Relations - Introductory Abstract Algebra (Undergraduate Advanced)
3
18 hrs
[University] Introduction to Abstract AlgebraMaster 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. Introduction82
Introduction to ordered pairs, Cartesian products and relations.
Chapter lessons
2. Reflexive Relations12
3. Symmetric Relations12
4. Transitive Relations12
Meaning and identification of transitive relations.
Chapter lessons
4-1. Definition13:17
Meaning of transitive relations.
5. Equivalence Relations13
Meaning and identification of equivalence relations.
Chapter lessons
5-1. Definition14:52
Meaning of equivalence relations.
6. Empty Relation2
Meaning and properties of the empty relation.
Chapter lessons
6-1. Definition4:15
Meaning of the empty relation.
6-2. Properties22:46
Reflexivity, symmetry and transitivity of the empty relation.
7. Equivalence Classes24
Meaning and calculation of equivalence classes for a given equivalence relation.
Chapter lessons
7-1. Equivalence class15:08
Meaning of equivalence class for each element in the domain of a relation.
7-2. Quotient set10:38
Meaning of the quotient set of a given set with respect to an equivalence relation.
8. Congruence and Residue Classes3
Meaning and examples of congruence relations and residue classes.
Chapter lessons
8-1. Modular arithmetics28:30
How to carry out modular arithmetic operations.
8-2. Congruence16:43
Meaning of the congruence modulo m relation.
8-3. Residue classes26:31
Meaning and evaluation of residue classes modulo m.
9. Partially-Ordered Sets73
Meaning and properties of partially-ordered sets.
Chapter lessons
9-1. Definition20:25
Meaning of a partially-ordered set (POSET).
9-2. Precedence and dominance10:25
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. Comparability8:29
When are two elements of a poset said to be comparable?
9-6. Totally-ordered sets10:51
Meaning and examples of totally-ordered sets.
9-7. Well-ordered sets20:20
Meaning and examples of well-ordered sets.
10. Lattices4
Meaning and properties of lattices.
Chapter lessons
10-1. Definition43:57
Meaning and examples of a lattice.
10-2. Sub-lattice9:17
Meaning and examples of a sub-lattice.
10-3. Distributivity11:56
When is a lattice said to be distributive?
10-4. Properties8:02
General properties of lattices.