Foundations of Mathematical Proof

Department: 
MATH
Course Number: 
2106
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Fall and Spring semesters, some summers

An introduction to proofs in advanced mathematics, intended as a transition to upper division courses including MATH 4107, 4150 and 4317. Fundamentals of mathematical abstraction including sets, logic, equivalence relations, and functions. Thorough development of the basic proof techniques: direct, contrapositive, existence, contradiction, and induction. Introduction to proofs in analysis and algebra.

Prerequisites: 

(MATH 1552 or MATH 1X52) and (MATH 1553 or MATH 1554 or MATH 1564 or MATH 1X53 or MATH 1X54))

Course Text: 

At the level of:

Book of Proof (3rd edition), by Richard Hammack 

Abstract Algebra: Theory and Applications (2019 edition), by Thomas Judson 

Elementary Analysis: The Theory of Calculus, by Kenneth Ross

Topic Outline: 

The following chapters and sections from all three books:

From Book of Proof (3rd edition), by Richard Hammack 
•    Sets  (Chapter 1) 
•    Logic (Chapter 2) 
•    Direct Proof (Chapter 4) 
•    Contrapositive Proof (Chapter 5) 
•    Proof by Contradiction (Chapter 6) 
•    Proving Non-Conditional Statements (Chapter 7) 
•    Proof Involving Sets (Chapter 8) 
•    Disproof (Chapter 9) 
•    Mathematical Induction (Chapter 10) 
•    Relations (Chapter 11) 
•    Functions (Chapter 12) 
•    Cardinality of Sets (Chapter 14) 

From Abstract Algebra: Theory and Applications, by Thomas Judson 
•    Groups (Chapter 3) 
•    Cosets and Lagrange theorem (Sections 6.1 and 6.2) 

From Elementary Analysis: The Theory of Calculus, by Kenneth Ross
•    The Completeness Axiom (Section 4 from Chapter 1)
•    Sequences (Sections 7, 9, 10, 11 from Chapter 2)
•    Continuity (Section 17 from Chapter 3) 

Texts and topics may vary slightly according to time availability and instructor’s interest.