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.

Logic

30 topics

1.1. Statements

	1.1.1.	Statements and Predicates
	1.1.2.	The "And" and "Or" Connectives
	1.1.3.	The "Not" Connective
	1.1.4.	Logical Equivalence
	1.1.5.	The Associative and Commutative Laws
	1.1.6.	The Distributive Laws
	1.1.7.	The Absorption Laws
	1.1.8.	De Morgan's Laws

1.2. Implications and Biconditionals

	1.2.1.	Conditional Statements
	1.2.2.	Logical Equivalence with Conditional Statements
	1.2.3.	Biconditional Statements
	1.2.4.	Tautologies and Contradictions
	1.2.5.	Converse, Inverse, and Contrapositive

1.3. Predicates

	1.3.1.	Truth Sets of Predicates
	1.3.2.	The "And" and "Or" Connectives With Predicates
	1.3.3.	The "Not" Connective With Predicates
	1.3.4.	Simplifying Predicate Expressions Using De Morgan's Laws
	1.3.5.	Conditional Statements With Predicates
	1.3.6.	Necessary and Sufficient Conditions
	1.3.7.	Grammatical Constructions for Conditional Statements
	1.3.8.	Translating Between Logical and Set Operations

1.4. Quantifiers

	1.4.1.	Universal and Existential Quantifiers
	1.4.2.	Negating Quantified Statements
	1.4.3.	Nested Quantifiers
	1.4.4.	Formal and Informal Language
	1.4.5.	Negating Statements With Nested Quantifiers
	1.4.6.	Prenex Normal Form

1.5. Logical Inference

	1.5.1.	Implication Elimination and Denying the Consequent
	1.5.2.	Disjunctive Syllogism and Transitivity of Implication
	1.5.3.	Additional Rules of Logical Inference

Set Theory

25 topics

2.6. Introduction to Set Theory

	2.6.1.	Sets
	2.6.2.	Special Sets
	2.6.3.	Equivalent Sets
	2.6.4.	Set-Builder Notation
	2.6.5.	Cardinality of Finite Sets
	2.6.6.	The Maximum and Minimum of a Set

2.7. Subsets

	2.7.1.	Subsets
	2.7.2.	Power Sets
	2.7.3.	Partitions of Sets
	2.7.4.	Indicator Functions
	2.7.5.	Proving Subset Relations

2.8. Set Operations

	2.8.1.	The Union of Sets
	2.8.2.	The Intersection of Sets
	2.8.3.	The Difference of Sets
	2.8.4.	Set Complements
	2.8.5.	The Cartesian Product
	2.8.6.	Indexed Sets
	2.8.7.	Disjoint Sets

2.9. Properties of Set Operations

	2.9.1.	Elementary Properties of Set Operations
	2.9.2.	Proving Elementary Properties of Set Operations
	2.9.3.	De Morgan's Laws for Sets
	2.9.4.	Proving De Morgan's Laws for Sets
	2.9.5.	Distributive Properties of Set Operations
	2.9.6.	Proving Distributive Properties of Union and Intersection
	2.9.7.	Proving Distributive Properties of the Cartesian Product

Discrete Mathematics & Number Theory

11 topics

3.10. Integer Divisibility

	3.10.1.	Parity
	3.10.2.	Properties of Divisibility
	3.10.3.	The Division Algorithm
	3.10.4.	The Euclidian Algorithm
	3.10.5.	The Extended Euclidean Algorithm
	3.10.6.	Linear Diophantine Equations

3.11. Proof by Induction

	3.11.1.	Mathematical Induction
	3.11.2.	Proving Inequalities Using Induction
	3.11.3.	Proving Divisibility Using Induction
	3.11.4.	Proving Matrix Identities Using Induction
	3.11.5.	Strong Induction and Recurrence Relations

Mathematical Proof

15 topics

4.12. Introduction to Mathematical Proof

	4.12.1.	Direct Proof
	4.12.2.	Proving Parity
	4.12.3.	Proving Divisibility
	4.12.4.	Proof by Cases
	4.12.5.	Disproving Universal Statements
	4.12.6.	Disproving Implications
	4.12.7.	Trivial and Vacuous Proofs
	4.12.8.	Proving Biconditional Statements

4.13. Proof by Contrapositive and Contradiction

	4.13.1.	Proof by Contrapositive
	4.13.2.	Proving Parity by Contrapositive
	4.13.3.	Proving Divisibility by Contrapositive
	4.13.4.	Proof by Contradiction
	4.13.5.	Proving Irrationality by Contradiction
	4.13.6.	Proving Properties of Irrationals by Contradiction
	4.13.7.	Proving Divisibility by Contradiction

Sequences & Functions

20 topics

5.14. Sequences

	5.14.1.	The Limit of a Null Sequence
	5.14.2.	Proving the Limit of a Null Sequence
	5.14.3.	Proving the Finite Limit of a Sequence
	5.14.4.	Infinite Limits of Sequences
	5.14.5.	Proving a Sequence Has an Infinite Limit

5.15. Functions

	5.15.1.	Sets and Functions
	5.15.2.	Injections
	5.15.3.	Proving Injectivity
	5.15.4.	Surjections
	5.15.5.	Proving Surjectivity
	5.15.6.	Into Functions
	5.15.7.	Bijections
	5.15.8.	The Floor and Ceiling Functions

5.16. Cardinality

	5.16.1.	Discrete Infinite Sets With Equal Cardinality
	5.16.2.	Continuous Infinite Sets With Equal Cardinality
	5.16.3.	Cardinality of the Natural Numbers, Integers, and Rationals
	5.16.4.	Cantor's Diagonal Argument
	5.16.5.	The Cantor-Bernstein-Schröder Theorem
	5.16.6.	The Cardinality of the Power Set of Natural Numbers
	5.16.7.	Cantor's Theorem

Congruence of Integers

12 topics

6.17. Modular Congruence

	6.17.1.	Introduction to Modular Congruence
	6.17.2.	The Addition Property of Modular Arithmetic
	6.17.3.	Residues
	6.17.4.	The Multiplication Property of Modular Arithmetic
	6.17.5.	The Division Property of Modular Arithmetic
	6.17.6.	Proving Properties of Congruence
	6.17.7.	Proving Divisibility Using Congruence
	6.17.8.	Proving Congruence by Contrapositive

6.18. Linear Congruences

	6.18.1.	Solving Linear Congruences
	6.18.2.	Advanced Linear Congruences
	6.18.3.	The Chinese Remainder Theorem
	6.18.4.	Proving Properties of Linear Congruences

Equivalence Relations

14 topics

7.19. Relations

	7.19.1.	Introduction to Relations
	7.19.2.	The Domain and Range of a Relation
	7.19.3.	Graphical Representations of Relations
	7.19.4.	Operations on Relations
	7.19.5.	N-ary Relations
	7.19.6.	Functional Relations

7.20. Equivalence Relations

	7.20.1.	Reflexive and Irreflexive Relations
	7.20.2.	Symmetric and Antisymmetric Relations
	7.20.3.	Transitive Relations
	7.20.4.	Equivalence Relations
	7.20.5.	Proving a Relation Is an Equivalence Relation
	7.20.6.	Residue Classes
	7.20.7.	Equivalence Classes
	7.20.8.	The Integers Modulo N

Methods of Proof

Overview

Outcomes

Content

Sets and Relations

Modular Arithmetic

Logic

Proofs