Logic for Mathematics
University Mathematics
Individually-sequenced IMACS courses are suitable for diligent, capable middle and high school students who are sufficiently motivated and talented. Topics covered in depth include propositional and predicate logic, number theory, axiomatic set theory, and applications of a formal set theory into fundamental mathematical objects such as mappings.
Students who complete the full IMACS Logic for Mathematics curriculum typically will have an "unfair advantage" with a mathematical foundation that will make all technical classes vastly easier. Former students remark on this effect in courses ranging from physics to philosophy to computer science to pre-law.
This four-part curriculum deals in the first place with the subject matter of the logic courses that are a required part of a college major in mathematics, computer science or philosophy, and goes on to introduce the techniques in logic and reasoning that underpin research and development in mathematics.
Students are introduced first to branches of mathematics called "propositional logic", "predicate logic" and "set theory". The emphasis throughout is on developing a true understanding for the logical underpinning of mathematics.
LM1: Introduction to Logic I
This course introduces students to the Propositional Logic, a branch of
modern mathematics which provides a rigorous mathematical
analysis of the process of rational argument. Students earn
one high school honors credit through the Kaplan College Preparatory School.
Introduction
Logic Puzzles
Logical Thinking
Problem Set LM1.1
The Formal Language
When Is An Argument A Proof?
Analyzing An Argument
Truth and Falsity: Simple Statements
Truth and Falsity: Compound Statements
The Language of the Propositional Calculus
Well-formed Formulas
Interpreting Well-formed Formulas
Decomposing Well-formed Formulas
Instances of Well-formed Formulas
Introduction to Truth Tables
The Truth Table for Conjunction
Building Truth Tables
The Truth Table for Disjunction
The Truth Table for Implication
Problem Set LM1.2
Tautologies
Equivalent Well-formed Formulas
The Biconditional
Problem Set LM1.3
Transitivity of Equivalence
Converse and Contrapositive
Equivalence and Tautologies
Wffs, Meta-wffs, and Instances
Lifting Well-formed Formulas
Truth Table Templates
The Tautology Principle
Problem Set LM1.4
Review for Test LM1.1
Introducing Demonstrations
Applying Modus Ponens
Examining an Argument
What is a Demonstration?
Demonstrations and Program Outlines
Problem Set LM1.5
Conjunctive Inference
Conjunctive Simplification
Problem Set LM1.6
Unnecessary Hypotheses
Contrapositive Inference and Modus Tollens
Applying Modus Tollens
Problem Set LM1.7
Syllogistic Inference
Inference by Cases
Problem Set LM1.8
Modus Ponens for the Biconditional
Commutativity and Transitivity of the Biconditional
Contrapositive Inference and Modus Tollens for the Biconditional
Biconditional Inference
The Substitution Principle
Problem Set LM1.9
Tautologies and Demonstrations
The Substitution Principle and Equivalence
Consequences of the Substitution Principle
Rules of Inference: A Summary
Problem Set LM1.10
Review for Test LM1.2
Working With Demonstrations
Using The Deduction Theorem
Problem Set LM1.11
More Uses of the Deduction Theorem
Problem Set LM1.12
Justifying the Deduction Theorem
Problem Set LM1.13
Reflecting on the Deduction Theorem
Language and Metalanguage
Indirect Inference
Problem Set LM1.14
Justifying Indirect Inference
Using Indirect Inference
Review for Test LM1.3
LM2: Introduction to Logic II
The course introduces students to the Predicate Logic, a so-called "first-order logic"
sufficient to formalize all of set theory and thereby a substantial part of all of mathematics. Students earn one high
school honors credit through the Kaplan College Preparatory School.
A Logic for Set Theory
A New Formal Language
Interpreting Terms and Formulas
Free and Bound Occurrences
Open and Closed Terms and Formulas
Problem Set LM2.1
Simplifying the Notation
Rebuilding Terms and Formulas
Problem Set LM2.2
Metaterms, Metaformulas, and Instances
Introducing Tautologies
Demonstrations
Rules of Inference: A Summary
A Sample Demonstration
Problem Set LM2.3
Axioms for the Predicate Calculus
Problem Set LM2.4
Using Axioms in Demonstrations
Rules of Inference for Axioms
Demonstrations Revisited
Using IU and PGU
Problem Set LM2.5
Review for Test LM2.1
Theorems and Metatheorems
Problem Set LM2.6
Using Metatheorems in Demonstrations
Streamlining Demonstration Outlines
The Existential Quantifier
Properties of the Existential Quantifier
Problem Set LM2.7
The Substitution Principle
Naming Objects that Exist
Inference from an Existential
Rules and Regulations
Problem Set LM2.8
Demonstrations Revisited Again
Review for Test LM2.2
LM3: Set Theory
This course introduces students to the axiomatic set theory of
John von Neumann, Paul Bernays and Kurt Godel ("NBG"). This theory has a central role in modern mathematics,
providing the basic language in which most of mathematics is expressed. Students earn
one high school honors credit through the Kaplan College Preparatory School.
The Theory of Sets and Classes
Properties of Equality
Using Theorems as Metatheorems
Substitution of Equals
Problem Set LM3.1
Subclasses
Unique Existence
Axioms for Described Terms
Problem Set LM3.2
Working With Described Terms
Sets
Class Symbols
The Comprehension Principle
Problem Set LM3.3
Using The Comprehension Principle
Some Special Classes
Russell's Paradox
Problem Set LM3.4
The Empty Set
Rules and Regulations
Review for Test LM3.1
Complements and Differences
Unions and Intersections
Power Classes
Problem Set LM3.5
Singletons and Doubletons
Manifold Union and Intersection
Problem Set LM3.6
The Whole Numbers
Mathematical Induction
The Peano Postulates
Ordering the Whole Numbers
The Well-ordering Principle
The Axiom of Regularity
Problem Set LM3.7
Review for Test LM3.2
LM4: Relations, Mappings and Functions
This one course introduces students to the concepts of relations,
mappings and functions. Students earn one high school honors credit
through the Kaplan College Preparatory School.
From Demonstrations to Proofs
First Steps
Problem Set LM4.1
Handling the Deduction Theorem
Arguing Indirectly
Problem Set LM4.2
Chains of Equalities and Biconditionals
Problem Set LM4.3
Writing Induction Proofs
Problem Set LM4.4
Relations
Properties of Ordered Pairs
Problem Set LM4.5
Some Notational Conventions
Cartesian Products
Classes of Ordered Pairs
Cartesian Products and Proper Classes
Relations
Problem Set LM4.6
Naming Relations
Restricting Relations
Image of a Class under a Relation
Problem Set LM4.7
Review for Test LM4.1
The Converse of a Relation
Composition of Relations
Problem Set LM4.8
Mappings
Function Values
Functional Relations and Sets
Introduction to Mappings
Composition of Mappings
Restrictions and Images
Problem Set LM4.9
Commutative Diagrams
Families
Piecewise Definition of a Mapping
Problem Set LM4.10
Injections
Surjections
Bijections
The Axiom of Choice
Problem Set LM4.11
Review for Test LM4.2
Procedures
Each student is assigned a principal IMACS instructor, and is given unlimited access during the enrollment period.
The principal instructor is available by phone or by email during regular business hours. Additionally, the principal instructor or other IMACS instructors are available via the online Help Desk, monitored seven days a week including evenings.
Students submit assignments and tests online. Scores for assignments and tests are accumulated into the student's online gradebook so that parents, students and instructors can monitor student progress.
