Relations - Introductory Abstract Algebra (Undergraduate Advanced)

This course builds up a key part of the foundation for abstract algebra—relations. We explore equivalence relations, partitions, orderings, and how they help us organize and understand mathematical structures. These ideas connect directly to how we define and work with algebraic systems like groups and rings. Clear, focused, and made for first-time learners stepping into abstract algebra.

3

18 hrs

Payment required for enrolment
Enrolment valid for 12 months
This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
[University] Introduction to Abstract Algebra
[University] Introduction to Abstract Algebra
Master the foundational structures of modern mathematics. This learning track provides a direct path through abstract algebra, from basic sets to groups, rings, and fields. It delivers the analytical framework essential for advanced theoretical work. This programme is for undergraduate students in mathematics, computer science, or theoretical physics. It is also essential for professionals requiring a rigorous grasp of algebraic structures for work in cryptography, algorithm design, or quantum computing. Construct rigorous proofs and analyse the properties of groups, rings, and fields. This programme directly prepares you for postgraduate studies in pure mathematics and for advanced technical roles in cryptography and algorithm theory.

Master the foundational structures of modern mathematics. This learning track provides a direct path through abstract algebra, from basic sets to groups, rings, and fields. It delivers the analytical framework essential for advanced theoretical work. This programme is for undergraduate students in mathematics, computer science, or theoretical physics. It is also essential for professionals requiring a rigorous grasp of algebraic structures for work in cryptography, algorithm design, or quantum computing. Construct rigorous proofs and analyse the properties of groups, rings, and fields. This programme directly prepares you for postgraduate studies in pure mathematics and for advanced technical roles in cryptography and algorithm theory.

Course Chapters

1. Introduction
8
2

Introduction to ordered pairs, Cartesian products and relations.

Chapter lessons

1-1. Welcome
12:34

Welcome to the course and outline of course.

1-2. Ordered pairs
8:24

Meaning and equality of ordered pairs.

1-3. Cartesian product
19:36

Meaning and cardinality of the Cartesian product of two sets.

1-4. Relations
30:08

Definition and examples of relations.

1-5. More examples
28:13

More examples of relations.

1-6. Domain and range
12:39

Meaning of domain and range of a relation.

1-7. Inverse relation
5:36

Meaning of the inverse of a relation.

1-8. Equality of relations
11:20

When are two relations said to be equal?

2. Reflexive Relations
1
2

Meaning and identification of reflexive relations.

Chapter lessons

2-1. Definition
13:22

When is a relation said to be reflexive?

3. Symmetric Relations
1
2

Meaning and identification of symmetric relations.

Chapter lessons

3-1. Definition
9:28

Meaning of symmetric relations.

4. Transitive Relations
1
2

Meaning and identification of transitive relations.

Chapter lessons

4-1. Definition
13:17

Meaning of transitive relations.

5. Equivalence Relations
1
3

Meaning and identification of equivalence relations.

Chapter lessons

5-1. Definition
14:52

Meaning of equivalence relations.

6. Empty Relation
2

Meaning and properties of the empty relation.

Chapter lessons

6-1. Definition
4:15

Meaning of the empty relation.

6-2. Properties
22:46

Reflexivity, symmetry and transitivity of the empty relation.

7. Equivalence Classes
2
4

Meaning and calculation of equivalence classes for a given equivalence relation.

Chapter lessons

7-1. Equivalence class
15:08

Meaning of equivalence class for each element in the domain of a relation.

7-2. Quotient set
10:38

Meaning of the quotient set of a given set with respect to an equivalence relation.

8. Congruence and Residue Classes
3

Meaning and examples of congruence relations and residue classes.

Chapter lessons

8-1. Modular arithmetics
28:30

How to carry out modular arithmetic operations.

8-2. Congruence
16:43

Meaning of the congruence modulo m relation.

8-3. Residue classes
26:31

Meaning and evaluation of residue classes modulo m.

9. Partially-Ordered Sets
7
3

Meaning and properties of partially-ordered sets.

Chapter lessons

9-1. Definition
20:25

Meaning of a partially-ordered set (POSET).

9-2. Precedence and dominance
10:25

Meaning of precedence and dominance in a partially-ordered set.

9-3. Properties (1)
15:28

Properties of partially-ordered sets - first or least element, last or greatest element, minimal and maximal elements.

9-4. Properties (2)
8:49

Properties of partially-ordered sets - upper and lower bounds.

9-5. Comparability
8:29

When are two elements of a poset said to be comparable?

9-6. Totally-ordered sets
10:51

Meaning and examples of totally-ordered sets.

9-7. Well-ordered sets
20:20

Meaning and examples of well-ordered sets.

10. Lattices
4

Meaning and properties of lattices.

Chapter lessons

10-1. Definition
43:57

Meaning and examples of a lattice.

10-2. Sub-lattice
9:17

Meaning and examples of a sub-lattice.

10-3. Distributivity
11:56

When is a lattice said to be distributive?

10-4. Properties
8:02

General properties of lattices.