Math Academy

Upon successful completion of this course, students will have mastered the following:

Sets and Relations

Construct sets using set-builder notation and demonstrate fluency with set operations and terminology.
Identify surjective, injective, and bijective functions.
Compute the cardinality of a set and determine whether an infinite set is countable.
Identify equivalence relations and reason about equivalence classes.

Modular Arithmetic

Define modular congruence in terms of equivalence classes.
Perform arithmetic operations on residue classes.
Use the extended euclidean algorithm to compute modular inverses and solve linear diophantine equations.

Logic

Translate between verbal and symbolic forms of mathematical statements.
Determine whether two statements are equivalent by constructing and comparing truth tables.
Understand that the contrapositive of a statement is logically equivalent to the original statement.
Apply rules of logical inference including distribution, absorption, and De Morgan’s laws.
Translate between logical operations and set operations.
Understand the difference between necessary and sufficient conditions.
Translate between formal and informal language using quantifiers.

Proofs

Construct direct proofs of statements involving numbers, sets, logical operations, relations, and functions.
Disprove false statements by finding counterexamples.
Prove statements using induction, including strong induction.
Leverage indirect proof techniques, including proof by contradiction and proof by contrapositive, to reformulate a proof statement in a way that is easier to prove.

Sets

23 topics

1.1. Introduction to Sets

	1.1.1.	Introduction to Sets
	1.1.2.	Special Sets
	1.1.3.	Equivalent Sets
	1.1.4.	Set-Builder Notation
	1.1.5.	Cardinality of Finite Sets
	1.1.6.	Subsets
	1.1.7.	Power Sets

1.2. Set Operations

	1.2.1.	The Complement of a Set
	1.2.2.	The Union of Sets
	1.2.3.	The Intersection of Sets
	1.2.4.	The Difference of Sets
	1.2.5.	De Morgan's Laws for Sets
	1.2.6.	The Cartesian Product
	1.2.7.	Elementary Properties of Union and Intersection
	1.2.8.	Distributive Properties of Set Operations
	1.2.9.	Disjoint Sets
	1.2.10.	Partitions of Sets
	1.2.11.	Indexed Sets
	1.2.12.	Indicator Functions
	1.2.13.	The Maximum and Minimum of a Set

1.3. Sets in the Plane

	1.3.1.	Interior and Boundary Points
	1.3.2.	Interiors and Boundaries of Sets
	1.3.3.	Open and Closed Sets

Logic

30 topics

2.4. Compound Statements

	2.4.1.	Mathematical Statements
	2.4.2.	The "And" and "Or" Connectives
	2.4.3.	The "Not" Connective
	2.4.4.	Logical Equivalence
	2.4.5.	Associative and Commutative Laws
	2.4.6.	Distributing Conjunctions and Disjunctions
	2.4.7.	The Absorption Laws
	2.4.8.	De Morgan's Laws for Logic
	2.4.9.	Translating Between Logical and Set Operations

2.5. Implications

	2.5.1.	Conditional Statements
	2.5.2.	Logical Equivalence with Conditional Statements
	2.5.3.	Converse, Inverse, and Contrapositive
	2.5.4.	Biconditional Statements
	2.5.5.	Tautologies and Contradictions

2.6. Predicates

	2.6.1.	Truth Sets of Predicates
	2.6.2.	The "And" and "Or" Connectives With Predicates
	2.6.3.	The "Not" Connective With Predicates
	2.6.4.	Simplifying Expressions With Predicates Using De Morgan's Laws
	2.6.5.	Conditional and Biconditional Statements With Predicates
	2.6.6.	Necessary and Sufficient Conditions
	2.6.7.	Grammatical Constructions for Implications and Biconditionals

2.7. Quantifiers

	2.7.1.	Universal and Existential Quantifiers
	2.7.2.	Negating Universal and Existential Statements
	2.7.3.	Formal and Informal Language
	2.7.4.	Nested Quantifiers
	2.7.5.	Negating Statements With Nested Quantifiers
	2.7.6.	Prenex Normal Form

2.8. Logical Inference

	2.8.1.	Implication Elimination and Denying the Consequent
	2.8.2.	Disjunctive Syllogism and Transitivity of Implication
	2.8.3.	Additional Rules of Logical Inference

Functions & Sequences

11 topics

3.9. Functions

	3.9.1.	Sets and Functions
	3.9.2.	Surjections
	3.9.3.	Injections
	3.9.4.	Bijections
	3.9.5.	Into Functions
	3.9.6.	Floor and Ceiling Functions

3.10. Sequences

	3.10.1.	The Limit of a Null Sequence
	3.10.2.	Proving a Sequence Converges to Zero
	3.10.3.	Proving a Sequence Converges to a Finite Limit
	3.10.4.	Infinite Limits of Sequences
	3.10.5.	Proving a Sequence Has an Infinite Limit

Relations

15 topics

4.11. Modular Congruence

	4.11.1.	Modular Congruence
	4.11.2.	The Addition Property of Modular Arithmetic
	4.11.3.	Residues
	4.11.4.	The Multiplication Property of Modular Arithmetic
	4.11.5.	The Division Property of Modular Arithmetic
	4.11.6.	Solving Linear Congruences
	4.11.7.	Solving Advanced Linear Congruences

4.12. Equivalence Relations

	4.12.1.	Relations
	4.12.2.	Equivalence Relations on Scalars
	4.12.3.	Residue Classes
	4.12.4.	Equivalence Classes with Scalars

4.13. Proving Statements Concerning Relations

	4.13.1.	Proving Relations in Modular Arithmetic
	4.13.2.	Proving Set Relations
	4.13.3.	Proving Relations With Matrices
	4.13.4.	Partial Order

Number Theory

5 topics

5.14. Divisibility of Integers

	5.14.1.	The Division Algorithm
	5.14.2.	The Euclidean Algorithm
	5.14.3.	The Extended Euclidean Algorithm
	5.14.4.	Properties of Divisibility
	5.14.5.	Linear Diophantine Equations

Cardinality

6 topics

6.15. Cardinality of Sets

	6.15.1.	Infinite Sets of the Same Cardinality
	6.15.2.	Cardinality of Natural Numbers, Integers, and Rationals
	6.15.3.	Cantor's Diagonal Argument
	6.15.4.	Cantor-Bernstein-Schröder Theorem
	6.15.5.	The Cardinality of the Power Set of Natural Numbers
	6.15.6.	Cantor's Theorem

Proof

49 topics

7.16. Direct Proof

	7.16.1.	Parity
	7.16.2.	Proving Parity Statements
	7.16.3.	Proving Two-Variable Parity Statements
	7.16.4.	Proof by Cases
	7.16.5.	Proving Divisibility
	7.16.6.	Proving Modular Congruence
	7.16.7.	Proving Divisibility Using Modular Congruence
	7.16.8.	Proving Elementary Results Involving Unions and Intersections
	7.16.9.	Direct Proofs Involving Set Complements and Differences
	7.16.10.	Direct Proofs Involving Cartesian Products
	7.16.11.	Direct Proofs of Inequalities
	7.16.12.	Proving the Existence of an Element in a Set
	7.16.13.	Direct Proofs of Prime Properties
	7.16.14.	Direct Proofs of Real Number Statements

7.17. Proof by Induction

	7.17.1.	Proving Sums of Series Using Induction
	7.17.2.	Proving Inequalities Using Induction
	7.17.3.	Proving Divisibility Using Induction
	7.17.4.	Proving Matrix Identities Using Induction
	7.17.5.	Proving Recurrence Relations Using Strong Induction

7.18. Proof by Counterexample

	7.18.1.	Proof by Counterexample
	7.18.2.	Counterexamples Involving Parity
	7.18.3.	Counterexamples Involving Divisibility
	7.18.4.	Counterexamples of Real Number Statements
	7.18.5.	Counterexamples Involving Modular Congruence
	7.18.6.	Counterexamples Involving Inequalities
	7.18.7.	Counterexamples Involving Set Equalities
	7.18.8.	Counterexamples Involving Set Operations
	7.18.9.	Counterexamples Involving Cartesian Products

7.19. Proof by Contrapositive

	7.19.1.	Introduction to Proof by Contrapositive
	7.19.2.	Proving Parity by Contrapositive
	7.19.3.	Proving Parity Statements With Multiple Variables by Contrapositive
	7.19.4.	Proving Divisibility by Contrapositive
	7.19.5.	Proving Modular Congruence by Contrapositive
	7.19.6.	Proving Real Number Statements by Contrapositive
	7.19.7.	Proving Inequalities by Contrapositive
	7.19.8.	Proving Set Inequalities by Contrapositive
	7.19.9.	Proving Statements Involving Set Operations by Contrapositive

7.20. Proof by Contradiction

	7.20.1.	Proof by Contradiction
	7.20.2.	Proving Parity by Contradiction
	7.20.3.	Proving Real Number Statements by Contradiction
	7.20.4.	Proving Divisibility by Contradiction
	7.20.5.	Proving Modular Congruence by Contradiction
	7.20.6.	Proving Set Equalities by Contradiction
	7.20.7.	Proving Statements Involving Set Operations by Contradiction
	7.20.8.	Proving Inequalities by Contradiction
	7.20.9.	Proving the Existence of Infinitely Many Primes by Contradiction
	7.20.10.	Proving the Division Algorithm

7.21. Trivial and Vacuous Proofs

	7.21.1.	Trivial Proofs
	7.21.2.	Vacuous Proofs

Methods of Proof

Overview

Outcomes

Content

Sets and Relations

Modular Arithmetic

Logic

Proofs