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MA111: Introduction to Mathematical Reasoning
Sections
Course Introduction
Unit 1: Logic
Unit 2: Sets
Unit 3: Introduction to Number Theory
Unit 4: Rational Numbers
Unit 5: Mathematical Induction
Unit 6: Relations and Functions
Unit 7: Sets, Part II
Unit 8: Combinatorics
Course Feedback Survey
Certificate Final Exam
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MA111: Introduction to Mathematical Reasoning
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Mathematics
MA111: Introduction to Mathematical Reasoning
Sections
Unit 7: Sets, Part II
7.2: The Rational Numbers Are Countable
7.2.1: The Proof
Theorem of the Week: "Theorem 18: The Rational Numbers Are Countable"
Back to '7.2.1: The Proof\'
Theorem of the Week: "Theorem 18: The Rational Numbers Are Countable"
Mark as completed
Study the proof on this webpage, which shows that rational numbers are countable.
Click
http://web.archive.org/web/20150709073158/https://theoremoftheweek.wordpress.com/2010/02/24/theorem-18-the-rational-numbers-are-countable/
link to open resource.
Mark as completed
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Course Syllabus
Course Terms of Use
Wikipedia: "Sudoku"
Tom Davis' "The Mathematics of Sudoku"
Wikipedia: "Latin Square"
Harold Reiter's "Introduction to Mathematical Reasoning"
The New York Times
: "Ken-Ken Puzzles"
Set Enterprises: "Daily Puzzle"
Khan Academy: "Brain Teasers"
University of Chicago: Antonio Montalban and Yannet Interian's "Module on Puzzles"
Alexander Bogomolny's "A Fake Among Eight Coins"
Hofstra University: Stefan Waner and Steven R. Costenoble's "Introduction to Logic"
Propositional Logic
Boolean Operators
Introduction to Propositional Logic Part I
Wikipedia: "Logical Connective"
University of Cincinnati, Blue Ash: Kenneth R. Koehler's "Logic and Set Theory"
Propositional Logic, Continued
Predicates and Qualifiers
Methods of Proof
California State University, San Bernardino: Peter Williams' "Notes on Methods of Proof"
Old Dominion University: Shunichi Toida's "Problem Solving"
Gowers' Weblog: "Basic Logical Relationships between Statements, Converses, and Contrapositives"
Donna Roberts' "Logic and Related Conditionals Quiz"
Unit 1 Assessment
Sets
Old Dominion University: Shunichi Toida's "Set Theory"
University of California, San Diego: Edward Bender and S. Williamson's "Arithmetic, Logic and Numbers, Unit SF: Sets and Functions"
Set Theory
University of California, San Diego: Edward Bender and S. Williamson's "Arithmetic, Logic and Numbers, Unit SF: Sets and Functions"
Old Dominion University: Shunichi Toida's "Set Operations"
Old Dominion University: Shunichi Toida's "Properties of Set Operation"
Simpson College: Lydia Sinapova's "Boolean Algebra"
The University of Western Australia: Greg Gamble's "Set Theory, Logic, and Boolean Algebra"
Jerusalem College of Technology: Dr. Dana-Picard's "The Characteristic Function of a Set"
Jerusalem College of Technology: Dr. Dana-Picard's "The Characteristic Function of a Set"
Disjoint Sets
Nikos Drakos and Ross Moore's "Cartesian Product of Sets"
Equivalent Sets
University of Hawaii: G.N. Hile's "Set Cardinality"
Unit 2 Assessment
Harold Reiter's "Fusing Dots"
Wisconsin Technical College System: Laurie Jarvis' "Understanding Place Value"
University of St. Andrews: J.J. O'Connor and E.F. Robertson's "Prime Numbers"
Wikipedia: "Prime Number"
The University of Tennessee at Martin: Chris K. Caldwell's "Euclid's Proof of the Infinitude of Primes"
The University of Utah: Peter Alfeld's "Prime Number Problems"
Plus Magazine
: "Mathematical Mysteries: Twin Primes"
Mike James' "Goldbach Conjecture: Closer to Solved?"
Wikipedia: "Riemann Hypothesis"
The Fundamental Theorem of Arithmetic
Alexander Bogomolny's "Euclid's Algorithm" and "GCD and the Fundamental Theorem of Arithmetic"
Wikipedia: "Fundamental Theorem of Arithmetic"
University of California, Berkeley: Zvezdelina Stankova-Frenkel's "Unique and Nonunique Factorization"
Modular Arithmetic
Divisibility by 3, 9, and 11
Harold Reiter's "Building the Rings to Z
6
and Z
7
"
The Art of Problem Solving: "2000 AMC 12 Problems"
The University of Western Australia: Greg Gamble's "The Floor or Integer Part Function" and "Number Theory 1"
Andy Schultz's "GCD and LCM"
Wikipedia: "Divisor Function"
Harold Reiter's "Just the Factors, Ma'am"
Alexander Bogomolny's "The Euclidean Algorithm"
Michael Slone, Kimberly Lloyd, and Chi Woo's "Proof of the Fundamental Theorem of Arithmetic"
Massachusetts Institute of Technology: Dr. Srini Devadas and Dr. Eric Lehman's "Number Theory I"
Harold Reiter's "Decanting"
The University of Western Australia: Greg Gamble's "Number Theory 1"
DavData: "Solving Ax + By = C"
Carnegie Mellon University: Victor Adamchick's "Integer Divisibility"
Harold Reiter's "Fractions"
Rational vs. Irrational Numbers
Recurring Decimals to Fractions
Proof: The Square Root of 2 Is Irrational
New York University: Lawrence Tsang's "Real Numbers"
Dan Sewell Ward's "Transcendental Numbers"
New York University: Lawrence Tsang's "Real Numbers"
Induction
Mathematical Induction
Proof by Induction
Mathematical Induction, Divisibility Proof
Old Dominion University: Shunichi Toida's "Recursive Definition"
Hard Inequality
Unit 5 Assessment
Binary Relations
Relations and Functions
Injections, Surjections, and Bijections
Binary Relations
Old Dominion University: Shunichi Toida's "Equivalence Relations"
MathVids: "Equivalence Relations and Partial Orders"
Unit 6 Assessment, Part 1
Unit 6 Assessment, Part 2
Cardinality
American Public University: "Equivalent Sets", "Infinite Sets and Cardinality", and "Subset and Proper Subset"
Indian Institute of Technology, Madras: Arindama Singh's "Cantor's Little Theorem"
Proof: There Are More Real Numbers than Natural Numbers
How to Count Infinity
The Cantor Set Is Uncountable
Brown University: Rich Schwartz's "Countable and Uncountable Sets"
Proof: There Are the Same Number of Rational Numbers as Natural Numbers
Alex Youcis' "Algebraic Numbers Are Countable"
University of St. Andrews: John O'Connor's "Infinity and Infinites"
Florida State University: Dr. Penelope Kirby's "Property of Functions"
Unit 7 Assessment, Part 1
Unit 7 Assessment, Part 2
Permutations and Combinations
Harold Reiter's "Counting"
Inclusion/Exclusion
Formula for the Union of Sets - Two Sets and Three Sets
Wikipedia: "Inclusion-Exclusion Principle"
Inclusion/Exclusion Examples
Pigeon-Hole Principle
Pigeon Hole Principle
Pigeon-hole Principle Problem Examples
Basic Pigeon-Hole Principle Problems
Course Feedback Survey
MA111: Certificate Final Exam
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