Relations - Introductory Abstract Algebra
1
18 hrs
[UI, Ibadan] MAT 211/213: Abstract AlgebraThis 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
1Introduction
Introduction to ordered pairs, Cartesian products and relations.
Chapter lessons
2Reflexive Relations
3Symmetric Relations
Meaning and identification of symmetric relations.
Chapter lessons
4Transitive 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.
5Equivalence 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.
6Empty 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.
7Equivalence 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)27:23
Worked examples on calculating equivalence classes and quotient sets.
4.Worked examples (2)27:49
More worked examples on calculating equivalence classes and quotient sets.
5.Worked examples (3)52:46
More worked examples on calculating equivalence classes and quotient sets.
6.Worked examples (4)20:03
More worked examples on calculating equivalence classes and quotient sets.
8Congruence and Residue Classes
Meaning and examples of congruence relations and residue classes.
Chapter lessons
1.Modular arithmetics28:30
How to carry out modular arithmetic operations.
2.Congruence16:43
Meaning of the congruence modulo m relation.
3.Residue classes26:31
Meaning and evaluation of residue classes modulo m.
9Partially-Ordered Sets
Meaning and properties of partially-ordered sets.
Chapter lessons
1.Definition20:25
Meaning of a partially-ordered set (POSET).
2.Precedence and dominance10:25
Meaning of precedence and dominance in a partially-ordered set.
3.Properties (1)15:28
Properties of partially-ordered sets - first or least element, last or greatest element, minimal and maximal elements.
4.Worked examples (1)20:37
Worked examples on identification and properties of partially-ordered sets.
5.Worked examples (2)16:14
More worked examples on identification and properties of partially-ordered sets.
6.Properties (2)8:49
Properties of partially-ordered sets - upper and lower bounds.
7.Worked examples (3)21:02
More worked examples on identification and properties of partially-ordered sets.
8.Comparability8:29
When are two elements of a poset said to be comparable?
9.Totally-ordered sets10:51
Meaning and examples of totally-ordered sets.
10.Well-ordered sets20:20
Meaning and examples of well-ordered sets.
10Lattices
Meaning and properties of lattices.
Chapter lessons
1.Definition43:57
Meaning and examples of a lattice.
2.Sub-lattice9:17
Meaning and examples of a sub-lattice.
3.Distributivity11:56
When is a lattice said to be distributive?
4.Properties8:02
General properties of lattices.