New York City College of Technology
The City University of New York
DEPARTMENT: Mathematics
PREPARED BY: Professors Victoria Gitman, Jonas Reitz,
COURSE: MAT 2071
TITLE: Introduction to Proofs and Logic
DESCRIPTION: The course is designed to prepare students for an advanced mathematics
curriculum by providing a transition from Calculus to abstract mathematics. The course
focuses on the processes of mathematical reasoning, argument, and discovery. Topics
include propositional and first order logic, learning proofs through puzzles and games,
axiomatic approach to group theory, number theory, and set theory, abstract properties of
relations and functions, elementary graph theory, sets of different cardinalities, and the
construction and properties of real numbers.
TEXT: Hammack, R. (2009). Book of Proof. Edition 1.3. Creative Commons Attribution - No
Derivative Works 3.0
CREDIT HOURS: 4 cl hrs, 0 lab hrs, 4 cr
PRE- or COREQUISITES: MAT 1575
LEARNING OUTCOMES:
1 Students will be able to evaluate truth of statements in propositional and first
order logic.
2 Students will be able to understand and use formal reasoning methods.
3 Students will be able to recognize the role of sets in mathematics.
INSTRUCTIONAL OBJECTIVES
ASSESSMENT
For successful completion of the course,
students should be able to:
Instructional Activity, Evaluation Methods
and Criteria
Reason in accordance with laws of
propositional and first-order logic
Lecture, group work, homework
assignments, examinations
Evaluate truth of statements in
propositional and first-order logic
Use the axiomatic method in establishing
the truth of mathematical statements
Analyze and prove elementary statements
about group theory, number theory, set
theory, and graph theory
View mathematics from the perspective of
its constituent blocks – sets
Construct real numbers and derive their
properties starting from the natural
numbers
GENERAL EDUCATION LEARNING
OUTCOMES
Gather, interpret, evaluate, and apply
information discerningly from a variety of
sources.
Employ scientific reasoning and logical
thinking.
Acquire tools for lifelong learning.
Use creativity to solve problems.
Show curiosity and the desire to learn.
Lecture, group work, homework
assignments, examinations
Lecture, group work, homework
assignments, examinations
Lecture, group work, homework
assignments, examinations
Lecture, group work, homework
assignments, examinations
Lecture, group work, homework
assignments, examinations
ASSESSMENT
Classroom discussion, writing assignments,
student presentations, problem sets, tests,
exams.
Classroom discussion, writing assignments,
student presentations, problem sets, tests,
exams.
Classroom discussion, writing assignments,
student presentations, tests, exams.
Problem sets, group work.
Classroom discussion, groups work,
presentations.
GRADING PROCEDURE:
● Homework assignments and oral presentations
● Midterm
● Final Exam
TEACHING/LEARNING METHODS:
● Lecture and guided discussion
● Student presentations
● Use of online resources
● Writing intensive assignment
30%
35%
35%
WEEKLY COURSE OUTLINE:
WEEK
TOPIC
CHAPTERS/SECTIONS
1
Sets
sets, cartesian products, subsets
3
Logic
propositional logic, statements, logical
connectives, truth tables, logical
equivalence
2
Sets
4
Logic
Counting
6
Proof
5
7
8
9
10
11
Counting
Proof
Proof
Midterm
Proof
Proof
Proof
Relations
Relations
12
Functions
13
Functions
15
Final exam
14
Cardinality
set operations, collections of sets
first order logic, quantifiers, inference,
lists, factorials
counting subsets, binomial theorem,
direct proof, examples from number
theory
contrapositive proof
proof by contradiction
if-and-only-if proofs, existence proofs,
proofs involving sets
disproof, counterexamples, proof by
induction, logic puzzles
strong induction, minimal
counterexamples, relations and their
properties
equivalence relations and partitions,
integers modulo n, real numbers, graph
theory, relations between sets
injective and surjective functions,
pigeonhole principle, composition of
functions
inverse functions, image and preimage,
cardinalities
countable and uncountable sets,
comparing cardinalities