6.6.9), the Cartesian product of
finitely many countable sets is countable (Theorem 6.6.10), every
infinite set has a countably infinite
subset (Theorem 6.6.11), the set of rational numbers is countably
infinite (Theorem 6.7.1), the set of real numbers is uncountable
(Theorem 6.7.3), any finite Cartesian power of the
set of real numbers has the same cardinality as the set of real
numbers (Theorem 6.7.5).
- Statements whose proof techniques you should be
able to reproduce to prove similar statements:
Summation formula for an arithmetic sequence (Proposition 6.3.3), every
Boolean function may be computed using switching circuits (Example
6.3.5), inequalities like Proposition 6.3.7, every positive integer
is one, a prime, or a product of finitely many primes (Theorem
6.3.10), closed form formula for the Fibonacci numbers (formula
(6.4.1), I would provide it, you would need to prove it by induction).
- Statements you should be able to state (without
proof):
Properties of infinite unions and intersections (Theorem 3.4.5),
Division Algorithm (Theorem A.5), equivalence in terms of
equivalence classes (Corollary
5.3.5), if a function is constant on the equivalence classes, it
factors through the canonical map (Lemma 5.3.9), definition by
recursion (Theorem 6.2.3), basic properties of addition,
multiplication, and the "greater than" relation (Theorem
6.2.4), Well-Ordering Principle (Theorem 6.2.5), Principle of
Mathematical Induction and variants (Theorems 6.3.1, 6.3.6, 6.3.8,
6.3.9), properties of functions on finite
sets (Theorem 6.3.11/ parts 1, 2 and 3), the power set of the natural
numbers is uncountable (Corollary 6.5.8), "smaller or
equal" is antisymmetric on sets (Schroeder-Bernstein Theorem,
Theorem 6.5.10), Trichotomy Law for Sets (Theorem 6.5.13), two finite sets have the same
cardinality exactly when they are associated with the same natural
number (Lemma 6.6.2 and Corollary 6.6.3), rules
on cardinalities of finite sets (Theorem 6.6.5), a set is infinite iff
it contains an infinite subset (Corollary 6.6.6), equivalent descriptions
of a countable set (Theorem 6.6.8), the
set of algebraic numbers is countably infinite (Theorem 6.7.2),
the set if irrational numbers has the same cardinality as the set
of real numbers (Theorem 6.7.4).
- What to expect
The exam will be closed book.
The above guide is meant to help with the mandatory part. For the
optional part prepare as if it was another midterm. The mandatory part
will be as long as the midterm, the optional part will have only about
5 questions. On the mandatory part, 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!