Undergraduate (UG) Roadmap | BMLabs Mathematics

Level curriculum

Mapped learning units

Every unit below is an existing subsection whose level is UG in content-data.json.

Academic domain

Discrete Mathematics

DOMAIN 01

Chapter 01

Mathematical Logic

5 sections

Academic section

Basic Logical Connectives and Truth Tables

Academic section

Logical Equivalence and the Laws of Logic

Equivalent Forms of Compound Statements

1 elementsOpen module

De Morgan's Laws

1 elementsOpen module

Distributive Laws for Logical Connectives

1 elementsOpen module

Laws of Logic

1 elementsOpen module

Duality in Logical Equivalence

1 elementsOpen module

Substitution Rules for Logical Equivalence

1 elementsOpen module

Negation and Duals of Conditional Statements

1 elementsOpen module

Converse, Inverse, and Contrapositive

1 elementsOpen module

Nested Conditionals and Program Logic

1 elementsOpen module

Simplification of Compound Statements

1 elementsOpen module

Switching Networks and Logical Equivalence

1 elementsOpen module

NAND and NOR Connectives

1 elementsOpen module

Academic section

Logical Implication and Rules of Inference

Valid Arguments and Logical Implication

1 elementsOpen module

Direct Rules of Inference

1 elementsOpen module

Invalid Conditional Reasoning

1 elementsOpen module

Conjunction and Disjunction Rules

1 elementsOpen module

Contradiction and Indirect Proof

1 elementsOpen module

Proofs Using Several Rules of Inference

1 elementsOpen module

Invalid Arguments and Counterexamples

1 elementsOpen module

Academic section

Predicates and Quantifiers

Open Statements and Universe of Discourse

1 elementsOpen module

Existential and Universal Quantifiers

1 elementsOpen module

Implicit Quantification and Universe Dependence

1 elementsOpen module

Quantifiers in Programming Statements

1 elementsOpen module

Truth Conditions for Quantified Statements

1 elementsOpen module

Logical Equivalence and Implication of Predicates

1 elementsOpen module

Converse, Inverse, and Contrapositive of Quantified Implications

1 elementsOpen module

Logical Laws for Quantified Statements

1 elementsOpen module

Negation of Quantified Statements

1 elementsOpen module

Multiple and Nested Quantifiers

1 elementsOpen module

Negation of Nested Quantified Statements

1 elementsOpen module

Unique Existence

1 elementsOpen module

Academic section

Quantifiers Definitions And Proofs Of Theorems

Biconditional Definitions with Quantifiers

1 elementsOpen module

Proof by Exhaustion and Theorem Terminology

1 elementsOpen module

Universal Specification in Quantified Arguments

1 elementsOpen module

Universal Generalization in Theorem Proofs

1 elementsOpen module

Quantifier Laws and Existential Rules

1 elementsOpen module

Integer Parity Definitions and Direct Proofs

1 elementsOpen module

Indirect Proof Methods for Theorems

1 elementsOpen module

Academic domain

Abstract Algebra

DOMAIN 02

Chapter 01

Introduction to Groups

8 sections

Academic section

Definition and Examples of Groups

Additive Groups Number Systems

1 elementsOpen module

Binary Operations on Sets

1 elementsOpen module

First Nonexamples of Groups

1 elementsOpen module

Four Group Axioms

1 elementsOpen module

Group Definition

1 elementsOpen module

Multiplicative Groups Nonzero Numbers

1 elementsOpen module

Verify Group Axioms

1 elementsOpen module

Academic section

Standard Examples of Groups

Additive Groups Modulo N

1 elementsOpen module

Alternating Groups

1 elementsOpen module

Groups From Number Systems

1 elementsOpen module

Klein Four Group

1 elementsOpen module

Permutation Groups

1 elementsOpen module

Roots of Unity Groups

1 elementsOpen module

Academic section

Semigroup Conditions for Groups

Finite Cancellative Semigroups

1 elementsOpen module

Idempotents in Finite Semigroups

1 elementsOpen module

Semigroup Equation Criterion for Groups

1 elementsOpen module

Semigroups

1 elementsOpen module

Unique Idempotent Semigroups

1 elementsOpen module

Academic section

Elementary Properties of Groups

Cancellation Laws in Groups

1 elementsOpen module

Equations in Groups

1 elementsOpen module

Idempotent Elements in Groups

1 elementsOpen module

Identity Element in Groups

1 elementsOpen module

Inverse Element in Groups

1 elementsOpen module

Properties of Inverses

1 elementsOpen module

Academic section

Abelian Groups

Abelian Group

1 elementsOpen module

Elements of Order Two

1 elementsOpen module

Groups of Exponent Two

1 elementsOpen module

Inverse Product Criterion

1 elementsOpen module

Power Criteria for Abelian Groups

1 elementsOpen module

Academic section

Cyclic Groups

Cyclic Group

1 elementsOpen module

Cyclic Groups Are Abelian

1 elementsOpen module

Even Cyclic Groups

1 elementsOpen module

Finite and Infinite Cyclic Groups

1 elementsOpen module

Generators of Finite Cyclic Groups

1 elementsOpen module

Prime Order Cyclic Groups

1 elementsOpen module

Academic section

Order of an Element

Finite and Infinite Order

1 elementsOpen module

Order of Inverse Element

1 elementsOpen module

Order of Powers

1 elementsOpen module

Orders of Products and Conjugates

1 elementsOpen module

Powers Equal to Identity

1 elementsOpen module

Prime Order Elements

1 elementsOpen module

Academic section

Powers of Elements in Groups

Conjugation and Powers

1 elementsOpen module

Integral Power Laws

1 elementsOpen module

Laws of Powers in Groups

1 elementsOpen module

Positive and Negative Powers

1 elementsOpen module

Power Equations in Groups

1 elementsOpen module

Chapter 02

Subgroups and Normal Subgroups

8 sections

Academic section

Subgroups

Finite Subgroups

1 elementsOpen module

Intersections of Subgroups

1 elementsOpen module

Products of Subgroups

1 elementsOpen module

Subgroup Criteria

1 elementsOpen module

Subgroups

1 elementsOpen module

Unions of Subgroups

1 elementsOpen module

Academic section

Special Subgroups

Centraliser of an Element

1 elementsOpen module

Centre and Centralisers

1 elementsOpen module

Centre of a Group

1 elementsOpen module

Academic section

Cosets of Subgroups

Cardinality and Correspondence of Cosets

1 elementsOpen module

Coset Membership and Equality Criteria

1 elementsOpen module

Cosets and Intersections of Subgroups

1 elementsOpen module

Cosets as Equivalence Classes

1 elementsOpen module

Cosets as Partitions

1 elementsOpen module

Cosets Determined by Elements of a Subgroup

1 elementsOpen module

Equal and Disjoint Cosets

1 elementsOpen module

Left and Right Cosets

1 elementsOpen module

Academic section

Index and Lagrange Theorem

Divisors and Subgroup Orders

1 elementsOpen module

Groups of Prime Order

1 elementsOpen module

Index of a Subgroup

1 elementsOpen module

Lagrange Theorem

1 elementsOpen module

Order of an Element in a Finite Group

1 elementsOpen module

Powers in a Finite Group

1 elementsOpen module

Academic section

Applications of Lagrange Theorem

Index and Intersections of Subgroups

1 elementsOpen module

Index Multiplication Formula

1 elementsOpen module

Product Formula for Finite Subgroups

1 elementsOpen module

Relatively Prime Subgroup Orders

1 elementsOpen module

Subgroups of Prime Index

1 elementsOpen module

Subgroups of Prime Power Order Groups

1 elementsOpen module

Academic section

Double Cosets

Cardinality of Double Cosets

1 elementsOpen module

Double Coset Definition

1 elementsOpen module

Double Cosets as Equivalence Classes

1 elementsOpen module

Academic section

Normal Subgroups

Central and Cyclic Sources of Normal Subgroups

1 elementsOpen module

Definition and First Examples of Normal Subgroups

1 elementsOpen module

Index Two and Coset Criteria for Normality

1 elementsOpen module

Normalisers of Subgroups

1 elementsOpen module

Normality Inside Related Subgroups

1 elementsOpen module

Products and Intersections of Normal Subgroups

1 elementsOpen module

Academic section

Quotient Groups

Abelian and Cyclic Quotient Groups

1 elementsOpen module

Abelian Quotients and Commutators

1 elementsOpen module

Construction of Quotient Groups

1 elementsOpen module

Normal Subgroups in Quotient Groups

1 elementsOpen module

Order of a Quotient Group

1 elementsOpen module

Quotients by Central Subgroups

1 elementsOpen module

Academic domain

Real Analysis

DOMAIN 03

Chapter 01

FUNCTIONS OF A SINGLE VARIABLE II

1 sections

Academic section

Higher Order Derivatives

Foundations of Repeated Differentiation

1 elementsOpen module

Standard nth Derivatives

1 elementsOpen module

Quadratic Reciprocal Higher Derivatives

1 elementsOpen module

Cyclic Exponential and Trigonometric Derivatives

1 elementsOpen module

Phase-Amplitude Exponential-Trigonometric Products

1 elementsOpen module

Second-Order Differentiation

1 elementsOpen module

Implicit Second Derivatives

1 elementsOpen module

Parametric Second Derivatives

1 elementsOpen module

Second-Order Composite Functions

1 elementsOpen module

Derivatives of Inverse Functions

1 elementsOpen module

Quadratic Trigonometric Differential Identities

1 elementsOpen module

Rational Decomposition for Successive Differentiation

1 elementsOpen module

Linear-Factor Partial Fractions in Higher Derivatives

1 elementsOpen module

Quadratic-Factor Partial Fractions in Higher Derivatives

1 elementsOpen module

Leibniz Formula for Successive Derivatives

1 elementsOpen module

Advanced Problems in Repeated Differentiation

1 elementsOpen module

Inductive Proof of the Leibniz Formula

1 elementsOpen module

Operator Methods and Exponential Shifts

1 elementsOpen module

Structured Higher-Derivative Identities

1 elementsOpen module

Logarithmic Derivative Recurrences

1 elementsOpen module

Chapter 02

FUNCTIONS

1 sections

Academic section

Hyperbolic and Inverse Hyperbolic Functions

Exponential Construction of Hyperbolic Functions

1 elementsOpen module

Fundamental Identities of Hyperbolic Functions

1 elementsOpen module

Geometry and Range of Hyperbolic Functions

1 elementsOpen module

Differentiation of Hyperbolic Functions

1 elementsOpen module

Applications of Hyperbolic Differentiation

1 elementsOpen module

Logarithmic Forms of Inverse Hyperbolic Functions

1 elementsOpen module

Composition Laws for Inverse Hyperbolic Functions

1 elementsOpen module

Differentiation of Inverse Hyperbolic Functions

1 elementsOpen module

Chain Rules and Ranges for Inverse Hyperbolic Functions

1 elementsOpen module

Applications of Inverse Hyperbolic Differentiation

1 elementsOpen module

Integration Formulas for Hyperbolic Functions

1 elementsOpen module

Standard Integrals with Inverse Hyperbolic Forms

1 elementsOpen module

Applications of Hyperbolic Integration

1 elementsOpen module

Academic domain

Functional Analysis

DOMAIN 04

Chapter 01

Metric Spaces

5 sections

Academic section

Introduction to Metric Spaces

Metric Axioms and Subspace Metrics

1 elementsOpen module

Standard Metrics on Euclidean Spaces

1 elementsOpen module

Bounded Metrics from Existing Metrics

1 elementsOpen module

Pseudometrics and Quotient Metrics

1 elementsOpen module

Metrics on Extended Spaces

1 elementsOpen module

Metrics on Sequence Spaces

1 elementsOpen module

Metrics on Function Spaces

1 elementsOpen module

Limits of Sequences in Metric Spaces

1 elementsOpen module

Subsequences and Cauchy Criteria

1 elementsOpen module

Cauchy Sequences in Metric Spaces

1 elementsOpen module

Complete and Incomplete Metric Spaces

1 elementsOpen module

Completeness of Standard Metric Spaces

1 elementsOpen module

Coordinatewise and Uniform Convergence

1 elementsOpen module

Complete Function Spaces and Sequence Problems

1 elementsOpen module

Completion and Isometric Embedding

1 elementsOpen module

Constructed Metrics and Practice Problems

1 elementsOpen module

Academic section

Topological Structure of Metric Spaces

Metric Balls and Neighbourhoods

1 elementsOpen module

Open Sets and Interior

1 elementsOpen module

Closed Sets and Complementation

1 elementsOpen module

Limit Points and Closure

1 elementsOpen module

Boundary and Exterior of Sets

1 elementsOpen module

Bounded Sets and Diameters

1 elementsOpen module

Relative Topology of Subspaces

1 elementsOpen module

Completeness of Subspaces

1 elementsOpen module

Nested Closed Sets in Complete Spaces

1 elementsOpen module

Separable Metric Spaces

1 elementsOpen module

Countability Bases in Metric Spaces

1 elementsOpen module

Lindelof and Dense Metric Spaces

1 elementsOpen module

Category and Nowhere Dense Sets

1 elementsOpen module

Baire Category Theorem and Applications

1 elementsOpen module

Oscillation and Sets of Discontinuity

1 elementsOpen module

Academic section

Continuous Mappings and Fixed Point Principles

Pointwise Continuity and Sequential Tests

1 elementsOpen module

Limits of Mappings

1 elementsOpen module

Open and Closed Set Tests for Continuity

1 elementsOpen module

Composition and Closure Tests

1 elementsOpen module

Coordinate Mappings and Elementary Operators

1 elementsOpen module

Continuous Extensions from Dense and Closed Sets

1 elementsOpen module

Algebra of Scalar-Valued Continuous Functions

1 elementsOpen module

Uniform Continuity and Distance Functions

1 elementsOpen module

Uniform Continuity and Cauchy Sequences

1 elementsOpen module

Topological Equivalence of Metric Spaces

1 elementsOpen module

Isometries and Metric Invariants

1 elementsOpen module

Equivalent Metrics

1 elementsOpen module

Pointwise Limits of Function Sequences

1 elementsOpen module

Uniform Limits of Function Sequences

1 elementsOpen module

Uniformly Convergent Function Series

1 elementsOpen module

Uniform Series Tests and Examples

1 elementsOpen module

Extension Theorems for Real-Valued Functions

1 elementsOpen module

Contractions and Fixed Points

1 elementsOpen module

Linear Systems by Fixed Point Methods

1 elementsOpen module

Differential and Integral Equations by Contractions

1 elementsOpen module

Iterated Contractions

1 elementsOpen module

Academic section

Connectedness and Path Structure in Metric Spaces

Separation Criteria for Connected Spaces

1 elementsOpen module

Intervals and Connected Subsets of the Real Line

1 elementsOpen module

Two-Point Tests for Connectedness

1 elementsOpen module

Continuous Images and Intermediate Values

1 elementsOpen module

Closures and Enlargements of Connected Sets

1 elementsOpen module

Unions of Connected Subspaces

1 elementsOpen module

Components of Metric Spaces

1 elementsOpen module

Totally Separated Metric Spaces

1 elementsOpen module

Local Forms of Connectedness

1 elementsOpen module

Paths and Path-Connected Sets

1 elementsOpen module

Path-Connectedness and Ordinary Connectedness

1 elementsOpen module

Path-Connected Images and Unions

1 elementsOpen module

Open Connected Sets in the Plane

1 elementsOpen module

Academic section

Compactness and Its Consequences in Metric Spaces

Open Covers and Compactness

1 elementsOpen module

Finite Intersection Tests for Compactness

1 elementsOpen module

Closedness and Boundedness of Compact Sets

1 elementsOpen module

Epsilon Nets and Total Boundedness

1 elementsOpen module

Total Boundedness and Cauchy Subsequences

1 elementsOpen module

Completeness and Compactness

1 elementsOpen module

Sequential Forms of Compactness

1 elementsOpen module

Compact Subspaces and Countable Covers

1 elementsOpen module

Lebesgue Numbers for Compact Covers

1 elementsOpen module

Compactness Under Continuous Mappings

1 elementsOpen module

Extreme Values on Compact Domains

1 elementsOpen module

Compact Domains and Homeomorphisms

1 elementsOpen module

Uniform Continuity on Compact Domains

1 elementsOpen module

Local Compactness

1 elementsOpen module

Compact Neighbourhoods and Exhaustions

1 elementsOpen module

Equicontinuous Families in Function Spaces

1 elementsOpen module

Uniform Boundedness in Function Spaces

1 elementsOpen module

Arzela-Ascoli Compactness Criterion

1 elementsOpen module

Compactness Criteria in Sequence Spaces

1 elementsOpen module

Compactness in Discrete and Euclidean Spaces

1 elementsOpen module

Continue learning

Curated pathways

Metric Spaces

Comprehensive module covering 5 sections in Functional Analysis.

Explore pathway

Introduction To Discrete Mathematics

Comprehensive module covering 6 sections in Discrete Mathematics.

Explore pathway

Mapped learning units

Basic Logical Connectives and Truth Tables

Statements and Primitive Statements

Negation and Compound Statements

Conjunction and Disjunction

Implication and Biconditional

Open Sentences and Truth Values

Truth Tables for Basic Connectives

Symbolic Translation of Compound Statements

Conditional Statements and Programming Decisions

Parentheses in Compound Truth Tables

Tautologies, Contradictions, and Valid Arguments

Truth-Table Applications

Logical Equivalence and the Laws of Logic

Equivalent Forms of Compound Statements

De Morgan's Laws

Distributive Laws for Logical Connectives

Laws of Logic

Duality in Logical Equivalence

Substitution Rules for Logical Equivalence

Negation and Duals of Conditional Statements

Converse, Inverse, and Contrapositive

Nested Conditionals and Program Logic

Simplification of Compound Statements

Switching Networks and Logical Equivalence

NAND and NOR Connectives

Logical Implication and Rules of Inference

Valid Arguments and Logical Implication

Direct Rules of Inference

Invalid Conditional Reasoning

Conjunction and Disjunction Rules

Contradiction and Indirect Proof

Proofs Using Several Rules of Inference

Invalid Arguments and Counterexamples

Predicates and Quantifiers

Open Statements and Universe of Discourse

Existential and Universal Quantifiers

Implicit Quantification and Universe Dependence

Quantifiers in Programming Statements

Truth Conditions for Quantified Statements

Logical Equivalence and Implication of Predicates

Converse, Inverse, and Contrapositive of Quantified Implications

Logical Laws for Quantified Statements

Negation of Quantified Statements

Multiple and Nested Quantifiers

Negation of Nested Quantified Statements

Unique Existence

Quantifiers Definitions And Proofs Of Theorems

Biconditional Definitions with Quantifiers

Proof by Exhaustion and Theorem Terminology

Universal Specification in Quantified Arguments

Universal Generalization in Theorem Proofs

Quantifier Laws and Existential Rules

Integer Parity Definitions and Direct Proofs

Indirect Proof Methods for Theorems

Definition and Examples of Groups

Additive Groups Number Systems

Binary Operations on Sets

First Nonexamples of Groups

Four Group Axioms

Group Definition

Multiplicative Groups Nonzero Numbers

Verify Group Axioms

Standard Examples of Groups

Additive Groups Modulo N

Alternating Groups

Groups From Number Systems

Klein Four Group

Permutation Groups

Roots of Unity Groups

Semigroup Conditions for Groups

Finite Cancellative Semigroups

Idempotents in Finite Semigroups

Semigroup Equation Criterion for Groups

Semigroups

Unique Idempotent Semigroups

Elementary Properties of Groups

Cancellation Laws in Groups

Equations in Groups

Idempotent Elements in Groups