Relations - Introductory Abstract Algebra

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.

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$ 10.00

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.
[UI, Ibadan] MAT 211/213: Abstract Algebra
[UI, Ibadan] MAT 211/213: Abstract Algebra
This learning track brings together the essential building blocks of abstract algebra in one clear, structured path. We begin with the fundamentals—set theory, relations, and mappings—to build the logical foundation for working with algebraic systems. Then we move into binary operations, groups, subgroups, and homomorphisms, helping you develop the tools to recognize structure and symmetry. The track wraps up with rings, fields, and key ideas from elementary number theory that tie everything together. Curated for second-year undergraduates in engineering and physical sciences at the University of Ibadan, but equally valuable for any learner stepping into abstract algebra for the first time. Clear, focused, and paced to guide you with both depth and intuition.

This learning track brings together the essential building blocks of abstract algebra in one clear, structured path. We begin with the fundamentals—set theory, relations, and mappings—to build the logical foundation for working with algebraic systems. Then we move into binary operations, groups, subgroups, and homomorphisms, helping you develop the tools to recognize structure and symmetry. The track wraps up with rings, fields, and key ideas from elementary number theory that tie everything together. Curated for second-year undergraduates in engineering and physical sciences at the University of Ibadan, but equally valuable for any learner stepping into abstract algebra for the first time. Clear, focused, and paced to guide you with both depth and intuition.

Course Chapters

1
Introduction

Introduction to ordered pairs, Cartesian products and relations.

Chapter lessons

1.Welcome12:34

Welcome to the course and outline of course.

2.Ordered pairs8:24

Meaning and equality of ordered pairs.

3.Cartesian product19:36

Meaning and cardinality of the Cartesian product of two sets.

4.Relations30:08

Definition and examples of relations.

5.More examples28:13

More examples of relations.

6.Domain and range12:39

Meaning of domain and range of a relation.

7.Worked examples (1)29:25

Worked examples on the domain and range of a relation.

8.Inverse relation5:36

Meaning of the inverse of a relation.

9.Worked examples (2)18:15

Worked examples on the inverse of relations.

10.Equality of relations11:20

When are two relations said to be equal?

2
Reflexive Relations

Meaning and identification of reflexive relations.

Chapter lessons

1.Definition13:22

When is a relation said to be reflexive?

2.Worked examples (1)13:55

Worked examples on identifying reflexive relations.

3.Worked examples (2)16:44

More worked examples on identifying reflexive relations.

3
Symmetric Relations

Meaning and identification of symmetric relations.

Chapter lessons

1.Definition9:28

Meaning of symmetric relations.

2.Worked examples (1)14:24

Worked examples on identifying symmetric and anti-symmetric relations.

3.Worked examples (2)14:45

More worked examples on identifying symmetric and anti-symmetric relations.

4
Transitive Relations

Meaning and identification of transitive relations.

Chapter lessons

1.Definition13:17

Meaning of transitive relations.

2.Worked examples (1)14:10

Worked examples on identifying transitive relations.

3.Worked examples (2)16:30

More worked examples on identifying transitive relations.

5
Equivalence Relations

Meaning and identification of equivalence relations.

Chapter lessons

1.Definition14:52

Meaning of equivalence relations.

2.Worked examples (1)13:31

Worked examples on identifying equivalence relations.

3.Worked examples (2)19:37

More worked examples on identifying equivalence relations.

4.Worked examples (3)26:31

More worked examples on identifying equivalence relations.

6
Empty Relation

Meaning and properties of the empty relation.

Chapter lessons

1.Definition4:15

Meaning of the empty relation.

2.Properties22:46

Reflexivity, symmetry and transitivity of the empty relation.

7
Equivalence Classes

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

Chapter lessons

1.Equivalence class15:08

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

2.Quotient set10:38

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

3.Worked examples (1)

Worked examples on calculating equivalence classes and quotient sets.

4.Worked examples (2)

More worked examples on calculating equivalence classes and quotient sets.

8
Congruence and Residue Classes

Meaning and examples of congruence relations and residue classes.

Chapter lessons

1.Modular arithmetics

How to carry out modular arithmetic operations.

2.Congruence

Meaning of the congruence modulo m relation.

3.Residue classes

Meaning and evaluation of residue classes modulo m.

9
Partially-Ordered Sets

Meaning and properties of partially-ordered sets.

Chapter lessons

1.Definition

Meaning of a partially-ordered set (POSET).

2.Precedence and dominance

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

3.Properties (1)

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

4.Properties (2)

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

5.Totally-ordered sets

Meaning and examples of totally-ordered sets.

6.Well-ordered sets

Meaning and examples of well-ordered sets.

7.Worked examples (1)

Worked examples on identification and properties of partially-ordered sets.

8.Worked examples (2)

More worked examples on identification and properties of partially-ordered sets.

10
Lattices

Meaning and properties of lattices.

Chapter lessons

1.Definition

Meaning and examples of a lattice.

2.Sub-lattice

Meaning and examples of a sub-lattice.

3.Properties

Properties of lattices.

4.Worked examples (1)

Worked examples on identification and properties of lattices.

5.Worked examples (2)

Worked examples on identification and properties of lattices.